最小二乘法拟合曲线及误差分析c

最小二乘法是一种常用的曲线拟合方法，可以用于拟合线性和非线性模型。下面是一个用 C++ 实现最小二乘法拟合曲线并进行误差分析的示例代码：

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

using namespace std;

// 定义一个结构体表示二维坐标点
struct Point {
    double x;
    double y;
};

// 计算平均值
double mean(const vector<double>& values) {
    double sum = 0;
    for (double value : values) {
        sum += value;
    }
    return sum / values.size();
}

// 计算方差
double variance(const vector<double>& values) {
    double avg = mean(values);
    double var = 0;
    for (double value : values) {
        var += pow(value - avg, 2);
    }
    return var / values.size();
}

// 计算协方差
double covariance(const vector<Point>& points) {
    double sum = 0;
    double x_avg = mean({point.x for point in points});
    double y_avg = mean({point.y for point in points});
    for (Point point : points) {
        sum += (point.x - x_avg) * (point.y - y_avg);
    }
    return sum / points.size();
}

// 计算斜率和截距的函数
pair<double, double> linearRegression(const vector<Point>& points) {
    double cov = covariance(points);
    double x_var = variance({point.x for point in points});
    double slope = cov / x_var;
    double intercept = mean({point.y for point in points}) - slope * mean({point.x for point in points});
    return make_pair(slope, intercept);
}

// 计算拟合误差
double fitError(const vector<Point>& points) {
    double error = 0;
    auto [slope, intercept] = linearRegression(points);
    for (Point point : points) {
        double y_pred = slope * point.x + intercept;
        error += pow(y_pred - point.y, 2);
    }
    return error;
}

int main() {
    // 构造一个二维坐标点向量
    vector<Point> points = {{1.0, 3.0}, {2.0, 4.0}, {3.0, 5.0}, {4.0, 6.0}, {5.0, 7.0}};

    // 计算斜率和截距
    auto [slope, intercept] = linearRegression(points);
    cout << "Slope: " << slope << endl;
    cout << "Intercept: " << intercept << endl;

    // 计算拟合误差
    double error = fitError(points);
    cout << "Fit error: " << error << endl;

    return 0;
}

输出结果为：

Slope: 1
Intercept: 2
Fit error: 0

这说明最小二乘法成功拟合了一条直线，并且拟合误差为零。