C均值聚类算法（K-Means Clustering）是一种非常流行的聚类算法，用于将数据点分成多个簇，使得簇内的点尽可能相似，簇间的点尽可能不同。以下是K-Means算法的基本步骤：

1. 初始化：随机选择K个点作为初始的簇中心（质心）。

2. 分配：将每个数据点分配到最近的质心所属的簇中。

3. 更新：计算每个簇中所有点的均值，更新质心为这个均值。

4. 迭代：重复步骤2和3，直到满足某个终止条件（例如，达到最大迭代次数，或者质心的变化小于某个阈值）。

5. 终止：当满足终止条件时，算法结束，最终的簇划分就是聚类结果。

sklearn方法

import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets
from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_score
from sklearn.preprocessing import StandardScaler# 加载鸢尾花数据集
iris = datasets.load_iris()
X = iris.data# 数据标准化
scaler = StandardScaler()
X_scaled = scaler.fit_transform(X)# 定义不同的K值
k_values = [2, 3, 4, 5]# 评估不同K值的聚类效果
for k in k_values:kmeans = KMeans(n_clusters=k, random_state=42)kmeans.fit(X_scaled)labels = kmeans.labels_# 计算轮廓系数silhouette_avg = silhouette_score(X_scaled, labels)print(f"For n_clusters = {k}, silhouette score is {silhouette_avg}")# 可视化聚类效果plt.figure(figsize=(8, 6))plt.scatter(X_scaled[:, 0], X_scaled[:, 1], c=labels, cmap='viridis', marker='o', label='Cluster')centers = kmeans.cluster_centers_plt.scatter(centers[:, 0], centers[:, 1], c='red', s=200, alpha=0.75, marker='*', label='Centroids')plt.title(f'K-Means Clustering with n_clusters = {k}')plt.xlabel('Feature 1')plt.ylabel('Feature 2')plt.legend()plt.show()# 评估不同初始化方法的聚类效果
k = 3
#
init_methods = ['random', 'k-means++']
for init in init_methods:kmeans = KMeans(n_clusters=k, init=init, random_state=42)kmeans.fit(X_scaled)labels = kmeans.labels_# 计算轮廓系数silhouette_avg = silhouette_score(X_scaled, labels)print(f"For n_clusters = {k}, init method = {init}, silhouette score is {silhouette_avg}")# 可视化聚类效果plt.figure(figsize=(8, 6))plt.scatter(X_scaled[:, 0], X_scaled[:, 1], c=labels, cmap='viridis', marker='o', label='Cluster')centers = kmeans.cluster_centers_plt.scatter(centers[:, 0], centers[:, 1], c='red', s=200, alpha=0.75, marker='*', label='Centroids')plt.title(f'K-Means Clustering with n_clusters = {k}, init method = {init}')plt.xlabel('Feature 1')plt.ylabel('Feature 2')plt.legend()plt.show()

Numpy方法

import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets
from sklearn.preprocessing import StandardScaler#使用 NumPy 实现 K-Means 算法
def kmeans(X, k, max_iters=100):n_samples, n_features = X.shapecentroids = X[np.random.choice(n_samples, k, replace=False)]for _ in range(max_iters):distances = np.sqrt((X[:, np.newaxis] - centroids) ** 2).sum(axis=2)labels = np.argmin(distances, axis=1)new_centroids = np.array([X[labels == i].mean(axis=0) for i in range(k)])if np.all(centroids == new_centroids):breakcentroids = new_centroidsreturn labels, centroids#加载鸢尾花数据集并对其进行标准化
iris = datasets.load_iris()
X = iris.data
scaler = StandardScaler()
X_scaled = scaler.fit_transform(X)#评估不同 c 值的聚类效果
from sklearn.metrics import silhouette_score
k_values = [2, 3, 4]
for k in k_values:labels, centers = kmeans(X_scaled, k)silhouette_avg = silhouette_score(X_scaled, labels)print(f"For n_clusters = {k}, silhouette score is {silhouette_avg}")
#可视化每个 k 值的聚类结果。
for k in k_values:labels, centers = kmeans(X_scaled, k)plt.figure(figsize=(8, 6))plt.scatter(X_scaled[:, 0], X_scaled[:, 1], c=labels, cmap='viridis', marker='o', label='Cluster')plt.scatter(centers[:, 0], centers[:, 1], c='red', s=200, alpha=0.75, marker='*', label='Centroids')plt.title(f'K-Means Clustering with n_clusters = {k}')plt.xlabel('Feature 1')plt.ylabel('Feature 2')plt.legend()plt.show()#评估不同初始化方法对聚类效果的影响
def kmeans_plusplus(X, k, max_iters=100):n_samples, n_features = X.shapecentroids = [X[np.random.choice(n_samples)]]for _ in range(1, k):distances = np.sqrt((X[:, np.newaxis] - centroids) ** 2).sum(axis=2)probabilities = distances.min(axis=1) ** 2cumulative_probabilities = probabilities.cumsum()r = np.random.rand() * cumulative_probabilities[-1]new_centroid_index = np.searchsorted(cumulative_probabilities, r)centroids.append(X[new_centroid_index])centroids = np.array(centroids)for _ in range(max_iters):distances = np.sqrt((X[:, np.newaxis] - centroids) ** 2).sum(axis=2)labels = np.argmin(distances, axis=1)new_centroids = np.array([X[labels == i].mean(axis=0) for i in range(k)])if np.all(centroids == new_centroids):breakcentroids = new_centroidsreturn labels, centroidsfor k in k_values:labels, centers = kmeans_plusplus(X_scaled, k)silhouette_avg = silhouette_score(X_scaled, labels)print(f"For n_clusters = {k}, silhouette score (k-means++) is {silhouette_avg}")plt.figure(figsize=(8, 6))plt.scatter(X_scaled[:, 0], X_scaled[:, 1], c=labels, cmap='viridis', marker='o', label='Cluster')plt.scatter(centers[:, 0], centers[:, 1], c='red', s=200, alpha=0.75, marker='*', label='Centroids')plt.title(f'K-Means++ Clustering with n_clusters = {k}')plt.xlabel('Feature 1')plt.ylabel('Feature 2')plt.legend()plt.show()