前言： 纸上得来终觉浅，绝知此事要躬行
机器学习之逻辑回归python实现 理论基础python实现模型假设：
H(W;X)=11+e−(wx+b) 损失函数(通过最大似然推导)： L(W;X)=−1m∏i=0m[yilog(H(W;X))+(1−yi)log(1−H(W;X)] 权值更新： wj=wj−α(H(w;x)−y)xj bj=bj−α(H(w;x)−y)最难最关键的地方就是梯度下降，下面贴出最重要的一行代码：
self.theta = self.theta - 1.0 / m * self.alpha * np.transpose(x_train) * (self.sigmoid(x_train * self.theta) - y_train)这里theta是增广向量。对照着上面的权值更新看懂这行代码就行了。 梯度下降的函数如下：
def __gradient_decent__(self, x_train, y_train): [m, n] = x_train.shape for i in xrange(self.iterator_num): print "step : %d" % i self.theta = self.theta - 1.0 / m * self.alpha * np.transpose(x_train) * (self.sigmoid(x_train * self.theta) - y_train)随机梯度下降的函数如下：
def __stochastic_gradient_decent__(self, x_train, y_train): [m, n] = x_train.shape for j in xrange(self.iterator_num): data_index = range(m) print "step : ", j for i in xrange(m): #动态调整学习率 self.alpha = 4 / (1.0 + j + i) + 0.01 #随机选取一个样本，随后将其从dataIndex中删除 rand_index = int(np.random.uniform(0, len(data_index))) error = self.sigmoid(np.dot(x_train[rand_index, :], self.theta)) - y_train[rand_index] self.theta = self.theta - np.multiply(self.alpha, np.multiply(error, x_train[rand_index].T)) del data_index[rand_index]随机梯度下降相对于梯度下降，主要改动： - 每次使用一个样本更新，减少计算，加快运行速度 - 动态调整学习率，缓解数据波动或者高频波动。学习率随着i与j的增大而减小 - 每次更新使用的样本是随机选取的，可以减少系数更新的周期性波动
完整代码 LogisticRegression.py #encoding=utf-8 ''' Copyright : CNIC Author : LiuYao Date : 2017-8-31 Description : Define the LogisticRegression class ''' import numpy as np class LogisticRegression(object): ''' implement the lr relative functions ''' def __init__(self, alpha=0.1, iterator_num=100, optimization='sgd'): ''' lr parameters init Args: alpha: the learning rate, default is 0.1. iterator_num: the count of iteration, default is 100. optimization: the optimization method, such as 'sgd', 'gd', default is 'sgd'. ''' self.alpha = alpha self.iterator_num = iterator_num self.optimization = optimization def train(self, x_train, y_train): ''' lr train function Args: x_train: the train data, shape = (m, n), m is the count of the samples, n is the count of the features y_train: the train labels, shape = (m, 1), m is the count of the samples ''' m, n = x_train.shape x_train = np.mat(x_train) self.theta = np.mat(np.random.ranf(n + 1, 1)) x_train = np.hstack((x_train, np.ones((m, 1)))) y_train = np.mat(np.reshape(y_train, (m, 1))) if(self.optimization == 'gd'): self.__gradient_decent__(x_train, y_train) elif(self.optimization == 'sgd'): self.__stochastic_gradient_decent__(x_train, y_train) def sigmoid(self, x): return 1.0 / (1.0 + np.exp(-x)) def __gradient_decent__(self, x_train, y_train): [m, n] = x_train.shape for i in xrange(self.iterator_num): print "step : %d" % i self.theta = self.theta - 1.0 / m * self.alpha * np.transpose(x_train) * (self.sigmoid(x_train * self.theta) - y_train) def __stochastic_gradient_decent__(self, x_train, y_train): [m, n] = x_train.shape for j in xrange(self.iterator_num): data_index = range(m) print "step : ", j for i in xrange(m): #动态调整学习率 self.alpha = 4 / (1.0 + j + i) + 0.01 #随机选取一个样本，随后将其从dataIndex中删除 rand_index = int(np.random.uniform(0, len(data_index))) error = self.sigmoid(np.dot(x_train[rand_index, :], self.theta)) - y_train[rand_index] self.theta = self.theta - np.multiply(self.alpha, np.multiply(error, x_train[rand_index].T)) del data_index[rand_index] def predict(self, x_test): ''' lr predict function Args: x_test: the test data, shape = (m, 1), m is the count of the test data ''' [m, n] = x_test.shape x_test = np.mat(x_test) x_test = np.hstack((x_test, np.ones((m, 1)))) return self.sigmoid(x_test * self.theta)test.py(这里只是画出分界线)
#encoding=utf-8 ''' Copyright : CNIC Author : LiuYao Date : 2017-8-31 Description : test my algorithm ''' import pandas as pd import numpy as np from marchine_learning.linear_model import LogisticRegression import matplotlib.pyplot as plt def load_data(): ''' load data ''' data = pd.read_csv('./data.csv') x = data[['x', 'y']] y = data['label'] return x, y def plot(x_train, y_train, theta): [m, n] = x_train.shape plt.scatter(x_train.values[:, 0], x_train.values[:, 1], c=y_train) x1 = np.random.rand(100, 1) * 25 x2 = (-theta[2] - x1 * theta[0]) / theta[1] plt.plot(x1, x2) plt.show() def main(): ''' program entry ''' x, y = load_data() #这里有两个选项：'gd'（梯度下降）， 'sgd'（随机梯度下降） lr = LogisticRegression.LogisticRegression(iterator_num=10, optimization='sgd') lr.train(x.values, y.values.T) plot(x, y, lr.theta) if __name__ == '__main__': main()运行结果：
梯度下降-10次迭代 随机梯度下降-10次迭代 梯度下降-5次迭代 随机梯度下降-5次迭代总结：随机梯度下降可以减少计算量并加速计算，当数据集很大时，使用随机梯度下降会更好。
git地址：https://github.com/YaoLiuDL/MarchineLearning
