梯度下降法做线性回归：
估计函数：
误差函数：
在选定线性回归模型后，只需要确定参数θ，就可以将模型用来预测。然而θ需要在J(θ)最小的情况下才能确定。因此问题归结为求极小值问题，使用梯度下降法。梯度下降法最大的问题是求得有可能是全局极小值，这与初始点的选取有关。
梯度下降法是按面的流程进行：
1）首先对 θ 赋值，这个可以是随机的也让 θ 是一个全零的向量。
2）改变 θ 的值，使得 的值，使得 J(θ) 按梯度下降的方向减少
多变量实现：
# coding: utf-8 import pandas as pd import numpy as np import matplotlib.pyplot as plt def plotData(x, y): plt.title('mytest') plt.xlabel('Population of City in 10,000s') plt.ylabel('Profit in $10,000s') plt.plot(x, y, 'rx') plt.show() def gradientDescent(x, y, theta, alpha, num_iters): m = len(y) # 用来存储历史损失 J_history = np.zeros((num_iters, 2)) for i in range(num_iters): # 梯度下降法求解 theta -= (alpha / m) * (np.dot(x.T, (np.dot(x, theta) - y))) J_history[i] = computeCost(x, y, theta) return theta, J_history def computeCost(x, y, theta): m = len(y) # 计算损失均方误差 return sum(np.square(np.dot(x, theta) - y)) / (2 * m) def featureNormalize(x): # 求均值与方差，注意 axis的设置，控制求均值的方向（维度），如果不设置则为整体的均值/方差 mu = x.mean(axis=0) sigma = x.std(axis=0) return (x - mu)/sigma def main(): # 读入数据 mydata = pd.read_csv("ex1data2.txt", header=None) # 选取前两列（Dataframe如何切片） x = np.array(mydata.ix[:, 0:1]) y = np.array(mydata[2]) m = len(y) # 画出数据图像 # plotData(x, y) theta = np.zeros((3, 1)) x = featureNormalize(x) # numpy添加一列 x = np.c_[np.ones((m, 1)), x] # 变成列向量 y = y.reshape((len(y), 1)) J = computeCost(x, y, theta) iterations = 400 alpha = 0.01 theta, J_history = gradientDescent(x, y, theta, alpha, iterations) print theta plt.plot(J_history, 'r') plt.show() if __name__ == "__main__": main() 单变量实现： # coding: utf-8 import pandas as pd import numpy as np import matplotlib.pyplot as plt def plotData(x, y): plt.title('mytest') plt.xlabel('Population of City in 10,000s') plt.ylabel('Profit in $10,000s') plt.plot(x, y, 'rx') plt.show() def gradientDescent(x, y, theta, alpha, num_iters): m = len(y) J_history = np.zeros((num_iters, 2)) for i in range(num_iters): theta -= (alpha/m)*(np.dot(x.T, (np.dot(x, theta) - y))) J_history[i] = computeCost(x, y, theta) return theta, J_history def computeCost(x, y, theta): m = len(y) return sum(np.square(np.dot(x, theta)-y))/(2*m) def main(): # 读入数据 mydata = pd.read_csv("ex1data1.txt", header=None) x = np.array(mydata[0]) y = np.array(mydata[1]) m = len(y) # 画出数据图像 plotData(x,y) theta = np.zeros((2, 1)) x = np.c_[np.ones((m, 1)), x] y = y.reshape((len(y), 1)) J = computeCost(x, y, theta) iterations = 1500 alpha = 0.01 theta, J_history = gradientDescent(x, y, theta, alpha, iterations) print theta plt.plot(J_history,'r') plt.show() if __name__ == "__main__": main()
