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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. 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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  1. Prediction of a nonlinear time series with feedforward neural networks Mats Nikus Process Control Laboratory

  2. The time series

  3. A closer look

  4. Another look

  5. Studying the time series • Some features seem to reapeat themselves over and over, but not totally ”deterministically” • Lets study the autocovariance function

  6. The autocovariance function

  7. 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

  8. Phase plot

  9. 3D phase plot

  10. 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.

  11. 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)

  12. Prediction results

  13. Residuals (on test data)

  14. 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.

  15. Phase plot for a noisy timevariant case

  16. Residuals with the model

  17. 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

  18. Kalman-filter • Use the following recursive equations The gradient needed in Ck is fairly simple to calculate for a sigmoidal network

  19. Residuals

  20. Neural network parameters

  21. Henon series • The timeseries is actually described by

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