导入所需要的库 import numpy as npimport pandas as pdfrom matplotlib import font_manager as fm, rcParamsimport matplotlib.pyplot as pltfrom sklearn.model_selection import train_test_split # 用于分割数据集 将sklearn中的data数据转换成csv import pandas as pdfrom sklearn.datasets import load_bostonboston = load_boston()df = pd.DataFrame(boston.data, columns=boston.feature_names)df['MEDV'] = boston['target']df.to_csv('./boston.csv', index=None) # 存储为csv文件 读取csv文件 data = pd.read_csv(r'./boston.csv') 梯度缩放
提高梯度下降的收敛速度，使得data处于[0, 1]之间
data = (data - data.mean()) / data.std() # 特征放缩 (x-平均值)/标准差
公式如下：
将X和y从数据集中分离出来 cols = data.shape[1] # 表示data的列数X = data.iloc[:,0:cols-1]y = data.iloc[:,cols-1:cols] 数据集的分割 X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=10) 梯度下降 将数据转换成矩阵 # 将数据转换成numpy矩阵X_train = np.matrix(X_train.values)y_train = np.matrix(y_train.values)X_test = np.matrix(X_test.values)y_test = np.matrix(y_test.values) 初始化θ矩阵 theta = np.matrix([0,0,0,0,0,0,0,0,0,0,0,0,0,0]) 添加偏置列 # 添加偏置列，值为1，axis=1 添加列X_train = np.insert(X_train,0,1,axis=1)X_test = np.insert(X_test,0,1,axis=1) 构建代价函数 # 代价函数def CostFunction(X, y, theta): inner = np.power(X*theta.T-y,2) return np.sum(inner)/(2*len(X))
公式如下：
构建正则化代价函数 # 正则化代价函数def regularizedcost(X, y, theta, l): reg = (l/(2*len(X))) * (np.power(theta, 2).sum()) return CostFunction(X,y,theta) + reg
公式如下：
设置相关函数 alpha = 0.01 # 设置学习率epoch = 1000 # 设置迭代步数l = 50 # 正则化参数 构建梯度下降函数 # 梯度下降def GradientDescent(X, y, theta, l, alpha, epoch): temp = np.matrix(np.zeros(np.shape(theta))) # 定义临时矩阵存储theta parameters = int(theta.flatten().shape[1]) # 参数θ的数量 cost = np.zeros(epoch) # 初始化一个ndarray，包含每次epoch的cost m = X.shape[0] # 样本数量m for i in range(epoch): # 利用向量化一步求解 temp = theta - (alpha / m) * (X * theta.T - y).T * X - (alpha * l / m) * theta theta = temp cost[i] = regularizedcost(X, y, theta, l) # 记录每次迭代后的代价函数值 return theta, cost
θ的更新函数：
预测模型 # 运行梯度下降算法，并得出最终拟合的theta值 代价函数J(θ)final_theta, cost = GradientDescent(X_train, y_train, theta, l, alpha, epoch)print(final_theta) 模型评估 y_hat_train = X_train * final_theta.Ty_hat_test = X_test * final_theta.Tmse = np.sum(np.power(y_hat_test - y_test, 2)) / len(X_test)rmse = np.sqrt(mse)R2_train = 1 - np.sum(np.power(y_hat_train - y_train,2)) / np.sum(np.power(np.mean(y_train) - y_train,2))R2_test = 1 - np.sum(np.power(y_hat_test - y_test,2)) / np.sum(np.power(np.mean(y_test) - y_test,2))print('MSE = ',mse)print('RMSE = ',rmse)print('R2_train = ',R2_train)print('R2_test = ',R2_test)
Result:
绘制图形观察梯度下降的情况 fig, ax = plt.subplots(figsize=(20,10))ax.plot(np.arange(epoch), cost, 'r') # np.arange()返回等差数组ax.set_xlabel('Iterations')ax.set_ylabel('Cost')ax.set_title('The effect of Training Epoch')plt.show()
Result:
希望这篇文章对大家的学习有所帮助！