Matlab tips (22) matrix analysis -- stepwise 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 ！

In statistics , regression analysis （regression analysis) It refers to a statistical analysis method to determine the quantitative relationship between two or more variables . Regression analysis according to the number of variables involved , It is divided into univariate regression and multiple regression analysis ; According to the number of dependent variables , It can be divided into simple regression analysis and multiple regression analysis ; According to the type of relationship between independent variables and dependent variables , It can be divided into linear regression analysis and nonlinear regression analysis .

In big data analysis , Regression analysis is a predictive modeling technique , It studies the dependent variable （ The goal is ） And independent variables （ predictor ） The relationship between . This technique is usually used for predictive analysis , Time series model and find the causal relationship between variables . for example , The relationship between reckless driving of drivers and the number of road traffic accidents , The best research method is regression .

The main methods of regression analysis ：

1. Linear Regression Linear regression
   It is one of the most well-known modeling techniques . Linear regression is usually one of the preferred techniques for learning prediction models . In this technology , The dependent variable is continuous , Independent variables can be continuous or discrete , The property of regression line is linear .

2. Logistic Regression Logical regression
   Logistic regression is used to calculate “ event =Success” and “ event =Failure” Probability . When the type of dependent variable is binary （1 / 0, really / false , yes / no ） variable , Logistic regression should be used . here ,Y The value of is 0 or 1.

3. Polynomial Regression Polynomial regression
   For a regression equation , If the exponent of the independent variable is greater than 1, So it's a polynomial regression equation .

4. Stepwise Regression Stepwise regression
   When dealing with multiple arguments , This form of regression can be used . In this technology , The selection of independent variables is completed in an automatic process , This includes non-human operations .

5. Ridge Regression Ridge return
   When there is multicollinearity between data （ The independent variables are highly correlated ） when , We need to use ridge regression analysis . When there is multicollinearity , Although the least square method （OLS） There is no deviation in the measured estimate , Their variances will also be large , Thus, the observed value is far from the true value . Ridge regression adds a bias value to the regression estimate , To reduce the standard error .

6. Lasso Regression Lasso return
   It's similar to ridge regression ,Lasso （Least Absolute Shrinkage and Selection Operator） The penalty term will also be given for the regression coefficient vector . Besides , It can reduce the degree of change and improve the accuracy of linear regression model

7. ElasticNet Return to
   ElasticNet yes Lasso and Ridge A hybrid of regression techniques . It USES L1 To train and L2 Priority as regularization matrix . When there are multiple related features ,ElasticNet It's very useful .Lasso Will pick one of them at random , and ElasticNet Then two... Will be selected .
   Lasso and Ridge The practical advantage between is , It allows the ElasticNet In the state of inheritance loop Ridge Some stability of .

Of the simulation example MATLAB Version is MATLAB2015b.

One . MATLAB Simulation

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% function ： matrix analysis -- regression analysis 
% Environmental Science ：Win7,Matlab2015b
%Modi: C.S
% Time ：2022-06-27
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% I.  Clear environment variables 
clear all
clc

tic

X=[7,26,6,60;
   1,29,15,52;
   11,56,8,20;
   11,31,8,47;
   7,52,6,33;
   11,55,9,22;
   3,71,17,6;
   1,31,22,44;
   2,54,18,22;
   21,47,4,26;
   1,40,23,34;
   11,66,9,12];  
% # Argument data ,12 Samples , Each sample 4 dimension 
Y=[78.5,74.3,104.3,87.6,95.9,109.2,102.7,72.5,93.1,115.9,83.8,113.3]';  % Dependent variable data   12 Two sample values 

xy =[X Y];

% Multiple stepwise regression analysis 
%xy For the raw data to be input , According to first x after y An array arranged in columns 
% Such as ：x1 x2 x3 x4 y wait 


% Calculate the dispersion matrix R(m,m)
[n,m]=size(xy);

%F1=0;F2=0;
%disp(' The mean for :')
xy_aver=mean(xy)% Calculating mean 
for i=1:m
    for j=1:i
        R(i,j)=0;
        for k=1:n
            R(i,j)=R(i,j)+(xy(k,i)-xy_aver(i))*(xy(k,j)-xy_aver(j));
        end
        R(j,i)=R(i,j);
    end
    SR(i)=sqrt(R(i,i));% Calculate the square root of the diagonal element 
end
%disp('************ Deviation Matrix & Value of SR ( Dispersion matrix R&SR) ***********') % Output dispersion matrix R, And SR
%[R  SR'] 
% Calculate the correlation coefficient R(m,m)
for i=1:m
    for j=1:i
        R(i,j)=R(i,j)/(SR(i)*SR(j));
        R(j,i)=R(i,j);
    end
end
%disp('********** Correlation Coefficient Matrix ( Correlation coefficient matrix R) **********')% Output correlation coefficient matrix R
%R
flag=1;% Whether the stepwise regression is repeated 
while(flag)
    disp('******** Stepwise Regression Analysis Start *************')
    F1=input(' Remove threshold value :F1=');
    F2=input(' Introduce threshold value :F2=');
    S=0;% Calculation steps 
    L=0;% The number of independent variables of the equation is introduced 
    FQ=n-1;% Degrees of freedom of the sum of squares of residuals     
    disp('************** Discriminant Value of Contribution V **************')
    Imin(1)=0;Imax=1:m-1;% Definition has been introduced ( Minimum ) And not introduced ( Maximum ) The ordinal number of the variable 
    inn=0;outt=0;% The sequence number of the introduced and eliminated variables 
    while(1)
    %    pause
        VN=1E+08;% The minimum value of the independent variable contribution of the equation has been introduced 
        VX=0;% The maximum value of the independent variable contribution without introducing the equation 
        IN=0;% The sequence number of the introduced argument with the least contribution 
        IX=0;% The sequence number of the uninitialized argument with the greatest contribution 
        S=S+1;
        disp(['--------- step = ' int2str(S) '------------'])% Number of output steps 
        for i=1:m-1
            if R(i,i)<1E-08
                continue
            end
    %        disp(['VMAX=' int2str(VX) '; IMAX=' int2str(IX)]) % Output Vmax=VX;Imax=IX;
            V(i)=R(m,i)^2/R(i,i);% Calculate the variance contribution of the introduced variables 
            if V(i)>=0
                if V(i)>VX % Find the maximum value of variance contribution of uninitialized variables 
                    for in=1:length(Imax)
                        if i==Imax(in)
                             VX=V(i);IX=i;
                         end
                     end
                end
            end
            if abs(V(i))<VN % Find the minimum value of variance contribution of the introduced variable 
                for out=1:length(Imin)
                    if i==Imin(out)
                        VN=abs(V(i));IN=i;
                     end
                 end                
            end        
    %disp([' Variance contribution :V=' num2str(V(i)) 'VX=' num2str(VX) 'IX=' int2str(IX) 'VN=' num2str(VN) 'IN=' int2str(IN)])
        end
    %    Imax(inn+1)=IX;inn=inn+1;
        t=find(Imax==IX);
        Imax(t)=[];
        disp(['********  Variance contribution V **********' num2str(V)])
        disp(['VMAX=' num2str(VX) '; IMAX=' int2str(IX)]) % Output Vmax=VX;Imax=IX;
    %    disp(['VMIN=' num2str(VN) '; IMIN=' int2str(IN)]) % Output Vmin=VN;Imin=IN;
        if S==1
            disp(['S=' int2str(S)]) % Output S=1
        else
            disp(['VMIN=' num2str(VN) '; IMIN=' int2str(IN)]) % Output Vmin=VN;Imin=IN;
        end
        if S==1%||S==2||S==3
            FE=VX*(n-L-2)/(R(m,m)-VX);
            disp(['FE=' num2str(FE)]) % Output  FE
            if FE<F1
                if L~=0
                    disp('Neither Delete Out Nor Select In!')
                else
                    disp('May Be Smaller F1 And F2')
                    disp('The Stepwise Regression Analysis End!')
                    break;% Program end 
                end
            else
                L=L+1;FQ=FQ-1;K=IX;
                disp(['X' int2str(K) ' Be Selected In'])
                Imin(outt+1)=IX;outt=outt+1;
                disp(['L = ' int2str(L) ])
                R=xiaoqu(R,K) % Call subfunction , Perform elimination transformation 
                if L~=m-1
                    continue;
                end
                disp('Already Selecting End')
                break;
            end
        else
            % Calculate the value of the excluded variable F Test value 
            FT=VN*(n-L-1)/R(m,m);
            disp([' Eliminate variables F Test value ' num2str(FT)])
            if FT>=F2
                FE=VX*(n-L-2)/(R(m,m)-VX);
                disp(['***FE=' num2str(FE)]) % Output  FE
                if FE<F1
                    if L~=0
                        disp('Neither Delete Out Nor Select In!')
                        disp('The Stepwise Regression Analysis End!')
                        break;% Program end 
                    else
                        disp('May Be Smaller F1 And F2')
                        disp('The Stepwise Regression Analysis End!')
                        break;% Program end 
                    end
                else
                    L=L+1;FQ=FQ-1;K=IX;
                    disp(['X' int2str(K) ' Be Selected In'])
                    disp(['L = ' int2str(L) ])
                    Imin(outt+1)=IX;outt=outt+1;
                    R=xiaoqu(R,K) % Call subfunction , Perform elimination transformation 
                    if L~=m-1
                        continue;
                    end
                    disp('Already Selecting End')
                    break;
                end
            else
                 L=L-1;FQ=FQ+1;K=IN;
                 disp(['X' int2str(K) ' Be Deleted Out'])
                 disp(['L = ' int2str(L) ' (No. of Variable Selected)'])
                 R=xiaoqu(R,K) % Call subfunction 
                 continue
            end
        end
    end
    % Output the corresponding calculation results 
    for i=1:m-1
        kk=R(i,m)*R(m,i);
        if kk<0
            B(i)=R(i,m)*SR(m)/SR(i);
        else
            B(i)=0;
        end
    end
    B0=xy_aver(m); 
    for i=1:m-1
        B0=B0-B(i)*xy_aver(i);
    end
    disp([' The regression coefficient is :' num2str(B0) ' ' num2str(B)])
    disp(' The regression equation is ：')
    disp(['Y=' num2str(B0)])
    for i=1:m-1
        if B(i)~=0
            if B(i)>0
                disp(['+' num2str(B(i)) 'X' int2str(i)]);
            else
                disp([num2str(B(i)) 'X' int2str(i)]);
            end
        end    
    end        

    Q=SR(m)^2*R(m,m);% The sum of squared residuals 
    disp(['Sum of SQuares of Residual Error( The sum of squared residuals ) Q = ' num2str(Q)])
    S=SR(m)*sqrt(R(m,m)/FQ);% Residual standard deviation 
    disp(['Standard Deviation( Residual standard deviation , That is, the root mean square of the model error ) S = ' num2str(S)])
    RR=sqrt(1-R(m,m));% Complex correlation coefficient 
    disp(['Multiple Correlation Coefficient( Complex correlation coefficient ) R = ' num2str(RR)])
    FF=FQ*(1-R(m,m))/(L*R(m,m));% Significance test of regression equation F value 
    disp(['F Value for Test of Regression( Significance test of regression equation , Statistics of regression model ) F = ' num2str(FF)])
    %F=SH*(m-n-1)/(SX*n);%F- statistic 
    %PROB = 1 - fcdf(FF,m,n-length(Imin)-1)% And statistics F The corresponding probability P

    for i=1:m-1
        CC=R(i,i)*R(m,m);
        T(i)=R(i,m)/sqrt(CC/FQ);% Of each regression coefficient t Test value 
        R1(i)=R(i,m)/sqrt(CC+R(i,m)^2);% Partial correlation coefficient of each variable 
    end
    disp(['t Test Value of Argument( Of each regression coefficient t Test value ):' num2str(T)])
    disp(['Partial Corre.Coeffi.Ofargu.( Partial correlation coefficient of each variable ):' num2str(R1)])

    
    flag=input(' Whether to conduct stepwise regression analysis again (1: yes ;0: no )：');
end
toc

Two . Simulation results

xy_aver =

    7.2500   46.5000   12.0833   31.5000   94.2583

******** Stepwise Regression Analysis Start *************
 Remove threshold value :F1=10
 Introduce threshold value :F2=100
************** Discriminant Value of Contribution V **************
--------- step = 1------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0.64727; IMAX=4
S=1
FE=18.35
X4 Be Selected In
L = 1
--------- step = 2------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0.32314; IMAX=1
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 18.35
X4 Be Deleted Out
L = 0 (No. of Variable Selected)
--------- step = 3------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0.6395; IMAX=2
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 7.1199
X4 Be Deleted Out
L = -1 (No. of Variable Selected)
--------- step = 4------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0.28898; IMAX=3
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 22.02
X4 Be Deleted Out
L = -2 (No. of Variable Selected)
--------- step = 5------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 8.4145
X4 Be Deleted Out
L = -3 (No. of Variable Selected)
--------- step = 6------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 25.69
X4 Be Deleted Out
L = -4 (No. of Variable Selected)
--------- step = 7------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 9.709
X4 Be Deleted Out
L = -5 (No. of Variable Selected)
--------- step = 8------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 29.36
X4 Be Deleted Out
L = -6 (No. of Variable Selected)
--------- step = 9------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 11.0035
X4 Be Deleted Out
L = -7 (No. of Variable Selected)
--------- step = 10------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 33.03
X4 Be Deleted Out
L = -8 (No. of Variable Selected)
--------- step = 11------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 12.2981
X4 Be Deleted Out
L = -9 (No. of Variable Selected)
--------- step = 12------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 36.7
X4 Be Deleted Out
L = -10 (No. of Variable Selected)
--------- step = 13------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 13.5926
X4 Be Deleted Out
L = -11 (No. of Variable Selected)
--------- step = 14------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 40.37
X4 Be Deleted Out
L = -12 (No. of Variable Selected)
--------- step = 15------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 14.8871
X4 Be Deleted Out
L = -13 (No. of Variable Selected)
--------- step = 16------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 44.04
X4 Be Deleted Out
L = -14 (No. of Variable Selected)
--------- step = 17------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 16.1817
X4 Be Deleted Out
L = -15 (No. of Variable Selected)
--------- step = 18------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 47.71
X4 Be Deleted Out
L = -16 (No. of Variable Selected)
--------- step = 19------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 17.4762
X4 Be Deleted Out
L = -17 (No. of Variable Selected)
--------- step = 20------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 51.38
X4 Be Deleted Out
L = -18 (No. of Variable Selected)
--------- step = 21------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 18.7707
X4 Be Deleted Out
L = -19 (No. of Variable Selected)
--------- step = 22------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 55.05
X4 Be Deleted Out
L = -20 (No. of Variable Selected)
--------- step = 23------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 20.0653
X4 Be Deleted Out
L = -21 (No. of Variable Selected)
--------- step = 24------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 58.72
X4 Be Deleted Out
L = -22 (No. of Variable Selected)
--------- step = 25------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 21.3598
X4 Be Deleted Out
L = -23 (No. of Variable Selected)
--------- step = 26------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 62.39
X4 Be Deleted Out
L = -24 (No. of Variable Selected)
--------- step = 27------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 22.6543
X4 Be Deleted Out
L = -25 (No. of Variable Selected)
--------- step = 28------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 66.06
X4 Be Deleted Out
L = -26 (No. of Variable Selected)
--------- step = 29------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 23.9488
X4 Be Deleted Out
L = -27 (No. of Variable Selected)
--------- step = 30------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 69.73
X4 Be Deleted Out
L = -28 (No. of Variable Selected)
--------- step = 31------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 25.2434
X4 Be Deleted Out
L = -29 (No. of Variable Selected)
--------- step = 32------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 73.4
X4 Be Deleted Out
L = -30 (No. of Variable Selected)
--------- step = 33------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 26.5379
X4 Be Deleted Out
L = -31 (No. of Variable Selected)
--------- step = 34------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 77.07
X4 Be Deleted Out
L = -32 (No. of Variable Selected)
--------- step = 35------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 27.8324
X4 Be Deleted Out
L = -33 (No. of Variable Selected)
--------- step = 36------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 80.74
X4 Be Deleted Out
L = -34 (No. of Variable Selected)
--------- step = 37------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 29.127
X4 Be Deleted Out
L = -35 (No. of Variable Selected)
--------- step = 38------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 84.41
X4 Be Deleted Out
L = -36 (No. of Variable Selected)
--------- step = 39------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 30.4215
X4 Be Deleted Out
L = -37 (No. of Variable Selected)
--------- step = 40------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 88.08
X4 Be Deleted Out
L = -38 (No. of Variable Selected)
--------- step = 41------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 31.716
X4 Be Deleted Out
L = -39 (No. of Variable Selected)
--------- step = 42------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 91.75
X4 Be Deleted Out
L = -40 (No. of Variable Selected)
--------- step = 43------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 33.0106
X4 Be Deleted Out
L = -41 (No. of Variable Selected)
--------- step = 44------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 95.42
X4 Be Deleted Out
L = -42 (No. of Variable Selected)
--------- step = 45------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 34.3051
X4 Be Deleted Out
L = -43 (No. of Variable Selected)
--------- step = 46------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 99.09
X4 Be Deleted Out
L = -44 (No. of Variable Selected)
--------- step = 47------------
********  Variance contribution V **********0.53207      0.6395     0.26368     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 35.5996
X4 Be Deleted Out
L = -45 (No. of Variable Selected)
--------- step = 48------------
********  Variance contribution V **********0.32314   0.0058487     0.28898     0.64727
VMAX=0; IMAX=0
VMIN=0.64727; IMIN=4
 Eliminate variables F Test value 102.7599
***FE=0
Neither Delete Out Nor Select In!
The Stepwise Regression Analysis End!
 The regression coefficient is :117.3697 0           0           0    -0.73369
 The regression equation is ：
Y=117.3697
-0.73369X4
Sum of SQuares of Residual Error( The sum of squared residuals ) Q = 883.291
Standard Deviation( Residual standard deviation , That is, the root mean square of the model error ) S = 3.9715
Multiple Correlation Coefficient( Complex correlation coefficient ) R = 0.80453
F Value for Test of Regression( Significance test of regression equation , Statistics of regression model ) F = -2.2836
t Test Value of Argument( Of each regression coefficient t Test value ):7.16248     0.963604     -6.77331     -10.1371
Partial Corre.Coeffi.Ofargu.( Partial correlation coefficient of each variable ):0.69145     0.12771    -0.67106    -0.80453
 Whether to conduct stepwise regression analysis again (1: yes ;0: no )：0
 Time has passed  14.692745  second .

3、 ... and . Summary

Examples of stepwise regression analysis , It may be used later , Take a note here . Learn one every day MATLAB Little knowledge , Let's learn and make progress together ！