多元多项式回归 C++带类详细正确完整实现及案例 变量形式为：vector<vector<double>> X vector<double> y

以下是一个基于类的多元多项式回归的实现，实现了最小二乘法（Ordinary Least Squares）来拟合模型。

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

class PolynomialRegression {
private:
    int degree; // 多项式的次数
    std::vector<std::vector<double>> X;
    std::vector<double> y;
    std::vector<double> coefficients; // 多项式系数

public:
    PolynomialRegression(int degree) {
        this->degree = degree;
    }

    // 添加样本
    void addSample(std::vector<double> x, double y) {
        X.push_back(x);
        this->y.push_back(y);
    }

    // 多项式回归
    void fit() {
        int n = X.size();
        int m = X[0].size();
        std::vector<std::vector<double>> X_T(m, std::vector<double>(n));
        std::vector<std::vector<double>> X_T_X(m, std::vector<double>(m));

        // 计算 X 的转置矩阵
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                X_T[j][i] = X[i][j];
            }
        }

        // 计算 X_T * X 矩阵
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < m; j++) {
                double sum = 0.0;
                for (int k = 0; k < n; k++) {
                    sum += X_T[i][k] * X[k][j];
                }
                X_T_X[i][j] = sum;
            }
        }

        // 计算 X_T * y 向量
        std::vector<double> X_T_y(m);
        for (int i = 0; i < m; i++) {
            double sum = 0.0;
            for (int j = 0; j < n; j++) {
                sum += X_T[i][j] * y[j];
            }
            X_T_y[i] = sum;
        }

        // 解方程 X_T * X * coefficients = X_T * y
        for (int i = 0; i < m; i++) {
            for (int j = i + 1; j < m; j++) {
                double factor = X_T_X[j][i] / X_T_X[i][i];
                for (int k = i; k < m; k++) {
                    X_T_X[j][k] -= factor * X_T_X[i][k];
                }
                X_T_y[j] -= factor * X_T_y[i];
            }
        }

        coefficients.resize(m);
        for (int i = m - 1; i >= 0; i--) {
            double sum = 0.0;
            for (int j = i + 1; j < m; j++) {
                sum += X_T_X[i][j] * coefficients[j];
            }
            coefficients[i] = (X_T_y[i] - sum) / X_T_X[i][i];
        }
    }

    // 预测
    double predict(std::vector<double> x) {
        double y_pred = 0.0;
        for (int i = 0; i <= degree; i++) {
            y_pred += coefficients[i] * std::pow(x[i], i);
        }
        return y_pred;
    }
};

int main() {
    // 创建多项式回归模型，指定多项式的次数
    PolynomialRegression model(2);

    // 添加样本
    std::vector<double> x1 = {1.0, 1.0};
    model.addSample(x1, 2.0);

    std::vector<double> x2 = {1.0, 2.0};
    model.addSample(x2, 3.0);

    std::vector<double> x3 = {1.0, 3.0};
    model.addSample(x3, 4.0);

    // 训练模型
    model.fit();

    // 预测
    std::vector<double> x_test = {1.0, 4.0};
    double y_pred = model.predict(x_test);
    std::cout << "Predicted value: " << y_pred << std::endl;

    return 0;
}

这个类实现了 `addSample` 方法来添加样本， `fit` 方法来训练模型， `predict` 方法来预测新的样本点。

在这个例子中，我们创建了一个二次多项式回归模型，并添加了三个样本点（x1, x2 和 x3），然后训练模型并预测了一个新的样本点（x_test）的结果。