Abstract
We introduce a method for quantifying the inherent unpredictability of a continuous-valued time series via an extension of the differential Shannon entropy rate. Our extension, the specific entropy rate, quantifies the amount of predictive uncertainty associated with a specific state, rather than averaged over all states. We provide a data-driven approach for estimating the specific entropy rate of an observed time series. Finally, we consider three case studies of estimating the specific entropy rate from synthetic and physiological data relevant to the analysis of heart rate variability.
Highlights
The analysis of time series resulting from complex systems must often be performed “blind”: in many cases, mechanistic or phenomenological models are not available because of the inherent difficulty in formulating accurate models for complex systems
Via a decomposition of the entropy rate of a discrete-time, continuous-valued stochastic dynamical system, we have proposed a measure of state-specific uncertainty: the specific entropy rate
We have shown how to estimate the specific entropy rate from finite data using kernel density estimators and provided a data-driven method for choosing the free parameters in the kernel density estimation
Summary
The analysis of time series resulting from complex systems must often be performed “blind”: in many cases, mechanistic or phenomenological models are not available because of the inherent difficulty in formulating accurate models for complex systems. In this case, a typical analysis might treat the data as the model, in the spirit of nonparametric statistics, and attempt to generalize from the available data to the system more generally. On a more practical level, it is common to compare two time series from similar systems, in which case one wants to meaningfully ask: is the phenomenon resulting from System A more or less complex than the phenomenon resulting from System B?
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