线性回归算法原理

线性回归的基本思想是通过一条直线来拟合数据点，使得数据点到这条直线的距离平方和最小。其数学表达式为：

$$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\cdots +{\beta }_n{x}_n$$

其中，${\beta }_0$是偏置项（intercept），${\beta }_1,{\beta }_2,\cdots ,{\beta }_n$是各个特征的系数（coefficients）。

Java实现线性回归

以下是一个简单的Java实现，分为以下几个部分：

• 添加偏置项
• 计算系数
• 预测
• 矩阵运算

1. 添加偏置项

首先，我们需要在特征矩阵X中添加一列全为1的偏置项。

private double[][] addIntercept(double[][] X) {
    int nSamples = X.length;
    int nFeatures = X[0].length;
    double[][] X_with_intercept = new double[nSamples][nFeatures + 1];

    for (int i = 0; i < nSamples; i++) {
        X_with_intercept[i][0] = 1;  // intercept
        System.arraycopy(X[i], 0, X_with_intercept[i], 1, nFeatures);
    }

    return X_with_intercept;
}

2. 计算系数

接下来，我们使用最小二乘法来计算系数。通过矩阵运算，我们可以得到以下公式：

$$\beta =(X^{T}X{)}^{-1}{X}^{T}y$$

private double[] calculateCoefficients(double[][] X, double[] y) {
    int nFeatures = X[0].length;
    double[][] XtX = new double[nFeatures][nFeatures];
    double[] XtY = new double[nFeatures];

    for (int i = 0; i < X.length; i++) {
        for (int j = 0; j < nFeatures; j++) {
            for (int k = 0; k < nFeatures; k++) {
                XtX[j][k] += X[i][j] * X[i][k];
            }
            XtY[j] += X[i][j] * y[i];
        }
    }

    return solveLinearEquation(XtX, XtY);
}

3. 预测

根据计算出的系数，我们可以对新的数据进行预测：

public double[] predict(double[][] X) {
    if (coefficients == null) {
        throw new IllegalStateException("模型尚未训练，请先调用fit方法进行训练。");
    }

    double[][] X_with_intercept = addIntercept(X);
    double[] predictions = calculatePredictions(X_with_intercept);
    return predictions;
}

private double[] calculatePredictions(double[][] X) {
    double[] predictions = new double[X.length];
    for (int i = 0; i < X.length; i++) {
        for (int j = 0; j < coefficients.length; j++) {
            predictions[i] += X[i][j] * coefficients[j];
        }
    }
    return predictions;
}

4. 矩阵运算

我们使用Jama库来解决线性方程：

private double[] solveLinearEquation(double[][] A, double[] b) {
    Matrix matrixA = new Matrix(A);
    Matrix matrixB = new Matrix(b, b.length);
    Matrix solution = matrixA.solve(matrixB);
    double[] result = new double[solution.getRowDimension()];
    for (int i = 0; i < result.length; i++) {
        result[i] = solution.get(i, 0);
    }
    return result;
}

5. 完整代码

以下是完整的代码实现：

package cn.intana.business.sdk.utils;

import Jama.Matrix;

public class LinearRegression {
    private double[] coefficients;

    public void fit(double[][] X, double[] y) {
        double[][] X_with_intercept = addIntercept(X);
        coefficients = calculateCoefficients(X_with_intercept, y);
    }

    public double[] predict(double[][] X) {
        if (coefficients == null) {
            throw new IllegalStateException("模型尚未训练，请先调用fit方法进行训练。");
        }

        double[][] X_with_intercept = addIntercept(X);
        double[] predictions = calculatePredictions(X_with_intercept);
        return predictions;
    }

    private double[][] addIntercept(double[][] X) {
        int nSamples = X.length;
        int nFeatures = X[0].length;
        double[][] X_with_intercept = new double[nSamples][nFeatures + 1];

        for (int i = 0; i < nSamples; i++) {
            X_with_intercept[i][0] = 1;
            System.arraycopy(X[i], 0, X_with_intercept[i], 1, nFeatures);
        }

        return X_with_intercept;
    }

    private double[] calculateCoefficients(double[][] X, double[] y) {
        int nFeatures = X[0].length;
        double[][] XtX = new double[nFeatures][nFeatures];
        double[] XtY = new double[nFeatures];

        for (int i = 0; i < X.length; i++) {
            for (int j = 0; j < nFeatures; j++) {
                for (int k = 0; k < nFeatures; k++) {
                    XtX[j][k] += X[i][j] * X[i][k];
                }
                XtY[j] += X[i][j] * y[i];
            }
        }

        return solveLinearEquation(XtX, XtY);
    }

    private double[] solveLinearEquation(double[][] A, double[] b) {
        Matrix matrixA = new Matrix(A);
        Matrix matrixB = new Matrix(b, b.length);
        Matrix solution = matrixA.solve(matrixB);
        double[] result = new double[solution.getRowDimension()];
        for (int i = 0; i < result.length; i++) {
            result[i] = solution.get(i, 0);
        }
        return result;
    }

    private double[] calculatePredictions(double[][] X) {
        double[] predictions = new double[X.length];
        for (int i = 0; i < X.length; i++) {
            for (int j = 0; j < coefficients.length; j++) {
                predictions[i] += X[i][j] * coefficients[j];
            }
        }
        return predictions;
    }
}

pom

<dependency>
            <groupId>gov.nist.math</groupId>
            <artifactId>jama</artifactId>
            <version>1.0.3</version>
</dependency>