Prediction of a nonlinear time series with feedforward neural networks

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19 Οκτ 2013 (πριν από 3 χρόνια και 5 μήνες)

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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”

Lets study the autocovariance function

The autocovariance function

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

3D phase plot

The phase plots tell

Use two lagged values

The first lagged value describes a

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

The neural network



y(k) y(k

Lets try with 3 hidden nodes

2 for the ”parabola”

and one for the ”rest”

Prediction results

Residuals (on test data)

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

Residuals with the model

Use a Kalman
filter to update
the weights

We can improve the predictions by
using a Kalman

Assume that the process we want to
predict is described by


Use the following recursive equations

The gradient needed in


is fairly simple to

calculate for a sigmoidal



Neural network parameters

Henon series

The timeseries is actually described by