[MLIA] Logistic Regression

Logistic Regression

本代码来自Machine Learning in Action。

想要了解更多的朋友可以参考此书。

Sigmoid函数

$$\sigma(z) = \frac{1}{(1+e^{-z})}$$

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import numpy as np
import matplotlib.pyplot as plt

def sigmoid(inX):
return 1.0/(1+np.exp(-inX))

z = np.linspace(-5, 5, 100)
y = sigmoid(z)
plt.plot(z, y)
plt.show()

png

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z = np.linspace(-60, 60, 100)
y = sigmoid(z)
plt.plot(z, y)
plt.show()

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Sigmoid函数类似一个单位阶跃函数。当x=0时,Sigmoid函数值为0.5;随着x增大,Sigmoid函数值将逼近于1;随着x减小,Sigmoid函数将逼近于0。利用这个性质可以对它的输入做一个二分类。

为了实现Logistic回归分类器,我们可以在每个特征上都乘以一个回归系数,然后把它的所有的结果值相加,将这个总和带入Sigmoid函数中,进而得到一个范围在0~1之间是数值。当大于0.5的时候,将数据分类为1;当小于0.5的时候,将数据分类为0。

Sigmoid函数的输入记为z:

$$z=w_0x_0 + w_1x_1 + w_2x_2 + \cdot \cdot \cdot + w_n x_n$$

Sigmoid函数的导数

Sigmoid导数具体推导过程如下:

$$
\begin{align}
f^{\prime}(z) &= (\frac{1}{1+e^{-z}})^{\prime}\\
&=\frac{e^{-z}}{(1+e^{-z})^2}\\
&=\frac{1+e^{-z}-1}{(1+e^{-z})^2}\\
&=\frac{1}{(1+e^{-z})}(1-\frac{1}{(1+e^{-z})})\\
&=f(z)(1-f(z))
\end{align}
$$

梯度上升法

梯度上升法:顾名思义就是利用梯度方向,寻找到某函数的最大值。

梯度上升算法迭代公式:
$$w:=w+\alpha \nabla_w f(w)$$

梯度下降法:和梯度上升想法,利用梯度方法,寻找某个函数的最小值。
梯度下降算法迭代公式:
$$w:=w-\alpha \nabla_w f(w)$$

梯度上升算法每次更新之后,都会重新估计移动的方法,即梯度。

Logistic 回归梯度上升优化法

加载数据

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def loadDataSet():
dataMat = []; labelMat = []
fr = open('testSet.txt')
for line in fr.readlines():
lineArr = line.strip().split()
dataMat.append([1.0, float(lineArr[0]), float(lineArr[1])])
labelMat.append(int(lineArr[2]))
return dataMat,labelMat
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dataArray, labelMat = loadDataSet()
print("Total: ", len(dataArray))
print("The first sample: ", dataArray[0])
print("The second sample: ", dataArray[1])
print("Label: ", labelMat)
('Total: ', 100)
('The first sample: ', [1.0, -0.017612, 14.053064])
('The second sample: ', [1.0, -1.395634, 4.662541])
('Label: ', [0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0])

数据集梯度上升

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def sigmoid(inX):
return 1.0/(1+np.exp(-inX))

def gradAscent(dataMatIn, classLabels):
dataMatrix = np.mat(dataMatIn) #convert to NumPy matrix
labelMat = np.mat(classLabels).transpose() #convert to NumPy matrix
m,n = np.shape(dataMatrix)
alpha = 0.001
maxCycles = 500
weights = np.ones((n,1))
for k in range(maxCycles): #heavy on matrix operations
h = sigmoid(dataMatrix*weights) #matrix mult
error = (labelMat - h) #vector subtraction
weights = weights + alpha * dataMatrix.transpose()* error #matrix mult
return weights
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gradAscent(dataArray, labelMat)
matrix([[ 4.12414349],
        [ 0.48007329],
        [-0.6168482 ]])

绘制数据和决策边界

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def plotBestFit(weights):
import matplotlib.pyplot as plt
dataMat,labelMat=loadDataSet()
dataArr = np.array(dataMat)
n = np.shape(dataArr)[0]
xcord1 = []; ycord1 = []
xcord2 = []; ycord2 = []
for i in range(n):
if int(labelMat[i])== 1:
xcord1.append(dataArr[i,1]); ycord1.append(dataArr[i,2])
else:
xcord2.append(dataArr[i,1]); ycord2.append(dataArr[i,2])
fig = plt.figure()
ax = fig.add_subplot(111)
ax.scatter(xcord1, ycord1, s=30, c='red', marker='s')
ax.scatter(xcord2, ycord2, s=30, c='green')
x = np.arange(-3.0, 3.0, 0.1)
y = (-weights[0]-weights[1]*x)/weights[2]
ax.plot(x, y)
plt.xlabel('X1'); plt.ylabel('X2');
plt.show()
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weights = gradAscent(dataArray, labelMat)
plotBestFit(weights.getA())

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1个epoch的随机梯度上升

梯度上升算法在每次更新系数的时候都需要便利整个数据集,如果数据集的样本比较大,该方法的复杂度和计算代价就很高。有一种改进的方法叫做随机梯度上升方法。该方法的思想是选取一个样本,计算该样本的梯度,更新系数,再选取下一个样本。

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def stocGradAscent0(dataMatrix, classLabels):
m,n = np.shape(dataMatrix)
alpha = 0.01
weights = np.ones(n) #initialize to all ones
for i in range(m):
h = sigmoid(sum(dataMatrix[i]*weights))
error = classLabels[i] - h
weights = weights + alpha * error * dataMatrix[i]
return weights
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weights = stocGradAscent0(np.array(dataArray), labelMat)
plotBestFit(weights)

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上图之后遍历了一次数据集,这样的模型还处于欠拟合状态。需要多次遍历数据集才能优化好模型,接下来我们会运行200次迭代。

200个epoch的随机梯度上升

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def stocGradAscent0(dataMatrix, classLabels):
X0, X1, X2 = [], [], []
m,n = np.shape(dataMatrix)
alpha = 0.01
weights = np.ones(n) #initialize to all ones
for j in range(200):
for i in range(m):
h = sigmoid(sum(dataMatrix[i]*weights))
error = classLabels[i] - h
weights = weights + alpha * error * dataMatrix[i]
X0.append(weights[0])
X1.append(weights[1])
X2.append(weights[2])
return weights, X0, X1, X2
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weights, X0, X1, X2 = stocGradAscent0(np.array(dataArray), labelMat)
plotBestFit(weights)

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可视化权重(weights)的变化

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fig, ax = plt.subplots(3, 1, figsize=(10, 5))
ax[0].plot(np.arange(len(X0)), np.array(X0))
ax[1].plot(np.arange(len(X1)), np.array(X1))
ax[2].plot(np.arange(len(X2)), np.array(X2))
plt.show()

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从上图可以看出,算法正在逐渐收敛。由于数据集并不是线性可分的,所以存在一些不能正确分类的样本点,每次更新权重引起了周期的变化。

更新过后的随机梯度上升算法

  1. 学习率alpha会在每次迭代之后调整。
  2. 采用随机选取样本的更新策略,减少周期性的波动。
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def stocGradAscent1(dataMatrix, classLabels, numIter=150):
X0, X1, X2 = [], [], []
m,n = np.shape(dataMatrix)
weights = np.ones(n) #initialize to all ones
for j in range(numIter):
dataIndex = range(m)
for i in range(m):
alpha = 4/(1.0+j+i)+0.0001 #apha decreases with iteration, does not
randIndex = int(np.random.uniform(0,len(dataIndex)))#go to 0 because of the constant
h = sigmoid(sum(dataMatrix[randIndex]*weights))
error = classLabels[randIndex] - h
weights = weights + alpha * error * dataMatrix[randIndex]
X0.append(weights[0])
X1.append(weights[1])
X2.append(weights[2])
del(dataIndex[randIndex])
return weights, X0, X1, X2
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weights, X0, X1, X2 = stocGradAscent1(np.array(dataArray), labelMat)
plotBestFit(weights)

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可视化权重(weights)的变化

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fig, ax = plt.subplots(3, 1, figsize=(10, 5))
ax[0].plot(np.arange(len(X0)), np.array(X0))
ax[1].plot(np.arange(len(X1)), np.array(X1))
ax[2].plot(np.arange(len(X2)), np.array(X2))
plt.show()

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示例:从疝气病症预测病马的死亡率

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def classifyVector(inX, weights):
prob = sigmoid(sum(inX*weights))
if prob > 0.5: return 1.0
else: return 0.0

def colicTest():
frTrain = open('horseColicTraining.txt', 'r'); frTest = open('horseColicTest.txt', 'r')
trainingSet = []; trainingLabels = []
for line in frTrain.readlines():
currLine = line.strip().split('\t')
lineArr =[]
for i in range(21):
lineArr.append(float(currLine[i]))
trainingSet.append(lineArr)
trainingLabels.append(float(currLine[21]))
trainWeights, X0, X1, X2 = stocGradAscent1(np.array(trainingSet), trainingLabels, 1000)
errorCount = 0; numTestVec = 0.0
for line in frTest.readlines():
numTestVec += 1.0
currLine = line.strip().split('\t')
lineArr =[]
for i in range(21):
lineArr.append(float(currLine[i]))
if int(classifyVector(np.array(lineArr), trainWeights))!= int(currLine[21]):
errorCount += 1
errorRate = (float(errorCount)/numTestVec)
print "the error rate of this test is: %f" % errorRate
return errorRate

def multiTest():
numTests = 10; errorSum=0.0
for k in range(numTests):
errorSum += colicTest()
print "after %d iterations the average error rate is: %f" % (numTests, errorSum/float(numTests))
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multiTest()
/home/tianliang/anaconda2/lib/python2.7/site-packages/ipykernel_launcher.py:2: RuntimeWarning: overflow encountered in exp



the error rate of this test is: 0.328358
the error rate of this test is: 0.432836
the error rate of this test is: 0.388060
the error rate of this test is: 0.373134
the error rate of this test is: 0.373134
the error rate of this test is: 0.447761
the error rate of this test is: 0.343284
the error rate of this test is: 0.313433
the error rate of this test is: 0.328358
the error rate of this test is: 0.462687
after 10 iterations the average error rate is: 0.379104