斯坦福机器学习 ex1 Python实现

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()