Matlab tips (20) matrix analysis -- principal component regression

Preface

MATLAB Learning about image processing is very friendly , You can start from scratch , There are many encapsulated functions that can be called directly for basic image processing , This series of articles is mainly to introduce some of you in MATLAB Some concept functions are commonly used in routine demonstration ！

Principal component regression , Is a kind of regression analysis . When the independent variable has a complex collinear stiffness , Statistical analysis method for improving least square regression . Hotelling 1933 In, the principal component analysis was first used to analyze the related structures ,1965 Marcy proposed principal component regression in . Basic steps ：（1） Convert argument to standard score ;（2） Find out the principal component of this standard score , Remove the small principal component of the characteristic root ;（3） The least square method is used to make the regression of the dependent variable to the retained principal component ;（4） Replace the principal components in the regression equation with the linear combination of standard components , Get the regression equation given by the standard score .

Principal component method is through linear transformation , Combine the original multiple indicators into a few independent indicators that can fully reflect the overall information , Thus, the problem of collinearity between variables can be avoided without losing important information , For further analysis . Each principal component extracted from the principal component analysis is a linear combination of the original multiple indicators . This is also an example encountered when searching for data , Here to share ,MATLAB Version is MATLAB2015b.

One . MATLAB Simulation

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% function ： matrix analysis -- Principal component regression 
% Environmental Science ：Win7,Matlab2015b
%Modi: C.S
% Time ：2022-06-25
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% I.  Clear environment variables 
clear all
clc

tic

%% II.  Import data 
load spectra;

%% III.  Randomly divide the training set and the test set 
temp = randperm(size(NIR, 1));
% temp = 1:60;
%%
% 1.  Training set ——50 Samples 
P_train = NIR(temp(1:50),:);
T_train = octane(temp(1:50),:);
%%
% 2.  Test set ——10 Samples 
P_test = NIR(temp(51:end),:);
T_test = octane(temp(51:end),:);

%% IV.  Principal component analysis 
%%
% 1.  Principal component contribution rate analysis 
[PCALoadings,PCAScores,PCAVar] = princomp(NIR);
figure
percent_explained = 100 * PCAVar / sum(PCAVar);
pareto(percent_explained)
xlabel(' The principal components ')
ylabel(' Contribution rate (%)')
title(' Principal component contribution rate ')

%%
% 2.  First principal component vs. Second principal component 
[PCALoadings,PCAScores,PCAVar] = princomp(P_train);
figure
plot(PCAScores(:,1),PCAScores(:,2),'r+')
hold on
[PCALoadings_test,PCAScores_test,PCAVar_test] = princomp(P_test);
plot(PCAScores_test(:,1),PCAScores_test(:,2),'o')
xlabel('1st Principal Component')
ylabel('2nd Principal Component')
legend('Training Set','Testing Set','location','best')

%% V.  Principal component regression model 
%%
% 1.  Creating models 
k = 4;
betaPCR = regress(T_train-mean(T_train),PCAScores(:,1:k));
betaPCR = PCALoadings(:,1:k) * betaPCR;
betaPCR = [mean(T_train)-mean(P_train) * betaPCR;betaPCR];
%%
% 2.  Prediction fitting 
N = size(P_test,1);
T_sim = [ones(N,1) P_test] * betaPCR;

%% VI.  Result analysis and mapping 
%%
% 1.  Relative error error
error = abs(T_sim - T_test) ./ T_test;
%%
% 2.  Coefficient of determination R^2
R2 = (N * sum(T_sim .* T_test) - sum(T_sim) * sum(T_test))^2 / ((N * sum((T_sim).^2) - (sum(T_sim))^2) * (N * sum((T_test).^2) - (sum(T_test))^2)); 
%%
% 3.  Results contrast 
result = [T_test T_sim error]

%% 
% 4.  mapping 
figure
plot(1:N,T_test,'b:*',1:N,T_sim,'r-o')
legend(' True value ',' Predictive value ','location','best')
xlabel(' Prediction samples ')
ylabel(' Octane number ')
string = {
    ' Comparison of prediction results of octane number content in test set ';['R^2=' num2str(R2)]};
title(string)

toc

Two . Simulation results

result =

   87.0000   86.8053    0.0022
   85.5000   85.3947    0.0012
   88.6500   88.0991    0.0062
   88.4000   87.9918    0.0046
   84.7000   84.6877    0.0001
   88.2500   87.7173    0.0060
   88.0000   88.1574    0.0018
   84.6000   84.4274    0.0020
   88.8500   88.8830    0.0004
   85.2500   84.8463    0.0047

 Time has passed  1.935602  second .

3、 ... and . Summary

Examples of principal component regression , It is also the first to determine the data with high contribution value through principal component analysis , Then the model is obtained by training the training set , Compare with test set results , It turned out pretty well . Learn one every day MATLAB Little knowledge , Let's learn and make progress together ！