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Content-Transfer-Encoding: 8bit --- assignment-1/submission/18340986009/img/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 assignment-1/submission/18340986009/img/.keep diff --git a/assignment-1/submission/18340986009/img/.keep b/assignment-1/submission/18340986009/img/.keep deleted file mode 100644 index e69de29..0000000 -- Gitee From fa1101e031a94d473167bab8c78e5c262e9d8da9 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 29 Mar 2021 23:32:33 +0800 Subject: [PATCH 05/68] (): --- assignment-1/submission/18340986009/README.md | 133 ++++++++++++++++++ 1 file changed, 133 insertions(+) create mode 100644 assignment-1/submission/18340986009/README.md diff --git a/assignment-1/submission/18340986009/README.md b/assignment-1/submission/18340986009/README.md new file mode 100644 index 0000000..ea3d71a --- /dev/null +++ b/assignment-1/submission/18340986009/README.md @@ -0,0 +1,133 @@ +# KNN Classification + +This report includes two parts: +1. Find a model that maximize accuracy rate with given dataset. (Distribution type of each class = Gaussian, distribution parameters chosen at random) +2. Assess how distribution parameters affects model accuracy using the model built in part 1. + +*Data used in this report is under ./data folder. One can reproduce the result using the saved data.* + +## 1. Model Generation + +### 1.1 Overview of Mock Data + +Generate 3 classes of 2-dimension Gaussian Distribution. + +$ + N_0 = 150 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\50\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}60 & -50\\-50 & 140\end{bmatrix})\, +$ + +$ + N_1 = 250 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}60\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}130 & 10\\10 & 100\end{bmatrix})\, +$ + +$ + N_2 = 100 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\60\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}120 & 20\\20 & 90\end{bmatrix})\, +$ + +Mock Data 1 [D1.csv] Overview: + + + +500 points are then split randomly into training set [train_data1.csv] (80%) and testing set [test_data1.csv] (20%). + +### 1.2 Model Accuracy with Different K and Distance Method + +Since a rule of thumb is to let $K = \sqrt{N}$, where $ N = N_0 + N_1 + N_2$, we first try some Ks around $\sqrt{400} = 20$ using both Euclidean and Manhattan distance. + +| \ | K = 10 | K = 15 | K = 20 | K = 25 | K = 30 | +| ------------ |:------:|:------:|:------:|:------:|:------:| +| **Euclidean** |83.0|82.0|83.0|81.0|80.0| +| **Manhattan** |83.0|82.0|81.0|81.0|81.0| + +The KNN model with $K = 10$ gives the best prediction result of 83% for both distance methods, so we consider choosing $K_{0} = 10$ as a starting point for model optimization. Below is a scatter plot showing the prediction result of the chosen model ($K = 10$, Euclidean Distance). Each red dot represents a mis-classification. + + + +### 1.3 Model Optimization + +General Idea: $K_{i+1} = \lceil{K_{i} + Step_{i+1}}\rceil$ + +Detailed steps: + + - For each $K_{i+1}$, calculate its accuracy rate $R_{i+1}$. + - If $R_{i+1} > R_{0}$, a better model is find. End our optimization. Else: + - If $R_{i+1} > R_{i}$, let $Step_{i+1} = \frac{1}{C} Step_{i} $, where $C = (R_{i+1} - R_{i}) / R_{i}$. + Which is, if improvements on accuracy is more than 1%, continue in this direction with a smaller step; If accuracy improves by less than 1%, continue in this direction with a larger step. + - If $R_{i+1} <= R_{i}$, let $Step_{i+1} = - \frac{1}{2} Step_{i}$. + Which is, if the new K does not improve model accuracy, try a smaller step in reverse direction. + +The model from 1.2 gives K = 10 and Euclidean distance. Using this model as the starting point, define the first step $Step_{0} = \frac{1}{100}N = 5$. + +Optimization process: + +| \ | K = 10 | K = 5 | K = 8 | +| ------------ |:------:|:------:|:------:| +| **Accuracy rate (%)** |83.0|83.0|85.0| + + After three iterations, a higher accuracy rate of 85% is reached when K is adjusted to 8. Thus, our final KNN model will use K = 8 and Euclidean distance. + +Prediction result evaluation: + + +Compared with the model before optimization, two points on the top is now classified correctly. + +## 2. Distribution Parameters & Model Accuracy + +From inuition, we hypothesis that any change that results in a more balanced mixture of all classes will make classification harder, thereby decrease model accuracy. Below, we modify the parameters of Gaussian distributions to test our hypothesis. + +### 2.1 Change of Variance and Covariance + +Let the means stay the same. Modify the variance-covariance matrix for each class to increase overlapping between each class: + +$ + N_0 = 150 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\50\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}300 & 0\\0 & 200\end{bmatrix})\, +$ + +$ + N_1 = 250 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}60\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}250 & 0\\0 & 150\end{bmatrix})\, +$ + +$ + N_2 = 100 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\60\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}150 & 0\\0 & 150\end{bmatrix})\, +$ + +Mock Data 2 [D2.csv] Overview: + + +Prediction result evaluation: + + +Accuracy of our model drop from 85% to 79% as expected. + +### 2.2 Change of Mean + +Let other parameters stay the same, decrease the distance between the means of each class to increase overlapping: + +$ + N_0 = 150 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\50\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}60 & -50\\-50 & 140\end{bmatrix})\, +$ + +$ + N_1 = 250 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\40\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}130 & 10\\10 & 100\end{bmatrix})\, +$ + +$ + N_2 = 100 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}40\\60\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}120 & 20\\20 & 90\end{bmatrix})\, +$ + +Mock Data 3 [D3.csv] Overview: + + +Prediction result evaluation: + + +Accuracy of our model drop from 85% to 73% as expected. -- Gitee From 7d8dcaf24f5386c39d311c23298f974f70097523 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 29 Mar 2021 23:44:05 +0800 Subject: [PATCH 06/68] =?UTF-8?q?=E5=88=A0=E9=99=A4=E6=96=87=E4=BB=B6=20as?= =?UTF-8?q?signment-1/submission/18340986009/README.md?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- assignment-1/submission/18340986009/README.md | 133 ------------------ 1 file changed, 133 deletions(-) delete mode 100644 assignment-1/submission/18340986009/README.md diff --git a/assignment-1/submission/18340986009/README.md b/assignment-1/submission/18340986009/README.md deleted file mode 100644 index ea3d71a..0000000 --- a/assignment-1/submission/18340986009/README.md +++ /dev/null @@ -1,133 +0,0 @@ -# KNN Classification - -This report includes two parts: -1. Find a model that maximize accuracy rate with given dataset. (Distribution type of each class = Gaussian, distribution parameters chosen at random) -2. Assess how distribution parameters affects model accuracy using the model built in part 1. - -*Data used in this report is under ./data folder. One can reproduce the result using the saved data.* - -## 1. Model Generation - -### 1.1 Overview of Mock Data - -Generate 3 classes of 2-dimension Gaussian Distribution. - -$ - N_0 = 150 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\50\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}60 & -50\\-50 & 140\end{bmatrix})\, -$ - -$ - N_1 = 250 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}60\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}130 & 10\\10 & 100\end{bmatrix})\, -$ - -$ - N_2 = 100 \hspace{1cm} - C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\60\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}120 & 20\\20 & 90\end{bmatrix})\, -$ - -Mock Data 1 [D1.csv] Overview: - - - -500 points are then split randomly into training set [train_data1.csv] (80%) and testing set [test_data1.csv] (20%). - -### 1.2 Model Accuracy with Different K and Distance Method - -Since a rule of thumb is to let $K = \sqrt{N}$, where $ N = N_0 + N_1 + N_2$, we first try some Ks around $\sqrt{400} = 20$ using both Euclidean and Manhattan distance. - -| \ | K = 10 | K = 15 | K = 20 | K = 25 | K = 30 | -| ------------ |:------:|:------:|:------:|:------:|:------:| -| **Euclidean** |83.0|82.0|83.0|81.0|80.0| -| **Manhattan** |83.0|82.0|81.0|81.0|81.0| - -The KNN model with $K = 10$ gives the best prediction result of 83% for both distance methods, so we consider choosing $K_{0} = 10$ as a starting point for model optimization. Below is a scatter plot showing the prediction result of the chosen model ($K = 10$, Euclidean Distance). Each red dot represents a mis-classification. - - - -### 1.3 Model Optimization - -General Idea: $K_{i+1} = \lceil{K_{i} + Step_{i+1}}\rceil$ - -Detailed steps: - - - For each $K_{i+1}$, calculate its accuracy rate $R_{i+1}$. - - If $R_{i+1} > R_{0}$, a better model is find. End our optimization. Else: - - If $R_{i+1} > R_{i}$, let $Step_{i+1} = \frac{1}{C} Step_{i} $, where $C = (R_{i+1} - R_{i}) / R_{i}$. - Which is, if improvements on accuracy is more than 1%, continue in this direction with a smaller step; If accuracy improves by less than 1%, continue in this direction with a larger step. - - If $R_{i+1} <= R_{i}$, let $Step_{i+1} = - \frac{1}{2} Step_{i}$. - Which is, if the new K does not improve model accuracy, try a smaller step in reverse direction. - -The model from 1.2 gives K = 10 and Euclidean distance. Using this model as the starting point, define the first step $Step_{0} = \frac{1}{100}N = 5$. - -Optimization process: - -| \ | K = 10 | K = 5 | K = 8 | -| ------------ |:------:|:------:|:------:| -| **Accuracy rate (%)** |83.0|83.0|85.0| - - After three iterations, a higher accuracy rate of 85% is reached when K is adjusted to 8. Thus, our final KNN model will use K = 8 and Euclidean distance. - -Prediction result evaluation: - - -Compared with the model before optimization, two points on the top is now classified correctly. - -## 2. Distribution Parameters & Model Accuracy - -From inuition, we hypothesis that any change that results in a more balanced mixture of all classes will make classification harder, thereby decrease model accuracy. Below, we modify the parameters of Gaussian distributions to test our hypothesis. - -### 2.1 Change of Variance and Covariance - -Let the means stay the same. Modify the variance-covariance matrix for each class to increase overlapping between each class: - -$ - N_0 = 150 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\50\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}300 & 0\\0 & 200\end{bmatrix})\, -$ - -$ - N_1 = 250 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}60\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}250 & 0\\0 & 150\end{bmatrix})\, -$ - -$ - N_2 = 100 \hspace{1cm} - C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\60\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}150 & 0\\0 & 150\end{bmatrix})\, -$ - -Mock Data 2 [D2.csv] Overview: - - -Prediction result evaluation: - - -Accuracy of our model drop from 85% to 79% as expected. - -### 2.2 Change of Mean - -Let other parameters stay the same, decrease the distance between the means of each class to increase overlapping: - -$ - N_0 = 150 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\50\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}60 & -50\\-50 & 140\end{bmatrix})\, -$ - -$ - N_1 = 250 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\40\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}130 & 10\\10 & 100\end{bmatrix})\, -$ - -$ - N_2 = 100 \hspace{1cm} - C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}40\\60\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}120 & 20\\20 & 90\end{bmatrix})\, -$ - -Mock Data 3 [D3.csv] Overview: - - -Prediction result evaluation: - - -Accuracy of our model drop from 85% to 73% as expected. -- Gitee From b6a89aaad29775ad6d4a4d29c8e2546e9b619253 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 29 Mar 2021 23:44:27 +0800 Subject: [PATCH 07/68] (): --- assignment-1/submission/18340986009/README.md | 138 ++++++++++++++++++ 1 file changed, 138 insertions(+) create mode 100644 assignment-1/submission/18340986009/README.md diff --git a/assignment-1/submission/18340986009/README.md b/assignment-1/submission/18340986009/README.md new file mode 100644 index 0000000..319ff2e --- /dev/null +++ b/assignment-1/submission/18340986009/README.md @@ -0,0 +1,138 @@ +# KNN Classification + +This report includes two parts: +1. Find a model that maximize accuracy rate with given dataset. (Distribution type of each class = Gaussian, distribution parameters chosen at random) +2. Assess how distribution parameters affects model accuracy using the model built in part 1. + +*Data used in this report is under ./data folder. One can reproduce the result using the saved data.* + +## 1. Model Generation + +### 1.1 Overview of Mock Data + +Generate 3 classes of 2-dimension Gaussian Distribution. + +$ + N_0 = 150 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}60 & -50\\\\-50 & 140\end{bmatrix})\, +$ + +$ + N_1 = 250 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}60\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}130 & 10\\\\10 & 100\end{bmatrix})\, +$ + +$ + N_2 = 100 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\60\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}120 & 20\\\\20 & 90\end{bmatrix})\, +$ + +Mock Data 1 [D1.csv] Overview: + + + +500 points are then split randomly into training set [train_data1.csv] (80%) and testing set [test_data1.csv] (20%). + +### 1.2 Model Accuracy with Different K and Distance Method + +Since a rule of thumb is to let $K = \sqrt{N}$, where $ N = N_0 + N_1 + N_2$, we first try some Ks around $\sqrt{400} = 20$ using both Euclidean and Manhattan distance. + +| \ | K = 10 | K = 15 | K = 20 | K = 25 | K = 30 | +| ------------ |:------:|:------:|:------:|:------:|:------:| +| **Euclidean** |83.0|82.0|83.0|81.0|80.0| +| **Manhattan** |83.0|82.0|81.0|81.0|81.0| + +The KNN model with $K = 10$ gives the best prediction result of 83% for both distance methods, so we consider choosing $K_{0} = 10$ as a starting point for model optimization. Below is a scatter plot showing the prediction result of the chosen model ($K = 10$, Euclidean Distance). Each red dot represents a mis-classification. + + + +### 1.3 Model Optimization + +General Idea: $K_{i+1} = \lceil{K_{i} + Step_{i+1}}\rceil$ + +Detailed steps: + + - For each $K_{i+1}$, calculate its accuracy rate $R_{i+1}$. + - If $R_{i+1} > R_{0}$, a better model is find. End our optimization. Else: + - If $R_{i+1} > R_{i}$, let $Step_{i+1} = \frac{1}{C} Step_{i} $, where $C = (R_{i+1} - R_{i}) / R_{i}$. + Which is, if improvements on accuracy is more than 1%, continue in this direction with a smaller step; If accuracy improves by less than 1%, continue in this direction with a larger step. + - If $R_{i+1} <= R_{i}$, let $Step_{i+1} = - \frac{1}{2} Step_{i}$. + Which is, if the new K does not improve model accuracy, try a smaller step in reverse direction. + +The model from 1.2 gives K = 10 and Euclidean distance. Using this model as the starting point, define the first step $Step_{0} = \frac{1}{100}N = 5$. + +Optimization process: + +| \ | K = 10 | K = 5 | K = 8 | +| ------------ |:------:|:------:|:------:| +| **Accuracy rate (%)** |83.0|83.0|85.0| + + After three iterations, a higher accuracy rate of 85% is reached when K is adjusted to 8. Thus, our final KNN model will use K = 8 and Euclidean distance. + +Prediction result evaluation: + + + +Compared with the model before optimization, two points on the top is now classified correctly. + +## 2. Distribution Parameters & Model Accuracy + +From inuition, we hypothesis that any change that results in a more balanced mixture of all classes will make classification harder, thereby decrease model accuracy. Below, we modify the parameters of Gaussian distributions to test our hypothesis. + +### 2.1 Change of Variance and Covariance + +Let the means stay the same. Modify the variance-covariance matrix for each class to increase overlapping between each class: + +$ + N_0 = 150 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}300 & 0\\\\0 & 200\end{bmatrix})\, +$ + +$ + N_1 = 250 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}60\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}250 & 0\\\\0 & 150\end{bmatrix})\, +$ + +$ + N_2 = 100 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\60\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}150 & 0\\\\0 & 150\end{bmatrix})\, +$ + +Mock Data 2 [D2.csv] Overview: + + + +Prediction result evaluation: + + + +Accuracy of our model drop from 85% to 79% as expected. + +### 2.2 Change of Mean + +Let other parameters stay the same, decrease the distance between the means of each class to increase overlapping: + +$ + N_0 = 150 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}60 & -50\\\\-50 & 140\end{bmatrix})\, +$ + +$ + N_1 = 250 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\40\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}130 & 10\\\\10 & 100\end{bmatrix})\, +$ + +$ + N_2 = 100 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}40\\\\60\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}120 & 20\\\\20 & 90\end{bmatrix})\, +$ + +Mock Data 3 [D3.csv] Overview: + + + +Prediction result evaluation: + + + +Accuracy of our model drop from 85% to 73% as expected. -- Gitee From 0fdddb62e43005f46f035372d5717c2c5d981d44 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 29 Mar 2021 23:49:55 +0800 Subject: [PATCH 08/68] =?UTF-8?q?=E5=88=A0=E9=99=A4=E6=96=87=E4=BB=B6=20as?= =?UTF-8?q?signment-1/submission/18340986009/README.md?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- assignment-1/submission/18340986009/README.md | 138 ------------------ 1 file changed, 138 deletions(-) delete mode 100644 assignment-1/submission/18340986009/README.md diff --git a/assignment-1/submission/18340986009/README.md b/assignment-1/submission/18340986009/README.md deleted file mode 100644 index 319ff2e..0000000 --- a/assignment-1/submission/18340986009/README.md +++ /dev/null @@ -1,138 +0,0 @@ -# KNN Classification - -This report includes two parts: -1. Find a model that maximize accuracy rate with given dataset. (Distribution type of each class = Gaussian, distribution parameters chosen at random) -2. Assess how distribution parameters affects model accuracy using the model built in part 1. - -*Data used in this report is under ./data folder. One can reproduce the result using the saved data.* - -## 1. Model Generation - -### 1.1 Overview of Mock Data - -Generate 3 classes of 2-dimension Gaussian Distribution. - -$ - N_0 = 150 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}60 & -50\\\\-50 & 140\end{bmatrix})\, -$ - -$ - N_1 = 250 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}60\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}130 & 10\\\\10 & 100\end{bmatrix})\, -$ - -$ - N_2 = 100 \hspace{1cm} - C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\60\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}120 & 20\\\\20 & 90\end{bmatrix})\, -$ - -Mock Data 1 [D1.csv] Overview: - - - -500 points are then split randomly into training set [train_data1.csv] (80%) and testing set [test_data1.csv] (20%). - -### 1.2 Model Accuracy with Different K and Distance Method - -Since a rule of thumb is to let $K = \sqrt{N}$, where $ N = N_0 + N_1 + N_2$, we first try some Ks around $\sqrt{400} = 20$ using both Euclidean and Manhattan distance. - -| \ | K = 10 | K = 15 | K = 20 | K = 25 | K = 30 | -| ------------ |:------:|:------:|:------:|:------:|:------:| -| **Euclidean** |83.0|82.0|83.0|81.0|80.0| -| **Manhattan** |83.0|82.0|81.0|81.0|81.0| - -The KNN model with $K = 10$ gives the best prediction result of 83% for both distance methods, so we consider choosing $K_{0} = 10$ as a starting point for model optimization. Below is a scatter plot showing the prediction result of the chosen model ($K = 10$, Euclidean Distance). Each red dot represents a mis-classification. - - - -### 1.3 Model Optimization - -General Idea: $K_{i+1} = \lceil{K_{i} + Step_{i+1}}\rceil$ - -Detailed steps: - - - For each $K_{i+1}$, calculate its accuracy rate $R_{i+1}$. - - If $R_{i+1} > R_{0}$, a better model is find. End our optimization. Else: - - If $R_{i+1} > R_{i}$, let $Step_{i+1} = \frac{1}{C} Step_{i} $, where $C = (R_{i+1} - R_{i}) / R_{i}$. - Which is, if improvements on accuracy is more than 1%, continue in this direction with a smaller step; If accuracy improves by less than 1%, continue in this direction with a larger step. - - If $R_{i+1} <= R_{i}$, let $Step_{i+1} = - \frac{1}{2} Step_{i}$. - Which is, if the new K does not improve model accuracy, try a smaller step in reverse direction. - -The model from 1.2 gives K = 10 and Euclidean distance. Using this model as the starting point, define the first step $Step_{0} = \frac{1}{100}N = 5$. - -Optimization process: - -| \ | K = 10 | K = 5 | K = 8 | -| ------------ |:------:|:------:|:------:| -| **Accuracy rate (%)** |83.0|83.0|85.0| - - After three iterations, a higher accuracy rate of 85% is reached when K is adjusted to 8. Thus, our final KNN model will use K = 8 and Euclidean distance. - -Prediction result evaluation: - - - -Compared with the model before optimization, two points on the top is now classified correctly. - -## 2. Distribution Parameters & Model Accuracy - -From inuition, we hypothesis that any change that results in a more balanced mixture of all classes will make classification harder, thereby decrease model accuracy. Below, we modify the parameters of Gaussian distributions to test our hypothesis. - -### 2.1 Change of Variance and Covariance - -Let the means stay the same. Modify the variance-covariance matrix for each class to increase overlapping between each class: - -$ - N_0 = 150 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}300 & 0\\\\0 & 200\end{bmatrix})\, -$ - -$ - N_1 = 250 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}60\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}250 & 0\\\\0 & 150\end{bmatrix})\, -$ - -$ - N_2 = 100 \hspace{1cm} - C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\60\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}150 & 0\\\\0 & 150\end{bmatrix})\, -$ - -Mock Data 2 [D2.csv] Overview: - - - -Prediction result evaluation: - - - -Accuracy of our model drop from 85% to 79% as expected. - -### 2.2 Change of Mean - -Let other parameters stay the same, decrease the distance between the means of each class to increase overlapping: - -$ - N_0 = 150 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}60 & -50\\\\-50 & 140\end{bmatrix})\, -$ - -$ - N_1 = 250 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\40\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}130 & 10\\\\10 & 100\end{bmatrix})\, -$ - -$ - N_2 = 100 \hspace{1cm} - C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}40\\\\60\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}120 & 20\\\\20 & 90\end{bmatrix})\, -$ - -Mock Data 3 [D3.csv] Overview: - - - -Prediction result evaluation: - - - -Accuracy of our model drop from 85% to 73% as expected. -- Gitee From 9707af357ef22d43dd16cdbc22158f1ebc191f9d Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 29 Mar 2021 23:50:26 +0800 Subject: [PATCH 09/68] (): --- assignment-1/submission/18340986009/README.md | 138 ++++++++++++++++++ 1 file changed, 138 insertions(+) create mode 100644 assignment-1/submission/18340986009/README.md diff --git a/assignment-1/submission/18340986009/README.md b/assignment-1/submission/18340986009/README.md new file mode 100644 index 0000000..3c4e8a1 --- /dev/null +++ b/assignment-1/submission/18340986009/README.md @@ -0,0 +1,138 @@ +# KNN Classification + +This report includes two parts: +1. Find a model that maximize accuracy rate with given dataset. (Distribution type of each class = Gaussian, distribution parameters chosen at random) +2. Assess how distribution parameters affects model accuracy using the model built in part 1. + +*Data used in this report is under ./data folder. One can reproduce the result using the saved data.* + +## 1. Model Generation + +### 1.1 Overview of Mock Data + +Generate 3 classes of 2-dimension Gaussian Distribution. + +$ + N_0 = 150 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\sigma^{2} = \begin{bmatrix}60 & -50\\\\-50 & 140\end{bmatrix}) +$ + +$ + N_1 = 250 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}60\\\\20\end{bmatrix},\sigma^{2} = \begin{bmatrix}130 & 10\\\\10 & 100\end{bmatrix}) +$ + +$ + N_2 = 100 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\60\end{bmatrix},\sigma^{2} = \begin{bmatrix}120 & 20\\\\20 & 90\end{bmatrix}) +$ + +Mock Data 1 [D1.csv] Overview: + + + +500 points are then split randomly into training set [train_data1.csv] (80%) and testing set [test_data1.csv] (20%). + +### 1.2 Model Accuracy with Different K and Distance Method + +Since a rule of thumb is to let $K = \sqrt{N}$, where $ N = N_0 + N_1 + N_2$, we first try some Ks around $\sqrt{400} = 20$ using both Euclidean and Manhattan distance. + +| \ | K = 10 | K = 15 | K = 20 | K = 25 | K = 30 | +| ------------ |:------:|:------:|:------:|:------:|:------:| +| **Euclidean** |83.0|82.0|83.0|81.0|80.0| +| **Manhattan** |83.0|82.0|81.0|81.0|81.0| + +The KNN model with $K = 10$ gives the best prediction result of 83% for both distance methods, so we consider choosing $K_{0} = 10$ as a starting point for model optimization. Below is a scatter plot showing the prediction result of the chosen model ($K = 10$, Euclidean Distance). Each red dot represents a mis-classification. + + + +### 1.3 Model Optimization + +General Idea: $K_{i+1} = \lceil{K_{i} + Step_{i+1}}\rceil$ + +Detailed steps: + + - For each $K_{i+1}$, calculate its accuracy rate $R_{i+1}$. + - If $R_{i+1} > R_{0}$, a better model is find. End our optimization. Else: + - If $R_{i+1} > R_{i}$, let $Step_{i+1} = \frac{1}{C} Step_{i} $, where $C = (R_{i+1} - R_{i}) / R_{i}$. + Which is, if improvements on accuracy is more than 1%, continue in this direction with a smaller step; If accuracy improves by less than 1%, continue in this direction with a larger step. + - If $R_{i+1} <= R_{i}$, let $Step_{i+1} = - \frac{1}{2} Step_{i}$. + Which is, if the new K does not improve model accuracy, try a smaller step in reverse direction. + +The model from 1.2 gives K = 10 and Euclidean distance. Using this model as the starting point, define the first step $Step_{0} = \frac{1}{100}N = 5$. + +Optimization process: + +| \ | K = 10 | K = 5 | K = 8 | +| ------------ |:------:|:------:|:------:| +| **Accuracy rate (%)** |83.0|83.0|85.0| + + After three iterations, a higher accuracy rate of 85% is reached when K is adjusted to 8. Thus, our final KNN model will use K = 8 and Euclidean distance. + +Prediction result evaluation: + + + +Compared with the model before optimization, two points on the top is now classified correctly. + +## 2. Distribution Parameters & Model Accuracy + +From inuition, we hypothesis that any change that results in a more balanced mixture of all classes will make classification harder, thereby decrease model accuracy. Below, we modify the parameters of Gaussian distributions to test our hypothesis. + +### 2.1 Change of Variance and Covariance + +Let the means stay the same. Modify the variance-covariance matrix for each class to increase overlapping between each class: + +$ + N_0 = 150 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\sigma^{2} = \begin{bmatrix}300 & 0\\\\0 & 200\end{bmatrix}) +$ + +$ + N_1 = 250 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}60\\\\20\end{bmatrix},\sigma^{2} = \begin{bmatrix}250 & 0\\\\0 & 150\end{bmatrix}) +$ + +$ + N_2 = 100 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\60\end{bmatrix},\sigma^{2} = \begin{bmatrix}150 & 0\\\\0 & 150\end{bmatrix}) +$ + +Mock Data 2 [D2.csv] Overview: + + + +Prediction result evaluation: + + + +Accuracy of our model drop from 85% to 79% as expected. + +### 2.2 Change of Mean + +Let other parameters stay the same, decrease the distance between the means of each class to increase overlapping: + +$ + N_0 = 150 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\sigma^{2} = \begin{bmatrix}60 & -50\\\\-50 & 140\end{bmatrix}) +$ + +$ + N_1 = 250 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\40\end{bmatrix},\sigma^{2} = \begin{bmatrix}130 & 10\\\\10 & 100\end{bmatrix}) +$ + +$ + N_2 = 100 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}40\\\\60\end{bmatrix},\sigma^{2} = \begin{bmatrix}120 & 20\\\\20 & 90\end{bmatrix}) +$ + +Mock Data 3 [D3.csv] Overview: + + + +Prediction result evaluation: + + + +Accuracy of our model drop from 85% to 73% as expected. -- Gitee From 525979f82fd73dcb46e57f3bf66b4647ee321e19 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 29 Mar 2021 23:52:24 +0800 Subject: [PATCH 10/68] =?UTF-8?q?=E5=88=A0=E9=99=A4=E6=96=87=E4=BB=B6=20as?= =?UTF-8?q?signment-1/submission/18340986009/README.md?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- assignment-1/submission/18340986009/README.md | 138 ------------------ 1 file changed, 138 deletions(-) delete mode 100644 assignment-1/submission/18340986009/README.md diff --git a/assignment-1/submission/18340986009/README.md b/assignment-1/submission/18340986009/README.md deleted file mode 100644 index 3c4e8a1..0000000 --- a/assignment-1/submission/18340986009/README.md +++ /dev/null @@ -1,138 +0,0 @@ -# KNN Classification - -This report includes two parts: -1. Find a model that maximize accuracy rate with given dataset. (Distribution type of each class = Gaussian, distribution parameters chosen at random) -2. Assess how distribution parameters affects model accuracy using the model built in part 1. - -*Data used in this report is under ./data folder. One can reproduce the result using the saved data.* - -## 1. Model Generation - -### 1.1 Overview of Mock Data - -Generate 3 classes of 2-dimension Gaussian Distribution. - -$ - N_0 = 150 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\sigma^{2} = \begin{bmatrix}60 & -50\\\\-50 & 140\end{bmatrix}) -$ - -$ - N_1 = 250 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}60\\\\20\end{bmatrix},\sigma^{2} = \begin{bmatrix}130 & 10\\\\10 & 100\end{bmatrix}) -$ - -$ - N_2 = 100 \hspace{1cm} - C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\60\end{bmatrix},\sigma^{2} = \begin{bmatrix}120 & 20\\\\20 & 90\end{bmatrix}) -$ - -Mock Data 1 [D1.csv] Overview: - - - -500 points are then split randomly into training set [train_data1.csv] (80%) and testing set [test_data1.csv] (20%). - -### 1.2 Model Accuracy with Different K and Distance Method - -Since a rule of thumb is to let $K = \sqrt{N}$, where $ N = N_0 + N_1 + N_2$, we first try some Ks around $\sqrt{400} = 20$ using both Euclidean and Manhattan distance. - -| \ | K = 10 | K = 15 | K = 20 | K = 25 | K = 30 | -| ------------ |:------:|:------:|:------:|:------:|:------:| -| **Euclidean** |83.0|82.0|83.0|81.0|80.0| -| **Manhattan** |83.0|82.0|81.0|81.0|81.0| - -The KNN model with $K = 10$ gives the best prediction result of 83% for both distance methods, so we consider choosing $K_{0} = 10$ as a starting point for model optimization. Below is a scatter plot showing the prediction result of the chosen model ($K = 10$, Euclidean Distance). Each red dot represents a mis-classification. - - - -### 1.3 Model Optimization - -General Idea: $K_{i+1} = \lceil{K_{i} + Step_{i+1}}\rceil$ - -Detailed steps: - - - For each $K_{i+1}$, calculate its accuracy rate $R_{i+1}$. - - If $R_{i+1} > R_{0}$, a better model is find. End our optimization. Else: - - If $R_{i+1} > R_{i}$, let $Step_{i+1} = \frac{1}{C} Step_{i} $, where $C = (R_{i+1} - R_{i}) / R_{i}$. - Which is, if improvements on accuracy is more than 1%, continue in this direction with a smaller step; If accuracy improves by less than 1%, continue in this direction with a larger step. - - If $R_{i+1} <= R_{i}$, let $Step_{i+1} = - \frac{1}{2} Step_{i}$. - Which is, if the new K does not improve model accuracy, try a smaller step in reverse direction. - -The model from 1.2 gives K = 10 and Euclidean distance. Using this model as the starting point, define the first step $Step_{0} = \frac{1}{100}N = 5$. - -Optimization process: - -| \ | K = 10 | K = 5 | K = 8 | -| ------------ |:------:|:------:|:------:| -| **Accuracy rate (%)** |83.0|83.0|85.0| - - After three iterations, a higher accuracy rate of 85% is reached when K is adjusted to 8. Thus, our final KNN model will use K = 8 and Euclidean distance. - -Prediction result evaluation: - - - -Compared with the model before optimization, two points on the top is now classified correctly. - -## 2. Distribution Parameters & Model Accuracy - -From inuition, we hypothesis that any change that results in a more balanced mixture of all classes will make classification harder, thereby decrease model accuracy. Below, we modify the parameters of Gaussian distributions to test our hypothesis. - -### 2.1 Change of Variance and Covariance - -Let the means stay the same. Modify the variance-covariance matrix for each class to increase overlapping between each class: - -$ - N_0 = 150 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\sigma^{2} = \begin{bmatrix}300 & 0\\\\0 & 200\end{bmatrix}) -$ - -$ - N_1 = 250 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}60\\\\20\end{bmatrix},\sigma^{2} = \begin{bmatrix}250 & 0\\\\0 & 150\end{bmatrix}) -$ - -$ - N_2 = 100 \hspace{1cm} - C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\60\end{bmatrix},\sigma^{2} = \begin{bmatrix}150 & 0\\\\0 & 150\end{bmatrix}) -$ - -Mock Data 2 [D2.csv] Overview: - - - -Prediction result evaluation: - - - -Accuracy of our model drop from 85% to 79% as expected. - -### 2.2 Change of Mean - -Let other parameters stay the same, decrease the distance between the means of each class to increase overlapping: - -$ - N_0 = 150 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\sigma^{2} = \begin{bmatrix}60 & -50\\\\-50 & 140\end{bmatrix}) -$ - -$ - N_1 = 250 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\40\end{bmatrix},\sigma^{2} = \begin{bmatrix}130 & 10\\\\10 & 100\end{bmatrix}) -$ - -$ - N_2 = 100 \hspace{1cm} - C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}40\\\\60\end{bmatrix},\sigma^{2} = \begin{bmatrix}120 & 20\\\\20 & 90\end{bmatrix}) -$ - -Mock Data 3 [D3.csv] Overview: - - - -Prediction result evaluation: - - - -Accuracy of our model drop from 85% to 73% as expected. -- Gitee From 91607208b011d0f36b7933619abb70981ba02f12 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 29 Mar 2021 23:52:40 +0800 Subject: [PATCH 11/68] (): --- assignment-1/submission/18340986009/README.md | 138 ++++++++++++++++++ 1 file changed, 138 insertions(+) create mode 100644 assignment-1/submission/18340986009/README.md diff --git a/assignment-1/submission/18340986009/README.md b/assignment-1/submission/18340986009/README.md new file mode 100644 index 0000000..a62a1df --- /dev/null +++ b/assignment-1/submission/18340986009/README.md @@ -0,0 +1,138 @@ +# KNN Classification + +This report includes two parts: +1. Find a model that maximize accuracy rate with given dataset. (Distribution type of each class = Gaussian, distribution parameters chosen at random) +2. Assess how distribution parameters affects model accuracy using the model built in part 1. + +*Data used in this report is under ./data folder. One can reproduce the result using the saved data.* + +## 1. Model Generation + +### 1.1 Overview of Mock Data + +Generate 3 classes of 2-dimension Gaussian Distribution. + +$ + N_0 = 150 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\sigma^{2} = \begin{bmatrix}60 & -50\\\\-50 & 140\end{bmatrix}) +$ + +$ + N_1 = 250 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}60\\\\20\end{bmatrix},\sigma^{2} = \begin{bmatrix}130 & 10\\\\10 & 100\end{bmatrix}) +$ + +$ + N_2 = 100 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\60\end{bmatrix},\sigma^{2} = \begin{bmatrix}120 & 20\\\\20 & 90\end{bmatrix}) +$ + +Mock Data 1 [D1.csv] Overview: + + + +500 points are then split randomly into training set [train_data1.csv] (80%) and testing set [test_data1.csv] (20%). + +### 1.2 Model Accuracy with Different K and Distance Method + +Since a rule of thumb is to let $K = \sqrt{N}$, where $ N = N_0 + N_1 + N_2$, we first try some Ks around $\sqrt{400} = 20$ using both Euclidean and Manhattan distance. + +| \ | K = 10 | K = 15 | K = 20 | K = 25 | K = 30 | +| ------------ |:------:|:------:|:------:|:------:|:------:| +| **Euclidean** |83.0|82.0|83.0|81.0|80.0| +| **Manhattan** |83.0|82.0|81.0|81.0|81.0| + +The KNN model with $K = 10$ gives the best prediction result of 83% for both distance methods, so we consider choosing $K_{0} = 10$ as a starting point for model optimization. Below is a scatter plot showing the prediction result of the chosen model ($K = 10$, Euclidean Distance). Each red dot represents a mis-classification. + + + +### 1.3 Model Optimization + +General Idea: $K_{i+1} = \lceil{K_{i} + Step_{i+1}}\rceil$ + +Detailed steps: + + - For each $K_{i+1}$, calculate its accuracy rate $R_{i+1}$. + - If $R_{i+1} > R_{0}$, a better model is find. End our optimization. Else: + - If $R_{i+1} > R_{i}$, let $Step_{i+1} = \frac{1}{C} Step_{i} $, where $C = (R_{i+1} - R_{i}) / R_{i}$. + Which is, if improvements on accuracy is more than 1%, continue in this direction with a smaller step; If accuracy improves by less than 1%, continue in this direction with a larger step. + - If $R_{i+1} <= R_{i}$, let $Step_{i+1} = - \frac{1}{2} Step_{i}$. + Which is, if the new K does not improve model accuracy, try a smaller step in reverse direction. + +The model from 1.2 gives K = 10 and Euclidean distance. Using this model as the starting point, define the first step $Step_{0} = \frac{1}{100}N = 5$. + +Optimization process: + +| \ | K = 10 | K = 5 | K = 8 | +| ------------ |:------:|:------:|:------:| +| **Accuracy rate (%)** |83.0|83.0|85.0| + + After three iterations, a higher accuracy rate of 85% is reached when K is adjusted to 8. Thus, our final KNN model will use K = 8 and Euclidean distance. + +Prediction result evaluation: + + + +Compared with the model before optimization, two points on the top is now classified correctly. + +## 2. Distribution Parameters & Model Accuracy + +From inuition, we hypothesis that any change that results in a more balanced mixture of all classes will make classification harder, thereby decrease model accuracy. Below, we modify the parameters of Gaussian distributions to test our hypothesis. + +### 2.1 Change of Variance and Covariance + +Let the means stay the same. Modify the variance-covariance matrix for each class to increase overlapping between each class: + +$ + N_0 = 150 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\sigma^{2} = \begin{bmatrix}300 & 0\\\\0 & 200\end{bmatrix}) +$ + +$ + N_1 = 250 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}60\\\\20\end{bmatrix},\sigma^{2} = \begin{bmatrix}250 & 0\\\\0 & 150\end{bmatrix}) +$ + +$ + N_2 = 100 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\60\end{bmatrix},\sigma^{2} = \begin{bmatrix}150 & 0\\\\0 & 150\end{bmatrix}) +$ + +Mock Data 2 [D2.csv] Overview: + + + +Prediction result evaluation: + + + +Accuracy of our model drop from 85% to 79% as expected. + +### 2.2 Change of Mean + +Let other parameters stay the same, decrease the distance between the means of each class to increase overlapping: + +$ + N_0 = 150 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\sigma^{2} = \begin{bmatrix}60 & -50\\\\-50 & 140\end{bmatrix}) +$ + +$ + N_1 = 250 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\40\end{bmatrix},\sigma^{2} = \begin{bmatrix}130 & 10\\\\10 & 100\end{bmatrix}) +$ + +$ + N_2 = 100 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}40\\\\60\end{bmatrix},\sigma^{2} = \begin{bmatrix}120 & 20\\\\20 & 90\end{bmatrix}) +$ + +Mock Data 3 [D3.csv] Overview: + + + +Prediction result evaluation: + + + +Accuracy of our model drop from 85% to 73% as expected. -- Gitee From 61245b0352f56df31292088897347b72b9a9c9c4 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Tue, 30 Mar 2021 00:01:28 +0800 Subject: [PATCH 12/68] =?UTF-8?q?=E5=88=A0=E9=99=A4=E6=96=87=E4=BB=B6=20as?= =?UTF-8?q?signment-1/submission/18340986009/img/Figure=205.png?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../submission/18340986009/img/Figure 5.png | Bin 23949 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 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=?UTF-8?q?=E5=88=A0=E9=99=A4=E6=96=87=E4=BB=B6=20as?= =?UTF-8?q?signment-2/submission/18340986009/README.md?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- assignment-2/submission/18340986009/README.md | 265 ------------------ 1 file changed, 265 deletions(-) delete mode 100644 assignment-2/submission/18340986009/README.md diff --git a/assignment-2/submission/18340986009/README.md b/assignment-2/submission/18340986009/README.md deleted file mode 100644 index 3352820..0000000 --- a/assignment-2/submission/18340986009/README.md +++ /dev/null @@ -1,265 +0,0 @@ -# FNN - -## 1. Activation Function - -The data used in this report is from the **torch** library. The data is a collection of 28$\small{\times}$28 matrices each representing a picture. The goal is to train the FNN model to identify which digit from 0 to 9 (c = 10) is in the picture. - -With a batch size of 128 (N = 128), in each epoch, our training data is a 3-order tensor of shape (128, 28, 28) and our label after one-hot encoding is a tensor of shape (128, 10). In the forwarding step, each matrix $\underset{28\times 28}{\mathrm{X}}$ is flattened to a vector $\underset{1\times 28^{2}}{\mathrm{x}}$ for computational purpose. There are 3 layers in total, each layer reshapes its input by the weight matrix $\mathrm{W}$. - -_For the sake of consistency, we'll use numerator layout throughout, -use subscript $X_{(i)}$ to denote the $ith$ row of $X$, superscript $X^{(j)}$ to denote the $jth$ column of $X$, and use $X_{ij}$ to denote the $(i,j)$ element of $X$_ - -### 1.1 Matmul - - - - - -```python - def backward(self, grad_y): - """ - grad_y: shape(N, d') - """ - x = self.memory['x'] - W = self.memory['W'] - - grad_x = np.matmul(grad_y, W.T) - grad_W = np.matmul(x.T, grad_y) - return grad_x, grad_W -``` - -### 1.2 Relu: - - - -```python - def backward(self, grad_y): - """ - grad_y: same shape as x - """ - x = self.memory['x'] - grad_x = np.where(x <= 0, 0, grad_y) - - return grad_x -``` - -### 1.3 Log - - - -```python - def backward(self, grad_y): - """ - grad_y: same shape as x - """ - x = self.memory['x'] - grad_x = (x + self.epsilon)**(-1) * grad_y - - return grad_x -``` - -### 1.4 Softmax: - - - -```python - def forward(self, x): - """ - x: shape(N, c) - """ - exp = np.exp(x - self.epsilon) - out = exp/exp.sum(axis=1, keepdims=True) - self.memory['x'] = x - self.memory['s'] = out - - return out - - def backward(self, grad_y): - """ - grad_y: same shape as x - """ - s = self.memory['s'] - n = s.shape[1] - - grad_x = np.zeros(s.shape) - # calculate Jacobian for ith row - for i in range(s.shape[0]): - jacob = np.zeros((n, n)) - - for j in range(n): - for k in range(n): - if j == k: - jacob[k, j] = s[i, k] * (1-s[i, j]) - else: - jacob[k, j] = -1 * s[i, k] * s[i, j] - # apply Jacobian to the ith row of grad_y - grad_x[i, :] = np.matmul(jacob, grad_y[i, :]) - - return grad_x -``` - -## 2. Model Training - -### 2.1 Change mini_batch: - -From documentation, we can see that mini_batch is a function that shuffles data and create small batches with user-specified batch size. The return is a batch such that the code: **for x, y in mini_batch(train_dataset):** in function **numpy_run()** can retrive appropriate training data and training label in the batch. The suffling method implemented by the code below uses ordered sampling without replacement. If the remaining data is insufficient to form a batch of the specified size, we'll drop the remaining ones. - -```python -def mini_batch(dataset, batch_size=128): - - # get total number of observations to batch from - size = dataset.train_data.shape[0] - - # fetch data and label - data = [] - label = [] - for one_obs in dataset: - data.append(np.array(one_obs[0])) - label.append(np.array(one_obs[1])) - data = np.array(data) - label = np.array(label) - - # random shuffle - idx = np.arange(size) - np.random.shuffle(idx) - data = data[idx] - label = label[idx] - - # split to batch - # discard the last batch if remaining size < batch_size - data_batches = [] - label_batches = [] - batched = 0 - while batched+batch_size <= size: - data_batches.append(data[batched:batched + batch_size]) - label_batches.append(label[batched:batched + batch_size]) - batched += batch_size - - data_batches = np.array(data_batches) - label_batches = np.array(label_batches) - - return zip(data_batches, label_batches) -``` - -### 2.2 Training Result: - - - Hyper-parameters: - * epoch number = 3 - * learning rate = 0.1 - - - - Model structure: - * 3 layers - * dimension reduced in each layer using a weight matrix - * softmax as the final activation function - * loss calculated as $-y^{T}\log{\hat{y}}$ - -```python -def forward(self, x): - x = x.reshape(-1, 28 * 28) - a0 = x - - #First Layer - z1 = self.matmul_1.forward(a0, self.W1) - a1 = self.relu_1.forward(z1) - - #Second Layer - z2 = self.matmul_2.forward(a1, self.W2) - a2 = self.relu_2.forward(z2) - - #Third Layer - z3 = self.matmul_3.forward(a2, self.W3) - a3 = self.softmax.forward(z3) - x = self.log.forward(a3) - - return x -``` - -```python -def backward(self, y): - - #Third Layer - self.log_grad = self.log.backward(y) - self.softmax_grad = self.softmax.backward(self.log_grad) - self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) - - #Second Layer - self.relu_2_grad = self.relu_2.backward(self.x3_grad) - self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) - - #First Layer - self.relu_1_grad = self.relu_1.backward(self.x2_grad) - self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) - - pass -``` - -Result: -```python -[0] Accuracy: 0.9482 -[1] Accuracy: 0.9643 -[2] Accuracy: 0.9735 -``` - - - -## 3. Change of Parameters - -**Learning Rate [0.2, 0.4, 0.6]** - - - - - -**Batch Size [20, 40, 60]** - - - - - -## 4. Optimization - -**Momentum:** - - - -```python -#Extends NumpyModel -#Save step size -self.delta1 = 0 -self.delta2 = 0 -self.delta3 = 0 - -#Extends the optimize function defined in numpy_fnn.py -if method == "Momentum": - self.delta1 = momentum * self.delta1 - learning_rate * self.W1_grad - self.W1 += self.delta1 - - self.delta2 = momentum * self.delta2 - learning_rate * self.W2_grad - self.W2 += self.delta2 - - self.delta3 = momentum * self.delta3 - learning_rate * self.W3_grad - self.W3 += self.delta3 -``` - -Result: -```python -[0] Accuracy: 0.9654 -[1] Accuracy: 0.9712 -[2] Accuracy: 0.9726 -``` - - - -Discussion: - -Momentum method differs from gradient descent in that it adds a short term memory for step size. Momentum, a number between 0 and 1, is the weight given to the previous step size. Momentum smoothes the path (visual demonstration below, [ref](https://dominikschmidt.xyz/nesterov-momentum/)). A greater momentum results in a more direct path with less changes in direction. Or, from another perspective, accelarates towards the optimum. If momentum = 0, the method reduces to Gradient Descent. - - - -Momentum $\beta$ can be thought of the weight of the previous force, while learning rate $\alpha$ can be thought of the weight of the current force. Intuitively, the optimal training model needs an optimal combination of $\alpha$ and $\beta$ depending on the eigenvalues of a convex function. Inappropriate values could lead to divergence. - - -```python - -``` -- Gitee From 444b700d6fb09666372a23a5846870bbd46ff552 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Sun, 2 May 2021 22:42:13 +0800 Subject: [PATCH 31/68] (): --- assignment-2/submission/18340986009/README.md | 265 ++++++++++++++++++ 1 file changed, 265 insertions(+) create mode 100644 assignment-2/submission/18340986009/README.md diff --git a/assignment-2/submission/18340986009/README.md b/assignment-2/submission/18340986009/README.md new file mode 100644 index 0000000..3352820 --- /dev/null +++ b/assignment-2/submission/18340986009/README.md @@ -0,0 +1,265 @@ +# FNN + +## 1. Activation Function + +The data used in this report is from the **torch** library. The data is a collection of 28$\small{\times}$28 matrices each representing a picture. The goal is to train the FNN model to identify which digit from 0 to 9 (c = 10) is in the picture. + +With a batch size of 128 (N = 128), in each epoch, our training data is a 3-order tensor of shape (128, 28, 28) and our label after one-hot encoding is a tensor of shape (128, 10). In the forwarding step, each matrix $\underset{28\times 28}{\mathrm{X}}$ is flattened to a vector $\underset{1\times 28^{2}}{\mathrm{x}}$ for computational purpose. There are 3 layers in total, each layer reshapes its input by the weight matrix $\mathrm{W}$. + +_For the sake of consistency, we'll use numerator layout throughout, +use subscript $X_{(i)}$ to denote the $ith$ row of $X$, superscript $X^{(j)}$ to denote the $jth$ column of $X$, and use $X_{ij}$ to denote the $(i,j)$ element of $X$_ + +### 1.1 Matmul + + + + + +```python + def backward(self, grad_y): + """ + grad_y: shape(N, d') + """ + x = self.memory['x'] + W = self.memory['W'] + + grad_x = np.matmul(grad_y, W.T) + grad_W = np.matmul(x.T, grad_y) + return grad_x, grad_W +``` + +### 1.2 Relu: + + + +```python + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + x = self.memory['x'] + grad_x = np.where(x <= 0, 0, grad_y) + + return grad_x +``` + +### 1.3 Log + + + +```python + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + x = self.memory['x'] + grad_x = (x + self.epsilon)**(-1) * grad_y + + return grad_x +``` + +### 1.4 Softmax: + + + +```python + def forward(self, x): + """ + x: shape(N, c) + """ + exp = np.exp(x - self.epsilon) + out = exp/exp.sum(axis=1, keepdims=True) + self.memory['x'] = x + self.memory['s'] = out + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + s = self.memory['s'] + n = s.shape[1] + + grad_x = np.zeros(s.shape) + # calculate Jacobian for ith row + for i in range(s.shape[0]): + jacob = np.zeros((n, n)) + + for j in range(n): + for k in range(n): + if j == k: + jacob[k, j] = s[i, k] * (1-s[i, j]) + else: + jacob[k, j] = -1 * s[i, k] * s[i, j] + # apply Jacobian to the ith row of grad_y + grad_x[i, :] = np.matmul(jacob, grad_y[i, :]) + + return grad_x +``` + +## 2. Model Training + +### 2.1 Change mini_batch: + +From documentation, we can see that mini_batch is a function that shuffles data and create small batches with user-specified batch size. The return is a batch such that the code: **for x, y in mini_batch(train_dataset):** in function **numpy_run()** can retrive appropriate training data and training label in the batch. The suffling method implemented by the code below uses ordered sampling without replacement. If the remaining data is insufficient to form a batch of the specified size, we'll drop the remaining ones. + +```python +def mini_batch(dataset, batch_size=128): + + # get total number of observations to batch from + size = dataset.train_data.shape[0] + + # fetch data and label + data = [] + label = [] + for one_obs in dataset: + data.append(np.array(one_obs[0])) + label.append(np.array(one_obs[1])) + data = np.array(data) + label = np.array(label) + + # random shuffle + idx = np.arange(size) + np.random.shuffle(idx) + data = data[idx] + label = label[idx] + + # split to batch + # discard the last batch if remaining size < batch_size + data_batches = [] + label_batches = [] + batched = 0 + while batched+batch_size <= size: + data_batches.append(data[batched:batched + batch_size]) + label_batches.append(label[batched:batched + batch_size]) + batched += batch_size + + data_batches = np.array(data_batches) + label_batches = np.array(label_batches) + + return zip(data_batches, label_batches) +``` + +### 2.2 Training Result: + + - Hyper-parameters: + * epoch number = 3 + * learning rate = 0.1 + + + - Model structure: + * 3 layers + * dimension reduced in each layer using a weight matrix + * softmax as the final activation function + * loss calculated as $-y^{T}\log{\hat{y}}$ + +```python +def forward(self, x): + x = x.reshape(-1, 28 * 28) + a0 = x + + #First Layer + z1 = self.matmul_1.forward(a0, self.W1) + a1 = self.relu_1.forward(z1) + + #Second Layer + z2 = self.matmul_2.forward(a1, self.W2) + a2 = self.relu_2.forward(z2) + + #Third Layer + z3 = self.matmul_3.forward(a2, self.W3) + a3 = self.softmax.forward(z3) + x = self.log.forward(a3) + + return x +``` + +```python +def backward(self, y): + + #Third Layer + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + + #Second Layer + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + + #First Layer + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + pass +``` + +Result: +```python +[0] Accuracy: 0.9482 +[1] Accuracy: 0.9643 +[2] Accuracy: 0.9735 +``` + + + +## 3. Change of Parameters + +**Learning Rate [0.2, 0.4, 0.6]** + + + + + +**Batch Size [20, 40, 60]** + + + + + +## 4. Optimization + +**Momentum:** + + + +```python +#Extends NumpyModel +#Save step size +self.delta1 = 0 +self.delta2 = 0 +self.delta3 = 0 + +#Extends the optimize function defined in numpy_fnn.py +if method == "Momentum": + self.delta1 = momentum * self.delta1 - learning_rate * self.W1_grad + self.W1 += self.delta1 + + self.delta2 = momentum * self.delta2 - learning_rate * self.W2_grad + self.W2 += self.delta2 + + self.delta3 = momentum * self.delta3 - learning_rate * self.W3_grad + self.W3 += self.delta3 +``` + +Result: +```python +[0] Accuracy: 0.9654 +[1] Accuracy: 0.9712 +[2] Accuracy: 0.9726 +``` + + + +Discussion: + +Momentum method differs from gradient descent in that it adds a short term memory for step size. Momentum, a number between 0 and 1, is the weight given to the previous step size. Momentum smoothes the path (visual demonstration below, [ref](https://dominikschmidt.xyz/nesterov-momentum/)). A greater momentum results in a more direct path with less changes in direction. Or, from another perspective, accelarates towards the optimum. If momentum = 0, the method reduces to Gradient Descent. + + + +Momentum $\beta$ can be thought of the weight of the previous force, while learning rate $\alpha$ can be thought of the weight of the current force. Intuitively, the optimal training model needs an optimal combination of $\alpha$ and $\beta$ depending on the eigenvalues of a convex function. Inappropriate values could lead to divergence. + + +```python + +``` -- Gitee From e6cb680abf49c3715a7419e79420bfaf29d6f46b Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Sun, 2 May 2021 22:54:57 +0800 Subject: [PATCH 32/68] =?UTF-8?q?=E5=88=A0=E9=99=A4=E6=96=87=E4=BB=B6=20as?= =?UTF-8?q?signment-2/submission/18340986009/README.md?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- assignment-2/submission/18340986009/README.md | 265 ------------------ 1 file changed, 265 deletions(-) delete mode 100644 assignment-2/submission/18340986009/README.md diff --git a/assignment-2/submission/18340986009/README.md b/assignment-2/submission/18340986009/README.md deleted file mode 100644 index 3352820..0000000 --- a/assignment-2/submission/18340986009/README.md +++ /dev/null @@ -1,265 +0,0 @@ -# FNN - -## 1. Activation Function - -The data used in this report is from the **torch** library. The data is a collection of 28$\small{\times}$28 matrices each representing a picture. The goal is to train the FNN model to identify which digit from 0 to 9 (c = 10) is in the picture. - -With a batch size of 128 (N = 128), in each epoch, our training data is a 3-order tensor of shape (128, 28, 28) and our label after one-hot encoding is a tensor of shape (128, 10). In the forwarding step, each matrix $\underset{28\times 28}{\mathrm{X}}$ is flattened to a vector $\underset{1\times 28^{2}}{\mathrm{x}}$ for computational purpose. There are 3 layers in total, each layer reshapes its input by the weight matrix $\mathrm{W}$. - -_For the sake of consistency, we'll use numerator layout throughout, -use subscript $X_{(i)}$ to denote the $ith$ row of $X$, superscript $X^{(j)}$ to denote the $jth$ column of $X$, and use $X_{ij}$ to denote the $(i,j)$ element of $X$_ - -### 1.1 Matmul - - - - - -```python - def backward(self, grad_y): - """ - grad_y: shape(N, d') - """ - x = self.memory['x'] - W = self.memory['W'] - - grad_x = np.matmul(grad_y, W.T) - grad_W = np.matmul(x.T, grad_y) - return grad_x, grad_W -``` - -### 1.2 Relu: - - - -```python - def backward(self, grad_y): - """ - grad_y: same shape as x - """ - x = self.memory['x'] - grad_x = np.where(x <= 0, 0, grad_y) - - return grad_x -``` - -### 1.3 Log - - - -```python - def backward(self, grad_y): - """ - grad_y: same shape as x - """ - x = self.memory['x'] - grad_x = (x + self.epsilon)**(-1) * grad_y - - return grad_x -``` - -### 1.4 Softmax: - - - -```python - def forward(self, x): - """ - x: shape(N, c) - """ - exp = np.exp(x - self.epsilon) - out = exp/exp.sum(axis=1, keepdims=True) - self.memory['x'] = x - self.memory['s'] = out - - return out - - def backward(self, grad_y): - """ - grad_y: same shape as x - """ - s = self.memory['s'] - n = s.shape[1] - - grad_x = np.zeros(s.shape) - # calculate Jacobian for ith row - for i in range(s.shape[0]): - jacob = np.zeros((n, n)) - - for j in range(n): - for k in range(n): - if j == k: - jacob[k, j] = s[i, k] * (1-s[i, j]) - else: - jacob[k, j] = -1 * s[i, k] * s[i, j] - # apply Jacobian to the ith row of grad_y - grad_x[i, :] = np.matmul(jacob, grad_y[i, :]) - - return grad_x -``` - -## 2. Model Training - -### 2.1 Change mini_batch: - -From documentation, we can see that mini_batch is a function that shuffles data and create small batches with user-specified batch size. The return is a batch such that the code: **for x, y in mini_batch(train_dataset):** in function **numpy_run()** can retrive appropriate training data and training label in the batch. The suffling method implemented by the code below uses ordered sampling without replacement. If the remaining data is insufficient to form a batch of the specified size, we'll drop the remaining ones. - -```python -def mini_batch(dataset, batch_size=128): - - # get total number of observations to batch from - size = dataset.train_data.shape[0] - - # fetch data and label - data = [] - label = [] - for one_obs in dataset: - data.append(np.array(one_obs[0])) - label.append(np.array(one_obs[1])) - data = np.array(data) - label = np.array(label) - - # random shuffle - idx = np.arange(size) - np.random.shuffle(idx) - data = data[idx] - label = label[idx] - - # split to batch - # discard the last batch if remaining size < batch_size - data_batches = [] - label_batches = [] - batched = 0 - while batched+batch_size <= size: - data_batches.append(data[batched:batched + batch_size]) - label_batches.append(label[batched:batched + batch_size]) - batched += batch_size - - data_batches = np.array(data_batches) - label_batches = np.array(label_batches) - - return zip(data_batches, label_batches) -``` - -### 2.2 Training Result: - - - Hyper-parameters: - * epoch number = 3 - * learning rate = 0.1 - - - - Model structure: - * 3 layers - * dimension reduced in each layer using a weight matrix - * softmax as the final activation function - * loss calculated as $-y^{T}\log{\hat{y}}$ - -```python -def forward(self, x): - x = x.reshape(-1, 28 * 28) - a0 = x - - #First Layer - z1 = self.matmul_1.forward(a0, self.W1) - a1 = self.relu_1.forward(z1) - - #Second Layer - z2 = self.matmul_2.forward(a1, self.W2) - a2 = self.relu_2.forward(z2) - - #Third Layer - z3 = self.matmul_3.forward(a2, self.W3) - a3 = self.softmax.forward(z3) - x = self.log.forward(a3) - - return x -``` - -```python -def backward(self, y): - - #Third Layer - self.log_grad = self.log.backward(y) - self.softmax_grad = self.softmax.backward(self.log_grad) - self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) - - #Second Layer - self.relu_2_grad = self.relu_2.backward(self.x3_grad) - self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) - - #First Layer - self.relu_1_grad = self.relu_1.backward(self.x2_grad) - self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) - - pass -``` - -Result: -```python -[0] Accuracy: 0.9482 -[1] Accuracy: 0.9643 -[2] Accuracy: 0.9735 -``` - - - -## 3. Change of Parameters - -**Learning Rate [0.2, 0.4, 0.6]** - - - - - -**Batch Size [20, 40, 60]** - - - - - -## 4. Optimization - -**Momentum:** - - - -```python -#Extends NumpyModel -#Save step size -self.delta1 = 0 -self.delta2 = 0 -self.delta3 = 0 - -#Extends the optimize function defined in numpy_fnn.py -if method == "Momentum": - self.delta1 = momentum * self.delta1 - learning_rate * self.W1_grad - self.W1 += self.delta1 - - self.delta2 = momentum * self.delta2 - learning_rate * self.W2_grad - self.W2 += self.delta2 - - self.delta3 = momentum * self.delta3 - learning_rate * self.W3_grad - self.W3 += self.delta3 -``` - -Result: -```python -[0] Accuracy: 0.9654 -[1] Accuracy: 0.9712 -[2] Accuracy: 0.9726 -``` - - - -Discussion: - -Momentum method differs from gradient descent in that it adds a short term memory for step size. Momentum, a number between 0 and 1, is the weight given to the previous step size. Momentum smoothes the path (visual demonstration below, [ref](https://dominikschmidt.xyz/nesterov-momentum/)). A greater momentum results in a more direct path with less changes in direction. Or, from another perspective, accelarates towards the optimum. If momentum = 0, the method reduces to Gradient Descent. - - - -Momentum $\beta$ can be thought of the weight of the previous force, while learning rate $\alpha$ can be thought of the weight of the current force. Intuitively, the optimal training model needs an optimal combination of $\alpha$ and $\beta$ depending on the eigenvalues of a convex function. Inappropriate values could lead to divergence. - - -```python - -``` -- Gitee From cf7e7cef0b9e43ab5b0fecb513978bb3dc439a1b Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Sun, 2 May 2021 22:55:16 +0800 Subject: [PATCH 33/68] (): --- assignment-2/submission/18340986009/README.md | 260 ++++++++++++++++++ 1 file changed, 260 insertions(+) create mode 100644 assignment-2/submission/18340986009/README.md diff --git a/assignment-2/submission/18340986009/README.md b/assignment-2/submission/18340986009/README.md new file mode 100644 index 0000000..2c6df32 --- /dev/null +++ b/assignment-2/submission/18340986009/README.md @@ -0,0 +1,260 @@ +# FNN + +## 1. Activation Function + + + +### 1.1 Matmul + + + + + +```python + def backward(self, grad_y): + """ + grad_y: shape(N, d') + """ + x = self.memory['x'] + W = self.memory['W'] + + grad_x = np.matmul(grad_y, W.T) + grad_W = np.matmul(x.T, grad_y) + return grad_x, grad_W +``` + +### 1.2 Relu: + + + +```python + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + x = self.memory['x'] + grad_x = np.where(x <= 0, 0, grad_y) + + return grad_x +``` + +### 1.3 Log + + + +```python + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + x = self.memory['x'] + grad_x = (x + self.epsilon)**(-1) * grad_y + + return grad_x +``` + +### 1.4 Softmax: + + + +```python + def forward(self, x): + """ + x: shape(N, c) + """ + exp = np.exp(x - self.epsilon) + out = exp/exp.sum(axis=1, keepdims=True) + self.memory['x'] = x + self.memory['s'] = out + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + s = self.memory['s'] + n = s.shape[1] + + grad_x = np.zeros(s.shape) + # calculate Jacobian for ith row + for i in range(s.shape[0]): + jacob = np.zeros((n, n)) + + for j in range(n): + for k in range(n): + if j == k: + jacob[k, j] = s[i, k] * (1-s[i, j]) + else: + jacob[k, j] = -1 * s[i, k] * s[i, j] + # apply Jacobian to the ith row of grad_y + grad_x[i, :] = np.matmul(jacob, grad_y[i, :]) + + return grad_x +``` + +## 2. Model Training + +### 2.1 Change mini_batch: + +From documentation, we can see that mini_batch is a function that shuffles data and create small batches with user-specified batch size. The return is a batch such that the code: **for x, y in mini_batch(train_dataset):** in function **numpy_run()** can retrive appropriate training data and training label in the batch. The suffling method implemented by the code below uses ordered sampling without replacement. If the remaining data is insufficient to form a batch of the specified size, we'll drop the remaining ones. + +```python +def mini_batch(dataset, batch_size=128): + + # get total number of observations to batch from + size = dataset.train_data.shape[0] + + # fetch data and label + data = [] + label = [] + for one_obs in dataset: + data.append(np.array(one_obs[0])) + label.append(np.array(one_obs[1])) + data = np.array(data) + label = np.array(label) + + # random shuffle + idx = np.arange(size) + np.random.shuffle(idx) + data = data[idx] + label = label[idx] + + # split to batch + # discard the last batch if remaining size < batch_size + data_batches = [] + label_batches = [] + batched = 0 + while batched+batch_size <= size: + data_batches.append(data[batched:batched + batch_size]) + label_batches.append(label[batched:batched + batch_size]) + batched += batch_size + + data_batches = np.array(data_batches) + label_batches = np.array(label_batches) + + return zip(data_batches, label_batches) +``` + +### 2.2 Training Result: + + - Hyper-parameters: + * epoch number = 3 + * learning rate = 0.1 + + + - Model structure: + * 3 layers + * dimension reduced in each layer using a weight matrix + * softmax as the final activation function + * loss calculated as $-y^{T}\log{\hat{y}}$ + +```python +def forward(self, x): + x = x.reshape(-1, 28 * 28) + a0 = x + + #First Layer + z1 = self.matmul_1.forward(a0, self.W1) + a1 = self.relu_1.forward(z1) + + #Second Layer + z2 = self.matmul_2.forward(a1, self.W2) + a2 = self.relu_2.forward(z2) + + #Third Layer + z3 = self.matmul_3.forward(a2, self.W3) + a3 = self.softmax.forward(z3) + x = self.log.forward(a3) + + return x +``` + +```python +def backward(self, y): + + #Third Layer + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + + #Second Layer + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + + #First Layer + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + pass +``` + +Result: +```python +[0] Accuracy: 0.9482 +[1] Accuracy: 0.9643 +[2] Accuracy: 0.9735 +``` + + + +## 3. Change of Parameters + +**Learning Rate [0.2, 0.4, 0.6]** + + + + + +**Batch Size [20, 40, 60]** + + + + + +## 4. Optimization + +**Momentum:** + + + +```python +#Extends NumpyModel +#Save step size +self.delta1 = 0 +self.delta2 = 0 +self.delta3 = 0 + +#Extends the optimize function defined in numpy_fnn.py +if method == "Momentum": + self.delta1 = momentum * self.delta1 - learning_rate * self.W1_grad + self.W1 += self.delta1 + + self.delta2 = momentum * self.delta2 - learning_rate * self.W2_grad + self.W2 += self.delta2 + + self.delta3 = momentum * self.delta3 - learning_rate * self.W3_grad + self.W3 += self.delta3 +``` + +Result: +```python +[0] Accuracy: 0.9654 +[1] Accuracy: 0.9712 +[2] Accuracy: 0.9726 +``` + + + +Discussion: + +Momentum method differs from gradient descent in that it adds a short term memory for step size. Momentum, a number between 0 and 1, is the weight given to the previous step size. Momentum smoothes the path (visual demonstration below, [ref](https://dominikschmidt.xyz/nesterov-momentum/)). A greater momentum results in a more direct path with less changes in direction. Or, from another perspective, accelarates towards the optimum. If momentum = 0, the method reduces to Gradient Descent. + + + +Momentum $\beta$ can be thought of the weight of the previous force, while learning rate $\alpha$ can be thought of the weight of the current force. Intuitively, the optimal training model needs an optimal combination of $\alpha$ and $\beta$ depending on the eigenvalues of a convex function. Inappropriate values could lead to divergence. + + +```python + +``` -- Gitee From cec8ba62e7d5ba407e011135a0f35179e38954af Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 3 May 2021 08:52:26 +0800 Subject: [PATCH 34/68] (): --- assignment-2/submission/numpy_fnn.py | 198 +++++++++++++++++++++++++ assignment-2/submission/numpy_mnist.py | 144 ++++++++++++++++++ 2 files changed, 342 insertions(+) create mode 100644 assignment-2/submission/numpy_fnn.py create mode 100644 assignment-2/submission/numpy_mnist.py diff --git a/assignment-2/submission/numpy_fnn.py b/assignment-2/submission/numpy_fnn.py new file mode 100644 index 0000000..7d8bac7 --- /dev/null +++ b/assignment-2/submission/numpy_fnn.py @@ -0,0 +1,198 @@ +import numpy as np + + +class NumpyOp: + + def __init__(self): + self.memory = {} + self.epsilon = 1e-12 + + +class Matmul(NumpyOp): + + def forward(self, x, W): + """ + x: shape(N, d) + w: shape(d, d') + """ + self.memory['x'] = x + self.memory['W'] = W + h = np.matmul(x, W) + + return h + + def backward(self, grad_y): + """ + grad_y: shape(N, d') + """ + x = self.memory['x'] + W = self.memory['W'] + + grad_x = np.matmul(grad_y, W.T) + grad_W = np.matmul(x.T, grad_y) + return grad_x, grad_W + + +class Relu(NumpyOp): + + def forward(self, x): + self.memory['x'] = x + return np.where(x > 0, x, np.zeros_like(x)) + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + x = self.memory['x'] + grad_x = np.where(x <= 0, 0, grad_y) + + return grad_x + + +class Log(NumpyOp): + + def forward(self, x): + """ + x: shape(N, c) + """ + out = np.log(x + self.epsilon) + self.memory['x'] = x + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + x = self.memory['x'] + grad_x = (x + self.epsilon)**(-1) * grad_y + + return grad_x + + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + exp = np.exp(x + self.epsilon) + out = exp/exp.sum(axis=1, keepdims=True) + self.memory['x'] = x + self.memory['s'] = out + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + s = self.memory['s'] + n = s.shape[1] + + grad_x = np.zeros(s.shape) + # calculate Jacobian for ith row + for i in range(s.shape[0]): + jacob = np.zeros((n, n)) + + for j in range(n): + for k in range(n): + if j == k: + jacob[k, j] = s[i, k] * (1-s[i, j]) + else: + jacob[k, j] = -1 * s[i, k] * s[i, j] + # apply Jacobian to the ith row of grad_y + grad_x[i, :] = np.matmul(jacob, grad_y[i, :]) + + return grad_x + + +class NumpyLoss: + + def __init__(self): + self.target = None + + def get_loss(self, pred, target): + self.target = target + return (-1 * pred * target).sum(axis=1).mean() + + def backward(self): + return -1 * self.target / self.target.shape[0] + + +class NumpyModel: + def __init__(self): + self.epsilon = 1e-12 + + self.W1 = np.random.normal(size=(28 * 28, 256)) + self.W2 = np.random.normal(size=(256, 64)) + self.W3 = np.random.normal(size=(64, 10)) + + # 以下算子会在 forward 和 backward 中使用 + self.matmul_1 = Matmul() + self.relu_1 = Relu() + self.matmul_2 = Matmul() + self.relu_2 = Relu() + self.matmul_3 = Matmul() + self.softmax = Softmax() + self.log = Log() + + # 以下变量需要在 backward 中更新。 softmax_grad, log_grad 等为算子反向传播的梯度( loss 关于算子输入的偏导) + self.x1_grad, self.W1_grad = None, None + self.relu_1_grad = None + self.x2_grad, self.W2_grad = None, None + self.relu_2_grad = None + self.x3_grad, self.W3_grad = None, None + self.softmax_grad = None + self.log_grad = None + + # 如需要,以下变量在optimize中更新 + self.delta1 = 0 + self.delta2 = 0 + self.delta3 = 0 + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + a0 = x + z1 = self.matmul_1.forward(a0, self.W1) + a1 = self.relu_1.forward(z1) + z2 = self.matmul_2.forward(a1, self.W2) + a2 = self.relu_2.forward(z2) + z3 = self.matmul_3.forward(a2, self.W3) + a3 = self.softmax.forward(z3) + x = self.log.forward(a3) + + return x + + def backward(self, y): + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + pass + + def optimize(self, learning_rate, method="GD", momentum=0.9, gamma=0.9): + if method == "GD": + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad + + # idea: want different learning rate on different dimensions + if method == "Momentum": + self.delta1 = momentum * self.delta1 - learning_rate * self.W1_grad + self.W1 += self.delta1 + + self.delta2 = momentum * self.delta2 - learning_rate * self.W2_grad + self.W2 += self.delta2 + + self.delta3 = momentum * self.delta3 - learning_rate * self.W3_grad + self.W3 += self.delta3 + + diff --git a/assignment-2/submission/numpy_mnist.py b/assignment-2/submission/numpy_mnist.py new file mode 100644 index 0000000..6d2ae6d --- /dev/null +++ b/assignment-2/submission/numpy_mnist.py @@ -0,0 +1,144 @@ +import numpy as np +import torch +from numpy_fnn import NumpyModel, NumpyLoss +from matplotlib import pyplot as plt + + +def plot_curve(data): + plt.plot(range(len(data)), data, color='blue') + plt.legend(['loss_value'], loc='upper right') + plt.xlabel('step') + plt.ylabel('value') + plt.show() + + +def download_mnist(): + from torchvision import datasets, transforms + + transform = transforms.Compose([ + transforms.ToTensor(), + transforms.Normalize(mean=(0.1307,), std=(0.3081,)) + ]) + + train_dataset = datasets.MNIST(root="./data/", transform=transform, train=True, download=True) + test_dataset = datasets.MNIST(root="./data/", transform=transform, train=False, download=True) + + return train_dataset, test_dataset + + +def one_hot(y, numpy=True): + if numpy: + y_ = np.zeros((y.shape[0], 10)) + y_[np.arange(y.shape[0], dtype=np.int32), y] = 1 + return y_ + else: + y_ = torch.zeros((y.shape[0], 10)) + y_[torch.arange(y.shape[0], dtype=torch.long), y] = 1 + return y_ + + +def batch(dataset, numpy=True): + data = [] + label = [] + for each in dataset: + data.append(each[0]) + label.append(each[1]) + data = torch.stack(data) + label = torch.LongTensor(label) + if numpy: + return [(data.numpy(), label.numpy())] + else: + return [(data, label)] + + +def mini_batch(dataset, batch_size=128): + size = dataset.train_data.shape[0] + + # fetch data and label from dataset MNIST + data = [] + label = [] + for one_obs in dataset: + data.append(np.array(one_obs[0])) + label.append(np.array(one_obs[1])) + + data = np.array(data) + label = np.array(label) + + # shuffle + idx = np.arange(size) + np.random.shuffle(idx) + data = data[idx] + label = label[idx] + + # split to batch, discard the last batch if remaining size < batch_size + data_batches = [] + label_batches = [] + batched = 0 + while batched+batch_size <= size: + data_batches.append(data[batched:batched + batch_size]) + label_batches.append(label[batched:batched + batch_size]) + batched += batch_size + + data_batches = np.array(data_batches) + label_batches = np.array(label_batches) + + return zip(data_batches, label_batches) + + +def get_torch_initialization(numpy=True): + fc1 = torch.nn.Linear(28 * 28, 256) + fc2 = torch.nn.Linear(256, 64) + fc3 = torch.nn.Linear(64, 10) + + if numpy: + W1 = fc1.weight.T.detach().clone().numpy() + W2 = fc2.weight.T.detach().clone().numpy() + W3 = fc3.weight.T.detach().clone().numpy() + else: + W1 = fc1.weight.T.detach().clone().data + W2 = fc2.weight.T.detach().clone().data + W3 = fc3.weight.T.detach().clone().data + + return W1, W2, W3 + + +def numpy_run(epoch_number=3, learning_rate=0.1, batch_size=128): + train_dataset, test_dataset = download_mnist() + + model = NumpyModel() + numpy_loss = NumpyLoss() + model.W1, model.W2, model.W3 = get_torch_initialization() + + train_loss = [] + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset, batch_size): + y = one_hot(y) + + y_pred = model.forward(x) + loss = numpy_loss.get_loss(y_pred, y) + + model.backward(numpy_loss.backward()) + model.optimize(learning_rate) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset)[0] + accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + numpy_run() + +# for lr in [2, 4, 6]: +# l_rate = lr/10 +# numpy_run(epoch_number=3, learning_rate=l_rate, batch_size=128) + +# for bs in [20, 40, 60]: +# numpy_run(epoch_number=3, learning_rate=0.1, batch_size=bs) + + + -- Gitee From eac80e5a04cad528b545daebaba1ed4089698615 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 3 May 2021 08:52:42 +0800 Subject: [PATCH 35/68] =?UTF-8?q?=E5=88=A0=E9=99=A4=E6=96=87=E4=BB=B6=20as?= =?UTF-8?q?signment-2/submission/.keep?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- assignment-2/submission/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 assignment-2/submission/.keep diff --git a/assignment-2/submission/.keep b/assignment-2/submission/.keep deleted file mode 100644 index e69de29..0000000 -- Gitee From 8c93f17cf3f0d9f079da79358f2d35f6b9701eec Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 3 May 2021 08:53:05 +0800 Subject: [PATCH 36/68] =?UTF-8?q?=E5=88=A0=E9=99=A4=E6=96=87=E4=BB=B6=20as?= =?UTF-8?q?signment-2/submission/numpy=5Ffnn.py?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- assignment-2/submission/numpy_fnn.py | 198 --------------------------- 1 file changed, 198 deletions(-) delete mode 100644 assignment-2/submission/numpy_fnn.py diff --git a/assignment-2/submission/numpy_fnn.py b/assignment-2/submission/numpy_fnn.py deleted file mode 100644 index 7d8bac7..0000000 --- a/assignment-2/submission/numpy_fnn.py +++ /dev/null @@ -1,198 +0,0 @@ -import numpy as np - - -class NumpyOp: - - def __init__(self): - self.memory = {} - self.epsilon = 1e-12 - - -class Matmul(NumpyOp): - - def forward(self, x, W): - """ - x: shape(N, d) - w: shape(d, d') - """ - self.memory['x'] = x - self.memory['W'] = W - h = np.matmul(x, W) - - return h - - def backward(self, grad_y): - """ - grad_y: shape(N, d') - """ - x = self.memory['x'] - W = self.memory['W'] - - grad_x = np.matmul(grad_y, W.T) - grad_W = np.matmul(x.T, grad_y) - return grad_x, grad_W - - -class Relu(NumpyOp): - - def forward(self, x): - self.memory['x'] = x - return np.where(x > 0, x, np.zeros_like(x)) - - def backward(self, grad_y): - """ - grad_y: same shape as x - """ - x = self.memory['x'] - grad_x = np.where(x <= 0, 0, grad_y) - - return grad_x - - -class Log(NumpyOp): - - def forward(self, x): - """ - x: shape(N, c) - """ - out = np.log(x + self.epsilon) - self.memory['x'] = x - - return out - - def backward(self, grad_y): - """ - grad_y: same shape as x - """ - x = self.memory['x'] - grad_x = (x + self.epsilon)**(-1) * grad_y - - return grad_x - - -class Softmax(NumpyOp): - """ - softmax over last dimension - """ - - def forward(self, x): - """ - x: shape(N, c) - """ - exp = np.exp(x + self.epsilon) - out = exp/exp.sum(axis=1, keepdims=True) - self.memory['x'] = x - self.memory['s'] = out - - return out - - def backward(self, grad_y): - """ - grad_y: same shape as x - """ - s = self.memory['s'] - n = s.shape[1] - - grad_x = np.zeros(s.shape) - # calculate Jacobian for ith row - for i in range(s.shape[0]): - jacob = np.zeros((n, n)) - - for j in range(n): - for k in range(n): - if j == k: - jacob[k, j] = s[i, k] * (1-s[i, j]) - else: - jacob[k, j] = -1 * s[i, k] * s[i, j] - # apply Jacobian to the ith row of grad_y - grad_x[i, :] = np.matmul(jacob, grad_y[i, :]) - - return grad_x - - -class NumpyLoss: - - def __init__(self): - self.target = None - - def get_loss(self, pred, target): - self.target = target - return (-1 * pred * target).sum(axis=1).mean() - - def backward(self): - return -1 * self.target / self.target.shape[0] - - -class NumpyModel: - def __init__(self): - self.epsilon = 1e-12 - - self.W1 = np.random.normal(size=(28 * 28, 256)) - self.W2 = np.random.normal(size=(256, 64)) - self.W3 = np.random.normal(size=(64, 10)) - - # 以下算子会在 forward 和 backward 中使用 - self.matmul_1 = Matmul() - self.relu_1 = Relu() - self.matmul_2 = Matmul() - self.relu_2 = Relu() - self.matmul_3 = Matmul() - self.softmax = Softmax() - self.log = Log() - - # 以下变量需要在 backward 中更新。 softmax_grad, log_grad 等为算子反向传播的梯度( loss 关于算子输入的偏导) - self.x1_grad, self.W1_grad = None, None - self.relu_1_grad = None - self.x2_grad, self.W2_grad = None, None - self.relu_2_grad = None - self.x3_grad, self.W3_grad = None, None - self.softmax_grad = None - self.log_grad = None - - # 如需要,以下变量在optimize中更新 - self.delta1 = 0 - self.delta2 = 0 - self.delta3 = 0 - - def forward(self, x): - x = x.reshape(-1, 28 * 28) - a0 = x - z1 = self.matmul_1.forward(a0, self.W1) - a1 = self.relu_1.forward(z1) - z2 = self.matmul_2.forward(a1, self.W2) - a2 = self.relu_2.forward(z2) - z3 = self.matmul_3.forward(a2, self.W3) - a3 = self.softmax.forward(z3) - x = self.log.forward(a3) - - return x - - def backward(self, y): - self.log_grad = self.log.backward(y) - self.softmax_grad = self.softmax.backward(self.log_grad) - self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) - self.relu_2_grad = self.relu_2.backward(self.x3_grad) - self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) - self.relu_1_grad = self.relu_1.backward(self.x2_grad) - self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) - - pass - - def optimize(self, learning_rate, method="GD", momentum=0.9, gamma=0.9): - if method == "GD": - self.W1 -= learning_rate * self.W1_grad - self.W2 -= learning_rate * self.W2_grad - self.W3 -= learning_rate * self.W3_grad - - # idea: want different learning rate on different dimensions - if method == "Momentum": - self.delta1 = momentum * self.delta1 - learning_rate * self.W1_grad - self.W1 += self.delta1 - - self.delta2 = momentum * self.delta2 - learning_rate * self.W2_grad - self.W2 += self.delta2 - - self.delta3 = momentum * self.delta3 - learning_rate * self.W3_grad - self.W3 += self.delta3 - - -- Gitee From 6451bd8ccd18ba5ad187e80dd458897358e3ee6f Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 3 May 2021 08:53:14 +0800 Subject: [PATCH 37/68] =?UTF-8?q?=E5=88=A0=E9=99=A4=E6=96=87=E4=BB=B6=20as?= =?UTF-8?q?signment-2/submission/numpy=5Fmnist.py?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- assignment-2/submission/numpy_mnist.py | 144 ------------------------- 1 file changed, 144 deletions(-) delete mode 100644 assignment-2/submission/numpy_mnist.py diff --git a/assignment-2/submission/numpy_mnist.py b/assignment-2/submission/numpy_mnist.py deleted file mode 100644 index 6d2ae6d..0000000 --- a/assignment-2/submission/numpy_mnist.py +++ /dev/null @@ -1,144 +0,0 @@ -import numpy as np -import torch -from numpy_fnn import NumpyModel, NumpyLoss -from matplotlib import pyplot as plt - - -def plot_curve(data): - plt.plot(range(len(data)), data, color='blue') - plt.legend(['loss_value'], loc='upper right') - plt.xlabel('step') - plt.ylabel('value') - plt.show() - - -def download_mnist(): - from torchvision import datasets, transforms - - transform = transforms.Compose([ - transforms.ToTensor(), - transforms.Normalize(mean=(0.1307,), std=(0.3081,)) - ]) - - train_dataset = datasets.MNIST(root="./data/", transform=transform, train=True, download=True) - test_dataset = datasets.MNIST(root="./data/", transform=transform, train=False, download=True) - - return train_dataset, test_dataset - - -def one_hot(y, numpy=True): - if numpy: - y_ = np.zeros((y.shape[0], 10)) - y_[np.arange(y.shape[0], dtype=np.int32), y] = 1 - return y_ - else: - y_ = torch.zeros((y.shape[0], 10)) - y_[torch.arange(y.shape[0], dtype=torch.long), y] = 1 - return y_ - - -def batch(dataset, numpy=True): - data = [] - label = [] - for each in dataset: - data.append(each[0]) - label.append(each[1]) - data = torch.stack(data) - label = torch.LongTensor(label) - if numpy: - return [(data.numpy(), label.numpy())] - else: - return [(data, label)] - - -def mini_batch(dataset, batch_size=128): - size = dataset.train_data.shape[0] - - # fetch data and label from dataset MNIST - data = [] - label = [] - for one_obs in dataset: - data.append(np.array(one_obs[0])) - label.append(np.array(one_obs[1])) - - data = np.array(data) - label = np.array(label) - - # shuffle - idx = np.arange(size) - np.random.shuffle(idx) - data = data[idx] - label = label[idx] - - # split to batch, discard the last batch if remaining size < batch_size - data_batches = [] - label_batches = [] - batched = 0 - while batched+batch_size <= size: - data_batches.append(data[batched:batched + batch_size]) - label_batches.append(label[batched:batched + batch_size]) - batched += batch_size - - data_batches = np.array(data_batches) - label_batches = np.array(label_batches) - - return zip(data_batches, label_batches) - - -def get_torch_initialization(numpy=True): - fc1 = torch.nn.Linear(28 * 28, 256) - fc2 = torch.nn.Linear(256, 64) - fc3 = torch.nn.Linear(64, 10) - - if numpy: - W1 = fc1.weight.T.detach().clone().numpy() - W2 = fc2.weight.T.detach().clone().numpy() - W3 = fc3.weight.T.detach().clone().numpy() - else: - W1 = fc1.weight.T.detach().clone().data - W2 = fc2.weight.T.detach().clone().data - W3 = fc3.weight.T.detach().clone().data - - return W1, W2, W3 - - -def numpy_run(epoch_number=3, learning_rate=0.1, batch_size=128): - train_dataset, test_dataset = download_mnist() - - model = NumpyModel() - numpy_loss = NumpyLoss() - model.W1, model.W2, model.W3 = get_torch_initialization() - - train_loss = [] - - for epoch in range(epoch_number): - for x, y in mini_batch(train_dataset, batch_size): - y = one_hot(y) - - y_pred = model.forward(x) - loss = numpy_loss.get_loss(y_pred, y) - - model.backward(numpy_loss.backward()) - model.optimize(learning_rate) - - train_loss.append(loss.item()) - - x, y = batch(test_dataset)[0] - accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) - print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) - - plot_curve(train_loss) - - -if __name__ == "__main__": - numpy_run() - -# for lr in [2, 4, 6]: -# l_rate = lr/10 -# numpy_run(epoch_number=3, learning_rate=l_rate, batch_size=128) - -# for bs in [20, 40, 60]: -# numpy_run(epoch_number=3, learning_rate=0.1, batch_size=bs) - - - -- Gitee From 83cf1531dc8bb4c92a7b6907dd7e0cd00fb294d6 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 3 May 2021 08:53:35 +0800 Subject: [PATCH 38/68] =?UTF-8?q?=E5=88=A0=E9=99=A4=E6=96=87=E4=BB=B6=20as?= =?UTF-8?q?signment-2/submission/18340986009/README.md?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- assignment-2/submission/18340986009/README.md | 260 ------------------ 1 file changed, 260 deletions(-) delete mode 100644 assignment-2/submission/18340986009/README.md diff --git a/assignment-2/submission/18340986009/README.md b/assignment-2/submission/18340986009/README.md deleted file mode 100644 index 2c6df32..0000000 --- a/assignment-2/submission/18340986009/README.md +++ /dev/null @@ -1,260 +0,0 @@ -# FNN - -## 1. Activation Function - - - -### 1.1 Matmul - - - - - -```python - def backward(self, grad_y): - """ - grad_y: shape(N, d') - """ - x = self.memory['x'] - W = self.memory['W'] - - grad_x = np.matmul(grad_y, W.T) - grad_W = np.matmul(x.T, grad_y) - return grad_x, grad_W -``` - -### 1.2 Relu: - - - -```python - def backward(self, grad_y): - """ - grad_y: same shape as x - """ - x = self.memory['x'] - grad_x = np.where(x <= 0, 0, grad_y) - - return grad_x -``` - -### 1.3 Log - - - -```python - def backward(self, grad_y): - """ - grad_y: same shape as x - """ - x = self.memory['x'] - grad_x = (x + self.epsilon)**(-1) * grad_y - - return grad_x -``` - -### 1.4 Softmax: - - - -```python - def forward(self, x): - """ - x: shape(N, c) - """ - exp = np.exp(x - self.epsilon) - out = exp/exp.sum(axis=1, keepdims=True) - self.memory['x'] = x - self.memory['s'] = out - - return out - - def backward(self, grad_y): - """ - grad_y: same shape as x - """ - s = self.memory['s'] - n = s.shape[1] - - grad_x = np.zeros(s.shape) - # calculate Jacobian for ith row - for i in range(s.shape[0]): - jacob = np.zeros((n, n)) - - for j in range(n): - for k in range(n): - if j == k: - jacob[k, j] = s[i, k] * (1-s[i, j]) - else: - jacob[k, j] = -1 * s[i, k] * s[i, j] - # apply Jacobian to the ith row of grad_y - grad_x[i, :] = np.matmul(jacob, grad_y[i, :]) - - return grad_x -``` - -## 2. Model Training - -### 2.1 Change mini_batch: - -From documentation, we can see that mini_batch is a function that shuffles data and create small batches with user-specified batch size. The return is a batch such that the code: **for x, y in mini_batch(train_dataset):** in function **numpy_run()** can retrive appropriate training data and training label in the batch. The suffling method implemented by the code below uses ordered sampling without replacement. If the remaining data is insufficient to form a batch of the specified size, we'll drop the remaining ones. - -```python -def mini_batch(dataset, batch_size=128): - - # get total number of observations to batch from - size = dataset.train_data.shape[0] - - # fetch data and label - data = [] - label = [] - for one_obs in dataset: - data.append(np.array(one_obs[0])) - label.append(np.array(one_obs[1])) - data = np.array(data) - label = np.array(label) - - # random shuffle - idx = np.arange(size) - np.random.shuffle(idx) - data = data[idx] - label = label[idx] - - # split to batch - # discard the last batch if remaining size < batch_size - data_batches = [] - label_batches = [] - batched = 0 - while batched+batch_size <= size: - data_batches.append(data[batched:batched + batch_size]) - label_batches.append(label[batched:batched + batch_size]) - batched += batch_size - - data_batches = np.array(data_batches) - label_batches = np.array(label_batches) - - return zip(data_batches, label_batches) -``` - -### 2.2 Training Result: - - - Hyper-parameters: - * epoch number = 3 - * learning rate = 0.1 - - - - Model structure: - * 3 layers - * dimension reduced in each layer using a weight matrix - * softmax as the final activation function - * loss calculated as $-y^{T}\log{\hat{y}}$ - -```python -def forward(self, x): - x = x.reshape(-1, 28 * 28) - a0 = x - - #First Layer - z1 = self.matmul_1.forward(a0, self.W1) - a1 = self.relu_1.forward(z1) - - #Second Layer - z2 = self.matmul_2.forward(a1, self.W2) - a2 = self.relu_2.forward(z2) - - #Third Layer - z3 = self.matmul_3.forward(a2, self.W3) - a3 = self.softmax.forward(z3) - x = self.log.forward(a3) - - return x -``` - -```python -def backward(self, y): - - #Third Layer - self.log_grad = self.log.backward(y) - self.softmax_grad = self.softmax.backward(self.log_grad) - self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) - - #Second Layer - self.relu_2_grad = self.relu_2.backward(self.x3_grad) - self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) - - #First Layer - self.relu_1_grad = self.relu_1.backward(self.x2_grad) - self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) - - pass -``` - -Result: -```python -[0] Accuracy: 0.9482 -[1] Accuracy: 0.9643 -[2] Accuracy: 0.9735 -``` - - - -## 3. Change of Parameters - -**Learning Rate [0.2, 0.4, 0.6]** - - - - - -**Batch Size [20, 40, 60]** - - - - - -## 4. Optimization - -**Momentum:** - - - -```python -#Extends NumpyModel -#Save step size -self.delta1 = 0 -self.delta2 = 0 -self.delta3 = 0 - -#Extends the optimize function defined in numpy_fnn.py -if method == "Momentum": - self.delta1 = momentum * self.delta1 - learning_rate * self.W1_grad - self.W1 += self.delta1 - - self.delta2 = momentum * self.delta2 - learning_rate * self.W2_grad - self.W2 += self.delta2 - - self.delta3 = momentum * self.delta3 - learning_rate * self.W3_grad - self.W3 += self.delta3 -``` - -Result: -```python -[0] Accuracy: 0.9654 -[1] Accuracy: 0.9712 -[2] Accuracy: 0.9726 -``` - - - -Discussion: - -Momentum method differs from gradient descent in that it adds a short term memory for step size. Momentum, a number between 0 and 1, is the weight given to the previous step size. Momentum smoothes the path (visual demonstration below, [ref](https://dominikschmidt.xyz/nesterov-momentum/)). A greater momentum results in a more direct path with less changes in direction. Or, from another perspective, accelarates towards the optimum. If momentum = 0, the method reduces to Gradient Descent. - - - -Momentum $\beta$ can be thought of the weight of the previous force, while learning rate $\alpha$ can be thought of the weight of the current force. Intuitively, the optimal training model needs an optimal combination of $\alpha$ and $\beta$ depending on the eigenvalues of a convex function. Inappropriate values could lead to divergence. - - -```python - -``` -- Gitee From 3ef016e9d3ed15ed4822f5061b923d9faeb991bf Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 3 May 2021 08:55:06 +0800 Subject: [PATCH 39/68] (): --- assignment-2/submission/18340986009/img/md0.png | Bin 0 -> 91682 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 assignment-2/submission/18340986009/img/md0.png diff --git a/assignment-2/submission/18340986009/img/md0.png b/assignment-2/submission/18340986009/img/md0.png new file mode 100644 index 0000000000000000000000000000000000000000..21513b1a3ced81649b595e1a879352b845ee1452 GIT binary patch literal 91682 zcmeFY1y@_`7Bxy~fffo>kODSS1WIw2;1G%j zEgBN+OW*f=XPo;N?!9}AjPUFvgT0?M*IIMU%zV{QdrwSAO^AboL#(JEtA&Gur-y@s z`|LhG_RQy@#$fEz9Z#+I?{F$do+Gh~dv?;Q(l|KPF+^9Ec-Zv=Hw8ma930y4zmGdH z8%H)cIJX6gveG(!W_#DJ_NM&-TQP(J&YLVDzv`>(dp8@LC$@8|yZp#xdl?#2Ov;YV z-0X?~V$$m~JItMrN<5|gt1UrT@aR%8jbkc>F);qdhpP=q~s|7k$=$4gc24ogx>Vh%% 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qYkEF$14};7%_jA^Zj(T%kP)959^bhOWY~{BP>^&eA;;G@snSlgP3;7 zPJW|f;=(XFzwYQ%JM&cG6`*x8^2t*ck7siYcLDdlR61P$v3KFsj}NR5 zo!#o>b-=1F?&(QY)j3(AI)@(s*VAUmP6Tdg0D}f){>IREn|Int{y9~(BG66sghqO+ z)@G$*nODb8l{)wy^q0KQdv3v07lzBX|9G){JvD#!9hOzIdp`OKUgsNkH_HVsWBa%@&wl31XZJ;Ab~2xYg?9n) z#)4PDW)WID?k(dmTGjtp`!x59=6RV@BA31j32e1nuzFJV9JjPB(}2PHSSxv!=4la^ zrd>T89PZQAPds|Z63@MT`>}O%dO!UIdhfi2wKwp(fz+MAbK27){p7D+;XPtEb79{9 z?Vpr4a!SDLlriO4^Xn4xj54nYg%zDgSUghAG8guTrcIr7jZxIt$@leYt?W5`tAmZb zfnmC1)%71X`zmKLNec9xPw`bx6JO367Ju$C>woJ<*&9#so)D7G5)Dnieu{h5w$R9` zW965-^w}Q(7e1)@O@>ug4YlBKHMnIKR3v}Ngnxo7bF0ybin6e||26)v%gYdbZa3$d zu~hQwTB-8K{F=r_>|!3j9ri}_f6;OETl;2b=YciO$8IbU+SAKYDrtDw+-|Zd^G=b5 z+-J}Ae!qOU?$5@rp`h#K=7--)<=ApZch%R*pT~i9(zLQ~dB6ePi`GUA{4l?}^D(Wy z^KNdST*I16M`C?XoW8qoYN6bU;*lZiupr`7V7?@SFPr4kp7Ul!zy3|QHt*d} zV=2&@@Pv<6b60(`UBZ*?`~E_USoe}B`8 z%a;uJxi-Hx3;~`GeDLU^z2f{CAAbR-`TjrqyH0B5>A6);+QNQLSM2!E^j0{w&LQr* zZD7&b+Yb&D!2*!M9eAG&uxM>;`@!S;Hqg>N`1MK4&Tn&n$KHyP5ZbI>>i+bOh^oEW z;b#Gl4FAuO4(HUhF~7WPv)$AChYoLXTjL2F$2ix1Fz@a|;N@eLRdYJ7-D*A?X}WFt z)P(99PL8?Ip$g+_6Y%wUAZz&qfrs`>-QE(m+vzWGL+gc0@Wue}B;NB-`SXe#qfVakh oLJI1mSKh$V`Y;X+@kRf|ndUS8n^5!W6(}Kiy85}Sb4q9e0ONJhBLDyZ literal 0 HcmV?d00001 -- Gitee From eecd9aa6ee78150f6b22b41eae531f634bfee99d Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 3 May 2021 08:57:50 +0800 Subject: [PATCH 40/68] (): --- assignment-2/submission/18340986009/README.md | 260 ++++++++++++++++++ .../submission/18340986009/numpy_fnn.py | 198 +++++++++++++ .../submission/18340986009/numpy_mnist.py | 144 ++++++++++ 3 files changed, 602 insertions(+) create mode 100644 assignment-2/submission/18340986009/README.md create mode 100644 assignment-2/submission/18340986009/numpy_fnn.py create mode 100644 assignment-2/submission/18340986009/numpy_mnist.py diff --git a/assignment-2/submission/18340986009/README.md b/assignment-2/submission/18340986009/README.md new file mode 100644 index 0000000..2c6df32 --- /dev/null +++ b/assignment-2/submission/18340986009/README.md @@ -0,0 +1,260 @@ +# FNN + +## 1. Activation Function + + + +### 1.1 Matmul + + + + + +```python + def backward(self, grad_y): + """ + grad_y: shape(N, d') + """ + x = self.memory['x'] + W = self.memory['W'] + + grad_x = np.matmul(grad_y, W.T) + grad_W = np.matmul(x.T, grad_y) + return grad_x, grad_W +``` + +### 1.2 Relu: + + + +```python + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + x = self.memory['x'] + grad_x = np.where(x <= 0, 0, grad_y) + + return grad_x +``` + +### 1.3 Log + + + +```python + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + x = self.memory['x'] + grad_x = (x + self.epsilon)**(-1) * grad_y + + return grad_x +``` + +### 1.4 Softmax: + + + +```python + def forward(self, x): + """ + x: shape(N, c) + """ + exp = np.exp(x - self.epsilon) + out = exp/exp.sum(axis=1, keepdims=True) + self.memory['x'] = x + self.memory['s'] = out + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + s = self.memory['s'] + n = s.shape[1] + + grad_x = np.zeros(s.shape) + # calculate Jacobian for ith row + for i in range(s.shape[0]): + jacob = np.zeros((n, n)) + + for j in range(n): + for k in range(n): + if j == k: + jacob[k, j] = s[i, k] * (1-s[i, j]) + else: + jacob[k, j] = -1 * s[i, k] * s[i, j] + # apply Jacobian to the ith row of grad_y + grad_x[i, :] = np.matmul(jacob, grad_y[i, :]) + + return grad_x +``` + +## 2. Model Training + +### 2.1 Change mini_batch: + +From documentation, we can see that mini_batch is a function that shuffles data and create small batches with user-specified batch size. The return is a batch such that the code: **for x, y in mini_batch(train_dataset):** in function **numpy_run()** can retrive appropriate training data and training label in the batch. The suffling method implemented by the code below uses ordered sampling without replacement. If the remaining data is insufficient to form a batch of the specified size, we'll drop the remaining ones. + +```python +def mini_batch(dataset, batch_size=128): + + # get total number of observations to batch from + size = dataset.train_data.shape[0] + + # fetch data and label + data = [] + label = [] + for one_obs in dataset: + data.append(np.array(one_obs[0])) + label.append(np.array(one_obs[1])) + data = np.array(data) + label = np.array(label) + + # random shuffle + idx = np.arange(size) + np.random.shuffle(idx) + data = data[idx] + label = label[idx] + + # split to batch + # discard the last batch if remaining size < batch_size + data_batches = [] + label_batches = [] + batched = 0 + while batched+batch_size <= size: + data_batches.append(data[batched:batched + batch_size]) + label_batches.append(label[batched:batched + batch_size]) + batched += batch_size + + data_batches = np.array(data_batches) + label_batches = np.array(label_batches) + + return zip(data_batches, label_batches) +``` + +### 2.2 Training Result: + + - Hyper-parameters: + * epoch number = 3 + * learning rate = 0.1 + + + - Model structure: + * 3 layers + * dimension reduced in each layer using a weight matrix + * softmax as the final activation function + * loss calculated as $-y^{T}\log{\hat{y}}$ + +```python +def forward(self, x): + x = x.reshape(-1, 28 * 28) + a0 = x + + #First Layer + z1 = self.matmul_1.forward(a0, self.W1) + a1 = self.relu_1.forward(z1) + + #Second Layer + z2 = self.matmul_2.forward(a1, self.W2) + a2 = self.relu_2.forward(z2) + + #Third Layer + z3 = self.matmul_3.forward(a2, self.W3) + a3 = self.softmax.forward(z3) + x = self.log.forward(a3) + + return x +``` + +```python +def backward(self, y): + + #Third Layer + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + + #Second Layer + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + + #First Layer + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + pass +``` + +Result: +```python +[0] Accuracy: 0.9482 +[1] Accuracy: 0.9643 +[2] Accuracy: 0.9735 +``` + + + +## 3. Change of Parameters + +**Learning Rate [0.2, 0.4, 0.6]** + + + + + +**Batch Size [20, 40, 60]** + + + + + +## 4. Optimization + +**Momentum:** + + + +```python +#Extends NumpyModel +#Save step size +self.delta1 = 0 +self.delta2 = 0 +self.delta3 = 0 + +#Extends the optimize function defined in numpy_fnn.py +if method == "Momentum": + self.delta1 = momentum * self.delta1 - learning_rate * self.W1_grad + self.W1 += self.delta1 + + self.delta2 = momentum * self.delta2 - learning_rate * self.W2_grad + self.W2 += self.delta2 + + self.delta3 = momentum * self.delta3 - learning_rate * self.W3_grad + self.W3 += self.delta3 +``` + +Result: +```python +[0] Accuracy: 0.9654 +[1] Accuracy: 0.9712 +[2] Accuracy: 0.9726 +``` + + + +Discussion: + +Momentum method differs from gradient descent in that it adds a short term memory for step size. Momentum, a number between 0 and 1, is the weight given to the previous step size. Momentum smoothes the path (visual demonstration below, [ref](https://dominikschmidt.xyz/nesterov-momentum/)). A greater momentum results in a more direct path with less changes in direction. Or, from another perspective, accelarates towards the optimum. If momentum = 0, the method reduces to Gradient Descent. + + + +Momentum $\beta$ can be thought of the weight of the previous force, while learning rate $\alpha$ can be thought of the weight of the current force. Intuitively, the optimal training model needs an optimal combination of $\alpha$ and $\beta$ depending on the eigenvalues of a convex function. Inappropriate values could lead to divergence. + + +```python + +``` diff --git a/assignment-2/submission/18340986009/numpy_fnn.py b/assignment-2/submission/18340986009/numpy_fnn.py new file mode 100644 index 0000000..7d8bac7 --- /dev/null +++ b/assignment-2/submission/18340986009/numpy_fnn.py @@ -0,0 +1,198 @@ +import numpy as np + + +class NumpyOp: + + def __init__(self): + self.memory = {} + self.epsilon = 1e-12 + + +class Matmul(NumpyOp): + + def forward(self, x, W): + """ + x: shape(N, d) + w: shape(d, d') + """ + self.memory['x'] = x + self.memory['W'] = W + h = np.matmul(x, W) + + return h + + def backward(self, grad_y): + """ + grad_y: shape(N, d') + """ + x = self.memory['x'] + W = self.memory['W'] + + grad_x = np.matmul(grad_y, W.T) + grad_W = np.matmul(x.T, grad_y) + return grad_x, grad_W + + +class Relu(NumpyOp): + + def forward(self, x): + self.memory['x'] = x + return np.where(x > 0, x, np.zeros_like(x)) + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + x = self.memory['x'] + grad_x = np.where(x <= 0, 0, grad_y) + + return grad_x + + +class Log(NumpyOp): + + def forward(self, x): + """ + x: shape(N, c) + """ + out = np.log(x + self.epsilon) + self.memory['x'] = x + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + x = self.memory['x'] + grad_x = (x + self.epsilon)**(-1) * grad_y + + return grad_x + + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + exp = np.exp(x + self.epsilon) + out = exp/exp.sum(axis=1, keepdims=True) + self.memory['x'] = x + self.memory['s'] = out + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + s = self.memory['s'] + n = s.shape[1] + + grad_x = np.zeros(s.shape) + # calculate Jacobian for ith row + for i in range(s.shape[0]): + jacob = np.zeros((n, n)) + + for j in range(n): + for k in range(n): + if j == k: + jacob[k, j] = s[i, k] * (1-s[i, j]) + else: + jacob[k, j] = -1 * s[i, k] * s[i, j] + # apply Jacobian to the ith row of grad_y + grad_x[i, :] = np.matmul(jacob, grad_y[i, :]) + + return grad_x + + +class NumpyLoss: + + def __init__(self): + self.target = None + + def get_loss(self, pred, target): + self.target = target + return (-1 * pred * target).sum(axis=1).mean() + + def backward(self): + return -1 * self.target / self.target.shape[0] + + +class NumpyModel: + def __init__(self): + self.epsilon = 1e-12 + + self.W1 = np.random.normal(size=(28 * 28, 256)) + self.W2 = np.random.normal(size=(256, 64)) + self.W3 = np.random.normal(size=(64, 10)) + + # 以下算子会在 forward 和 backward 中使用 + self.matmul_1 = Matmul() + self.relu_1 = Relu() + self.matmul_2 = Matmul() + self.relu_2 = Relu() + self.matmul_3 = Matmul() + self.softmax = Softmax() + self.log = Log() + + # 以下变量需要在 backward 中更新。 softmax_grad, log_grad 等为算子反向传播的梯度( loss 关于算子输入的偏导) + self.x1_grad, self.W1_grad = None, None + self.relu_1_grad = None + self.x2_grad, self.W2_grad = None, None + self.relu_2_grad = None + self.x3_grad, self.W3_grad = None, None + self.softmax_grad = None + self.log_grad = None + + # 如需要,以下变量在optimize中更新 + self.delta1 = 0 + self.delta2 = 0 + self.delta3 = 0 + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + a0 = x + z1 = self.matmul_1.forward(a0, self.W1) + a1 = self.relu_1.forward(z1) + z2 = self.matmul_2.forward(a1, self.W2) + a2 = self.relu_2.forward(z2) + z3 = self.matmul_3.forward(a2, self.W3) + a3 = self.softmax.forward(z3) + x = self.log.forward(a3) + + return x + + def backward(self, y): + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + pass + + def optimize(self, learning_rate, method="GD", momentum=0.9, gamma=0.9): + if method == "GD": + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad + + # idea: want different learning rate on different dimensions + if method == "Momentum": + self.delta1 = momentum * self.delta1 - learning_rate * self.W1_grad + self.W1 += self.delta1 + + self.delta2 = momentum * self.delta2 - learning_rate * self.W2_grad + self.W2 += self.delta2 + + self.delta3 = momentum * self.delta3 - learning_rate * self.W3_grad + self.W3 += self.delta3 + + diff --git a/assignment-2/submission/18340986009/numpy_mnist.py b/assignment-2/submission/18340986009/numpy_mnist.py new file mode 100644 index 0000000..6d2ae6d --- /dev/null +++ b/assignment-2/submission/18340986009/numpy_mnist.py @@ -0,0 +1,144 @@ +import numpy as np +import torch +from numpy_fnn import NumpyModel, NumpyLoss +from matplotlib import pyplot as plt + + +def plot_curve(data): + plt.plot(range(len(data)), data, color='blue') + plt.legend(['loss_value'], loc='upper right') + plt.xlabel('step') + plt.ylabel('value') + plt.show() + + +def download_mnist(): + from torchvision import datasets, transforms + + transform = transforms.Compose([ + transforms.ToTensor(), + transforms.Normalize(mean=(0.1307,), std=(0.3081,)) + ]) + + train_dataset = datasets.MNIST(root="./data/", transform=transform, train=True, download=True) + test_dataset = datasets.MNIST(root="./data/", transform=transform, train=False, download=True) + + return train_dataset, test_dataset + + +def one_hot(y, numpy=True): + if numpy: + y_ = np.zeros((y.shape[0], 10)) + y_[np.arange(y.shape[0], dtype=np.int32), y] = 1 + return y_ + else: + y_ = torch.zeros((y.shape[0], 10)) + y_[torch.arange(y.shape[0], dtype=torch.long), y] = 1 + return y_ + + +def batch(dataset, numpy=True): + data = [] + label = [] + for each in dataset: + data.append(each[0]) + label.append(each[1]) + data = torch.stack(data) + label = torch.LongTensor(label) + if numpy: + return [(data.numpy(), label.numpy())] + else: + return [(data, label)] + + +def mini_batch(dataset, batch_size=128): + size = dataset.train_data.shape[0] + + # fetch data and label from dataset MNIST + data = [] + label = [] + for one_obs in dataset: + data.append(np.array(one_obs[0])) + label.append(np.array(one_obs[1])) + + data = np.array(data) + label = np.array(label) + + # shuffle + idx = np.arange(size) + np.random.shuffle(idx) + data = data[idx] + label = label[idx] + + # split to batch, discard the last batch if remaining size < batch_size + data_batches = [] + label_batches = [] + batched = 0 + while batched+batch_size <= size: + data_batches.append(data[batched:batched + batch_size]) + label_batches.append(label[batched:batched + batch_size]) + batched += batch_size + + data_batches = np.array(data_batches) + label_batches = np.array(label_batches) + + return zip(data_batches, label_batches) + + +def get_torch_initialization(numpy=True): + fc1 = torch.nn.Linear(28 * 28, 256) + fc2 = torch.nn.Linear(256, 64) + fc3 = torch.nn.Linear(64, 10) + + if numpy: + W1 = fc1.weight.T.detach().clone().numpy() + W2 = fc2.weight.T.detach().clone().numpy() + W3 = fc3.weight.T.detach().clone().numpy() + else: + W1 = fc1.weight.T.detach().clone().data + W2 = fc2.weight.T.detach().clone().data + W3 = fc3.weight.T.detach().clone().data + + return W1, W2, W3 + + +def numpy_run(epoch_number=3, learning_rate=0.1, batch_size=128): + train_dataset, test_dataset = download_mnist() + + model = NumpyModel() + numpy_loss = NumpyLoss() + model.W1, model.W2, model.W3 = get_torch_initialization() + + train_loss = [] + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset, batch_size): + y = one_hot(y) + + y_pred = model.forward(x) + loss = numpy_loss.get_loss(y_pred, y) + + model.backward(numpy_loss.backward()) + model.optimize(learning_rate) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset)[0] + accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + numpy_run() + +# for lr in [2, 4, 6]: +# l_rate = lr/10 +# numpy_run(epoch_number=3, learning_rate=l_rate, batch_size=128) + +# for bs in [20, 40, 60]: +# numpy_run(epoch_number=3, learning_rate=0.1, batch_size=bs) + + + -- Gitee From f66f54c52e4b5e4571561bbde6ceb06818b2c944 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 15:59:00 +0800 Subject: [PATCH 41/68] Create assignment-3 --- assignment-3/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 assignment-3/.keep diff --git a/assignment-3/.keep b/assignment-3/.keep new file mode 100644 index 0000000..e69de29 -- Gitee From 597e139f9c945e5d7360367d439104e5a67e864f Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 15:59:22 +0800 Subject: [PATCH 42/68] Create img --- 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Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 18:02:11 +0800 Subject: [PATCH 44/68] Remove assignment-3/img/.keep --- assignment-3/img/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 assignment-3/img/.keep diff --git a/assignment-3/img/.keep b/assignment-3/img/.keep deleted file mode 100644 index e69de29..0000000 -- Gitee From 38670f9df6f5007349733382cc07b51449add44b Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 18:02:19 +0800 Subject: [PATCH 45/68] Remove assignment-3/.keep --- assignment-3/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 assignment-3/.keep diff --git a/assignment-3/.keep b/assignment-3/.keep deleted file mode 100644 index e69de29..0000000 -- Gitee From 2c625fa8ef02123a723e3befea6fa2102b5b8279 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 18:05:23 +0800 Subject: [PATCH 46/68] source --- assignment-3/source.py | 358 +++++++++++++++++++++++++++++++++++++++++ 1 file changed, 358 insertions(+) create mode 100644 assignment-3/source.py diff --git a/assignment-3/source.py b/assignment-3/source.py new file mode 100644 index 0000000..32b86e8 --- /dev/null +++ b/assignment-3/source.py @@ -0,0 +1,358 @@ +import numpy as np +import matplotlib.pyplot as plt + +import sklearn.metrics + + + +def shuffle(*datas): + data = np.concatenate(datas) + label = np.concatenate([ + np.ones((d.shape[0],), dtype=int) * i + for (i, d) in enumerate(datas) + ]) + N = data.shape[0] + idx = np.arange(N) + np.random.shuffle(idx) + data = data[idx] + label = label[idx] + return data, label + + +def data_2d(ns, means, covs, train_pct=.8): + # Generate data sets and labels + d_list = [] + l_list = [] + for i in range(len(ns)): + d = np.random.multivariate_normal(means[i], covs[i], (ns[i],)) + d_list.append(d) + l_list.append(np.repeat(i, ns[i])) + data = np.concatenate(d_list) + label = np.concatenate(l_list) + + # Shuffle + N = np.sum(ns) + cut = int(np.ceil(N * train_pct)) + idx = np.arange(N) + np.random.shuffle(idx) + + data = data[idx] + label = label[idx] + + return (data, data), len(ns), label + + +def find_dist(x, y): + return np.sqrt(np.matmul(x - y, x - y)) + + +def plot_cluster(test_data, result, assigned_means, true_means): + # Plot model result + plt.scatter(test_data[:, 0], test_data[:, 1], + c=result, alpha=.6) + # Plot model assigned means + plt.scatter(assigned_means[:, 0], assigned_means[:, 1], + c='red', marker='^', s=70) + # Plot true means + plt.scatter(true_means[:, 0], true_means[:, 1], + c='black', marker='^', s=70) + plt.show() + + +class KMeans: + + def __init__(self, n_clusters, n_iter=10): + self.k = n_clusters + self.iter = n_iter + self.centroids = None + + def fit(self, train_data): + # INITIALIZE centroids + idx = np.random.choice(len(train_data), self.k, replace=False) + self.centroids = train_data[idx] + + # ITERATE for a specified number of times to update the centroids + centroids_record = [self.centroids] + for _ in range(self.iter): + # ASSIGN the nearest centroid to every point + groups = [] + for point in train_data: + dists = [] + for centroid in self.centroids: + dist = find_dist(point, centroid) + dists.append(dist) + group = np.argmin(dists) + groups.append(group) + + # UPDATE centroids based on assigned points + new_centroids = [] + for group in np.unique(groups): + idx = [v == group for v in groups] + new_centroid = np.mean(train_data[idx], axis=0) + new_centroids.append(new_centroid) + self.centroids = new_centroids + centroids_record.append(np.array(new_centroids)) + + def predict(self, test_data): + # ASSIGN test data points to their nearest centroid + groups = [] + for point in test_data: + dists = [] + for centroid in self.centroids: + dist = find_dist(point, centroid) + dists.append(dist) + group = np.argmin(dists) + groups.append(group) + + return np.array(groups) + + +class GaussianMixture: + + def __init__(self, n_clusters, n_iter=5): + self.k = n_clusters + self.centroids = None + self.covs = None + self.pis = None + self.iter = n_iter + self.epsilon = 10 ** (-5) + + def __Normal__(self, x, miu, cov): + # If data is 1-D + if cov.shape == (1, 1): + c = 1 / (np.sqrt(2 * np.pi) * np.sqrt(cov)) + md = ((x - miu) / np.sqrt(cov)) ** 2 / (-2) + + # If data is > 1-D + else: + c = 1 / np.sqrt((2 * np.pi) ** cov.shape[0] * np.linalg.det(cov)) + md = np.dot(np.dot((x - miu).T, np.linalg.inv(cov)), (x - miu)) / (-2) + + return np.max([c * np.exp(md), self.epsilon]) + + def fit(self, train_data): + # SPLIT train_data into k sub-data sets + N = len(train_data) + sub_ds = np.array_split(train_data, self.k) + + # Randomly add another point if there exists only one number in an array + for i in range(len(sub_ds)): + if len(sub_ds[i]) == 1: + idx = np.random.choice(N - 1, 1) + dum = sub_ds[i].tolist() + dum.append(train_data[idx].tolist()[0]) + sub_ds[i] = np.array(dum) + + # INITIALIZE means, covariances, and pis for sub-data sets + self.pis = [1 / self.k for _ in range(self.k)] + self.centroids = [np.mean(v, axis=0) for v in sub_ds] + self.covs = [] + for i in range(self.k): + # If data is 1-D + if sub_ds[i].shape[0] != 1 and sub_ds[i].shape[1] == 1: + cov = np.array([[np.var(sub_ds[i])]]) + # If data is > 1-D + else: + cov = np.cov(sub_ds[i].T) + self.covs.append(cov) + + # ITERATE for a specified number of times to update parameters + for _ in range(self.iter): + # E STEP + # CALCULATE r matrix + r_mat = np.zeros((N, self.k)) + for n in range(N): + for k in range(self.k): + num = self.pis[k] * self.__Normal__(train_data[n], self.centroids[k], self.covs[k]) + dum = [self.pis[k] * self.__Normal__(train_data[n], self.centroids[k], self.covs[k]) + for k in range(self.k)] + denom = np.sum(dum) + r_mat[n][k] = num / denom + Nks = np.sum(r_mat, axis=0) + + # M STEP + # UPDATE pis + self.pis = [Nks[k] / N for k in range(self.k)] + + # UPDATE mean parameter + self.centroids = np.zeros((self.k, len(train_data[0]))) + for k in range(self.k): + for n in range(N): + self.centroids[k] += r_mat[n][k] * train_data[n] + self.centroids = [self.centroids[k] / Nks[k] for k in range(self.k)] + + # UPDATE covariance parameter + self.covs = [np.zeros((len(train_data[0]), len(train_data[0]))) for _ in range(self.k)] + for k in range(self.k): + for n in range(N): + v = (train_data[n] - self.centroids[k]) + self.covs[k] += r_mat[n][k] * np.outer(v, v) + self.covs = [self.covs[k] / Nks[k] for k in range(self.k)] + + def predict(self, test_data): + groups = [] + # ASSIGN test data points to centroids with their maximum conditional probability + for n in range(len(test_data)): + ps = [self.__Normal__(test_data[n], self.centroids[k], self.covs[k]) for k in range(self.k)] + group = np.argmax(ps) + groups.append(group) + + return np.array(groups) + + +class ClusteringAlgorithm: + + def __init__(self, Ks, method='KMeans'): + self.Ks = Ks + self.method = method + self.optimum_K = None + self.centroids = None + + def __plot__(self, K, S): + plt.figure(figsize=(16, 8)) + plt.plot(K, S, 'bx-') + plt.axvline(x=self.optimum_K, linestyle='--', color='r') + plt.xlabel('K') + plt.title('Elbow Method') + plt.show() + + def fit(self, train_data): + Scores_of_Ks = [] + + # ITERATE over all K candidates + for j in range(len(self.Ks)): + # Train Model & Obtain result + if self.method == 'KMeans': + model = KMeans(self.Ks[j]) + if self.method == 'GMM': + model = GaussianMixture(self.Ks[j]) + model.fit(train_data) + result = model.predict(train_data) + + # CALCULATE cumulative sum for each K + cumsum = 0 + for i in range(len(train_data)): + centroid = model.centroids[result[i]] + cumsum += find_dist(centroid, train_data[i]) + Scores_of_Ks.append(cumsum) + + # FIND Optimum K + d = np.zeros((len(self.Ks), 1)) + for i in range(len(self.Ks)): + y = (Scores_of_Ks[-1] - Scores_of_Ks[0]) / (self.Ks[-1] - self.Ks[0]) * \ + (self.Ks[i] - self.Ks[0]) + Scores_of_Ks[0] + d[i] = y - Scores_of_Ks[i] + + self.optimum_K = self.Ks[np.argmax(d)] + + # VISUALIZE + self.__plot__(self.Ks, Scores_of_Ks) + + def predict(self, test_data): + + if self.method == 'KMeans': + model = KMeans(self.optimum_K) + if self.method == 'GMM': + model = GaussianMixture(self.optimum_K) + + model.fit(test_data) + self.centroids = model.centroids + result = model.predict(test_data) + + return result + + def eval(self, test_data, true_label, true_means, assigned_label): + N = len(self.centroids) + + # FIND the corresponding groups + corresponds = dict.fromkeys(np.unique(true_label)) + for g in range(N): + flt = true_label[assigned_label == g] + score = [len(flt[flt == m]) for m in range(N)] + corresponds[g] = np.argmax(score) + pred_label = np.array([corresponds[v] for v in assigned_label]) + + # Plot model result + wrong_pts = test_data[true_label != pred_label] + plt.scatter(test_data[:, 0], test_data[:, 1], + c='grey', alpha=.6) + # Plot wrong points + plt.scatter(wrong_pts[:, 0], wrong_pts[:, 1], + c='red', alpha=.8) + # Plot model assigned means + plt.scatter([v[0] for v in self.centroids], [v[1] for v in self.centroids], + c='purple', marker='^', s=70) + # Plot true means + plt.scatter(true_means[:, 0], true_means[:, 1], + c='black', marker='^', s=70) + plt.show() + + return assigned_label + + +if __name__ == '__main__': + # 基础实验==================================================== + ns = np.array([300, 300, 300]) + + means = np.array([ + [20, 20], + [35, 25], + [25, 35]]) + + covs = np.array([ + [[20, 5], [5, 35]], + [[45, 0], [0, 20]], + [[20, 0], [0, 15]]]) + + (train_data, test_data), n_clusters, test_label = data_2d(ns, means, covs) + + # K-Means + model = KMeans(n_clusters, n_iter=10) + model.fit(train_data) + res = model.predict(test_data) + plot_cluster(test_data, res, + np.array([v.tolist() for v in model.centroids]), + means) + + # GMM + model = GaussianMixture(n_clusters, n_iter=10) + model.fit(train_data) + res = model.predict(test_data) + plot_cluster(test_data, res, + np.array([v.tolist() for v in model.centroids]), + means) + + # 自动选择聚簇数量实验=========================================== + Ks = [1, 2, 3, 4, 5, 6, 7, 8] + + table = np.zeros((len(Ks), 2)) + for k in range(len(Ks)): + print(k) + cut = int(800 / (1 + k)) + + ns = np.array([cut, 800 - cut]) + + means = np.array([ + [20, 20], + [40, 20]]) + + covs = np.array([ + [[20, 5], [5, 35]], + [[45, 0], [0, 20]]]) + + (train_data, test_data), n_clusters, test_label = data_2d(ns, means, covs) + + model = ClusteringAlgorithm(Ks, method='KMeans') + model.fit(train_data) + print('optimum K') + print(model.optimum_K) + res = model.predict(test_data) + # plot_cluster(test_data, res, + # np.array([v.tolist() for v in model.centroids]), + # means) + + assigned_label = model.eval(test_data, test_label, means, res) + rand_s = sklearn.metrics.adjusted_rand_score(test_label, assigned_label) + silhouette_s = sklearn.metrics.silhouette_score(test_data, assigned_label) + table[k, 0] = rand_s + table[k, 1] = silhouette_s -- Gitee From da704487148e766949eb0997a985d74c6daa5c43 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 18:17:06 +0800 Subject: [PATCH 47/68] README --- assignment-3/README.ipynb | 407 ++++++++++++++++++++++++++++++++++++++ 1 file changed, 407 insertions(+) create mode 100644 assignment-3/README.ipynb diff --git a/assignment-3/README.ipynb b/assignment-3/README.ipynb new file mode 100644 index 0000000..8307339 --- /dev/null +++ b/assignment-3/README.ipynb @@ -0,0 +1,407 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Clustering Algorithm (K-Means & GMM)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Below is true for all the following analysis:\n", + " - Data used: 2-D, Gaussian distributions with given means and variance-covariance matrix.\n", + " - Default number of iteration is set to 10.\n", + " - Centroids are initialized by randomly selecting k points from the train dataset." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 1. Algorithm Realization" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Main purpose of this section is to validify the models used for clustering. Dataset below is used for model training." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$\n", + " N_0 = 300 \\hspace{1cm}\n", + " C_0 \\sim \\mathcal{N}(\\mu = \\begin{bmatrix}20\\\\\\\\20\\end{bmatrix},\\,\\sigma^{2} = \\begin{bmatrix}20 & 5\\\\\\\\5 & 35\\end{bmatrix})\\, \n", + "$\n", + "\n", + "$\n", + " N_1 = 300 \\hspace{1cm}\n", + " C_1 \\sim \\mathcal{N}(\\mu = \\begin{bmatrix}35\\\\\\\\25\\end{bmatrix},\\,\\sigma^{2} = \\begin{bmatrix}45 & 0\\\\\\\\0 & 20\\end{bmatrix})\\, \n", + "$\n", + "\n", + "$\n", + " N_2 = 300 \\hspace{1cm}\n", + " C_2 \\sim \\mathcal{N}(\\mu = \\begin{bmatrix}25\\\\\\\\35\\end{bmatrix},\\,\\sigma^{2} = \\begin{bmatrix}20 & 0\\\\\\\\0 & 15\\end{bmatrix})\\, \n", + "$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### 1.1 K-Means" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Steps:\n", + "\n", + " 1. Initialize centroids by random selection\n", + " For n_iter times:\n", + " 2. Assign every point to their nearest centroid\n", + " 3. Recalculate Means based on the assigned points " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Result:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Red points indicate centroids assigned by our model. They are very close the the actual mean in this example." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### 1.2 GMM" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Steps:\n", + "\n", + " 1. Initialize distribution parameters by spliting data into k sub-datasets and perform calculation\n", + " For n_iter times:\n", + " 2. E-Step, calculate matrix\n", + " 3. M-Step, update parameters (means, variance-covariance, pis)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Red points indicate centroids assigned by our model. They are further away from the actual means compared with that of K-Means." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 2. K-optimizating Experiment" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Elbow method:\n", + "Choose K by evaluating the cost (measured by the sum of squared distance from the assigned centroid for every point) of adding every new centroid." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### 2.1 Elbow Method Demonstration" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Dataset below is used in this section:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$\n", + " N_0 = 400 \\hspace{1cm}\n", + " C_0 \\sim \\mathcal{N}(\\mu = \\begin{bmatrix}20\\\\\\\\20\\end{bmatrix},\\,\\sigma^{2} = \\begin{bmatrix}20 & 5\\\\\\\\5 & 35\\end{bmatrix})\\, \n", + "$\n", + "\n", + "$\n", + " N_1 = 400 \\hspace{1cm}\n", + " C_1 \\sim \\mathcal{N}(\\mu = \\begin{bmatrix}40\\\\\\\\20\\end{bmatrix},\\,\\sigma^{2} = \\begin{bmatrix}45 & 0\\\\\\\\0 & 20\\end{bmatrix})\\, \n", + "$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Candicates of K = [1, 2, 3, 4, 5, 6].\n", + "\n", + "For each candidate, run a K-Means model, calculate the sum of squared distance from its centroid for every cluster, and plot the sum of squares to help determine the optimum K.\n", + "\n", + "The optimum K is found automatically by finding the maximum distance between the line of sum-of-squares and the straight line passing through the starting and ending point of the sum-of-squares." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Elbow Line Plot**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Result From Model**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Misclassified Points**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "*A limitation of elbow method is that the optimum K will never be the first and last candidate of K, so make sure to be less restricting on choosing the range of K." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### 2.2 Effect of data balancing on Clustering Algorithm" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**The effect of data-balancing on Elbow mthod**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Using the same means and variance-covariance matrix in 2.1, we'll investigate if changing the ratios of the number of points between two clusters has any effect on our algorithm.\n", + "\n", + "ratios tested = [1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7]" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Metrics for model evaluation:**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Adjusted Rand Index:\n", + "\n", + " Let A be a clustering result with clusters {a1, a2, ..., ar}\n", + " Let B be a clustering result with clusters {b1, b2, ..., bs}\n", + " Then given a (r,n) contingency table where the values of each cell in the ith, jth position is nij, the Adjusted Rand Index (ARI) is calculated as:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Silhouette Coefficient:\n", + " \n", + " Let a be The mean distance between a sample and all other points in the same class\n", + " Let b be the mean distance between a sample and all other points in the next nearest cluster\n", + " Then for every point: (Silhouette Coefficient for the whole dataset is taking the mean of all s)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Result**:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "| \\ | 1:1 | 1:2 | 1:3 | 1:4 | 1:5 | 1:6 | 1:7 |\n", + "| ------------ |:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n", + "| opt K | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |\n", + "| Adjusted Rand Index| .796|.845|.386|.346|.305|.287|.277|\n", + "| Silhouette Index| .558| .557| .390| .400| .385| .356| .358|" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The optimum K found using Elbow Method is 3 for the last 5 cases, which is not the actual number of cluster. Thus, for data 1 ratio less than 1/2, ARI, a measure similar to accuracy drops significantly. Clustering result became less similar to the actual clustering pattern. Silhouette index is roughly equal for 1:1 and 1:2 and equal for the rest of the cases. This difference can also be related to the difference on the choice of K. As two centroids are assigned to a cluster of points which actually shares the same centroid, the separateness between the two clusters becomes small." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.5" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} -- Gitee From 380e34b28ce764177f83b5fc695ccc0571dfa8f7 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 18:17:16 +0800 Subject: [PATCH 48/68] Remove assignment-3/README.ipynb --- assignment-3/README.ipynb | 407 -------------------------------------- 1 file changed, 407 deletions(-) delete mode 100644 assignment-3/README.ipynb diff --git a/assignment-3/README.ipynb b/assignment-3/README.ipynb deleted file mode 100644 index 8307339..0000000 --- a/assignment-3/README.ipynb +++ /dev/null @@ -1,407 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Clustering Algorithm (K-Means & GMM)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Below is true for all the following analysis:\n", - " - Data used: 2-D, Gaussian distributions with given means and variance-covariance matrix.\n", - " - Default number of iteration is set to 10.\n", - " - Centroids are initialized by randomly selecting k points from the train dataset." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## 1. Algorithm Realization" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Main purpose of this section is to validify the models used for clustering. Dataset below is used for model training." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$\n", - " N_0 = 300 \\hspace{1cm}\n", - " C_0 \\sim \\mathcal{N}(\\mu = \\begin{bmatrix}20\\\\\\\\20\\end{bmatrix},\\,\\sigma^{2} = \\begin{bmatrix}20 & 5\\\\\\\\5 & 35\\end{bmatrix})\\, \n", - "$\n", - "\n", - "$\n", - " N_1 = 300 \\hspace{1cm}\n", - " C_1 \\sim \\mathcal{N}(\\mu = \\begin{bmatrix}35\\\\\\\\25\\end{bmatrix},\\,\\sigma^{2} = \\begin{bmatrix}45 & 0\\\\\\\\0 & 20\\end{bmatrix})\\, \n", - "$\n", - "\n", - "$\n", - " N_2 = 300 \\hspace{1cm}\n", - " C_2 \\sim \\mathcal{N}(\\mu = \\begin{bmatrix}25\\\\\\\\35\\end{bmatrix},\\,\\sigma^{2} = \\begin{bmatrix}20 & 0\\\\\\\\0 & 15\\end{bmatrix})\\, \n", - "$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 1.1 K-Means" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Steps:\n", - "\n", - " 1. Initialize centroids by random selection\n", - " For n_iter times:\n", - " 2. Assign every point to their nearest centroid\n", - " 3. Recalculate Means based on the assigned points " - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Result:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Red points indicate centroids assigned by our model. They are very close the the actual mean in this example." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 1.2 GMM" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Steps:\n", - "\n", - " 1. Initialize distribution parameters by spliting data into k sub-datasets and perform calculation\n", - " For n_iter times:\n", - " 2. E-Step, calculate matrix\n", - " 3. M-Step, update parameters (means, variance-covariance, pis)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Red points indicate centroids assigned by our model. They are further away from the actual means compared with that of K-Means." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## 2. K-optimizating Experiment" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Elbow method:\n", - "Choose K by evaluating the cost (measured by the sum of squared distance from the assigned centroid for every point) of adding every new centroid." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 2.1 Elbow Method Demonstration" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Dataset below is used in this section:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$\n", - " N_0 = 400 \\hspace{1cm}\n", - " C_0 \\sim \\mathcal{N}(\\mu = \\begin{bmatrix}20\\\\\\\\20\\end{bmatrix},\\,\\sigma^{2} = \\begin{bmatrix}20 & 5\\\\\\\\5 & 35\\end{bmatrix})\\, \n", - "$\n", - "\n", - "$\n", - " N_1 = 400 \\hspace{1cm}\n", - " C_1 \\sim \\mathcal{N}(\\mu = \\begin{bmatrix}40\\\\\\\\20\\end{bmatrix},\\,\\sigma^{2} = \\begin{bmatrix}45 & 0\\\\\\\\0 & 20\\end{bmatrix})\\, \n", - "$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Candicates of K = [1, 2, 3, 4, 5, 6].\n", - "\n", - "For each candidate, run a K-Means model, calculate the sum of squared distance from its centroid for every cluster, and plot the sum of squares to help determine the optimum K.\n", - "\n", - "The optimum K is found automatically by finding the maximum distance between the line of sum-of-squares and the straight line passing through the starting and ending point of the sum-of-squares." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "**Elbow Line Plot**" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "**Result From Model**" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "**Misclassified Points**" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "*A limitation of elbow method is that the optimum K will never be the first and last candidate of K, so make sure to be less restricting on choosing the range of K." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 2.2 Effect of data balancing on Clustering Algorithm" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "**The effect of data-balancing on Elbow mthod**" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Using the same means and variance-covariance matrix in 2.1, we'll investigate if changing the ratios of the number of points between two clusters has any effect on our algorithm.\n", - "\n", - "ratios tested = [1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "**Metrics for model evaluation:**" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Adjusted Rand Index:\n", - "\n", - " Let A be a clustering result with clusters {a1, a2, ..., ar}\n", - " Let B be a clustering result with clusters {b1, b2, ..., bs}\n", - " Then given a (r,n) contingency table where the values of each cell in the ith, jth position is nij, the Adjusted Rand Index (ARI) is calculated as:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Silhouette Coefficient:\n", - " \n", - " Let a be The mean distance between a sample and all other points in the same class\n", - " Let b be the mean distance between a sample and all other points in the next nearest cluster\n", - " Then for every point: (Silhouette Coefficient for the whole dataset is taking the mean of all s)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "**Result**:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "| \\ | 1:1 | 1:2 | 1:3 | 1:4 | 1:5 | 1:6 | 1:7 |\n", - "| ------------ |:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n", - "| opt K | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |\n", - "| Adjusted Rand Index| .796|.845|.386|.346|.305|.287|.277|\n", - "| Silhouette Index| .558| .557| .390| .400| .385| .356| .358|" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The optimum K found using Elbow Method is 3 for the last 5 cases, which is not the actual number of cluster. Thus, for data 1 ratio less than 1/2, ARI, a measure similar to accuracy drops significantly. Clustering result became less similar to the actual clustering pattern. Silhouette index is roughly equal for 1:1 and 1:2 and equal for the rest of the cases. This difference can also be related to the difference on the choice of K. As two centroids are assigned to a cluster of points which actually shares the same centroid, the separateness between the two clusters becomes small." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.5" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} -- Gitee From 0e1d68388242e81cc5a93d684bfecab31bd42a3b Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 18:18:05 +0800 Subject: [PATCH 49/68] README --- assignment-3/README.md | 149 +++++++++++++++++++++++++++++++++++++++++ 1 file changed, 149 insertions(+) create mode 100644 assignment-3/README.md diff --git a/assignment-3/README.md b/assignment-3/README.md new file mode 100644 index 0000000..49eaa42 --- /dev/null +++ b/assignment-3/README.md @@ -0,0 +1,149 @@ +# Clustering Algorithm (K-Means & GMM) + +Below is true for all the following analysis: + - Data used: 2-D, Gaussian distributions with given means and variance-covariance matrix. + - Default number of iteration is set to 10. + - Centroids are initialized by randomly selecting k points from the train dataset. + +## 1. Algorithm Realization + +Main purpose of this section is to validify the models used for clustering. Dataset below is used for model training. + +$ + N_0 = 300 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}20 & 5\\\\5 & 35\end{bmatrix})\, +$ + +$ + N_1 = 300 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}35\\\\25\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}45 & 0\\\\0 & 20\end{bmatrix})\, +$ + +$ + N_2 = 300 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}25\\\\35\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}20 & 0\\\\0 & 15\end{bmatrix})\, +$ + + + +### 1.1 K-Means + +Steps: + + 1. Initialize centroids by random selection + For n_iter times: + 2. Assign every point to their nearest centroid + 3. Recalculate Means based on the assigned points + +Result: + + + +Red points indicate centroids assigned by our model. They are very close the the actual mean in this example. + +### 1.2 GMM + +Steps: + + 1. Initialize distribution parameters by spliting data into k sub-datasets and perform calculation + For n_iter times: + 2. E-Step, calculate matrix + 3. M-Step, update parameters (means, variance-covariance, pis) + + + +Red points indicate centroids assigned by our model. They are further away from the actual means compared with that of K-Means. + +## 2. K-optimizating Experiment + +Elbow method: +Choose K by evaluating the cost (measured by the sum of squared distance from the assigned centroid for every point) of adding every new centroid. + +### 2.1 Elbow Method Demonstration + +Dataset below is used in this section: + +$ + N_0 = 400 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}20 & 5\\\\5 & 35\end{bmatrix})\, +$ + +$ + N_1 = 400 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}40\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}45 & 0\\\\0 & 20\end{bmatrix})\, +$ + +Candicates of K = [1, 2, 3, 4, 5, 6]. + +For each candidate, run a K-Means model, calculate the sum of squared distance from its centroid for every cluster, and plot the sum of squares to help determine the optimum K. + +The optimum K is found automatically by finding the maximum distance between the line of sum-of-squares and the straight line passing through the starting and ending point of the sum-of-squares. + +**Elbow Line Plot** + + + +**Result From Model** + + + +**Misclassified Points** + + + +*A limitation of elbow method is that the optimum K will never be the first and last candidate of K, so make sure to be less restricting on choosing the range of K. + +### 2.2 Effect of data balancing on Clustering Algorithm + +**The effect of data-balancing on Elbow mthod** + +Using the same means and variance-covariance matrix in 2.1, we'll investigate if changing the ratios of the number of points between two clusters has any effect on our algorithm. + +ratios tested = [1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7] + +**Metrics for model evaluation:** + +Adjusted Rand Index: + + Let A be a clustering result with clusters {a1, a2, ..., ar} + Let B be a clustering result with clusters {b1, b2, ..., bs} + Then given a (r,n) contingency table where the values of each cell in the ith, jth position is nij, the Adjusted Rand Index (ARI) is calculated as: + + + +Silhouette Coefficient: + + Let a be The mean distance between a sample and all other points in the same class + Let b be the mean distance between a sample and all other points in the next nearest cluster + Then for every point: (Silhouette Coefficient for the whole dataset is taking the mean of all s) + + + +**Result**: + +| \ | 1:1 | 1:2 | 1:3 | 1:4 | 1:5 | 1:6 | 1:7 | +| ------------ |:---:|:---:|:---:|:---:|:---:|:---:|:---:| +| opt K | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | +| Adjusted Rand Index| .796|.845|.386|.346|.305|.287|.277| +| Silhouette Index| .558| .557| .390| .400| .385| .356| .358| + +The optimum K found using Elbow Method is 3 for the last 5 cases, which is not the actual number of cluster. Thus, for data 1 ratio less than 1/2, ARI, a measure similar to accuracy drops significantly. Clustering result became less similar to the actual clustering pattern. Silhouette index is roughly equal for 1:1 and 1:2 and equal for the rest of the cases. This difference can also be related to the difference on the choice of K. As two centroids are assigned to a cluster of points which actually shares the same centroid, the separateness between the two clusters becomes small. + + + + + + + + + + + + + + + + +```python + +``` -- Gitee From 82e023723e807646b88a37f48f41d5b8cc0042b9 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 18:23:11 +0800 Subject: [PATCH 50/68] Remove assignment-3/README.md --- assignment-3/README.md | 149 ----------------------------------------- 1 file changed, 149 deletions(-) delete mode 100644 assignment-3/README.md diff --git a/assignment-3/README.md b/assignment-3/README.md deleted file mode 100644 index 49eaa42..0000000 --- a/assignment-3/README.md +++ /dev/null @@ -1,149 +0,0 @@ -# Clustering Algorithm (K-Means & GMM) - -Below is true for all the following analysis: - - Data used: 2-D, Gaussian distributions with given means and variance-covariance matrix. - - Default number of iteration is set to 10. - - Centroids are initialized by randomly selecting k points from the train dataset. - -## 1. Algorithm Realization - -Main purpose of this section is to validify the models used for clustering. Dataset below is used for model training. - -$ - N_0 = 300 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}20 & 5\\\\5 & 35\end{bmatrix})\, -$ - -$ - N_1 = 300 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}35\\\\25\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}45 & 0\\\\0 & 20\end{bmatrix})\, -$ - -$ - N_2 = 300 \hspace{1cm} - C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}25\\\\35\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}20 & 0\\\\0 & 15\end{bmatrix})\, -$ - - - -### 1.1 K-Means - -Steps: - - 1. Initialize centroids by random selection - For n_iter times: - 2. Assign every point to their nearest centroid - 3. Recalculate Means based on the assigned points - -Result: - - - -Red points indicate centroids assigned by our model. They are very close the the actual mean in this example. - -### 1.2 GMM - -Steps: - - 1. Initialize distribution parameters by spliting data into k sub-datasets and perform calculation - For n_iter times: - 2. E-Step, calculate matrix - 3. M-Step, update parameters (means, variance-covariance, pis) - - - -Red points indicate centroids assigned by our model. They are further away from the actual means compared with that of K-Means. - -## 2. K-optimizating Experiment - -Elbow method: -Choose K by evaluating the cost (measured by the sum of squared distance from the assigned centroid for every point) of adding every new centroid. - -### 2.1 Elbow Method Demonstration - -Dataset below is used in this section: - -$ - N_0 = 400 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}20 & 5\\\\5 & 35\end{bmatrix})\, -$ - -$ - N_1 = 400 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}40\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}45 & 0\\\\0 & 20\end{bmatrix})\, -$ - -Candicates of K = [1, 2, 3, 4, 5, 6]. - -For each candidate, run a K-Means model, calculate the sum of squared distance from its centroid for every cluster, and plot the sum of squares to help determine the optimum K. - -The optimum K is found automatically by finding the maximum distance between the line of sum-of-squares and the straight line passing through the starting and ending point of the sum-of-squares. - -**Elbow Line Plot** - - - -**Result From Model** - - - -**Misclassified Points** - - - -*A limitation of elbow method is that the optimum K will never be the first and last candidate of K, so make sure to be less restricting on choosing the range of K. - -### 2.2 Effect of data balancing on Clustering Algorithm - -**The effect of data-balancing on Elbow mthod** - -Using the same means and variance-covariance matrix in 2.1, we'll investigate if changing the ratios of the number of points between two clusters has any effect on our algorithm. - -ratios tested = [1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7] - -**Metrics for model evaluation:** - -Adjusted Rand Index: - - Let A be a clustering result with clusters {a1, a2, ..., ar} - Let B be a clustering result with clusters {b1, b2, ..., bs} - Then given a (r,n) contingency table where the values of each cell in the ith, jth position is nij, the Adjusted Rand Index (ARI) is calculated as: - - - -Silhouette Coefficient: - - Let a be The mean distance between a sample and all other points in the same class - Let b be the mean distance between a sample and all other points in the next nearest cluster - Then for every point: (Silhouette Coefficient for the whole dataset is taking the mean of all s) - - - -**Result**: - -| \ | 1:1 | 1:2 | 1:3 | 1:4 | 1:5 | 1:6 | 1:7 | -| ------------ |:---:|:---:|:---:|:---:|:---:|:---:|:---:| -| opt K | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | -| Adjusted Rand Index| .796|.845|.386|.346|.305|.287|.277| -| Silhouette Index| .558| .557| .390| .400| .385| .356| .358| - -The optimum K found using Elbow Method is 3 for the last 5 cases, which is not the actual number of cluster. Thus, for data 1 ratio less than 1/2, ARI, a measure similar to accuracy drops significantly. Clustering result became less similar to the actual clustering pattern. Silhouette index is roughly equal for 1:1 and 1:2 and equal for the rest of the cases. This difference can also be related to the difference on the choice of K. As two centroids are assigned to a cluster of points which actually shares the same centroid, the separateness between the two clusters becomes small. - - - - - - - - - - - - - - - - -```python - -``` -- Gitee From 1a3036961de9b77d8535c141b34d24b5cdf885d3 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 18:23:33 +0800 Subject: [PATCH 51/68] README --- assignment-3/README.md | 150 +++++++++++++++++++++++++++++++++++++++++ 1 file changed, 150 insertions(+) create mode 100644 assignment-3/README.md diff --git a/assignment-3/README.md b/assignment-3/README.md new file mode 100644 index 0000000..74bae1a --- /dev/null +++ b/assignment-3/README.md @@ -0,0 +1,150 @@ +# Clustering Algorithm (K-Means & GMM) + +Below is true for all the following analysis: + - Data used: 2-D, Gaussian distributions with given means and variance-covariance matrix. + - Default number of iteration is set to 10. + - Centroids are initialized by randomly selecting k points from the train dataset. + +## 1. Algorithm Realization + +Main purpose of this section is to validify the models used for clustering. Dataset below is used for model training. + +$ + N_0 = 300 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\20\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}20 & 5\\\\5 & 35\end{bmatrix})\, +$ + +$ + N_1 = 300 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}35\\\\25\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}45 & 0\\\\0 & 20\end{bmatrix})\, +$ + +$ + N_2 = 300 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}25\\\\35\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}20 & 0\\\\0 & 15\end{bmatrix})\, +$ + + + +### 1.1 K-Means + +Steps: + + 1. Initialize centroids by random selection + For n_iter times: + 2. Assign every point to their nearest centroid + 3. Recalculate Means based on the assigned points + +Result: + + + +Red points indicate centroids assigned by our model. They are very close the the actual mean in this example. + +### 1.2 GMM + +Steps: + + 1. Initialize distribution parameters by spliting data into k sub-datasets and perform calculation + For n_iter times: + 2. E-Step, calculate matrix + 3. M-Step, update parameters (means, variance-covariance, pis) + + + +Red points indicate centroids assigned by our model. They are further away from the actual means compared with that of K-Means. + +## 2. K-optimizating Experiment + +Elbow method: +Choose K by evaluating the cost (measured by the sum of squared distance from the assigned centroid for every point) of adding every new centroid. + +### 2.1 Elbow Method Demonstration + +Dataset below is used in this section: + +$ + N_0 = 400 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}20 & 5\\\\5 & 35\end{bmatrix})\, +$ + +$ + N_1 = 400 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}40\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}45 & 0\\\\0 & 20\end{bmatrix})\, +$ + +Candicates of K = [1, 2, 3, 4, 5, 6]. + +For each candidate, run a K-Means model, calculate the sum of squared distance from its centroid for every cluster, and plot the sum of squares to help determine the optimum K. + +The optimum K is found automatically by finding the maximum distance between the line of sum-of-squares and the straight line passing through the starting and ending point of the sum-of-squares. + +**Elbow Line Plot** + + + +**Result From Model** + + + +**Misclassified Points** + + + +*A limitation of elbow method is that the optimum K will never be the first and last candidate of K, so make sure to be less restricting on choosing the range of K. + +### 2.2 Effect of data balancing on Clustering Algorithm + +**The effect of data-balancing on Elbow mthod** + +Using the same means and variance-covariance matrix in 2.1, we'll investigate if changing the ratios of the number of points between two clusters has any effect on our algorithm. + +ratios tested = [1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7] + +**Metrics for model evaluation:** + +Adjusted Rand Index: + + Let A be a clustering result with clusters {a1, a2, ..., ar} + Let B be a clustering result with clusters {b1, b2, ..., bs} + Then given a (r,n) contingency table where the values of each cell in the ith, jth position is nij, + the Adjusted Rand Index (ARI) is calculated as: + + + +Silhouette Coefficient: + + Let a be The mean distance between a sample and all other points in the same class + Let b be the mean distance between a sample and all other points in the next nearest cluster + Then for every point: (Silhouette Coefficient for the whole dataset is taking the mean of all s) + + + +**Result**: + +| \ | 1:1 | 1:2 | 1:3 | 1:4 | 1:5 | 1:6 | 1:7 | +| ------------ |:---:|:---:|:---:|:---:|:---:|:---:|:---:| +| opt K | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | +| Adjusted Rand Index| .796|.845|.386|.346|.305|.287|.277| +| Silhouette Index| .558| .557| .390| .400| .385| .356| .358| + +The optimum K found using Elbow Method is 3 for the last 5 cases, which is not the actual number of cluster. Thus, for data 1 ratio less than 1/2, ARI, a measure similar to accuracy drops significantly. Clustering result became less similar to the actual clustering pattern. Silhouette index is roughly equal for 1:1 and 1:2 and equal for the rest of the cases. This difference can also be related to the difference on the choice of K. As two centroids are assigned to a cluster of points which actually shares the same centroid, the separateness between the two clusters becomes small. + + + + + + + + + + + + + + + + +```python + +``` -- Gitee From 4a5c2b52d8090a3218aca74aea69028c2cec741e Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 18:25:59 +0800 Subject: [PATCH 52/68] Remove assignment-3/README.md --- assignment-3/README.md | 150 ----------------------------------------- 1 file changed, 150 deletions(-) delete mode 100644 assignment-3/README.md diff --git a/assignment-3/README.md b/assignment-3/README.md deleted file mode 100644 index 74bae1a..0000000 --- a/assignment-3/README.md +++ /dev/null @@ -1,150 +0,0 @@ -# Clustering Algorithm (K-Means & GMM) - -Below is true for all the following analysis: - - Data used: 2-D, Gaussian distributions with given means and variance-covariance matrix. - - Default number of iteration is set to 10. - - Centroids are initialized by randomly selecting k points from the train dataset. - -## 1. Algorithm Realization - -Main purpose of this section is to validify the models used for clustering. Dataset below is used for model training. - -$ - N_0 = 300 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\20\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}20 & 5\\\\5 & 35\end{bmatrix})\, -$ - -$ - N_1 = 300 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}35\\\\25\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}45 & 0\\\\0 & 20\end{bmatrix})\, -$ - -$ - N_2 = 300 \hspace{1cm} - C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}25\\\\35\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}20 & 0\\\\0 & 15\end{bmatrix})\, -$ - - - -### 1.1 K-Means - -Steps: - - 1. Initialize centroids by random selection - For n_iter times: - 2. Assign every point to their nearest centroid - 3. Recalculate Means based on the assigned points - -Result: - - - -Red points indicate centroids assigned by our model. They are very close the the actual mean in this example. - -### 1.2 GMM - -Steps: - - 1. Initialize distribution parameters by spliting data into k sub-datasets and perform calculation - For n_iter times: - 2. E-Step, calculate matrix - 3. M-Step, update parameters (means, variance-covariance, pis) - - - -Red points indicate centroids assigned by our model. They are further away from the actual means compared with that of K-Means. - -## 2. K-optimizating Experiment - -Elbow method: -Choose K by evaluating the cost (measured by the sum of squared distance from the assigned centroid for every point) of adding every new centroid. - -### 2.1 Elbow Method Demonstration - -Dataset below is used in this section: - -$ - N_0 = 400 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}20 & 5\\\\5 & 35\end{bmatrix})\, -$ - -$ - N_1 = 400 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}40\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}45 & 0\\\\0 & 20\end{bmatrix})\, -$ - -Candicates of K = [1, 2, 3, 4, 5, 6]. - -For each candidate, run a K-Means model, calculate the sum of squared distance from its centroid for every cluster, and plot the sum of squares to help determine the optimum K. - -The optimum K is found automatically by finding the maximum distance between the line of sum-of-squares and the straight line passing through the starting and ending point of the sum-of-squares. - -**Elbow Line Plot** - - - -**Result From Model** - - - -**Misclassified Points** - - - -*A limitation of elbow method is that the optimum K will never be the first and last candidate of K, so make sure to be less restricting on choosing the range of K. - -### 2.2 Effect of data balancing on Clustering Algorithm - -**The effect of data-balancing on Elbow mthod** - -Using the same means and variance-covariance matrix in 2.1, we'll investigate if changing the ratios of the number of points between two clusters has any effect on our algorithm. - -ratios tested = [1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7] - -**Metrics for model evaluation:** - -Adjusted Rand Index: - - Let A be a clustering result with clusters {a1, a2, ..., ar} - Let B be a clustering result with clusters {b1, b2, ..., bs} - Then given a (r,n) contingency table where the values of each cell in the ith, jth position is nij, - the Adjusted Rand Index (ARI) is calculated as: - - - -Silhouette Coefficient: - - Let a be The mean distance between a sample and all other points in the same class - Let b be the mean distance between a sample and all other points in the next nearest cluster - Then for every point: (Silhouette Coefficient for the whole dataset is taking the mean of all s) - - - -**Result**: - -| \ | 1:1 | 1:2 | 1:3 | 1:4 | 1:5 | 1:6 | 1:7 | -| ------------ |:---:|:---:|:---:|:---:|:---:|:---:|:---:| -| opt K | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | -| Adjusted Rand Index| .796|.845|.386|.346|.305|.287|.277| -| Silhouette Index| .558| .557| .390| .400| .385| .356| .358| - -The optimum K found using Elbow Method is 3 for the last 5 cases, which is not the actual number of cluster. Thus, for data 1 ratio less than 1/2, ARI, a measure similar to accuracy drops significantly. Clustering result became less similar to the actual clustering pattern. Silhouette index is roughly equal for 1:1 and 1:2 and equal for the rest of the cases. This difference can also be related to the difference on the choice of K. As two centroids are assigned to a cluster of points which actually shares the same centroid, the separateness between the two clusters becomes small. - - - - - - - - - - - - - - - - -```python - -``` -- Gitee From 872a621952ea42d1eeec564d4cea4a7dcfc8c688 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 18:26:16 +0800 Subject: [PATCH 53/68] README --- assignment-3/README.md | 150 +++++++++++++++++++++++++++++++++++++++++ 1 file changed, 150 insertions(+) create mode 100644 assignment-3/README.md diff --git a/assignment-3/README.md b/assignment-3/README.md new file mode 100644 index 0000000..989fd32 --- /dev/null +++ b/assignment-3/README.md @@ -0,0 +1,150 @@ +# Clustering Algorithm (K-Means & GMM) + +Below is true for all the following analysis: + - Data used: 2-D, Gaussian distributions with given means and variance-covariance matrix. + - Default number of iteration is set to 10. + - Centroids are initialized by randomly selecting k points from the train dataset. + +## 1. Algorithm Realization + +Main purpose of this section is to validify the models used for clustering. Dataset below is used for model training. + +$ + N_0 = 300 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\20\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}20 & 5\\\\5 & 35\end{bmatrix})\, +$ + +$ + N_1 = 300 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}35\\\\25\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}45 & 0\\\\0 & 20\end{bmatrix})\, +$ + +$ + N_2 = 300 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}25\\\\35\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}20 & 0\\\\0 & 15\end{bmatrix})\, +$ + + + +### 1.1 K-Means + +Steps: + + 1. Initialize centroids by random selection + For n_iter times: + 2. Assign every point to their nearest centroid + 3. Recalculate Means based on the assigned points + +Result: + + + +Red points indicate centroids assigned by our model. They are very close the the actual mean in this example. + +### 1.2 GMM + +Steps: + + 1. Initialize distribution parameters by spliting data into k sub-datasets and perform calculation + For n_iter times: + 2. E-Step, calculate matrix + 3. M-Step, update parameters (means, variance-covariance, pis) + + + +Red points indicate centroids assigned by our model. They are further away from the actual means compared with that of K-Means. + +## 2. K-optimizating Experiment + +Elbow method: +Choose K by evaluating the cost (measured by the sum of squared distance from the assigned centroid for every point) of adding every new centroid. + +### 2.1 Elbow Method Demonstration + +Dataset below is used in this section: + +$ + N_0 = 400 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}20 & 5\\\\5 & 35\end{bmatrix})\, +$ + +$ + N_1 = 400 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}40\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}45 & 0\\\\0 & 20\end{bmatrix})\, +$ + +Candicates of K = [1, 2, 3, 4, 5, 6]. + +For each candidate, run a K-Means model, calculate the sum of squared distance from its centroid for every cluster, and plot the sum of squares to help determine the optimum K. + +The optimum K is found automatically by finding the maximum distance between the line of sum-of-squares and the straight line passing through the starting and ending point of the sum-of-squares. + +**Elbow Line Plot** + + + +**Result From Model** + + + +**Misclassified Points** + + + +*A limitation of elbow method is that the optimum K will never be the first and last candidate of K, so make sure to be less restricting on choosing the range of K. + +### 2.2 Effect of data balancing on Clustering Algorithm + +**The effect of data-balancing on Elbow mthod** + +Using the same means and variance-covariance matrix in 2.1, we'll investigate if changing the ratios of the number of points between two clusters has any effect on our algorithm. + +ratios tested = [1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7] + +**Metrics for model evaluation:** + +Adjusted Rand Index: + + Let A be a clustering result with clusters {a1, a2, ..., ar} + Let B be a clustering result with clusters {b1, b2, ..., bs} + Then given a (r,n) contingency table where the values of each cell in the ith, jth position is nij, + the Adjusted Rand Index (ARI) is calculated as: + + + +Silhouette Coefficient: + + Let a be The mean distance between a sample and all other points in the same class + Let b be the mean distance between a sample and all other points in the next nearest cluster + Then for every point: (Silhouette Coefficient for the whole dataset is taking the mean of all s) + + + +**Result**: + +| \ | 1:1 | 1:2 | 1:3 | 1:4 | 1:5 | 1:6 | 1:7 | +| ------------ |:---:|:---:|:---:|:---:|:---:|:---:|:---:| +| opt K | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | +| Adjusted Rand Index| .796|.845|.386|.346|.305|.287|.277| +| Silhouette Index| .558| .557| .390| .400| .385| .356| .358| + +The optimum K found using Elbow Method is 3 for the last 5 cases, which is not the actual number of cluster. Thus, for data 1 ratio less than 1/2, ARI, a measure similar to accuracy drops significantly. Clustering result became less similar to the actual clustering pattern. Silhouette index is roughly equal for 1:1 and 1:2 and equal for the rest of the cases. This difference can also be related to the difference on the choice of K. As two centroids are assigned to a cluster of points which actually shares the same centroid, the separateness between the two clusters becomes small. + + + + + + + + + + + + + + + + +```python + +``` -- Gitee From f28adbc92d6ddff484e712ed82e0049e0041bfe6 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 18:27:46 +0800 Subject: [PATCH 54/68] Remove assignment-3/README.md --- assignment-3/README.md | 150 ----------------------------------------- 1 file changed, 150 deletions(-) delete mode 100644 assignment-3/README.md diff --git a/assignment-3/README.md b/assignment-3/README.md deleted file mode 100644 index 989fd32..0000000 --- a/assignment-3/README.md +++ /dev/null @@ -1,150 +0,0 @@ -# Clustering Algorithm (K-Means & GMM) - -Below is true for all the following analysis: - - Data used: 2-D, Gaussian distributions with given means and variance-covariance matrix. - - Default number of iteration is set to 10. - - Centroids are initialized by randomly selecting k points from the train dataset. - -## 1. Algorithm Realization - -Main purpose of this section is to validify the models used for clustering. Dataset below is used for model training. - -$ - N_0 = 300 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\20\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}20 & 5\\\\5 & 35\end{bmatrix})\, -$ - -$ - N_1 = 300 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}35\\\\25\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}45 & 0\\\\0 & 20\end{bmatrix})\, -$ - -$ - N_2 = 300 \hspace{1cm} - C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}25\\\\35\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}20 & 0\\\\0 & 15\end{bmatrix})\, -$ - - - -### 1.1 K-Means - -Steps: - - 1. Initialize centroids by random selection - For n_iter times: - 2. Assign every point to their nearest centroid - 3. Recalculate Means based on the assigned points - -Result: - - - -Red points indicate centroids assigned by our model. They are very close the the actual mean in this example. - -### 1.2 GMM - -Steps: - - 1. Initialize distribution parameters by spliting data into k sub-datasets and perform calculation - For n_iter times: - 2. E-Step, calculate matrix - 3. M-Step, update parameters (means, variance-covariance, pis) - - - -Red points indicate centroids assigned by our model. They are further away from the actual means compared with that of K-Means. - -## 2. K-optimizating Experiment - -Elbow method: -Choose K by evaluating the cost (measured by the sum of squared distance from the assigned centroid for every point) of adding every new centroid. - -### 2.1 Elbow Method Demonstration - -Dataset below is used in this section: - -$ - N_0 = 400 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}20 & 5\\\\5 & 35\end{bmatrix})\, -$ - -$ - N_1 = 400 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}40\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}45 & 0\\\\0 & 20\end{bmatrix})\, -$ - -Candicates of K = [1, 2, 3, 4, 5, 6]. - -For each candidate, run a K-Means model, calculate the sum of squared distance from its centroid for every cluster, and plot the sum of squares to help determine the optimum K. - -The optimum K is found automatically by finding the maximum distance between the line of sum-of-squares and the straight line passing through the starting and ending point of the sum-of-squares. - -**Elbow Line Plot** - - - -**Result From Model** - - - -**Misclassified Points** - - - -*A limitation of elbow method is that the optimum K will never be the first and last candidate of K, so make sure to be less restricting on choosing the range of K. - -### 2.2 Effect of data balancing on Clustering Algorithm - -**The effect of data-balancing on Elbow mthod** - -Using the same means and variance-covariance matrix in 2.1, we'll investigate if changing the ratios of the number of points between two clusters has any effect on our algorithm. - -ratios tested = [1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7] - -**Metrics for model evaluation:** - -Adjusted Rand Index: - - Let A be a clustering result with clusters {a1, a2, ..., ar} - Let B be a clustering result with clusters {b1, b2, ..., bs} - Then given a (r,n) contingency table where the values of each cell in the ith, jth position is nij, - the Adjusted Rand Index (ARI) is calculated as: - - - -Silhouette Coefficient: - - Let a be The mean distance between a sample and all other points in the same class - Let b be the mean distance between a sample and all other points in the next nearest cluster - Then for every point: (Silhouette Coefficient for the whole dataset is taking the mean of all s) - - - -**Result**: - -| \ | 1:1 | 1:2 | 1:3 | 1:4 | 1:5 | 1:6 | 1:7 | -| ------------ |:---:|:---:|:---:|:---:|:---:|:---:|:---:| -| opt K | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | -| Adjusted Rand Index| .796|.845|.386|.346|.305|.287|.277| -| Silhouette Index| .558| .557| .390| .400| .385| .356| .358| - -The optimum K found using Elbow Method is 3 for the last 5 cases, which is not the actual number of cluster. Thus, for data 1 ratio less than 1/2, ARI, a measure similar to accuracy drops significantly. Clustering result became less similar to the actual clustering pattern. Silhouette index is roughly equal for 1:1 and 1:2 and equal for the rest of the cases. This difference can also be related to the difference on the choice of K. As two centroids are assigned to a cluster of points which actually shares the same centroid, the separateness between the two clusters becomes small. - - - - - - - - - - - - - - - - -```python - -``` -- Gitee From 91412fd9aef8d38a096a5fe807bab89ebf01d20a Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 19:20:12 +0800 Subject: [PATCH 55/68] README --- assignment-3/README.md | 150 +++++++++++++++++++++++++++++++++++++++++ 1 file changed, 150 insertions(+) create mode 100644 assignment-3/README.md diff --git a/assignment-3/README.md b/assignment-3/README.md new file mode 100644 index 0000000..1ce62e2 --- /dev/null +++ b/assignment-3/README.md @@ -0,0 +1,150 @@ +# Clustering Algorithm (K-Means & GMM) + +Below is true for all the following analysis: + - Data used: 2-D, Gaussian distributions with given means and variance-covariance matrix. + - Default number of iteration is set to 10. + - Centroids are initialized by randomly selecting k points from the train dataset. + +## 1. Algorithm Realization + +Main purpose of this section is to validify the models used for clustering. Dataset below is used for model training. + +$ + N_0 = 300 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\20\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}20 & 5\\\\5 & 35\end{bmatrix})\, +$ + +$ + N_1 = 300 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}35\\\\25\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}45 & 0\\\\0 & 20\end{bmatrix})\, +$ + +$ + N_2 = 300 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}25\\\\35\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}20 & 0\\\\0 & 15\end{bmatrix})\, +$ + + + +### 1.1 K-Means + +Steps: + + 1. Initialize centroids by random selection + For n_iter times: + 2. Assign every point to their nearest centroid + 3. Recalculate Means based on the assigned points + +Result: + + + +Red points indicate centroids assigned by our model. They are very close the the actual mean in this example. + +### 1.2 GMM + +Steps: + + 1. Initialize distribution parameters by spliting data into k sub-datasets and perform calculation + For n_iter times: + 2. E-Step, calculate matrix + 3. M-Step, update parameters (means, variance-covariance, pis) + + + +Red points indicate centroids assigned by our model. They are further away from the actual means compared with that of K-Means. + +## 2. K-optimizating Experiment + +Elbow method: +Choose K by evaluating the cost (measured by the sum of squared distance from the assigned centroid for every point) of adding every new centroid. + +### 2.1 Elbow Method Demonstration + +Dataset below is used in this section: + +$ + N_0 = 400 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}20 & 5\\\\5 & 35\end{bmatrix})\, +$ + +$ + N_1 = 400 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}40\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}45 & 0\\\\0 & 20\end{bmatrix})\, +$ + +Candicates of K = [1, 2, 3, 4, 5, 6]. + +For each candidate, run a K-Means model, calculate the sum of squared distance from its centroid for every cluster, and plot the sum of squares to help determine the optimum K. + +The optimum K is found automatically by finding the maximum distance between the line of sum-of-squares and the straight line passing through the starting and ending point of the sum-of-squares. + +**Elbow Line Plot** + + + +**Result From Model** + + + +**Misclassified Points** + + + +*A limitation of elbow method is that the optimum K will never be the first and last candidate of K, so make sure to be less restricting on choosing the range of K. + +### 2.2 Effect of data balancing on Clustering Algorithm + +**The effect of data-balancing on Elbow mthod** + +Using the same means and variance-covariance matrix in 2.1, we'll investigate if changing the ratios of the number of points between two clusters has any effect on our algorithm. + +ratios tested = [1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7] + +**Metrics for model evaluation:** + +Adjusted Rand Index: + + Let A be a clustering result with clusters {a1, a2, ..., ar} + Let B be a clustering result with clusters {b1, b2, ..., bs} + Then given a (r,n) contingency table where the values of each cell in the ith, jth position is nij, + the Adjusted Rand Index (ARI) is calculated as: + + + +Silhouette Coefficient: + + Let a be The mean distance between a sample and all other points in the same class + Let b be the mean distance between a sample and all other points in the next nearest cluster + Then for every point: (Silhouette Coefficient for the whole dataset is taking the mean of all s) + + + +**Result**: + +| \ | 1:1 | 1:2 | 1:3 | 1:4 | 1:5 | 1:6 | 1:7 | +| ------------ |:---:|:---:|:---:|:---:|:---:|:---:|:---:| +| opt K | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | +| Adjusted Rand Index| .796|.845|.386|.346|.305|.287|.277| +| Silhouette Index| .558| .557| .390| .400| .385| .356| .358| + +The optimum K found using Elbow Method is 3 for the last 5 cases, which is not the actual number of cluster. Thus, for data 1 ratio less than 1/2, ARI, a measure similar to accuracy drops significantly. Clustering result became less similar to the actual clustering pattern. Silhouette index is roughly equal for 1:1 and 1:2 and equal for the rest of the cases. This difference can also be related to the difference on the choice of K. As two centroids are assigned to a cluster of points which actually shares the same centroid, the separateness between the two clusters becomes small. + + + + + + + + + + + + + + + + +```python + +``` -- Gitee From 7553a23373fb6d0a3af26d449bdf5e645a1b9080 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 19:25:42 +0800 Subject: [PATCH 56/68] Create submission --- assignment-3/submission/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 assignment-3/submission/.keep diff --git a/assignment-3/submission/.keep b/assignment-3/submission/.keep new file mode 100644 index 0000000..e69de29 -- Gitee From 9f78bd25cb536627e5f8c9e6b949fbf58ddbb3a3 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 19:26:12 +0800 Subject: [PATCH 57/68] Create 18340986009 --- assignment-3/submission/18340986009/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 assignment-3/submission/18340986009/.keep diff --git a/assignment-3/submission/18340986009/.keep b/assignment-3/submission/18340986009/.keep new file mode 100644 index 0000000..e69de29 -- Gitee From 9f09657e3f23b02db1d84aac5cb81a2a1e374805 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 19:28:58 +0800 Subject: [PATCH 58/68] README source --- assignment-3/submission/18340986009/README.md | 150 +++++++++ assignment-3/submission/18340986009/source.py | 298 ++++++++++++++++++ 2 files changed, 448 insertions(+) create mode 100644 assignment-3/submission/18340986009/README.md create mode 100644 assignment-3/submission/18340986009/source.py diff --git a/assignment-3/submission/18340986009/README.md b/assignment-3/submission/18340986009/README.md new file mode 100644 index 0000000..1ce62e2 --- /dev/null +++ b/assignment-3/submission/18340986009/README.md @@ -0,0 +1,150 @@ +# Clustering Algorithm (K-Means & GMM) + +Below is true for all the following analysis: + - Data used: 2-D, Gaussian distributions with given means and variance-covariance matrix. + - Default number of iteration is set to 10. + - Centroids are initialized by randomly selecting k points from the train dataset. + +## 1. Algorithm Realization + +Main purpose of this section is to validify the models used for clustering. Dataset below is used for model training. + +$ + N_0 = 300 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\20\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}20 & 5\\\\5 & 35\end{bmatrix})\, +$ + +$ + N_1 = 300 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}35\\\\25\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}45 & 0\\\\0 & 20\end{bmatrix})\, +$ + +$ + N_2 = 300 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}25\\\\35\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}20 & 0\\\\0 & 15\end{bmatrix})\, +$ + + + +### 1.1 K-Means + +Steps: + + 1. Initialize centroids by random selection + For n_iter times: + 2. Assign every point to their nearest centroid + 3. Recalculate Means based on the assigned points + +Result: + + + +Red points indicate centroids assigned by our model. They are very close the the actual mean in this example. + +### 1.2 GMM + +Steps: + + 1. Initialize distribution parameters by spliting data into k sub-datasets and perform calculation + For n_iter times: + 2. E-Step, calculate matrix + 3. M-Step, update parameters (means, variance-covariance, pis) + + + +Red points indicate centroids assigned by our model. They are further away from the actual means compared with that of K-Means. + +## 2. K-optimizating Experiment + +Elbow method: +Choose K by evaluating the cost (measured by the sum of squared distance from the assigned centroid for every point) of adding every new centroid. + +### 2.1 Elbow Method Demonstration + +Dataset below is used in this section: + +$ + N_0 = 400 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}20 & 5\\\\5 & 35\end{bmatrix})\, +$ + +$ + N_1 = 400 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}40\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}45 & 0\\\\0 & 20\end{bmatrix})\, +$ + +Candicates of K = [1, 2, 3, 4, 5, 6]. + +For each candidate, run a K-Means model, calculate the sum of squared distance from its centroid for every cluster, and plot the sum of squares to help determine the optimum K. + +The optimum K is found automatically by finding the maximum distance between the line of sum-of-squares and the straight line passing through the starting and ending point of the sum-of-squares. + +**Elbow Line Plot** + + + +**Result From Model** + + + +**Misclassified Points** + + + +*A limitation of elbow method is that the optimum K will never be the first and last candidate of K, so make sure to be less restricting on choosing the range of K. + +### 2.2 Effect of data balancing on Clustering Algorithm + +**The effect of data-balancing on Elbow mthod** + +Using the same means and variance-covariance matrix in 2.1, we'll investigate if changing the ratios of the number of points between two clusters has any effect on our algorithm. + +ratios tested = [1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7] + +**Metrics for model evaluation:** + +Adjusted Rand Index: + + Let A be a clustering result with clusters {a1, a2, ..., ar} + Let B be a clustering result with clusters {b1, b2, ..., bs} + Then given a (r,n) contingency table where the values of each cell in the ith, jth position is nij, + the Adjusted Rand Index (ARI) is calculated as: + + + +Silhouette Coefficient: + + Let a be The mean distance between a sample and all other points in the same class + Let b be the mean distance between a sample and all other points in the next nearest cluster + Then for every point: (Silhouette Coefficient for the whole dataset is taking the mean of all s) + + + +**Result**: + +| \ | 1:1 | 1:2 | 1:3 | 1:4 | 1:5 | 1:6 | 1:7 | +| ------------ |:---:|:---:|:---:|:---:|:---:|:---:|:---:| +| opt K | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | +| Adjusted Rand Index| .796|.845|.386|.346|.305|.287|.277| +| Silhouette Index| .558| .557| .390| .400| .385| .356| .358| + +The optimum K found using Elbow Method is 3 for the last 5 cases, which is not the actual number of cluster. Thus, for data 1 ratio less than 1/2, ARI, a measure similar to accuracy drops significantly. Clustering result became less similar to the actual clustering pattern. Silhouette index is roughly equal for 1:1 and 1:2 and equal for the rest of the cases. This difference can also be related to the difference on the choice of K. As two centroids are assigned to a cluster of points which actually shares the same centroid, the separateness between the two clusters becomes small. + + + + + + + + + + + + + + + + +```python + +``` diff --git a/assignment-3/submission/18340986009/source.py b/assignment-3/submission/18340986009/source.py new file mode 100644 index 0000000..683183c --- /dev/null +++ b/assignment-3/submission/18340986009/source.py @@ -0,0 +1,298 @@ +import numpy as np +import matplotlib.pyplot as plt + + +def shuffle(*datas): + data = np.concatenate(datas) + label = np.concatenate([ + np.ones((d.shape[0],), dtype=int)*i + for (i, d) in enumerate(datas) + ]) + N = data.shape[0] + idx = np.arange(N) + np.random.shuffle(idx) + data = data[idx] + label = label[idx] + return data, label + + +def data_2d(ns, means, covs, train_pct=.8): + # Generate data sets and labels + d_list = [] + l_list = [] + for i in range(len(ns)): + d = np.random.multivariate_normal(means[i], covs[i], (ns[i],)) + d_list.append(d) + l_list.append(np.repeat(i, ns[i])) + data = np.concatenate(d_list) + label = np.concatenate(l_list) + + # Shuffle + N = np.sum(ns) + cut = int(np.ceil(N * train_pct)) + idx = np.arange(N) + np.random.shuffle(idx) + + data = data[idx] + label = label[idx] + + return (data[:cut], data[cut:]), len(ns), label[cut:] + + +def find_dist(x, y): + return np.sqrt(np.matmul(x - y, x - y)) + + +def evaluate(test_data, result, assigned_means, true_means): + # Plot model result + plt.scatter(test_data[:, 0], test_data[:, 1], + c=result, alpha=.6) + # Plot model assigned means + plt.scatter(assigned_means[:, 0], assigned_means[:, 1], + c='red', marker='^', s=70) + # Plot true means + plt.scatter(true_means[:, 0], true_means[:, 1], + c='black', marker='^', s=70) + plt.show() + + +class KMeans: + + def __init__(self, n_clusters, n_iter=10): + self.k = n_clusters + self.iter = n_iter + self.centroids = None + + def fit(self, train_data): + # INITIALIZE centroids + idx = np.random.choice(len(train_data), self.k, replace=False) + self.centroids = train_data[idx] + + # ITERATE for a specified number of times to update the centroids + centroids_record = [self.centroids] + for _ in range(self.iter): + # ASSIGN the nearest centroid to every point + groups = [] + for point in train_data: + dists = [] + for centroid in self.centroids: + dist = find_dist(point, centroid) + dists.append(dist) + group = np.argmin(dists) + groups.append(group) + + # UPDATE centroids based on assigned points + new_centroids = [] + for group in np.unique(groups): + idx = [v == group for v in groups] + new_centroid = np.mean(train_data[idx], axis=0) + new_centroids.append(new_centroid) + self.centroids = new_centroids + centroids_record.append(np.array(new_centroids)) + + def predict(self, test_data): + # ASSIGN test data points to their nearest centroid + groups = [] + for point in test_data: + dists = [] + for centroid in self.centroids: + dist = find_dist(point, centroid) + dists.append(dist) + group = np.argmin(dists) + groups.append(group) + + return np.array(groups) + + +class GaussianMixture: + + def __init__(self, n_clusters, n_iter=5): + self.k = n_clusters + self.centroids = None + self.covs = None + self.pis = None + self.iter = n_iter + self.epsilon = 10 ** (-5) + + def __Normal__(self, x, miu, cov): + # If data is 1-D + if cov.shape == (1, 1): + c = 1 / (np.sqrt(2 * np.pi) * np.sqrt(cov)) + md = ((x - miu) / np.sqrt(cov)) ** 2 / (-2) + + # If data is > 1-D + else: + c = 1 / np.sqrt((2 * np.pi) ** cov.shape[0] * np.linalg.det(cov)) + md = np.dot(np.dot((x - miu).T, np.linalg.inv(cov)), (x - miu)) / (-2) + + return np.max([c * np.exp(md), self.epsilon]) + + def fit(self, train_data): + # SPLIT train_data into k sub-data sets + N = len(train_data) + sub_ds = np.array_split(train_data, self.k) + + # Randomly add another point if there exists only one number in an array + for i in range(len(sub_ds)): + if len(sub_ds[i]) == 1: + idx = np.random.choice(N - 1, 1) + dum = sub_ds[i].tolist() + dum.append(train_data[idx].tolist()[0]) + sub_ds[i] = np.array(dum) + + # INITIALIZE means, covariances, and pis for sub-data sets + self.pis = [1 / self.k for _ in range(self.k)] + self.centroids = [np.mean(v, axis=0) for v in sub_ds] + self.covs = [] + for i in range(self.k): + # If data is 1-D + if sub_ds[i].shape[0] != 1 and sub_ds[i].shape[1] == 1: + cov = np.array([[np.var(sub_ds[i])]]) + # If data is > 1-D + else: + cov = np.cov(sub_ds[i].T) + self.covs.append(cov) + + # ITERATE for a specified number of times to update parameters + for _ in range(self.iter): + # E STEP + # CALCULATE r matrix + r_mat = np.zeros((N, self.k)) + for n in range(N): + for k in range(self.k): + num = self.pis[k] * self.__Normal__(train_data[n], self.centroids[k], self.covs[k]) + dum = [self.pis[k] * self.__Normal__(train_data[n], self.centroids[k], self.covs[k]) + for k in range(self.k)] + denom = np.sum(dum) + r_mat[n][k] = num / denom + Nks = np.sum(r_mat, axis=0) + + # M STEP + # UPDATE pis + self.pis = [Nks[k] / N for k in range(self.k)] + + # UPDATE mean parameter + self.centroids = np.zeros((self.k, len(train_data[0]))) + for k in range(self.k): + for n in range(N): + self.centroids[k] += r_mat[n][k] * train_data[n] + self.centroids = [self.centroids[k] / Nks[k] for k in range(self.k)] + + # UPDATE covariance parameter + self.covs = [np.zeros((len(train_data[0]), len(train_data[0]))) for _ in range(self.k)] + for k in range(self.k): + for n in range(N): + v = (train_data[n] - self.centroids[k]) + self.covs[k] += r_mat[n][k] * np.outer(v, v) + self.covs = [self.covs[k] / Nks[k] for k in range(self.k)] + + def predict(self, test_data): + groups = [] + # ASSIGN test data points to centroids with their maximum conditional probability + for n in range(len(test_data)): + ps = [self.__Normal__(test_data[n], self.centroids[k], self.covs[k]) for k in range(self.k)] + group = np.argmax(ps) + groups.append(group) + + return np.array(groups) + + +class ClusteringAlgorithm: + + def __init__(self, Ks, method='KMeans'): + self.Ks = Ks + self.method = method + self.optimum_K = None + + def __plot__(self, K, S): + plt.figure(figsize=(16, 8)) + plt.plot(K, S, 'bx-') + plt.axvline(x=self.optimum_K, linestyle='--', color='r') + plt.xlabel('K') + plt.title('Elbow Method') + plt.show() + + def fit(self, train_data): + Scores_of_Ks = [] + + # ITERATE over all K candidates + for j in range(len(self.Ks)): + # Train Model & Obtain result + if self.method == 'KMeans': + model = KMeans(self.Ks[j]) + if self.method == 'GMM': + model = GaussianMixture(self.Ks[j]) + model.fit(train_data) + result = model.predict(train_data) + + # CALCULATE cumulative sum for each K + cumsum = 0 + for i in range(len(train_data)): + centroid = model.centroids[result[i]] + cumsum += find_dist(centroid, train_data[i]) + Scores_of_Ks.append(cumsum) + + # FIND Optimum K + d = np.zeros((len(self.Ks), 1)) + for i in range(len(self.Ks)): + y = (Scores_of_Ks[-1] - Scores_of_Ks[0]) / (self.Ks[-1] - self.Ks[0]) * \ + (self.Ks[i] - self.Ks[0]) + Scores_of_Ks[0] + d[i] = y - Scores_of_Ks[i] + + self.optimum_K = self.Ks[np.argmax(d)] + + # VISUALIZE + self.__plot__(self.Ks, Scores_of_Ks) + + def predict(self, test_data): + + if self.method == 'KMeans': + model = KMeans(self.optimum_K) + if self.method == 'GMM': + model = GaussianMixture(self.optimum_K) + + model.fit(test_data) + result = model.predict(test_data) + + return result + + +if __name__ == '__main__': + ns = np.array([300, 300, 300]) + + means = np.array([ + [20, 20], + [35, 25], + [25, 35]]) + + covs = np.array([ + [[20, 5], [5, 35]], + [[45, 0], [0, 20]], + [[20, 0], [0, 15]]]) + + (train_data, test_data), n_clusters, test_label = data_2d(ns, means, covs) + + # 基础实验 + # K-Means + model = KMeans(n_clusters, n_iter=10) + model.fit(train_data) + res = model.predict(test_data) + evaluate(test_data, res, + np.array([v.tolist() for v in model.centroids]), + means) + + # GMM + model = GaussianMixture(n_clusters, n_iter=10) + model.fit(train_data) + res = model.predict(test_data) + evaluate(test_data, res, + np.array([v.tolist() for v in model.centroids]), + means) + + # 自动选择聚簇数量实验 + Ks = [2, 3, 4, 5] + finder = ClusteringAlgorithm(Ks, method='KMeans') + res = finder.predict(test_data) + evaluate(test_data, res, + np.array([v.tolist() for v in model.centroids]), + means) -- Gitee From d772ca9f316823846fd340b85fc461a84822bf68 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 19:29:41 +0800 Subject: [PATCH 59/68] Remove assignment-3/submission/18340986009/source.py --- assignment-3/submission/18340986009/source.py | 298 ------------------ 1 file changed, 298 deletions(-) delete mode 100644 assignment-3/submission/18340986009/source.py diff --git a/assignment-3/submission/18340986009/source.py b/assignment-3/submission/18340986009/source.py deleted file mode 100644 index 683183c..0000000 --- a/assignment-3/submission/18340986009/source.py +++ /dev/null @@ -1,298 +0,0 @@ -import numpy as np -import matplotlib.pyplot as plt - - -def shuffle(*datas): - data = np.concatenate(datas) - label = np.concatenate([ - np.ones((d.shape[0],), dtype=int)*i - for (i, d) in enumerate(datas) - ]) - N = data.shape[0] - idx = np.arange(N) - np.random.shuffle(idx) - data = data[idx] - label = label[idx] - return data, label - - -def data_2d(ns, means, covs, train_pct=.8): - # Generate data sets and labels - d_list = [] - l_list = [] - for i in range(len(ns)): - d = np.random.multivariate_normal(means[i], covs[i], (ns[i],)) - d_list.append(d) - l_list.append(np.repeat(i, ns[i])) - data = np.concatenate(d_list) - label = np.concatenate(l_list) - - # Shuffle - N = np.sum(ns) - cut = int(np.ceil(N * train_pct)) - idx = np.arange(N) - np.random.shuffle(idx) - - data = data[idx] - label = label[idx] - - return (data[:cut], data[cut:]), len(ns), label[cut:] - - -def find_dist(x, y): - return np.sqrt(np.matmul(x - y, x - y)) - - -def evaluate(test_data, result, assigned_means, true_means): - # Plot model result - plt.scatter(test_data[:, 0], test_data[:, 1], - c=result, alpha=.6) - # Plot model assigned means - plt.scatter(assigned_means[:, 0], assigned_means[:, 1], - c='red', marker='^', s=70) - # Plot true means - plt.scatter(true_means[:, 0], true_means[:, 1], - c='black', marker='^', s=70) - plt.show() - - -class KMeans: - - def __init__(self, n_clusters, n_iter=10): - self.k = n_clusters - self.iter = n_iter - self.centroids = None - - def fit(self, train_data): - # INITIALIZE centroids - idx = np.random.choice(len(train_data), self.k, replace=False) - self.centroids = train_data[idx] - - # ITERATE for a specified number of times to update the centroids - centroids_record = [self.centroids] - for _ in range(self.iter): - # ASSIGN the nearest centroid to every point - groups = [] - for point in train_data: - dists = [] - for centroid in self.centroids: - dist = find_dist(point, centroid) - dists.append(dist) - group = np.argmin(dists) - groups.append(group) - - # UPDATE centroids based on assigned points - new_centroids = [] - for group in np.unique(groups): - idx = [v == group for v in groups] - new_centroid = np.mean(train_data[idx], axis=0) - new_centroids.append(new_centroid) - self.centroids = new_centroids - centroids_record.append(np.array(new_centroids)) - - def predict(self, test_data): - # ASSIGN test data points to their nearest centroid - groups = [] - for point in test_data: - dists = [] - for centroid in self.centroids: - dist = find_dist(point, centroid) - dists.append(dist) - group = np.argmin(dists) - groups.append(group) - - return np.array(groups) - - -class GaussianMixture: - - def __init__(self, n_clusters, n_iter=5): - self.k = n_clusters - self.centroids = None - self.covs = None - self.pis = None - self.iter = n_iter - self.epsilon = 10 ** (-5) - - def __Normal__(self, x, miu, cov): - # If data is 1-D - if cov.shape == (1, 1): - c = 1 / (np.sqrt(2 * np.pi) * np.sqrt(cov)) - md = ((x - miu) / np.sqrt(cov)) ** 2 / (-2) - - # If data is > 1-D - else: - c = 1 / np.sqrt((2 * np.pi) ** cov.shape[0] * np.linalg.det(cov)) - md = np.dot(np.dot((x - miu).T, np.linalg.inv(cov)), (x - miu)) / (-2) - - return np.max([c * np.exp(md), self.epsilon]) - - def fit(self, train_data): - # SPLIT train_data into k sub-data sets - N = len(train_data) - sub_ds = np.array_split(train_data, self.k) - - # Randomly add another point if there exists only one number in an array - for i in range(len(sub_ds)): - if len(sub_ds[i]) == 1: - idx = np.random.choice(N - 1, 1) - dum = sub_ds[i].tolist() - dum.append(train_data[idx].tolist()[0]) - sub_ds[i] = np.array(dum) - - # INITIALIZE means, covariances, and pis for sub-data sets - self.pis = [1 / self.k for _ in range(self.k)] - self.centroids = [np.mean(v, axis=0) for v in sub_ds] - self.covs = [] - for i in range(self.k): - # If data is 1-D - if sub_ds[i].shape[0] != 1 and sub_ds[i].shape[1] == 1: - cov = np.array([[np.var(sub_ds[i])]]) - # If data is > 1-D - else: - cov = np.cov(sub_ds[i].T) - self.covs.append(cov) - - # ITERATE for a specified number of times to update parameters - for _ in range(self.iter): - # E STEP - # CALCULATE r matrix - r_mat = np.zeros((N, self.k)) - for n in range(N): - for k in range(self.k): - num = self.pis[k] * self.__Normal__(train_data[n], self.centroids[k], self.covs[k]) - dum = [self.pis[k] * self.__Normal__(train_data[n], self.centroids[k], self.covs[k]) - for k in range(self.k)] - denom = np.sum(dum) - r_mat[n][k] = num / denom - Nks = np.sum(r_mat, axis=0) - - # M STEP - # UPDATE pis - self.pis = [Nks[k] / N for k in range(self.k)] - - # UPDATE mean parameter - self.centroids = np.zeros((self.k, len(train_data[0]))) - for k in range(self.k): - for n in range(N): - self.centroids[k] += r_mat[n][k] * train_data[n] - self.centroids = [self.centroids[k] / Nks[k] for k in range(self.k)] - - # UPDATE covariance parameter - self.covs = [np.zeros((len(train_data[0]), len(train_data[0]))) for _ in range(self.k)] - for k in range(self.k): - for n in range(N): - v = (train_data[n] - self.centroids[k]) - self.covs[k] += r_mat[n][k] * np.outer(v, v) - self.covs = [self.covs[k] / Nks[k] for k in range(self.k)] - - def predict(self, test_data): - groups = [] - # ASSIGN test data points to centroids with their maximum conditional probability - for n in range(len(test_data)): - ps = [self.__Normal__(test_data[n], self.centroids[k], self.covs[k]) for k in range(self.k)] - group = np.argmax(ps) - groups.append(group) - - return np.array(groups) - - -class ClusteringAlgorithm: - - def __init__(self, Ks, method='KMeans'): - self.Ks = Ks - self.method = method - self.optimum_K = None - - def __plot__(self, K, S): - plt.figure(figsize=(16, 8)) - plt.plot(K, S, 'bx-') - plt.axvline(x=self.optimum_K, linestyle='--', color='r') - plt.xlabel('K') - plt.title('Elbow Method') - plt.show() - - def fit(self, train_data): - Scores_of_Ks = [] - - # ITERATE over all K candidates - for j in range(len(self.Ks)): - # Train Model & Obtain result - if self.method == 'KMeans': - model = KMeans(self.Ks[j]) - if self.method == 'GMM': - model = GaussianMixture(self.Ks[j]) - model.fit(train_data) - result = model.predict(train_data) - - # CALCULATE cumulative sum for each K - cumsum = 0 - for i in range(len(train_data)): - centroid = model.centroids[result[i]] - cumsum += find_dist(centroid, train_data[i]) - Scores_of_Ks.append(cumsum) - - # FIND Optimum K - d = np.zeros((len(self.Ks), 1)) - for i in range(len(self.Ks)): - y = (Scores_of_Ks[-1] - Scores_of_Ks[0]) / (self.Ks[-1] - self.Ks[0]) * \ - (self.Ks[i] - self.Ks[0]) + Scores_of_Ks[0] - d[i] = y - Scores_of_Ks[i] - - self.optimum_K = self.Ks[np.argmax(d)] - - # VISUALIZE - self.__plot__(self.Ks, Scores_of_Ks) - - def predict(self, test_data): - - if self.method == 'KMeans': - model = KMeans(self.optimum_K) - if self.method == 'GMM': - model = GaussianMixture(self.optimum_K) - - model.fit(test_data) - result = model.predict(test_data) - - return result - - -if __name__ == '__main__': - ns = np.array([300, 300, 300]) - - means = np.array([ - [20, 20], - [35, 25], - [25, 35]]) - - covs = np.array([ - [[20, 5], [5, 35]], - [[45, 0], [0, 20]], - [[20, 0], [0, 15]]]) - - (train_data, test_data), n_clusters, test_label = data_2d(ns, means, covs) - - # 基础实验 - # K-Means - model = KMeans(n_clusters, n_iter=10) - model.fit(train_data) - res = model.predict(test_data) - evaluate(test_data, res, - np.array([v.tolist() for v in model.centroids]), - means) - - # GMM - model = GaussianMixture(n_clusters, n_iter=10) - model.fit(train_data) - res = model.predict(test_data) - evaluate(test_data, res, - np.array([v.tolist() for v in model.centroids]), - means) - - # 自动选择聚簇数量实验 - Ks = [2, 3, 4, 5] - finder = ClusteringAlgorithm(Ks, method='KMeans') - res = finder.predict(test_data) - evaluate(test_data, res, - np.array([v.tolist() for v in model.centroids]), - means) -- Gitee From 91786412023b8dae5b48e6d2376a7434cb0188e5 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 19:30:09 +0800 Subject: [PATCH 60/68] source --- assignment-3/submission/18340986009/source.py | 358 ++++++++++++++++++ 1 file changed, 358 insertions(+) create mode 100644 assignment-3/submission/18340986009/source.py diff --git a/assignment-3/submission/18340986009/source.py b/assignment-3/submission/18340986009/source.py new file mode 100644 index 0000000..32b86e8 --- /dev/null +++ b/assignment-3/submission/18340986009/source.py @@ -0,0 +1,358 @@ +import numpy as np +import matplotlib.pyplot as plt + +import sklearn.metrics + + + +def shuffle(*datas): + data = np.concatenate(datas) + label = np.concatenate([ + np.ones((d.shape[0],), dtype=int) * i + for (i, d) in enumerate(datas) + ]) + N = data.shape[0] + idx = np.arange(N) + np.random.shuffle(idx) + data = data[idx] + label = label[idx] + return data, label + + +def data_2d(ns, means, covs, train_pct=.8): + # Generate data sets and labels + d_list = [] + l_list = [] + for i in range(len(ns)): + d = np.random.multivariate_normal(means[i], covs[i], (ns[i],)) + d_list.append(d) + l_list.append(np.repeat(i, ns[i])) + data = np.concatenate(d_list) + label = np.concatenate(l_list) + + # Shuffle + N = np.sum(ns) + cut = int(np.ceil(N * train_pct)) + idx = np.arange(N) + np.random.shuffle(idx) + + data = data[idx] + label = label[idx] + + return (data, data), len(ns), label + + +def find_dist(x, y): + return np.sqrt(np.matmul(x - y, x - y)) + + +def plot_cluster(test_data, result, assigned_means, true_means): + # Plot model result + plt.scatter(test_data[:, 0], test_data[:, 1], + c=result, alpha=.6) + # Plot model assigned means + plt.scatter(assigned_means[:, 0], assigned_means[:, 1], + c='red', marker='^', s=70) + # Plot true means + plt.scatter(true_means[:, 0], true_means[:, 1], + c='black', marker='^', s=70) + plt.show() + + +class KMeans: + + def __init__(self, n_clusters, n_iter=10): + self.k = n_clusters + self.iter = n_iter + self.centroids = None + + def fit(self, train_data): + # INITIALIZE centroids + idx = np.random.choice(len(train_data), self.k, replace=False) + self.centroids = train_data[idx] + + # ITERATE for a specified number of times to update the centroids + centroids_record = [self.centroids] + for _ in range(self.iter): + # ASSIGN the nearest centroid to every point + groups = [] + for point in train_data: + dists = [] + for centroid in self.centroids: + dist = find_dist(point, centroid) + dists.append(dist) + group = np.argmin(dists) + groups.append(group) + + # UPDATE centroids based on assigned points + new_centroids = [] + for group in np.unique(groups): + idx = [v == group for v in groups] + new_centroid = np.mean(train_data[idx], axis=0) + new_centroids.append(new_centroid) + self.centroids = new_centroids + centroids_record.append(np.array(new_centroids)) + + def predict(self, test_data): + # ASSIGN test data points to their nearest centroid + groups = [] + for point in test_data: + dists = [] + for centroid in self.centroids: + dist = find_dist(point, centroid) + dists.append(dist) + group = np.argmin(dists) + groups.append(group) + + return np.array(groups) + + +class GaussianMixture: + + def __init__(self, n_clusters, n_iter=5): + self.k = n_clusters + self.centroids = None + self.covs = None + self.pis = None + self.iter = n_iter + self.epsilon = 10 ** (-5) + + def __Normal__(self, x, miu, cov): + # If data is 1-D + if cov.shape == (1, 1): + c = 1 / (np.sqrt(2 * np.pi) * np.sqrt(cov)) + md = ((x - miu) / np.sqrt(cov)) ** 2 / (-2) + + # If data is > 1-D + else: + c = 1 / np.sqrt((2 * np.pi) ** cov.shape[0] * np.linalg.det(cov)) + md = np.dot(np.dot((x - miu).T, np.linalg.inv(cov)), (x - miu)) / (-2) + + return np.max([c * np.exp(md), self.epsilon]) + + def fit(self, train_data): + # SPLIT train_data into k sub-data sets + N = len(train_data) + sub_ds = np.array_split(train_data, self.k) + + # Randomly add another point if there exists only one number in an array + for i in range(len(sub_ds)): + if len(sub_ds[i]) == 1: + idx = np.random.choice(N - 1, 1) + dum = sub_ds[i].tolist() + dum.append(train_data[idx].tolist()[0]) + sub_ds[i] = np.array(dum) + + # INITIALIZE means, covariances, and pis for sub-data sets + self.pis = [1 / self.k for _ in range(self.k)] + self.centroids = [np.mean(v, axis=0) for v in sub_ds] + self.covs = [] + for i in range(self.k): + # If data is 1-D + if sub_ds[i].shape[0] != 1 and sub_ds[i].shape[1] == 1: + cov = np.array([[np.var(sub_ds[i])]]) + # If data is > 1-D + else: + cov = np.cov(sub_ds[i].T) + self.covs.append(cov) + + # ITERATE for a specified number of times to update parameters + for _ in range(self.iter): + # E STEP + # CALCULATE r matrix + r_mat = np.zeros((N, self.k)) + for n in range(N): + for k in range(self.k): + num = self.pis[k] * self.__Normal__(train_data[n], self.centroids[k], self.covs[k]) + dum = [self.pis[k] * self.__Normal__(train_data[n], self.centroids[k], self.covs[k]) + for k in range(self.k)] + denom = np.sum(dum) + r_mat[n][k] = num / denom + Nks = np.sum(r_mat, axis=0) + + # M STEP + # UPDATE pis + self.pis = [Nks[k] / N for k in range(self.k)] + + # UPDATE mean parameter + self.centroids = np.zeros((self.k, len(train_data[0]))) + for k in range(self.k): + for n in range(N): + self.centroids[k] += r_mat[n][k] * train_data[n] + self.centroids = [self.centroids[k] / Nks[k] for k in range(self.k)] + + # UPDATE covariance parameter + self.covs = [np.zeros((len(train_data[0]), len(train_data[0]))) for _ in range(self.k)] + for k in range(self.k): + for n in range(N): + v = (train_data[n] - self.centroids[k]) + self.covs[k] += r_mat[n][k] * np.outer(v, v) + self.covs = [self.covs[k] / Nks[k] for k in range(self.k)] + + def predict(self, test_data): + groups = [] + # ASSIGN test data points to centroids with their maximum conditional probability + for n in range(len(test_data)): + ps = [self.__Normal__(test_data[n], self.centroids[k], self.covs[k]) for k in range(self.k)] + group = np.argmax(ps) + groups.append(group) + + return np.array(groups) + + +class ClusteringAlgorithm: + + def __init__(self, Ks, method='KMeans'): + self.Ks = Ks + self.method = method + self.optimum_K = None + self.centroids = None + + def __plot__(self, K, S): + plt.figure(figsize=(16, 8)) + plt.plot(K, S, 'bx-') + plt.axvline(x=self.optimum_K, linestyle='--', color='r') + plt.xlabel('K') + plt.title('Elbow Method') + plt.show() + + def fit(self, train_data): + Scores_of_Ks = [] + + # ITERATE over all K candidates + for j in range(len(self.Ks)): + # Train Model & Obtain result + if self.method == 'KMeans': + model = KMeans(self.Ks[j]) + if self.method == 'GMM': + model = GaussianMixture(self.Ks[j]) + model.fit(train_data) + result = model.predict(train_data) + + # CALCULATE cumulative sum for each K + cumsum = 0 + for i in range(len(train_data)): + centroid = model.centroids[result[i]] + cumsum += find_dist(centroid, train_data[i]) + Scores_of_Ks.append(cumsum) + + # FIND Optimum K + d = np.zeros((len(self.Ks), 1)) + for i in range(len(self.Ks)): + y = (Scores_of_Ks[-1] - Scores_of_Ks[0]) / (self.Ks[-1] - self.Ks[0]) * \ + (self.Ks[i] - self.Ks[0]) + Scores_of_Ks[0] + d[i] = y - Scores_of_Ks[i] + + self.optimum_K = self.Ks[np.argmax(d)] + + # VISUALIZE + self.__plot__(self.Ks, Scores_of_Ks) + + def predict(self, test_data): + + if self.method == 'KMeans': + model = KMeans(self.optimum_K) + if self.method == 'GMM': + model = GaussianMixture(self.optimum_K) + + model.fit(test_data) + self.centroids = model.centroids + result = model.predict(test_data) + + return result + + def eval(self, test_data, true_label, true_means, assigned_label): + N = len(self.centroids) + + # FIND the corresponding groups + corresponds = dict.fromkeys(np.unique(true_label)) + for g in range(N): + flt = true_label[assigned_label == g] + score = [len(flt[flt == m]) for m in range(N)] + corresponds[g] = np.argmax(score) + pred_label = np.array([corresponds[v] for v in assigned_label]) + + # Plot model result + wrong_pts = test_data[true_label != pred_label] + plt.scatter(test_data[:, 0], test_data[:, 1], + c='grey', alpha=.6) + # Plot wrong points + plt.scatter(wrong_pts[:, 0], wrong_pts[:, 1], + c='red', alpha=.8) + # Plot model assigned means + plt.scatter([v[0] for v in self.centroids], [v[1] for v in self.centroids], + c='purple', marker='^', s=70) + # Plot true means + plt.scatter(true_means[:, 0], true_means[:, 1], + c='black', marker='^', s=70) + plt.show() + + return assigned_label + + +if __name__ == '__main__': + # 基础实验==================================================== + ns = np.array([300, 300, 300]) + + means = np.array([ + [20, 20], + [35, 25], + [25, 35]]) + + covs = np.array([ + [[20, 5], [5, 35]], + [[45, 0], [0, 20]], + [[20, 0], [0, 15]]]) + + (train_data, test_data), n_clusters, test_label = data_2d(ns, means, covs) + + # K-Means + model = KMeans(n_clusters, n_iter=10) + model.fit(train_data) + res = model.predict(test_data) + plot_cluster(test_data, res, + np.array([v.tolist() for v in model.centroids]), + means) + + # GMM + model = GaussianMixture(n_clusters, n_iter=10) + model.fit(train_data) + res = model.predict(test_data) + plot_cluster(test_data, res, + np.array([v.tolist() for v in model.centroids]), + means) + + # 自动选择聚簇数量实验=========================================== + Ks = [1, 2, 3, 4, 5, 6, 7, 8] + + table = np.zeros((len(Ks), 2)) + for k in range(len(Ks)): + print(k) + cut = int(800 / (1 + k)) + + ns = np.array([cut, 800 - cut]) + + means = np.array([ + [20, 20], + [40, 20]]) + + covs = np.array([ + [[20, 5], [5, 35]], + [[45, 0], [0, 20]]]) + + (train_data, test_data), n_clusters, test_label = data_2d(ns, means, covs) + + model = ClusteringAlgorithm(Ks, method='KMeans') + model.fit(train_data) + print('optimum K') + print(model.optimum_K) + res = model.predict(test_data) + # plot_cluster(test_data, res, + # np.array([v.tolist() for v in model.centroids]), + # means) + + assigned_label = model.eval(test_data, test_label, means, res) + rand_s = sklearn.metrics.adjusted_rand_score(test_label, assigned_label) + silhouette_s = sklearn.metrics.silhouette_score(test_data, assigned_label) + table[k, 0] = rand_s + table[k, 1] = silhouette_s -- Gitee From 860bb464e9fbf340247450739ba8cd41961cbc3d Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 19:30:24 +0800 Subject: [PATCH 61/68] Create img --- assignment-3/submission/18340986009/img/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 assignment-3/submission/18340986009/img/.keep diff --git a/assignment-3/submission/18340986009/img/.keep b/assignment-3/submission/18340986009/img/.keep new file mode 100644 index 0000000..e69de29 -- Gitee From 91e95efa6903c19b5a0724b5fddc945aada748cb Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 19:30:42 +0800 Subject: [PATCH 62/68] Remove assignment-3/submission/18340986009/.keep --- 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From: Yating Zou Date: Mon, 14 Jun 2021 19:31:42 +0800 Subject: [PATCH 66/68] Remove assignment-3/README.md --- assignment-3/README.md | 150 ----------------------------------------- 1 file changed, 150 deletions(-) delete mode 100644 assignment-3/README.md diff --git a/assignment-3/README.md b/assignment-3/README.md deleted file mode 100644 index 1ce62e2..0000000 --- a/assignment-3/README.md +++ /dev/null @@ -1,150 +0,0 @@ -# Clustering Algorithm (K-Means & GMM) - -Below is true for all the following analysis: - - Data used: 2-D, Gaussian distributions with given means and variance-covariance matrix. - - Default number of iteration is set to 10. - - Centroids are initialized by randomly selecting k points from the train dataset. - -## 1. Algorithm Realization - -Main purpose of this section is to validify the models used for clustering. Dataset below is used for model training. - -$ - N_0 = 300 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\20\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}20 & 5\\\\5 & 35\end{bmatrix})\, -$ - -$ - N_1 = 300 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}35\\\\25\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}45 & 0\\\\0 & 20\end{bmatrix})\, -$ - -$ - N_2 = 300 \hspace{1cm} - C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}25\\\\35\end{bmatrix}\,\sigma^{2} = \begin{bmatrix}20 & 0\\\\0 & 15\end{bmatrix})\, -$ - - - -### 1.1 K-Means - -Steps: - - 1. Initialize centroids by random selection - For n_iter times: - 2. Assign every point to their nearest centroid - 3. Recalculate Means based on the assigned points - -Result: - - - -Red points indicate centroids assigned by our model. They are very close the the actual mean in this example. - -### 1.2 GMM - -Steps: - - 1. Initialize distribution parameters by spliting data into k sub-datasets and perform calculation - For n_iter times: - 2. E-Step, calculate matrix - 3. M-Step, update parameters (means, variance-covariance, pis) - - - -Red points indicate centroids assigned by our model. They are further away from the actual means compared with that of K-Means. - -## 2. K-optimizating Experiment - -Elbow method: -Choose K by evaluating the cost (measured by the sum of squared distance from the assigned centroid for every point) of adding every new centroid. - -### 2.1 Elbow Method Demonstration - -Dataset below is used in this section: - -$ - N_0 = 400 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}20 & 5\\\\5 & 35\end{bmatrix})\, -$ - -$ - N_1 = 400 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}40\\\\20\end{bmatrix},\,\sigma^{2} = \begin{bmatrix}45 & 0\\\\0 & 20\end{bmatrix})\, -$ - -Candicates of K = [1, 2, 3, 4, 5, 6]. - -For each candidate, run a K-Means model, calculate the sum of squared distance from its centroid for every cluster, and plot the sum of squares to help determine the optimum K. - -The optimum K is found automatically by finding the maximum distance between the line of sum-of-squares and the straight line passing through the starting and ending point of the sum-of-squares. - -**Elbow Line Plot** - - - -**Result From Model** - - - -**Misclassified Points** - - - -*A limitation of elbow method is that the optimum K will never be the first and last candidate of K, so make sure to be less restricting on choosing the range of K. - -### 2.2 Effect of data balancing on Clustering Algorithm - -**The effect of data-balancing on Elbow mthod** - -Using the same means and variance-covariance matrix in 2.1, we'll investigate if changing the ratios of the number of points between two clusters has any effect on our algorithm. - -ratios tested = [1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7] - -**Metrics for model evaluation:** - -Adjusted Rand Index: - - Let A be a clustering result with clusters {a1, a2, ..., ar} - Let B be a clustering result with clusters {b1, b2, ..., bs} - Then given a (r,n) contingency table where the values of each cell in the ith, jth position is nij, - the Adjusted Rand Index (ARI) is calculated as: - - - -Silhouette Coefficient: - - Let a be The mean distance between a sample and all other points in the same class - Let b be the mean distance between a sample and all other points in the next nearest cluster - Then for every point: (Silhouette Coefficient for the whole dataset is taking the mean of all s) - - - -**Result**: - -| \ | 1:1 | 1:2 | 1:3 | 1:4 | 1:5 | 1:6 | 1:7 | -| ------------ |:---:|:---:|:---:|:---:|:---:|:---:|:---:| -| opt K | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | -| Adjusted Rand Index| .796|.845|.386|.346|.305|.287|.277| -| Silhouette Index| .558| .557| .390| .400| .385| .356| .358| - -The optimum K found using Elbow Method is 3 for the last 5 cases, which is not the actual number of cluster. Thus, for data 1 ratio less than 1/2, ARI, a measure similar to accuracy drops significantly. Clustering result became less similar to the actual clustering pattern. Silhouette index is roughly equal for 1:1 and 1:2 and equal for the rest of the cases. This difference can also be related to the difference on the choice of K. As two centroids are assigned to a cluster of points which actually shares the same centroid, the separateness between the two clusters becomes small. - - - - - - - - - - - - - - - - -```python - -``` -- Gitee From 70b447f60a6fb180995bcd974565f7a780f8f149 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 19:31:50 +0800 Subject: [PATCH 67/68] Remove assignment-3/source.py --- assignment-3/source.py | 358 ----------------------------------------- 1 file changed, 358 deletions(-) delete mode 100644 assignment-3/source.py diff --git a/assignment-3/source.py b/assignment-3/source.py deleted file mode 100644 index 32b86e8..0000000 --- a/assignment-3/source.py +++ /dev/null @@ -1,358 +0,0 @@ -import numpy as np -import matplotlib.pyplot as plt - -import sklearn.metrics - - - -def shuffle(*datas): - data = np.concatenate(datas) - label = np.concatenate([ - np.ones((d.shape[0],), dtype=int) * i - for (i, d) in enumerate(datas) - ]) - N = data.shape[0] - idx = np.arange(N) - np.random.shuffle(idx) - data = data[idx] - label = label[idx] - return data, label - - -def data_2d(ns, means, covs, train_pct=.8): - # Generate data sets and labels - d_list = [] - l_list = [] - for i in range(len(ns)): - d = np.random.multivariate_normal(means[i], covs[i], (ns[i],)) - d_list.append(d) - l_list.append(np.repeat(i, ns[i])) - data = np.concatenate(d_list) - label = np.concatenate(l_list) - - # Shuffle - N = np.sum(ns) - cut = int(np.ceil(N * train_pct)) - idx = np.arange(N) - np.random.shuffle(idx) - - data = data[idx] - label = label[idx] - - return (data, data), len(ns), label - - -def find_dist(x, y): - return np.sqrt(np.matmul(x - y, x - y)) - - -def plot_cluster(test_data, result, assigned_means, true_means): - # Plot model result - plt.scatter(test_data[:, 0], test_data[:, 1], - c=result, alpha=.6) - # Plot model assigned means - plt.scatter(assigned_means[:, 0], assigned_means[:, 1], - c='red', marker='^', s=70) - # Plot true means - plt.scatter(true_means[:, 0], true_means[:, 1], - c='black', marker='^', s=70) - plt.show() - - -class KMeans: - - def __init__(self, n_clusters, n_iter=10): - self.k = n_clusters - self.iter = n_iter - self.centroids = None - - def fit(self, train_data): - # INITIALIZE centroids - idx = np.random.choice(len(train_data), self.k, replace=False) - self.centroids = train_data[idx] - - # ITERATE for a specified number of times to update the centroids - centroids_record = [self.centroids] - for _ in range(self.iter): - # ASSIGN the nearest centroid to every point - groups = [] - for point in train_data: - dists = [] - for centroid in self.centroids: - dist = find_dist(point, centroid) - dists.append(dist) - group = np.argmin(dists) - groups.append(group) - - # UPDATE centroids based on assigned points - new_centroids = [] - for group in np.unique(groups): - idx = [v == group for v in groups] - new_centroid = np.mean(train_data[idx], axis=0) - new_centroids.append(new_centroid) - self.centroids = new_centroids - centroids_record.append(np.array(new_centroids)) - - def predict(self, test_data): - # ASSIGN test data points to their nearest centroid - groups = [] - for point in test_data: - dists = [] - for centroid in self.centroids: - dist = find_dist(point, centroid) - dists.append(dist) - group = np.argmin(dists) - groups.append(group) - - return np.array(groups) - - -class GaussianMixture: - - def __init__(self, n_clusters, n_iter=5): - self.k = n_clusters - self.centroids = None - self.covs = None - self.pis = None - self.iter = n_iter - self.epsilon = 10 ** (-5) - - def __Normal__(self, x, miu, cov): - # If data is 1-D - if cov.shape == (1, 1): - c = 1 / (np.sqrt(2 * np.pi) * np.sqrt(cov)) - md = ((x - miu) / np.sqrt(cov)) ** 2 / (-2) - - # If data is > 1-D - else: - c = 1 / np.sqrt((2 * np.pi) ** cov.shape[0] * np.linalg.det(cov)) - md = np.dot(np.dot((x - miu).T, np.linalg.inv(cov)), (x - miu)) / (-2) - - return np.max([c * np.exp(md), self.epsilon]) - - def fit(self, train_data): - # SPLIT train_data into k sub-data sets - N = len(train_data) - sub_ds = np.array_split(train_data, self.k) - - # Randomly add another point if there exists only one number in an array - for i in range(len(sub_ds)): - if len(sub_ds[i]) == 1: - idx = np.random.choice(N - 1, 1) - dum = sub_ds[i].tolist() - dum.append(train_data[idx].tolist()[0]) - sub_ds[i] = np.array(dum) - - # INITIALIZE means, covariances, and pis for sub-data sets - self.pis = [1 / self.k for _ in range(self.k)] - self.centroids = [np.mean(v, axis=0) for v in sub_ds] - self.covs = [] - for i in range(self.k): - # If data is 1-D - if sub_ds[i].shape[0] != 1 and sub_ds[i].shape[1] == 1: - cov = np.array([[np.var(sub_ds[i])]]) - # If data is > 1-D - else: - cov = np.cov(sub_ds[i].T) - self.covs.append(cov) - - # ITERATE for a specified number of times to update parameters - for _ in range(self.iter): - # E STEP - # CALCULATE r matrix - r_mat = np.zeros((N, self.k)) - for n in range(N): - for k in range(self.k): - num = self.pis[k] * self.__Normal__(train_data[n], self.centroids[k], self.covs[k]) - dum = [self.pis[k] * self.__Normal__(train_data[n], self.centroids[k], self.covs[k]) - for k in range(self.k)] - denom = np.sum(dum) - r_mat[n][k] = num / denom - Nks = np.sum(r_mat, axis=0) - - # M STEP - # UPDATE pis - self.pis = [Nks[k] / N for k in range(self.k)] - - # UPDATE mean parameter - self.centroids = np.zeros((self.k, len(train_data[0]))) - for k in range(self.k): - for n in range(N): - self.centroids[k] += r_mat[n][k] * train_data[n] - self.centroids = [self.centroids[k] / Nks[k] for k in range(self.k)] - - # UPDATE covariance parameter - self.covs = [np.zeros((len(train_data[0]), len(train_data[0]))) for _ in range(self.k)] - for k in range(self.k): - for n in range(N): - v = (train_data[n] - self.centroids[k]) - self.covs[k] += r_mat[n][k] * np.outer(v, v) - self.covs = [self.covs[k] / Nks[k] for k in range(self.k)] - - def predict(self, test_data): - groups = [] - # ASSIGN test data points to centroids with their maximum conditional probability - for n in range(len(test_data)): - ps = [self.__Normal__(test_data[n], self.centroids[k], self.covs[k]) for k in range(self.k)] - group = np.argmax(ps) - groups.append(group) - - return np.array(groups) - - -class ClusteringAlgorithm: - - def __init__(self, Ks, method='KMeans'): - self.Ks = Ks - self.method = method - self.optimum_K = None - self.centroids = None - - def __plot__(self, K, S): - plt.figure(figsize=(16, 8)) - plt.plot(K, S, 'bx-') - plt.axvline(x=self.optimum_K, linestyle='--', color='r') - plt.xlabel('K') - plt.title('Elbow Method') - plt.show() - - def fit(self, train_data): - Scores_of_Ks = [] - - # ITERATE over all K candidates - for j in range(len(self.Ks)): - # Train Model & Obtain result - if self.method == 'KMeans': - model = KMeans(self.Ks[j]) - if self.method == 'GMM': - model = GaussianMixture(self.Ks[j]) - model.fit(train_data) - result = model.predict(train_data) - - # CALCULATE cumulative sum for each K - cumsum = 0 - for i in range(len(train_data)): - centroid = model.centroids[result[i]] - cumsum += find_dist(centroid, train_data[i]) - Scores_of_Ks.append(cumsum) - - # FIND Optimum K - d = np.zeros((len(self.Ks), 1)) - for i in range(len(self.Ks)): - y = (Scores_of_Ks[-1] - Scores_of_Ks[0]) / (self.Ks[-1] - self.Ks[0]) * \ - (self.Ks[i] - self.Ks[0]) + Scores_of_Ks[0] - d[i] = y - Scores_of_Ks[i] - - self.optimum_K = self.Ks[np.argmax(d)] - - # VISUALIZE - self.__plot__(self.Ks, Scores_of_Ks) - - def predict(self, test_data): - - if self.method == 'KMeans': - model = KMeans(self.optimum_K) - if self.method == 'GMM': - model = GaussianMixture(self.optimum_K) - - model.fit(test_data) - self.centroids = model.centroids - result = model.predict(test_data) - - return result - - def eval(self, test_data, true_label, true_means, assigned_label): - N = len(self.centroids) - - # FIND the corresponding groups - corresponds = dict.fromkeys(np.unique(true_label)) - for g in range(N): - flt = true_label[assigned_label == g] - score = [len(flt[flt == m]) for m in range(N)] - corresponds[g] = np.argmax(score) - pred_label = np.array([corresponds[v] for v in assigned_label]) - - # Plot model result - wrong_pts = test_data[true_label != pred_label] - plt.scatter(test_data[:, 0], test_data[:, 1], - c='grey', alpha=.6) - # Plot wrong points - plt.scatter(wrong_pts[:, 0], wrong_pts[:, 1], - c='red', alpha=.8) - # Plot model assigned means - plt.scatter([v[0] for v in self.centroids], [v[1] for v in self.centroids], - c='purple', marker='^', s=70) - # Plot true means - plt.scatter(true_means[:, 0], true_means[:, 1], - c='black', marker='^', s=70) - plt.show() - - return assigned_label - - -if __name__ == '__main__': - # 基础实验==================================================== - ns = np.array([300, 300, 300]) - - means = np.array([ - [20, 20], - [35, 25], - [25, 35]]) - - covs = np.array([ - [[20, 5], [5, 35]], - [[45, 0], [0, 20]], - [[20, 0], [0, 15]]]) - - (train_data, test_data), n_clusters, test_label = data_2d(ns, means, covs) - - # K-Means - model = KMeans(n_clusters, n_iter=10) - model.fit(train_data) - res = model.predict(test_data) - plot_cluster(test_data, res, - np.array([v.tolist() for v in model.centroids]), - means) - - # GMM - model = GaussianMixture(n_clusters, n_iter=10) - model.fit(train_data) - res = model.predict(test_data) - plot_cluster(test_data, res, - np.array([v.tolist() for v in model.centroids]), - means) - - # 自动选择聚簇数量实验=========================================== - Ks = [1, 2, 3, 4, 5, 6, 7, 8] - - table = np.zeros((len(Ks), 2)) - for k in range(len(Ks)): - print(k) - cut = int(800 / (1 + k)) - - ns = np.array([cut, 800 - cut]) - - means = np.array([ - [20, 20], - [40, 20]]) - - covs = np.array([ - [[20, 5], [5, 35]], - [[45, 0], [0, 20]]]) - - (train_data, test_data), n_clusters, test_label = data_2d(ns, means, covs) - - model = ClusteringAlgorithm(Ks, method='KMeans') - model.fit(train_data) - print('optimum K') - print(model.optimum_K) - res = model.predict(test_data) - # plot_cluster(test_data, res, - # np.array([v.tolist() for v in model.centroids]), - # means) - - assigned_label = model.eval(test_data, test_label, means, res) - rand_s = sklearn.metrics.adjusted_rand_score(test_label, assigned_label) - silhouette_s = sklearn.metrics.silhouette_score(test_data, assigned_label) - table[k, 0] = rand_s - table[k, 1] = silhouette_s -- Gitee From b1747fd5398102c43b8758bd0d96a3233d2a4dbf Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Mon, 14 Jun 2021 19:32:07 +0800 Subject: [PATCH 68/68] Remove assignment-3/submission/.keep --- assignment-3/submission/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 assignment-3/submission/.keep diff --git a/assignment-3/submission/.keep b/assignment-3/submission/.keep deleted file mode 100644 index e69de29..0000000 -- Gitee

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1847e28325f7a971d9da2b61c8c73abf9a75b525 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Thu, 1 Apr 2021 20:29:13 +0800 Subject: [PATCH 17/68] =?UTF-8?q?=E5=88=A0=E9=99=A4=E6=96=87=E4=BB=B6=20as?= =?UTF-8?q?signment-1/submission/18340986009/README.md?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- assignment-1/submission/18340986009/README.md | 138 ------------------ 1 file changed, 138 deletions(-) delete mode 100644 assignment-1/submission/18340986009/README.md diff --git a/assignment-1/submission/18340986009/README.md b/assignment-1/submission/18340986009/README.md deleted file mode 100644 index a62a1df..0000000 --- a/assignment-1/submission/18340986009/README.md +++ /dev/null @@ -1,138 +0,0 @@ -# KNN Classification - -This report includes two parts: -1. Find a model that maximize accuracy rate with given dataset. (Distribution type of each class = Gaussian, distribution parameters chosen at random) -2. Assess how distribution parameters affects model accuracy using the model built in part 1. - -*Data used in this report is under ./data folder. One can reproduce the result using the saved data.* - -## 1. Model Generation - -### 1.1 Overview of Mock Data - -Generate 3 classes of 2-dimension Gaussian Distribution. - -$ - N_0 = 150 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\sigma^{2} = \begin{bmatrix}60 & -50\\\\-50 & 140\end{bmatrix}) -$ - -$ - N_1 = 250 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}60\\\\20\end{bmatrix},\sigma^{2} = \begin{bmatrix}130 & 10\\\\10 & 100\end{bmatrix}) -$ - -$ - N_2 = 100 \hspace{1cm} - C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\60\end{bmatrix},\sigma^{2} = \begin{bmatrix}120 & 20\\\\20 & 90\end{bmatrix}) -$ - -Mock Data 1 [D1.csv] Overview: - - - -500 points are then split randomly into training set [train_data1.csv] (80%) and testing set [test_data1.csv] (20%). - -### 1.2 Model Accuracy with Different K and Distance Method - -Since a rule of thumb is to let $K = \sqrt{N}$, where $ N = N_0 + N_1 + N_2$, we first try some Ks around $\sqrt{400} = 20$ using both Euclidean and Manhattan distance. - -| \ | K = 10 | K = 15 | K = 20 | K = 25 | K = 30 | -| ------------ |:------:|:------:|:------:|:------:|:------:| -| **Euclidean** |83.0|82.0|83.0|81.0|80.0| -| **Manhattan** |83.0|82.0|81.0|81.0|81.0| - -The KNN model with $K = 10$ gives the best prediction result of 83% for both distance methods, so we consider choosing $K_{0} = 10$ as a starting point for model optimization. Below is a scatter plot showing the prediction result of the chosen model ($K = 10$, Euclidean Distance). Each red dot represents a mis-classification. - - - -### 1.3 Model Optimization - -General Idea: $K_{i+1} = \lceil{K_{i} + Step_{i+1}}\rceil$ - -Detailed steps: - - - For each $K_{i+1}$, calculate its accuracy rate $R_{i+1}$. - - If $R_{i+1} > R_{0}$, a better model is find. End our optimization. Else: - - If $R_{i+1} > R_{i}$, let $Step_{i+1} = \frac{1}{C} Step_{i} $, where $C = (R_{i+1} - R_{i}) / R_{i}$. - Which is, if improvements on accuracy is more than 1%, continue in this direction with a smaller step; If accuracy improves by less than 1%, continue in this direction with a larger step. - - If $R_{i+1} <= R_{i}$, let $Step_{i+1} = - \frac{1}{2} Step_{i}$. - Which is, if the new K does not improve model accuracy, try a smaller step in reverse direction. - -The model from 1.2 gives K = 10 and Euclidean distance. Using this model as the starting point, define the first step $Step_{0} = \frac{1}{100}N = 5$. - -Optimization process: - -| \ | K = 10 | K = 5 | K = 8 | -| ------------ |:------:|:------:|:------:| -| **Accuracy rate (%)** |83.0|83.0|85.0| - - After three iterations, a higher accuracy rate of 85% is reached when K is adjusted to 8. Thus, our final KNN model will use K = 8 and Euclidean distance. - -Prediction result evaluation: - - - -Compared with the model before optimization, two points on the top is now classified correctly. - -## 2. Distribution Parameters & Model Accuracy - -From inuition, we hypothesis that any change that results in a more balanced mixture of all classes will make classification harder, thereby decrease model accuracy. Below, we modify the parameters of Gaussian distributions to test our hypothesis. - -### 2.1 Change of Variance and Covariance - -Let the means stay the same. Modify the variance-covariance matrix for each class to increase overlapping between each class: - -$ - N_0 = 150 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\sigma^{2} = \begin{bmatrix}300 & 0\\\\0 & 200\end{bmatrix}) -$ - -$ - N_1 = 250 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}60\\\\20\end{bmatrix},\sigma^{2} = \begin{bmatrix}250 & 0\\\\0 & 150\end{bmatrix}) -$ - -$ - N_2 = 100 \hspace{1cm} - C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\60\end{bmatrix},\sigma^{2} = \begin{bmatrix}150 & 0\\\\0 & 150\end{bmatrix}) -$ - -Mock Data 2 [D2.csv] Overview: - - - -Prediction result evaluation: - - - -Accuracy of our model drop from 85% to 79% as expected. - -### 2.2 Change of Mean - -Let other parameters stay the same, decrease the distance between the means of each class to increase overlapping: - -$ - N_0 = 150 \hspace{1cm} - C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\sigma^{2} = \begin{bmatrix}60 & -50\\\\-50 & 140\end{bmatrix}) -$ - -$ - N_1 = 250 \hspace{1cm} - C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\40\end{bmatrix},\sigma^{2} = \begin{bmatrix}130 & 10\\\\10 & 100\end{bmatrix}) -$ - -$ - N_2 = 100 \hspace{1cm} - C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}40\\\\60\end{bmatrix},\sigma^{2} = \begin{bmatrix}120 & 20\\\\20 & 90\end{bmatrix}) -$ - -Mock Data 3 [D3.csv] Overview: - - - -Prediction result evaluation: - - - -Accuracy of our model drop from 85% to 73% as expected. -- Gitee From f18a534fcde29f2b0512f8fe89f1a8f6aa19c5a1 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Thu, 1 Apr 2021 20:29:25 +0800 Subject: [PATCH 18/68] (): --- assignment-1/submission/18340986009/README.md | 159 ++++++++++++++++++ 1 file changed, 159 insertions(+) create mode 100644 assignment-1/submission/18340986009/README.md diff --git a/assignment-1/submission/18340986009/README.md b/assignment-1/submission/18340986009/README.md new file mode 100644 index 0000000..fa66f00 --- /dev/null +++ b/assignment-1/submission/18340986009/README.md @@ -0,0 +1,159 @@ +# KNN Classification + +This report includes two parts: +1. Find a KNN model that maximize accuracy rate with given dataset. (Distribution type of each class = Gaussian, distribution parameters chosen at random) +2. Assess how distribution parameters affects model accuracy using the model built in part 1. + + +## 1. Model Generation + +### 1.1 Overview of Mock Data + +Generate 3 classes of 2-dimension Gaussian Distribution. + +$ + N_0 = 150 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\sigma^{2} = \begin{bmatrix}60 & -50\\\\-50 & 140\end{bmatrix}) +$ + +$ + N_1 = 250 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}60\\\\20\end{bmatrix},\sigma^{2} = \begin{bmatrix}130 & 10\\\\10 & 100\end{bmatrix}) +$ + +$ + N_2 = 100 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\60\end{bmatrix},\sigma^{2} = \begin{bmatrix}120 & 20\\\\20 & 90\end{bmatrix}) +$ + +Mock Data 1 Overview: + + + +500 points are then split randomly into training set (80%) and testing set (20%). + +### 1.2 Model Accuracy with Different K and Distance Method + +Since a rule of thumb is to let $K = \sqrt{N}$, where $ N = N_0 + N_1 + N_2$, we first try some Ks around $\sqrt{400} = 20$ using both Euclidean and Manhattan distance. + +| \ | K = 10 | K = 15 | K = 20 | K = 25 | K = 30 | +| ------------ |:------:|:------:|:------:|:------:|:------:| +| **Euclidean** |83.0|82.0|83.0|81.0|80.0| +| **Manhattan** |83.0|82.0|81.0|81.0|81.0| + +The KNN model with $K = 10$ gives the best prediction result of 83% for both distance methods, so we consider choosing $K_{0} = 10$ as a starting point for model optimization. Below is a scatter plot showing the prediction result of the chosen model ($K = 10$, Euclidean Distance). Each red dot represents a mis-classification. + +*Noticed model accuracy using different distance method doesn't show much difference for this dataset. + + + +### 1.3 Model Optimization + +General Idea: $K_{i+1} = \lceil{K_{i} + Step_{i+1}}\rceil$ + +Detailed steps: + + - For each $K_{i+1}$, calculate its accuracy rate $R_{i+1}$. + - If $R_{i+1} > R_{0}$, a better model is find. End our optimization. Else: + - If $R_{i+1} > R_{i}$, let $Step_{i+1} = \frac{1}{C} Step_{i} $, where $C = (R_{i+1} - R_{i}) / R_{i}$. + Which is, if model accuracy improves, continue in this direction with a smaller step. The step size is negatively related to the percentage of improvement. + - If $R_{i+1} <= R_{i}$, let $Step_{i+1} = - \frac{1}{2} Step_{i}$. + Which is, if the new K does not improve model accuracy, try a smaller step in reverse direction. + +The model from 1.2 gives K = 10 and Euclidean distance. Using this model as the starting point, define the first step $Step_{0} = \frac{1}{100}N = 5$. + +Optimization process: + +| \ | K = 10 | K = 5 | K = 8 | +| ------------ |:------:|:------:|:------:| +| **Accuracy rate (%)** |83.0|83.0|85.0| + + After three iterations, a higher accuracy rate of 85% is reached when K is adjusted to 8. Thus, our final KNN model will use K = 8 and Euclidean distance. + +Prediction result evaluation: + + + +Compared with the model before optimization, two points on the top is now classified correctly. + +## 2. Distribution Parameters & Model Accuracy + +From inuition, we hypothesis that any change that results in a more balanced mixture of all classes will make classification harder, thereby decrease model accuracy. Below, we modify the parameters of Gaussian distributions to test our hypothesis. + +### 2.1 Change of Variance and Covariance + +Let the means stay the same. Modify the variance-covariance matrix for each class to increase overlapping between each class: + +$ + N_0 = 150 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\sigma^{2} = \begin{bmatrix}300 & 0\\\\0 & 200\end{bmatrix}) +$ + +$ + N_1 = 250 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}60\\\\20\end{bmatrix},\sigma^{2} = \begin{bmatrix}250 & 0\\\\0 & 150\end{bmatrix}) +$ + +$ + N_2 = 100 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}20\\\\60\end{bmatrix},\sigma^{2} = \begin{bmatrix}150 & 0\\\\0 & 150\end{bmatrix}) +$ + +Mock Data 2 Overview: + + + +Prediction result evaluation: + + + +Accuracy of our model drop from 85% to 79% as expected. + +### 2.2 Change of Mean + +Let other parameters stay the same, decrease the distance between the means of each class to increase overlapping: + +$ + N_0 = 150 \hspace{1cm} + C_0 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\50\end{bmatrix},\sigma^{2} = \begin{bmatrix}60 & -50\\\\-50 & 140\end{bmatrix}) +$ + +$ + N_1 = 250 \hspace{1cm} + C_1 \sim \mathcal{N}(\mu = \begin{bmatrix}50\\\\40\end{bmatrix},\sigma^{2} = \begin{bmatrix}130 & 10\\\\10 & 100\end{bmatrix}) +$ + +$ + N_2 = 100 \hspace{1cm} + C_2 \sim \mathcal{N}(\mu = \begin{bmatrix}40\\\\60\end{bmatrix},\sigma^{2} = \begin{bmatrix}120 & 20\\\\20 & 90\end{bmatrix}) +$ + +Mock Data 3 Overview: + + + +Prediction result evaluation: + + + +Accuracy of our model drop from 85% to 73% as expected. + +### 2.3 N & Model Accuracy + +In attempts to increase model accuracy, we try double the Ns in proportion to Data 3. With $N_{total} = 1000$, we expect some increase on model accuracy. + +Mock Data 4 Overview: + + + +Prediction result evaluation: + + + +Model accuracy decreases from 73% to 62.5% even though our data size doubled. This suggests sample size contributes much less to model accuracy compared with distribution parameters. This makes sense because if the data labeled by different categories does indeed come from the same distribution, increasing N should provide more evidence of the similarity between these different categories. + +## Summary + +The main takeaways for this exercise: + +Model accuracy depends more on distribution parameters and the choice of K. Distance method have little influence on model accuracy, and whether an increase of N improves model accuracy or not depends on if the true distributions of all categories are significantly different (Might be able to use p-value from a statistical test to evaluate). -- Gitee From 064a9293aa6263ae0a8fb6bb735c3492471300b7 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Thu, 1 Apr 2021 20:30:20 +0800 Subject: [PATCH 19/68] (): --- assignment-1/submission/18340986009/source.py | 249 ++++++++++++++++++ 1 file changed, 249 insertions(+) create mode 100644 assignment-1/submission/18340986009/source.py diff --git a/assignment-1/submission/18340986009/source.py b/assignment-1/submission/18340986009/source.py new file mode 100644 index 0000000..410b588 --- /dev/null +++ b/assignment-1/submission/18340986009/source.py @@ -0,0 +1,249 @@ +#!/usr/bin/env python +# coding: utf-8 + +# In[1]: + + +import sys +import numpy as np +import matplotlib.pyplot as plt + + +# ## Define Global Functions + +# In[139]: + + +# Generate Training and Testing Sets +def generate(Ns, Means, Covs, train_frac): + + # Generate 2-D data of N class + data = list() + label = list() + + for i in range(0,len(Ns)): + Ci = np.random.multivariate_normal(Means[i], Covs[i], Ns[i]) + data.append(Ci) + label.append([i]*Ns[i]) + + data = np.array([v for subl in data for v in subl]) + label = np.array([v for subl in label for v in subl]) + + #Assign random number + idx = np.arange(sum(Ns)) + np.random.shuffle(idx) + + data = data[idx] + label = label[idx] + + # Split into training and testing set + split_point = int(label.size * train_frac) + train_data, test_data = data[:split_point,], data[split_point:,] + train_label, test_label = label[:split_point,], label[split_point:,] + + np.save("data.npy",((train_data, train_label), + (test_data, test_label))) + + return train_data, train_label, test_data, test_label + + +# Read in saved data +def read(): + (train_data, train_label), (test_data, test_label) = np.load( + "data.npy", allow_pickle = True) + return (train_data, train_label), (test_data, test_label) + + +# Create scatter plot of different categories +def display(data, colorby, name, title): + colors = ['red','grey','blue'] + datas =[[],[],[]] + + for i in range(len(data)): + datas[colorby[i]].append(data[i]) + + for i in range(len(datas)): + each = np.array(datas[i]) + if len(each) == 0: + continue + plt.scatter(each[:, 0], each[:, 1], + marker = 'o', + color = colors[i], + alpha = 0.7) + + plt.xlabel("X1") + plt.ylabel("X2") + plt.title(title) + plt.savefig(f'img/{name}') + plt.show() + + +# ## Define Class KNN + +# In[140]: + + +class KNN: + + def __init__(self): + + self.K = None + self.Dist = None + self.data = None + self.label = None + + + # Calculate distance between two given points + def get_distance(self, x, y, dist_type = "Euclidean"): + dist = 0.0 + if "Euclidean" == dist_type: + distance = 0.0 + for i in range(len(x)): + distance += (x[i] - y[i])**2 + dist = np.sqrt(distance) + + if "Manhattan" == dist_type: + distance = 0.0 + for i in range(len(x)): + distance += np.abs(x[i] - y[i]) + dist = distance + + return dist + + + # Make a prediction for one point + def predict_for_one(self, K, Dist, target, train_data, train_label): + # Calculate distances between target point and other points + dists = [] + neighbors = [] + + for i in range(len(train_data)): + dist = self.get_distance(target, train_data[i], Dist) + dists.append((train_data[i], train_label[i], dist)) + + # Get the K nearest neighbors + dists.sort(key = lambda e: e[-1]) + neighbors = dists[1:K+1] + + # Make prediction based on conditional probabilities + neighbors_class = [e[-2] for e in neighbors] + prediction = max(neighbors_class, key = neighbors_class.count) + + return prediction + + + # Calculate model accuracy + def calc_accuracy(self, K, Dist, train_data, train_label): + predictions = [] + # Make predictions for the training data + for i in range(len(train_label)): + target = train_data[i] + prediction = self.predict_for_one( + K, Dist, target, train_data, train_label + ) + predictions.append(prediction) + + correct = 0 + for i in range(len(predictions)): + if train_label[i] == predictions[i]: + correct += 1 + accuracy = correct / len(predictions) * 100 + + return accuracy + + + # Find the Optimal K & Distance combination + def fit(self, K_list, Dist_list, train_data, train_label): + + # Loop through the given options for K and distance methods + accuracy_list = [] + for i in range(len(Dist_list)): + Dist = Dist_list[i] + dum_list = [] + for j in range(len(K_list)): + K = K_list[j] + accuracy = self.calc_accuracy( + K, Dist, train_data, train_label + ) + dum_list.append(accuracy) + accuracy_list.append(dum_list) + + # Find the K & Distance method that gives the highest accuracy + ac_array = np.array(accuracy_list) + global_max = max([max(subl) for subl in accuracy_list]) + params = np.where(ac_array == global_max) + + # Assign the optimal parameters to KNN object + # Randomly choice one if there exist more than one highest accuracy + Dist_idx = np.random.choice(np.array(params[0])) + K_idx = np.random.choice(np.array(params[1])) + + self.Dist = Dist_list[Dist_idx] + self.K = K_list[K_idx] + self.data = train_data + self.label = train_label + + return ac_array + + + def predict(self, test_data): + # If test data has been inputed & Model has been obtained + predictions = [] + # For every point(target) in test data + for i in range(len(test_data)): + target = test_data[i] + prediction = self.predict_for_one( + self.K, self.Dist, + target, + self.data, + self.label) + predictions.append(prediction) + + return np.array(predictions) + + +# ## Start of Program + +# In[143]: + + +if __name__ == '__main__': + + if len(sys.argv) > 1 and sys.argv[1] == "g": + generate( + Ns = [100, 250, 150], + + Means = [[50,50], + [60,20], + [20,60]], + + Covs = [[[60,-50],[-50,140]], + [[130,10],[10,100]], + [[120,20],[20,90]]], + + train_frac = 0.8 + ) + + elif len(sys.argv) > 1 and sys.argv[1] == "d": + (train_data, train_label), (test_data, test_label) = read() + + display(train_data, train_label, + 'train', 'Scatter Plot of Training Data') + display(test_data, test_label, + 'test', 'Scatter Plot of Testing Data') + else: + (train_data, train_label), (test_data, test_label) = read() + + model = KNN() + + model.fit( + K_list = [15, 20, 25], + Dist_list = ["Euclidean", "Manhattan"], + train_data = train_data, + train_label = train_label) + + res = model.predict(test_data) + + print("acc =",np.mean(np.equal(res, test_label))) + + -- Gitee From 0b62be3171ed2941e92832806d41463317c92b10 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Sun, 2 May 2021 22:18:02 +0800 Subject: [PATCH 20/68] =?UTF-8?q?=E6=96=B0=E5=BB=BA=20assignment-2?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- assignment-2/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 assignment-2/.keep diff --git a/assignment-2/.keep b/assignment-2/.keep new file mode 100644 index 0000000..e69de29 -- Gitee From a9d3ccb4420c641e842715676ae0a8d455caee0b Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Sun, 2 May 2021 22:19:37 +0800 Subject: [PATCH 21/68] =?UTF-8?q?=E6=96=B0=E5=BB=BA=20submission?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- assignment-2/submission/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 assignment-2/submission/.keep diff --git a/assignment-2/submission/.keep b/assignment-2/submission/.keep new file mode 100644 index 0000000..e69de29 -- Gitee From 413d110d64e021a7926fb53fdde2cffb3fb1f95b Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Sun, 2 May 2021 22:19:55 +0800 Subject: [PATCH 22/68] =?UTF-8?q?=E6=96=B0=E5=BB=BA=2018340986009?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- assignment-2/submission/18340986009/.keep | 0 1 file changed, 0 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z(8J=y>3@MyeqAWcIK z+paSNrzh%#^QN`^cMA0b3+ z+b-^8OJjBf?INNa7wl>eGGKB(8Vut{)yDd1!K#Cb(V!Ju{af5uU|SRrxz&R>3fKv~ zC7NK*2_wu}UAO4OZHVt9ci)Ly9 zTL#CToe1`xPyAkVIG!m0NFloG(H2T#eB6~kNVLG0k*1%0{2AtjApuD#3`TP=DmM@~ z$k7PK7>M3^`da(IjJGUhphqP@xemqwmHW@UF8+l*gti+%@%roH_b}M!`(hV=fZc{< z6;Q-J01*i)FaNoJ@$dgu^1pNPzjgv#;eUPS|2rtaiFSR*8-MV6^a%uY(h~CG1 Date: Sun, 2 May 2021 22:22:59 +0800 Subject: [PATCH 25/68] (): --- assignment-2/submission/18340986009/README.md | 458 ++++++++++++++++++ 1 file changed, 458 insertions(+) create mode 100644 assignment-2/submission/18340986009/README.md diff --git a/assignment-2/submission/18340986009/README.md b/assignment-2/submission/18340986009/README.md new file mode 100644 index 0000000..02dc703 --- /dev/null +++ b/assignment-2/submission/18340986009/README.md @@ -0,0 +1,458 @@ +# FNN + +## 1. Activation Function + +The data used in this report is from the **torch** library. The data is a collection of 28$\small{\times}$28 matrices each representing a picture. The goal is to train the FNN model to identify which digit from 0 to 9 (c = 10) is in the picture. + +With a batch size of 128 (N = 128), in each epoch, our training data is a 3-order tensor of shape (128, 28, 28) and our label after one-hot encoding is a tensor of shape (128, 10). In the forwarding step, each matrix $\underset{28\times 28}{\mathrm{X}}$ is flattened to a vector $\underset{1\times 28^{2}}{\mathrm{x}}$ for computational purpose. There are 3 layers in total, each layer reshapes its input by the weight matrix $\mathrm{W}$. + +_For the sake of consistency, we'll use numerator layout throughout, +use subscript $X_{(i)}$ to denote the $ith$ row of $X$, superscript $X^{(j)}$ to denote the $jth$ column of $X$, and use $X_{ij}$ to denote the $(i,j)$ element of $X$_ + +### 1.1 Matmul + +$ +\underset{N\times d_{new}}{Y} = matmul({X}, {W}) = +[\underset{N\times d_{prev}}{\mathrm{X}}] \hspace{0.3cm} +[\underset{d_{prev}\times d_{new}}{\mathrm{W}}] +$ + +$ +{\mathrm{Y_{ij}}} = {\sum\limits_{i=1}^{d}}{X_{ik}}{W_{kj}} +$, +where each element ${\mathrm{Y_{ij}}}$ is related only to $X_{(i)}$ and $W^{(j)}$ + + +**Grad X:** + +Since Y is an $N \times d_{new}$ matrix and X is an $N \times d_{prev}$ matrix, + +$ +{J} = \frac{\partial Y}{\partial X} +$ +will be an $(N \times N) \times (d_{new} \times d_{prev})$ 4-d tensor $J$ where +$J_{ijkl} = +\frac{\partial Y_{ij}}{\partial X_{kl}} = +\begin{cases}W_{lj} & i=k \\ 0 & otherwise \end{cases} +$ + +Expanding $J$, we have + +$J = +\begin{bmatrix} + J_{(1)} = \frac{\partial Y_{i=1j}}{\partial X_{kl}}\\ + \vdots \\ + J_{(N)} = \frac{\partial Y_{i=Nj}}{\partial X_{kl}}\\ +\end{bmatrix} +$, where + +$J_{(i)} = +\begin{bmatrix} + \frac{\partial Y_{ij=1}}{\partial X_{kl}} + \cdots + \frac{\partial Y_{ij=d_{new}}}{\partial X_{kl}} +\end{bmatrix} +$, for i = 1 to N, where + +$J_{(i)}^{(j)} = +\begin{bmatrix} + W_{11}, W_{12}, \cdots, W_{1d_{prev}}\\ + 0, \cdots, 0 +\end{bmatrix} +$, for j = 1 to $d_{new}$, with non-trivial values only in the ith row. + +Thus, collecting terms, $J_{(i)}$ reduces to +$ +\begin{bmatrix} + W_{11}, W_{21}, \cdots, W_{d_{prev}1}\\ + W_{12}, W_{22}, \cdots, W_{d_{prev}2}\\ + W_{1d_{new}}, W_{2d_{new}}, \cdots, W_{d_{new}d_{prev}}\\ +\end{bmatrix} = W^{T} +$, and + +$ +grad X_{(i)} = grad Y_{(i)} J_{(i)} = grad Y_{(i)}W^{T} +$ for i = 1 to N + +**Grad W:** + +Since Y is a $N \times d_{new}$ matrix and W is an $d_{prev} \times d_{new}$ matrix, + +$ +{J} = \frac{\partial Y}{\partial W} +$ +will be an $(N \times d_{prev}) \times (d_{new} \times d_{new})$ 4-d tensor $J$ where +$J_{ijkl} = +\frac{\partial Y_{ij}}{\partial W_{kl}} = +\begin{cases}X_{ik} & j=l \\ 0 & otherwise \end{cases} +$ + +Expanding $J$, we have + +$J = +\begin{bmatrix} + J^{(1)} = \frac{\partial Y_{ij=1}}{\partial W_{kl}}, + \cdots , + J^{(d_{new})} = \frac{\partial Y_{ij=d_{new}}}{\partial W_{kl}} +\end{bmatrix} +$, where + +$J^{*} = J^{(j)} = +\begin{bmatrix} + \frac{\partial Y_{i=1j}}{\partial W_{kl}}\\ + \vdots \\ + \frac{\partial Y_{i=Nj}}{\partial W_{kl}}\\ +\end{bmatrix} +$, for j = 1 to $d_{new}$, where + +$ J^{*}_{(i)} = J^{(j)}_{(i)} = +\begin{bmatrix} + X_{11} \vdots\\ + \vdots \vdots\\ + X_{d_{prev}1} \vdots\\ +\end{bmatrix} +$, with non-trivial values only in the jth column. + +Thus, collecting terms, $J^{*}$ reduces to +$ +\begin{bmatrix} + X_{11}, X_{12}, \cdots, X_{1N}\\ + X_{21}, X_{22}, \cdots, X_{2N}\\ + X_{d_{prev}1}, X_{d_{prev}2}, \cdots, X_{d_{prev}N}\\ +\end{bmatrix} = X^{T} +$. Thus, + +$ +grad W^{(j)} = J^{(j)}gradY^{(j)} = X^{T}gradY^{(j)} +$ for j = 1 to $d_{new}$ + +```python + def backward(self, grad_y): + """ + grad_y: shape(N, d') + """ + x = self.memory['x'] + W = self.memory['W'] + + grad_x = np.zeros(x.shape) + # apply Jacobian to every row of grad_y + for i in range(grad_y.shape[0]): + grad_x[i, :] = np.matmul(grad_y[i, :], W.T) + grad_x = np.array(grad_x) + + grad_W = np.zeros(W.shape) + # apply Jacobian to every column of grad_y + for j in range(grad_y.shape[1]): + grad_W[:, j] = np.matmul(x.T, grad_y[:, j]) + grad_W = np.array(grad_W) + + return grad_x, grad_W +``` + +### 1.2 Relu: + +Relu is an element-wise function. For every $Y_{ij}$, + +$ +{\mathrm{Y}_{ij}} = +relu(\mathrm{X}_{ij}) = +\begin{cases}0 & X_{ij} <= 0 \\ +X_{ij} & otherwise +\end{cases} +$ + +$ +\frac{\partial relu(X_{ij})}{\partial X_{ij}} = +\begin{cases}0 & X_{ij} <= 0 \\ 1 & otherwise \end{cases} +$ + +$ +grad X_{ij} = \begin{cases}0 & X_{ij} <= 0 \\ grad Y_{ij} & otherwise \end{cases} +$ +,by the chain rule + +```python + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + x = self.memory['x'] + grad_x = np.where(x <= 0, 0, grad_y) + + return grad_x +``` + +### 1.3 Log + +Log is an element-wise function. For every $Y_{ij}$, + +$ +{\mathrm{Y_{ij}}} = +\log{\big{(}{\mathrm{X}_{ij}}\big{)}} +$ + +$ +\frac{\partial \log{X}_{ij}}{\partial X_{ij}} = \frac{1}{X_{ij}} +$ + +$ +grad X_{ij} = \frac{1}{X_{ij +}} grad Y_{ij} +$ +, by the chain-rule + +```python + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + x = self.memory['x'] + grad_x = (x + self.epsilon)**(-1) * grad_y + + return grad_x +``` + +### 1.4 Softmax: + +$ +\underset{N\times c}{Y} = +softmax(\underset{N\times c}{X}) +$ +,where +$ +Y_{ij} = \dfrac{e^{X_{ij}}}{\sum\limits_{j=1}^{c}e^{X_{ij}}} +$(denominator sums along each row) + +Since each element in $Y_{(i)}$ is related to $X_{ij}$ for all j, we'll need a jacobian matrix $J_{(i)} = +\dfrac{\partial{Y_{(i)}}}{\partial{X_{ij}}}$ for each row of Y. + +$\hspace{0.5cm}$ Denote ${\sum\limits_{j=1}^{c}e^{X_{ij}}}$ as $\sum$. $J_{(i)jk}$ as the (j,k) element of $J_{i}$. Then for every $J_{(i)}$, + +$\hspace{0.5cm}$ When j = k, +$ +J_{(i)jk} = +\dfrac{e^{X_{ij}}\sum - e^{X_{ij}}e^{X_{ik}}}{\sum^{2}} = +\dfrac{e^{X_{ij}}}{\sum} +\dfrac{\sum - e^{X_{ij}}}{\sum} = +Y_{ij}(1-Y_{ik}) +$ + +$\hspace{0.5cm}$ Otherwise, +$ +J_{(i)jk} = +\dfrac{0 - e^{X_{ij}}e^{X_{ik}}}{\sum^{2}} = +\dfrac{e^{X_{ij}}}{\sum} +\dfrac{e^{X_{ik}}}{\sum} = +-Y_{ij}Y_{ik} +$ + + +$J_{(i)jk} = +\begin{cases}Y_{ik}(1 - Y_{ij}) & j=k \\ -Y_{ik}Y_{ij} & otherwise \end{cases} +$. Thus, + +$ +{grad X_{(i)}} = {J_{(i)}} \hspace{0.3cm} +{grad Y_{(i)}} +$ for $i = 1$ to $128$ + +```python + def forward(self, x): + """ + x: shape(N, c) + """ + exp = np.exp(x - self.epsilon) + out = exp/exp.sum(axis=1, keepdims=True) + self.memory['x'] = x + self.memory['s'] = out + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + s = self.memory['s'] + n = s.shape[1] + + grad_x = np.zeros(s.shape) + # calculate Jacobian for ith row + for i in range(s.shape[0]): + jacob = np.zeros((n, n)) + + for j in range(n): + for k in range(n): + if j == k: + jacob[k, j] = s[i, k] * (1-s[i, j]) + else: + jacob[k, j] = -1 * s[i, k] * s[i, j] + # apply Jacobian to the ith row of grad_y + grad_x[i, :] = np.matmul(jacob, grad_y[i, :]) + + return grad_x +``` + +## 2. Model Training + +### 2.1 Change mini_batch: + +From documentation, we can see that mini_batch is a function that shuffles data and create small batches with user-specified batch size. The return is a batch such that the code: **for x, y in mini_batch(train_dataset):** in function **numpy_run()** can retrive appropriate training data and training label in the batch. The suffling method implemented by the code below uses ordered sampling without replacement. If the remaining data is insufficient to form a batch of the specified size, we'll drop the remaining ones. + +```python +def mini_batch(dataset, batch_size=128): + + # get total number of observations to batch from + size = dataset.train_data.shape[0] + + # fetch data and label + data = [] + label = [] + for one_obs in dataset: + data.append(np.array(one_obs[0])) + label.append(np.array(one_obs[1])) + data = np.array(data) + label = np.array(label) + + # random shuffle + idx = np.arange(size) + np.random.shuffle(idx) + data = data[idx] + label = label[idx] + + # split to batch + # discard the last batch if remaining size < batch_size + data_batches = [] + label_batches = [] + batched = 0 + while batched+batch_size <= size: + data_batches.append(data[batched:batched + batch_size]) + label_batches.append(label[batched:batched + batch_size]) + batched += batch_size + + data_batches = np.array(data_batches) + label_batches = np.array(label_batches) + + return zip(data_batches, label_batches) +``` + +### 2.2 Training Result: + + - Hyper-parameters: + * epoch number = 3 + * learning rate = 0.1 + + + - Model structure: + * 3 layers + * dimension reduced in each layer using a weight matrix + * softmax as the final activation function + * loss calculated as $-y^{T}\log{\hat{y}}$ + +```python +def forward(self, x): + x = x.reshape(-1, 28 * 28) + a0 = x + + #First Layer + z1 = self.matmul_1.forward(a0, self.W1) + a1 = self.relu_1.forward(z1) + + #Second Layer + z2 = self.matmul_2.forward(a1, self.W2) + a2 = self.relu_2.forward(z2) + + #Third Layer + z3 = self.matmul_3.forward(a2, self.W3) + a3 = self.softmax.forward(z3) + x = self.log.forward(a3) + + return x +``` + +```python +def backward(self, y): + + #Third Layer + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + + #Second Layer + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + + #First Layer + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + pass +``` + +Result: +```python +[0] Accuracy: 0.9482 +[1] Accuracy: 0.9643 +[2] Accuracy: 0.9735 +``` + + + +## 3. Change of Parameters + +### 3.1 Learning Rate [0.2, 0.4, 0.6] + +![](img/lr1.png) ![](img/lr2.png) ![](img/lr3.png) + +### 3.2 Batch Size [20, 40, 60] + +![](img/bs1.png) ![](img/bs2.png) ![](img/bs3.png) + +## 4. Optimization + +**Momentum:** + +$ +v_{t+1} = \mu v_t-\eta\nabla l(\theta)\\\theta_{t+1} = \theta_t + v_{t+1} +$ +**referenced from Algorithms for Optimization, 2019, by Mykel J., K & Tim A.,W* + +```python +#Extends NumpyModel +#Save step size +self.delta1 = 0 +self.delta2 = 0 +self.delta3 = 0 + +#Extends the optimize function defined in numpy_fnn.py +if method == "Momentum": + self.delta1 = momentum * self.delta1 - learning_rate * self.W1_grad + self.W1 += self.delta1 + + self.delta2 = momentum * self.delta2 - learning_rate * self.W2_grad + self.W2 += self.delta2 + + self.delta3 = momentum * self.delta3 - learning_rate * self.W3_grad + self.W3 += self.delta3 +``` + +Result: +```python +[0] Accuracy: 0.9654 +[1] Accuracy: 0.9712 +[2] Accuracy: 0.9726 +``` + + + +Discussion: + +Momentum method differs from gradient descent in that it adds a short term memory for step size. Momentum, a number between 0 and 1, is the weight given to the previous step size. Momentum smoothes the path (visual demonstration below, [ref](https://dominikschmidt.xyz/nesterov-momentum/)). A greater momentum results in a more direct path with less changes in direction. Or, from another perspective, accelarates towards the optimum. If momentum = 0, the method reduces to Gradient Descent. + + + +Momentum $\beta$ can be thought of the weight of the previous force, while learning rate $\alpha$ can be thought of the weight of the current force. Intuitively, the optimal training model needs an optimal combination of $\alpha$ and $\beta$ depending on the eigenvalues of a convex function. Inappropriate values could lead to divergence. + + +```python + +``` -- Gitee From 2205c42429a33ea7f2387cf2df5687e6e5e04e88 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Sun, 2 May 2021 22:32:06 +0800 Subject: [PATCH 26/68] =?UTF-8?q?=E5=88=A0=E9=99=A4=E6=96=87=E4=BB=B6=20as?= =?UTF-8?q?signment-2/submission/18340986009/README.md?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- assignment-2/submission/18340986009/README.md | 458 ------------------ 1 file changed, 458 deletions(-) delete mode 100644 assignment-2/submission/18340986009/README.md diff --git a/assignment-2/submission/18340986009/README.md b/assignment-2/submission/18340986009/README.md deleted file mode 100644 index 02dc703..0000000 --- a/assignment-2/submission/18340986009/README.md +++ /dev/null @@ -1,458 +0,0 @@ -# FNN - -## 1. Activation Function - -The data used in this report is from the **torch** library. The data is a collection of 28$\small{\times}$28 matrices each representing a picture. The goal is to train the FNN model to identify which digit from 0 to 9 (c = 10) is in the picture. - -With a batch size of 128 (N = 128), in each epoch, our training data is a 3-order tensor of shape (128, 28, 28) and our label after one-hot encoding is a tensor of shape (128, 10). In the forwarding step, each matrix $\underset{28\times 28}{\mathrm{X}}$ is flattened to a vector $\underset{1\times 28^{2}}{\mathrm{x}}$ for computational purpose. There are 3 layers in total, each layer reshapes its input by the weight matrix $\mathrm{W}$. - -_For the sake of consistency, we'll use numerator layout throughout, -use subscript $X_{(i)}$ to denote the $ith$ row of $X$, superscript $X^{(j)}$ to denote the $jth$ column of $X$, and use $X_{ij}$ to denote the $(i,j)$ element of $X$_ - -### 1.1 Matmul - -$ -\underset{N\times d_{new}}{Y} = matmul({X}, {W}) = -[\underset{N\times d_{prev}}{\mathrm{X}}] \hspace{0.3cm} -[\underset{d_{prev}\times d_{new}}{\mathrm{W}}] -$ - -$ -{\mathrm{Y_{ij}}} = {\sum\limits_{i=1}^{d}}{X_{ik}}{W_{kj}} -$, -where each element ${\mathrm{Y_{ij}}}$ is related only to $X_{(i)}$ and $W^{(j)}$ - - -**Grad X:** - -Since Y is an $N \times d_{new}$ matrix and X is an $N \times d_{prev}$ matrix, - -$ -{J} = \frac{\partial Y}{\partial X} -$ -will be an $(N \times N) \times (d_{new} \times d_{prev})$ 4-d tensor $J$ where -$J_{ijkl} = -\frac{\partial Y_{ij}}{\partial X_{kl}} = -\begin{cases}W_{lj} & i=k \\ 0 & otherwise \end{cases} -$ - -Expanding $J$, we have - -$J = -\begin{bmatrix} - J_{(1)} = \frac{\partial Y_{i=1j}}{\partial X_{kl}}\\ - \vdots \\ - J_{(N)} = \frac{\partial Y_{i=Nj}}{\partial X_{kl}}\\ -\end{bmatrix} -$, where - -$J_{(i)} = -\begin{bmatrix} - \frac{\partial Y_{ij=1}}{\partial X_{kl}} - \cdots - \frac{\partial Y_{ij=d_{new}}}{\partial X_{kl}} -\end{bmatrix} -$, for i = 1 to N, where - -$J_{(i)}^{(j)} = -\begin{bmatrix} - W_{11}, W_{12}, \cdots, W_{1d_{prev}}\\ - 0, \cdots, 0 -\end{bmatrix} -$, for j = 1 to $d_{new}$, with non-trivial values only in the ith row. - -Thus, collecting terms, $J_{(i)}$ reduces to -$ -\begin{bmatrix} - W_{11}, W_{21}, \cdots, W_{d_{prev}1}\\ - W_{12}, W_{22}, \cdots, W_{d_{prev}2}\\ - W_{1d_{new}}, W_{2d_{new}}, \cdots, W_{d_{new}d_{prev}}\\ -\end{bmatrix} = W^{T} -$, and - -$ -grad X_{(i)} = grad Y_{(i)} J_{(i)} = grad Y_{(i)}W^{T} -$ for i = 1 to N - -**Grad W:** - -Since Y is a $N \times d_{new}$ matrix and W is an $d_{prev} \times d_{new}$ matrix, - -$ -{J} = \frac{\partial Y}{\partial W} -$ -will be an $(N \times d_{prev}) \times (d_{new} \times d_{new})$ 4-d tensor $J$ where -$J_{ijkl} = -\frac{\partial Y_{ij}}{\partial W_{kl}} = -\begin{cases}X_{ik} & j=l \\ 0 & otherwise \end{cases} -$ - -Expanding $J$, we have - -$J = -\begin{bmatrix} - J^{(1)} = \frac{\partial Y_{ij=1}}{\partial W_{kl}}, - \cdots , - J^{(d_{new})} = \frac{\partial Y_{ij=d_{new}}}{\partial W_{kl}} -\end{bmatrix} -$, where - -$J^{*} = J^{(j)} = -\begin{bmatrix} - \frac{\partial Y_{i=1j}}{\partial W_{kl}}\\ - \vdots \\ - \frac{\partial Y_{i=Nj}}{\partial W_{kl}}\\ -\end{bmatrix} -$, for j = 1 to $d_{new}$, where - -$ J^{*}_{(i)} = J^{(j)}_{(i)} = -\begin{bmatrix} - X_{11} \vdots\\ - \vdots \vdots\\ - X_{d_{prev}1} \vdots\\ -\end{bmatrix} -$, with non-trivial values only in the jth column. - -Thus, collecting terms, $J^{*}$ reduces to -$ -\begin{bmatrix} - X_{11}, X_{12}, \cdots, X_{1N}\\ - X_{21}, X_{22}, \cdots, X_{2N}\\ - X_{d_{prev}1}, X_{d_{prev}2}, \cdots, X_{d_{prev}N}\\ -\end{bmatrix} = X^{T} -$. Thus, - -$ -grad W^{(j)} = J^{(j)}gradY^{(j)} = X^{T}gradY^{(j)} -$ for j = 1 to $d_{new}$ - -```python - def backward(self, grad_y): - """ - grad_y: shape(N, d') - """ - x = self.memory['x'] - W = self.memory['W'] - - grad_x = np.zeros(x.shape) - # apply Jacobian to every row of grad_y - for i in range(grad_y.shape[0]): - grad_x[i, :] = np.matmul(grad_y[i, :], W.T) - grad_x = np.array(grad_x) - - grad_W = np.zeros(W.shape) - # apply Jacobian to every column of grad_y - for j in range(grad_y.shape[1]): - grad_W[:, j] = np.matmul(x.T, grad_y[:, j]) - grad_W = np.array(grad_W) - - return grad_x, grad_W -``` - -### 1.2 Relu: - -Relu is an element-wise function. For every $Y_{ij}$, - -$ -{\mathrm{Y}_{ij}} = -relu(\mathrm{X}_{ij}) = -\begin{cases}0 & X_{ij} <= 0 \\ -X_{ij} & otherwise -\end{cases} -$ - -$ -\frac{\partial relu(X_{ij})}{\partial X_{ij}} = -\begin{cases}0 & X_{ij} <= 0 \\ 1 & otherwise \end{cases} -$ - -$ -grad X_{ij} = \begin{cases}0 & X_{ij} <= 0 \\ grad Y_{ij} & otherwise \end{cases} -$ -,by the chain rule - -```python - def backward(self, grad_y): - """ - grad_y: same shape as x - """ - x = self.memory['x'] - grad_x = np.where(x <= 0, 0, grad_y) - - return grad_x -``` - -### 1.3 Log - -Log is an element-wise function. For every $Y_{ij}$, - -$ -{\mathrm{Y_{ij}}} = -\log{\big{(}{\mathrm{X}_{ij}}\big{)}} -$ - -$ -\frac{\partial \log{X}_{ij}}{\partial X_{ij}} = \frac{1}{X_{ij}} -$ - -$ -grad X_{ij} = \frac{1}{X_{ij -}} grad Y_{ij} -$ -, by the chain-rule - -```python - def backward(self, grad_y): - """ - grad_y: same shape as x - """ - x = self.memory['x'] - grad_x = (x + self.epsilon)**(-1) * grad_y - - return grad_x -``` - -### 1.4 Softmax: - -$ -\underset{N\times c}{Y} = -softmax(\underset{N\times c}{X}) -$ -,where -$ -Y_{ij} = \dfrac{e^{X_{ij}}}{\sum\limits_{j=1}^{c}e^{X_{ij}}} -$(denominator sums along each row) - -Since each element in $Y_{(i)}$ is related to $X_{ij}$ for all j, we'll need a jacobian matrix $J_{(i)} = -\dfrac{\partial{Y_{(i)}}}{\partial{X_{ij}}}$ for each row of Y. - -$\hspace{0.5cm}$ Denote ${\sum\limits_{j=1}^{c}e^{X_{ij}}}$ as $\sum$. $J_{(i)jk}$ as the (j,k) element of $J_{i}$. Then for every $J_{(i)}$, - -$\hspace{0.5cm}$ When j = k, -$ -J_{(i)jk} = -\dfrac{e^{X_{ij}}\sum - e^{X_{ij}}e^{X_{ik}}}{\sum^{2}} = -\dfrac{e^{X_{ij}}}{\sum} -\dfrac{\sum - e^{X_{ij}}}{\sum} = -Y_{ij}(1-Y_{ik}) -$ - -$\hspace{0.5cm}$ Otherwise, -$ -J_{(i)jk} = -\dfrac{0 - e^{X_{ij}}e^{X_{ik}}}{\sum^{2}} = -\dfrac{e^{X_{ij}}}{\sum} -\dfrac{e^{X_{ik}}}{\sum} = --Y_{ij}Y_{ik} -$ - - -$J_{(i)jk} = -\begin{cases}Y_{ik}(1 - Y_{ij}) & j=k \\ -Y_{ik}Y_{ij} & otherwise \end{cases} -$. Thus, - -$ -{grad X_{(i)}} = {J_{(i)}} \hspace{0.3cm} -{grad Y_{(i)}} -$ for $i = 1$ to $128$ - -```python - def forward(self, x): - """ - x: shape(N, c) - """ - exp = np.exp(x - self.epsilon) - out = exp/exp.sum(axis=1, keepdims=True) - self.memory['x'] = x - self.memory['s'] = out - - return out - - def backward(self, grad_y): - """ - grad_y: same shape as x - """ - s = self.memory['s'] - n = s.shape[1] - - grad_x = np.zeros(s.shape) - # calculate Jacobian for ith row - for i in range(s.shape[0]): - jacob = np.zeros((n, n)) - - for j in range(n): - for k in range(n): - if j == k: - jacob[k, j] = s[i, k] * (1-s[i, j]) - else: - jacob[k, j] = -1 * s[i, k] * s[i, j] - # apply Jacobian to the ith row of grad_y - grad_x[i, :] = np.matmul(jacob, grad_y[i, :]) - - return grad_x -``` - -## 2. Model Training - -### 2.1 Change mini_batch: - -From documentation, we can see that mini_batch is a function that shuffles data and create small batches with user-specified batch size. The return is a batch such that the code: **for x, y in mini_batch(train_dataset):** in function **numpy_run()** can retrive appropriate training data and training label in the batch. The suffling method implemented by the code below uses ordered sampling without replacement. If the remaining data is insufficient to form a batch of the specified size, we'll drop the remaining ones. - -```python -def mini_batch(dataset, batch_size=128): - - # get total number of observations to batch from - size = dataset.train_data.shape[0] - - # fetch data and label - data = [] - label = [] - for one_obs in dataset: - data.append(np.array(one_obs[0])) - label.append(np.array(one_obs[1])) - data = np.array(data) - label = np.array(label) - - # random shuffle - idx = np.arange(size) - np.random.shuffle(idx) - data = data[idx] - label = label[idx] - - # split to batch - # discard the last batch if remaining size < batch_size - data_batches = [] - label_batches = [] - batched = 0 - while batched+batch_size <= size: - data_batches.append(data[batched:batched + batch_size]) - label_batches.append(label[batched:batched + batch_size]) - batched += batch_size - - data_batches = np.array(data_batches) - label_batches = np.array(label_batches) - - return zip(data_batches, label_batches) -``` - -### 2.2 Training Result: - - - Hyper-parameters: - * epoch number = 3 - * learning rate = 0.1 - - - - Model structure: - * 3 layers - * dimension reduced in each layer using a weight matrix - * softmax as the final activation function - * loss calculated as $-y^{T}\log{\hat{y}}$ - -```python -def forward(self, x): - x = x.reshape(-1, 28 * 28) - a0 = x - - #First Layer - z1 = self.matmul_1.forward(a0, self.W1) - a1 = self.relu_1.forward(z1) - - #Second Layer - z2 = self.matmul_2.forward(a1, self.W2) - a2 = self.relu_2.forward(z2) - - #Third Layer - z3 = self.matmul_3.forward(a2, self.W3) - a3 = self.softmax.forward(z3) - x = self.log.forward(a3) - - return x -``` - -```python -def backward(self, y): - - #Third Layer - self.log_grad = self.log.backward(y) - self.softmax_grad = self.softmax.backward(self.log_grad) - self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) - - #Second Layer - self.relu_2_grad = self.relu_2.backward(self.x3_grad) - self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) - - #First Layer - self.relu_1_grad = self.relu_1.backward(self.x2_grad) - self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) - - pass -``` - -Result: -```python -[0] Accuracy: 0.9482 -[1] Accuracy: 0.9643 -[2] Accuracy: 0.9735 -``` - - - -## 3. Change of Parameters - -### 3.1 Learning Rate [0.2, 0.4, 0.6] - -![](img/lr1.png) ![](img/lr2.png) ![](img/lr3.png) - -### 3.2 Batch Size [20, 40, 60] - -![](img/bs1.png) ![](img/bs2.png) ![](img/bs3.png) - -## 4. Optimization - -**Momentum:** - -$ -v_{t+1} = \mu v_t-\eta\nabla l(\theta)\\\theta_{t+1} = \theta_t + v_{t+1} -$ -**referenced from Algorithms for Optimization, 2019, by Mykel J., K & Tim A.,W* - -```python -#Extends NumpyModel -#Save step size -self.delta1 = 0 -self.delta2 = 0 -self.delta3 = 0 - -#Extends the optimize function defined in numpy_fnn.py -if method == "Momentum": - self.delta1 = momentum * self.delta1 - learning_rate * self.W1_grad - self.W1 += self.delta1 - - self.delta2 = momentum * self.delta2 - learning_rate * self.W2_grad - self.W2 += self.delta2 - - self.delta3 = momentum * self.delta3 - learning_rate * self.W3_grad - self.W3 += self.delta3 -``` - -Result: -```python -[0] Accuracy: 0.9654 -[1] Accuracy: 0.9712 -[2] Accuracy: 0.9726 -``` - - - -Discussion: - -Momentum method differs from gradient descent in that it adds a short term memory for step size. Momentum, a number between 0 and 1, is the weight given to the previous step size. Momentum smoothes the path (visual demonstration below, [ref](https://dominikschmidt.xyz/nesterov-momentum/)). A greater momentum results in a more direct path with less changes in direction. Or, from another perspective, accelarates towards the optimum. If momentum = 0, the method reduces to Gradient Descent. - - - -Momentum $\beta$ can be thought of the weight of the previous force, while learning rate $\alpha$ can be thought of the weight of the current force. Intuitively, the optimal training model needs an optimal combination of $\alpha$ and $\beta$ depending on the eigenvalues of a convex function. Inappropriate values could lead to divergence. - - -```python - -``` -- Gitee From 54ef0c27172e35c96a5373feec9b653b0311cbec Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Sun, 2 May 2021 22:32:14 +0800 Subject: [PATCH 27/68] =?UTF-8?q?=E5=88=A0=E9=99=A4=E6=96=87=E4=BB=B6=20as?= =?UTF-8?q?signment-2/submission/18340986009/.keep?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- assignment-2/submission/18340986009/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 assignment-2/submission/18340986009/.keep diff --git a/assignment-2/submission/18340986009/.keep b/assignment-2/submission/18340986009/.keep deleted file mode 100644 index e69de29..0000000 -- Gitee From 7267161a859ebc70bce7a670be8ae6a8dc547674 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Sun, 2 May 2021 22:39:50 +0800 Subject: [PATCH 28/68] (): --- assignment-2/submission/18340986009/README.md | 265 ++++++++++++++++++ 1 file changed, 265 insertions(+) create mode 100644 assignment-2/submission/18340986009/README.md diff --git a/assignment-2/submission/18340986009/README.md b/assignment-2/submission/18340986009/README.md new file mode 100644 index 0000000..3352820 --- /dev/null +++ b/assignment-2/submission/18340986009/README.md @@ -0,0 +1,265 @@ +# FNN + +## 1. Activation Function + +The data used in this report is from the **torch** library. The data is a collection of 28$\small{\times}$28 matrices each representing a picture. The goal is to train the FNN model to identify which digit from 0 to 9 (c = 10) is in the picture. + +With a batch size of 128 (N = 128), in each epoch, our training data is a 3-order tensor of shape (128, 28, 28) and our label after one-hot encoding is a tensor of shape (128, 10). In the forwarding step, each matrix $\underset{28\times 28}{\mathrm{X}}$ is flattened to a vector $\underset{1\times 28^{2}}{\mathrm{x}}$ for computational purpose. There are 3 layers in total, each layer reshapes its input by the weight matrix $\mathrm{W}$. + +_For the sake of consistency, we'll use numerator layout throughout, +use subscript $X_{(i)}$ to denote the $ith$ row of $X$, superscript $X^{(j)}$ to denote the $jth$ column of $X$, and use $X_{ij}$ to denote the $(i,j)$ element of $X$_ + +### 1.1 Matmul + + + + + +```python + def backward(self, grad_y): + """ + grad_y: shape(N, d') + """ + x = self.memory['x'] + W = self.memory['W'] + + grad_x = np.matmul(grad_y, W.T) + grad_W = np.matmul(x.T, grad_y) + return grad_x, grad_W +``` + +### 1.2 Relu: + + + +```python + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + x = self.memory['x'] + grad_x = np.where(x <= 0, 0, grad_y) + + return grad_x +``` + +### 1.3 Log + + + +```python + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + x = self.memory['x'] + grad_x = (x + self.epsilon)**(-1) * grad_y + + return grad_x +``` + +### 1.4 Softmax: + + + +```python + def forward(self, x): + """ + x: shape(N, c) + """ + exp = np.exp(x - self.epsilon) + out = exp/exp.sum(axis=1, keepdims=True) + self.memory['x'] = x + self.memory['s'] = out + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + s = self.memory['s'] + n = s.shape[1] + + grad_x = np.zeros(s.shape) + # calculate Jacobian for ith row + for i in range(s.shape[0]): + jacob = np.zeros((n, n)) + + for j in range(n): + for k in range(n): + if j == k: + jacob[k, j] = s[i, k] * (1-s[i, j]) + else: + jacob[k, j] = -1 * s[i, k] * s[i, j] + # apply Jacobian to the ith row of grad_y + grad_x[i, :] = np.matmul(jacob, grad_y[i, :]) + + return grad_x +``` + +## 2. Model Training + +### 2.1 Change mini_batch: + +From documentation, we can see that mini_batch is a function that shuffles data and create small batches with user-specified batch size. The return is a batch such that the code: **for x, y in mini_batch(train_dataset):** in function **numpy_run()** can retrive appropriate training data and training label in the batch. The suffling method implemented by the code below uses ordered sampling without replacement. If the remaining data is insufficient to form a batch of the specified size, we'll drop the remaining ones. + +```python +def mini_batch(dataset, batch_size=128): + + # get total number of observations to batch from + size = dataset.train_data.shape[0] + + # fetch data and label + data = [] + label = [] + for one_obs in dataset: + data.append(np.array(one_obs[0])) + label.append(np.array(one_obs[1])) + data = np.array(data) + label = np.array(label) + + # random shuffle + idx = np.arange(size) + np.random.shuffle(idx) + data = data[idx] + label = label[idx] + + # split to batch + # discard the last batch if remaining size < batch_size + data_batches = [] + label_batches = [] + batched = 0 + while batched+batch_size <= size: + data_batches.append(data[batched:batched + batch_size]) + label_batches.append(label[batched:batched + batch_size]) + batched += batch_size + + data_batches = np.array(data_batches) + label_batches = np.array(label_batches) + + return zip(data_batches, label_batches) +``` + +### 2.2 Training Result: + + - Hyper-parameters: + * epoch number = 3 + * learning rate = 0.1 + + + - Model structure: + * 3 layers + * dimension reduced in each layer using a weight matrix + * softmax as the final activation function + * loss calculated as $-y^{T}\log{\hat{y}}$ + +```python +def forward(self, x): + x = x.reshape(-1, 28 * 28) + a0 = x + + #First Layer + z1 = self.matmul_1.forward(a0, self.W1) + a1 = self.relu_1.forward(z1) + + #Second Layer + z2 = self.matmul_2.forward(a1, self.W2) + a2 = self.relu_2.forward(z2) + + #Third Layer + z3 = self.matmul_3.forward(a2, self.W3) + a3 = self.softmax.forward(z3) + x = self.log.forward(a3) + + return x +``` + +```python +def backward(self, y): + + #Third Layer + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + + #Second Layer + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + + #First Layer + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + pass +``` + +Result: +```python +[0] Accuracy: 0.9482 +[1] Accuracy: 0.9643 +[2] Accuracy: 0.9735 +``` + + + +## 3. Change of Parameters + +**Learning Rate [0.2, 0.4, 0.6]** + + + + + +**Batch Size [20, 40, 60]** + + + + + +## 4. Optimization + +**Momentum:** + + + +```python +#Extends NumpyModel +#Save step size +self.delta1 = 0 +self.delta2 = 0 +self.delta3 = 0 + +#Extends the optimize function defined in numpy_fnn.py +if method == "Momentum": + self.delta1 = momentum * self.delta1 - learning_rate * self.W1_grad + self.W1 += self.delta1 + + self.delta2 = momentum * self.delta2 - learning_rate * self.W2_grad + self.W2 += self.delta2 + + self.delta3 = momentum * self.delta3 - learning_rate * self.W3_grad + self.W3 += self.delta3 +``` + +Result: +```python +[0] Accuracy: 0.9654 +[1] Accuracy: 0.9712 +[2] Accuracy: 0.9726 +``` + + + +Discussion: + +Momentum method differs from gradient descent in that it adds a short term memory for step size. Momentum, a number between 0 and 1, is the weight given to the previous step size. Momentum smoothes the path (visual demonstration below, [ref](https://dominikschmidt.xyz/nesterov-momentum/)). A greater momentum results in a more direct path with less changes in direction. Or, from another perspective, accelarates towards the optimum. If momentum = 0, the method reduces to Gradient Descent. + + + +Momentum $\beta$ can be thought of the weight of the previous force, while learning rate $\alpha$ can be thought of the weight of the current force. Intuitively, the optimal training model needs an optimal combination of $\alpha$ and $\beta$ depending on the eigenvalues of a convex function. Inappropriate values could lead to divergence. + + +```python + +``` -- Gitee From fc2d73211392a0b1e1d49d13d85ef793b77d54a5 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Sun, 2 May 2021 22:41:17 +0800 Subject: [PATCH 29/68] (): --- .../submission/18340986009/img/md1.png | Bin 0 -> 121657 bytes .../submission/18340986009/img/md2.png | Bin 0 -> 99956 bytes .../submission/18340986009/img/md3.png | Bin 0 -> 38740 bytes .../submission/18340986009/img/md4.png | Bin 0 -> 21471 bytes .../submission/18340986009/img/md5.png | Bin 0 -> 85628 bytes .../submission/18340986009/img/md6.png | Bin 0 -> 18290 bytes 6 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 assignment-2/submission/18340986009/img/md1.png create mode 100644 assignment-2/submission/18340986009/img/md2.png create mode 100644 assignment-2/submission/18340986009/img/md3.png create mode 100644 assignment-2/submission/18340986009/img/md4.png create mode 100644 assignment-2/submission/18340986009/img/md5.png create 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