Ensemble Neural Network in Matlab

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Yulius on 21 Jan 2013

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Dear All, I am creating ensemble neural network comprises 3 component neural networks (NNs) with different number of hidden neurons. There are 3 input neurons and 2 output neurons with 1 hidden layer where hidden neurons are varied set as 3, 2, and 1 for each component NN. The 3 input neurons correspond with on-site displacement measurement at 3 different locations while the 2 output neurons correspond with Young’s modulus (E) and horizontal to vertical stress ratio (k). Total training data is 39 and testing data is 10.

These are the testing results:

E = [77.78224 42.33231 59.33344 53.02192 49.52091 85.40332 56.09939 53.33929 56.81571 16.95089];

target E = [80 70 60 30 50 80 40 70 30 20];

error E = [-2.21776 -27.6677 -0.66656 23.02192 -0.47909 5.403321 16.09939 -16.6607 26.81571 -3.04911];

k = [2.104783 2.32816 2.943708 3.00457 2.723133 2.291231 3.009603 3.024749 3.021532 3.002399];

target k = [1 1.5 2 2 2.5 2.5 3 3 3.5 4];

error k = [1.104783 0.82816 0.943708 1.00457 0.223133 -0.20877 0.009603 0.024749 -0.47847 -0.9976];

error E = E – target E

The results are not good especially for k value.

I worried that the correlation between input and output is not that good; hence I checked using simple model. I used 3 inputs (x1, x2, x3) and 2 outputs (y, z) and they have these relationships: y = x1^3-x2^2+x3; z = 1/x1-1/x2^2+1/(x3^2)

Total training data is 10 and testing data is 5. But again the testing result is not good especially for z.

y = [2831.213 5304.791 6902.318 12844.01 24449.48];

target y = [2088 4758 6650 11896 24048];

error y = [743.2125 546.7907 252.318 948.0119 401.4825];

z = [-0.18233 -0.18192 -0.18165 -0.16924 -0.16051];

target z = [-0.00704 -0.013 -0.01013 -0.01226 -0.01565];

error z = [-0.17528 -0.16893 -0.17153 -0.15698 -0.14486]

Please advise how to improve the ensemble neural networks.

Below is the code of my neural network for your reference:

clc % clear the command window clear; rand('seed', 1); randn('seed', 1);

ensemblesize =3; maxepoch=30;

for runno=1:1:10

 fprintf('ENN  dda 3 different number of hidden nodes       epoch = %g    hidden node no = var', maxepoch);
 fprintf('\n');

% training data set

E=[20 30 40 50 60 70 20 30 40 50 60 80 20 40 50 70 80 20 30 40 60 70 20 30 50 60 80 20 40 50 60 70 80 30 40 50 60 70 80]; k=[1 1 1 1 1 1 1.5 1.5 1.5 1.5 1.5 1.5 2 2 2 2 2 2.5 2.5 2.5 2.5 2.5 3 3 3 3 3 3.5 3.5 3.5 3.5 3.5 3.5 4 4 4 4 4 4];

ux1=[0.82 0.76 0.75 0.69 0.74 0.83 1.31 1.35 1.1 1.14 1.39 1.09 1.84 1.62 1.66 1.6 1.56 2 1.95 1.96 1.98 1.89 2.79 2.68 2.41 2.51 2.55 3.63 2.92 2.91 2.93 3 2.86 3.38 3.41 4.16 3.31 3.32 3.27]; ux2=[-0.83 -0.83 -0.79 -0.77 -0.77 -0.89 -1.3 -1.6 -1.28 -1.3 -1.6 -1.26 -1.93 -1.84 -1.91 -1.86 -1.78 -2.28 -2.29 -2.28 -2.27 -2.29 -2.88 -2.97 -2.78 -2.81 -2.92 -3.51 -3.44 -3.38 -3.43 -3.46 -3.36 -3.92 -3.97 -3.83 -3.97 -3.9 -3.89]; ux3=[-0.53 -0.58 -0.53 -0.53 -0.52 -0.62 -0.9 -1.11 -0.86 -0.88 -1.09 -0.85 -1.39 -1.26 -1.32 -1.27 -1.2 -1.56 -1.6 -1.61 -1.55 -1.59 -2.06 -2.1 -1.89 -1.94 -2.06 -2.5 -2.44 -2.35 -2.38 -2.38 -2.36 -2.75 -2.75 -2.65 -2.74 -2.72 -2.72]; ux4=[-0.36 -0.39 -0.37 -0.36 -0.36 -0.42 -0.61 -0.81 -0.6 -0.61 -0.78 -0.6 -0.96 -0.89 -0.94 -0.9 -0.86 -1.12 -1.14 -1.14 -1.11 -1.12 -1.46 -1.5 -1.37 -1.42 -1.5 -1.76 -1.72 -1.69 -1.7 -1.74 -1.69 -1.97 -1.99 -1.9 -1.94 -1.94 -2]; ux5=[-0.25 -0.27 -0.25 -0.25 -0.25 -0.3 -0.42 -0.57 -0.42 -0.43 -0.56 -0.41 -0.69 -0.61 -0.66 -0.63 -0.6 -0.79 -0.81 -0.79 -0.78 -0.79 -1.02 -1.06 -0.96 -1 -1.05 -1.25 -1.21 -1.19 -1.19 -1.24 -1.19 -1.39 -1.42 -1.35 -1.36 -1.37 -1.41]; ux6=[-0.17 -0.19 -0.17 -0.17 -0.17 -0.2 -0.28 -0.39 -0.29 -0.29 -0.38 -0.28 -0.48 -0.41 -0.46 -0.44 -0.41 -0.55 -0.57 -0.56 -0.54 -0.54 -0.7 -0.73 -0.66 -0.69 -0.73 -0.87 -0.85 -0.82 -0.83 -0.85 -0.82 -0.97 -1 -0.92 -0.95 -0.96 -0.96]; ux7=[-0.1 -0.11 -0.11 -0.11 -0.11 -0.13 -0.18 -0.25 -0.18 -0.17 -0.23 -0.16 -0.3 -0.25 -0.28 -0.26 -0.24 -0.33 -0.35 -0.34 -0.33 -0.33 -0.43 -0.46 -0.41 -0.43 -0.45 -0.54 -0.53 -0.5 -0.51 -0.53 -0.49 -0.61 -0.62 -0.55 -0.6 -0.59 -0.58]; ux8=[-0.04 -0.05 -0.05 -0.05 -0.05 -0.05 -0.07 -0.1 -0.07 -0.07 -0.1 -0.07 -0.12 -0.11 -0.12 -0.12 -0.1 -0.14 -0.14 -0.15 -0.14 -0.14 -0.17 -0.19 -0.17 -0.18 -0.19 -0.23 -0.23 -0.21 -0.22 -0.22 -0.21 -0.26 -0.26 -0.23 -0.26 -0.25 -0.25]; ux9=[-0.01 0 -0.01 0 -0.01 -0.01 -0.01 -0.02 -0.02 -0.02 -0.02 -0.02 -0.03 -0.02 -0.02 -0.02 -0.02 -0.03 -0.03 -0.03 -0.03 -0.03 -0.04 -0.04 -0.04 -0.04 -0.04 -0.05 -0.05 -0.04 -0.04 -0.04 -0.04 -0.05 -0.05 -0.05 -0.05 -0.05 -0.05];

uy10=[-2.83 -0.93 -0.55 -0.58 -0.47 -1.24 -2.56 -2.55 -0.54 -0.49 -2.75 0.93 -3.5 -0.36 -1.92 -1.23 0.54 -1.33 -0.74 -0.62 -0.46 -0.43 -2.64 -2.91 -0.5 -1.45 0.45 -2.47 -1.65 -1.48 -1.48 -1.85 -1.5 -1.57 -1.36 -1.57 -1.41 -1.47 -1.43]; uy11=[-2.68 -0.77 -0.37 -0.38 -0.31 -0.25 -2.4 -0.45 -0.35 -0.3 -0.14 1.07 -1.56 0.41 -0.57 -0.07 0.67 -1.09 -0.48 -0.36 -0.23 -0.2 -2.35 -0.21 -0.23 -0.02 1.95 0.13 0.2 0.21 0.13 0.36 0.25 0.17 0.2 -0.23 0.21 0.25 0.2]; uy12=[-2.65 -0.65 -0.27 -0.28 -0.23 -0.18 -2.34 -0.38 -0.27 -0.21 -0.08 1.09 -1.51 0.36 -0.46 -0.03 0.64 -1 -0.38 -0.21 -0.14 -0.09 -2.13 -0.17 -0.11 0.03 1.85 0.25 0.24 0.21 0.18 0.33 0.24 0.22 0.24 -0.1 0.25 0.28 0.23]; uy13=[-2.63 -0.57 -0.17 -0.21 -0.16 -0.13 -2.3 -0.29 -0.18 -0.15 -0.03 1.09 -1.53 0.31 -0.36 -0.04 0.6 -0.93 -0.26 -0.15 -0.06 -0.03 -1.91 -0.19 -0.04 0.08 1.46 0.3 0.23 0.21 0.19 0.29 0.24 0.22 0.24 -0.02 0.24 0.26 0.23]; uy14=[-2.64 -0.47 -0.05 -0.2 -0.09 -0.07 -2.28 -0.19 -0.11 -0.04 -0.03 1.05 -1.53 0.22 -0.22 0.03 0.46 -0.86 -0.18 -0.09 -0.02 0.01 -1.13 -0.2 0.01 0.07 1.01 0.28 0.17 0.17 0.15 0.22 0.18 0.18 0.18 0.02 0.18 0.2 0.17]; uy15=[-2.67 -0.38 -0.04 -0.07 -0.13 -0.05 -2.08 -0.09 -0.04 -0.08 0.02 0.82 -1.37 0.09 -0.09 0.07 0.21 -0.7 -0.08 -0.05 0.02 -0.01 -0.81 0 -0.03 0.05 0.41 0.26 0.08 0.08 0.07 0.11 0.08 0.08 0.1 0.02 0.1 0.1 0.1]; uy16=[-2.62 -0.34 0.01 -0.03 0.01 -0.05 -2.03 -0.02 -0.01 0.02 -0.05 0.71 -1.4 0.04 -0.05 0 0.09 -0.65 -0.01 -0.03 0.01 0 -0.64 -0.03 -0.01 0 0.03 0.23 0 0 0 0.01 0.01 0 0.01 -0.03 0 0.01 0];

% test data set

teE=[80 70 60 30 50 80 40 70 30 20]; tek=[1 1.5 2 2 2.5 2.5 3 3 3.5 4]; teux1=[0.77 1.16 1.61 1.54 2.25 2.02 2.48 2.53 2.92 3.56]; teux2=[-0.79 -1.31 -1.93 -1.78 -2.56 -2.29 -2.87 -2.89 -3.38 -3.91]; teux3=[-0.52 -0.88 -1.31 -1.22 -1.79 -1.61 -2.01 -2 -2.33 -2.79]; teux4=[-0.37 -0.62 -0.93 -0.89 -1.29 -1.12 -1.47 -1.43 -1.69 -2.02]; teux5=[-0.25 -0.44 -0.64 -0.62 -0.91 -0.77 -1.05 -1.01 -1.19 -1.41]; teux6=[-0.17 -0.3 -0.44 -0.43 -0.64 -0.53 -0.74 -0.68 -0.83 -0.97]; teux7=[-0.11 -0.18 -0.27 -0.27 -0.4 -0.33 -0.45 -0.42 -0.52 -0.59]; teux8=[-0.04 -0.07 -0.12 -0.11 -0.18 -0.14 -0.18 -0.18 -0.22 -0.27]; teux9=[-0.01 -0.02 -0.03 -0.02 -0.03 -0.02 -0.04 -0.04 -0.05 -0.05];

teuy10=[1.12 -0.97 -0.83 -1.49 -2.5 0.86 -1.69 -1.91 -1.93 -2.08]; teuy11=[1.27 -0.19 0.25 -1.2 0.08 1.71 -0.05 0.15 0.15 0.28]; teuy12=[1.33 -0.16 0.29 -1.02 0.1 1.66 0.02 0.17 0.19 0.29]; teuy13=[1.34 -0.13 0.23 -0.81 0.09 1.53 0.03 0.15 0.19 0.25]; teuy14=[1.31 -0.1 0.18 -0.61 0.08 1.25 0.03 0.1 0.16 0.17]; teuy15=[1.22 -0.02 0.04 -0.31 0.04 0.71 0.07 0.06 0.09 0.08]; teuy16=[1.11 0.01 0.01 -0.1 0 0.43 0.01 0 0 -0.03];

% generate the component neural networks

 for i = 1:ensemblesize
     % generate the component training sets
    input = [ux1;ux2;uy10];
    comptarget = [E;k];
    testinput = [teux1;teux2;teuy10];
    testtarget = [teE;tek];
        if i==1   % hidden nodes=3
        hiddenno1=3;
        fprintf('ENN  dda component NN 1       epoch = %g    hidden node no=%g ', maxepoch, hiddenno1);
        fprintf('\n');
         net = newff(input,comptarget,hiddenno1,{'tansig' 'purelin'});
    net.divideFcn='';     
    net.trainParam.epochs = maxepoch;
    net.trainParam.goal = 0.0;
    net.trainParam.mc=0.7;
    net.trainParam.lr=0.05;
    net.trainParam.show=NaN;
   net = train(net,input,comptarget);
       f1 = sim(net,input);
     output1 = sim(net,testinput);   % now 'output' stores the real-va
    elseif i==2  % hidden nodes=2
        hiddenno2=2;
        fprintf('ENN  dda component NN 2       epoch = %g    hidden node no=%g ', maxepoch, hiddenno2);
        fprintf('\n');
         net = newff(input,comptarget,hiddenno2,{'tansig' 'purelin'});
    net.divideFcn='';     
    net.trainParam.epochs = maxepoch;
    net.trainParam.goal = 0.0;
    net.trainParam.mc=0.7;
    net.trainParam.lr=0.05;
    net.trainParam.show=NaN;
   net = train(net,input,comptarget);
    f2 = sim(net,input);
    output2 = sim(net,testinput); 
    elseif i==3   % hidden nodes=1
        hiddenno3=1;
        fprintf('ENN  dda component NN 3       epoch = %g    hidden node no=%g ', maxepoch, hiddenno3);
        fprintf('\n');
         net = newff(input,comptarget,hiddenno3,{'tansig' 'purelin'});
    net.divideFcn='';     
    net.trainParam.epochs = maxepoch;
    net.trainParam.goal = 0.0;
    net.trainParam.mc=0.7;
    net.trainParam.lr=0.05;
    net.trainParam.show=NaN;
   net = train(net,input,comptarget);
     f3 = sim(net,input);
     output3 = sim(net,testinput); 
    else
    fprintf('wrong input--ensemble size ');
    fprintf('\n');   
    end
 end

enoutput_training= (f1+f2+f3)/ensemblesize; enoutput_tes= (output1+output2+output3)/ensemblesize;

 fprintf('ENN  dda 3 different number of hidden neurons       epoch = %g    hidden node no = var', maxepoch);
 fprintf('\n');   
       mse_training = mse(enoutput_training - comptarget)  ;
       mse_tes = mse(enoutput_tes - testtarget)  ;
   fprintf('dda ENN   simple average        mse_training = %12.5g   \n', mse_training );       
   fprintf('dda ENN   simple average        mse_testing = %12.5g   \n\n', mse_tes );

% % fprintf(' dda E value = \n'); % % fprintf(' target = '); % % fprintf(1,' %12.4g', teE1(1,:)); % % fprintf(' \n '); % % % % fprintf(' simple enn = '); % % fprintf(1,' %12.4g',enoutput_tes(1,:)); % % fprintf(' \n '); % % % % fprintf(' better enn = '); % % fprintf(1,' %12.4g',enoutput_better(1,:)); % % fprintf(' \n\n ');

end % end of function

1 Comment
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Greg Heath on 23 Jan 2013

I don't understand why you are posting data for ux1 to ux9 and uy10 to uy16

when your input is only 3-dimensional.

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Answer 1

Greg Heath on 23 Jan 2013

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Edited: Greg Heath on 23 Jan 2013

Open in MATLAB Online

I did not look closely at your code. However, I have several comments:

1. If you are going to post data and/or code, make sure that they will cause no errors when cut and pasted into the command line.

2. If you are not going to use validation stopping or regularization (msereg and/ or trainbr) then choose H just as large as necessary to obtain a practical MSE training goal (e.g., ~ 0.01 or 0.005 * mean(var(target',1)). Adjustments for loss of degrees of freedom due to the estimation of Nw weights is considered below.

3. Convergence to a good local minimum depends on the random initial weights. Therefore, design many nets for each trial value of H and choose a good design using either nontraining validation error or adjusted training error (as shown below).

4. Your guesses of H=3 and maxepoch = 30 are woefully inadequate. Study my structured search over 130 designs (10 random weight initialiations each for 13 candidate values of H) where I discover that 9 <= H <= 12 and maxepoch = 1000(default) are more appropriate.

5. I recommend accepting default training values unless you have a specific reason for using your own. The only ones that I overwrote were net.divide, net.trainParam.goal and H.

6. If you are going to use an ensemble, you might be able to use fewer hidden nodes and larger training goals. However, it probably is not worth the effort.

 close all, clear all,clc, plt=0;
 tic
 %TRAINING DATA
 ux1 = [ 0.82  0.76  0.75  0.69  0.74  0.83  1.31  1.35  1.1 1.14  ...
    1.39  1.09  1.84  1.62  1.66  1.6    1.56   2     1.95 1.96 ...
    1.98  1.89  2.79  2.68  2.41  2.51  2.55  3.63   2.92 2.91 ...
    2.93   3      2.86  3.38  3.41  4.16   3.31 3.32   3.27 ];
 ux2 = [ -0.83  -0.83  -0.79  -0.77 -0.77  -0.89  -1.3  -1.6  -1.28  -1.3 ...
    -1.6    -1.26  -1.93  -1.84  -1.91  -1.86  -1.78  -2.28  -2.29  -2.28 ...
    -2.27  -2.29  -2.88  -2.97  -2.78  -2.81  -2.92  -3.51  -3.44  -3.38 ...
    -3.43  -3.46  -3.36  -3.92  -3.97  -3.83  -3.97  -3.9    -3.89 ];
 uy10 = [-2.83 -0.93 -0.55 -0.58  -0.47  -1.24  -2.56  -2.55  -0.54  -0.49 ...
    -2.75    0.93  -3.5    -0.36  -1.92  -1.23   0.54  -1.33  -0.74  -0.62 ...
    -0.46   -0.43  -2.64  -2.91  -0.5    -1.45   0.45  -2.47  -1.65  -1.48 ...
    -1.48   -1.85  -1.5    -1.57  -1.36  -1.57  -1.41  -1.47  -1.43 ];
 E = [ 20 30 40 50 60 70 20 30 40 50 60 80 20 40 50 70 80 20 30 40 ...
    60 70 20 30 50 60 80 20 40 50 60 70 80 30 40 50 60 70 80];
 k = [ 1  1  1  1  1  1  1.5 1.5 1.5 1.5 1.5 1.5 2 2 2 2 2 2.5 2.5  2.5 ...
      2.5  2.5 3 3 3 3  3  3.5  3.5  3.5  3.5  3.5  3.5  4  4  4  4  4 4];
 xtrn  = [ ux1; ux2 ;uy10 ];
 ttrn  = [ E; k ];
 % TEST DATA
 teux1 = [ 0.77 1.16 1.61 1.54 2.25 2.02 2.48 2.53 2.92 3.56 ];
 teux2 = [ -0.79 -1.31 -1.93 -1.78 -2.56 -2.29 -2.87 -2.89 -3.38 -3.91 ];
 teuy10 = [ 1.12 -0.97 -0.83 -1.49 -2.5 0.86 -1.69 -1.91 -1.93 -2.08 ];
 teE = [ 80 70 60 30 50 80 40 70 30 20 ];
 tek = [ 1 1.5 2 2 2.5 2.5 3 3 3.5 4 ];
 xtst  = [ teux1; teux2; teuy10 ];
 ttst  = [ teE; tek ];
 [ I Ntrn   ] = size(xtrn)                    %  [ 3 39 ]
 [ O Ntrn ] = size(ttrn)                      % [ 2 39 ]
 [ I Ntst   ] = size(xtst)                    % [ 2 10 ]
 [ O Ntst ] = size(ttst)                      % [ 2 10 ]
 Ntrneq = Ntrn*O                              % 78     Training Equations
 % NAIVE CONSTANT MODEL ( "a" => Degree of freedom "a"djustment)
 y00 = repmat(mean(ttrn,2),1,Ntrn);
 Nw00 = O                                     % 2 weights
 e00 = ttrn-y00;
 MSE00 = sse(e00)/Ntrneq                      % 193.8259
 MSE00a = sse(e00)/(Ntrneq-Nw00)              % 198.9266   DOFA
 MSE00 = mean(var(ttrn',1))                   % 193.8259   Target Variance
 MSE00a = mean(var(ttrn'))                    % 198.9266
 %LINEAR MODEL (R2 => Fraction of target variance modeled)
 W =  ttrn / [ ones(1,Ntrn) ; xtrn ] % 52.8493 -23.1829 -24.9930 10.7487
                                     %  0.2567   0.0956  -0.8929  0.0556
 Nw0    = numel(W)                   % 8 weights
 y0     = W*[ ones(1,Ntrn) ; xtrn ];
 e0     = ttrn-y0;
 MSE0   = sse(e0)/Ntrneq            % 115.9305
 MSE0a  = sse(e0)/(Ntrneq-Nw0)      % 129.1797
 NMSE0   = MSE0/MSE00               % 0.5981
 NMSE0a = MSE0a/MSE00a              % 0.6494
 R20    = 1-NMSE0                   % 0.4019    Same as H = 0 below
 R20a   = 1-NMSE0a                  % 0.3506
 %NEURAL NET MODEL
 % Nw = (I+1)*H+(H+1)*O             % No.of estimated weights
 Hub = (Ntrneq-O)/(I+O+1)           % 12.6667   Ntrneq = Nw
 Hmax = ceil(Hub)-1                 % 12        Ntrneq > Nw
 Ntrials = 10
 rng(0)
 j=0
 for h = 0:Hmax
    j = j+1
    for i = 1:Ntrials
        if h==0
            net = fitnet([]);
            Nw = (I+1)*O         
        else
            net = fitnet(h);
            Nw = (I+1)*h+(h+1)*O
        end
        net.divideFcn       = '';     % No val or test data
        net.trainParam.goal = 0.01*(Ntrneq-Nw)*MSE00a/Ntrneq; %R2a >= 0.99
        [ net tr Ytrn Etrn] = train( net, xtrn, ttrn);
        Nepochs(i,j) = tr.epoch(end);
        MSE         = tr.perf(end);
        MSEa        = Ntrneq*MSE/(Ntrneq-Nw);
        R2(i,j)     = 1-MSE/MSE00;
        R2a(i,j)    = 1- MSEa/MSE00a;
    end
 end
 time    = toc       %  694.84 sec  (11.58 min)
 H       = 0:Hmax
 Nepochs = Nepochs
 R2      = R2
 R2a     = R2a
 % H =   
 %
 %  0     1     2    3     4     5      6    7    8     9   10   11    12
 %  
 % Nepochs =
 % 
 %  2  1000    29  1000  1000  1000  1000  1000   685  230  123    78   67
 %  2  1000  1000     9  1000  1000  1000  1000  1000  177  876    72  133
 %  2     8   166   345  1000  1000  1000  1000  1000   97  295    16  263
 %  2  1000  1000  1000  1000  1000  1000  1000  1000 1000  322   112  167
 %  2  1000  1000  1000    48  1000  1000  1000  1000  145  189   117  335
 %  2  1000   46   1000   262   229  1000  1000  1000  119  353   494  303
 %  2  1000  1000  1000  1000  1000  1000  1000  1000  214  195   378  102
 %  2  1000  1000  1000  1000  1000  1000  1000  1000  509  134   542  104
 %  2  1000   208  1000   168  1000  1000  1000  1000 1000  175    83  167
 %  2  1000  1000  1000   931  1000  1000   389  1000 1000   78   211  244
 %  
 % R2 =
 % 
 % 0.4019 0.4054 0.5996 0.7503 0.8250 0.7970 0.8678 0.9506 0.9802 0.9971 0.9979    0.9987 0.9995
 % 0.4019 0.4054 0.4314 0.3411 0.6335 0.7708 0.9052 0.9904 0.9873 0.9971    0.9979    0.9988 0.9996
 % 0.4019 0.2417 0.5181 0.5998 0.7224 0.8452 0.8749 0.8640 0.9646 0.9974    0.9979    0.9989 0.9995
 % 0.4019 0.4054 0.5956 0.6478 0.7224 0.7115 0.8095 0.9645 0.9429 0.9531    0.9979    0.9987 0.9995
 % 0.4019 0.4054 0.4054 0.6313 0.7001 0.7655 0.8609 0.8560 0.9807 0.9971    0.9979    0.9988 0.9995
 % 0.4019 0.4054 0.5995 0.5212 0.6768 0.7132 0.8032 0.9620 0.9692 0.9971    0.9980    0.9987 0.9995
 % 0.4019 0.4054 0.5989 0.5687 0.6967 0.7421 0.8060 0.8933 0.9898 0.9971 0.9983    0.9987 0.9995
 % 0.4019 0.4043 0.5281 0.5770 0.6837 0.7449 0.8574 0.9774 0.8911 0.9973    0.9979    0.9987 0.9995
 % 0.4019 0.4054 0.5181 0.5994 0.6960 0.8263 0.8377 0.9766 0.9675 0.8793    0.9988    0.9987 0.9995
 % 0.4019 0.4054 0.5961 0.5701 0.6809 0.7450 0.8698 0.9327 0.9606 0.9941    0.9980    0.9987 0.9995
%  
% R2a =
% 
%     0.3506    0.3545    0.5245    0.6727    0.7442    0.6647    0.7489    0.8895    0.9462    0.9900    0.9900    0.9901    0.9911
%     0.3506    0.3545    0.3247    0.1366    0.4643    0.6213    0.8198    0.9785    0.9656    0.9900    0.9900    0.9908    0.9915
%     0.3506    0.1767    0.4277    0.4756    0.5942    0.7442    0.7624    0.6961    0.9040    0.9908    0.9900    0.9914    0.9903
%     0.3506    0.3545    0.5197    0.5385    0.5942    0.5234    0.6381    0.9206    0.8451    0.8379    0.9902    0.9901    0.9904
%     0.3506    0.3545    0.2940    0.5169    0.5617    0.6126    0.7358    0.6781    0.9476    0.9901    0.9901    0.9908    0.9900
%     0.3506    0.3545    0.5244    0.3726    0.5276    0.5262    0.6260    0.9150    0.9164    0.9900    0.9903    0.9901    0.9907
%     0.3506    0.3545    0.5237    0.4349    0.5568    0.5740    0.6314    0.7616    0.9722    0.9900    0.9919    0.9900    0.9902
%     0.3506    0.3532    0.4397    0.4458    0.5377    0.5785    0.7291    0.9495    0.7044    0.9907    0.9900    0.9902    0.9908
%     0.3506    0.3545    0.4277    0.4751    0.5557    0.7130    0.6916    0.9477    0.9117    0.5832    0.9944    0.9901    0.9913
%     0.3506    0.3545    0.5203    0.4367    0.5336    0.5788    0.7525    0.8497    0.8931    0.9796    0.9907    0.9900    0.9907

Ensemble Neural Network in Matlab

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Ensemble Neural Network in Matlab

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