From bd329765db25d414ead694b469900f28f58d4746 Mon Sep 17 00:00:00 2001
From: obangw <18307130116@fudan.edu.cn>
Date: Sat, 5 Jun 2021 00:02:39 +0800
Subject: [PATCH 1/7] assignment3
---
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+# Assignment3实验报告
+
+[toc]
+
+## 注释
+
+本次实验中所有函数均配有详尽的google风格的代码注释，因此除重点函数外，将略去实验报告中对API的详细介绍
+
+## 模型实现
+
+### 数据预处理
+
+为了避免极端值的出现，笔者将原先的数据各个维度投影到[-10, 10]区间，且等比例投影在欧式距离的情况下并不会影响到聚类结果，从而避免在计算概率值时出现指数运算溢出的情况
+
+### Kmeans
+
+#### 原理与收敛条件
+
+Kmeans的原理相对较为简单，在初始随机选中一些点作为聚类中心后，计算每个点对于聚类中心的距离，并归到最近的类中，更新聚类中心后重复该过程直至收敛。
+
+本次实验的收敛条件采用聚类中心是否变化作为最终条件，当某一轮更新后，聚类中心不发生改变，认为收敛。由于Kmeans对初值敏感，为避免某些随机初始化情况下在一些点上震荡，同时给定循环轮数的上界。
+
+#### 模型结构
+
+除给定的`init`,`fit`和`predict`三个API外，为方便使用，增添了`get_class`，`get_distance`，`update_center`三个API方便使用，`get_class`给定某个坐标点，在当前的聚类中心下，获得对应的类，`get_distance`:给定两个坐标点，获得对应的距离，`update_center`:根据聚类情况更新聚类中心
+
+### GMM
+
+#### 原理和问题
+
+高斯混合模型的基本假设有如下两点
+
+* 认为点的分布为高斯分布的叠加
+
+* 各个维度的高斯分布相互独立
+
+其中第一点指导了整个模型的构建，第二点简化了模型的计算，具体表现为分布的协方差矩阵为对角阵。
+
+GMM同样**对于初值非常敏感**，另一方面GMM对于数据量要求也较高，可以想见在极端小的数据规模下，高斯分布拟合出的结果会是一个瘦高的类似于冲激函数的形态，这将意味着，较小的扰动都会带来较大的概率值变化，从而影响对数似然的敛散性，因此对于初始值选取，有较高的要求。
+
+另一方面，GMM模型更新采用的为EM算法，E步在固定协方差和均值后，计算高维高斯概率值，并加和归一化后获得最终结果，其目标是在E步利用变分方法，获得一个对后验概率的拟合，使得似然和证据下界相同，并在M步给定了后验估计后，优化高斯分布的参数，从而优化似然值
+
+#### 初始值选取
+
+前面提到，GMM算法对初始值非常敏感，尤其是在少样本的情况下，GMM算法会在自变量小范围震荡时，引发较大的似然值改变。实际代码实现时，若采用似然改变阈值作为收敛条件，会出现无法跑出结果的情况。
+
+笔者阅读SKlearn中GMM的源码，以及查找了相关资料，总结出了两种初始值选取方法
+
+* 采用抽样法，对训练数据进行抽样，并采取平均结果作为初始值
+* 采用Kmeans预先训练，采用聚类中心作为高斯分布的均值
+
+在本次实验中，采用第二种实现方法，经过实际试验测试，较好的改进了原先震荡的情况
+
+#### 模型结构
+
+除了给定的接口外，额外添加了`point_probability`和`calculate`两个api，其中`point_probabilty`用于计算点对于所有类的联合概率，`calculate`则计算点在某一类下的概率值
+
+### ClusteringAlgorithm
+
+#### 实现原理
+
+以Kmeans聚类方法为基础，clusteringAlgorithm主要完成的任务是筛选一个合适的聚类数K，将在后文采用两种方法完成
+
+#### K值选取方法实现
+
+##### 经验法
+
+经验法为K值的经验公式，一般为$\sqrt{n/2}$，在本轮实验中，该数值作为了Elbow法的上界
+
+##### Elbow
+
+Elbow方法主要寻找整个变化曲线中的拐点，对于如下图像，拐点处被认为是最合理的点，图中该点的类别为4，正确为3，大致在正确值附近
+
+
+
+在该点处，类别切分没有太细，且误差较小，一般考虑为较好的K值。在本次实验中，为了自动化寻找拐点，我们用数组`dis`记录下每个k的阈值，并且认为前一次下降的幅度如果小于后一次下降幅度的两倍，即达到了拐点处，具体选择情况见实验部分[自动化聚类算法实验结果](###自动化聚类算法实验结果)
+
+##### Canopy
+
+###### 原理
+
+Canopy为一种粗粒度的聚类方法，其原理为给定一个两个阈值t1, t2，满足t1>t2，在初始选定点后，按照某种距离度量，与初始点距离小于t2的点有较大概率不会成为类中心，将其从数据集中删去，小于t1的类型的点则有较大概率为同类型点，本次实验中因为只需要c运行后的类别数量，对同类型的有哪些不做考虑。重复上述过程直到整个数据集为空
+
+###### 实现与阈值设定
+
+Canopy的实现相对较为简单，这里主要讨论t1和t2值的给定，出于自动化获取k值的考量，这里选取t1,t2也通过获得数据维度的方法直接给定，考虑到真实数据分布为[-10, 10]，认为$\sqrt{维度数}*2$为t2，而t1选2*t2，且考虑到在聚类中，一直聚类到所有点全部被划分完成，一定程度上会受到离群点影响，且Canopy预处理过程只需要获取最终的K值，对每个点的聚类情况并不关心。这里更改终止条件为，**当80%的点都完成聚类后，将终止Canopy算法并返回**，80%的阈值设定为考虑到二八法则，即认为80%的点能够较真实的反映数据的总体情况，最终结果参见实验部分[自动化聚类算法实验结果](###自动化聚类算法实验结果)
+
+## 实验部分
+
+### 数据生成与可视化
+
+数据生成部分重复利用了[Assignment1](https://gitee.com/fnlp/prml-21-spring/blob/master/assignment-1/submission/18307130116/source.py#L94)的API，并在此基础上添加了代码注释以及method接口，可以生成对数正态分布和高维高斯，可视化部分除了在[Assignment1](https://gitee.com/fnlp/prml-21-spring/blob/master/assignment-1/submission/18307130116/source.py#L129)基础上添加了代码注释以外，还添加了新的color参数，因为在考虑到实际聚类问题实现并不清楚各个类的分布情况，不着色绘图有时是合理的
+
+对应的API接口`data_generate_and_save`负责数据生成和保存，`data_load`负责数据载入，`visualize`负责数据可视化
+
+### 聚类效果评估方法
+
+除了直接观察法外，我们采用Silhouette Coefficient(轮廓系数，简称SC)作为聚类效果的衡量指标，SC的度量方式为计算一个点到其簇中其他点的平均距离，同该点到其他类所有点的平均距离最小值的比值，该比值介于[-1,1]，且越趋近于1，表现越优
+
+`compute_SC`接口只需给定所有的点和其标签，即可完成计算
+
+### 基础实验
+
+#### 常规情况
+
+在这部分实验中，我们将简单生成一组数据，可视化Kmeans和GMM聚类结果，并和真实值相对比
+
+参数如下表所示
+
+| 类别 | 均值 | 方差 | 点数 |
+| ---- | ------- | ---------------- | ---- |
+| 1 | (1, -7) | [[1, 0], [0, 1]] | 800 |
+| 2 | (1, -4) | [[1, 0], [0, 1]] | 800 |
+| 3 | (1, 0) | [[1, 0], [0, 1]] | 800 |
+
+无着色图例和真实图例如下
+
+
+
+Kmeans获取的结果(左) 和 GMM获取的结果(右)如下图所示
+
+
+
+从图中可见，大体上模型都将其正确聚类，在细微表现上有所差异
+
+计算其SC值，Kmeans = 0.497，GMM = 0.497，两者相差不大，Kmeans略优于GMM
+
+#### 数据中存在极端值
+
+当数据中存在极端值时，即考虑某个维度较大，进行相关实验，参数如下
+
+| 类别 | 均值 | 方差 | 点数 |
+| ---- | --------- | ------------------ | ---- |
+| 1 | (1, -700) | [[1, 0], [0, 100]] | 800 |
+| 2 | (1, -400) | [[1, 0], [0, 100]] | 800 |
+| 3 | (1, 0) | [[1, 0], [0, 1]] | 800 |
+
+该部分实际测试为测试极端值是否会影响模型计算，实际效果上，属于各类区别很大的情况
+
+无着色图例和真实图例如下
+
+
+
+Kmeans获取的结果(左) 和 GMM获取的结果(右)如下图所示
+
+
+
+计算其SC值，Kmeans = 0.973，GMM = 0.973，两者完全相同
+
+#### 高度重叠下的聚类结果
+
+以上我们已经测试了重叠不大，无重叠两组数据，进一步比较，我们将测试高度重叠下的聚类结果
+
+采用如下参数
+
+| 类别 | 均值 | 方差 | 点数 |
+| ---- | ------ | ---------------- | ---- |
+| 1 | (1, 1) | [[2, 0], [0, 2]] | 800 |
+| 2 | (1, 2) | [[2, 0], [0, 2]] | 800 |
+| 3 | (1, 0) | [[2, 0], [0, 2]] | 800 |
+
+结果如下图所示
+
+无着色图例和真实图例如下
+
+
+
+Kmeans获取的结果(左) 和 GMM获取的结果(右)如下图所示
+
+
+
+计算其SC值，Kmeans = 0.323，GMM = 0.310，Kmeans略优于GMM，我们同时计算了高度重叠下的原始数据SC，为-0.003，聚类器聚类结果相比较原始类别更优，但是在实际操作过程中，这样可能具有欺诈性，因此在使用聚类器时，应该首先对数据分布有所了解，而不应该直接使用聚类器尝试，否则可能得到“评价较优”的错误结果
+
+### GMM模型补充实验
+
+我们GMM在进行拟合时，初值使用了Kmeans获取，考虑到GMM的意义不止在于聚类，更在于能够估算叠加形式的高斯分布参数，我们进一步尝试了非高斯分布在GMM下的分类情况，仅作为补充了解，非高斯分布在GMM下会有怎样的表现
+
+
+
+
+
+Kmeans获取的结果(左) 和 GMM获取的结果(右)如下图所示
+
+
+
+计算其SC值，Kmeans = 0.677，GMM = 0.62009476，原数据的SC为-0.0348，对数正态会被GMM理解成高度重叠的高斯分布，并没有对实际效果产生过大的影响
+
+### 自动化聚类算法实验结果
+
+ClusteringAlgorithm自动化实现是基于Kmeans完成的，因此在以下实验中，我们仅关注K值选取，而不重复进行Kmeans聚类效果实验
+
+我们首先根据某一组参数，多轮测试基于ELBOW算法和Canopy算法的结果，真实值为K = 3，数据参数见表格，选取某一轮可视化如下图，加号为测试数据，有类别版本见右侧
+
+
+
+| 类别 | 均值 | 方差 | 点数 |
+| ---- | -------- | ---------------------- | ---- |
+| 1 | (1, 2) | [[73, 0], [0, 22]] | 800 |
+| 2 | (16, -5) | [[21.2, 0], [0, 32.1]] | 200 |
+| 3 | (10, 22) | [[10, 5], [5, 10]] | 1000 |
+
+| 轮数 | ELBOW自动获取的K | Canopy自动获取的K |
+| ---- | ---------------- | ----------------- |
+| 1 | 5 | 4 |
+| 2 | 4 | 4 |
+| 3 | 4 | 5 |
+| 4 | 3 | 5 |
+| 5 | 4 | 4 |
+
+可以看出，ELBOW和Canopy算法获得K都和真实值相差不大，较好的完成了自动化的工作，相比之下，ELBOW需要跑$\sqrt{n/2}$轮Kmeans，而Canopy算法只需要跑一轮即可，速度相对较快，ELBOW算法的波动情况更突出，能否取得最优结果取决于Kmeans过程中的随机初值，但总的来说完成情况较好
+
+### Canopy鲁棒性测试
+
+根据上文结果，ELBOW和Canopy算法获得的K值结果相差不大，且Canopy算法速度远快于ELBOW，但是由于依赖于阈值t1,t2，而在本次实验，阈值是对于同一维度数据完全相同的，因此增添鲁棒性测试。而ELBOW算法为暴力枚举K，以获得最好的结果，相对而言可以预想到鲁棒性较好，这里略去Kmeans的测试
+
+为保证阈值相同，笔者测试了多组二维数据，参数和结果如下表格所示，增添第三组几乎不相交组和第四组高度重叠组，测试实际效果
+
+| 轮数 | 均值 | 协方差 | 各类别点数 | 真实类别数 | 结果 |
+| ---- | ------------------------------ | ----------------------------------------------------------- | --------------- | ---------- | ---- |
+| 1 | [(1, -7), (1, -4), (1, 0)] | 单位阵 | [800, 800, 800] | 3 | 3 |
+| 2 | [(1, -7), (1, -4), (1, 0)] | [[10, 0],[0, 1]] [[1, 0], [0, 10]] [[2, 0], [6, 5]] | [800, 800, 800] | 3 | 3 |
+| 3 | [(10, 10), (-10, -10), (5, 0)] | 单位阵 | [800, 800, 800] | 3 | 3 |
+| 4 | [(1, 1), (1, 2), (1, 0)] | [[2, 0],[0, 2]] [[2, 0], [0, 2]] [[2, 0], [0, 2]] | [800, 800, 800] | 3 | 4 |
+
+高度重叠组和完全分离数据可视化如下：
+
+
+
+在高度重叠的数据下，Canopy方法仍表现出较好的性能，模型对分布鲁棒
+
+紧接着，我们测试模型对分布点数的鲁棒性，全部采取上表中的第一组参数，仅更改各类别点数
+
+| 轮数 | 点数 | 结果 |
+| ---- | ---------------- | ---- |
+| 1 | [80, 1000, 80] | 1 |
+| 2 | [200, 1000, 10] | 2 |
+| 3 | [200, 1000, 100] | 3 |
+
+可以看到点数对Canopy的影响较大，考虑到实际实现过程中，采用二八法则忽略少量离群点对于整体聚类数量的影响，在某个类别显著高于其他类时，其他类别的点将会被忽略，且距离越近忽略的可能性越大，这点也符合实际聚类时的特征
+
+进一步的，我们测试多类情况下的K值选取,[a,b]表示多组测试下，K值最终落入区间[a, b]
+
+| 轮数 | 均值 | 协方差 | 各类别点数 | 真实类别数 | 结果 |
+| ---- | ----------------------------------- | ------------------------------------------------------------ | --------------- | ---------- | ----- |
+| 1 | [(1, -7), (1, -4), (1, 0), (-2, 0)] | 单位阵 | [800, 800, 800] | 4 | [3,4] |
+| 2 | [(1, -7), (1, -4), (1, 0), (-2, 0)] | [[10, 0],[0, 1]] [[1, 0], [0, 10]] [[2, 0], [6, 5]] [[3, 0], [1, 5]] | [800, 800, 800] | 4 | [3,4] |
+
+#### 总结
+
+在这部分中，我们看到，Canopy的鲁棒性对于分布情况较好，在重叠程度较大和较小时都有较好的性能，但是对于各类点数鲁棒性较差，容易忽略一些少量点，这与实现方式相关，也符合预期
\ No newline at end of file
diff --git "a/assignment-3/submission/18307130116/img/GMM\345\237\272\347\241\200\345\256\236\351\252\214\347\273\223\346\236\234.png" "b/assignment-3/submission/18307130116/img/GMM\345\237\272\347\241\200\345\256\236\351\252\214\347\273\223\346\236\234.png"
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