Dynamical Complexity Of Short and Noisy Time Series

Shannon Entropy has been extensively used for characterizing complexity of time series arising from chaotic dynamical systems and stochastic processes such as Markov chains. However, for short and noisy time series, Shannon entropy performs poorly. Complexity measures which are based on lossless compression algorithms are a good substitute in such scenarios. We evaluate the performance of two such Compression-Complexity Measures namely Lempel-Ziv complexity ($LZ$) and Effort-To-Compress ($ETC$) on short time series from chaotic dynamical systems in the presence of noise. Both $LZ$ and $ETC$ outperform Shannon entropy ($H$) in accurately characterizing the dynamical complexity of such systems. For very short binary sequences (which arise in neuroscience applications), $ETC$ has higher number of distinct complexity values than $LZ$ and $H$, thus enabling a finer resolution. For two-state ergodic Markov chains, we empirically show that $ETC$ converges to a steady state value faster than $LZ$. Compression-Complexity Measures are promising for applications which involve short and noisy time series.