logistic regression model
logistic regression model
逻辑回归模型一般指的是二项分类的逻辑回归模型,也是非常经典的模型,它主要的决策函数是,给定数据的情况下,来求取Y属于1或者0的概率。具体的,我们可以做如下表示:
这里, 是输入, 是输出,和是模型参数,是内积。
当然这里的决策函数化简后就是众所周知的sigmoid函数。这里我们使用更为通用的hypothesis来表示函数,具体如下:
loss fuction
当我们的假设函数确定后,我们就需要来确定真实值与假设函数值之间能量损失。通俗的讲,我们的目标就是要求假设函数的值无限逼近真实值,是的二者的差值达到最小。从而得到最有的参数。
我们可以使用极大似然函数来求解我们的模型参数。0, 1综合起来得到:
求最大似然估计:
求取对数:
得到之后,我们就可以通过梯度下降法或者牛顿法来求得参数,也就是
这里我们假设loss function 为可得到:
使用梯度下降法来求取模型参数:
梯度公式更新参数求解:
这样子就可以求解的模型参数了。当模型参数确定后就可以用来对数据进行分类了。
softmax
多元逻辑回归,顾名思义就是考虑多分类问题的逻辑回归方法。
它的假设函数为:
不过这个看着貌似有点繁琐,在Andrew Ng的文章中给出了一个更直观的表现形式:
对于给定的测试输入,我们想用假设函数针对每一个类别j估算出概率值。也就是说,我们想估计的每一种分类结果出现的概率。因此,我们的假设函数将要输出一个维的向量(向量元素的和为1)来表示这个估计的概率值。 具体地说,我们的假设函数形式如下:
这里需要注意的是这一项对概率分布进行归一化,是的概率之和为1.
具体的代价函数如下:
以看到,Softmax代价函数与logistic 代价函数在形式上非常类似,只是在Softmax损失函数中对类标记的个可能值进行了累加。注意在Softmax回归中将分类为类别的概率为:
基于python的logistic regression代码
代码来自《机器学习实战》这本书
#!/usr/bin/python
# encoding: utf-8
"""
@version: 1.0
@author: Fly
@license: Apache Licence
@contact: luyfuyu@gmail.com
@site: https://www.flyuuu.cn
@software: PyCharm
@file: logistic_regression.py
"""
import numpy as np
import math
def load_dataset(filename="./data/lr_testSet.txt"):
data_matrix = []
label_matrix = []
with open(filename, 'r') as f:
for line in f.readlines():
line_list = line.strip().split()
data_matrix.append([1.0, float(line_list[0]), float(line_list[1])])
label_matrix.append(int(line_list[2]))
return data_matrix, label_matrix
def sigmoid(x):
return 1/(1 + np.exp(-x))
def grandient_ascent(intput_data, class_labels, maxiter=500):
data_matrix = np.mat(intput_data)
label_matrix = np.mat(class_labels).transpose()
m,n = data_matrix.shape
alpha = 0.001
weights = np.ones((n,1))
for k in range(maxiter):
h = sigmoid(data_matrix * weights)
error = label_matrix - h
weights = weights + alpha * data_matrix.transpose() * error
return weights
def stoc_ascent(intput_data, class_lables):
data_matrix = np.mat(intput_data)
label_matrix = class_lables
m, n = data_matrix.shape
alpha = 0.001
weights = np.ones(n)
for i in range(m):
h = sigmoid(data_matrix[i] * weights)
error = label_matrix[i] - h
weights = weights + alpha * data_matrix[i] * error
return weights
def plotBestFit(weights):
import matplotlib.pyplot as plt
dataMat, labelMat=load_dataset()
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]).tolist()[0]
ax.plot(x, y)
plt.xlabel('X1')
plt.ylabel('X2')
plt.show()
def stocGradAscent1(dataMatrix, classLabels, numIter=150):
m,n = np.shape(dataMatrix)
weights = np.ones(n) #initialize to all ones
for j in range(numIter):
dataIndex = list(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]
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('./data/horseColicTraining.txt'); frTest = open('./data/horseColicTest.txt')
trainingSet = []
trainingLabels = []
for line in frTrain.readlines():
currLine = line.strip().split(' ')
lineArr =[]
for i in range(21):
lineArr.append(float(currLine[i]))
trainingSet.append(lineArr)
trainingLabels.append(float(currLine[21]))
trainWeights = stocGradAscent1(np.array(trainingSet), trainingLabels, 1000)
errorCount = 0
numTestVec = 0.0
for line in frTest.readlines():
numTestVec += 1.0
currLine = line.strip().split(' ')
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)))
if __name__ == "__main__":
filename = "./data/lr_testSet.txt"
x, y = load_dataset(filename)
weights = grandient_ascent(x, y)
plotBestFit(weights)
multiTest()