1.sigmoid函數應用
- logistics迴歸是用來分類的，並且屬於監督學習，分類也是僅限於二分類，就是結果非0即1 (這種函數通常稱作躍階函數)
- 這個時候就出現問題了 01之間的分界點怎麼處理？
- 引入sigmoid函數 圖像見下圖
2.算法中的數學思想
舉個引例：求 函數y = -x^2+3x+1 的最大值
很簡單 求得導數 y’ = -2x+3
當且僅當x=1.5時函數y取得最大值
然而並不是所有的函數都可以這麼一下求出來
所以 就利用了梯度上升的方法來求極值(最大值)
# y = -x^2+3x+1
# y'= -2x+3
def fun(inx):
return (-2*inx+3)
if __name__ == '__main__':
xNew = 2
xOld = 0
eps = 0.00001
alpha = 0.01
while abs(xOld - xNew) > eps: # 梯度上升 步長*'方向'
xOld = xNew
xNew = xNew + alpha * fun(xNew)
print xNew- 拓展到矩陣 就有了不一樣的表達式 解釋logistic中的梯度上升
- 求出最佳迴歸係數的目的是爲了帶入到sigmoid函數
- 每個數據點都乘上一個迴歸係數 帶入sigmoid函數 得到0-1之間的值 實現分類目的
3.算法僞代碼–梯度上升算法
每個迴歸係數置爲1
for 0-loopNum
帶入sigmoid計算估計值
更新迴歸係數的向量
返回係數矩陣
4.缺點不足
要想得到好的結果時間花費大
容易欠擬合 精度不太高
5.改進算法-隨機梯度上升算法
- 隨機選取樣本來更新迴歸係數
- 步長時刻改變
- 減少了運算複雜程度
6.僞代碼-隨機梯度上升算法
每個迴歸係數置爲1
for 0-loopNum
for 0-n所有的點
更新步長
隨機選取數據點帶入sigmoid計算估計值
更新迴歸係數的向量
標記已經隨機使用的點
返回係數矩陣
# -*- coding:utf-8 -*-
from numpy import *
def classify(inX, theta): # 分類器0/1二分類
res = sigm(sum(theta * inX))
if res > 0.5:
return 1
else:
return 0
# 加載文件 返回list
def loadFile(fileName):
txtFile = open(fileName, "rb")
dataArr = []
labelArr = []
for line in txtFile.readlines():
lineArr = line.split()
dataArr.append([2.0, float(lineArr[0]), float(lineArr[1])])
labelArr.append(int(lineArr[-1]))
txtFile.close()
return dataArr, labelArr
# sigmoid函數
def sigm(inX):
return 1 / (1 + exp(-inX))
# 繪圖 打印出點以及直線
def plotPrint(theta):
import matplotlib.pyplot as plt
dataMat, labelMat = loadFile("testSet.txt")
dataArr = array(dataMat) # list 轉換爲數組
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(-3.5, 3.5, 0.1)
y = (-2 * theta[0] - theta[1] * x) / theta[2]
# theta[0]*2 + theta[1]*x + theta[2]*y = 0
# print theta[0], theta[1], theta[2]
ax.plot(x, y)
plt.xlabel('x')
plt.ylabel('y')
plt.show()
# 梯度上升算法 初始版
def stocGradAscent0(dataMat, labelMat, alpha, loopNum):
m, n = shape(dataMat)
theta = ones((n, 1)) # 初始爲n*1的迴歸參數矩陣
for i in range(loopNum):
h = sigm(dataMat * theta) # z = w0x0+w1x1+...+wnxn
loss = labelMat - h # y與y拔之差 預測值與真實值的差
theta = theta + alpha * dataMat.T * loss # alpha * 3*1 (3*100 * 100*1) 矩陣的運算 不是一步運算
return theta
# 隨即梯度上升算法 改進版 一維array數值運算 不涉及矩陣的運算
def stocGradAscent(dataMat, labelMat, loopNum=150):
m, n = shape(dataMat)
theta = ones(n)
for i in range(loopNum): # 循環次數
dataIndex = range(m) # 下標存放的數組
for j in range(m):
alpha = 4 / (1.0 + j + i) + 0.01
randomIndex = int(random.uniform(0, len(dataIndex))) # 隨機下標
h = sigm(sum(dataMat[randomIndex] * theta)) # sum(3L*3L) = 3L
loss = labelMat[randomIndex] - h # 誤差
theta = theta + alpha * loss * dataMat[randomIndex]
del (dataIndex[randomIndex]) # 用過之後刪除該下標
return theta
if __name__ == '__main__':
Dir = "testSet.txt"
# 步長 循環次數
alpha = 0.001
loopNum = 1000
# 數據矩陣 類別(標籤)矩陣
data, label = loadFile(Dir)
data = mat(data)
label = mat(label)
# th就是迴歸參數矩陣
th = stocGradAscent0(data, label.T, alpha, 500)
plotPrint(th.A) # th.getA() th.A得到ndarray數組
print "----" * 10
data, label = loadFile(Dir)
theta = stocGradAscent(array(data), label, 500)
plotPrint(theta)
print theta
testPoint = mat(ones((3, 1)))
print classify(testPoint, theta)
附上程序運行結果
'''
----------------------------------------
[ 9.37024571 1.33192559 -2.62149254]
1
'''配上數據集 100*3
'dataSet.txt'
"""
-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
"""