  The search for patterns in time series is a very common task when dealing
with complex systems. This is usually accomplished by employing a complexity
measure such as entropies and fractal dimensions. However, such measures
usually only capture a single aspect of the system dynamics. Here, we propose a
family of complexity measures for time series based on a generalization of the
complexity-entropy causality plane. By replacing the Shannon entropy by a
mono-parametric entropy (Tsallis $q$-entropy) and after considering the proper
generalization of the statistical complexity ($q$-complexity), we build up a
parametric curve (the $q$-complexity-entropy curve) that is used for
characterizing/classifying time series. Based on simple exact results and
numerical simulations of stochastic processes, we show that these curves can
distinguish among different long-range, short-range and oscillating correlated
behaviors. Also, we verify that simulated chaotic and stochastic time series
can be distinguished based on whether these curves are open or closed. We
further test this technique in experimental scenarios related to chaotic laser
intensity, stock price, sunspot, and geomagnetic dynamics, confirming its
usefulness. Finally, we prove that these curves enhance the automatic
classification of time series with long-range correlations and interbeat
intervals of healthy subjects and patients with heart disease.
