An empirical dynamic modeling framework for missing or irregular samples

Empirical dynamic modeling (EDM) is a powerful method for forecasting and analyzing nonlinear dynamics. However, typical applications of EDM assume that samples are evenly spaced over time. This presents problems in ecology, in which data are often missing or sampled irregularly. Standard methods for handling irregularity in EDM suffer under conditions that are common in ecology, such as short time series and large dynamic fluctuations, so there is a need to adapt the framework to cope with these challenges more effectively. Here we consider a variable step-size extension of EDM, which incorporates the temporal spacing between samples into EDM delay-coordinate vectors and circumvents the challenges faced by other approaches. We evaluated the forecast accuracy of the variable step-size method along with that of two other methods: (1) exclusion of delay-coordinate vectors with missing data and (2) linear interpolation along with ordinary EDM. We tested these methods using simulated data from three chaotic ecological models with various amounts and patterns of missing data. We also evaluated them using two empirical datasets: laboratory rotifer dynamics and aphid dynamics from the field. Results showed that while exclusion and linear interpolation can produce accurate forecasts in some scenarios, the variable step-size method consistently gives accurate forecasts in a wide range of scenarios. Our analysis demonstrates that variable step-size EDM is an effective method for coping with missing or irregular samples and expands the number of datasets to which EDM can be applied. Furthermore, EDM can be extended to estimate Lyapunov exponents from irregularly sampled time series and approximate continuous dynamics from discrete-time data.