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Prediction of a nonlinear time series with feedforward neural networksPowerPoint Presentation

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 neural networks

A closer look neural networks

Another look neural networks

Studying the time series neural networks

- Some features seem to reapeat themselves over and over, but not totally ”deterministically”
- Lets study the autocovariance function

The autocovariance function neural networks

Studying the time series neural networks

- 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 neural networks

3D phase plot neural networks

The phase plots tell neural networks

- 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 neural networks

^

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 neural networks

Residuals (on test data) neural networks

A more difficult case neural networks

- 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 neural networks

Residuals with the model neural networks

Use a Kalman-filter to update the weights neural networks

- We can improve the predictions by using a Kalman-filter
- Assume that the process we want to predict is described by

Kalman-filter neural networks

- Use the following recursive equations

The gradient needed in

Ck is fairly simple to

calculate for a sigmoidal

network

Residuals neural networks

Neural network parameters neural networks

Henon series neural networks

- The timeseries is actually described by

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