Generalized entropy and multifractality of time-series: relationship between order and intermittency

The intermittency of a time-series is its tendency to have large departures from its characteristic dynamics. The quantification of intermittency has applications to the study of physical, biological, and economic phenomena. Intermittency has been quantified by multifractality, the extent to which generalized Hurst exponents differ. As an alternative descriptor of intermittent processes, we present a nonextensive measure of order, based on the Tsallis entropy of a sequence of symbols corresponding to the time-series. Like multifractality, nonextensive order increases with intermittency. Nonextensive order has the advantage that it does not assume scaling in the time-series, whereas a scaling region has to be identified in order to estimate multifractality. However, unlike multifractality, nonextensive order requires the selection of parameters used to generate subsequences of symbols from the time-series. Both nonextensive order and multifractality can distinguish time-series that have different levels of intermittency. In distinguishing simulated point processes of D=0.1 from those of D=0.5, nonextensive order and multifractality performed about equally well and nonextensive order performed better than its extensive counterpart. Multifractality more accurately distinguished processes with D=0.5 from those of D=0.9. Which statistic better describes a time-series depends on the specific application.