Matlab tips (21) matrix analysis -- partial least squares 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 ！

Partial least squares is a mathematical optimization technique , It finds the best function matching of a set of data by minimizing the sum of squares of errors . Use the simplest method to find some absolutely unknown truth values , And minimize the sum of the squares of the errors . Many other optimization problems can also be expressed in the form of least squares by minimizing energy or maximizing entropy .

Compared with the traditional multiple linear regression model , The characteristics of partial least squares are ：

1、 Be able to carry out regression modeling under the condition of serious multiple correlation of independent variables .

2、 Regression modeling is allowed when the number of sample points is less than the number of variables .

3、 Partial least squares regression will include all original independent variables in the final model .

4、 Partial least squares regression model is easier to identify system information and noise .

5、 In the partial least squares regression model , The regression coefficient of each independent variable will be easier to explain .

When calculating variance and covariance , There are two ways to find the coefficient before the sum sign ： When the set of sample points is randomly selected , We should take 1/(n-1); If it's not random , This coefficient can be taken as 1/n. This example I saw when searching for data , Here to share ,MATLAB Version is MATLAB2015b.

One . MATLAB Simulation

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% function ： matrix analysis -- Partial least squares regression 
% Environmental Science ：Win7,Matlab2015b
%Modi: C.S
% Time ：2022-06-26
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% 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. PLS The regression model 
%%
% 1.  Creating models 
k = 2;
[Xloadings,Yloadings,Xscores,Yscores,betaPLS,PLSPctVar,MSE,stats] = plsregress(P_train,T_train,k);

%%
% 2.  Principal component contribution rate analysis 
figure
percent_explained = 100 * PLSPctVar(2,:) / sum(PLSPctVar(2,:));
pareto(percent_explained)
xlabel(' The principal components ')
ylabel(' Contribution rate (%)')
title(' Principal component contribution rate ')

%%
% 3.  Prediction fitting 
N = size(P_test,1);
T_sim = [ones(N,1) P_test] * betaPLS;

%% V.  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 =

   88.9000   88.6958    0.0023
   88.7500   88.3616    0.0044
   87.0500   87.0516    0.0000
   88.3500   88.4744    0.0014
   88.8500   89.1747    0.0037
   88.5500   88.6033    0.0006
   87.3000   87.4689    0.0019
   88.7000   88.9760    0.0031
   84.6000   84.6464    0.0005
   89.6000   88.6938    0.0101

 Time has passed  1.684586  second .

3、 ... and . Summary

Examples of partial least squares regression for predictive analysis , Least squares is a common basic algorithm in all kinds of algorithms , Students interested in algorithms , It is suggested to follow the principle , Formula derivation, understand and learn well , This example may be used later , So here's a note . Learn one every day MATLAB Little knowledge , Let's learn and make progress together ！