Striving to better forecast the dynamics of ecological systems is often touted as important for practical purposes, and as a potential driver and indicator of the state of understanding of a system. But if such dynamics are intrinsically unpredictable, as some recent research and commentaries suggest, such efforts are wasted. One approach for assessing the predictability of a system, more specifically a time series from that system, is to assess the predictive power of a suite of forecast models. Assessing predictability while avoiding model assumptions or other limitations of specific forecast models requires a model free method. Weighted permutation entropy (WPE) is such a model free measure of intrinsic predictability that has rarely been applied to ecological dynamics. We will describe how weighted permutation entropy quantifies the information in, and therefore the intrinsic predictability of, an arbitrary time series, and then give some results from simulated and real ecological data.
As shown previously, simulations of the single species logistic map show an increase in WPE (i.e. decrease in information content and intrinsic predictability) as intrinsic growth rate increases, interrupted by strong decreases in WPE at growth rates associated with cyclic dynamics. Furthermore, this variation in WPE is associated with the predictive skill of a forecast model (specifically, an empirical dynamic model, sensu Ye et al 2015 PNAS). The results suggest that WPE can provide a model-free estimate of the intrinsic predictability of a time series of ecological dynamics.
Analyses of time series from microbial microcosm experiments highlight the challenges of application to real ecological dynamics. For time series of population dynamics, rarity posed a considerable problem, as it resulted in long series of apparently zero abundance. This and other types of observation error may result in WPE underestimating intrinsic predictability. Commonly required transformations, such as interpolation (WPE assumes temporally regular observations) and rescaling (e.g. log transformation) also influence estimates of WPE, though it remains unclear if they create bias or just add error.
The increasing quantity of data available to ecological modellers provides some reasons for optimism regarding the future of ecological forecasting. Our findings suggest, however, that efforts to increase data quality are essential, if we are to avoid fundamental problems associated with application of forecasting tools, such as WPE, to ecological data.