+
+在保证准确率不变的条件下,选择较小的数值k=5,预测的准确率达83.3%,对应数据可视化如下图
+
+
+
+### 对比实验1:减少点集重叠
+
+上图能较为清晰的看到,三种颜色的点集分布基本分离开,但是仍存在一部分重叠,推测重叠部分会使得KNN效果变差,下面通过改变均值和协方差验证这一结论
+
+首先将协方差对应更改成为
+
+|         | 均值    | 协方差矩阵      |
+| ------- | ------- | --------------- |
+| class 1 | (1, 2)  | [[1,0], [0, 1]] |
+| class 2 | (4, 5)  | [[1,0], [0, 1]] |
+| class 3 | (-2, 6) | [[1,0], [0, 1]] |
+
+对应K的曲线和点集分布图如下
+
+
+
+此时选择K = 3,对应的KNN准确率已经提高到了96.7%符合预期
+
+同样的,我们更改对应的均值大小,使得高斯分布尽可能分开
+
+|         | 均值      | 协方差矩阵        |
+| ------- | --------- | ----------------- |
+| class 1 | (-10, 2)  | [[10, 0], [0, 2]] |
+| class 2 | (4, 5)    | [[7, 3], [15, 1]] |
+| class 3 | (-2, -16) | [[0, 1], [1, 2]]  |
+
+对应曲线如下,准确率达到1.0,此时K=1已经达到了最大值
+
+
+
+#### 结论
+
+从该对比实验中,我们能够较为清晰的看到点集分布对于KNN准确率的影响,当类之间重合度较低时,KNN的准确率显著提升
+
+### 对比实验2:距离选择
+
+在上述实验中,我们采用的距离为欧式距离,下面将更改距离计算方式为曼哈顿距离,考察对应的影响
+
+当点集区分较开时,曼哈顿距离与欧式距离在准确率上差别不大,这里不做展示,当点集重叠程度较高时,对以下分布生成了多组数据
+
+|         | 均值   | 协方差矩阵        |
+| ------- | ------ | ----------------- |
+| class 1 | (1, 4) | [[10, 0], [0, 2]] |
+| class 2 | (2, 5) | [[7, 3], [15, 1]] |
+| class 3 | (2, 6) | [[0, 1], [1, 2]]  |
+
+对应的k值选取和准确率(acc)如下表所示
+
+| 欧氏距离          | 曼哈顿距离         |
+| ----------------- | ------------------ |
+| k = 3, acc = 0.7  | k=3, acc= 0.683    |
+| k = 1, acc = 0.53 | k = 1, acc = 0.483 |
+| k = 7, acc = 0.63 | k = 8, acc = 0.567 |
+
+综合来看点集分布重叠程度较高时,欧氏距离优于曼哈顿距离,推测以高斯分布生成的点,欧式距离对某一维度上较大差距的惩罚大于曼哈顿距离,较符合高斯分布点生成方式,较好拟合当前位置的概率密度,从而准确率更高
+
+#### 结论
+
+当点集区分较开时,曼哈顿距离和欧式距离差别不大,点集重合较大时,欧式距离由于曼哈顿距离
+
+### 对比实验3:点集数量
+
+对于如下分布
+
+|         | 均值    | 协方差矩阵        |
+| ------- | ------- | ----------------- |
+| class 1 | (1, 4)  | [[10, 0], [0, 2]] |
+| class 2 | (2, -3) | [[7, 3], [15, 1]] |
+| class 3 | (2, 5)  | [[0, 1], [1, 2]]  |
+
+分别生成了[100, 100, 100], [100, 10, 100], [100, 50, 200],[200, 200, 200]四组,每组多次避免偶然误差
+
+结果如下表格所示
+
+|      | [100, 100, 100] | [100, 10, 100] | [100, 50, 200] | [200, 200, 200] |
+| ---- | --------------- | -------------- | -------------- | --------------- |
+| 1    | 0.867           | 0.809          | 0.886          | 0.875           |
+| 2    | 0.800           | 0.809          | 0.843          | 0.825           |
+| 3    | 0.867           | 0.809          | 0.857          | 0.9             |
+| 4    | 0.917           | 0.761          | 0.886          | 0.792           |
+| 平均 | 0.862           | 0.797          | 0.868          | 0.848           |
+
+#### 结论
+
+当点集数量上升时,增大重叠面积,准确率相应下降,当某组点数量显著小于其他点集时,将会较大影响到准确率,当差距过大时,将会一定程度上退化成N-1分类问题,反而导致准确率提升
+
+### 对比实验4:各维度尺度
+
+当各个维度的尺度并不匹配时,例如(年龄,财产)二元组,基于空间上欧式距离相当于退化成为闵式距离,为进一步对比其影响,生成了如下数据
+
+|         | 均值     | 协方差矩阵            |
+| ------- | -------- | --------------------- |
+| class 1 | (1, 400) | [[10, 0], [0, 20000]] |
+| class 2 | (2, 300) | [[7, 0], [0, 10000]]  |
+| class 3 | (2, 300) | [[1, 0], [0, 10000]]  |
+
+其中一组对应k和点集分布如下图所示,多次测量的平均准确率为0.399
+
+
+
+为了对比其影响,我们等比例放缩对应的维度100倍,
+
+|         | 均值   | 协方差矩阵        |
+| ------- | ------ | ----------------- |
+| class 1 | (1, 4) | [[10, 0], [0, 2]] |
+| class 2 | (2, 3) | [[7, 0], [15, 1]] |
+| class 3 | (2, 3) | [[1, 0], [0, 1]]  |
+
+对应的k和点集可视化如下图
+
+
+
+多次测量的平均准确率为0.539
+
+#### 结论
+
+尺度归一化较大程度的影响了准确率的大小,通过等比例尺度放缩,准确率有了较大提升,但是,结合前面点集分布的表现,推测当点集自身区分较开时,归一化的影响不大
\ No newline at end of file
diff --git a/assignment-1/submission/18307130116/img/Figure_1.png b/assignment-1/submission/18307130116/img/Figure_1.png
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