吴恩达机器学习作业2---logistic regression

logistic regression

代码分析

测试1

首先导入将要用到的类库

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
from scipy import optimize
from scipy.special import expit #Vectorized sigmoid function

%matplotlib inline#可选

读入数据，并进行处理

datafile = 'data/ex2data1.txt'

#读入数据，逗号分开，转置
cols = np.loadtxt(datafile,delimiter=',',usecols=(0,1,2),unpack=True) 

#X为前两列，y为最后一列，m为数据集大小
X = np.transpose(np.array(cols[:-1]))
y = np.transpose(np.array(cols[-1:]))
m = y.size 

#在X矩阵前加上全为1的一列，作为theta_0
X = np.insert(X,0,1,axis=1)

将样本集分为两个部分

#将样本集分为两部分，一部分为1，一部分为0
pos = np.array([X[i] for i in range(X.shape[0]) if y[i] == 1])
neg = np.array([X[i] for i in range(X.shape[0]) if y[i] == 0])

数据集的可视化函数

#样本点的可视化函数
def plotData():
    plt.figure(figsize=(10,6))
    plt.plot(pos[:,1],pos[:,2],'k+',label='Admitted')
    plt.plot(neg[:,1],neg[:,2],'yo',label='Not admitted')
    plt.xlabel('Exam 1 score')
    plt.ylabel('Exam 2 score')
    plt.legend()#网格线
    plt.grid(True)

可视化

#可视化
plotData()

测试一下expit函数

#检查一下expit函数

#expit(x) = 1/(1+exp(-x))
#myx为-10到10，以0.1为间隔的array
myx = np.arange(-10,10,.1)
plt.plot(myx,expit(myx))
plt.title("Woohoo this looks like a sigmoid function to me.")
plt.grid(True)

$$h_{\theta }(x)=g({\theta }^{T}x),g(x)=\frac{1}{1+e^{-z}}$$

#logistic regression的假设函数
def h(mytheta,myX): #Logistic hypothesis function
    return expit(np.dot(myX,mytheta))

$$J(\theta )=-[\frac{1}{m}\sum _{i=1}^m{y}^{(i)}log{h}_{\theta }(x^{(i)})+(1-y^{(i)})log(1-h_{\theta }(x^{(i)}))+\frac{\lambda }{2m}\sum _{j=1}^n{\theta }_j^2]$$

#损失函数，默认lambda为0，无正则化
def computeCost(mytheta,myX,myy,mylambda = 0.): 
    term1 = np.dot(-np.array(myy).T,np.log(h(mytheta,myX)))
    term2 = np.dot((1-np.array(myy)).T,np.log(1-h(mytheta,myX)))
    #正则化参数，跳过theta0
    regterm = (mylambda/2) * np.sum(np.dot(mytheta[1:].T,mytheta[1:]))
    return float( (1./m) * ( np.sum(term1 - term2) + regterm ) )

测试

#测试
#Check that with theta as zeros, cost returns about 0.693:
initial_theta = np.zeros((X.shape[1],1))
computeCost(initial_theta,X,y)

输出：0.6931471805599452

最优化θ的函数，代替梯度下降

#优化θ的函数，代替梯度下降
def optimizeTheta(mytheta,myX,myy,mylambda=0.):
    result = optimize.fmin(computeCost, x0=mytheta, args=(myX, myy, mylambda), maxiter=400, full_output=True)
    return result[0], result[1]

调用优化函数，得到优化后的参数theta和最小cost值

theta, mincost = optimizeTheta(initial_theta,X,y)

Optimization terminated successfully.
Current function value: 0.203498
Iterations: 157
Function evaluations: 287

得到参数theta后画出决策界限

#测试1的成绩
boundary_xs = np.array([np.min(X[:,1]), np.max(X[:,1])])
#自变量为测试1成绩x的直线
boundary_ys = (-1./theta[2])*(theta[0] + theta[1]*boundary_xs)
#打印数据点
plotData()
plt.plot(boundary_xs,boundary_ys,'b-',label='Decision Boundary')
plt.legend()

#测试
给出一个ex1 45分，ex2 85分的学生，计算成功的概率

print (h(theta,np.array([1, 45.,85.])))

输出：0.7762915904112411

预测函数

#预测函数，概率大于0.5则返回True，否则返回False
def makePrediction(mytheta, myx):
    return h(mytheta,myx) >= 0.5

计算分类模型的准确率

#通过优化后的theta，测试样本集上预测的正确率

#成功预测为1的个数
pos_correct = float(np.sum(makePrediction(theta,pos)))
#成功预测为0的个数
neg_correct = float(np.sum(np.invert(makePrediction(theta,neg))))
tot = len(pos)+len(neg)
prcnt_correct = float(pos_correct+neg_correct)/tot
print("Fraction of training samples correctly predicted: %f." % prcnt_correct)

Fraction of training samples correctly predicted: 0.890000.

测试2

导入数据

datafile = 'data/ex2data2.txt'


cols = np.loadtxt(datafile,delimiter=',',usecols=(0,1,2),unpack=True) #Read in comma separated data

X = np.transpose(np.array(cols[:-1]))
y = np.transpose(np.array(cols[-1:]))
m = y.size # number of training examples

X = np.insert(X,0,1,axis=1)

分为两部分

pos = np.array([X[i] for i in range(X.shape[0]) if y[i] == 1])
neg = np.array([X[i] for i in range(X.shape[0]) if y[i] == 0])

可视化数据的函数

#可视化数据函数
def plotData():
    plt.plot(pos[:,1],pos[:,2],'k+',label='y=1')
    plt.plot(neg[:,1],neg[:,2],'yo',label='y=0')
    plt.xlabel('Microchip Test 1')
    plt.ylabel('Microchip Test 2')
    plt.legend()
    plt.grid(True)

可视化

#可视化数据
plt.figure(figsize=(6,6))
plotData()

原本的输入X有两个自变量，为了更好的拟合决策边界，我们打算使用六阶的多项式来进行拟合

于是我们使用x1，x2这两个变量创造出了一个28维列向量作为输入

转换函数

def mapFeature( x1col, x2col ):
    degrees = 6
    #返回全为1的array
    out = np.ones( (x1col.shape[0], 1) )

    for i in range(1, degrees+1):#1-6
        for j in range(0, i+1):
            term1 = x1col ** (i-j)
            term2 = x2col ** (j)
            term  = (term1 * term2).reshape( term1.shape[0], 1 ) #（118，1）
            #把out和term水平叠加在一起（即把term水平加到out的右边）
            out = np.hstack(( out, term ))
    return out

测试

mappedX = mapFeature(X[:,1],X[:,2])
print(mappedX.shape)

输出：(118, 28)

初始化参数

#初始化参数
initial_theta = np.zeros((mappedX.shape[1],1))

采用minimize函数（BFGS算法），正则化方法对参数theta进行优化

#采用minimize函数（BFGS算法），正则化方法对参数theta进行优化
def optimizeRegularizedTheta(mytheta,myX,myy,mylambda=0.):
    result = optimize.minimize(computeCost, mytheta, args=(myX, myy, mylambda),  method='BFGS', options={
     "maxiter":500, "disp":False} )
    return np.array([result.x]), result.fun

计算得出优化后的参数theta和最小cost

#计算得出优化后的参数theta和最小cost
theta, mincost = optimizeRegularizedTheta(initial_theta,mappedX,y)

好了，得到参数θ后，我们画出决策界限
（采用等高线画法）

def plotBoundary(mytheta, myX, myy, mylambda=0.):
    theta, mincost = optimizeRegularizedTheta(mytheta,myX,myy,mylambda)
    xvals = np.linspace(-1,1.5,50)
    yvals = np.linspace(-1,1.5,50)
    zvals = np.zeros((len(xvals),len(yvals)))
    #zvals为决策界限
    for i in range(len(xvals)):
        for j in range(len(yvals)):
            myfeaturesij = mapFeature(np.array([xvals[i]]),np.array([yvals[j]]))
            zvals[i][j] = np.dot(theta,myfeaturesij.T)
    zvals = zvals.transpose()
    #转换为网格形数据
    u, v = np.meshgrid( xvals, yvals )
    #画等高线图
    mycontour = plt.contour( u, v, zvals, [0])
    #给等高线图加上标签
    myfmt = {
      0:'Lambda = %d'%mylambda}
    plt.clabel(mycontour, inline=1, fontsize=15, fmt=myfmt)
    plt.title("Decision Boundary")

绘制

#Build a figure showing contours for various values of regularization parameter, lambda
#It shows for lambda=0 we are overfitting, and for lambda=100 we are underfitting

#依次画出不同正则化参数的决策边界的图像

plt.figure(figsize=(12,10))
#加一个小图
#221表示两行两列索引为1
plt.subplot(221)
plotData()
plotBoundary(theta,mappedX,y,0.)

plt.subplot(222)
plotData()
plotBoundary(theta,mappedX,y,1.)

plt.subplot(223)
plotData()
plotBoundary(theta,mappedX,y,10.)

plt.subplot(224)
plotData()
plotBoundary(theta,mappedX,y,100.)

可见λ为0时，决策边界存在过拟合，λ为100时欠拟合

测试集1

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80.36675600171273,90.96014789746954,1
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测试集2

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