Solving XOR problem with a multilayer perceptron

PROBLEM DESCRIPTION: 4 clusters of data (A,B,C,D) are defined in a 2-dimensional input space. (A,C) and (B,D) clusters represent XOR classification problem. The task is to define a neural network for solving the XOR problem.

Define 4 clusters of input data
Define output coding for XOR problem
Prepare inputs & outputs for network training
Create and train a multilayer perceptron
plot targets and network response to see how good the network learns the data
Plot classification result for the complete input space

Define 4 clusters of input data

close all, clear all, clc, format compact

% number of samples of each class
K = 100;
% define 4 clusters of input data
q = .6; % offset of classes
A = [rand(1,K)-q; rand(1,K)+q];
B = [rand(1,K)+q; rand(1,K)+q];
C = [rand(1,K)+q; rand(1,K)-q];
D = [rand(1,K)-q; rand(1,K)-q];
% plot clusters
figure(1)
plot(A(1,:),A(2,:),'k+')
hold on
grid on
plot(B(1,:),B(2,:),'bd')
plot(C(1,:),C(2,:),'k+')
plot(D(1,:),D(2,:),'bd')
% text labels for clusters
text(.5-q,.5+2*q,'Class A')
text(.5+q,.5+2*q,'Class B')
text(.5+q,.5-2*q,'Class A')
text(.5-q,.5-2*q,'Class B')

Define output coding for XOR problem

% encode clusters a and c as one class, and b and d as another class
a = -1; %  a  |  b
c = -1; %  -------
b =  1; %  d  |  c
d =  1; %

Prepare inputs & outputs for network training

% define inputs (combine samples from all four classes)
P = [A B C D];
% define targets
T = [repmat(a,1,length(A)) repmat(b,1,length(B)) ...
     repmat(c,1,length(C)) repmat(d,1,length(D)) ];
% view inputs |outputs
%[P' T']

Create and train a multilayer perceptron

% create a neural network
net = feedforwardnet([5 3]);

% train net
net.divideParam.trainRatio = 1; % training set [%]
net.divideParam.valRatio   = 0; % validation set [%]
net.divideParam.testRatio  = 0; % test set [%]

% train a neural network
[net,tr,Y,E] = train(net,P,T);

% show network
view(net)

plot targets and network response to see how good the network learns the data

figure(2)
plot(T','linewidth',2)
hold on
plot(Y','r--')
grid on
legend('Targets','Network response','location','best')
ylim([-1.25 1.25])

Plot classification result for the complete input space

% generate a grid
span = -1:.005:2;
[P1,P2] = meshgrid(span,span);
pp = [P1(:) P2(:)]';

% simulate neural network on a grid
aa = net(pp);

% translate output into [-1,1]
%aa = -1 + 2*(aa>0);

% plot classification regions
figure(1)
mesh(P1,P2,reshape(aa,length(span),length(span))-5);
colormap cool
view(2)