LinearRegression_GD.py

import numpy as np
import matplotlib.pyplot as plt


class LinearRegression_GradDesc:
    """
    线性回归，梯度下降法求解模型系数
    1、数据的预处理：是否训练偏置项fit_intercept（默认True），是否标准化normalized（默认True）
    2、模型的训练：闭式解公式，fit(self, x_train, y_train)
    3、模型的预测，predict(self, x_test)
    4、均方误差，判决系数
    5、模型预测可视化
    """

    def __init__(self, fit_intercept=True, normalize=True, alpha=0.05, max_epochs=300, batch_size=20):
        """
        :param fit_intercept: 是否训练偏置项
        :param normalize: 是否标准化
        :param alpha: 学习率
        :param max_epochs: 最大迭代次数
        :param batch_size: 批量大小，若为1，则为随机梯度，若为训练集样本量，则为批量梯度，否则为小批量梯度
        """
        self.fit_intercept = fit_intercept  # 线性模型的常数项。也即偏置bias，模型中的theta0
        self.normalize = normalize  # 是否标准化数据
        self.alpha = alpha  # 学习率
        self.max_epochs = max_epochs
        self.batch_size = batch_size
        self.theta = None  # 训练权重系数
        if normalize:
            self.feature_mean, self.feature_std = None, None  # 特征的均值，标准方差
        self.mse = np.infty  # 训练样本的均方误差
        self.r2, self.r2_adj = 0.0, 0.0  # 判定系数和修正判定系数
        self.n_samples, self.n_features = 0, 0  # 样本量和特征数
        self.train_loss, self.test_loss = [], []  # 存储训练过程中的训练损失和测试损失

    def init_params(self, n_features):
        """
        初始化参数
        如果训练偏置项，也包含了bias的初始化
        :return:
        """
        self.theta = np.random.randn(n_features, 1) * 0.1

    def fit(self, x_train, y_train, x_test=None, y_test=None):
        """
        模型训练，根据是否标准化与是否拟合偏置项分类讨论
        :param x_train: 训练样本集
        :param y_train: 训练目标集
        :param x_test: 测试样本集
        :param y_test: 测试目标集
        :return:
        """
        if self.normalize:
            self.feature_mean = np.mean(x_train, axis=0)  # 样本均值
            self.feature_std = np.std(x_train, axis=0) + 1e-8  # 样本方差
            x_train = (x_train - self.feature_mean) / self.feature_std  # 标准化
            if x_test is not None:
                x_test = (x_test - self.feature_mean) / self.feature_std  # 标准化
        if self.fit_intercept:
            x_train = np.c_[x_train, np.ones_like(y_train)]  # 添加一列1，即偏置项样本
            if x_test is not None and y_test is not None:
                x_test = np.c_[x_test, np.ones_like(y_test)]  # 添加一列1，即偏置项样本
        self.init_params(x_train.shape[1])  # 初始化参数
        self._fit_gradient_desc(x_train, y_train, x_test, y_test)  # 梯度下降法训练模型

    def _fit_gradient_desc(self, x_train, y_train, x_test=None, y_test=None):
        """
        三种梯度下降求解：
        （1）如果batch_size为1，则为随机梯度下降法
        （2）如果batch_size为样本量，则为批量梯度下降法
        （3）如果batch_size小于样本量，则为小批量梯度下降法
        :return:
        """
        train_sample = np.c_[x_train, y_train]  # 组合训练集和目标集，以便随机打乱样本
        # np.c_水平方向连接数组，np.r_竖直方向连接数组
        # 按batch_size更新theta，三种梯度下降法取决于batch_size的大小
        best_theta, best_mse = None, np.infty  # 最佳训练权重与验证均方误差
        for i in range(self.max_epochs):
            self.alpha *= 0.95
            np.random.shuffle(train_sample)  # 打乱样本顺序，模拟随机化
            batch_nums = train_sample.shape[0] // self.batch_size  # 批次
            for idx in range(batch_nums):
                # 取小批量样本，可以是随机梯度（1），批量梯度（n）或者是小批量梯度（<n）
                batch_xy = train_sample[self.batch_size * idx: self.batch_size * (idx + 1)]
                # 分取训练样本和目标样本，并保持维度
                batch_x, batch_y = batch_xy[:, :-1], batch_xy[:, -1:]
                # 计算权重更新增量
                delta = batch_x.T.dot(batch_x.dot(self.theta) - batch_y) / self.batch_size
                self.theta = self.theta - self.alpha * delta
            train_mse = ((x_train.dot(self.theta) - y_train.reshape(-1, 1)) ** 2).mean()
            self.train_loss.append(train_mse)
            if x_test is not None and y_test is not None:
                test_mse = ((x_test.dot(self.theta) - y_test.reshape(-1, 1)) ** 2).mean()
                self.test_loss.append(test_mse)

    def get_params(self):
        """
        返回线性模型训练的系数
        :return:
        """
        if self.fit_intercept:  # 存在偏置项
            weight, bias = self.theta[:-1], self.theta[-1]
        else:
            weight, bias = self.theta, np.array([0])
        if self.normalize:  # 标准化后的系数
            weight = weight / self.feature_std.reshape(-1, 1)  # 还原模型系数
            bias = bias - weight.T.dot(self.feature_mean)
        return weight.reshape(-1), bias

    def predict(self, x_test):
        """
        测试数据预测
        :param x_test: 待预测样本集，不包括偏置项
        :return:
        """
        try:
            self.n_samples, self.n_features = x_test.shape[0], x_test.shape[1]
        except IndexError:
            self.n_samples, self.n_features = x_test.shape[0], 1  # 测试样本数和特征数
        if self.normalize:
            x_test = (x_test - self.feature_mean) / self.feature_std  # 测试数据标准化
        if self.fit_intercept:
            # 存在偏置项，加一列1
            x_test = np.c_[x_test, np.ones(shape=x_test.shape[0])]
        y_pred = x_test.dot(self.theta).reshape(-1, 1)
        return y_pred

    def cal_mse_r2(self, y_test, y_pred):
        """
        计算均方误差，计算拟合优度的判定系数R方和修正判定系数
        :param y_pred: 模型预测目标真值
        :param y_test: 测试目标真值
        :return:
        """
        self.mse = ((y_test.reshape(-1, 1) - y_pred.reshape(-1, 1)) ** 2).mean()  # 均方误差
        # 计算测试样本的判定系数和修正判定系数
        self.r2 = 1 - ((y_test.reshape(-1, 1) - y_pred.reshape(-1, 1)) ** 2).sum() / \
                  ((y_test.reshape(-1, 1) - y_test.mean()) ** 2).sum()
        self.r2_adj = 1 - (1 - self.r2) * (self.n_samples - 1) / \
                      (self.n_samples - self.n_features - 1)
        return self.mse, self.r2, self.r2_adj

    def plt_predict(self, y_test, y_pred, is_show=True, is_sort=True):
        """
        绘制预测值与真实值对比图
        :return:
        """
        if self.mse is np.infty:
            self.cal_mse_r2(y_pred, y_test)
        if is_show:
            plt.figure(figsize=(8, 6))
        if is_sort:
            idx = np.argsort(y_test)  # 升序排列，获得排序后的索引
            plt.plot(y_test[idx], "k--", lw=1.5, label="Test True Val")
            plt.plot(y_pred[idx], "r:", lw=1.8, label="Predictive Val")
        else:
            plt.plot(y_test, "ko-", lw=1.5, label="Test True Val")
            plt.plot(y_pred, "r*-", lw=1.8, label="Predictive Val")
        plt.xlabel("Test sample observation serial number", fontdict={"fontsize": 12})
        plt.ylabel("Predicted sample value", fontdict={"fontsize": 12})
        plt.title("The predictive values of test samples \n MSE = %.5e, R2 = %.5f, R2_adj = %.5f"
                  % (self.mse, self.r2, self.r2_adj), fontdict={"fontsize": 14})
        plt.legend(frameon=False)
        plt.grid(ls=":")
        if is_show:
            plt.show()

    def plt_loss_curve(self, is_show=True):
        """
        可视化均方损失下降曲线
        :param is_show: 是否可视化
        :return:
        """
        if is_show:
            plt.figure(figsize=(8, 6))
        plt.plot(self.train_loss, "k-", lw=1, label="Train Loss")
        if self.test_loss:
            plt.plot(self.test_loss, "r--", lw=1.2, label="Test Loss")
        plt.xlabel("Epochs", fontdict={"fontsize": 12})
        plt.ylabel("Loss values", fontdict={"fontsize": 12})
        plt.title("Gradient Descent Method and Test Loss MSE = %.5f"
                  % (self.test_loss[-1]), fontdict={"fontsize": 14})
        plt.legend(frameon=False)
        plt.grid(ls=":")
        # plt.axis([0, 300, 20, 30])
        if is_show:
            plt.show()

test_linear_regression_gd.py

import numpy as np
from LinearRegression_GD import LinearRegression_GradDesc
from sklearn.model_selection import train_test_split


np.random.seed(42)
X = np.random.rand(1000, 6)  # 随机样本值，6个特征
coeff = np.array([4.2, -2.5, 7.8, 3.7, -2.9, 1.87])  # 模型参数
y = coeff.dot(X.T) + 0.5 * np.random.randn(1000)  # 目标函数值

X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state=0, shuffle=True)

lr_gd = LinearRegression_GradDesc(alpha=0.1, batch_size=1)
lr_gd.fit(X_train, y_train, X_test, y_test)
theta = lr_gd.get_params()
print(theta)
y_test_pred = lr_gd.predict(X_test)
lr_gd.plt_predict(y_test, y_test_pred)
lr_gd.plt_loss_curve()