N          = 2000;       % number of samples

train_frac = 0.75;       % percentage of values to use for training



u          = 0.9;        % constant for Ikeda map

x          = 0.3;        % initial x value for Ikeda map

y          = 0.3;        % initial y value for Ikeda map



epochs     = 500;        % number of epochs to train the network for

goal       = 1e-10;      % performance goal



% end index of training values, train_index + 1 is where testing begins

train_index = ceil (N * train_frac);



% 2-column input matrix

P = zeros (2, N);



P(:, 1) = [x y];



for n = 2:N

    t = 0.4 - 6 / (1 + x^2 + y^2);



    x_tmp = 1 + u * (x * cos (t) - y * sin (t)) ;

    y_tmp =     u * (x * sin (t) + y * cos (t)) ;



    x = x_tmp;

    y = y_tmp;

    

    P(:, n) = [x y];

end



% normalized version of the time series, now mapped between -1 and 1

P(1, :) = mapminmax (P(1, :));

P(2, :) = mapminmax (P(2, :));



% build a new feed forward back propagation network

net = newff ([-1 1; -1 1], [10 10 10 2], {'tansig' 'tansig' 'tansig' 'purelin'});



% train the network

net.trainParam.epochs = epochs;

net.trainParam.goal   = goal;

net = train (net, P(:, 1:train_index), P(:, 2:train_index + 1));



P_out = P;



% test the network

P_out(:, train_index + 1:N) = sim (net, P(:, train_index:N - 1));



% get mean square error

e = P(:, train_index + 1:N) - P_out(:, train_index + 1:N);

mse (e)



figure (2);

plot (P(1, train_index + 1:N), P(2, train_index + 1:N), P_out(1, train_index + 1:N), P_out(2, train_index + 1:N), '.-');

xlabel ('x');

ylabel ('y');

title  ('Target vs. Experimental Neural Network Ikeda Series');

legend ('Target', 'Experimental');