机器学习与数据挖掘实验二
（使用梯度下降法训练多元线性回归模型）
实验目的：
熟练掌握线性回归的基本原理以及梯度下降法和最小二乘法
实验环境：
Anaconda/Jupyter notebook/Pycharm
实验内容：
编码实现基于梯度下降的多变量线性回归算法，包括梯度的计算与验证；画数据散点图，以及得到的直线；画梯度下降过程中损失的变化图；基于训练得到的参数，输入新的样本数据，输出预测值；实验步骤：
import numpy as npimport matplotlib.pyplot as pltimport pandas as pdimport seaborn as snsdf_ads = np.loadtxt('./data2.txt',delimiter=',')
x = np.array(df_ads)y = np.array(df_ads[:,2])print("张量x的阶：",x.ndim)print("张量x的形状",x.shape)print(x)
y = y.reshape(-1,1)print(y.shape)
from sklearn import datasetsfrom mpl_toolkits.mplot3d import Axes3Dx1 = x[:,0]#自变量1x2 = x[:,1]#自变量2x3 = x[:,-1]#因变量fig = plt.figure()ax = Axes3D(fig)ax.scatter(x1,x2,x3)plt.show()
# 将数据继进行80%（训练集）和20%（验证集）的分割from sklearn.model_selection import train_test_splitx_train, x_test, y_train, y_test = train_test_split(x, y,test_size=0.2, random_state=0)
def scaler(train,test): # 定义归一化函数。进行数据压缩#数据的压缩min = train.min(axis=0) # 训练集最小值max = train.max(axis=0) # 训练集最大值gap = max - min # 最大值和最小值的差train -= min # 所有数据减最小值train /= gap # 所有数据除以大小值差test -= min #把训练集最小值应用于测试集test /= gap #把训练集大小值差应用于测试集return train, test # 返回压缩后的数据
def min_max_gap(train): # 计算训练集最大，最小值以及他们的差，用于后面反归一化过程min = train.min(axis=0) # 训练集最小值max = train.max(axis=0) # 训练集最大值gap = max - min # 最大值和最小值的差return min, max, gap y_min, y_max, y_gap = min_max_gap(y_train)
x_train_original = x_train.copy() # 保留一份训练集数据副本，用于对要预测数据归一化
x_train,x_test = scaler(x_train,x_test) # 对特征归一化y_train,y_test = scaler(y_train,y_test) # 对标签也归一化
x0_train = np.ones((len(x_train),1)) # 构造X_train长度的全1数组配合对Bias的点积x_train = np.append(x0_train, x_train, axis=1) #把X增加一系列的1x0_test = np.ones((len(x_test),1)) # 构造X_test长度的全1数组配合对Bias的点积x_test = np.append(x0_test, x_test, axis=1) #把X增加一系列的1print ("张量X的形状:", x_train.shape)print (x_train)
def loss_function(x, y, W): # 手工定义一个MSE均方误差函数,W此时是一个向量y_hat = x.dot(W.T) # 点积运算 h(x)=w_0*x_0 + w_1*x_1 + w_2*x_2 + w_3*x_3 print("x.shape",x.shape)print("W.shape",W.shape)loss = y_hat.reshape((len(y_hat),1))-y # 中间过程,求出当前W和真值的差异cost = np.sum(loss**2)/(2*len(x)) # 这是平方求和过程, 均方误差函数的代码实现return cost # 返回当前模型的均方误差值
def gradient_descent(x, y, W, lr, iterations): # 定义梯度下降函数l_history = np.zeros(iterations) # 初始化记录梯度下降过程中损失的数组W_history = np.zeros((iterations,len(W))) # 初始化权重数组 for iter in range(iterations): # 进行梯度下降的迭代，就是下多少级台阶y_hat = x.dot(W) # 这个是向量化运行实现的假设函数 loss = y_hat.reshape((len(y_hat),1))-y # 中间过程, y_hat和y真值的差derivative_W = x.T.dot(loss)/(2*len(x)) #求出多项式的梯度向量derivative_W = derivative_W.reshape(len(W)) W = W - alpha*derivative_W # 结合下降速率更新权重l_history[iter] = loss_function(x, y, W) # 损失的历史记录 W_history[iter] = W # 梯度下降过程中权重的历史记录return l_history, W_history # 返回梯度下降过程数据
#首先确定参数的初始值iterations = 300; # 迭代300次alpha = 0.15; #学习速率设为0.15weight = np.array([0.5,1,1,1]) # 权重向量，w[0] = bias#计算一下初始值的损失print ('当前损失：',loss_function(x_train, y_train, weight)
# 定义线性回归模型def linear_regression(x, y, weight, alpha, iterations): loss_history, weight_history = gradient_descent(x, y, weight, alpha, iterations)print("训练最终损失:", loss_history[-1]) # 打印最终损失y_pred = x.dot(weight_history[-1]) # 进行预测traning_acc = 100 - np.mean(np.abs(y_pred - y))*100 # 计算准确率print("线性回归训练准确率: {:.2f}%".format(traning_acc)) # 打印准确率return loss_history, weight_history # 返回训练历史记录
# 调用刚才定义的线性回归模型loss_history, weight_history = linear_regression(x_train, y_train,weight, alpha, iterations) #训练机器
print("权重历史记录：", weight_history)print("损失历史记录：", loss_history)
plt.plot(loss_history,'g--',label='Loss Curve')plt.xlabel('Iterations') # x轴Labelplt.ylabel('Loss') # y轴Labelplt.legend() # 显示图例plt.show() # 显示损失曲线print(alpha)
x_plan = [250,50,50] # 要预测的X特征数据x_train,x_plan = scaler(x_train_original,x_plan) # 对预测数据也要归一化缩放x_plan = np.append([1], x_plan ) # 加一个哑特征X0 = 1y_plan = np.dot(weight_history[-1],x_plan) # [-1] 即模型收敛时的权重# 对预测结果要做反向缩放，才能得到与原始广告费用对应的预测值y_value = y_plan*y_gap + y_min # y_gap是当前y_train中最大值和最小值的差，y_min是最小值print ("预计商品销售额： ",y_value, "千元")
x_p=x.copy()x_t_o=x_train_original.copy()x_t,x_p=scaler(x_t_o,x_p)c=np.ones(x_p.shape[0]).transpose()x_p=np.insert(x_p,0,values=c,axis=1)y_p=x_p.dot(weight_history[-1])y_v=y_p*y_gap+y_minfig=plt.figure()ax=Axes3D(fig)x_p0=np.array(x_p[:,0])x_p1,x_p2=np.meshgrid(x_p[:,1],x_p[:,2])y_p1=weight_history[-1][0]+x_p1.dot(weight_history[-1][1])+x_p2.dot(weight_history[-1][2])y_v1=y_p1*y_gap+y_minx_p1,x_p2=np.meshgrid(x[:,0],x[:,1])ax.plot_surface(x_p1,x_p2,y_v1,cmap='hot')
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