数据描述
导入python包
import numpy as np import matplotlib import matplotlib.pyplot as plt 子函数描述
#将文本记录转化为Numpy #filename为文件名，k为属性维数 def file2matrix(filename,k): fr = open(filename) numberOfLines = len(fr.readlines()) #get the number of lines in the file returnMat = np.zeros((numberOfLines,k)) #prepare matrix to return classLabelVector = np.zeros((numberOfLines,1)) #prepare labels return fr = open(filename) index = 0 for line in fr.readlines(): line = line.strip() listFromLine = line.split(',') returnMat[index,:] = listFromLine[0:k] classLabelVector[index,:] = float(listFromLine[-1]) index += 1 return returnMat,classLabelVector #特征标准化 def featureNormalize(X): X_norm = np.array(X) #mu = np.zeros((1,X.shape[1])) #sigma = np.zeros((1, X.shape[1])) mu = np.mean(X, axis = 0)#对每列求均值 sigma = np.std(X, axis = 0)#对每列求标准差 X_norm = (X_norm - np.tile(mu, (X.shape[0],1)))/np.tile(sigma, (X.shape[0],1)) return X_norm, mu, sigma # 损失函数 def computeCostMulti(X, y, theta): m = len(y) J = np.sum((y - np.dot(X, theta))**2) / (2 * m) return J #梯度下降 def gradientDescentMulti(X, y, theta, alpha, numb_iters): # Initialize some useful values m = len(y) # number of training examples J_history = [] for i in range(numb_iters): theta -= alpha / m * np.dot(X.T, (np.dot(X, theta) - y)) J_history.append(computeCostMulti(X, y, theta)) return theta, J_history # Normal Equation # theta = (X'X)^(-1)X'y def normalEqn(X, y): from numpy.linalg import inv return np.dot(np.dot(inv(np.dot(X.T, X)), X.T), y)读入数据
returnMat, LabelVector = file2matrix('ex1data2.txt',2) 这里，我们对文件进行读取。首先，open文件并读取其行数，定义存放特征数据和预测数据的变量（这里，我们使用的类型是np.array。其实我们也可以使用np.mat，都是大同小异）。然后，利用strip移除字符串头尾指定的字符(默认为空格或换行符)或字符序列，利用split(',')指定分隔符(这里为',')对字符串进行切片，将前两列存入特征数据中，最后一列存入预测数据中。这里，如果预测数据是英文字母，我们还需要对其进行类型的转换。returnMat, LabelVector分别为存放特征数据和预测数据的变量。
特征标准化
# Part 1: Feature Normalization X, mu, sigma = featureNormalize(returnMat) 我们对returnMat的数据求每一列的均值和标准差，这里使用到的函数为numpy自带的np.mean和np.std，具体使用详情可以查阅官方文件。然后使用np.tile对变量内容复制成输入矩阵同大小的矩阵。最后，(returnMat-mean)/std为标准化之后的结果，这里的‘/’是点除。
输入数据扩充
# Add intercept term to X X = np.column_stack((np.ones((X.shape[0],1)), X)) 我们在特征数据的第一列插入一列1矩阵，方便与之后delta的常数项系数相乘。
梯度下降
# Part 2: Gradient Descent # Choose some alpha value alpha = 0.01 num_iters = 400 # Init Theta and Run Gradient Descent theta = np.zeros((X.shape[1],1)) theta, J_history = gradientDescentMulti(X, LabelVector, theta, alpha, num_iters) 梯度下降法的公式参见下图
theta -= alpha / m * np.dot(X.T, (np.dot(X, theta) - y))
迭代误差结果图
fig = plt.figure() ax = fig.add_subplot(111) ax.scatter(range(len(J_history)), J_history) plt.show() 
预测
# 预测 # Estimate the price of a 1650 sq-ft, 3 br house # Recall that the first column of X is all-ones. Thus, it does not need to be normalized. price = np.dot(np.column_stack((np.array([[1]]), ((np.array([[1650, 3]]) - mu) / sigma))), theta) print("Predicted price of a 1650 sq-ft, 3 br house is $", price[0][0]) Predicted price of a 1650 sq-ft, 3 br house is $ 289221.547371
这里，我们需要注意的是数据刚开始就标准化，所以这里的数据也需要进行标准化。
Normal Equations
# Part 3: Normal Equations X, y = file2matrix('ex1data2.txt',2) # Add intercept term to X X = np.column_stack((np.ones((X.shape[0],1)), X)) theta1 = normalEqn(X, LabelVector) # 预测 # Estimate the price of a 1650 sq-ft, 3 br house # Recall that the first column of X is all-ones. Thus, it does not need to be normalized. price1 = np.dot(np.column_stack((np.array([[1]]), np.array([[1650, 3]]))), theta1) print("Predicted price of a 1650 sq-ft, 3 br house is $", price1[0][0]) 最小二乘法的delta用矩阵的形式表示为：
结果：Predicted price of a 1650 sq-ft, 3 br house is $ 293081.464335。这里，因为刚开始没有对数据进行标准化处理，所以在预测时候也不需要进行标准化处理。