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<!--This HTML is auto-generated from an m-file.Your changes will be overwritten.--><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="color:#990000; font-weight:bold; font-size:x-large">Linear Fit of Nonlinear Problem</p><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd">A linear neuron is trained to find the minimum sum-squared error linear fit toa nonlinear input/output problem.</p><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd">Copyright 1992-2002 The MathWorks, Inc.$Revision: 1.16 $ $Date: 2002/03/29 19:36:17 $</p><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">P defines four 1-element input patterns (column vectors). T definesassociated 1-element targets (column vectors). Note that the relationshipbetween values in P and in T is nonlinear. I.e. No W and B exist such thatP*W+B = T for all of four sets of P and T values above.</p><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="position: relative; left:30px">P = [+1.0 +1.5 +3.0 -1.2];T = [+0.5 +1.1 +3.0 -1.0];</pre><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">ERRSURF calculates errors for a neuron with a range of possible weight andbias values. PLOTES plots this error surface with a contour plot underneath.</p><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd">The best weight and bias values are those that result in the lowest point onthe error surface. Note that because a perfect linear fit is not possible,the minimum has an error greater than 0.</p><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="position: relative; left:30px">w_range =-2:0.4:2; b_range = -2:0.4:2;ES = errsurf(P,T,w_range,b_range,<span style="color:#B20000">'purelin'</span>);plotes(w_range,b_range,ES);</pre><img xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" src="demolin4_img03.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">MAXLINLR finds the fastest stable learning rate for training a linear network.NEWLIN creates a linear neuron. NEWLIN takes these arguments: 1) Rx2 matrixof min and max values for R input elements, 2) Number of elements in theoutput vector, 3) Input delay vector, and 4) Learning rate.</p><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="position: relative; left:30px">maxlr = maxlinlr(P,<span style="color:#B20000">'bias'</span>);net = newlin([-2 2],1,[0],maxlr);</pre><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">Override the default training parameters by setting the maximum number ofepochs. This ensures that training will stop.</p><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="position: relative; left:30px">net.trainParam.epochs = 15;</pre><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">To show the path of the training we will train only one epoch at a time andcall PLOTEP every epoch (code not shown here). The plot shows a history ofthe training. Each dot represents an epoch and the blue lines show eachchange made by the learning rule (Widrow-Hoff by default).</p><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="position: relative; left:30px"><span style="color:green">% [net,tr] = train(net,P,T);</span>net.trainParam.epochs = 1;net.trainParam.show = NaN;h=plotep(net.IW{1},net.b{1},mse(T-sim(net,P))); [net,tr] = train(net,P,T); r = tr;epoch = 1;<span style="color:blue">while</span> epoch < 15 epoch = epoch+1; [net,tr] = train(net,P,T); <span style="color:blue">if</span> length(tr.epoch) > 1 h = plotep(net.IW{1,1},net.b{1},tr.perf(2),h); r.epoch=[r.epoch epoch]; r.perf=[r.perf tr.perf(2)]; r.vperf=[r.vperf NaN]; r.tperf=[r.tperf NaN]; <span style="color:blue">else</span> <span style="color:blue">break</span> <span style="color:blue">end</span><span style="color:blue">end</span>tr=r;</pre><img xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" src="demolin4_img06.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">The train function outputs the trained network and a history of the trainingperformance (tr). Here the errors are plotted with respect to trainingepochs.</p><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd">Note that the error never reaches 0. This problem is nonlinear and thereforea zero error linear solution is not possible.</p><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="position: relative; left:30px">plotperf(tr,net.trainParam.goal);</pre><img xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" src="demolin4_img07.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">Now use SIM to test the associator with one of the original inputs, -1.2, andsee if it returns the target, 1.0.</p><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd">The result is not very close to 0.5! This is because the network is the bestlinear fit to a nonlinear problem.</p><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="position: relative; left:30px">p = -1.2;a = sim(net, p)</pre><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="color:gray; font-style:italic;">a = -1.1803</pre><originalCode xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" code="%% Linear Fit of Nonlinear Problem
% A linear neuron is trained to find the minimum sum-squared error linear fit to
% a nonlinear input/output problem.
%
% Copyright 1992-2002 The MathWorks, Inc.
% $Revision: 1.16 $ $Date: 2002/03/29 19:36:17 $

%%
% P defines four 1-element input patterns (column vectors). T defines
% associated 1-element targets (column vectors). Note that the relationship
% between values in P and in T is nonlinear. I.e. No W and B exist such that
% P*W+B = T for all of four sets of P and T values above.

P = [+1.0 +1.5 +3.0 -1.2];
T = [+0.5 +1.1 +3.0 -1.0];

%%
% ERRSURF calculates errors for a neuron with a range of possible weight and
% bias values. PLOTES plots this error surface with a contour plot underneath.
% 
% The best weight and bias values are those that result in the lowest point on
% the error surface. Note that because a perfect linear fit is not possible,
% the minimum has an error greater than 0.

w_range =-2:0.4:2; b_range = -2:0.4:2;
ES = errsurf(P,T,w_range,b_range,'purelin');
plotes(w_range,b_range,ES);

%%
% MAXLINLR finds the fastest stable learning rate for training a linear network.
% NEWLIN creates a linear neuron. NEWLIN takes these arguments: 1) Rx2 matrix
% of min and max values for R input elements, 2) Number of elements in the
% output vector, 3) Input delay vector, and 4) Learning rate.

maxlr = maxlinlr(P,'bias');
net = newlin([-2 2],1,[0],maxlr);


%%
% Override the default training parameters by setting the maximum number of
% epochs. This ensures that training will stop.

net.trainParam.epochs = 15;

%%
% To show the path of the training we will train only one epoch at a time and
% call PLOTEP every epoch (code not shown here). The plot shows a history of
% the training. Each dot represents an epoch and the blue lines show each
% change made by the learning rule (Widrow-Hoff by default).

% [net,tr] = train(net,P,T);
net.trainParam.epochs = 1;
net.trainParam.show = NaN;
h=plotep(net.IW{1},net.b{1},mse(T-sim(net,P))); 
[net,tr] = train(net,P,T); 
r = tr;
epoch = 1;
while epoch < 15
 epoch = epoch+1;
 [net,tr] = train(net,P,T);
 if length(tr.epoch) > 1
 h = plotep(net.IW{1,1},net.b{1},tr.perf(2),h);
 r.epoch=[r.epoch epoch]; 
 r.perf=[r.perf tr.perf(2)];
 r.vperf=[r.vperf NaN];
 r.tperf=[r.tperf NaN];
 else
 break
 end
end
tr=r;

%%
% The train function outputs the trained network and a history of the training
% performance (tr). Here the errors are plotted with respect to training
% epochs.
%
% Note that the error never reaches 0. This problem is nonlinear and therefore
% a zero error linear solution is not possible.

plotperf(tr,net.trainParam.goal);

%%
% Now use SIM to test the associator with one of the original inputs, -1.2, and
% see if it returns the target, 1.0.
%
% The result is not very close to 0.5! This is because the network is the best
% linear fit to a nonlinear problem.

p = -1.2;
a = sim(net, p)


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