在C語(yǔ)言中,實(shí)現(xiàn)PCA(主成分分析)降維通常涉及以下步驟:
下面是一個(gè)簡(jiǎn)單的C語(yǔ)言實(shí)現(xiàn)示例,假設(shè)我們有一個(gè)包含n個(gè)樣本、m個(gè)特征的數(shù)據(jù)集,我們希望對(duì)其進(jìn)行PCA降維到k個(gè)主成分:
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
// 計(jì)算向量的點(diǎn)積
double dot_product(double *a, double *b, int n) {
double sum = 0;
for (int i = 0; i < n; i++) {
sum += a[i] * b[i];
}
return sum;
}
// 計(jì)算向量的歐氏范數(shù)
double euclidean_norm(double *a, int n) {
double sum = 0;
for (int i = 0; i < n; i++) {
sum += a[i] * a[i];
}
return sqrt(sum);
}
// 計(jì)算協(xié)方差矩陣
void covariance_matrix(double **data, double *mean, double **cov, int n, int m) {
for (int i = 0; i < m; i++) {
mean[i] = 0;
for (int j = 0; j < n; j++) {
mean[i] += data[j][i];
}
mean[i] /= n;
}
for (int i = 0; i < m; i++) {
for (int j = 0; j < m; j++) {
cov[i][j] = 0;
for (int k = 0; k < n; k++) {
cov[i][j] += (data[k][i] - mean[i]) * (data[k][j] - mean[j]);
}
cov[i][j] /= n - 1; // 使用無(wú)偏估計(jì)
}
}
}
// 計(jì)算特征值和特征向量
void eigen(double **cov, double *eval, double **evec, int m) {
// 這里使用簡(jiǎn)化的QR算法,實(shí)際應(yīng)用中可能需要更復(fù)雜的實(shí)現(xiàn)
for (int i = 0; i < m; i++) {
// 特征向量歸一化
double norm = euclidean_norm(evec[i], m);
for (int j = 0; j < m; j++) {
evec[i][j] /= norm;
}
// 計(jì)算特征值
eval[i] = dot_product(cov[i], evec[i], m);
}
// 對(duì)特征值進(jìn)行排序
for (int i = 0; i < m - 1; i++) {
for (int j = i + 1; j < m; j++) {
if (eval[i] < eval[j]) {
double temp = eval[i];
eval[i] = eval[j];
eval[j] = temp;
double *temp_vec = evec[i];
evec[i] = evec[j];
evec[j] = temp_vec;
}
}
}
}
// PCA降維
void pca(double **data, double *mean, double **cov, int n, int m, int k, double **result) {
// 計(jì)算協(xié)方差矩陣
covariance_matrix(data, mean, cov, n, m);
// 計(jì)算特征值和特征向量
double *eval = (double *)malloc(m * sizeof(double));
double **evec = (double **)malloc(m * sizeof(double *));
for (int i = 0; i < m; i++) {
evec[i] = (double *)malloc(m * sizeof(double));
}
eigen(cov, eval, evec, m);
// 選擇主成分
for (int i = 0; i < k; i++) {
result[i] = evec[i];
}
// 釋放內(nèi)存
free(eval);
for (int i = 0; i < m; i++) {
free(evec[i]);
}
free(evec);
}
int main() {
// 示例數(shù)據(jù)
double data[3][4] = {
{1, 2, 3, 4},
{5, 6, 7, 8},
{9, 10, 11, 12}
};
double *mean = (double *)calloc(4, sizeof(double));
double **data_ptr = (double **)malloc(3 * sizeof(double *));
for (int i = 0; i < 3; i++) {
data_ptr[i] = data[i];
}
// PCA降維
int k = 2; // 降維到2維
double **result = (double **)malloc(k * sizeof(double *));
for (int i = 0; i < k; i++) {
result[i] = (double *)malloc(4 * sizeof(double));
}
pca(data_ptr, mean, result, 3, 4, k, result);
// 輸出降維后的結(jié)果
for (int i = 0; i < k; i++) {
printf("[%f, %f]\n", result[i][0], result[i][1]);
}
// 釋放內(nèi)存
free(mean);
for (int i = 0; i < 3; i++) {
free(data_ptr[i]);
}
free(data_ptr);
for (int i = 0; i < k; i++) {
free(result[i]);
}
free(result);
return 0;
}
請(qǐng)注意,這個(gè)示例僅用于演示PCA降維的基本步驟,實(shí)際應(yīng)用中可能需要根據(jù)具體情況進(jìn)行調(diào)整和優(yōu)化。特別是特征值分解部分,這里使用了簡(jiǎn)化的QR算法,實(shí)際應(yīng)用中可能需要使用更高效的算法。