Prediction of a nonlinear time series with feedforward neural networks
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Prediction of a nonlinear time series with feedforward neural networks. Mats Nikus Process Control Laboratory. The time series. A closer look. Another look. Studying the time series. Some features seem to reapeat themselves over and over, but not totally ”deterministically”

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Prediction of a nonlinear time series with feedforward neural networks

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Prediction of a nonlinear time series with feedforward neural networks

Prediction of a nonlinear time series with feedforward neural networks

Mats Nikus

Process Control Laboratory


The time series

The time series


A closer look

A closer look


Another look

Another look


Studying the time series

Studying the time series

  • Some features seem to reapeat themselves over and over, but not totally ”deterministically”

  • Lets study the autocovariance function


The autocovariance function

The autocovariance function


Studying the time series1

Studying the time series

  • The autocovariance function tells the same: There are certainly some dynamics in the data

  • Lets now make a phaseplot of the data

  • In a phaseplot the signal is plotted against itself with some lag

  • With one lag we get


Phase plot

Phase plot


3d phase plot

3D phase plot


The phase plots tell

The phase plots tell

  • Use two lagged values

  • The first lagged value describes a parabola

  • Lets make a neural network for prediction of the timeseries based on the findings.


The neural network

The neural network

^

y(k+1)

Lets try with 3 hidden nodes

2 for the ”parabola”

and one for the ”rest”

y(k) y(k-1)


Prediction results

Prediction results


Residuals on test data

Residuals (on test data)


A more difficult case

A more difficult case

  • If the time series is time variant (i.e. the dynamic behaviour changes over time) and the measurement data is noisy, the prediction task becomes more challenging.


Phase plot for a noisy timevariant case

Phase plot for a noisy timevariant case


Residuals with the model

Residuals with the model


Use a kalman filter to update the weights

Use a Kalman-filter to update the weights

  • We can improve the predictions by using a Kalman-filter

  • Assume that the process we want to predict is described by


Kalman filter

Kalman-filter

  • Use the following recursive equations

The gradient needed in

Ck is fairly simple to

calculate for a sigmoidal

network


Residuals

Residuals


Neural network parameters

Neural network parameters


Henon series

Henon series

  • The timeseries is actually described by


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