Finding unpredictability or uncertainty in a time series

Question

I am interested in finding a statistic that tracks the unpredictability of a time series. For simplicity sake, assume that each value in the time series is either 1 or 0. So for example, the following two time series are entirely predictable TS1: 1 1 1 1 1 1 1 1 TS2: 0 1 0 1 0 1 0 1 0 1 0 1

However, the following time series is not that predictable: TS3: 1 1 0 1 0 0 1 0 0 0 0 0 1 1 0 1 1 1

I am looking for a statistic that given a time series, would return a number between 0 and 1 with 0 indicating that the series is completely predictable and 1 indicating the series in completely unpredictable.

I looked at some entropy measures like Kolmogorov Complexity and Shannon entropy, but neither seem to fit my requirement. In Kolmogorov complexity, the statistic value changes depending on the length of the time series (as in "1 0 1 0 1" and "1 0 1 0" have different complexities, so its not possible to compare predictability of two time series with differing number of observations). In Shannon entropy, the order of observations didn't seem to matter.

Any pointers on what would be a good statistic for my requirement?

Answer 1

Since you have looked at Kolmogorov-Smirnov and Shannon entropy measures, I would like to suggest some other hopefully relevant options. First of all, you could take a look at the so-called approximate entropy $ApEn$. Other potential statistics include block entropy, T-complexity (T-entropy) as well as Tsallis entropy: http://members.noa.gr/anastasi/papers/B29.pdf

In addition to the above-mentioned potential measures, I would like to suggest to have a look at available statistics in Bayesian inference-based model of stochastic volatility in time series, implemented in R package stochvol: http://cran.r-project.org/web/packages/stochvol (see detailed vignette). Such statistics of uncertainty include overall level of volatility $\mu$, persistence $\phi$ and volatility of volatility $\sigma$: http://simpsonm.public.iastate.edu/BlogPosts/btcvol/KastnerFruwhirthSchnatterASISstochvol.pdf. A comprehensive example of using stochastic volatility model approach and stochvol package can be found in the excellent blog post "Exactly how volatile is bitcoin?" by Matt Simpson.