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# python
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from numpy import *
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# 数据集大小 即20个数据点
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m = 20
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# x的坐标以及对应的矩阵
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X0 = ones((m, 1)) # 生成一个m行1列的向量，也就是x0，全是1
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X1 = arange(1, m+1).reshape(m, 1) # 生成一个m行1列的向量，也就是x1，从1到m
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X = hstack((X0, X1)) # 按照列堆叠形成数组，其实就是样本数据
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# 对应的y坐标
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Y = array([
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3, 4, 5, 5, 2, 4, 7, 8, 11, 8, 12,
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11, 13, 13, 16, 17, 18, 17, 19, 21
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]).reshape(m, 1)
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# 学习率
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alpha = 0.01
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# 定义代价函数
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def cost_function(theta, X, Y):
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diff = dot(X, theta) - Y # dot() 数组需要像矩阵那样相乘，就需要用到dot()
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return (1/(2*m)) * dot(diff.transpose(), diff)
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# 定义代价函数对应的梯度函数
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def gradient_function(theta, X, Y):
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diff = dot(X, theta) - Y
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return (1/m) * dot(X.transpose(), diff)
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######Begin######
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# 梯度下降迭代
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def gradient_descent(X, Y, alpha):
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theta = array([1, 1]).reshape(2, 1)
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gradient = gradient_function(theta, X, Y)
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while not all(abs(gradient) <= 1e-5):
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theta = theta - alpha * gradient
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gradient = gradient_function(theta, X, Y)
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return theta
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optimal = gradient_descent(X, Y, alpha)
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print('optimal:', optimal)
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print('cost function:', cost_function(optimal, X, Y)[0][0])
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#######End#######
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# 根据数据画出对应的图像
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def plot(X, Y, theta):
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import matplotlib.pyplot as plt
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ax = plt.subplot(1,1,1)
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ax.scatter(X, Y, s=30, c="red", marker="s")
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plt.xlabel("X")
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plt.ylabel("Y")
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x = arange(0, 21, 0.2) # x的范围
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y = theta[0] + theta[1]*x
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ax.plot(x, y)
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plt.show()
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plot(X1, Y, optimal)
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