机器学习（一）线性回归、逻辑回归
看了ng的视频、机器学习实战和几篇博客，把逻辑回归彻底弄明白了，现在总结一下吧。
一、线性回归
线性回归无非就是训练得到线性函数的参数来回归出一个线性模型，学习《最优化方法》时中的最小二乘问题就是线性回归的问题。
关于线性回归，ng老师的视频里有讲，也可以看此博客单参数线性回归。简要说一下线性回归的原理。
假设拟合直线为h(x)=θ0+θ1*x, 记Cost Function为J(θ0,θ1)
这其实就是一个线性回归问题，上式也是一个最小二乘问题的模型。盗的图，我认为式中1/m完全没必要，还增加运算，完全可以删掉。线性回归就是要用数据训练出来参数θ，训练时要使目标函数最小，其中x(i)和y(i)中的i表示第i条样本数据，这就是一个无约束优化的问题，关于无约束优化问题的解决方法有很多，比如梯度下降，共轭梯度、牛顿法、步长加速法等等。关于无约束优化和最小二乘问题可以看《最优化方法》相关的书，都有很详细的讲解。这里用最简单的梯度下降。
需要朝着目标函数J梯度下降的方向迭代更行参数θ，
由此
以上就是线性回归了。
二、逻辑回归
线性回归能到一个模型来进行预测，要想用来分类，就要用到逻辑回归。
要分类，就要把h(x)的结果限制在0~1范围内,引入sigmoid函数
假设我们的样本是{ x, y}，y是0或者1，表示正类或者负类，x是我们的m维的样本特征向量。那么这个样本x属于正类，也就是y=1的“概率”可以通过下面的逻辑函数来表示：
所以说，LogisticRegression 就是一个被logistic方程归一化后的线性回归，仅此而已。
怎么来构造代价函数，就用到最大似然，关于最大似然估计，本科概率论有相关知识。
假设我们有n个独立的训练样本{(x1, y1) ,(x2, y2),…, (xn, yn)}，y={0, 1}。那每一个观察到的样本(xi, yi)出现的概率是：
这时候，用L(θ)对θ求导，得到：
三、优化
这一块可以看这篇博客逻辑回归，讲的很细。其实该篇博客主要内容来自《机器学习实战》
注意梯度下降和随机梯度下降的区别。梯度下降每一次迭代都要把整个数据来求导，随即梯度下降则是每一次迭代只用一个样本数据，梯度下降适合小数据，随即梯度下降适合大数据集。
四、总结
所以其实逻辑回归实现时真正用到的就是
每一次对上式迭代
关于matlab代码，上面博客作者的博客里有相关代码，但她是调用了无约束优化的相关函数，所以实现很简单，另外关于正则化的实现我认为她代码不对
上面的博客里有python的代码，《机器学习实战》也有。
#################################################
# logRegression: Logistic Regression
# Author : zouxy
# Date : 2014-03-02
# HomePage : http://blog.csdn.net/zouxy09
# Email : [email protected]
#################################################
from numpy import *
import matplotlib.pyplot as plt
import time
# calculate the sigmoid function
def sigmoid(inX):
return 1.0 / (1 + exp(-inX))
# train a logistic regression model using some optional optimize algorithm
# input: train_x is a mat datatype, each row stands for one sample
# train_y is mat datatype too, each row is the corresponding label
# opts is optimize option include step and maximum number of iterations
def trainLogRegres(train_x, train_y, opts):
# calculate training time
startTime = time.time()
numSamples, numFeatures = shape(train_x)
alpha = opts['alpha']; maxIter = opts['maxIter']
weights = ones((numFeatures, 1))
# optimize through gradient descent algorilthm
for k in range(maxIter):
if opts['optimizeType'] == 'gradDescent': # gradient descent algorilthm
output = sigmoid(train_x * weights)
error = train_y - output
weights = weights + alpha * train_x.transpose() * error
elif opts['optimizeType'] == 'stocGradDescent': # stochastic gradient descent#对于大数据集就不要让随即梯度下降在k循环下循环了，
for i in range(numSamples):
output = sigmoid(train_x[i, :] * weights)
error = train_y[i, 0] - output
weights = weights + alpha * train_x[i, :].transpose() * error
elif opts['optimizeType'] == 'smoothStocGradDescent': # smooth stochastic gradient descent
# randomly select samples to optimize for reducing cycle fluctuations
dataIndex = range(numSamples)
for i in range(numSamples):
alpha = 4.0 / (1.0 + k + i) + 0.01
randIndex = int(random.uniform(0, len(dataIndex)))
output = sigmoid(train_x[randIndex, :] * weights)
error = train_y[randIndex, 0] - output
weights = weights + alpha * train_x[randIndex, :].transpose() * error
del(dataIndex[randIndex]) # during one interation, delete the optimized sample
else:
raise NameError('Not support optimize method type!')
print 'Congratulations, training complete! Took %fs!' % (time.time() - startTime)
return weights
# test your trained Logistic Regression model given test set
def testLogRegres(weights, test_x, test_y):
numSamples, numFeatures = shape(test_x)
matchCount = 0
for i in xrange(numSamples):
predict = sigmoid(test_x[i, :] * weights)[0, 0] > 0.5
if predict == bool(test_y[i, 0]):
matchCount += 1
accuracy = float(matchCount) / numSamples
return accuracy
# show your trained logistic regression model only available with 2-D data
def showLogRegres(weights, train_x, train_y):
# notice: train_x and train_y is mat datatype
numSamples, numFeatures = shape(train_x)
if numFeatures != 3:
print "Sorry! I can not draw because the dimension of your data is not 2!"
return 1
# draw all samples
for i in xrange(numSamples):
if int(train_y[i, 0]) == 0:
plt.plot(train_x[i, 1], train_x[i, 2], 'or')
elif int(train_y[i, 0]) == 1:
plt.plot(train_x[i, 1], train_x[i, 2], 'ob')
# draw the classify line
min_x = min(train_x[:, 1])[0, 0]
max_x = max(train_x[:, 1])[0, 0]
weights = weights.getA() # convert mat to array
y_min_x = float(-weights[0] - weights[1] * min_x) / weights[2]
y_max_x = float(-weights[0] - weights[1] * max_x) / weights[2]
plt.plot([min_x, max_x], [y_min_x, y_max_x], '-g')
plt.xlabel('X1'); plt.ylabel('X2')
plt.show()
#################################################
# logRegression: Logistic Regression
# Author : zouxy
# Date : 2014-03-02
# HomePage : http://blog.csdn.net/zouxy09
# Email : [email protected]
#################################################
from numpy import *
import matplotlib.pyplot as plt
import time
from logregression import trainLogRegres,testLogRegres,showLogRegres
def loadData():
train_x = []
train_y = []
fileIn = open('E:/Python/Machine Learning in Action/testSet.txt')
for line in fileIn.readlines():
lineArr = line.strip().split()
train_x.append([1.0, float(lineArr[0]), float(lineArr[1])])#追加内容
train_y.append(float(lineArr[2]))
return mat(train_x), mat(train_y).transpose()
## step 1: load data
print "step 1: load data..."
train_x, train_y = loadData()
test_x = train_x; test_y = train_y
## step 2: training...
print "step 2: training..."
opts = {'alpha': 0.01, 'maxIter': 20, 'optimizeType': 'smoothStocGradDescent'}#字典
optimalWeights = trainLogRegres(train_x, train_y, opts)
## step 3: testing
print "step 3: testing..."#新版本要加括号
accuracy = testLogRegres(optimalWeights, test_x, test_y)
## step 4: show the result
print "step 4: show the result..."
print 'The classify accuracy is: %.3f%%' % (accuracy * 100)
showLogRegres(optimalWeights, train_x, train_y) 数据集
-0.017612 14.053064 0
-1.395634 4.662541 1
-0.752157 6.538620 0
-1.322371 7.152853 0
0.423363 11.054677 0
0.406704 7.067335 1
0.667394 12.741452 0
-2.460150 6.866805 1
0.569411 9.548755 0
-0.026632 10.427743 0
0.850433 6.920334 1
1.347183 13.175500 0
1.176813 3.167020 1
-1.781871 9.097953 0
-0.566606 5.749003 1
0.931635 1.589505 1
-0.024205 6.151823 1
-0.036453 2.690988 1
-0.196949 0.444165 1
1.014459 5.754399 1
1.985298 3.230619 1
-1.693453 -0.557540 1
-0.576525 11.778922 0
-0.346811 -1.678730 1
-2.124484 2.672471 1
1.217916 9.597015 0
-0.733928 9.098687 0
-3.642001 -1.618087 1
0.315985 3.523953 1
1.416614 9.619232 0
-0.386323 3.989286 1
0.556921 8.294984 1
1.224863 11.587360 0
-1.347803 -2.406051 1
1.196604 4.951851 1
0.275221 9.543647 0
0.470575 9.332488 0
-1.889567 9.542662 0
-1.527893 12.150579 0
-1.185247 11.309318 0
-0.445678 3.297303 1
1.042222 6.105155 1
-0.618787 10.320986 0
1.152083 0.548467 1
0.828534 2.676045 1
-1.237728 10.549033 0
-0.683565 -2.166125 1
0.229456 5.921938 1
-0.959885 11.555336 0
0.492911 10.993324 0
0.184992 8.721488 0
-0.355715 10.325976 0
-0.397822 8.058397 0
0.824839 13.730343 0
1.507278 5.027866 1
0.099671 6.835839 1
-0.344008 10.717485 0
1.785928 7.718645 1
-0.918801 11.560217 0
-0.364009 4.747300 1
-0.841722 4.119083 1
0.490426 1.960539 1
-0.007194 9.075792 0
0.356107 12.447863 0
0.342578 12.281162 0
-0.810823 -1.466018 1
2.530777 6.476801 1
1.296683 11.607559 0
0.475487 12.040035 0
-0.783277 11.009725 0
0.074798 11.023650 0
-1.337472 0.468339 1
-0.102781 13.763651 0
-0.147324 2.874846 1
0.518389 9.887035 0
1.015399 7.571882 0
-1.658086 -0.027255 1
1.319944 2.171228 1
2.056216 5.019981 1
-0.851633 4.375691 1
-1.510047 6.061992 0
-1.076637 -3.181888 1
1.821096 10.283990 0
3.010150 8.401766 1
-1.099458 1.688274 1
-0.834872 -1.733869 1
-0.846637 3.849075 1
1.400102 12.628781 0
1.752842 5.468166 1
0.078557 0.059736 1
0.089392 -0.715300 1
1.825662 12.693808 0
0.197445 9.744638 0
0.126117 0.922311 1
-0.679797 1.220530 1
0.677983 2.556666 1
0.761349 10.693862 0
-2.168791 0.143632 1
1.388610 9.341997 0
0.317029 14.739025 0五、在分布式系统中实现
关于lr在maprudece中的实现，logistic regression不支持并行，也就是mahout实现的也是单机的，运行在hadoop上面也没有意义（个人观点）。
看mahout中的源码分析：mahout源码分析之logistic regression（2）--RunLogistic
参考文献：
[1]. Ng机器学习视频
[2]. 机器学习实战
[3]. Stanford机器学习---第三讲. 逻辑回归和过拟合问题的解决 logistic Regression & Regularization
[4]. 机器学习算法与Python实践之（七）逻辑回归（Logistic Regression）