K-means聚类算法全解析

Week09-Clustering聚类&k-mean技术解析

聚类分析概述

聚类是一种无监督学习方法，旨在将数据划分为具有相似特征的组。与分类不同，聚类不需要预先标记的数据，而是通过数据的内在结构自动分组。常见的应用场景包括客户细分、图像分割和异常检测。

k-means算法原理

k-means是一种基于距离的聚类算法，通过迭代优化将数据划分为k个簇。其核心思想是最小化簇内平方误差（SSE），公式如下： $$SSE = \sum_{i=1}^{k}\sum_{x\in C_i}||x-\mu_i||^2$$ 其中$C_i$表示第i个簇，$\mu_i$是该簇的质心。

k-means实现步骤

1. 随机选择k个初始质心
2. 计算每个样本点到质心的距离，分配到最近的簇
3. 重新计算每个簇的质心（均值）
4. 重复步骤2-3直到质心不再变化或达到最大迭代次数

Python实现示例：

from sklearn.cluster import KMeans
import numpy as np

# 生成示例数据
X = np.random.rand(100, 2)

# 创建k-means模型
kmeans = KMeans(n_clusters=3, random_state=0)

# 训练模型
kmeans.fit(X)

# 获取聚类结果
labels = kmeans.labels_
centers = kmeans.cluster_centers_

算法优化与挑战

初始质心选择对结果影响显著，常用改进方法包括：

• k-means++：通过概率分布优化初始质心选择
• 二分k-means：通过层次分裂方式确定簇数
• Mini-Batch k-means：适用于大规模数据集

肘部法则可帮助确定最佳k值：

inertias = []
for k in range(1, 11):
    kmeans = KMeans(n_clusters=k)
    kmeans.fit(X)
    inertias.append(kmeans.inertia_)

评估指标

常用聚类评估方法包括：

• 轮廓系数：衡量簇内紧密度和簇间分离度
• Calinski-Harabasz指数：基于簇间和簇内离散度比值
• Davies-Bouldin指数：反映簇间相似度

应用实例

在客户细分中，k-means可根据购买行为将客户分组。特征工程阶段通常需要：

• 数据标准化（MinMaxScaler或StandardScaler）
• 特征选择（PCA降维）
• 异常值处理

扩展阅读

• 层次聚类：适合小规模数据集，可生成树状图
• DBSCAN：基于密度的聚类方法，能发现任意形状簇
• 谱聚类：结合图论和线性代数，适合非凸分布数据

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