机器学习–线性单元回归–单变量梯度下降的实现
【线性回归】
如果要用一句话来解释线性回归是什幺的话,那幺我的理解是这样子的: **线性回归,是从大量的数据中找出最优的线性(y=ax+b)拟合函数,通过数据确定函数中的未知参数,进而进行后续操作(预测) **回归的概念是从统计学的角度得出的,用抽样数据去预估整体(回归中,是通过数据去确定参数),然后再从确定的函数去预测样本。
【损失函数】
用线性函数去拟合数据,那幺问题来了,到底什幺样子的函数最能表现样本?对于这个问题,自然而然便引出了 损失函数 的概念,损失函数是一个用来评价样本数据与目标函数(此处为线性函数)拟合程度的一个指标。我们假设,线性函数模型为:
基于此函数模型,我们定义 损失函数 为:
从上式中我们不难看出,损失函数是一个累加和(统计量)用来记录预测值与真实值之间的1/2方差,从方差的概念我们知道,方差越小说明拟合的越好。那幺此问题进而演变称为求解损失函数最小值的问题,因为我们要通过样本来确定线性函数的中的参数θ_0和θ_1.
【梯度下降】
梯度下降算法是求解最小值的一种方法,但并不是唯一的方法。梯度下降法的核心思想就是对损失函数求偏导,从随机值(任一初始值)开始,沿着梯度下降的方向对 θ_0
和 θ_1
的迭代,最终确定 θ_0
和 θ_1
的值,注意,这里要同时迭代 θ_0
和 θ_1
(这一点在编程过程中很重要),具体迭代过程如下:
【Python代码实现】
那幺下面我们使用python代码来实现线性回归的梯度下降。
#此处数据集,采用吴恩达第一次作业的数据集:ex1data1.txt # -*- coding: utf-8 -*- import numpy as np import matplotlib.pyplot as plt # 读取数据 def readData(path): data = np.loadtxt(path, dtype=float, delimiter=',') return data # 损失函数,返回损失函数计算结果 def costFunction(theta_0, theta_1, x, y, m): predictValue = theta_0 + theta_1 * x return sum((predictValue - y) ** 2) / (2 * m) # 梯度下降算法 # data:数据 # theta_0、theta_1:参数θ_0、θ_1 # iterations:迭代次数 # alpha:步长(学习率) def gradientDescent(data, theta_0, theta_1, iterations, alpha): eachIterationValue = np.zeros((iterations, 1)) x = data[:, 0] y = data[:, 1] m = data.shape[0] for i in range(0, iterations): hypothesis = theta_0 + theta_1 * x temp_0 = theta_0 - alpha * ((1 / m) * sum(hypothesis - y)) temp_1 = theta_1 - alpha * (1 / m) * sum((hypothesis - y) * x) theta_0 = temp_0 theta_1 = temp_1 costFunction_temp = costFunction(theta_0, theta_1, x, y, m) eachIterationValue[i, 0] = costFunction_temp return theta_0, theta_1, eachIterationValue if __name__ == '__main__': data = readData('ex1data1.txt') iterations = 1500 plt.scatter(data[:, 0], data[:, 1], color='g', s=20) # plt.show() theta_0, theta_1, eachIterationValue = gradientDescent(data, 0, 0, iterations, 0.01) hypothesis = theta_0 + theta_1 * data[:, 0] plt.plot(data[:, 0], hypothesis) plt.title("Fittingcurve") plt.show() plt.plot(np.arange(iterations),eachIterationValue) plt.title('CostFunction') plt.show() # 在这里我们使用向量的知识来写代码 # -*- coding: utf-8 -*- import numpy as np import matplotlib.pyplot as plt """ 1.获取数据,并且将数据变为我们可以方便使用的数据格式 """ def LoadFile(filename): data = np.loadtxt(filename, delimiter=',', unpack=True, usecols=(0, 1)) X = np.transpose(np.array(data[:-1])) y = np.transpose(np.array(data[-1:])) X = np.insert(X, 0, 1, axis=1) m = y.size return X, y, m """ 定义线性关系:Linear hypothesis function """ def h(theta, X): return np.dot(X, theta) """ 定义CostFunction """ def CostFunction(theta, X, y, m): return float((1. / (2 * m)) * np.dot((h(theta, X) - y).T, (h(theta, X) - y))) iterations = 1500 alpha = 0.01 def descendGradient(X, y, m, theta_start=np.array(2)): theta = theta_start CostVector = [] theta_history = [] for i in range(0, iterations): tmptheta = theta CostVector.append(CostFunction(theta, X, y, m)) theta_history.append(list(theta[:, 0])) # 同步更新每一个theta的值 for j in range(len(tmptheta)): tmptheta[j] = theta[j] - (alpha / m) * np.sum((h(theta, X) - y) * np.array(X[:, j]).reshape(m, 1)) theta = tmptheta return theta, theta_history, CostVector if __name__ == '__main__': X, y, m = LoadFile('ex1data1.txt') plt.figure(figsize=(10, 6)) plt.scatter(X[:, 1], y[:, 0], color='red') theta = np.zeros((X.shape[1], 1)) theta, theta_history, CostVector = descendGradient(X, y, m, theta) predictValue = h(theta, X) plt.plot(X[:, 1], predictValue) plt.xlabel('the value of x') plt.ylabel('the value of y') plt.title('the liner gradient descend') plt.show() plt.plot(range(len(CostVector)), CostVector, 'bo') plt.grid(True) plt.title("Convergence of Cost Function") plt.xlabel("Iteration number") plt.ylabel("Cost function") plt.xlim([-0.05 * iterations, 1.05 * iterations]) plt.ylim([4, 7]) plt.title('CostFunction') plt.show() # 我们使用我们写好的线性模型去预测未知数据的情况,这样我们就可以得出一个属于我们自己的结果。 # 把我们线性模型预测的结果和实际的结果作一个对比,我们就可以看出实际结果是否真假性。 X, y, m = LoadFile('ex1data3.txt') predictValue = h(theta, X) print(predictValue) # 这里我们可以得到我们的预测值,我们用建立好的模型去预测未知的模型情况。 ‘’‘ [[1.16037866] [3.98169165]] ’‘’
项目运行的结果为:
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机器学习–多元线性回归–多变量梯度下降的实现
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梯度下降–特征缩放
通过特征缩放这个简单的方法,你将可以使得梯度下降的速度变得更快,收敛所迭代的次数变得更少。我们来看一下特征缩放的含义。
#Feature normalizing the columns (subtract mean, divide by standard deviation) #Store the mean and std for later use #Note don't modify the original X matrix, use a copy stored_feature_means, stored_feature_stds = [], [] Xnorm = X.copy() for icol in range(Xnorm.shape[1]): stored_feature_means.append(np.mean(Xnorm[:,icol])) stored_feature_stds.append(np.std(Xnorm[:,icol])) #Skip the first column if not icol: continue #Faster to not recompute the mean and std again, just used stored values Xnorm[:,icol] = (Xnorm[:,icol] - stored_feature_means[-1])/stored_feature_stds[-1]
学习率(alpha)
梯度下降的时候,我们有一个很重要的概念就是学习率的设定,在这里我们要明确一个概念,学习率可以反映梯度下降的情况。如果学习率太低,我们梯度下降的速率就会很慢。同时如果学习率太高,我们的梯度下降会错过最低点,theta的值不是最佳的,同时,可能不会收敛,一直梯度下去,值会越来越大。
那幺我们应该选择一个多少大小的学习率是比较合适的呢?这里吴恩达老师给了一个建议,我们不妨参考。
……0.01、0.03、006、009、0.1、0.3、0.6……。综上所述,我们应该选择一个合适大小的学习率。
特征和多项式回归
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正规方程法(区别与迭代方法的直接求解)
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【Python代码实现多元的线性回归】
# -*- coding: utf-8 -*- import numpy as np import matplotlib.pyplot as plt """ 1.获取数据的结果,使得数据是我们可以更好处理的数据 """ def LoadFile(filename): data = np.loadtxt(filename, delimiter=',', usecols=(0, 1, 2), unpack=True) X = np.transpose(np.array(data[:-1])) y = np.transpose(np.array(data[-1:])) X = np.insert(X, 0, 1, axis=1) m = y.shape[0] return X, y, m """ 2.构建线性函数 """ def h(theta, X): return np.dot(X, theta) """ 3.损失函数CostFunction """ def CostFunction(X, y, theta, m): return float((1. / (2 * m)) * np.dot((h(theta, X) - y).T, (h(theta, X) - y))) """ 4.定义特征缩放的函数 因为数据集之间的差别比较的大,所以我们这里用可梯度下降--特征缩放 """ def normal_feature(Xnorm): stored_feature_means, stored_feature_stds = [], [] for icol in range(Xnorm.shape[1]): # 求平均值 stored_feature_means.append(np.mean(Xnorm[:, icol])) # 求方差 stored_feature_stds.append(np.std(Xnorm[:, icol])) # Skip the first column if not icol: continue # Faster to not recompute the mean and std again, just used stored values Xnorm[:, icol] = (Xnorm[:, icol] - stored_feature_means[-1]) / stored_feature_stds[-1] return Xnorm, stored_feature_means, stored_feature_stds """ 5.定义梯度下降函数 """ iterations = 1500 alpha = 0.01 def descendGradient(X, y, m, theta): CostVector = [] theta_history = [] for i in range(iterations): tmptheta = theta theta_history.append(list(theta[:, 0])) CostVector.append(CostFunction(X, y, theta, m)) for j in range(len(tmptheta)): tmptheta[j, 0] = theta[j] - (alpha / m) * np.sum((h(theta, X) - y) * np.array(X[:, j]).reshape(m, 1)) theta = tmptheta return theta, theta_history, CostVector """ 6.定义绘图函数 """ def plotConvergence(jvec): plt.figure(figsize=(10, 6)) plt.plot(range(len(jvec)), jvec, 'bo') plt.grid(True) plt.title("Convergence of Cost Function") plt.xlabel("Iteration number") plt.ylabel("Cost function") plt.xlim([-0.05 * iterations, 1.05 * iterations]) plt.ylim([4, 7]) plt.show() if __name__ == '__main__': X, y, m = LoadFile('ex1data2.txt') plt.figure(figsize=(10, 6)) plt.grid(True) plt.xlim([-100, 5000]) plt.hist(X[:, 0], label='col1') plt.hist(X[:, 1], label='col2') plt.hist(X[:, 2], label='col3') plt.title('Clearly we need feature normalization.') plt.xlabel('Column Value') plt.ylabel('Counts') plt.legend() plt.show() Xnorm = X.copy() Xnorm, stored_feature_means, stored_feature_stds = normal_feature(Xnorm) plt.grid(True) plt.xlim([-5, 5]) plt.hist(Xnorm[:, 0], label='col1') plt.hist(Xnorm[:, 1], label='col2') plt.hist(Xnorm[:, 2], label='col3') plt.title('Feature Normalization Accomplished') plt.xlabel('Column Value') plt.ylabel('Counts') plt.legend() plt.show() theta = np.zeros((Xnorm.shape[1], 1)) theta, theta_history, CostVector = descendGradient(Xnorm, y, m, theta) plotConvergence(CostVector) print("Check of result: What is price of house with 1650 square feet and 3 bedrooms?") ytest = np.array([1650., 3.]) # To "undo" feature normalization, we "undo" 1650 and 3, then plug it into our hypothesis # 对于每次传来的一个数字我们读进行适当的特征缩放的功能 ytestscaled = [(ytest[x] - stored_feature_means[x + 1]) / stored_feature_stds[x + 1] for x in range(len(ytest))] ytestscaled.insert(0, 1) print(ytestscaled) # 预测未知的值,通过我们已经建立好的模型来预测未知的值。 print("$%0.2f" % float(h(theta, ytestscaled))) # 输出的结果为: [1, -0.4460438603276164, -0.2260933675776883] $293098.15 输出的截图这里就不截图了
【python代码实现一元和多元线性回归汇总】
%matplotlib inline import numpy as np import matplotlib.pyplot as plt cols = np.loadtxt(datafile,delimiter=',',usecols=(0,1),unpack=True) #Read in comma separated data #Form the usual "X" matrix and "y" vector X = np.transpose(np.array(cols[:-1])) y = np.transpose(np.array(cols[-1:])) m = y.size # number of training examples #Insert the usual column of 1's into the "X" matrix X = np.insert(X,0,1,axis=1) print(X) #Plot the data to see what it looks like plt.figure(figsize=(10,6)) plt.plot(X[:,1],y[:,0],'rx',markersize=10) plt.grid(True) #Always plot.grid true! plt.ylabel('Profit in $10,000s') plt.xlabel('Population of City in 10,000s') iterations = 1500 alpha = 0.01 def h(theta,X): #Linear hypothesis function return np.dot(X,theta) def computeCost(mytheta,X,y): #Cost function """ theta_start is an n- dimensional vector of initial theta guess X is matrix with n- columns and m- rows y is a matrix with m- rows and 1 column """ #note to self: *.shape is (rows, columns) return float((1./(2*m)) * np.dot((h(mytheta,X)-y).T,(h(mytheta,X)-y))) #Test that running computeCost with 0's as theta returns 32.07: initial_theta = np.zeros((X.shape[1],1)) #(theta is a vector with n rows and 1 columns (if X has n features) ) print(computeCost(initial_theta,X,y)) #Actual gradient descent minimizing routine def descendGradient(X, theta_start = np.zeros(2)): """ theta_start is an n- dimensional vector of initial theta guess X is matrix with n- columns and m- rows """ theta = theta_start jvec = [] #Used to plot cost as function of iteration thetahistory = [] #Used to visualize the minimization path later on for meaninglessvariable in range(iterations): tmptheta = theta jvec.append(computeCost(theta,X,y)) # Buggy line #thetahistory.append(list(tmptheta)) # Fixed line thetahistory.append(list(theta[:,0])) #Simultaneously updating theta values for j in range(len(tmptheta)): tmptheta[j] = theta[j] - (alpha/m)*np.sum((h(initial_theta,X) - y)*np.array(X[:,j]).reshape(m,1)) theta = tmptheta return theta, thetahistory, jvec #Actually run gradient descent to get the best-fit theta values initial_theta = np.zeros((X.shape[1],1)) theta, thetahistory, jvec = descendGradient(X,initial_theta) #Plot the convergence of the cost function def plotConvergence(jvec): plt.figure(figsize=(10,6)) plt.plot(range(len(jvec)),jvec,'bo') plt.grid(True) plt.title("Convergence of Cost Function") plt.xlabel("Iteration number") plt.ylabel("Cost function") dummy = plt.xlim([-0.05*iterations,1.05*iterations]) #dummy = plt.ylim([4,8]) plotConvergence(jvec) dummy = plt.ylim([4,7]) #Plot the line on top of the data to ensure it looks correct def myfit(xval): return theta[0] + theta[1]*xval plt.figure(figsize=(10,6)) plt.plot(X[:,1],y[:,0],'rx',markersize=10,label='Training Data') plt.plot(X[:,1],myfit(X[:,1]),'b-',label = 'Hypothesis: h(x) = %0.2f + %0.2fx'%(theta[0],theta[1])) plt.grid(True) #Always plot.grid true! plt.ylabel('Profit in $10,000s') plt.xlabel('Population of City in 10,000s') plt.legend() #Import necessary matplotlib tools for 3d plots from mpl_toolkits.mplot3d import axes3d, Axes3D from matplotlib import cm import itertools fig = plt.figure(figsize=(12,12)) ax = fig.gca(projection='3d') xvals = np.arange(-10,10,.5) yvals = np.arange(-1,4,.1) myxs, myys, myzs = [], [], [] for david in xvals: for kaleko in yvals: myxs.append(david) myys.append(kaleko) myzs.append(computeCost(np.array([[david], [kaleko]]),X,y)) scat = ax.scatter(myxs,myys,myzs,c=np.abs(myzs),cmap=plt.get_cmap('YlOrRd')) plt.xlabel(r'$\theta_0$',fontsize=30) plt.ylabel(r'$\theta_1$',fontsize=30) plt.title('Cost (Minimization Path Shown in Blue)',fontsize=30) plt.plot([x[0] for x in thetahistory],[x[1] for x in thetahistory],jvec,'bo-') plt.show() datafile = 'data/ex1data2.txt' #Read into the data file cols = np.loadtxt(datafile,delimiter=',',usecols=(0,1,2),unpack=True) #Read in comma separated data #Form the usual "X" matrix and "y" vector X = np.transpose(np.array(cols[:-1])) y = np.transpose(np.array(cols[-1:])) m = y.size # number of training examples #Insert the usual column of 1's into the "X" matrix X = np.insert(X,0,1,axis=1) #Quick visualize data plt.grid(True) plt.xlim([-100,5000]) dummy = plt.hist(X[:,0],label = 'col1') dummy = plt.hist(X[:,1],label = 'col2') dummy = plt.hist(X[:,2],label = 'col3') plt.title('Clearly we need feature normalization.') plt.xlabel('Column Value') plt.ylabel('Counts') dummy = plt.legend() #Feature normalizing the columns (subtract mean, divide by standard deviation) #Store the mean and std for later use #Note don't modify the original X matrix, use a copy stored_feature_means, stored_feature_stds = [], [] Xnorm = X.copy() for icol in range(Xnorm.shape[1]): stored_feature_means.append(np.mean(Xnorm[:,icol])) stored_feature_stds.append(np.std(Xnorm[:,icol])) #Skip the first column if not icol: continue #Faster to not recompute the mean and std again, just used stored values Xnorm[:,icol] = (Xnorm[:,icol] - stored_feature_means[-1])/stored_feature_stds[-1] #Quick visualize the feature-normalized data plt.grid(True) plt.xlim([-5,5]) dummy = plt.hist(Xnorm[:,0],label = 'col1') dummy = plt.hist(Xnorm[:,1],label = 'col2') dummy = plt.hist(Xnorm[:,2],label = 'col3') plt.title('Feature Normalization Accomplished') plt.xlabel('Column Value') plt.ylabel('Counts') dummy = plt.legend() #Run gradient descent with multiple variables, initial theta still set to zeros #(Note! This doesn't work unless we feature normalize! "overflow encountered in multiply") initial_theta = np.zeros((Xnorm.shape[1],1)) theta, thetahistory, jvec = descendGradient(Xnorm,initial_theta) #Plot convergence of cost function: plotConvergence(jvec) #print "Final result theta parameters: \n",theta print ("Check of result: What is price of house with 1650 square feet and 3 bedrooms?") ytest = np.array([1650.,3.]) #To "undo" feature normalization, we "undo" 1650 and 3, then plug it into our hypothesis # 对于每次传来的一个数字我们读进行适当的特征缩放的功能 ytestscaled = [(ytest[x]-stored_feature_means[x+1])/stored_feature_stds[x+1] for x in range(len(ytest))] ytestscaled.insert(0,1) print ("$%0.2f" % float(h(theta,ytestscaled))) from numpy.linalg import inv #Implementation of normal equation to find analytic solution to linear regression def normEqtn(X,y): #restheta = np.zeros((X.shape[1],1)) return np.dot(np.dot(inv(np.dot(X.T,X)),X.T),y) print ("Normal equation prediction for price of house with 1650 square feet and 3 bedrooms") print ("$%0.2f" % float(h(normEqtn(X,y),[1,1650.,3])))
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