逻辑回归总结

线性回归的鲁棒性很差，在整个是实数域内，其敏感性是一致的。
逻辑回归是一种减小预测范围，将预测值限制在[0,1]内。逻辑回归可以简单的理解为，在线性回归的基础上，套用了一个逻辑函数。其通常用于分类问题。例如，输出y>0.5时，认为是正类，否则为负类。由于将输出值限定在[0.1]内，所以可以认为输出值是一个概率。

1. 理论推导

线性回归：

y=θ0+θ1x1+⋯+θnxn=θTx

逻辑函数：

y=σ(x)=11+e−x

所以，逻辑回归核函数：

y=11+e−θTx=hθ(x)

该表达式一个对数几率的特性（逻辑回归也称为对数几率回归）
对数几率：

ln(y1−y)=θTx(1)

如果限定分类标签y为{0，1}，则概率函数：

P(y|x,θ)=(hθ(x))y(1−hθ(x))1−y

似然函数：

L(θ)=∏i=1NP(y(i)|x(i),θ)

取对数：

l(θ)=ln(L(θ)) =∑i=1Ny(i)ln(hθ(x(i)))+(1−y(i))ln(1−hθ(x(i))=∑i=1Ny(i)ln(hθ(x(i)1−hθ(x(i)))+ln(1−hθ(x(i))

代入等式（1）

l(θ)=∑i=1Ny(i)θTx+ln(1−hθ(x(i))

求梯度，首先sigmoid函数导数为：

σ′(x)=σ(1−σ)x

则：

∂l(θ)∂θ  =∑i=1Ny(i)x(i)−hθ(x(i))x(i)=∑i=1N(y(i)−y^(i))x(i)=∑i=1NError(x(i))x(i)

最后，更新模型权重：

θ:=θ+α∂l(θ)∂θ

2. 代码实现

我们采用python实现以上过程。其中，读取数据文件，效果评估用到了sklearn包提供 的一些工具，数据操作用到了numpy。

#!/usr/bin/env python
# -*- coding: utf-8 -*-  

# Copyright (c) 2017 - xiongjiezk 
from sklearn import datasets, metrics
import math
import numpy as np


class LogitRegression:

    def __init__(self, grad, learn_rate, epoch):
        self.weights = []
        self.bias = 0.0
        self.grad = grad
        self.learn_rate = learn_rate
        self.epoch = epoch

    def sigmoid(self, x):
        return 1.0 / (1 + math.exp(-x))

    def activate(self, x):
        poly_value = np.dot(self.weights, x) + self.bias
        return self.sigmoid(poly_value)

    def rmse(self, y, y_pred):
        return np.sum(np.square(np.subtract(y, y_pred)))

    def fit(self, X, y):
        self.weights = np.random.normal(size=np.shape(X[0]))
        self.bias = 1.0
        for iter in range(0, self.epoch):
            y_pred = []
            if self.grad == 'grad_decent':
                for i in range(0, len(X)):
                    y_ = self.activate(X[i])
                    y_pred.append(y_)
                delta = np.zeros(np.shape(self.weights))
                for i in range(0, len(X)):
                    delta += np.array(y[i] - y_pred[i]) * np.array(X[i])
                self.weights += self.learn_rate * delta
                self.bias += self.learn_rate * np.sum(np.subtract(y, y_pred))
            elif self.grad == 'stoc_grad_decent':
                for i in range(0, len(X)):
                    y_ = self.activate(X[i])
                    self.weights += self.learn_rate * (y[i] - y_) * np.array(X[i])
                    self.bias += self.learn_rate * (y[i] - y_)
                    y_pred.append(y_)
            else:
                pass
            loss = self.rmse(y, y_pred)
            if iter % 100 == 0:
                print('current epoch: %s, loss: %s' % (iter, loss))

    def predict(self, X):
        scores = []
        for i in range(0, len(X)):
            scores.append(self.activate(X[i]))

        class_ = np.array([0, 1])
        indices = (np.array(scores) > 0.5).astype(np.int)
        return np.array(class_[indices])

if __name__ == '__main__':
    data_and_labels = datasets.load_svmlight_file('E:/data/logit/train.txt')
    X_train = np.reshape(data_and_labels[0].data, data_and_labels[0].shape)
    y_train = data_and_labels[1]

    test_and_labels = datasets.load_svmlight_file('E:/data/logit/test.txt')
    X_test = np.reshape(test_and_labels[0].data, test_and_labels[0].shape)
    y_test = test_and_labels[1]

    logit = LogitRegression('grad_decent', 0.01, 1000)
    logit.fit(X_train, y_train)

    y_pred = logit.predict(X_test)

    print("Classification report for classifier %s:\n%s\n"
          % ([logit.weights, logit.bias], metrics.classification_report(y_test, y_pred)))
    print("Confusion matrix:\n%s" % metrics.confusion_matrix(y_test, np.array(y_pred)))

运行效果如下：

current epoch: 800, loss: 2.45524645114
current epoch: 900, loss: 2.45569467863
Classification report for classifier [array([ 0.70676676, -1.68913992]), 12.64359503469959]:
             precision    recall  f1-score   support

        0.0       1.00      0.88      0.93         8
        1.0       0.92      1.00      0.96        12

avg / total       0.95      0.95      0.95        20


Confusion matrix:
[[ 7  1]
 [ 0 12]]

训练数据文件

0    1:-0.017612    2:14.053064
1    1:-1.395634    2:4.662541
0    1:-0.752157    2:6.538620
0    1:-1.322371    2:7.152853
0    1:0.423363    2:11.054677
1    1:0.406704    2:7.067335
0    1:0.667394    2:12.741452
1    1:-2.460150    2:6.866805
0    1:0.569411    2:9.548755
0    1:-0.026632    2:10.427743
1    1:0.850433    2:6.920334
0    1:1.347183    2:13.175500
1    1:1.176813    2:3.167020
0    1:-1.781871    2:9.097953
1    1:-0.566606    2:5.749003
1    1:0.931635    2:1.589505
1    1:-0.024205    2:6.151823
1    1:-0.036453    2:2.690988
1    1:-0.196949    2:0.444165
1    1:1.014459    2:5.754399
1    1:1.985298    2:3.230619
1    1:-1.693453    2:-0.557540
0    1:-0.576525    2:11.778922
1    1:-0.346811    2:-1.678730
1    1:-2.124484    2:2.672471
0    1:1.217916    2:9.597015
0    1:-0.733928    2:9.098687
1    1:-3.642001    2:-1.618087
1    1:0.315985    2:3.523953
0    1:1.416614    2:9.619232
1    1:-0.386323    2:3.989286
1    1:0.556921    2:8.294984
0    1:1.224863    2:11.587360
1    1:-1.347803    2:-2.406051
1    1:1.196604    2:4.951851
0    1:0.275221    2:9.543647
0    1:0.470575    2:9.332488
0    1:-1.889567    2:9.542662
0    1:-1.527893    2:12.150579
0    1:-1.185247    2:11.309318
1    1:-0.445678    2:3.297303
1    1:1.042222    2:6.105155
0    1:-0.618787    2:10.320986
1    1:1.152083    2:0.548467
1    1:0.828534    2:2.676045
0    1:-1.237728    2:10.549033
1    1:-0.683565    2:-2.166125
1    1:0.229456    2:5.921938
0    1:-0.959885    2:11.555336
0    1:0.492911    2:10.993324
0    1:0.184992    2:8.721488
0    1:-0.355715    2:10.325976
0    1:-0.397822    2:8.058397
0    1:0.824839    2:13.730343
1    1:1.507278    2:5.027866
1    1:0.099671    2:6.835839
0    1:-0.344008    2:10.717485
1    1:1.785928    2:7.718645
0    1:-0.918801    2:11.560217
1    1:-0.364009    2:4.747300
1    1:-0.841722    2:4.119083
1    1:0.490426    2:1.960539
0    1:-0.007194    2:9.075792
0    1:0.356107    2:12.447863
0    1:0.342578    2:12.281162
1    1:-0.810823    2:-1.466018
1    1:2.530777    2:6.476801
0    1:1.296683    2:11.607559
0    1:0.475487    2:12.040035
0    1:-0.783277    2:11.009725
0    1:0.074798    2:11.023650
1    1:-1.337472    2:0.468339
0    1:-0.102781    2:13.763651
1    1:-0.147324    2:2.874846
0    1:0.518389    2:9.887035
0    1:1.015399    2:7.571882
1    1:-1.658086    2:-0.027255
1    1:1.319944    2:2.171228
1    1:2.056216    2:5.019981
1    1:-0.851633    2:4.375691

测试数据文件

0    1:-1.510047    2:6.061992    
1    1:-1.076637    2:-3.181888   
0    1:1.821096    2:10.283990    
1    1:3.010150    2:8.401766     
1    1:-1.099458    2:1.688274    
1    1:-0.834872    2:-1.733869   
1    1:-0.846637    2:3.849075    
0    1:1.400102    2:12.628781    
1    1:1.752842    2:5.468166     
1    1:0.078557    2:0.059736     
1    1:0.089392    2:-0.715300    
0    1:1.825662    2:12.693808    
0    1:0.197445    2:9.744638     
1    1:0.126117    2:0.922311     
1    1:-0.679797    2:1.220530    
1    1:0.677983    2:2.556666     
0    1:0.761349    2:10.693862    
1    1:-2.168791    2:0.143632    
0    1:1.388610    2:9.341997     
0    1:0.317029    2:14.739025    

参考资料：
https://www.cnblogs.com/sxron/p/5489214.html
http://blog.csdn.net/programmer_wei/article/details/52072939