有關邏輯回歸模型的理論知識： 邏輯回歸模型(logistic regression)
簡要回顧一下邏輯回歸進行分類任務：邏輯回歸使用函數y = sigmoid(wx+b) (通常把b包含在w向量内，直接寫成y = g(z)，z = wx)。 對于輸入樣本xi，若yi>0.5則判讀為正例，否則為反例。
是以最重要的就是确認模型參數w。常用方法是寫出代價函數，使用梯度下降法，核心公式如下：
《機器學習實戰》代碼 書中為梯度上升法，與梯度下降法原理相同，前者向正梯度方向修改參數，用來求極大值。後者向負梯度方向修改參數，用來求極小值。 PS： 書中使用梯度上升法求解的原因是因為其梯度公式中為（y-h(x)），而不像上圖中為（h(x)-y），是以“-”号變正号，梯度下降變梯度上升。
(1)普通梯度上升法
def gradAscent(dataMatIn, classLabels):
dataMatrix = mat(dataMatIn) #轉換為numpy矩陣
labelMat = mat(classLabels).transpose() #轉換為numpy矩陣并轉置
m,n = shape(dataMatrix)
alpha = 0.001 #更新步長
maxCycles = 500 #最大更新次數
weights = ones((n,1))
for k in range(maxCycles):
h = sigmoid(dataMatrix*weights) #邏輯回歸預測
error = (labelMat - h) #誤差，文中圖誤差為(h - labelMat)
weights = weights + alpha * dataMatrix.transpose()* error
#梯度更新公式（矩陣形式），和文中圖給出不同的就是差個-号
return weights
用梯度上升法訓練資料：
import log_reg
dataArr,labelMat = log_reg.loadDataSet()
weights = log_reg.gradAscent(dataArr,labelMat) #梯度上升算法
weights
matrix([[ 4.12414349],
[ 0.48007329],
[-0.6168482 ]])
log_reg.plotBestFit(weights.getA()) #繪制決策分界
（2）随機梯度上升法
從weights（或者叫theta）更新公式中可以看出，每一個weights更新都需要周遊所有樣本（1~m），當樣本資料巨大時（10^6以上），計算量是十分恐怖的，是以随機梯度法就是随機選取某一樣本計算梯度，計算量大大減少。
但很顯然，隻選擇一個樣本進行模型更新，模型會更加符合該樣本，而不一定符合所有樣本，是以更常用的做法是每次取出少量的資料樣本進行模型更新（例如16，32，64等，速度并不會比隻計算一個樣本慢太多）。 此外，在接近極值處，如果步長較大，可能出現反複振蕩，始終達不到要求的精度範圍，是以，還可以随着疊代次數的增加逐漸減小步長。
def stocGradAscent1(dataMatrix, classLabels, numIter=150):
m,n = shape(dataMatrix)
weights = ones(n) #全部初始化為1
for j in range(numIter):
dataIndex = range(m) #1~m随機數
for i in range(m):
alpha = 4/(1.0+j+i)+0.0001
#步長逐漸減小，但不會等于0（等于0将無法更新）
randIndex = int(random.uniform(0,len(dataIndex)))#從1~m中随機選取一個樣本
h = sigmoid(sum(dataMatrix[randIndex]*weights))
error = classLabels[randIndex] - h
weights = weights + alpha * error * dataMatrix[randIndex]
del(dataIndex[randIndex]) #已選過的樣本不會再選
return weights
（3）分類函數 這個簡單，不多說了
def classifyVector(inX, weights):
prob = sigmoid(sum(inX*weights))
if prob > 0.5: return 1.0
else: return 0.0
（4）分類實驗
預測患有‘疝’病的馬的存活問題（二分類任務），輸入的樣本資料（299*21），測試樣本為（67*21）,使用所給函數預測錯誤率0.37，效果不太好，具體問題還是得畫學習曲線分析：機器學習模型評價
完整代碼
'''
Created on Oct 27, 2010
Logistic Regression Working Module
@author: Peter
'''
from numpy import *
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
def sigmoid(inX):
return 1.0/(1+exp(-inX))
def gradAscent(dataMatIn, classLabels):
dataMatrix = mat(dataMatIn) #convert to NumPy matrix
labelMat = mat(classLabels).transpose() #convert to NumPy matrix
m,n = shape(dataMatrix)
alpha = 0.001
maxCycles = 500
weights = 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
def plotBestFit(weights):
import matplotlib.pyplot as plt
dataMat,labelMat=loadDataSet()
dataArr = array(dataMat)
n = 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 = 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()
def stocGradAscent0(dataMatrix, classLabels):
m,n = shape(dataMatrix)
alpha = 0.01
weights = 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
def stocGradAscent1(dataMatrix, classLabels, numIter=150):
m,n = shape(dataMatrix)
weights = 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(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]
del(dataIndex[randIndex])
return weights
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'); frTest = open('horseColicTest.txt')
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 = stocGradAscent1(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(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))