Em算法Java实现——深入解析期望最大化213

引言

期望最大化 (EM) 算法是一种广泛应用于机器学习和统计推断的迭代算法。它允许我们在观测数据不完整或存在潜在变量的情况下估计模型参数。本文提供了一个全面的 Java 实现示例，帮助读者深入理解 EM 算法的原理和应用。

算法原理

EM 算法通过迭代执行两个步骤来估计参数：
E 步 (期望步骤)：根据当前参数估计计算潜在变量的期望值。
M 步 (最大化步骤)：使用 E 步计算的期望值最大化模型的似然函数或后验概率。

Java 实现

以下 Java 代码实现了 EM 算法：```java
import .*;
public class EMAlgorithm {
private double[][] data;
private int numClusters;
private int numIterations;
private double[][] clusterMeans;
private double[][] clusterCovariances;
private double[] clusterPriors;
public EMAlgorithm(double[][] data, int numClusters, int numIterations) {
= data;
= numClusters;
= numIterations;
initializeParameters();
}
private void initializeParameters() {
// 随机初始化聚类中心
clusterMeans = new double[numClusters][data[0].length];
for (int i = 0; i < numClusters; i++) {
for (int j = 0; j < data[0].length; j++) {
clusterMeans[i][j] = () * (data[0].length - 1);
}
}
// 初始化聚类协方差为单位矩阵
clusterCovariances = new double[numClusters][data[0].length][data[0].length];
for (int i = 0; i < numClusters; i++) {
for (int j = 0; j < data[0].length; j++) {
for (int k = 0; k < data[0].length; k++) {
if (j == k) {
clusterCovariances[i][j][k] = 1.0;
} else {
clusterCovariances[i][j][k] = 0.0;
}
}
}
}
// 初始化聚类先验概率为相等
clusterPriors = new double[numClusters];
for (int i = 0; i < numClusters; i++) {
clusterPriors[i] = 1.0 / numClusters;
}
}
public void run() {
for (int iteration = 0; iteration < numIterations; iteration++) {
// E 步
double[][] responsibilities = calculateResponsibilities();
// M 步
updateParameters(responsibilities);
}
}
private double[][] calculateResponsibilities() {
double[][] responsibilities = new double[][numClusters];
for (int i = 0; i < ; i++) {
for (int j = 0; j < numClusters; j++) {
responsibilities[i][j] = calculateResponsibility(i, j);
}
}
return responsibilities;
}
private double calculateResponsibility(int dataPoint, int cluster) {
double numerator = calculateLikelihood(dataPoint, cluster) * clusterPriors[cluster];
double denominator = 0.0;
for (int k = 0; k < numClusters; k++) {
denominator += calculateLikelihood(dataPoint, k) * clusterPriors[k];
}
return numerator / denominator;
}
private double calculateLikelihood(int dataPoint, int cluster) {
// 计算数据点到聚类中心的马氏距离
double distance = calculateMahalanobisDistance(data[dataPoint], clusterMeans[cluster], clusterCovariances[cluster]);
// 计算高斯分布概率密度
double likelihood = (1 / ((2 * ) * (clusterCovariances[cluster]))) *
(-0.5 * distance);
return likelihood;
}
private double calculateMahalanobisDistance(double[] dataPoint, double[] clusterMean, double[][] clusterCovariance) {
// 计算协方差矩阵的逆
double[][] covarianceInverse = invertMatrix(clusterCovariance);
// 计算马氏距离
double distance = 0.0;
for (int i = 0; i < ; i++) {
for (int j = 0; j < ; j++) {
distance += (dataPoint[i] - clusterMean[i]) * covarianceInverse[i][j] * (dataPoint[j] - clusterMean[j]);
}
}
return distance;
}
private double[][] invertMatrix(double[][] matrix) {
// 使用 LU 分解计算矩阵逆
double[][] inverse = new double[][matrix[0].length];
for (int i = 0; i < ; i++) {
for (int j = 0; j < matrix[0].length; j++) {
if (i == j) {
inverse[i][j] = 1.0;
} else {
inverse[i][j] = 0.0;
}
}
}
// 执行 LU 分解
for (int i = 0; i < ; i++) {
for (int j = i + 1; j < ; j++) {
double factor = matrix[j][i] / matrix[i][i];
for (int k = 0; k < matrix[0].length; k++) {
matrix[j][k] -= factor * matrix[i][k];
inverse[j][k] -= factor * inverse[i][k];
}
}
}
// 执行反向代入
for (int i = - 1; i >= 0; i--) {
for (int j = i - 1; j >= 0; j--) {
double factor = matrix[j][i] / matrix[i][i];
for (int k = 0; k < matrix[0].length; k++) {
inverse[j][k] -= factor * inverse[i][k];
}
}
inverse[i][i] /= matrix[i][i];
}
return inverse;
}
private void updateParameters(double[][] responsibilities) {
// 更新聚类中心
for (int cluster = 0; cluster < numClusters; cluster++) {
for (int feature = 0; feature < data[0].length; feature++) {
double sum = 0.0;
double weightSum = 0.0;
for (int dataPoint = 0; dataPoint < ; dataPoint++) {
sum += responsibilities[dataPoint][cluster] * data[dataPoint][feature];
weightSum += responsibilities[dataPoint][cluster];
}
clusterMeans[cluster][feature] = sum / weightSum;
}
}
// 更新聚类协方差
for (int cluster = 0; cluster < numClusters; cluster++) {
for (int i = 0; i < data[0].length; i++) {
for (int j = 0; j < data[0].length; j++) {
double sum = 0.0;
double weightSum = 0.0;
for (int dataPoint = 0; dataPoint < ; dataPoint++) {
double diff = data[dataPoint][i] - clusterMeans[cluster][i];
sum += responsibilities[dataPoint][cluster] * diff * (data[dataPoint][j] - clusterMeans[cluster][j]);
weightSum += responsibilities[dataPoint][cluster];
}
clusterCovariances[cluster][i][j] = sum / weightSum;
}
}
}
// 更新聚类先验概率
for (int cluster = 0; cluster < numClusters; cluster++) {
double sum = 0.0;
for (int dataPoint = 0; dataPoint < ; dataPoint++) {
sum += responsibilities[dataPoint][cluster];
}
clusterPriors[cluster] = sum / ;
}
}
}
```