最小二乘法是一种常用的回归分析方法，用于估计一组数据的线性关系。在Java中，我们可以通过计算矩阵和向量，求解最小二乘法的系数，以达到预测新数据的目的。

下面就是一个简单的Java实现最小二乘法二阶的案例，实现该算法需要用到矩阵运算的基本知识，包括矩阵的转置、乘积、逆矩阵等操作。

代码实现：

import java.util.*;
public class LeastSquares {
  // 计算矩阵的转置
  public double[][] transpose(double[][] A) {
    int m = A.length, n = A[0].length;
    double[][] B = new double[n][m];
    for (int i = 0; i < m; i++)
      for (int j = 0; j < n; j++)
        B[j][i] = A[i][j];
    return B;
  }
  // 计算矩阵的乘积
  public double[][] multiply(double[][] A, double[][] B) {
    int m = A.length, n = B[0].length;
    double[][] C = new double[m][n];
    for (int i = 0; i < m; i++)
      for (int j = 0; j < n; j++)
        for (int k = 0; k < B.length; k++)
          C[i][j] += A[i][k] * B[k][j];
    return C;
  }
  // 计算矩阵的逆
  public double[][] inverse(double[][] A) {
    int n = A.length;
    double[][] B = new double[n][n];
    double[][] C = new double[n][n * 2];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < n; j++) {
        B[i][j] = A[i][j];
        if (i == j) C[i][j + n] = 1;
      }
    }
    for (int k = 0; k < n; k++) {
      double max = 0; int index = 0;
      for (int i = k; i < n; i++) {
        if (Math.abs(B[i][k]) > max) {
          max = Math.abs(B[i][k]);
          index = i;
        }
      }
      if (index != k) {
        for (int j = 0; j < n; j++) {
          double temp = B[k][j];
          B[k][j] = B[index][j];
          B[index][j] = temp;
          temp = C[k][j + n];
          C[k][j + n] = C[index][j + n];
          C[index][j + n] = temp;
        }
      }
      for (int j = k; j < n; j++) B[k][j] /= max;
      for (int j = 0; j < n; j++) C[k][j + n] /= max;
      for (int i = k + 1; i < n; i++) {
        double factor = B[i][k];
        for (int j = k + 1; j < n; j++) B[i][j] -= factor * B[k][j];
        for (int j = 0; j < n; j++) C[i][j + n] -= factor * C[k][j + n];
      }
    }
    for (int k = n - 1; k >= 0; k--) {
      for (int i = k - 1; i >= 0; i--) {
        double factor = B[i][k];
        for (int j = 0; j < n; j++) C[i][j + n] -= factor * C[k][j + n];
      }
    }
    double[][] D = new double[n][n];
    for (int i = 0; i < n; i++)
      for (int j = 0; j < n; j++)
        D[i][j] = C[i][j + n];
    return D;
  }
  // 计算最小二乘法的系数
  public double[] leastSquares(double[][] X, double[] y) {
    double[][] X1 = new double[X.length][3];
    for (int i = 0; i < X.length; i++) {
      X1[i][0] = 1;
      X1[i][1] = X[i][0];
      X1[i][2] = X[i][0] * X[i][0];
    }
    double[] y1 = Arrays.copyOf(y, y.length);
    double[][] X2 = transpose(X1);
    double[][] X3 = multiply(X2, X1);
    double[][] X4 = inverse(X3);
    double[][] X5 = multiply(X4, X2);
    double[] beta = multiply(X5, y1);
    return beta;
  }
  // 测试
  public static void main(String[] args) {
    double[][] X = {{1}, {2}, {3}, {4}, {5}};
    double[] y = {6, 9, 12, 15, 20};
    LeastSquares ls = new LeastSquares();
    double[] beta = ls.leastSquares(X, y);
    System.out.println("y = " + beta[0] + " + " + beta[1] + "x + " + beta[2] + "x^2");
  }
}

上述代码实现了最小二乘法二阶的算法，它包含了计算矩阵的转置、乘积、逆矩阵等基本操作，以及计算最小二乘法系数的方法leastSquares。

在测试中，我们输入了一个包含五个数据点的二次函数的点值，通过最小二乘法求得该二次函数的系数，最终输出该二次函数的表达式。

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