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

Prediction of a nonlinear time series with feedforward neural networks

Mats Nikus

Process Control Laboratory


The time series
The time series neural networks


A closer look
A closer look neural networks


Another look
Another look neural networks


Studying the time series
Studying the time series neural networks

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

  • Lets study the autocovariance function



Studying the time series1
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
Phase plot neural networks


3d phase plot
3D phase plot neural networks


The phase plots tell
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
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
Prediction results neural networks


Residuals on test data
Residuals (on test data) neural networks


A more difficult case
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.



Residuals with the model
Residuals with the model neural networks


Use a kalman filter to update the weights
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
Kalman-filter neural networks

  • Use the following recursive equations

The gradient needed in

Ck is fairly simple to

calculate for a sigmoidal

network


Residuals
Residuals neural networks


Neural network parameters
Neural network parameters neural networks


Henon series
Henon series neural networks

  • The timeseries is actually described by


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