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Local order and predictability of financial time series

In this series of posts I will discuss an implementation and tests of the paper Local order, entropy and predictability of financial time series by L. Molgedey and W. Ebeling. (pdf)

The paper presents an excellent application of information theory to time series analysis. The idea is simple: is it possible to find sub-trajectories in financial time series (here the daily returns of some indices or stock) where a "local order" exists with higher than average predictability.

I won't explain the paper in full, so please have a look at the pdf above for notation and details. However I will describe the most important concepts below. We consider one-dimensional, discretely partitioned time series. The authors use Shannon entropy H as basic tool to measure uncertainty or predictability of the probability distribution described by the time series. For a certain trajectory of length n the uncertainty of predicting the next state is the difference in Shannon entropies for trajectories of length n+1 and n: