ZhiQiXiaoYuan/utils/gpaPredictor.js

/**
 * GPA预测算法 - 多项式回归
 */

/**
 * 多项式回归预测
 * @param {Array} data - 历史GPA数据 [{semester: '2023-1', gpa: 3.5}, ...]
 * @param {Number} degree - 多项式阶数，默认2(二次)
 * @param {Number} future - 预测未来几个学期
 * @returns {Object} 预测结果
 */
function polynomialRegression(data, degree = 2, future = 3) {
  if (!data || data.length < 2) {
    return {
      predictions: [],
      trend: 0,
      confidence: 0
    };
  }

  // 提取数据点
  const x = data.map((_, index) => index); // 学期编号 [0, 1, 2, ...]
  const y = data.map(item => item.gpa); // GPA值

  // 构建范德蒙德矩阵并求解系数
  const coefficients = fitPolynomial(x, y, degree);

  // 预测未来学期
  const predictions = [];
  const startIndex = data.length;
  
  for (let i = 0; i < future; i++) {
    const xValue = startIndex + i;
    const yValue = evaluatePolynomial(coefficients, xValue);
    
    // 限制GPA在合理范围内 [0, 4.0]
    const predictedGPA = Math.max(0, Math.min(4.0, yValue));
    
    predictions.push({
      semester: generateSemesterName(data.length + i + 1),
      gpa: Number(predictedGPA.toFixed(2)),
      isPrediction: true
    });
  }

  // 计算趋势（与当前学期对比）
  const currentGPA = y[y.length - 1];
  const nextGPA = predictions[0].gpa;
  const trend = ((nextGPA - currentGPA) / currentGPA * 100).toFixed(1);

  // 计算置信度（基于R²）
  const rSquared = calculateRSquared(x, y, coefficients);
  const confidence = Math.round(rSquared * 100);

  return {
    predictions,
    trend: Number(trend),
    confidence,
    coefficients
  };
}

/**
 * 拟合多项式 - 使用最小二乘法
 */
function fitPolynomial(x, y, degree) {
  const n = x.length;
  const m = degree + 1;

  // 构建设计矩阵 X 和目标向量 Y
  const X = [];
  for (let i = 0; i < n; i++) {
    const row = [];
    for (let j = 0; j <= degree; j++) {
      row.push(Math.pow(x[i], j));
    }
    X.push(row);
  }

  // 计算 X^T * X
  const XTX = multiplyMatrices(transpose(X), X);
  
  // 计算 X^T * Y
  const XTY = multiplyMatrixVector(transpose(X), y);

  // 求解方程 (X^T * X) * coefficients = X^T * Y
  const coefficients = solveLinearSystem(XTX, XTY);

  return coefficients;
}

/**
 * 计算多项式值
 */
function evaluatePolynomial(coefficients, x) {
  let result = 0;
  for (let i = 0; i < coefficients.length; i++) {
    result += coefficients[i] * Math.pow(x, i);
  }
  return result;
}

/**
 * 计算R²（决定系数）
 */
function calculateRSquared(x, y, coefficients) {
  const n = x.length;
  const yMean = y.reduce((sum, val) => sum + val, 0) / n;

  let ssTotal = 0;
  let ssResidual = 0;

  for (let i = 0; i < n; i++) {
    const yPred = evaluatePolynomial(coefficients, x[i]);
    ssTotal += Math.pow(y[i] - yMean, 2);
    ssResidual += Math.pow(y[i] - yPred, 2);
  }

  return 1 - (ssResidual / ssTotal);
}

/**
 * 矩阵转置
 */
function transpose(matrix) {
  const rows = matrix.length;
  const cols = matrix[0].length;
  const result = [];

  for (let j = 0; j < cols; j++) {
    const row = [];
    for (let i = 0; i < rows; i++) {
      row.push(matrix[i][j]);
    }
    result.push(row);
  }

  return result;
}

/**
 * 矩阵乘法
 */
function multiplyMatrices(a, b) {
  const rowsA = a.length;
  const colsA = a[0].length;
  const colsB = b[0].length;
  const result = [];

  for (let i = 0; i < rowsA; i++) {
    const row = [];
    for (let j = 0; j < colsB; j++) {
      let sum = 0;
      for (let k = 0; k < colsA; k++) {
        sum += a[i][k] * b[k][j];
      }
      row.push(sum);
    }
    result.push(row);
  }

  return result;
}

/**
 * 矩阵向量乘法
 */
function multiplyMatrixVector(matrix, vector) {
  const rows = matrix.length;
  const result = [];

  for (let i = 0; i < rows; i++) {
    let sum = 0;
    for (let j = 0; j < vector.length; j++) {
      sum += matrix[i][j] * vector[j];
    }
    result.push(sum);
  }

  return result;
}

/**
 * 求解线性方程组 - 高斯消元法
 */
function solveLinearSystem(A, b) {
  const n = A.length;
  const augmented = A.map((row, i) => [...row, b[i]]);

  // 前向消元
  for (let i = 0; i < n; i++) {
    // 选主元
    let maxRow = i;
    for (let k = i + 1; k < n; k++) {
      if (Math.abs(augmented[k][i]) > Math.abs(augmented[maxRow][i])) {
        maxRow = k;
      }
    }

    // 交换行
    [augmented[i], augmented[maxRow]] = [augmented[maxRow], augmented[i]];

    // 消元
    for (let k = i + 1; k < n; k++) {
      const factor = augmented[k][i] / augmented[i][i];
      for (let j = i; j <= n; j++) {
        augmented[k][j] -= factor * augmented[i][j];
      }
    }
  }

  // 回代
  const x = new Array(n);
  for (let i = n - 1; i >= 0; i--) {
    x[i] = augmented[i][n];
    for (let j = i + 1; j < n; j++) {
      x[i] -= augmented[i][j] * x[j];
    }
    x[i] /= augmented[i][i];
  }

  return x;
}

/**
 * 生成学期名称
 */
function generateSemesterName(index) {
  const year = 2023 + Math.floor(index / 2);
  const term = index % 2 === 1 ? '1' : '2';
  return `${year}-${term}`;
}

/**
 * 计算GPA（根据分数）
 * @param {Number} score - 分数
 * @returns {Number} GPA绩点
 */
function calculateGPA(score) {
  if (score >= 60) {
    return (score / 10 - 5);
  }
  return 0;
}

/**
 * 批量计算GPA
 */
function batchCalculateGPA(courses) {
  return courses.map(course => ({
    ...course,
    gpa: calculateGPA(course.score)
  }));
}

/**
 * 计算平均GPA
 */
function calculateAverageGPA(courses) {
  if (!courses || courses.length === 0) return 0;
  
  const totalCredits = courses.reduce((sum, c) => sum + (c.credit || 1), 0);
  const weightedSum = courses.reduce((sum, c) => {
    const gpa = c.gpa || calculateGPA(c.score);
    return sum + gpa * (c.credit || 1);
  }, 0);

  return (weightedSum / totalCredits).toFixed(2);
}

module.exports = {
  polynomialRegression,
  calculateGPA,
  batchCalculateGPA,
  calculateAverageGPA
};