成本最小的学习方案: 了解公式推导指南，用计算机语言实现一遍，解决一个实例。
1. 公式推导
- 在线性回归中，我们研究的是连续量的变化情况，而在逻辑回归中研究对象则变成了离散量，简单来说，在线性回归中我们输入一个值 x ，然后输出它对应的数值 y ，而在逻辑回归中，我们输入一个值 x ，判断它是属于 0 还是 1 。
- 对于判断输入值属于 0 还是 1 ，我们通过的是概率的方法，概率大于 50% 判断为 1 ，概率小于 50% 判断为 0 。
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具体详见 https://zhuanlan.zhihu.com/p/37020923 -
通常，我们让0为“负类”（Negative class），1为“正类”（Positive class）：
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因为有一个outliner, 所以线形拟合不够好
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所以引入逻辑回归
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训练集用来拟合参数θ，然后θ就决定了决策边界。
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- 训练集用来拟合参数θ，然后θ就决定了决策边界。
代价函数（cost function）
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这样代价函数，是非凸函数，如果使用梯度下降，几乎不能收敛到最全局最小值，所以我们需要寻找其他的是凸函数的代价函数，这样就可以使用之前学过的算法了。
（只有凸函数可以到全局最小值，其他可能到局部最低点）
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当假设h(θ)=1时，如果y是1，那么cost=0；如果y=0，那么cost=∞。
当假设h(θ)=0时，如果y时1，那么cost=∞；如果y=0，那么cost=0。
简化代价函数和梯度下降
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- 这个代价函数它是凸的（convex），所以使用梯度下降可以获得全局最优解。
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梯度下降（gradient descent ）：
线性回归的特征缩放（提高梯度下降的速度）在逻辑回归中依然可以使用。
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python 实战
1. logRegression.py
#################################################
# logRegression: Logistic Regression
# Author : zouxy
# Date : 2014-03-02
# HomePage : http://blog.csdn.net/zouxy09
# Email : zouxy09@qq.com
#################################################
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
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 = list(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 range(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 range(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()
def loadData():
train_x = []
train_y = []
fileIn = open('testSet.txt')
for line in fileIn.readlines():
lineArr = line.strip().split()
train_x.append([1.0, float(lineArr[0]), float(lineArr[1])])
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'}
optimalWeights = trainLogRegres(train_x, train_y, opts)
## step 3: testing
print("step 3: testing...")
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)
data
-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
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