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