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<span style="color:green">% The initial MFs for training are shown in the plots.</span><span style="color:blue">for</span> input_index=1:4, subplot(2,2,input_index) [x,y]=plotmf(fismat,<span style="color:#B20000">'input'</span>,input_index); plot(x,y) axis([-inf inf 0 1.2]); xlabel([<span style="color:#B20000">'Input '</span> int2str(input_index)]);<span style="color:blue">end</span></pre><img xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" vspace="5" hspace="5" src="mgtsdemo_img_04_01.gif"><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="color:#990000; font-weight:bold; font-size:medium; page-break-before: auto;"><a name=""></a></p><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd">There are 2^4 = 16 rules in the generated FIS matrix and thenumber of fitting parameters is 108, including 24 nonlinearparameters and 80 linear parameters. This is a proper balancebetween number of fitting parameters and number of trainingdata (500). The ANFIS command looks like this:</p><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd">>> [trn_fismat,trn_error] = anfis(trn_data, fismat,[],[],chk_data)</pre><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd">To save time, we will load the training results directly.</p><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd">After ten epochs of training, the final MFs are shown in theplots. Note that these MFs after training do not changedrastically. Obviously most of the fitting is done by the linearparameters while the nonlinear parameters are mostly for fine-tuning for further improvement.</p><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="position: relative; left:30px"><span style="color:green">% load training results</span>load mganfis<span style="color:green">% plot final MF's on x, y, z, u</span><span style="color:blue">for</span> input_index=1:4, subplot(2,2,input_index) [x,y]=plotmf(trn_fismat,<span style="color:#B20000">'input'</span>,input_index); plot(x,y) axis([-inf inf 0 1.2]); xlabel([<span style="color:#B20000">'Input '</span> int2str(input_index)]);<span style="color:blue">end</span></pre><img xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" vspace="5" hspace="5" src="mgtsdemo_img_05_01.gif"><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="color:#990000; font-weight:bold; font-size:medium; page-break-before: auto;">Error curves<a name="Error curves"></a></p><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd">This plot displays error curves for both training andchecking data. Note that the training error is higher than thechecking error. This phenomenon is not uncommon in ANFISlearning or nonlinear regression in general; it could indicatethat the training process is not close to finished yet.</p><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="position: relative; left:30px"><span style="color:green">% error curves plot</span>epoch_n = 10;tmp = [trn_error chk_error];subplot(1,1,1)plot(tmp);hold on; plot(tmp, <span style="color:#B20000">'o'</span>); hold off;xlabel(<span style="color:#B20000">'Epochs'</span>);ylabel(<span style="color:#B20000">'RMSE (Root Mean Squared Error)'</span>);title(<span style="color:#B20000">'Error Curves'</span>);axis([0 epoch_n min(tmp(:)) max(tmp(:))]);</pre><img xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" vspace="5" hspace="5" src="mgtsdemo_img_06_01.gif"><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="color:#990000; font-weight:bold; font-size:medium; page-break-before: auto;">Comparisons<a name="Comparisons"></a></p><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd">This plot shows the original time series and the one predictedby ANFIS. The difference is so tiny that it is impossible to tellone from another by eye inspection. That is why you probablysee only the ANFIS prediction curve. The prediction errorsmust be viewed on another scale.</p><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="position: relative; left:30px">input = [trn_data(:, 1:4); chk_data(:, 1:4)];anfis_output = evalfis(input, trn_fismat);index = 125:1124;plot(time(index), [ts(index) anfis_output]);xlabel(<span style="color:#B20000">'Time (sec)'</span>);</pre><img xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" vspace="5" hspace="5" src="mgtsdemo_img_07_01.gif"><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="color:#990000; font-weight:bold; font-size:medium; page-break-before: auto;">Prediction errors of ANFIS<a name="Prediction errors of ANFIS"></a></p><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd">Prediction error of ANFIS is shown here. Note that the scaleis about a hundredth of the scale of the previous plot.Remember that we have only 10 epochs of training in this case;better performance is expected if we have extensive training.</p><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="position: relative; left:30px">diff = ts(index)-anfis_output;plot(time(index), diff);xlabel(<span style="color:#B20000">'Time (sec)'</span>);title(<span style="color:#B20000">'ANFIS Prediction Errors'</span>);</pre><img xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" vspace="5" hspace="5" src="mgtsdemo_img_08_01.gif"><originalCode xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" code="%% Chaotic Time Series Prediction
% Chaotic time series prediction using ANFIS.
%
% Copyright 1994-2002 The MathWorks, Inc. 
% $Revision: 1.15 $

%%
% The Mackey-Glass time-delay differential equation is defined by
% 
% dx(t)/dt = 0.2x(t-tau)/(1+x(t-tau)^10) - 0.1x(t)
% 
% When x(0) = 1.2 and tau = 17, we have a non-periodic and
% non-convergent time series that is very sensitive to
% initial conditions. (We assume x(t) = 0 when t < 0.)

load mgdata.dat
a=mgdata;
time = a(:, 1);
ts = a(:, 2);
plot(time, ts);
xlabel('Time (sec)'); ylabel('x(t)');
title('Mackey-Glass Chaotic Time Series');

%%
% Now we want to build an ANFIS that can predict x(t+6) from
% the past values of this time series, that is, x(t-18), x(t-12),
% x(t-6), and x(t). Therefore the training data format is
%
% [x(t-18), x(t-12), x(t-6), x(t); x(t+6]
%
% From t = 118 to 1117, we collect 1000 data pairs of the above
% format. The first 500 are used for training while the others
% are used for checking. The plot shows the segment of the time
% series where data pairs were extracted from.

trn_data = zeros(500, 5);
chk_data = zeros(500, 5);

% prepare training data
start = 101;
trn_data(:, 1) = ts(start:start+500-1);
start = start + 6;
trn_data(:, 2) = ts(start:start+500-1);
start = start + 6;
trn_data(:, 3) = ts(start:start+500-1);
start = start + 6;
trn_data(:, 4) = ts(start:start+500-1);
start = start + 6;
trn_data(:, 5) = ts(start:start+500-1);

% prepare checking data
start = 601;
chk_data(:, 1) = ts(start:start+500-1);
start = start + 6;
chk_data(:, 2) = ts(start:start+500-1);
start = start + 6;
chk_data(:, 3) = ts(start:start+500-1);
start = start + 6;
chk_data(:, 4) = ts(start:start+500-1);
start = start + 6;
chk_data(:, 5) = ts(start:start+500-1);

index = 118:1117+1; % ts starts with t = 0
plot(time(index), ts(index));
axis([min(time(index)) max(time(index)) min(ts(index)) max(ts(index))]);
xlabel('Time (sec)'); ylabel('x(t)');
title('Mackey-Glass Chaotic Time Series');

%%
% We use GENFIS1 to generate an initial FIS matrix from training
% data. The command is quite simple since default values for
% MF number (2) and MF type ('gbellmf') are used:
 
fismat = genfis1(trn_data);

% The initial MFs for training are shown in the plots.
for input_index=1:4,
 subplot(2,2,input_index)
 [x,y]=plotmf(fismat,'input',input_index);
 plot(x,y)
 axis([-inf inf 0 1.2]);
 xlabel(['Input ' int2str(input_index)]);
end

%%
% There are 2^4 = 16 rules in the generated FIS matrix and the
% number of fitting parameters is 108, including 24 nonlinear
% parameters and 80 linear parameters. This is a proper balance
% between number of fitting parameters and number of training
% data (500). The ANFIS command looks like this:
% 
% >> [trn_fismat,trn_error] = anfis(trn_data, fismat,[],[],chk_data)
% 
% To save time, we will load the training results directly.
%
% After ten epochs of training, the final MFs are shown in the
% plots. Note that these MFs after training do not change
% drastically. Obviously most of the fitting is done by the linear
% parameters while the nonlinear parameters are mostly for fine-
% tuning for further improvement.

% load training results
load mganfis

% plot final MF's on x, y, z, u
for input_index=1:4,
 subplot(2,2,input_index)
 [x,y]=plotmf(trn_fismat,'input',input_index);
 plot(x,y)
 axis([-inf inf 0 1.2]);
 xlabel(['Input ' int2str(input_index)]);
end

%% Error curves 
% This plot displays error curves for both training and
% checking data. Note that the training error is higher than the
% checking error. This phenomenon is not uncommon in ANFIS
% learning or nonlinear regression in general; it could indicate
% that the training process is not close to finished yet.

% error curves plot
epoch_n = 10;
tmp = [trn_error chk_error];
subplot(1,1,1)
plot(tmp);
hold on; plot(tmp, 'o'); hold off;
xlabel('Epochs');
ylabel('RMSE (Root Mean Squared Error)');
title('Error Curves');
axis([0 epoch_n min(tmp(:)) max(tmp(:))]);

%% Comparisons
% This plot shows the original time series and the one predicted
% by ANFIS. The difference is so tiny that it is impossible to tell
% one from another by eye inspection. That is why you probably
% see only the ANFIS prediction curve. The prediction errors
% must be viewed on another scale.

input = [trn_data(:, 1:4); chk_data(:, 1:4)];
anfis_output = evalfis(input, trn_fismat);
index = 125:1124;
plot(time(index), [ts(index) anfis_output]);
xlabel('Time (sec)');

%% Prediction errors of ANFIS
% Prediction error of ANFIS is shown here. Note that the scale
% is about a hundredth of the scale of the previous plot.
% Remember that we have only 10 epochs of training in this case;
% better performance is expected if we have extensive training.

diff = ts(index)-anfis_output;
plot(time(index), diff);
xlabel('Time (sec)');
title('ANFIS Prediction Errors');
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