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

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)=θ0x0+θ1x1+…+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）

在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：

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function [ jVal,gradient ] = costFunction2( theta )
%COSTFUNCTION2 Summary of this function goes here
% linear regression -> y=theta0 + theta1*x
% parameter: x:m*n theta:n*1 y:m*1 (m=4,n=1)
%

%Data
x=[1;2;3;4];
y=[1.1;2.2;2.7;3.8];
m=size(x,1);

hypothesis = h_func(x,theta);
delta = hypothesis - y;
jVal=sum(delta.^2);

gradient(1)=sum(delta)/m;
gradient(2)=sum(delta.*x)/m;

end

function [ jVal,gradient ] = costFunction2( theta )
%COSTFUNCTION2 Summary of this function goes here
%   linear regression -> y=theta0 + theta1*x
%   parameter: x:m*n  theta:n*1   y:m*1   (m=4,n=1)
%

%Data
x=[1;2;3;4];
y=[1.1;2.2;2.7;3.8];
m=size(x,1);

hypothesis = h_func(x,theta);
delta = hypothesis - y;
jVal=sum(delta.^2);

gradient(1)=sum(delta)/m;
gradient(2)=sum(delta.*x)/m;

end

其中，h_func是hypothesis的结果：

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function [res] = h_func(inputx,theta)
%H_FUNC Summary of this function goes here
% Detailed explanation goes here

%cost function 2
res= theta(1)+theta(2)*inputx;function [res] = h_func(inputx,theta)
end

function [res] = h_func(inputx,theta)
%H_FUNC Summary of this function goes here
%   Detailed explanation goes here

%cost function 2
res= theta(1)+theta(2)*inputx;function [res] = h_func(inputx,theta)
end

Gradient_descent：

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

function [optTheta,functionVal,exitFlag]=Gradient_descent( )
%GRADIENT_DESCENT Summary of this function goes here
%   Detailed explanation goes here

options = optimset('GradObj','on','MaxIter',100);
initialTheta = zeros(2,1);
[optTheta,functionVal,exitFlag] = fminunc(@costFunction2,initialTheta,options);

end

result：

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>> [optTheta,functionVal,exitFlag] = Gradient_descent() Local minimum found. Optimization completed because the size of the gradient is less than the default value of the function tolerance. <stopping criteria details> optTheta = 0.3000 0.8600 functionVal = 0.0720 exitFlag = 1

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

Local minimum found.

Optimization completed because the size of the gradient is less than
the default value of the function tolerance.

<stopping criteria details>

optTheta =

0.3000
0.8600

functionVal =

0.0720

exitFlag =

1

即得y=0.3+0.86x;
验证：

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function [ parameter ] = checkcostfunc( ) %CHECKC2 Summary of this function goes here % check if the cost function works well % check with the matlab fit function as standard %check cost function 2 x=[1;2;3;4]; y=[1.1;2.2;2.7;3.8]; EXPR= {'x','1'}; p=fittype(EXPR); parameter=fit(x,y,p); end

function [ parameter ] = checkcostfunc(  )
%CHECKC2 Summary of this function goes here
%   check if the cost function works well
%   check with the matlab fit function as standard

%check cost function 2
x=[1;2;3;4];
y=[1.1;2.2;2.7;3.8];

EXPR= {'x','1'};
p=fittype(EXPR);
parameter=fit(x,y,p);

end

运行结果：

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>> checkcostfunc()

ans =

Linear model:
ans(x) = a*x + b
Coefficients (with 95% confidence bounds):
a = 0.86 (0.4949, 1.225)
b = 0.3 (-0.6998, 1.3)

>> checkcostfunc()

ans =

Linear model:
ans(x) = a*x + b
Coefficients (with 95% confidence bounds):
a =        0.86  (0.4949, 1.225)
b =         0.3  (-0.6998, 1.3)

和我们的结果一样。下面画图：

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function PlotFunc( xstart,xend )
%PLOTFUNC Summary of this function goes here
% draw original data and the fitted

%===================cost function 2====linear regression
%original data
x1=[1;2;3;4];
y1=[1.1;2.2;2.7;3.8];
%plot(x1,y1,'ro-','MarkerSize',10);
plot(x1,y1,'rx','MarkerSize',10);
hold on;

%fitted line - 拟合曲线
x_co=xstart:0.1:xend;
y_co=0.3+0.86*x_co;
%plot(x_co,y_co,'g');
plot(x_co,y_co);

hold off;
end

function PlotFunc( xstart,xend )
%PLOTFUNC Summary of this function goes here
%   draw original data and the fitted

%===================cost function 2====linear regression
%original data
x1=[1;2;3;4];
y1=[1.1;2.2;2.7;3.8];
%plot(x1,y1,'ro-','MarkerSize',10);
plot(x1,y1,'rx','MarkerSize',10);
hold on;

%fitted line - 拟合曲线
x_co=xstart:0.1:xend;
y_co=0.3+0.86*x_co;
%plot(x_co,y_co,'g');
plot(x_co,y_co);

hold off;
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：

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function [ jVal,gradient ] = costFunction( theta )
%COSTFUNCTION Summary of this function goes here
% Detailed explanation goes here

jVal= (theta(1)-5)^2+(theta(2)-5)^2;

gradient = zeros(2,1);
%code to compute derivative to theta
gradient(1) = 2 * (theta(1)-5);
gradient(2) = 2 * (theta(2)-5);

end

function [ jVal,gradient ] = costFunction( theta )
%COSTFUNCTION Summary of this function goes here
%   Detailed explanation goes here

jVal= (theta(1)-5)^2+(theta(2)-5)^2;

gradient = zeros(2,1);
%code to compute derivative to theta
gradient(1) = 2 * (theta(1)-5);
gradient(2) = 2 * (theta(2)-5);

end

Gradient_descent，进行参数优化

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

function [optTheta,functionVal,exitFlag]=Gradient_descent( )
%GRADIENT_DESCENT Summary of this function goes here
%   Detailed explanation goes here

options = optimset('GradObj','on','MaxIter',100);
initialTheta = zeros(2,1)
[optTheta,functionVal,exitFlag] = fminunc(@costFunction,initialTheta,options);

end

matlab主窗口中调用，得到优化厚的参数(θ1,θ2)=(5,5)

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[optTheta,functionVal,exitFlag] = Gradient_descent()

initialTheta =

0
0

Local minimum found.

Optimization completed because the size of the gradient is less than
the default value of the function tolerance.

<stopping criteria details>

optTheta =

5
5

functionVal =

0

exitFlag =

1

[optTheta,functionVal,exitFlag] = Gradient_descent()

initialTheta =

0
0

Local minimum found.

Optimization completed because the size of the gradient is less than
the default value of the function tolerance.

<stopping criteria details>

optTheta =

5
5

functionVal =

0

exitFlag =

1

[b]第四部分：Y=1/(1+e^X)型---逻辑回归（sigmod 函数拟合）[/b]

hypothesis function:

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function [res] = h_func(inputx,theta)

%cost function 3
tmp=theta(1)+theta(2)*inputx;%m*1
res=1./(1+exp(-tmp));%m*1

end

function [res] = h_func(inputx,theta)

%cost function 3
tmp=theta(1)+theta(2)*inputx;%m*1
res=1./(1+exp(-tmp));%m*1

end

cost function:

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function [ jVal,gradient ] = costFunction3( theta )
%COSTFUNCTION3 Summary of this function goes here
% Logistic Regression

x=[-3; -2; -1; 0; 1; 2; 3];
y=[0.01; 0.05; 0.3; 0.45; 0.8; 1.1; 0.99];
m=size(x,1);

%hypothesis data
hypothesis = h_func(x,theta);

%jVal-cost function & gradient updating
jVal=-sum(log(hypothesis+0.01).*y + (1-y).*log(1-hypothesis+0.01))/m;
gradient(1)=sum(hypothesis-y)/m; %reflect to theta1
gradient(2)=sum((hypothesis-y).*x)/m; %reflect to theta 2

end

function [ jVal,gradient ] = costFunction3( theta )
%COSTFUNCTION3 Summary of this function goes here
%   Logistic Regression

x=[-3;      -2;     -1;     0;      1;      2;     3];
y=[0.01;    0.05;   0.3;    0.45;   0.8;    1.1;    0.99];
m=size(x,1);

%hypothesis  data
hypothesis = h_func(x,theta);

%jVal-cost function  &  gradient updating
jVal=-sum(log(hypothesis+0.01).*y + (1-y).*log(1-hypothesis+0.01))/m;
gradient(1)=sum(hypothesis-y)/m;   %reflect to theta1
gradient(2)=sum((hypothesis-y).*x)/m;    %reflect to theta 2

end

Gradient_descent:

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function [optTheta,functionVal,exitFlag]=Gradient_descent( ) options = optimset('GradObj','on','MaxIter',100); initialTheta = [0;0]; [optTheta,functionVal,exitFlag] = fminunc(@costFunction3,initialTheta,options); end

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

options = optimset('GradObj','on','MaxIter',100);
initialTheta = [0;0];
[optTheta,functionVal,exitFlag] = fminunc(@costFunction3,initialTheta,options);

end

运行结果：

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[optTheta,functionVal,exitFlag] = Gradient_descent()

Local minimum found.

Optimization completed because the size of the gradient is less than
the default value of the function tolerance.

<stopping criteria details>

optTheta =

0.3526
1.7573

functionVal =

0.2498

exitFlag =

1

[optTheta,functionVal,exitFlag] = Gradient_descent()

Local minimum found.

Optimization completed because the size of the gradient is less than
the default value of the function tolerance.

<stopping criteria details>

optTheta =

0.3526
1.7573

functionVal =

0.2498

exitFlag =

1

画图验证：

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function PlotFunc( xstart,xend ) %PLOTFUNC Summary of this function goes here % draw original data and the fitted %===================cost function 3=====logistic regression %original data x=[-3; -2; -1; 0; 1; 2; 3]; y=[0.01; 0.05; 0.3; 0.45; 0.8; 1.1; 0.99]; plot(x,y,'rx','MarkerSize',10); hold on %fitted line x_co=xstart:0.1:xend; theta = [0.3526,1.7573]; y_co=h_func(x_co,theta); plot(x_co,y_co); hold off end

function PlotFunc( xstart,xend )
%PLOTFUNC Summary of this function goes here
%   draw original data and the fitted

%===================cost function 3=====logistic regression

%original data
x=[-3;      -2;     -1;     0;      1;      2;     3];
y=[0.01;    0.05;   0.3;    0.45;   0.8;    1.1;    0.99];
plot(x,y,'rx','MarkerSize',10);
hold on

%fitted line
x_co=xstart:0.1:xend;
theta = [0.3526,1.7573];
y_co=h_func(x_co,theta);
plot(x_co,y_co);
hold off

end

有朋友问，这里就补充一下logistic regression中gradient的推导：
令



则有




由于cost function




可得




所以gradient = -J'(theta) = (z-y)x