本文详细介绍了一种混合高斯模型（GMM）算法的实现过程，包括初始化参数、计算概率、更新参数等关键步骤，并提供了完整的Java代码实现。
GMM算发步骤：
1. 初始化参数，包括Gauss分布个数、均值、协方差；
2. 计算每个节点属于每个分布的概率；
3. 计算每个分布产生每个节点的概率；
4. 更新每个分布的权值，均值和它们的协方差。
基本参数类：
public class Parameter {
private ArrayList<ArrayList<Double>> pMiu; // 均值参数k个分布的中心点，每个中心点d维
private ArrayList<Double> pPi; // k个GMM的权值
private ArrayList<ArrayList<ArrayList<Double>>> pSigma; // k类GMM的协方差矩阵,d*d*k
public ArrayList<ArrayList<Double>> getpMiu() {
return pMiu;
}
public void setpMiu(ArrayList<ArrayList<Double>> pMiu) {
this.pMiu = pMiu;
}
public ArrayList<Double> getpPi() {
return pPi;
}
public void setpPi(ArrayList<Double> pPi) {
this.pPi = pPi;
}
public ArrayList<ArrayList<ArrayList<Double>>> getpSigma() {
return pSigma;
}
public void setpSigma(ArrayList<ArrayList<ArrayList<Double>>> pSigma) {
this.pSigma = pSigma;
}
}
核心代码如下：
public class GMMAlgorithm {
/**
*
* @Title: GMMCluster
* @Description: GMM聚类算法的实现类，返回每条数据的类别(0~k-1)
* @return int[]
* @throws
*/
public int[] GMMCluster(ArrayList<ArrayList<Double>>dataSet, ArrayList<ArrayList<Double>> pMiu, int dataNum, int k, int dataDimen) {
Parameter parameter = iniParameters(dataSet, dataNum, k, dataDimen);
double Lpre = -1000000; // 上一次聚类的误差
double threshold = 0.0001;
while(true) {
ArrayList<ArrayList<Double>> px = computeProbablity(dataSet, pMiu, dataNum, k, dataDimen);
double[][] pGama = new double[dataNum][k];
for(int i = 0; i < dataNum; i++) {
for(int j = 0; j < k; j++) {
pGama[i][j] = px.get(i).get(j) * parameter.getpPi().get(j);
}
}
double[] sumPGama = GMMUtil.matrixSum(pGama, 2);
for(int i = 0; i < dataNum; i++) {
for(int j = 0; j < k; j++) {
pGama[i][j] = pGama[i][j] / sumPGama[i];
}
}
double[] NK = GMMUtil.matrixSum(pGama, 1); // 第k个高斯生成每个样本的概率的和，所有Nk的总和为N
// 更新pMiu
double[] NKReciprocal = new double[NK.length];
for(int i = 0; i < NK.length; i++) {
NKReciprocal[i] = 1 / NK[i];
}
double[][] pMiuTmp = GMMUtil.matrixMultiply(GMMUtil.matrixMultiply(GMMUtil.diag(NKReciprocal), GMMUtil.matrixReverse(pGama)), GMMUtil.toArray(dataSet));
// 更新pPie
double[][] pPie = new double[k][1];
for(int i = 0; i < NK.length; i++) {
pPie[i][1] = NK[i] / dataNum;
}
// 更新k个pSigma
double[][][] pSigmaTmp = new double[dataDimen][dataDimen][k];
for(int i = 0; i < k; i++) {
double[][] xShift = new double[dataNum][dataDimen];
for(int j = 0; j < dataNum; j++) {
for(int l = 0; l < dataDimen; l++) {
xShift[j][l] = pMiuTmp[i][l];
}
}
double[] pGamaK = new double[dataNum]; // 第k条pGama值
for(int j = 0; j < dataNum; j++) {
pGamaK[j] = pGama[j][i];
}
double[][] diagPGamaK = GMMUtil.diag(pGamaK);
double[][] pSigmaK = GMMUtil.matrixMultiply(GMMUtil.matrixReverse(xShift), (GMMUtil.matrixMultiply(diagPGamaK, xShift)));
for(int j = 0; j < dataDimen; j++) {
for(int l = 0; l < dataDimen; l++) {
pSigmaTmp[j][l][k] = pSigmaK[j][l] / NK[i];
}
}
}
// 判断是否迭代结束
double[][] a = GMMUtil.matrixMultiply(GMMUtil.toArray(px), pPie);
for(int i = 0; i < dataNum; i++) {
a[i][0] = Math.log(a[i][0]);
}
double L = GMMUtil.matrixSum(a, 1)[0];
if(L - Lpre < threshold) {
break;
}
Lpre = L;
}
return null;
}
/**
*
* @Title: computeProbablity
* @Description: 计算每个节点（共n个）属于每个分布（k个）的概率
* @return ArrayList<ArrayList<Double>>
* @throws
*/
public ArrayList<ArrayList<Double>> computeProbablity(ArrayList<ArrayList<Double>>dataSet, ArrayList<ArrayList<Double>> pMiu, int dataNum, int k, int dataDimen) {
double[][] px = new double[dataNum][k]; // 每条数据属于每个分布的概率
int[] type = getTypes(dataSet, pMiu, k, dataNum);
// 计算k个分布的协方差矩阵
ArrayList<ArrayList<ArrayList<Double>>> covList = new ArrayList<ArrayList<ArrayList<Double>>>();
for(int i = 0; i < k; i++) {
ArrayList<ArrayList<Double>> dataSetK = new ArrayList<ArrayList<Double>>();
for(int j = 0; j < dataNum; j++) {
if(type[k] == i) {
dataSetK.add(dataSet.get(i));
}
}
covList.set(i, GMMUtil.computeCov(dataSetK, dataDimen, dataSetK.size()));
}
// 计算每条数据属于每个分布的概率
for(int i = 0; i < dataNum; i++) {
for(int j = 0; j < k; j++) {
ArrayList<Double> offset = GMMUtil.matrixMinus(dataSet.get(i), pMiu.get(j));
ArrayList<ArrayList<Double>> invSigma = covList.get(k);
double[] tmp = GMMUtil.matrixSum(GMMUtil.matrixMultiply(GMMUtil.toArray1(offset), GMMUtil.toArray(invSigma)), 2);
double coef = Math.pow((2 * Math.PI), -(double)dataDimen / 2d) * Math.sqrt(GMMUtil.computeDet(invSigma, invSigma.size()));
px[i][j] = coef * Math.pow(Math.E, -0.5 * tmp[0]);
}
}
return GMMUtil.toList(px);
}
/**
*
* @Title: iniParameters
* @Description: 初始化参数Parameter
* @return Parameter
* @throws
*/
public Parameter iniParameters(ArrayList<ArrayList<Double>> dataSet, int dataNum, int k, int dataDimen) {
Parameter res = new Parameter();
ArrayList<ArrayList<Double>> pMiu = generateCentroids(dataSet, dataNum, k);
res.setpMiu(pMiu);
// 计算每个样本节点与每个中心节点的距离，以此为据对样本节点进行分类计数，进而初始化k个分布的权值
ArrayList<Double> pPi = new ArrayList<Double>();
int[] type = getTypes(dataSet, pMiu, k, dataNum);
int[] typeNum = new int[k];
for(int i = 0; i < dataNum; i++) {
typeNum[type[i]]++;
}
for(int i = 0; i < k; i++) {
pPi.add((double)(typeNum[i]) / (double)(dataNum));
}
res.setpPi(pPi);
// 计算k个分布的k个协方差
ArrayList<ArrayList<ArrayList<Double>>> pSigma = new ArrayList<ArrayList<ArrayList<Double>>>();
for(int i = 0; i < k; i++) {
ArrayList<ArrayList<Double>> tmp = new ArrayList<ArrayList<Double>>();
for(int j = 0; j < dataNum; j++) {
if(type[j] == i) {
tmp.add(dataSet.get(i));
}
}
pSigma.add(GMMUtil.computeCov(tmp, dataDimen, dataNum));
}
res.setpSigma(pSigma);
return res;
}
/**
*
* @Title: generateCentroids
* @Description: 获取随机的k个中心点
* @return ArrayList<ArrayList<Double>>
* @throws
*/
public ArrayList<ArrayList<Double>> generateCentroids(ArrayList<ArrayList<Double>> data, int dataNum, int k) {
ArrayList<ArrayList<Double>> res = null;
if(dataNum < k) {
return res;
} else {
res = new ArrayList<ArrayList<Double>>();
List<Integer> random = new ArrayList<Integer>();
// 随机产生不重复的k个数
while(k > 0) {
int index = (int)(Math.random() * dataNum);
if(!random.contains(index)) {
random.add(index);
k--;
res.add(data.get(index));
}
}
}
return res;
}
/**
*
* @Title: getTypes
* @Description: 返回每条数据的类别
* @return int[]
* @throws
*/
public int[] getTypes(ArrayList<ArrayList<Double>> dataSet, ArrayList<ArrayList<Double>> pMiu, int k, int dataNum) {
int[] type = new int[dataNum];
for(int j = 0; j < dataNum; j++) {
double minDistance = GMMUtil.computeDistance(dataSet.get(j), pMiu.get(0));
type[j] = 0; // 0作为该条数据的类别
for(int i = 1; i < k; i++) {
if(GMMUtil.computeDistance(dataSet.get(j), pMiu.get(0)) < minDistance) {
minDistance = GMMUtil.computeDistance(dataSet.get(j), pMiu.get(0));
type[j] = k;
}
}
}
return type;
}
public static void main(String[] args) {
ArrayList<Double> pPi = new ArrayList<Double>();
System.out.println(pPi.get(0));
}
}一些工具类：
public class GMMUtil {
/**
*
* @Title: computeDistance
* @Description: 计算任意两个节点间的距离
* @return double
* @throws
*/
public static double computeDistance(ArrayList<Double> d1, ArrayList<Double> d2) {
double squareSum = 0;
for(int i = 0; i < d1.size() - 1; i++) {
squareSum += (d1.get(i) - d2.get(i)) * (d1.get(i) - d2.get(i));
}
return Math.sqrt(squareSum);
}
/**
*
* @Title: computeCov
* @Description: 计算协方差矩阵
* @return ArrayList<ArrayList<Double>>
* @throws
*/
public static ArrayList<ArrayList<Double>> computeCov(ArrayList<ArrayList<Double>> dataSet, int dataDimen, int dataNum) {
ArrayList<ArrayList<Double>> res = new ArrayList<ArrayList<Double>>();
// 计算每一维数据的均值
double[] sum = new double[dataDimen];
for(ArrayList<Double> data : dataSet) {
for(int i = 0; i < dataDimen; i++) {
sum[i] += data.get(i);
}
}
for(int i = 0; i < dataDimen; i++) {
sum[i] = sum[i] / dataNum;
}
// 计算任意两组数据的协方差
for(int i = 0; i < dataDimen; i++) {
ArrayList<Double> tmp = new ArrayList<Double>();
for(int j = 0; j < dataDimen; j++) {
double cov = 0;
for(ArrayList<Double> data : dataSet) {
cov += (data.get(i) - sum[i]) * (data.get(j) - sum[j]);
}
tmp.add(cov);
}
res.add(tmp);
}
return res;
}
/**
*
* @Title: computeInv
* @Description: 计算矩阵的逆矩阵
* @return ArrayList<ArrayList<Double>>
* @throws
*/
public static double[][] computeInv(ArrayList<ArrayList<Double>> dataSet) {
int dataDimen = dataSet.size();
double[][] res = new double[dataDimen][dataDimen];
// 将list转化为array
double[][] a = toArray(dataSet);
// 计算伴随矩阵
double detA = computeDet(dataSet, dataDimen); // 整个矩阵的行列式
for (int i = 0; i < dataDimen; i++) {
for (int j = 0; j < dataDimen; j++) {
double num;
if ((i + j) % 2 == 0) {
num = computeDet(toList(computeAC(a, i + 1, j + 1)), dataDimen - 1);
} else {
num = -computeDet(toList(computeAC(a, i + 1, j + 1)), dataDimen - 1);
}
res[j][i] = num / detA;
}
}
return res;
}
/**
*
* @Title: computeAC
* @Description: 求指定行、列的代数余子式(algebraic complement)
* @return double[][]
* @throws
*/
public static double[][] computeAC(double[][] dataSet, int r, int c) {
int H = dataSet.length;
int V = dataSet[0].length;
double[][] newData = new double[H - 1][V - 1];
for (int i = 0; i < newData.length; i++) {
if (i < r - 1) {
for (int j = 0; j < newData[i].length; j++) {
if (j < c - 1) {
newData[i][j] = dataSet[i][j];
} else {
newData[i][j] = dataSet[i][j + 1];
}
}
} else {
for (int j = 0; j < newData[i].length; j++) {
if (j < c - 1) {
newData[i][j] = dataSet[i + 1][j];
} else {
newData[i][j] = dataSet[i + 1][j + 1];
}
}
}
}
return newData;
}
/**
*
* @Title: computeDet
* @Description: 计算行列式
* @return double
* @throws
*/
public static double computeDet(ArrayList<ArrayList<Double>> dataSet, int dataDimen) {
// 将list转化为array
double[][] a = toArray(dataSet);
if(dataDimen == 2) {
return a[0][0] * a[1][1] - a[0][1] * a[1][0];
}
double res = 0;
for(int i = 0; i < dataDimen; i++) {
if(i % 2 == 0) {
res += a[0][i] * computeDet(toList(computeAC(toArray(dataSet), 1, i + 1)), dataDimen - 1);
} else {
res += -a[0][i] * computeDet(toList(computeAC(toArray(dataSet), 1, i + 1)), dataDimen - 1);
}
}
return res;
}
/**
*
* @Title: toList
* @Description: 将array转化成list
* @return ArrayList<ArrayList<Double>>
* @throws
*/
public static ArrayList<ArrayList<Double>> toList(double[][] a) {
ArrayList<ArrayList<Double>> res = new ArrayList<ArrayList<Double>>();
for(int i = 0; i < a.length; i++) {
ArrayList<Double> tmp = new ArrayList<Double>();
for(int j = 0; j < a[i].length; j++) {
tmp.add(a[i][j]);
}
res.add(tmp);
}
return res;
}
public static double[][] matrixMultiply(double[][] a, double[][] b) {
double[][] res = new double[a.length][b[0].length];
for(int i = 0; i < a.length; i++) {
for(int j = 0; j < b[0].length; j++) {
for(int k = 0; k < a[0].length; k++) {
res[i][j] += a[i][k] * b[k][j];
}
}
}
return res;
}
/**
*
* @Title: dotMatrixMultiply
* @Description: 矩阵的点乘，即对应元素相乘
* @return double[][]
* @throws
*/
public static double[][] dotMatrixMultiply (double[][] a, double[][] b) {
double[][] res = new double[a.length][a[0].length];
for(int i = 0; i < a.length; i++) {
for(int j = 0; j < a[0].length; j++) {
res[i][j] = a[i][j] * b[i][j];
}
}
return res;
}
/**
*
* @Title: dotMatrixMultiply
* @Description: 矩阵的点除，即对应元素相除
* @return double[][]
* @throws
*/
public static double[][] dotMatrixDivide(double[][] a, double[][] b) {
double[][] res = new double[a.length][a[0].length];
for(int i = 0; i < a.length; i++) {
for(int j = 0; j < a[0].length; j++) {
res[i][j] = a[i][j] / b[i][j];
}
}
return res;
}
/**
*
* @Title: repmat
* @Description: 对应matlab的repmat的函数，对矩阵进行横向或纵向的平铺
* @return double[][]
* @throws
*/
public static double[][] repmat(double[][] a, int row, int clo) {
double[][] res = new double[a.length * row][a[0].length * clo];
return null;
}
/**
*
* @Title: matrixMinux
* @Description: 计算集合只差
* @return ArrayList<ArrayList<Double>>
* @throws
*/
public static ArrayList<Double> matrixMinus(ArrayList<Double> a1, ArrayList<Double> a2) {
ArrayList<Double> res = new ArrayList<Double>();
for(int i = 0; i < a1.size(); i++) {
res.add(a1.get(i) - a2.get(i));
}
return res;
}
/**
*
* @Title: matrixSum
* @Description: 返回矩阵每行之和(mark==2)或每列之和(mark==1)
* @return ArrayList<Double>
* @throws
*/
public static double[] matrixSum(double[][] a, int mark) {
double res[] = new double[a.length];
if(mark == 1) { // 计算每列之和，返回行向量
res = new double[a[0].length];
for(int i = 0; i < a[0].length; i++) {
for(int j = 0; j < a.length; j++) {
res[i] += a[j][i];
}
}
} else if (mark == 2) { // 计算每行之和， 返回列向量
for(int i = 0; i < a.length; i++) {
for(int j = 0; j < a[0].length; j++) {
res[i] += a[i][j];
}
}
}
return res;
}
public static double[][] toArray(ArrayList<ArrayList<Double>> a) {
int dataDimen = a.size();
double[][] res = new double[dataDimen][dataDimen];
for(int i = 0; i < dataDimen; i++) {
for(int j = 0; j < dataDimen; j++) {
res[i][j] = a.get(i).get(j);
}
}
return res;
}
public static double[][] toArray1(ArrayList<Double> a) {
int dataDimen = a.size();
double[][] res = new double[1][dataDimen];
for(int i = 0; i < dataDimen; i++) {
res[1][i] = a.get(i);
}
return res;
}
/**
*
* @Title: matrixReverse
* @Description: 矩阵专制
* @return double[][]
* @throws
*/
public static double[][] matrixReverse(double[][] a) {
double[][] res = new double[a[0].length][a.length];
for(int i = 0; i < a.length; i++) {
for(int j = 0; j < a[0].length; j++) {
res[j][i] = a[i][j];
}
}
return res;
}
/**
*
* @Title: diag
* @Description: 向量对角化
* @return double[][]
* @throws
*/
public static double[][] diag(double[] a) {
double[][] res = new double[a.length][a.length];
for(int i = 0; i < a.length; i++) {
for(int j = 0; j < a.length; j++) {
if(i == j) {
res[i][j] = a[i];
}
}
}
return res;
}
}
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