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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

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
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
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)

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.
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

henon series
Henon series
  • The timeseries is actually described by
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