PCA主成分分析
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不废话，先上代码。
package com.pca;public class Main { public static void main(String[] args) { // 原始数据 double[][] rawData = new double[][] { { 40.4, 24.7, 7.2, 6.1, 8.3, 8.7, 2.442, 20.0 }, { 25.0, 12.7, 11.2, 11.0, 12.9, 20.2, 3.542, 9.1 }, { 13.2, 3.3, 3.9, 4.3, 4.4, 5.5, 0.578, 3.6 }, { 22.3, 6.7, 5.6, 3.7, 6.0, 7.4, 0.176, 7.3 }, { 34.3, 11.8, 7.1, 7.1, 8.0, 8.9, 1.726, 27.5 }, { 35.6, 12.5, 16.4, 16.7, 22.8, 29.3, 3.017, 26.6 }, { 22.0, 7.8, 9.9, 10.2, 12.6, 17.6, 0.847, 10.6 }, { 48.4, 13.4, 10.9, 9.9, 10.9, 13.9, 1.772, 1.772 }, { 40.6, 19.1, 19.8, 19.0, 29.7, 39.6, 2.449, 35.8 }, { 24.8, 8.0, 9.8, 8.9, 11.9, 16.2, 0.789, 13.7 }, { 12.5, 9.7, 4.2, 4.2, 4.6, 6.5, 0.874, 3.9 }, { 1.8, 0.6, 0.7, 0.7, 0.8, 1.1, 0.056, 1.0 }, { 32.3, 13.9, 9.4, 8.3, 9.8, 13.3, 2.126, 17.1 }, { 38.5, 9.1, 11.3, 9.5, 12.2, 16.4, 1.327, 11.6 }, { 26.2, 10.1, 5.6, 15.6, 7.7, 30.1, 0.126, 25.9 } }; PrincipalComponentAnalysis pca = new PrincipalComponentAnalysis(); pca.buildPCA(rawData); int[] selected = pca.getSelected(); for (int i = 0; i < selected.length; i++) { System.out.print(selected[i] + " "); } System.out.println(); System.out.println("======"); double[] eigenValues = pca.getEigenValues(); for (int i = 0; i < eigenValues.length; i++) { System.out.print(eigenValues[i] + " "); } System.out.println(); System.out.println("======"); double[][] eigenVectors = pca.getEigenVectors(); for (int i = 0; i < eigenVectors.length; i++) { for (int j = 0; j < eigenVectors[0].length; j++) { System.out.print(eigenVectors[i][j] + " "); } System.out.println(); } System.out.println(); System.out.println("======"); double[] pEigenValues = pca.getPrincipalEigenValues(); for (int i = 0; i < pEigenValues.length; i++) { System.out.print(pEigenValues[i] + " "); } System.out.println(); System.out.println("======"); double[][] pEigenVectors = pca.getPrincipalEigenVectors(); for (int i = 0; i < pEigenVectors.length; i++) { for (int j = 0; j < pEigenVectors[0].length; j++) { System.out.print(pEigenVectors[i][j] + " "); } System.out.println(); } System.out.println(); System.out.println("======"); double[][] pData = pca.getPrincipalData(); for (int i = 0; i < pData.length; i++) { for (int j = 0; j < pData[0].length; j++) { System.out.print(pData[i][j] + " "); } System.out.println(); } }}package com.pca;import no.uib.cipr.matrix.DenseMatrix;import no.uib.cipr.matrix.EVD;import no.uib.cipr.matrix.Matrix;import no.uib.cipr.matrix.NotConvergedException;import org.apache.log4j.Logger;public class PrincipalComponentAnalysis { private static Logger logger = Logger .getLogger(PrincipalComponentAnalysis.class); private double[][] rawData = null; // 原始数据 private double[][] principalData = null; // 主成分提取后的数据 private double[] principalEigenValues = null; // 主要的特征值 private double[][] principalEigenVectors = null; // 主要的特征向量 private double[] eigenValues = null; // 特征值 private double[][] eigenVectors = null; // 特征向量 private int[] selected = null; // 选择的特征值序号 private double proportion = 0.9; // 主成分的比重，默认为90% /**************** 构造函数里配置一些参数 ****************/ public PrincipalComponentAnalysis() { } public PrincipalComponentAnalysis(double proportion) { this.proportion = proportion; } /**************** get方法 ****************/ public double[][] getRawData() { return rawData; } public double[][] getPrincipalData() { return principalData; } public double[] getPrincipalEigenValues() { return principalEigenValues; } public double[][] getPrincipalEigenVectors() { return principalEigenVectors; } public double[] getEigenValues() { return eigenValues; } public double[][] getEigenVectors() { return eigenVectors; } public int[] getSelected() { return selected; } public double getProportion() { return proportion; } /**************** PCA的内部方法 ****************/ // 将原始数据标准化 private double[][] calcStandardlizer(double[][] rawData) { double[][] standardData = null; if (rawData != null) { int N = rawData.length; // 二维矩阵的行数，样本个数 int p = rawData[0].length; // 二维矩阵的列数，属性个数 // 每个属性对应的列是该属性的一个采样，近似为该属性的分布 double[] average = new double[p]; // 每一列的平均值 double[] var = new double[p]; // 每一列的方差 standardData = new double[N][p]; // 标准化后的列向量组成的矩阵 // 取得平均值 for (int k = 0; k < p; k++) { double temp = 0; for (int i = 0; i < N; i++) { temp += rawData[i][k]; } average[k] = temp / N; } // 取得方差 for (int k = 0; k < p; k++) { double temp = 0; for (int i = 0; i < N; i++) { temp += (rawData[i][k] - average[k]) * (rawData[i][k] - average[k]); } var[k] = temp / (N - 1); } // 取得标准化的矩阵 期望为0，方差为1，简化相关系数的计算公式 for (int i = 0; i < N; i++) { for (int j = 0; j < p; j++) { standardData[i][j] = (double) ((rawData[i][j] - average[j]) / Math .sqrt(var[j])); } } } else { logger.info("There is no raw data."); } return standardData; } // 计算样本相关系数矩阵 各列之间相互计算（共p列），形成p*p的矩阵 // 输入为标准化之后的矩阵，利用期望=0，方差=1简化了相关系数的计算 private double[][] calcCoefficientOfAssociation(double[][] standardData) { double[][] assosiationMatrix = null; if (standardData != null) { int n = standardData.length; // 二维矩阵的行号 int p = standardData[0].length; // 二维矩阵的列号 assosiationMatrix = new double[p][p];// 相关系数矩阵 for (int i = 0; i < p; i++) { for (int j = 0; j < p; j++) { double temp = 0; for (int k = 0; k < n; k++) { temp += standardData[k][i] * standardData[k][j]; } assosiationMatrix[i][j] = temp / (n - 1); } } } return assosiationMatrix; } // 计算相关系数矩阵的特征值 private double[] calcEigenValue(double[][] assosiationMatrix) { // assosiationMatrix是一个方阵 double[] eigenValues = null; if (assosiationMatrix != null) { DenseMatrix Assosiation = new DenseMatrix(assosiationMatrix); int len = assosiationMatrix.length; EVD evd = new EVD(len); try { evd.factor(Assosiation); eigenValues = evd.getRealEigenvalues(); } catch (NotConvergedException e) { e.printStackTrace(); logger.error(e); } } return eigenValues; } // 计算相关系数矩阵的特征向量 private double[][] calcEigenVector(double[][] assosiationMatrix) { // assosiationMatrix是一个方阵 double[][] eigenVectors = null; if (assosiationMatrix != null) { DenseMatrix Assosiation = new DenseMatrix(assosiationMatrix); int len = assosiationMatrix.length; EVD evd = new EVD(len); try { evd.factor(Assosiation); DenseMatrix tempMatrix = evd.getLeftEigenvectors(); eigenVectors = new double[len][len]; for (int i = 0; i < len; i++) { for (int j = 0; j < len; j++) { eigenVectors[i][j] = tempMatrix.get(i, j); } } } catch (NotConvergedException e) { e.printStackTrace(); logger.error(e); } } return eigenVectors; } // 假设阈值是90%，选取最大的前几个特征值的序号 private int[] selectPrincipalComponent(double[] eigenValues) { int p = eigenValues.length; // 特征值个数 int[] sortedEigenValueIndex = new int[p]; // 特征值由大到小，序号排序 for (int i = 0; i < p; i++) { sortedEigenValueIndex[i] = i; } double[] temp = new double[p]; // 特征值副本 System.arraycopy(eigenValues, 0, temp, 0, p); // TODO 排序过程可以再优化 double tmp = 0.0; int pos = 0; for (int i = 1; i <= p; i++) { for (int j = 0; j < p - i - 1; j++) { if (temp[j] < temp[j + 1]) { tmp = temp[j]; temp[j] = temp[j + 1]; temp[j + 1] = tmp; pos = sortedEigenValueIndex[j]; sortedEigenValueIndex[j] = sortedEigenValueIndex[j + 1]; sortedEigenValueIndex[j + 1] = pos; } } } double total = 0.0; // 特征值的和 for (int i = 0; i < p; i++) { total += temp[i]; } int count = 0; double sum = 0.0; for (int i = 0; i < p; i++) { if (sum / total <= proportion) { sum += temp[i]; count++; } } int[] selected = new int[count]; System.arraycopy(sortedEigenValueIndex, 0, selected, 0, count); return selected; } // 取得主特征值 private double[] calcPrincipalEigenValues(double[] eigenValues, int[] selected) { int p = eigenValues.length; double[] principalEigenValues = new double[selected.length]; for (int i = 0; i < selected.length; i++) { principalEigenValues[i] = eigenValues[selected[i]]; } return principalEigenValues; } // 取得主特征向量，即变换矩阵 private double[][] calcPrincipalEigenVectors(double[][] eigenVectors, int[] selected) { int p = eigenVectors.length; double[][] principalEigenVectors = new double[p][selected.length]; for (int i = 0; i < selected.length; i++) { for (int j = 0; j < p; j++) { principalEigenVectors[j][i] = eigenVectors[j][selected[i]]; } } return principalEigenVectors; } // 原始数据的主成分数据 private double[][] calcPrincipalComponent(double[][] rawData) { Matrix A = new DenseMatrix(rawData); Matrix B = new DenseMatrix(principalEigenVectors); Matrix C = new DenseMatrix(rawData.length, principalEigenVectors[0].length); A.mult(B, C); // C=A*B double[][] principalData = new double[C.numRows()][C.numColumns()]; for (int i = 0; i < C.numRows(); i++) { for (int j = 0; j < C.numColumns(); j++) { principalData[i][j] = C.get(i, j); } } return principalData; } /**************** PCA的主流程 ****************/ public void buildPCA(double[][] rawData) { this.rawData = rawData; double[][] standardData = calcStandardlizer(rawData); double[][] assosiationMatrix = calcCoefficientOfAssociation(standardData); this.eigenValues = calcEigenValue(assosiationMatrix); this.eigenVectors = calcEigenVector(assosiationMatrix); this.selected = selectPrincipalComponent(eigenValues); this.principalEigenValues = calcPrincipalEigenValues(eigenValues, selected); this.principalEigenVectors = calcPrincipalEigenVectors(eigenVectors, selected); this.principalData = calcPrincipalComponent(rawData); }}PCA的原理和过程不难理解
1. 先计算原始数据的相关系数矩阵
2. 求相关系数矩阵的特征值和特征向量
3. 选择特征值最大的n个特征值对应的特征向量作为转换矩阵
4. 原始数据矩阵和转换矩阵相乘，得到结果
本质上，PCA是一个坐标转换。将原来不好的坐标系转换为好的坐标系。将每条样本的原始属性变为新属性。
第一，各个属性之间独立、不相关
第二，使得各个属性方差较大，数据具有区分度
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