用C++编写代码对地址"C:\Users\86139\Desktop\主成分作业 数据.csv"中的数据进行简单的主成分分析，避免使用第三方库

实现主成分分析的一般步骤如下：

1. 读取数据集
2. 对数据集进行中心化处理
3. 计算协方差矩阵
4. 对协方差矩阵进行特征值分解
5. 选择前k个最大的特征值及其对应的特征向量
6. 得到降维后的数据集

下面是一个简单的 C++ 程序实现：

#include <iostream>
#include <fstream>
#include <vector>
#include <cmath>

using namespace std;

// 读取数据集
vector<vector<double>> read_csv(string filename) {
    vector<vector<double>> data;
    ifstream file(filename);
    string line;
    while (getline(file, line)) {
        vector<double> row;
        size_t pos = 0;
        string token;
        while ((pos = line.find(",")) != string::npos) {
            token = line.substr(0, pos);
            row.push_back(stod(token));
            line.erase(0, pos + 1);
        }
        row.push_back(stod(line));
        data.push_back(row);
    }
    return data;
}

// 计算平均值
vector<double> mean(vector<vector<double>>& data) {
    vector<double> avg(data[0].size(), 0);
    int n = data.size();
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < data[0].size(); j++) {
            avg[j] += data[i][j];
        }
    }
    for (int j = 0; j < data[0].size(); j++) {
        avg[j] /= n;
    }
    return avg;
}

// 中心化处理
void center(vector<vector<double>>& data, vector<double>& avg) {
    int n = data.size();
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < data[0].size(); j++) {
            data[i][j] -= avg[j];
        }
    }
}

// 计算协方差矩阵
vector<vector<double>> covariance_matrix(vector<vector<double>>& data) {
    int n = data.size();
    int m = data[0].size();
    vector<vector<double>> cov(m, vector<double>(m, 0));
    for (int i = 0; i < m; i++) {
        for (int j = i; j < m; j++) {
            double sum = 0;
            for (int k = 0; k < n; k++) {
                sum += data[k][i] * data[k][j];
            }
            cov[i][j] = cov[j][i] = sum / (n - 1);
        }
    }
    return cov;
}

// 特征值分解
void eig(vector<vector<double>>& a, vector<double>& w, vector<vector<double>>& v) {
    int n = a.size();
    int max_iter = 1000;
    double eps = 1e-8;
    w.resize(n);
    v.resize(n, vector<double>(n, 0));
    for (int i = 0; i < n; i++) {
        v[i][i] = 1;
    }
    for (int iter = 0; iter < max_iter; iter++) {
        double s = 0;
        for (int i = 0; i < n; i++) {
            for (int j = i + 1; j < n; j++) {
                s += fabs(a[i][j]);
            }
        }
        if (s == 0) {
            break;
        }
        double t = 0.0;
        if (iter < 3) {
            t = s / (n * n);
        } else {
            double sum = 0;
            for (int i = 0; i < n; i++) {
                for (int j = 0; j < n; j++) {
                    sum += fabs(a[i][j]);
                }
            }
            t = eps * sum / n / n;
        }
        for (int i = 0; i < n; i++) {
            for (int j = i + 1; j < n; j++) {
                double g = 100 * fabs(a[i][j]);
                if (iter > 3 && fabs(w[i]) + g == fabs(w[i])
                    && fabs(w[j]) + g == fabs(w[j])) {
                    a[i][j] = 0;
                } else if (fabs(a[i][j]) > t) {
                    double h = w[j] - w[i];
                    double u = 0;
                    if (fabs(h) + g == fabs(h)) {
                        u = a[i][j] / h;
                    } else {
                        double theta = 0.5 * h / a[i][j];
                        u = 1 / (fabs(theta) + sqrt(1 + theta * theta));
                        if (theta < 0) {
                            u = -u;
                        }
                    }
                    double c = 1 / sqrt(1 + u * u);
                    double s = u * c;
                    double tau = s / (1 + c);
                    h = u * a[i][j];
                    w[i] -= h;
                    w[j] += h;
                    a[i][j] = 0;
                    for (int k = 0; k < i; k++) {
                        double x = a[k][i];
                        double y = a[k][j];
                        a[k][i] = x - s * (y + x * tau);
                        a[k][j] = y + s * (x - y * tau);
                    }
                    for (int k = i + 1; k < j; k++) {
                        double x = a[i][k];
                        double y = a[k][j];
                        a[i][k] = x - s * (y + x * tau);
                        a[k][j] = y + s * (x - y * tau);
                    }
                    for (int k = j + 1; k < n; k++) {
                        double x = a[i][k];
                        double y = a[j][k];
                        a[i][k] = x - s * (y + x * tau);
                        a[j][k] = y + s * (x - y * tau);
                    }
                    for (int k = 0; k < n; k++) {
                        double x = v[k][i];
                        double y = v[k][j];
                        v[k][i] = x - s * (y + x * tau);
                        v[k][j] = y + s * (x - y * tau);
                    }
                }
            }
        }
    }
}

// 主成分分析
vector<vector<double>> pca(vector<vector<double>>& data, int k) {
    vector<double> avg = mean(data);
    center(data, avg);
    vector<vector<double>> cov = covariance_matrix(data);
    vector<double> w;
    vector<vector<double>> v;
    eig(cov, w, v);
    vector<vector<double>> res(data.size(), vector<double>(k, 0));
    for (int i = 0; i < data.size(); i++) {
        for (int j = 0; j < k; j++) {
            for (int l = 0; l < data[0].size(); l++) {
                res[i][j] += data[i][l] * v[l][j];
            }
        }
    }
    return res;
}

int main() {
    vector<vector<double>> data = read_csv("C:\\Users\\86139\\Desktop\\主成分作业 数据.csv");
    vector<vector<double>> pca_data = pca(data, 2);
    for (auto& row : pca_data) {
        for (auto& col : row) {
            cout << col << " ";
        }
        cout << endl;
    }
    return 0;
}

在上述代码中，`read_csv()` 函数用于从 CSV 文件中读取数据集，`mean()` 函数用于计算平均值，`center()` 函数用于对数据集进行中心化处理，`covariance_matrix()` 函数用于计算协方差矩阵，`eig()` 函数用于进行特征值分解，`pca()` 函数用于进行主成分分析。在 `main()` 函数中，我们读取 CSV 文件并进行主成分分析，得到降维后的数据集并输出。