git:逻辑回归二分类器
已知数据集 testSet.txt 中数据格式如下:
设第一列特征为x1,第二列特征为x2,第三列标签为z
每一个特征都乘上一个回归系数w,则有
将z代入Sigmoid函数中,得:
Sigmoid函数:
import matplotlib.pyplot as plt
import numpy as np
def sigmoid(z):
return 1 / (1 + np.exp(-z))
nums = np.arange(-5, 5, step=0.3)
fig = plt.figure(figsize=(12, 4))
ax = fig.add_subplot(111)
ax.plot(nums, sigmoid(nums), 'r')
plt.show()
梯度上升法:
记梯度为
,则函数
的梯度为
梯度代表了函数变化的方向,记 为函数变化的大小,也称“步长”,则梯度上升算法的迭代公式为:
由此,我们就可以通过梯度上升法来寻找最佳的回归系数。
使用 Matplotlib 绘出数据点:
import TxtToNumpy
dataMat, labelList = TxtToNumpy.TxtToNumpy("testSet.txt")
type0_x = []; type0_y = []
type1_x = []; type1_y = []
for i in range(len(labelList)):
if labelList[i] == 0:
type0_x.append(dataMat[i][0])
type0_y.append(dataMat[i][1])
if labelList[i] == 1:
type1_x.append(dataMat[i][0])
type1_y.append(dataMat[i][1])
fig = plt.figure(figsize = (8, 4))
ax = fig.add_subplot(111)
type0 = ax.scatter(type0_x, type0_y, s = 30, c = 'r')
type1 = ax.scatter(type1_x, type1_y, s = 30, c = 'b')
ax.set_xlabel("X1")
ax.set_ylabel("X2")
ax.legend((type0, type1), ("Class 0", "Class 1"), loc=0)
plt.show()
TxtToNumpy.py 模块:
from numpy import *
def TxtToNumpy(filename):
file = open(filename)
file_lines_list = file.readlines()
number_of_file_lines = len(file_lines_list)
dataMat = zeros((number_of_file_lines, 3))
labelList = []
index = 0
for line in file_lines_list:
line = line.strip()
line_list = line.split('\t')
dataMat[index, :] = line_list[0:3]
labelList.append(int(line_list[-1]))
index += 1
return dataMat, labelList
if __name__ == "__main__":
print("Code Run As A Program")
画出决策边界:
①批处理梯度上升法求权重,进而画出决策边界:
批处理梯度上升法求权重时,每次更新回归系数都需要遍历整个数据集,因此准确度也最高,但计算复杂度也非常高。
BpGradientAscent.py 模块:
# coding: utf-8
#Batch Processing Gradient Ascent
import numpy as np
import matplotlib.pyplot as plt
#将txt文件中储存的数据和标签分别存储在列表dataMat和labelMat中
def loadDataSet(filename):
dataList = []
labelList = []
fr = open(filename)
for line in fr.readlines():
#将每一行的各个元素取出存放在列表lineArr中
lineArr = line.strip().split()
#[ , , ]中三个参数代表了公式 z = W^T X中的X,第一个X的值为1
dataList.append([1.0, float(lineArr[0]), float(lineArr[1])])
labelList.append(int(lineArr[2]))
return dataList, labelList
#sigmoid函数,用于分类
def sigmoid(z):
return 1.0 / (1 + np.exp(-z))
#batch Processing Gradient Ascent,批处理梯度上升求权重W; alpha表示步长, maxCycles表示梯度上升算法的最大迭代次数
def bpGradientAscent(filename, alpha=0.001, maxCycles=500):
dataList, labelList = loadDataSet(filename)
dataMatrix = np.mat(dataList)
#teanspose()用于矩阵转置
labelMatrix = np.mat(labelList).transpose()
m, n = np.shape(dataMatrix)
weights = np.ones((n, 1))
for i in range(maxCycles):
sig = sigmoid(dataMatrix * weights)
error = labelMatrix - sig
weights = weights + alpha * dataMatrix.transpose() * error
#getA()将矩阵转换为数组
return weights.getA()
#画出决策边界
def decisionBoundary(weights, filename):
dataMat, labelMat = loadDataSet(filename)
dataArr = np.array(dataMat)
n = np.shape(dataArr)[0]
type0_x = []; type0_y = []
type1_x = []; type1_y = []
for i in range(n):
if labelMat[i] == 0:
type0_x.append(dataMat[i][1])
type0_y.append(dataMat[i][2])
if labelMat[i] == 1:
type1_x.append(dataMat[i][1])
type1_y.append(dataMat[i][2])
fig = plt.figure(figsize = (8, 4))
ax = fig.add_subplot(111)
type0 = ax.scatter(type0_x, type0_y, s = 30, c = 'r')
type1 = ax.scatter(type1_x, type1_y, s = 30, c = 'b')
x1 = np.arange(-4.5, 4.5, 0.1)
x2 = (-weights[0]-weights[1]*x1) / weights[2]
ax.set_xlabel("X1")
ax.set_ylabel("X2")
ax.legend((type0, type1), ("Class 0", "Class 1"), loc=0)
ax.plot(x1, x2)
plt.show()
if __name__ == "__main__":
print("Code Run as a Program!")
调用该 BpGradientAscent.py 模块:
import matplotlib.pyplot as plt
import numpy as np
import BpGradientAscent
BpGradientAscent.decisionBoundary(BpGradientAscent.bpGradientAscent("testSet.txt"), "testSet.txt")
得到决策边界(蓝线):
②小批量随机梯度上升法求权重,进而画出决策边界:
小批量随机梯度上升法求权重时,每次更新回归系数只需要选取一部分数据,准确度相对于批处理梯度上升法有所降低,但计算复杂度相对也降低很多,可以通过调整步长和最大迭代次数来提供决策边界的准确度。
SbsGradientAscent.py 模块:
# coding: utf-8
#Small Batch Stochastic Gradient Ascent
import numpy as np
import matplotlib.pyplot as plt
#将txt文件中储存的数据和标签分别存储在列表dataMat和labelMat中
def loadDataSet(filename):
dataList = []
labelList = []
fr = open(filename)
for line in fr.readlines():
#将每一行的各个元素取出存放在列表lineArr中
lineArr = line.strip().split()
#[ , , ]中三个参数代表了公式 z = W^T X中的X,第一个X的值为1
dataList.append([1.0, float(lineArr[0]), float(lineArr[1])])
labelList.append(int(lineArr[2]))
return dataList, labelList
#sigmoid函数,用于分类
def sigmoid(z):
return 1.0 / (1 + np.exp(-z))
#small Batch Stochastic Gradient Ascent,小批量随机梯度上升求权重;maxCycles表示梯度上升算法的最大迭代次数
def sbsGradientAscent(filename, maxCycles = 300):
dataList, labelList = loadDataSet(filename)
m, n = np.shape(dataList)
weights = np.ones(n)
for i in range(maxCycles):
dataIndex = range(m)
for j in range(m):
#alpha表示步长
alpha = 4 / (1.0 + i + j) + 0.001
#uniform()表示在参数范围内随机取值
randomIndex = int(np.random.uniform(0, len(dataIndex)))
error = labelList[randomIndex] - sigmoid(sum(dataList[randomIndex] * weights))
weights = weights + alpha * error * np.array(dataList[randomIndex])
#从列表中移除刚刚被随机选取的值
del(list(dataIndex)[randomIndex])
return weights
def decisionBoundary(weights, filename):
dataMat, labelMat = loadDataSet(filename)
dataArr = np.array(dataMat)
n = np.shape(dataArr)[0]
type0_x = []; type0_y = []
type1_x = []; type1_y = []
for i in range(n):
if labelMat[i] == 0:
type0_x.append(dataMat[i][1])
type0_y.append(dataMat[i][2])
if labelMat[i] == 1:
type1_x.append(dataMat[i][1])
type1_y.append(dataMat[i][2])
fig = plt.figure(figsize = (8, 4))
ax = fig.add_subplot(111)
type0 = ax.scatter(type0_x, type0_y, s = 30, c = 'r')
type1 = ax.scatter(type1_x, type1_y, s = 30, c = 'b')
x1 = np.arange(-4.5, 4.5, 0.1)
x2 = (-weights[0]-weights[1]*x1) / weights[2]
ax.set_xlabel("X1")
ax.set_ylabel("X2")
ax.legend((type0, type1), ("Class 0", "Class 1"), loc=0)
ax.plot(x1, x2)
plt.show()
if __name__ == "__main__":
print("Code Run as a Program!")
调用 SbsGradientAscent.py 模块:
import matplotlib.pyplot as plt
import numpy as np
import SbsGradientAscent
SbsGradientAscent.decisionBoundary(SbsGradientAscent.sbsGradientAscent("testSet.txt"), "testSet.txt")
得到决策边界(蓝线):
BpGradientAscent.py 模块 和 SbsGradientAscent.py 模块的不同之处在于其中的 bpGradientAscent()函数和 sbsGradientAscent()函数 不同,分别表示 批处理梯度上升求权重 和 小批量随机梯度上升求权重
bpGradientAscent()函数:
#batch Processing Gradient Ascent,批处理梯度上升求权重W; alpha表示步长, maxCycles表示梯度上升算法的最大迭代次数
def bpGradientAscent(filename, alpha=0.001, maxCycles=500):
dataList, labelList = loadDataSet(filename)
dataMatrix = np.mat(dataList)
#teanspose()用于矩阵转置
labelMatrix = np.mat(labelList).transpose()
m, n = np.shape(dataMatrix)
weights = np.ones((n, 1))
for i in range(maxCycles):
sig = sigmoid(dataMatrix * weights)
error = labelMatrix - sig
weights = weights + alpha * dataMatrix.transpose() * error
#getA()将矩阵转换为数组
return weights.getA()
sbsGradientAscent()函数:
#small Batch Stochastic Gradient Ascent,小批量随机梯度上升求权重;maxCycles表示梯度上升算法的最大迭代次数
def sbsGradientAscent(filename, maxCycles = 300):
dataList, labelList = loadDataSet(filename)
m, n = np.shape(dataList)
weights = np.ones(n)
for i in range(maxCycles):
dataIndex = range(m)
for j in range(m):
#alpha表示步长
alpha = 4 / (1.0 + i + j) + 0.001
#uniform()表示在参数范围内随机取值
randomIndex = int(np.random.uniform(0, len(dataIndex)))
error = labelList[randomIndex] - sigmoid(sum(dataList[randomIndex] * weights))
weights = weights + alpha * error * np.array(dataList[randomIndex])
#从列表中移除刚刚被随机选取的值
del(list(dataIndex)[randomIndex])
return weights
最终得到的决策边界(蓝线)为:
bpGradientAscent()函数:
sbsGradientAscent()函数: