Chaotic Time Series

'Chaos' here is used in the mathematical sense. A chaotic process is one where positive feedback of some kind exists. Under some circumstances such processes can create time series that appear to be completely random - the corollary of this is that some seemingly random series are in fact chaotic, and thus to a certain extent predictable. Chaotic systems are never completely predictable; because of feedback the simulation and the real series will always rapidly diverge. This is effect is caused by small differences between the initial real state and the simulation growing geometrically as the simulation is advanced in time. Chaotic time series commonly occur in physics, biology, meteorology, engineering and finance.

An example of a chaotic time series

Logistic map: n(t+1) = 4 * n(t) * (1 – n(t)) , n(0) = 0.1

Subtract 0.5 from each sample and accumulate:

The time series above looks very much like a financial time series, or indeed series drawn from many disciplines that are normally considered incapable of prediction.

Detecting Chaos and estimating predictability

There are several measures that can be used to detect a chaotic series. One is the dimensionless Hurst exponent, which ranges between 0 and 1 where values close to zero indicate volatile, chaotic behavior, and values close to 1 represent smooth, mean-reverting behavior.

The Lyapunov exponent can be used to estimate the rate at which predictability recedes. It is scaled in bits per time period, and indicates how many bits of prediction accuracy are lost for each time series prediction increment into the future.

Embedding time series to detect structure

Using Taken’s embedding theorem you can create a multidimensional model of a time series. If there’s some structure to this you can use neighboring points to make limited predictions

An example embedded model (idealized!)	The prediction methodology used by ChaosKit