用梯度下降实现线性回归
y = mx + b 其中m是线性函数的斜率（Slope of the line），b是偏置（bias）
数据集：Swedish Insurance Dataset
代价函数/损失函数 Cost Function [J]= (1/2n) * sum((y_hat-y)^2)
其中:n为输入数据点的总个数
y_hat：使用从梯度下降获得的“m”和“b”值的y的预测值。
y：已经的标签数据y
梯度对参数求导：
Cost Function[J]= (1/2n) * sum(((m*X + b)-y)^2)
CostFunction对m求导：dJ/dm = (1/n)X((m*X+b)-y)
CostFunction对b求导：dJ/db = (1/n)((mX+b)-y) * 1
梯度更新：
m = m - alpha*(dJ/dm)
b = b - alpha*(dJ/db)
其中,alpha是斜率(Learning Rate)
code
# Import Dependenciesimport numpy as npimport pandas as pdimport matplotlib.pyplot as plt%matplotlib inlinedf = pd.read_excel('F:/datasets/slr06.xls', index_col=u'ID')#dataset txt format /RfHWAbI#dataset Swedish Auto Insurance Dataset #/mathematics/brase/understandable_statistics/7e/students/datasets/slr/excel/slr06.xls# convert data to arrayX = np.array(df['X'], dtype=np.float64)y = np.array(df['Y'], dtype=np.float64)fig,ax = plt.subplots()ax.scatter(X,y)def cost_Function(m,b,X,y):return sum(((m*X + b) - y)**2)/(2*float(len(X)))def gradientDescent(X,y,m,b,alpha,iters):# Initialize Values of Gradientsgradient_m = 0gradient_b = 0# n: Number of items in a rown = float(len(X))a = 0# Array to store values of error for analysishist = []# Perform Gradient Descent for itersfor _ in range(iters):# Perform Gradient Descentfor i in range(len(X)):gradient_m = (1/n) * X[i] * ((m*X[i] + b) - y[i])gradient_b = (1/n) * ((m*X[i] + b) - y[i])m = m - (alpha*gradient_m)b = b - (alpha*gradient_b)# Calculate the change in error with new values of "m" and "b"a = cost_Function(m,b,X,y)hist.append(a)return [m,b,hist]# Learning Ratelr = 0.0001# Initial Values of "m" and "b"initial_m = 0initial_b = 0# Number of Iterationsiterations = 1000print("Starting gradient descent...")# Check error with initial Values of m and bprint("Initial Error at m = {0} and b = {1} is error = {2}".format(initial_m, initial_b, cost_Function(initial_m, initial_b, X, y)))[m,b,hist] = gradientDescent(X, y, initial_m, initial_b, lr, iterations)print('Values obtained after {0} iterations are m = {1} and b = {2}'.format(iterations,m,b))y_hat = (m*X + b)print('y_hat: ',y_hat)fig,ax = plt.subplots()ax.scatter(X,y,c='r')ax.plot(X,y_hat,c='y')ax.set_xlabel('X')ax.set_ylabel('y')ax.set_title('Best Fit Line Plot')predict_X = 76predict_y = (m*predict_X + b)print('predict_y: ',predict_y)fig,ax = plt.subplots()ax.scatter(X,y)ax.scatter(predict_X,predict_y,c='r',s=100)ax.plot(X,y_hat,c='y')ax.set_xlabel('X')ax.set_ylabel('y')ax.set_title('Prediction Plot')fig,ax = plt.subplots()ax.plot(hist)ax.set_title('Cost Function Over Time')
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