  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.
