Matlab实现线性回归和逻辑回归: Linear Regression & Logistic Regression

原文:http://blog.csdn.net/abcjennifer/article/details/7732417

 

本文为Maching Learning 栏目补充内容,为上几章中所提到单参数线性回归多参数线性回归和 逻辑回归的总结版。旨在帮助大家更好地理解回归,所以我在Matlab中分别对他们予以实现,在本文中由易到难地逐个介绍。
 
 

本讲内容:

Matlab 实现各种回归函数

=========================

 

基本模型

 

Y=θ0+θ1X1型---线性回归(直线拟合)

解决过拟合问题---Regularization

 

Y=1/(1+e^X)型---逻辑回归(sigmod 函数拟合)

=========================

 

 
第一部分:基本模型

 

 

在解决拟合问题的解决之前,我们首先回忆一下线性回归和逻辑回归的基本模型。

 

设待拟合参数 θn*1 和输入参数[ xm*n, ym*1 ] 。

 

对于各类拟合我们都要根据梯度下降的算法,给出两部分:

①   cost function(指出真实值y与拟合值h<hypothesis>之间的距离):给出cost function 的表达式,每次迭代保证cost function的量减小;给出梯度gradient,即cost function对每一个参数θ的求导结果。

function [ jVal,gradient ] = costFunction ( theta )

 

②   Gradient_descent(主函数):用来运行梯度下降算法,调用上面的cost function进行不断迭代,直到最大迭代次数达到给定标准或者cost function返回值不再减小。

function [optTheta,functionVal,exitFlag]=Gradient_descent( )

 

线性回归:拟合方程为hθ(x)=θ0x01x1+…+θnxn,当然也可以有xn的幂次方作为线性回归项(如),这与普通意义上的线性不同,而是类似多项式的概念。

其cost function 为:

 

逻辑回归:拟合方程为hθ(x)=1/(1+e^(θTx)),其cost function 为:

 

cost function对各θj的求导请自行求取,看第三章最后一图,或者参见后文代码。

 

后面,我们分别对几个模型方程进行拟合,给出代码,并用matlab中的fit函数进行验证。

 

 

 

第二部分:Y=θ0+θ1X1型---线性回归(直线拟合)
 

Matlab 线性拟合 & 非线性拟合中我们已经讲过如何用matlab自带函数fit进行直线和曲线的拟合,非常实用。而这里我们是进行ML课程的学习,因此研究如何利用前面讲到的梯度下降法(gradient descent)进行拟合。

 

cost function:
 1 function [ jVal,gradient ] = costFunction2( theta )
 2 %COSTFUNCTION2 Summary of this function goes here
 3 %   linear regression -> y=theta0 + theta1*x
 4 %   parameter: x:m*n  theta:n*1   y:m*1   (m=4,n=1)
 5 %   
 6 
 7 %Data
 8 x=[1;2;3;4];
 9 y=[1.1;2.2;2.7;3.8];
10 m=size(x,1);
11 
12 hypothesis = h_func(x,theta);
13 delta = hypothesis - y;
14 jVal=sum(delta.^2);
15 
16 gradient(1)=sum(delta)/m;
17 gradient(2)=sum(delta.*x)/m;
18 
19 end

其中,h_func是hypothesis的结果:

1 function [res] = h_func(inputx,theta)
2 %H_FUNC Summary of this function goes here
3 %   Detailed explanation goes here
4 
5 
6 %cost function 2
7 res= theta(1)+theta(2)*inputx;function [res] = h_func(inputx,theta)
8 end

Gradient_descent:

1 function [optTheta,functionVal,exitFlag]=Gradient_descent( )
2 %GRADIENT_DESCENT Summary of this function goes here
3 %   Detailed explanation goes here
4 
5   options = optimset('GradObj','on','MaxIter',100);
6   initialTheta = zeros(2,1);
7   [optTheta,functionVal,exitFlag] = fminunc(@costFunction2,initialTheta,options);
8 
9 end

result:

 1 >> [optTheta,functionVal,exitFlag] = Gradient_descent()
 2 
 3 Local minimum found.
 4 
 5 Optimization completed because the size of the gradient is less than
 6 the default value of the function tolerance.
 7 
 8 <stopping criteria details>
 9 
10 
11 optTheta =
12 
13     0.3000
14     0.8600
15 
16 
17 functionVal =
18 
19     0.0720
20 
21 
22 exitFlag =
23 
24      1
即得y=0.3+0.86x;
验证:
 1 function [ parameter ] = checkcostfunc(  )
 2 %CHECKC2 Summary of this function goes here
 3 %   check if the cost function works well
 4 %   check with the matlab fit function as standard
 5 
 6 %check cost function 2
 7 x=[1;2;3;4];
 8 y=[1.1;2.2;2.7;3.8];
 9 
10 EXPR= {'x','1'};
11 p=fittype(EXPR);
12 parameter=fit(x,y,p);
13 
14 end

运行结果:

1 >> checkcostfunc()
2 
3 ans = 
4 
5      Linear model:
6      ans(x) = a*x + b
7      Coefficients (with 95% confidence bounds):
8        a =        0.86  (0.4949, 1.225)
9        b =         0.3  (-0.6998, 1.3)

和我们的结果一样。下面画图:

 1 function PlotFunc( xstart,xend )
 2 %PLOTFUNC Summary of this function goes here
 3 %   draw original data and the fitted 
 4 
 5 
 6 
 7 %===================cost function 2====linear regression
 8 %original data
 9 x1=[1;2;3;4];
10 y1=[1.1;2.2;2.7;3.8];
11 %plot(x1,y1,'ro-','MarkerSize',10);
12 plot(x1,y1,'rx','MarkerSize',10);
13 hold on;
14 
15 %fitted line - 拟合曲线
16 x_co=xstart:0.1:xend;
17 y_co=0.3+0.86*x_co;
18 %plot(x_co,y_co,'g');
19 plot(x_co,y_co);
20 
21 hold off;
22 end
 
 
第三部分:解决过拟合问题---Regularization
 
过拟合问题解决方法我们已在第三章中讲过,利用Regularization的方法就是在cost function中加入关于θ的项,使得部分θ的值偏小,从而达到fit效果。
例如定义costfunction J(θ): jVal=(theta(1)-5)^2+(theta(2)-5)^2;

在每次迭代中,按照gradient descent的方法更新参数θ:θ(i)-=gradient(i),其中gradient(i)是J(θ)对θi求导的函数式,在此例中就有gradient(1)=2*(theta(1)-5), gradient(2)=2*(theta(2)-5)。

 

函数costFunction, 定义jVal=J(θ)和对两个θ的gradient:

 1 function [ jVal,gradient ] = costFunction( theta )
 2 %COSTFUNCTION Summary of this function goes here
 3 %   Detailed explanation goes here
 4 
 5 jVal= (theta(1)-5)^2+(theta(2)-5)^2;
 6 
 7 gradient = zeros(2,1);
 8 %code to compute derivative to theta
 9 gradient(1) = 2 * (theta(1)-5);
10 gradient(2) = 2 * (theta(2)-5);
11 
12 end

Gradient_descent,进行参数优化

1 function [optTheta,functionVal,exitFlag]=Gradient_descent( )
2 %GRADIENT_DESCENT Summary of this function goes here
3 %   Detailed explanation goes here
4 
5  options = optimset('GradObj','on','MaxIter',100);
6  initialTheta = zeros(2,1)
7  [optTheta,functionVal,exitFlag] = fminunc(@costFunction,initialTheta,options);
8   
9 end

matlab主窗口中调用,得到优化厚的参数(θ1,θ2)=(5,5)

 1  [optTheta,functionVal,exitFlag] = Gradient_descent()
 2 
 3 initialTheta =
 4 
 5      0
 6      0
 7 
 8 
 9 Local minimum found.
10 
11 Optimization completed because the size of the gradient is less than
12 the default value of the function tolerance.
13 
14 <stopping criteria details>
15 
16 
17 optTheta =
18 
19      5
20      5
21 
22 
23 functionVal =
24 
25      0
26 
27 
28 exitFlag =
29 
30      1

第四部分:Y=1/(1+e^X)型---逻辑回归(sigmod 函数拟合)

 hypothesis function:

1 function [res] = h_func(inputx,theta)
2 
3 %cost function 3
4 tmp=theta(1)+theta(2)*inputx;%m*1
5 res=1./(1+exp(-tmp));%m*1
6 
7 end

cost function:

 1 function [ jVal,gradient ] = costFunction3( theta )
 2 %COSTFUNCTION3 Summary of this function goes here
 3 %   Logistic Regression
 4 
 5 x=[-3;      -2;     -1;     0;      1;      2;     3];
 6 y=[0.01;    0.05;   0.3;    0.45;   0.8;    1.1;    0.99];
 7 m=size(x,1);
 8 
 9 %hypothesis  data
10 hypothesis = h_func(x,theta);
11 
12 %jVal-cost function  &  gradient updating
13 jVal=-sum(log(hypothesis+0.01).*y + (1-y).*log(1-hypothesis+0.01))/m;
14 gradient(1)=sum(hypothesis-y)/m;   %reflect to theta1
15 gradient(2)=sum((hypothesis-y).*x)/m;    %reflect to theta 2
16 
17 end

Gradient_descent:

1 function [optTheta,functionVal,exitFlag]=Gradient_descent( )
2 
3  options = optimset('GradObj','on','MaxIter',100);
4  initialTheta = [0;0];
5  [optTheta,functionVal,exitFlag] = fminunc(@costFunction3,initialTheta,options);
6 
7 end

运行结果:

 1  [optTheta,functionVal,exitFlag] = Gradient_descent()
 2 
 3 Local minimum found.
 4 
 5 Optimization completed because the size of the gradient is less than
 6 the default value of the function tolerance.
 7 
 8 <stopping criteria details>
 9 
10 
11 optTheta =
12 
13     0.3526
14     1.7573
15 
16 
17 functionVal =
18 
19     0.2498
20 
21 
22 exitFlag =
23 
24      1

 

 
画图验证:
 
 1 function PlotFunc( xstart,xend )
 2 %PLOTFUNC Summary of this function goes here
 3 %   draw original data and the fitted 
 4 
 5 %===================cost function 3=====logistic regression
 6 
 7 %original data
 8 x=[-3;      -2;     -1;     0;      1;      2;     3];
 9 y=[0.01;    0.05;   0.3;    0.45;   0.8;    1.1;    0.99];
10 plot(x,y,'rx','MarkerSize',10);
11 hold on
12 
13 %fitted line
14 x_co=xstart:0.1:xend;
15 theta = [0.3526,1.7573];
16 y_co=h_func(x_co,theta);
17 plot(x_co,y_co);
18 hold off
19 
20 end

 

 
 
有朋友问,这里就补充一下logistic regression中gradient的推导:
则有
由于cost function
可得
所以gradient = -J'(theta) = (z-y)x

 

 

 

 

posted @ 2015-12-14 22:42  莫小  阅读(5414)  评论(0编辑  收藏