MATLAB Examples

## Mackey Glass Time Series Prediction Using Least Mean Square (LMS)

Author: SHUJAAT KHAN

```clc clear all close all ```

I generated a series x(t) for t = 0,1, . . . ,3000, using mackey glass series equation with the following configurations: b = 0.1, a = 0.2, Tau = 20, and the initial conditions x(t - Tau) = 0.

```load Dataset\Data.mat time_steps=2; teacher_forcing=1; % recurrent ARMA modelling with forced desired input after defined time steps ```

## Training and Testing datasets

For training

```Tr=Data(100:2500,1); % Selecting a Interval of series data t = 100~2500 Xr(Tr)=Data(Tr,2); % Selecting a chuck of series data x(t) % For testing Ts=Data(2500:3000,1); % Selecting a Interval of series data t = 2500~3000 Xs(Ts)=Data(Ts,2); % Selecting a chuck of series data x(t) ```

## LMS Parameters

```eta=5e-3; % Learning rate M=1; % Order of LMS filter U=zeros(1,M+1); % Initial values of taps W=randn(M+1,1); % Initial weight of LMS MSE=[]; % Initial mean squared error (MSE) ```

## Learning weights of LMS (Training)

```tic % start for i=Tr(1):Tr(end)-time_steps U(1:end-1)=U(2:end); % Shifting of tap window if (teacher_forcing==1) if rem(i,time_steps)==0 || (i==Tr(1)) U(end)=Xr(i); % Input (past/current samples) else U(end)=Y(i-1); % Input (past/current samples) end else U(end)=Xr(i); % Input (past/current samples) end Y(i)=W'*U'; % Predicted output e(i)=Xr(i+time_steps)-Y(i); % Error in predicted output W=W+eta*e(i)*U'; % Weight update rule of LMS % E(i)=mean(e(Tr(1):i).^2); % Current mean squared error (MSE) E(i)=e(i).^2; % Current mean squared error (MSE) end training_time=toc; % total time including training and calculation of MSE ```

## Prediction of a next outcome of series using previous samples (Testing)

```tic % start U=U*0; % Reinitialization of taps (optional) for i=Ts(1):Ts(end)-time_steps+1 U(1:end-1)=U(2:end); % Shifting of tap window if (teacher_forcing==1) if rem(i,time_steps)==0 || (i==Ts(1)) U(end)=Xs(i); % Input (past/current samples) else U(end)=Y(i-1); % Input (past/current samples) end else U(end)=Xs(i); % Input (past/current samples) end Y(i)=W'*U'; % Calculating output (future value) e(i)=Xs(i+time_steps-1)-Y(i); % Error in predicted output % E(i)=mean(e(Ts(1):i).^2); % Current mean squared error (MSE) E(i)=e(i).^2; % Current mean squared error (MSE) end testing_time=toc; % total time including testing and calculation of MSE ```

## Results

```plot(Tr,10*log10(E(Tr))); % MSE curve hold on plot(Ts(1:end-time_steps+1),10*log10(E(Ts(1:end-time_steps+1))),'r'); % MSE curve grid minor % legend('Training','Testing'); title('Cost Function'); xlabel('Iterations (samples)'); ylabel('Mean Squared Error (MSE)'); legend('Training Phase','Test Phase'); figure plot(Tr(2*M:end),Xr(Tr(2*M:end))); % Actual values of mackey glass series hold on plot(Tr(2*M:end),Y(Tr(2*M:end))','r') % Predicted values during training plot(Ts,Xs(Ts),'--b'); % Actual unseen data hold on plot(Ts(1:end-time_steps+1),Y(Ts(1:end-time_steps+1))','--r'); % Predicted values of mackey glass series (testing) xlabel('Time: t'); ylabel('Output: Y(t)'); % legend('Actual (Training)','Predicted (Training)','Actual (Testing)','Predicted (Testing)'); title('Mackey Glass Time Series Prediction Using Least Mean Square (LMS)') ylim([min(Xs)-0.5, max(Xs)+0.5]) legend('Training Phase (desired)','Training Phase (predicted)','Training Phase (desired)','Test Phase (predicted)'); mitr=10*log10(mean(E(Tr))); % Minimum MSE of training mits=10*log10(mean(E(Ts(1:end-time_steps+1)))); % Minimum MSE of testing display(sprintf('Total training time is %.5f, \nTotal testing time is %.5f \nMSE value during training %.3f (dB),\nMSE value during testing %.3f (dB)', ... training_time,testing_time,mitr,mits)); ```
```Total training time is 0.01034, Total testing time is 0.00505 MSE value during training -7.797 (dB), MSE value during testing -20.679 (dB) ```