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【聚类算法】K-Means算法及其Python实现

code • 2022-5-16 • python • 阅读 41

【聚类算法】K-Means算法及其Python实现,第1张

文章目录
  • 一：无监督学习
    • （1）什么是无监督学习
    • （2）聚类和分类
  • 二：K-Means算法
    • （1）算法简介
    • （2）算法流程
    • （3）Python实现
    • （4）聚类效果展示
      • A：鸢尾花数据集（效果好）
      • B：三个圆圈数据集（效果差）
  • 三：K-Means算法探讨
    • （1）距离度量
    • （2）聚类算法评价指标
      • A：inertia（簇内平方和）
      • B：轮廓系数
      • C： Calinski-Harabasz指数

一：无监督学习

K-Means算法是一种无监督的聚类算法

（1）什么是无监督学习

无监督学习：对于数据集，在训练的过程中，并没有告诉训练算法某一个数据应该属于哪一个类别；它是通过某种 *** 作，将一堆“相似”的数据聚集在一起然后当做一个类别

  • 例如图像分割，计算机面对图像数据时，并不认为这些数据有什么区别。但是通过聚类算法就可以实现简单的分类实现分割

（2）聚类和分类

分类和聚类是两类不同的机器学习算法， 简单来说

  • 分类：根据文本的特征或属性，划分到已有的类别中；因此这些类别是已知的
  • 聚类：分析时并不知道数据会分为几类，而是通过聚类算法进行归类

二：K-Means算法 （1）算法简介

专业解释：对于给定的样本集合 D = { x 1 , x 2 , . . . , x N } D=\{{x_{1},x_{2},...,x_{N}\}} D={x1​,x2​,...,xN​}，K-Means算法针对聚类所得到的簇为 C = { C 1 , C 2 , . . . , C k } C=\{{C_{1},C_{2},...,C_{k}\}} C={C1​,C2​,...,Ck​}，则划分的最小化平方误差为：

E = ∑ i = 1 k ∑ x ∈ C i ∣ ∣ x − μ i ∣ ∣ 2 2 E=\sum_{i=1}^{k}\sum_{x\in C_{i}}||x-\mu_{i}||_{2}^{2} E=i=1∑k​x∈Ci​∑​∣∣x−μi​∣∣22​

其中 μ i = 1 ∣ C i ∣ \mu_{i}=\frac{1}{|C_{i}|} μi​=∣Ci​∣1​ ∑ x ∈ C i x \underset{x\in C_{i}}{\sum}x x∈Ci​∑​x,如果 E E E越小则表示数据集样本相似度越高

通俗解释：K-Means算法将一组 N N N个样本划分为 K K K个无交集的 簇，簇中所有数据的均值称为这个簇的 质心

  • 簇：是聚类的结果表现，一个簇中的数据就认为是同一类
  • 质心：簇中所有数据的均值

在该算法中，簇个数 K K K是一个超参数，在算法开始前需要人为确定

  • 以图像分割为例，意思就是你认为这个图像大概可以分为多少块， K K K就设置为多少

因此，K-Means 的核心任务就是根据我们设定好的 K，找出 K 个最优的质心，并将离这些质心最近的数据分别分配到这些质心代表的簇中去

聚类算法认为，被分在同一个簇中的数据是有相似性的，而不同簇中的数据是不同的，因此K-Means以及其他的聚类算法所追求的目标就是 簇内差异小、簇外差异大，而这个差异由 样本点到其所在簇的质心的距离来衡量，对一个簇来说，所有样本点到质心的距离之和越小，就认为这个簇中的样本越相似，差异性也越小

K-Means算法在衡量距离时，大多会采用欧氏距离

d ( x , μ ) = ∑ i = 1 n ( x i − μ i ) 2 d(x,\mu)=\sqrt{\sum_{i=1}^{n}(x_{i}-\mu_{i})^{2}} d(x,μ)=i=1∑n​(xi​−μi​)2 ​

其中

  • x x x表示簇中的每个样本点
  • μ \mu μ表示该簇的质心
  • n n n表示每个样本点中的特征数目
  • i i i表示组成点 x x x的每个特征编号
（2）算法流程

K-Means算法流程：

  1. 选择初始化的 k k k个样本作为初始聚类中心，即 μ = μ 1 , μ 2 , . . . , μ k \mu=\mu_{1},\mu_{2},...,\mu_{k} μ=μ1​,μ2​,...,μk​
  2. 针对数据集中每个样本 x i x_{i} xi​，计算它到 k k k个聚类中心的距离并将其分配到距离最小的聚类中心所对应的类中
  3. 针对每个类别 μ j \mu_{j} μj​，重新计算属于该类的所有样本的质心： μ j = 1 ∣ C i ∣ \mu_{j}=\frac{1}{|C_{i}|} μj​=∣Ci​∣1​ ∑ x ∈ C i x \underset{x\in C_{i}}{\sum}x x∈Ci​∑​x
  4. 重复步骤2、3，知道到达某个中止条件（比如达到某个设定的迭代次数或者小于某个误差值等等）

K-Means算法流程演示：

1：假设有如下数据集，随机选取 k k k个点（ k k k=3，红绿蓝）

2：接着计算数据集样本中其它的点到质心的距离，然后选取最近质心的类别作为自己的类别

3：经过上面的步骤，分为了三个簇，然后在每个簇中重新选取质心

4：然后再进行划分


5：重复

（3）Python实现

KMeans.py：K-Means的代码实现

import numpy as np

class KMeans:
    def __init__(self, data_set, k):  # data_set：数据集  k：簇个数
        self.data_set = data_set
        self.k = k

    def kmeans(self, max_iterations):  # max_iterations：最大迭代次数
        examples_num = self.data_set.shape[0]  # 样本数据个数
        centroids = KMeans.centroids_init(self.data_set, self.k)  # 随机初始化k个质心
        closest_centroids_ids = np.empty((examples_num, 1))  # 用于保存各样本点距离哪个质心最近，例如closest_centroids_ids[0]=2表示0这个样本点和2这个执行离得最近

        #  不断迭代
        for _ in range(max_iterations):

            # 先计算各样本点到质心最短距离
            closest_centroids_ids = KMeans.centroids_find_closet(self.data_set, centroids)
            # 再更新质心
            centroids = KMeans.centroids_compute(self.data_set, closest_centroids_ids, self.k)
        return centroids, closest_centroids_ids

    @staticmethod
    def centroids_init(data_set, k):  # 初始化随机质心
        examples_num = data_set.shape[0]  # 样本数据个数
        random_ids = np.random.permutation(examples_num)  # 随机产生examples_num个数据
        centroids = data_set[random_ids[:k], :]  # 取前k个
        return centroids

    @staticmethod
    def centroids_find_closet(data_set, centroids):  # 每个样本点到质心距离
        examples_num = data_set.shape[0]  # 样本数据个数
        centroids_num = centroids.shape[0]  # 质心个数
        closet_centroids_ids = np.zeros((examples_num, 1))
        for examples_index in range(examples_num):
            distance = np.zeros((centroids_num, 1))  # 保存examples_index这个样本点到各质心的距离
            for centrodis_index in range(centroids_num):
                #  欧氏距离
                distance_diff = data_set[examples_index, :] - centroids[centrodis_index, :]  # 某点到质心差异值
                distance[centrodis_index] = np.sqrt(np.sum(np.power(distance_diff, 2)))
            closet_centroids_ids[examples_index] = np.argmin(distance)  # examples_index距离最近的那个质心（下标）
        return closet_centroids_ids

    @staticmethod
    def centroids_compute(data_set, closest_centroids_ids, k):
        features_num = data_set.shape[1]  # 特征格式，即属性
        centroids = np.zeros((k, features_num))  # k个簇每个都要计算，且每个簇的点都有features_num个属性，所以要对应计算
        for centroid_id in range(k):
            closest_ids = closest_centroids_ids == centroid_id  # 不懂的话可以查阅“Numpy布尔数组相关内容”
            centroids[centroid_id] = np.mean(data_set[closest_ids.flatten(), :], axis=0)
        return centroids
（4）聚类效果展示 A：鸢尾花数据集（效果好）

Iris.csv 文件：鸢尾花数据集

Id,SepalLengthCm,SepalWidthCm,PetalLengthCm,PetalWidthCm,Species
1,5.1,3.5,1.4,0.2,Iris-setosa
2,4.9,3.0,1.4,0.2,Iris-setosa
3,4.7,3.2,1.3,0.2,Iris-setosa
4,4.6,3.1,1.5,0.2,Iris-setosa
5,5.0,3.6,1.4,0.2,Iris-setosa
6,5.4,3.9,1.7,0.4,Iris-setosa
7,4.6,3.4,1.4,0.3,Iris-setosa
8,5.0,3.4,1.5,0.2,Iris-setosa
9,4.4,2.9,1.4,0.2,Iris-setosa
10,4.9,3.1,1.5,0.1,Iris-setosa
11,5.4,3.7,1.5,0.2,Iris-setosa
12,4.8,3.4,1.6,0.2,Iris-setosa
13,4.8,3.0,1.4,0.1,Iris-setosa
14,4.3,3.0,1.1,0.1,Iris-setosa
15,5.8,4.0,1.2,0.2,Iris-setosa
16,5.7,4.4,1.5,0.4,Iris-setosa
17,5.4,3.9,1.3,0.4,Iris-setosa
18,5.1,3.5,1.4,0.3,Iris-setosa
19,5.7,3.8,1.7,0.3,Iris-setosa
20,5.1,3.8,1.5,0.3,Iris-setosa
21,5.4,3.4,1.7,0.2,Iris-setosa
22,5.1,3.7,1.5,0.4,Iris-setosa
23,4.6,3.6,1.0,0.2,Iris-setosa
24,5.1,3.3,1.7,0.5,Iris-setosa
25,4.8,3.4,1.9,0.2,Iris-setosa
26,5.0,3.0,1.6,0.2,Iris-setosa
27,5.0,3.4,1.6,0.4,Iris-setosa
28,5.2,3.5,1.5,0.2,Iris-setosa
29,5.2,3.4,1.4,0.2,Iris-setosa
30,4.7,3.2,1.6,0.2,Iris-setosa
31,4.8,3.1,1.6,0.2,Iris-setosa
32,5.4,3.4,1.5,0.4,Iris-setosa
33,5.2,4.1,1.5,0.1,Iris-setosa
34,5.5,4.2,1.4,0.2,Iris-setosa
35,4.9,3.1,1.5,0.1,Iris-setosa
36,5.0,3.2,1.2,0.2,Iris-setosa
37,5.5,3.5,1.3,0.2,Iris-setosa
38,4.9,3.1,1.5,0.1,Iris-setosa
39,4.4,3.0,1.3,0.2,Iris-setosa
40,5.1,3.4,1.5,0.2,Iris-setosa
41,5.0,3.5,1.3,0.3,Iris-setosa
42,4.5,2.3,1.3,0.3,Iris-setosa
43,4.4,3.2,1.3,0.2,Iris-setosa
44,5.0,3.5,1.6,0.6,Iris-setosa
45,5.1,3.8,1.9,0.4,Iris-setosa
46,4.8,3.0,1.4,0.3,Iris-setosa
47,5.1,3.8,1.6,0.2,Iris-setosa
48,4.6,3.2,1.4,0.2,Iris-setosa
49,5.3,3.7,1.5,0.2,Iris-setosa
50,5.0,3.3,1.4,0.2,Iris-setosa
51,7.0,3.2,4.7,1.4,Iris-versicolor
52,6.4,3.2,4.5,1.5,Iris-versicolor
53,6.9,3.1,4.9,1.5,Iris-versicolor
54,5.5,2.3,4.0,1.3,Iris-versicolor
55,6.5,2.8,4.6,1.5,Iris-versicolor
56,5.7,2.8,4.5,1.3,Iris-versicolor
57,6.3,3.3,4.7,1.6,Iris-versicolor
58,4.9,2.4,3.3,1.0,Iris-versicolor
59,6.6,2.9,4.6,1.3,Iris-versicolor
60,5.2,2.7,3.9,1.4,Iris-versicolor
61,5.0,2.0,3.5,1.0,Iris-versicolor
62,5.9,3.0,4.2,1.5,Iris-versicolor
63,6.0,2.2,4.0,1.0,Iris-versicolor
64,6.1,2.9,4.7,1.4,Iris-versicolor
65,5.6,2.9,3.6,1.3,Iris-versicolor
66,6.7,3.1,4.4,1.4,Iris-versicolor
67,5.6,3.0,4.5,1.5,Iris-versicolor
68,5.8,2.7,4.1,1.0,Iris-versicolor
69,6.2,2.2,4.5,1.5,Iris-versicolor
70,5.6,2.5,3.9,1.1,Iris-versicolor
71,5.9,3.2,4.8,1.8,Iris-versicolor
72,6.1,2.8,4.0,1.3,Iris-versicolor
73,6.3,2.5,4.9,1.5,Iris-versicolor
74,6.1,2.8,4.7,1.2,Iris-versicolor
75,6.4,2.9,4.3,1.3,Iris-versicolor
76,6.6,3.0,4.4,1.4,Iris-versicolor
77,6.8,2.8,4.8,1.4,Iris-versicolor
78,6.7,3.0,5.0,1.7,Iris-versicolor
79,6.0,2.9,4.5,1.5,Iris-versicolor
80,5.7,2.6,3.5,1.0,Iris-versicolor
81,5.5,2.4,3.8,1.1,Iris-versicolor
82,5.5,2.4,3.7,1.0,Iris-versicolor
83,5.8,2.7,3.9,1.2,Iris-versicolor
84,6.0,2.7,5.1,1.6,Iris-versicolor
85,5.4,3.0,4.5,1.5,Iris-versicolor
86,6.0,3.4,4.5,1.6,Iris-versicolor
87,6.7,3.1,4.7,1.5,Iris-versicolor
88,6.3,2.3,4.4,1.3,Iris-versicolor
89,5.6,3.0,4.1,1.3,Iris-versicolor
90,5.5,2.5,4.0,1.3,Iris-versicolor
91,5.5,2.6,4.4,1.2,Iris-versicolor
92,6.1,3.0,4.6,1.4,Iris-versicolor
93,5.8,2.6,4.0,1.2,Iris-versicolor
94,5.0,2.3,3.3,1.0,Iris-versicolor
95,5.6,2.7,4.2,1.3,Iris-versicolor
96,5.7,3.0,4.2,1.2,Iris-versicolor
97,5.7,2.9,4.2,1.3,Iris-versicolor
98,6.2,2.9,4.3,1.3,Iris-versicolor
99,5.1,2.5,3.0,1.1,Iris-versicolor
100,5.7,2.8,4.1,1.3,Iris-versicolor
101,6.3,3.3,6.0,2.5,Iris-virginica
102,5.8,2.7,5.1,1.9,Iris-virginica
103,7.1,3.0,5.9,2.1,Iris-virginica
104,6.3,2.9,5.6,1.8,Iris-virginica
105,6.5,3.0,5.8,2.2,Iris-virginica
106,7.6,3.0,6.6,2.1,Iris-virginica
107,4.9,2.5,4.5,1.7,Iris-virginica
108,7.3,2.9,6.3,1.8,Iris-virginica
109,6.7,2.5,5.8,1.8,Iris-virginica
110,7.2,3.6,6.1,2.5,Iris-virginica
111,6.5,3.2,5.1,2.0,Iris-virginica
112,6.4,2.7,5.3,1.9,Iris-virginica
113,6.8,3.0,5.5,2.1,Iris-virginica
114,5.7,2.5,5.0,2.0,Iris-virginica
115,5.8,2.8,5.1,2.4,Iris-virginica
116,6.4,3.2,5.3,2.3,Iris-virginica
117,6.5,3.0,5.5,1.8,Iris-virginica
118,7.7,3.8,6.7,2.2,Iris-virginica
119,7.7,2.6,6.9,2.3,Iris-virginica
120,6.0,2.2,5.0,1.5,Iris-virginica
121,6.9,3.2,5.7,2.3,Iris-virginica
122,5.6,2.8,4.9,2.0,Iris-virginica
123,7.7,2.8,6.7,2.0,Iris-virginica
124,6.3,2.7,4.9,1.8,Iris-virginica
125,6.7,3.3,5.7,2.1,Iris-virginica
126,7.2,3.2,6.0,1.8,Iris-virginica
127,6.2,2.8,4.8,1.8,Iris-virginica
128,6.1,3.0,4.9,1.8,Iris-virginica
129,6.4,2.8,5.6,2.1,Iris-virginica
130,7.2,3.0,5.8,1.6,Iris-virginica
131,7.4,2.8,6.1,1.9,Iris-virginica
132,7.9,3.8,6.4,2.0,Iris-virginica
133,6.4,2.8,5.6,2.2,Iris-virginica
134,6.3,2.8,5.1,1.5,Iris-virginica
135,6.1,2.6,5.6,1.4,Iris-virginica
136,7.7,3.0,6.1,2.3,Iris-virginica
137,6.3,3.4,5.6,2.4,Iris-virginica
138,6.4,3.1,5.5,1.8,Iris-virginica
139,6.0,3.0,4.8,1.8,Iris-virginica
140,6.9,3.1,5.4,2.1,Iris-virginica
141,6.7,3.1,5.6,2.4,Iris-virginica
142,6.9,3.1,5.1,2.3,Iris-virginica
143,5.8,2.7,5.1,1.9,Iris-virginica
144,6.8,3.2,5.9,2.3,Iris-virginica
145,6.7,3.3,5.7,2.5,Iris-virginica
146,6.7,3.0,5.2,2.3,Iris-virginica
147,6.3,2.5,5.0,1.9,Iris-virginica
148,6.5,3.0,5.2,2.0,Iris-virginica
149,6.2,3.4,5.4,2.3,Iris-virginica
150,5.9,3.0,5.1,1.8,Iris-virginica

test_Iris.py：测试数据集

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from KMeans import KMeans

Iris_types = ['Iris-setosa', 'Iris-versicolor', 'Iris-virginica']  # 花类型
Iris_data = pd.read_csv('./Iris.csv')
x_axis = 'PetalLengthCm'  # 花瓣长度
y_axis = 'PetalWidthCm'   # 花瓣宽度

# x_axis = 'SepalLengthCm'  # 花萼长度
# y_axis = 'SepalWidthCm'  # 花萼宽度

examples_num = Iris_data.shape[0]  # 样本数量
train_data = Iris_data[[x_axis, y_axis]].values.reshape(examples_num, 2)  # 整理数据

#  训练参数
K = 3  # 簇数
max_iterations = 50  # 最大迭代次数
test = KMeans(train_data, K)
centroids, closest_centroids_ids = test.kmeans(max_iterations)

plt.figure(figsize=(12, 5), dpi=80)

#  第一幅图是已知标签
plt.subplot(1, 2, 1)

for Iris_type in Iris_types:
        plt.scatter(Iris_data[x_axis][Iris_data['Species'] == Iris_type], Iris_data[y_axis][Iris_data['Species'] == Iris_type], label=Iris_type)
plt.title('label know')
plt.legend()

# 第二幅图是聚类结果
plt.subplot(1, 2, 2)
for centroid_id, centroid in enumerate(centroids):
    current_examples_index = (closest_centroids_ids == centroid_id).flatten()
    plt.scatter(Iris_data[x_axis][current_examples_index],
                Iris_data[y_axis][current_examples_index], label=Iris_data['Species'][current_examples_index].values[0])

for centroid_id, centroid in enumerate(centroids):
    plt.scatter(centroid[0], centroid[1], c='red', marker='x')
plt.title('label kemans')
plt.legend()
plt.show()

效果

花瓣长度和花瓣宽度

花萼长度花萼宽度

B：三个圆圈数据集（效果差）

threecircles.csv：三个圆圈数据集

4.54E-01,5.15E-01,0.00E+00
4.55E-01,5.14E-01,0.00E+00
4.48E-01,5.18E-01,0.00E+00
4.36E-01,5.21E-01,0.00E+00
4.64E-01,5.26E-01,0.00E+00
4.46E-01,5.24E-01,0.00E+00
4.53E-01,5.30E-01,0.00E+00
4.54E-01,5.31E-01,0.00E+00
4.69E-01,5.27E-01,0.00E+00
4.40E-01,5.46E-01,0.00E+00
4.61E-01,5.36E-01,0.00E+00
4.29E-01,5.47E-01,0.00E+00
4.54E-01,5.45E-01,0.00E+00
4.37E-01,5.28E-01,0.00E+00
4.42E-01,5.46E-01,0.00E+00
4.30E-01,5.14E-01,0.00E+00
4.49E-01,5.19E-01,0.00E+00
4.36E-01,5.41E-01,0.00E+00
4.36E-01,5.28E-01,0.00E+00
4.35E-01,5.34E-01,0.00E+00
4.43E-01,5.31E-01,0.00E+00
4.42E-01,5.18E-01,0.00E+00
4.37E-01,5.34E-01,0.00E+00
4.32E-01,5.41E-01,0.00E+00
4.22E-01,5.27E-01,0.00E+00
4.44E-01,5.22E-01,0.00E+00
4.37E-01,5.12E-01,0.00E+00
4.33E-01,5.15E-01,0.00E+00
4.43E-01,5.28E-01,0.00E+00
4.38E-01,5.28E-01,0.00E+00
4.30E-01,5.16E-01,0.00E+00
4.05E-01,5.19E-01,0.00E+00
4.20E-01,5.18E-01,0.00E+00
4.25E-01,5.13E-01,0.00E+00
4.17E-01,5.06E-01,0.00E+00
4.28E-01,4.96E-01,0.00E+00
4.51E-01,5.11E-01,0.00E+00
4.15E-01,4.95E-01,0.00E+00
4.49E-01,5.20E-01,0.00E+00
4.47E-01,5.04E-01,0.00E+00
4.29E-01,4.93E-01,0.00E+00
4.31E-01,5.05E-01,0.00E+00
4.40E-01,5.02E-01,0.00E+00
4.30E-01,5.20E-01,0.00E+00
4.40E-01,4.99E-01,0.00E+00
4.48E-01,4.90E-01,0.00E+00
4.39E-01,4.96E-01,0.00E+00
4.54E-01,4.85E-01,0.00E+00
4.56E-01,4.98E-01,0.00E+00
4.50E-01,5.09E-01,0.00E+00
4.48E-01,4.91E-01,0.00E+00
4.52E-01,5.20E-01,0.00E+00
4.55E-01,5.17E-01,0.00E+00
4.63E-01,5.08E-01,0.00E+00
4.52E-01,5.17E-01,0.00E+00
4.47E-01,5.21E-01,0.00E+00
4.66E-01,5.06E-01,0.00E+00
4.65E-01,5.13E-01,0.00E+00
4.53E-01,5.23E-01,0.00E+00
4.54E-01,5.18E-01,0.00E+00
4.52E-01,5.14E-01,0.00E+00
5.76E-01,5.15E-01,1.00E+00
5.74E-01,5.13E-01,1.00E+00
5.74E-01,5.31E-01,1.00E+00
5.85E-01,5.08E-01,1.00E+00
5.79E-01,5.51E-01,1.00E+00
5.82E-01,5.49E-01,1.00E+00
5.88E-01,5.52E-01,1.00E+00
5.69E-01,5.72E-01,1.00E+00
5.71E-01,5.61E-01,1.00E+00
5.69E-01,5.81E-01,1.00E+00
5.54E-01,5.64E-01,1.00E+00
5.44E-01,5.68E-01,1.00E+00
5.47E-01,5.63E-01,1.00E+00
5.56E-01,5.94E-01,1.00E+00
5.54E-01,6.07E-01,1.00E+00
5.51E-01,6.02E-01,1.00E+00
5.42E-01,6.06E-01,1.00E+00
5.51E-01,6.04E-01,1.00E+00
5.48E-01,6.09E-01,1.00E+00
5.17E-01,6.48E-01,1.00E+00
5.44E-01,6.15E-01,1.00E+00
5.16E-01,6.36E-01,1.00E+00
5.07E-01,6.44E-01,1.00E+00
5.02E-01,6.36E-01,1.00E+00
5.11E-01,6.45E-01,1.00E+00
5.11E-01,6.31E-01,1.00E+00
4.95E-01,6.59E-01,1.00E+00
4.75E-01,6.45E-01,1.00E+00
4.82E-01,6.50E-01,1.00E+00
4.78E-01,6.50E-01,1.00E+00
4.72E-01,6.47E-01,1.00E+00
4.61E-01,6.21E-01,1.00E+00
4.62E-01,6.42E-01,1.00E+00
4.46E-01,6.45E-01,1.00E+00
4.43E-01,6.67E-01,1.00E+00
4.33E-01,6.54E-01,1.00E+00
4.47E-01,6.66E-01,1.00E+00
4.30E-01,6.55E-01,1.00E+00
4.20E-01,6.51E-01,1.00E+00
4.24E-01,6.51E-01,1.00E+00
4.24E-01,6.44E-01,1.00E+00
4.05E-01,6.42E-01,1.00E+00
4.01E-01,6.38E-01,1.00E+00
3.93E-01,6.45E-01,1.00E+00
4.03E-01,6.41E-01,1.00E+00
3.91E-01,6.26E-01,1.00E+00
3.60E-01,6.24E-01,1.00E+00
3.74E-01,6.14E-01,1.00E+00
3.79E-01,6.16E-01,1.00E+00
3.61E-01,6.19E-01,1.00E+00
3.68E-01,6.27E-01,1.00E+00
3.46E-01,6.21E-01,1.00E+00
3.38E-01,6.15E-01,1.00E+00
3.56E-01,6.09E-01,1.00E+00
3.52E-01,5.92E-01,1.00E+00
3.38E-01,6.05E-01,1.00E+00
3.36E-01,5.84E-01,1.00E+00
3.39E-01,5.88E-01,1.00E+00
3.15E-01,5.94E-01,1.00E+00
3.19E-01,5.84E-01,1.00E+00
3.23E-01,5.82E-01,1.00E+00
3.01E-01,5.66E-01,1.00E+00
3.26E-01,5.55E-01,1.00E+00
3.26E-01,5.54E-01,1.00E+00
3.07E-01,5.47E-01,1.00E+00
3.24E-01,5.32E-01,1.00E+00
3.04E-01,5.52E-01,1.00E+00
3.32E-01,5.25E-01,1.00E+00
3.07E-01,5.20E-01,1.00E+00
3.23E-01,5.40E-01,1.00E+00
3.17E-01,5.30E-01,1.00E+00
3.09E-01,5.02E-01,1.00E+00
3.08E-01,5.00E-01,1.00E+00
3.09E-01,4.97E-01,1.00E+00
3.10E-01,4.95E-01,1.00E+00
2.97E-01,4.63E-01,1.00E+00
3.23E-01,4.50E-01,1.00E+00
3.31E-01,4.64E-01,1.00E+00
3.19E-01,4.76E-01,1.00E+00
3.35E-01,4.59E-01,1.00E+00
3.37E-01,4.63E-01,1.00E+00
3.30E-01,4.63E-01,1.00E+00
3.35E-01,4.49E-01,1.00E+00
3.44E-01,4.34E-01,1.00E+00
3.43E-01,4.30E-01,1.00E+00
3.43E-01,4.42E-01,1.00E+00
3.51E-01,4.13E-01,1.00E+00
3.53E-01,4.20E-01,1.00E+00
3.49E-01,4.27E-01,1.00E+00
3.63E-01,4.15E-01,1.00E+00
3.61E-01,4.01E-01,1.00E+00
3.73E-01,4.05E-01,1.00E+00
3.86E-01,3.95E-01,1.00E+00
3.79E-01,3.99E-01,1.00E+00
4.09E-01,4.10E-01,1.00E+00
3.87E-01,4.20E-01,1.00E+00
3.93E-01,3.98E-01,1.00E+00
4.08E-01,4.25E-01,1.00E+00
3.94E-01,4.15E-01,1.00E+00
4.28E-01,4.04E-01,1.00E+00
4.42E-01,3.85E-01,1.00E+00
4.21E-01,3.97E-01,1.00E+00
4.23E-01,4.02E-01,1.00E+00
4.57E-01,3.97E-01,1.00E+00
4.47E-01,3.91E-01,1.00E+00
4.52E-01,3.97E-01,1.00E+00
4.57E-01,3.77E-01,1.00E+00
4.69E-01,3.94E-01,1.00E+00
4.82E-01,3.85E-01,1.00E+00
4.88E-01,3.98E-01,1.00E+00
4.75E-01,4.07E-01,1.00E+00
4.72E-01,3.90E-01,1.00E+00
4.88E-01,3.89E-01,1.00E+00
4.94E-01,3.97E-01,1.00E+00
4.93E-01,4.11E-01,1.00E+00
5.05E-01,4.08E-01,1.00E+00
5.21E-01,4.20E-01,1.00E+00
5.13E-01,4.06E-01,1.00E+00
5.22E-01,4.14E-01,1.00E+00
5.37E-01,4.09E-01,1.00E+00
5.45E-01,4.17E-01,1.00E+00
5.32E-01,4.22E-01,1.00E+00
5.61E-01,4.43E-01,1.00E+00
5.38E-01,4.45E-01,1.00E+00
5.53E-01,4.40E-01,1.00E+00
5.61E-01,4.48E-01,1.00E+00
5.49E-01,4.56E-01,1.00E+00
5.64E-01,4.48E-01,1.00E+00
5.60E-01,4.56E-01,1.00E+00
5.55E-01,4.77E-01,1.00E+00
5.53E-01,4.66E-01,1.00E+00
5.73E-01,4.80E-01,1.00E+00
5.92E-01,4.81E-01,1.00E+00
5.59E-01,4.92E-01,1.00E+00
5.79E-01,4.87E-01,1.00E+00
5.67E-01,5.07E-01,1.00E+00
5.77E-01,5.17E-01,1.00E+00
5.98E-01,5.07E-01,1.00E+00
5.95E-01,5.12E-01,1.00E+00
7.82E-01,5.14E-01,2.00E+00
7.79E-01,5.20E-01,2.00E+00
7.87E-01,5.79E-01,2.00E+00
7.88E-01,5.78E-01,2.00E+00
7.70E-01,5.90E-01,2.00E+00
7.45E-01,6.13E-01,2.00E+00
7.62E-01,6.43E-01,2.00E+00
7.51E-01,6.70E-01,2.00E+00
7.30E-01,6.77E-01,2.00E+00
7.06E-01,7.12E-01,2.00E+00
7.11E-01,7.20E-01,2.00E+00
6.90E-01,7.46E-01,2.00E+00
6.91E-01,7.46E-01,2.00E+00
6.59E-01,7.78E-01,2.00E+00
6.44E-01,7.61E-01,2.00E+00
6.21E-01,7.89E-01,2.00E+00
6.05E-01,8.07E-01,2.00E+00
6.08E-01,8.08E-01,2.00E+00
5.73E-01,8.07E-01,2.00E+00
5.60E-01,8.25E-01,2.00E+00
5.54E-01,8.28E-01,2.00E+00
5.27E-01,8.43E-01,2.00E+00
5.05E-01,8.63E-01,2.00E+00
4.57E-01,8.49E-01,2.00E+00
4.69E-01,8.60E-01,2.00E+00
4.24E-01,8.60E-01,2.00E+00
4.15E-01,8.43E-01,2.00E+00
4.00E-01,8.44E-01,2.00E+00
3.57E-01,8.46E-01,2.00E+00
3.49E-01,8.27E-01,2.00E+00
3.07E-01,8.33E-01,2.00E+00
3.02E-01,8.22E-01,2.00E+00
2.83E-01,8.14E-01,2.00E+00
2.82E-01,7.98E-01,2.00E+00
2.48E-01,7.83E-01,2.00E+00
2.26E-01,7.72E-01,2.00E+00
2.14E-01,7.53E-01,2.00E+00
1.87E-01,7.55E-01,2.00E+00
1.80E-01,7.52E-01,2.00E+00
1.81E-01,7.24E-01,2.00E+00
1.49E-01,7.03E-01,2.00E+00
1.50E-01,6.75E-01,2.00E+00
1.39E-01,6.46E-01,2.00E+00
1.41E-01,6.26E-01,2.00E+00
1.19E-01,6.17E-01,2.00E+00
1.35E-01,5.96E-01,2.00E+00
1.22E-01,5.70E-01,2.00E+00
1.04E-01,5.51E-01,2.00E+00
1.25E-01,5.34E-01,2.00E+00
9.68E-02,5.18E-01,2.00E+00
1.20E-01,4.85E-01,2.00E+00
1.11E-01,4.70E-01,2.00E+00
1.23E-01,4.78E-01,2.00E+00
1.19E-01,4.48E-01,2.00E+00
1.29E-01,4.14E-01,2.00E+00
1.24E-01,3.84E-01,2.00E+00
1.48E-01,3.59E-01,2.00E+00
1.69E-01,3.69E-01,2.00E+00
1.51E-01,3.25E-01,2.00E+00
1.83E-01,3.05E-01,2.00E+00
1.96E-01,3.01E-01,2.00E+00
1.95E-01,2.78E-01,2.00E+00
2.12E-01,2.65E-01,2.00E+00
2.35E-01,2.64E-01,2.00E+00
2.61E-01,2.40E-01,2.00E+00
2.63E-01,2.30E-01,2.00E+00
2.98E-01,2.23E-01,2.00E+00
2.97E-01,2.09E-01,2.00E+00
3.36E-01,2.06E-01,2.00E+00
3.56E-01,1.94E-01,2.00E+00
3.55E-01,1.70E-01,2.00E+00
3.96E-01,1.91E-01,2.00E+00
3.91E-01,1.79E-01,2.00E+00
4.21E-01,1.71E-01,2.00E+00
4.51E-01,1.90E-01,2.00E+00
4.72E-01,2.02E-01,2.00E+00
4.93E-01,1.82E-01,2.00E+00
5.03E-01,1.63E-01,2.00E+00
5.31E-01,1.99E-01,2.00E+00
5.55E-01,1.93E-01,2.00E+00
6.02E-01,2.06E-01,2.00E+00
5.83E-01,2.34E-01,2.00E+00
5.99E-01,2.23E-01,2.00E+00
6.25E-01,2.47E-01,2.00E+00
6.34E-01,2.67E-01,2.00E+00
6.63E-01,2.68E-01,2.00E+00
6.75E-01,2.76E-01,2.00E+00
6.97E-01,3.06E-01,2.00E+00
7.00E-01,3.00E-01,2.00E+00
7.36E-01,3.57E-01,2.00E+00
7.20E-01,3.65E-01,2.00E+00
7.49E-01,3.46E-01,2.00E+00
7.44E-01,3.93E-01,2.00E+00
7.44E-01,4.09E-01,2.00E+00
7.54E-01,4.18E-01,2.00E+00
7.76E-01,4.59E-01,2.00E+00
7.66E-01,4.88E-01,2.00E+00
7.87E-01,5.01E-01,2.00E+00
7.76E-01,5.25E-01,2.00E+00

test_threecircles.py：测试数据集

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from KMeans import KMeans

data_types = [0.0000000e+00, 1.0000000e+00, 2.0000000e+00]
raw_data = pd.read_csv('./threecircles.csv', header=None)
raw_data.columns = ['X', 'Y', 'Type']
x_axis = 'X'
y_axis = 'Y'

examples_num = raw_data.shape[0]
train_data = raw_data[[x_axis, y_axis]].values.reshape(examples_num, 2)

#  训练参数
K = 3
max_iterations = 50
test = KMeans(train_data, K)
centroids, closest_centroids_ids = test.kmeans(max_iterations)

plt.figure(figsize=(12, 5), dpi=80)

# 第一幅图已知标签
plt.subplot(1, 2, 1)

for data_types_index in data_types:
    plt.scatter(raw_data[x_axis][raw_data['Type'] == data_types_index], raw_data[y_axis][raw_data['Type'] == data_types_index], label=str(data_types_index))
plt.title('label know')
plt.legend()


# 第二幅图聚类结果
plt.subplot(1, 2, 2)
for centroid_id, centroid in enumerate(centroids):
    current_examples_index = (closest_centroids_ids == centroid_id).flatten()
    plt.scatter(raw_data[x_axis][current_examples_index],
                raw_data[y_axis][current_examples_index],
                label=raw_data['Type'][current_examples_index].values[0])

for centroid_id, centroid in enumerate(centroids):
    plt.scatter(centroid[0], centroid[1], c='red', marker='x')
plt.title('label kmeans')
plt.legend()
plt.show()

效果

三：K-Means算法探讨 （1）距离度量

K-Means算法中样本点到质心的距离度量公式有很多种，主要是以下三种

  • 欧几里得距离（本文采用）： d ( x , u ) = ∑ i = 1 n ( x i − u i ) 2 d(x,u)=\sum_{i=1}^{n}(x_{i}-u_{i})^{2} d(x,u)=∑i=1n​(xi​−ui​)2
  • 曼哈顿距离（计算速度快）： d ( x , u ) = ∑ i = 1 n ( ∣ x i − u i ∣ ) d(x,u)=\sum_{i=1}^{n}(|x_{i}-u_{i}|) d(x,u)=∑i=1n​(∣xi​−ui​∣)
  • 余弦距离（高纬更稳定）： c o s ( θ ) = A ▪ B ∣ ∣ A ∣ ∣ ∣ ∣ B ∣ ∣ = ∑ i = 1 n A i B i ∑ i = 1 n A i 2 ∑ i = 1 n B i 2 cos(\theta)=\frac{A▪B}{||A||||B||}=\frac{\sum_{i=1}^{n}A_{i}B_{i}}{\sqrt{\sum_{i=1}^{n}A_{i}^{2}}\sqrt{\sum_{i=1}^{n}B_{i}^{2}}} cos(θ)=∣∣A∣∣∣∣B∣∣A▪B​=∑i=1n​Ai2​ ​∑i=1n​Bi2​ ​∑i=1n​Ai​Bi​​

需要注意以下几点

①：关于余弦距离：

  • 余弦距离就是用1减去余弦相似度，所谓余弦相似度是指两个向量间的夹角的余弦值
  • 特别注意余弦距离取值范围为[0, 2]，这样才能满足非负的性质

②：余弦距离与欧氏距离的关系：

  • 当向量模长是经过归一化的，此时两种距离存在单调关系，即 ∣ ∣ A − B ∣ ∣ 2 = 2 ( 1 − c o s ( A , B ) ) ||A-B||_{2}=\sqrt{2(1-cos(A,B))} ∣∣A−B∣∣2​=2(1−cos(A,B)) ​；在此场景下，如果选择距离最小（相似度最大）的近邻，那么使用余弦相似度和欧氏距离的结果是相同的
  • 欧氏距离体现数值上的绝对差异，而余弦距离体现方向上的相对差异
  • 欧氏距离是计算空间中两个点的直线距离，所以它受不同维度上数值尺度的影响比较大，而余弦距离则是计算两者在空间中方向上的差异，它包含的信息更多，而且更稳定，用哪一个看具体场景

③：曼哈顿距离与欧氏距离的关系：

  • 红线：曼哈顿距离
  • 绿线：欧式距离
  • 蓝线和 黄线：等价于曼哈顿距离

④：利用Python实现这些距离：

# 欧氏距离
def distEclud(vecA, vecB):
	return np.sqrt(np.sum(np.power((vecA-vecB), 2)))

# 曼哈顿距离
def distManhattan(vecA, vecB):
	return sum(abs(vecA-vecB))


# 余弦距离（通过scipy计算）
a = [1, 2, 3, 4]
b = [5, 6, 7, 8]
def = 1-seatial.distance.cosine(a, b)
（2）聚类算法评价指标

聚类算法评价指标分为外部评估和内部评估，如下

  • 外部评估： 将结果与某个“参考模型”（reference model）进行比较
  • 内部评估： 直接考虑聚类结果而不利用任何参考模型

下面例子中均使用sklearn中的kmeans

A：inertia（簇内平方和）

当采用欧氏距离时，就可以得到一个簇中所有样本点到质心的距离的平方和(Cluster Sum Of Square(CSS))，也即簇内平方和(inertia)

C S S = ∑ j = 0 m ∑ i = 1 n ( x i − μ i ) 2 CSS=\sum_{j=0}^{m}\sum_{i=1}^{n}(x_{i}-\mu_{i})^{2} CSS=j=0∑m​i=1∑n​(xi​−μi​)2

其中

  • m m m表示一个簇中样本的个数
  • j j j表示每个样本的编号

同时，如果将一个数据集的所有簇的簇内平方和相加，就得到了整体平方和（Total Cluster Sum of Square） ，也即total inertia

T C S S = ∑ i = 1 k C S S l TCSS=\sum_{i=1}^{k}CSS_{l} TCSS=i=1∑k​CSSl​

total inertia越小代表每个簇内样本越相似，聚类效果也就越好，所以K-Means追求的是可以让簇内平方和最小化的质心。同时，在质心不断变化不断迭代的过程中，总体平方和是越来越小的，所以当整体平方和最小的时候，质心就不再发生变化了

这里使用sklearn中的KMeans，计算鸢尾花聚类后的inertia

# 导入所需模块
import matplotlib.pyplot as plt
import numpy as np
from sklearn.cluster import KMeans
from sklearn.datasets import load_iris

# 导入鸢尾花数据集
iris = load_iris()
X = iris.data[:]
# print(X)
print(X.shape)

# 建立簇为3的聚类器
estimator = KMeans(n_clusters=3, random_state=0)
estimator.fit(X)
# 获取标签
label_pred = estimator.labels_
# print(label_pred)
x0 = X[label_pred == 0]
x1 = X[label_pred == 1]
x2 = X[label_pred == 2]
# 绘制聚类结果
plt.scatter(x0[:, 0], x0[:, 2], c='red', marker='o', label='label0')
plt.scatter(x1[:, 0], x1[:, 2], c='green', marker='*', label='label1')
plt.scatter(x2[:, 0], x2[:, 2], c='blue', marker='v', label='label2')
centers = estimator.cluster_centers_
plt.scatter(centers[:, 0], centers[:, 2], marker='x', c='black', alpha=1, s=300)
plt.xlabel('petal length')
plt.ylabel('petal width')
plt.legend(loc=2)
print(estimator.inertia_)  # 计算inertia
plt.show()

B：轮廓系数

对于一个聚类任务，我们希望得到的类别簇中，簇内尽量紧密，簇间尽量远离，轮廓系数便是类的密集与分散程度的评价指标，公式如下

s ( i ) = b ( i ) − a ( i ) m a x { a ( i ) , b ( i ) } s(i)=\frac{b(i)-a(i)}{max\{{a(i),b(i)\}}} s(i)=max{a(i),b(i)}b(i)−a(i)​

其中

  • a ( i ) a(i) a(i)（越小越好）：表示样本 i i i到同簇其它样本的平均距离； a ( i ) a(i) a(i)说明样本 i i i就越应该被分到该簇，且将 a ( i ) a(i) a(i)称为样本 i i i的簇内不相似度
  • b ( i ) b(i) b(i)（越大越好）：表示样本 i i i到其他某簇 C j C_{j} Cj​的所有样本的平均距离 b i j b_{ij} bij​,称为样本 i i i与某簇 C j C_{j} Cj​的不相似度； b ( i ) = m i n { b i 1 , b i 2 , . . . , b i k } b(i)=min\{b_{i1},b_{i2},...,b_{ik}\} b(i)=min{bi1​,bi2​,...,bik​}

轮廓系数 s ( i ) s(i) s(i)的取值范围为[-1, 1]，且

  • 若 s ( i ) s(i) s(i)接近1（ a ( i ) a(i) a(i)< b ( i ) b(i) b(i)）：说明样本 i i i聚类合理（轮廓系数越大越好）
  • 若 s ( i ) s(i) s(i)接近-1（ a ( i ) a(i) a(i)> b ( i ) b(i) b(i)）：说明样本 i i i更应该分类到另外的簇上，聚类效果很差
  • 若 s ( i ) s(i) s(i)接近0（ a ( i ) a(i) a(i)= b ( i ) b(i) b(i)）：说明样本 i i i在两个簇的边界上
# 导入所需模块
import matplotlib.pyplot as plt
import numpy as np
from sklearn.cluster import KMeans
from sklearn.datasets import load_iris
from sklearn.metrics import silhouette_score

# 导入鸢尾花数据集
iris = load_iris()
X = iris.data[:]
# print(X)
print(X.shape)

# 建立簇为3的聚类器
estimator = KMeans(n_clusters=3, random_state=0)
estimator.fit(X)
print(silhouette_score(X, estimator.labels_))  # 打印轮廓系数
# 获取标签
label_pred = estimator.labels_
# print(label_pred)
x0 = X[label_pred == 0]
x1 = X[label_pred == 1]
x2 = X[label_pred == 2]
# 绘制聚类结果
plt.scatter(x0[:, 0], x0[:, 2], c='red', marker='o', label='label0')
plt.scatter(x1[:, 0], x1[:, 2], c='green', marker='*', label='label1')
plt.scatter(x2[:, 0], x2[:, 2], c='blue', marker='v', label='label2')
centers = estimator.cluster_centers_
plt.scatter(centers[:, 0], centers[:, 2], marker='x', c='black', alpha=1, s=300)
plt.xlabel('petal length')
plt.ylabel('petal width')
plt.legend(loc=2)
plt.show()

结果

C： Calinski-Harabasz指数

CH指标通过计算类中各点与类中心的距离平方和来度量类内的紧密度，通过计算各类中心点与数据集中心点距离平方和来度量数据集的分离度，公式为

C H ( K ) = B ( K ) ( N − K ) W ( K ) ( K − 1 ) CH(K)=\frac{B(K)(N-K)}{W(K)(K-1)} CH(K)=W(K)(K−1)B(K)(N−K)​

其中：

  • B ( K ) B(K) B(K)为类间散度（分散度）： B ( K ) = ∑ k = 1 K a k ∣ ∣ X k ‾ − X ‾ ∣ ∣ 2 B(K)=\sum_{k=1}^{K}a_{k}||\overline{X_{k}}-\overline{X}||^{2} B(K)=∑k=1K​ak​∣∣Xk​​−X∣∣2
  • W ( K ) W(K) W(K)为类内散度（紧密度）： W ( K ) = ∑ k = 1 K ∑ C j = k ∣ ∣ X j − X k ‾ ∣ ∣ 2 W(K)=\sum_{k=1}^{K}\sum_{C_{j}=k}||X_{j}-\overline{X_{k}}||^{2} W(K)=∑k=1K​∑Cj​=k​∣∣Xj​−Xk​​∣∣2

因此，CH指标由分离度与紧密度的比值得到，所以，CH越大代表着类自身越紧密，类与类之间越分散，即更优的聚类结果

# 导入所需模块
import matplotlib.pyplot as plt
import numpy as np
from sklearn.cluster import KMeans
from sklearn.datasets import load_iris
from sklearn.metrics import silhouette_score
from sklearn.metrics import calinski_harabasz_score

# 导入鸢尾花数据集
iris = load_iris()
X = iris.data[:]
# print(X)
print(X.shape)

# 建立簇为3的聚类器
estimator = KMeans(n_clusters=3, random_state=0)
estimator.fit(X)
print(calinski_harabasz_score(X, estimator.labels_))  # 打印CH指标
# 获取标签
label_pred = estimator.labels_
# print(label_pred)
x0 = X[label_pred == 0]
x1 = X[label_pred == 1]
x2 = X[label_pred == 2]
# 绘制聚类结果
plt.scatter(x0[:, 0], x0[:, 2], c='red', marker='o', label='label0')
plt.scatter(x1[:, 0], x1[:, 2], c='green', marker='*', label='label1')
plt.scatter(x2[:, 0], x2[:, 2], c='blue', marker='v', label='label2')
centers = estimator.cluster_centers_
plt.scatter(centers[:, 0], centers[:, 2], marker='x', c='black', alpha=1, s=300)
plt.xlabel('petal length')
plt.ylabel('petal width')
plt.legend(loc=2)
plt.show()

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