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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 05/19] (): --- 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 06/19] =?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 07/19] (): --- 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 08/19] =?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 09/19] (): --- 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 10/19] =?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 11/19] (): --- 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 12/19] =?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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charset=UTF-8 Content-Transfer-Encoding: 8bit --- assignment-1/submission/18340986009/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 assignment-1/submission/18340986009/.keep diff --git a/assignment-1/submission/18340986009/.keep b/assignment-1/submission/18340986009/.keep deleted file mode 100644 index e69de29..0000000 -- Gitee From dffbd1d79ff35e5e8cc87127b63e9d5a995a9cc9 Mon Sep 17 00:00:00 2001 From: Yating Zou Date: Thu, 1 Apr 2021 20:26:37 +0800 Subject: [PATCH 16/19] (): --- .../submission/18340986009/img/Figure 8.png | Bin 0 -> 58545 bytes .../submission/18340986009/img/Figure 9.png | Bin 0 -> 34785 bytes 2 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 assignment-1/submission/18340986009/img/Figure 8.png create mode 100644 assignment-1/submission/18340986009/img/Figure 9.png diff --git a/assignment-1/submission/18340986009/img/Figure 8.png b/assignment-1/submission/18340986009/img/Figure 8.png new file mode 100644 index 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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 17/19] =?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 18/19] (): --- 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 19/19] (): --- 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