機器學習——線性迴歸（吳恩達老師視頻總結和練習代碼）_機器學習

線性迴歸的代價函數： 線性迴歸的迭代過程： 特徵值縮放：

學習率： 如果學習率 α 過小， 則達到收斂所需的迭代次數會非常高；如果學習率 α 過大，每次迭代可能不會減小代價函數，可能會越過局部最小值導致無法收斂。 特徵和多項式迴歸： 正規方程：
擴充知識： 這裡知乎上有個問題很好的解釋最小二乘法，這個跟正規方程有很多相關的地方，值得一看： https://www.zhihu.com/question/37031188/answer/111336809 還有一個知乎問題關於泰勒公式的，數學知識忘記得差不多的可以看看： https://www.zhihu.com/question/21149770

下面是作業的代碼：

# -*- coding: utf-8 -*-"""__author__ = 'ljyn4180'"""import numpy as npimport pandas as pdimport matplotlib.pyplot as plt# 代價函數def CostFunction(matrixX, matrixY, matrixTheta):    Inner = np.power(((matrixX * matrixTheta.T) - matrixY), 2)    return np.sum(Inner) / (2 * len(matrixX))# 梯度下降迭代函數def GradientDescent(matrixX, matrixY, matrixTheta, fAlpha, nIterCounts):    matrixThetaTemp = np.matrix(np.zeros(matrixTheta.shape))    nParameters = int(matrixTheta.ravel().shape[1])    arrayCost = np.zeros(nIterCounts)    for i in xrange(nIterCounts):        matrixError = (matrixX * matrixTheta.T) - matrixY        for j in xrange(nParameters):            matrixSumTerm = np.multiply(matrixError, matrixX[:, j])            matrixThetaTemp[0, j] = matrixTheta[0, j] - fAlpha / len(matrixX) * np.sum(matrixSumTerm)        matrixTheta = matrixThetaTemp        arrayCost[i] = CostFunction(matrixX, matrixY, matrixTheta)    return matrixTheta, arrayCost# 顯示線性迴歸結果def ShowLineRegressionResult(dataFrame, matrixTheta):    x = np.linspace(dataFrame.Population.min(), dataFrame.Population.max(), 100)    f = matrixTheta[0, 0] + (matrixTheta[0, 1] * x)    plt.subplot(221)    plt.plot(x, f, 'r', label='Prediction')    plt.scatter(dataFrame.Population, dataFrame.Profit, label='Training Data')    plt.legend(loc=2)    plt.xlabel('Population')    plt.ylabel('Profit')    plt.title('Predicted Profit vs. Population Size')# 顯示代價函數的值的變化情況def ShowCostChange(arrayCost, nIterCounts):    plt.subplot(222)    plt.plot(np.arange(nIterCounts), arrayCost, 'r')    plt.xlabel('Iterations')    plt.ylabel('Cost')    plt.title('Error vs. Training Epoch')# 顯示不同Alpha值得學習曲線def ShowLearningRateChange(dictCost, nIterCounts):    plt.subplot(223)    for fAlpha, arrayCost in dictCost.iteritems():        plt.plot(np.arange(nIterCounts), arrayCost, label=fAlpha)    plt.xlabel('Iterations')    plt.ylabel('Cost')    plt.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.)    plt.title('learning rate')# 正規方程def NormalEquation(matrixX, matrixY):    matrixTheta = np.linalg.inv(matrixX.T * matrixX) * matrixX.T * matrixY  # Python2.7    # matrixTheta = np.linalg.inv(matrixX.T @ matrixX) @ matrixX.T @ matrixY  # Python3    return matrixTheta# 標準化特徵值def NormalizeFeature(dataFrame):    return dataFrame.apply(lambda column: (column - column.mean()) / column.std())def ExerciseOne():    path = 'ex1data1.txt'    dataFrame = pd.read_csv(path, header=None, names=['Population', 'Profit'])    # 補項    dataFrame.insert(0, 'Ones', 1)    nColumnCount = dataFrame.shape[1]    dataFrameX = dataFrame.iloc[:, 0:nColumnCount - 1]    dataFrameY = dataFrame.iloc[:, nColumnCount - 1:nColumnCount]    # 初始化資料    matrixX = np.matrix(dataFrameX.values)    matrixY = np.matrix(dataFrameY.values)    matrixOriginTheta = np.matrix(np.zeros(dataFrameX.shape[1]))    # 設定學習率和迭代次數    fAlpha = 0.01    nIterCounts = 1000    matrixTheta, arrayCost = GradientDescent(matrixX, matrixY, matrixOriginTheta, fAlpha, nIterCounts)    print matrixTheta    print NormalEquation(matrixX, matrixY)    # 設定不同的學習率    arrayAlpha = [0.000001, 0.00001, 0.0001, 0.001, 0.01]    dictCost = {}    for fAlpha in arrayAlpha:        _, arrayCostTemp = GradientDescent(matrixX, matrixY, matrixOriginTheta, fAlpha, nIterCounts)        dictCost[fAlpha] = arrayCostTemp    # 顯示不同圖表    plt.figure(figsize=(12, 12))    ShowLineRegressionResult(dataFrame, matrixTheta)    ShowCostChange(arrayCost, nIterCounts)    ShowLearningRateChange(dictCost, nIterCounts)    plt.show()ExerciseOne()

# -*- coding: utf-8 -*-"""__author__ = 'ljyn4180'"""import numpy as npimport pandas as pdimport matplotlib.pyplot as pltimport mpl_toolkits.mplot3d.axes3d as Axes3Ddef CostFunction(matrixX, matrixY, matrixTheta):    Inner = np.power(((matrixX * matrixTheta.T) - matrixY), 2)    return np.sum(Inner) / (2 * len(matrixX))def GradientDescent(matrixX, matrixY, matrixTheta, fAlpha, nIterCounts):    matrixThetaTemp = np.matrix(np.zeros(matrixTheta.shape))    nParameters = int(matrixTheta.ravel().shape[1])    arrayCost = np.zeros(nIterCounts)    for i in xrange(nIterCounts):        matrixError = (matrixX * matrixTheta.T) - matrixY        for j in xrange(nParameters):            matrixSumTerm = np.multiply(matrixError, matrixX[:, j])            matrixThetaTemp[0, j] = matrixTheta[0, j] - fAlpha / len(matrixX) * np.sum(matrixSumTerm)        matrixTheta = matrixThetaTemp        arrayCost[i] = CostFunction(matrixX, matrixY, matrixTheta)    return matrixTheta, arrayCostdef ShowLineRegressionResult(dataFrame, matrixTheta):    x = np.array(dataFrame.square)    y = np.array(dataFrame.bedrooms)    z = matrixTheta[0, 0] + (matrixTheta[0, 1] * x) + (matrixTheta[0, 2] * y)    print z    ax = plt.subplot(211, projection='3d')    ax.plot_trisurf(x, y, z)    ax.scatter(dataFrame.square, dataFrame.bedrooms, dataFrame.price, label='Training Data')    ax.set_xlabel('square')    ax.set_ylabel('bedrooms')    ax.set_zlabel('price')    # plt.title('Predicted Profit vs. Population Size')def ShowCostChange(arrayCost, nIterCounts):    plt.subplot(212)    plt.plot(np.arange(nIterCounts), arrayCost, 'r')    plt.xlabel('Iterations')    plt.ylabel('Cost')    plt.title('Error vs. Training Epoch')def NormalizeFeature(dataFrame):    return dataFrame.apply(lambda column: (column - column.mean()) / column.std())def ExerciseTwo():    path = 'ex1data2.txt'    dataFrame = pd.read_csv(path, header=None, names=['square', 'bedrooms', 'price'])    dataFrame = NormalizeFeature(dataFrame)    # data['square'] = data['square'] / 1000    # data['price'] = data['price'] / 100000    # 補項    dataFrame.insert(0, 'Ones', 1)    nColumnCount = dataFrame.shape[1]    dataFrameX = dataFrame.iloc[:, 0:nColumnCount - 1]    dataFrameY = dataFrame.iloc[:, nColumnCount - 1:nColumnCount]    matrixX = np.matrix(dataFrameX.values)    matrixY = np.matrix(dataFrameY.values)    matrixTheta = np.matrix(np.zeros(dataFrameX.shape[1]))    fAlpha = 0.01    nIterCounts = 1000    matrixTheta, arrayCost = GradientDescent(matrixX, matrixY, matrixTheta, fAlpha, nIterCounts)    print matrixTheta    plt.figure(figsize=(8, 10))    ShowLineRegressionResult(dataFrame, matrixTheta)    ShowCostChange(arrayCost, nIterCounts)    plt.show()ExerciseTwo()