Linear Prediction Design

Defining a Wave Form

Here time is defined from 0 to 5 seconds in steps of 1/40 of a second.

time = 0:0.025:5;

We can define a signal with respect to time.

signal = sin(time*4*pi);
plot(time,signal)
xlabel('Time');
ylabel('Signal');
title('Signal to be Predicted');

Setting up the Problem for a Neural Network

The signal convert is then converted to a cell array. Neural Networks represent timesteps as columns of a cell array, do distinguish them from different samples at a given time, which are represented with columns of matrices.

signal = con2seq(signal);

To set up the problem we will use the first four values of the signal as initial input delay states, and the rest except for the last step as inputs.

Xi = signal(1:4);
X = signal(5:(end-1));
timex = time(5:(end-1));

The targets are now defined to match the inputs, but shifted earlier by one timestep.

T = signal(6:end);

Designing the Linear Layer

The function newlind will now design a linear layer with a single neuron which predicts the next timestep of the signal given the current and four past values.

net = newlind(X,T,Xi);
view(net)

Testing the Linear Layer

The network can now be called like a function on the inputs and delayed states to get its time response.

Y = net(X,Xi);

The output signal is plotted with the targets.

figure
plot(timex,cell2mat(Y),timex,cell2mat(T),'+')
xlabel('Time');
ylabel('Output -  Target +');
title('Output and Target Signals');

The error can also be plotted.

figure
E = cell2mat(T)-cell2mat(Y);
plot(timex,E,'r')
hold off
xlabel('Time');
ylabel('Error');
title('Error Signal');

Notice how small the error is!

This example illustrated how to design a dynamic linear network which can predict a signal's next value from current and past values.