Explaining empirical dynamic modelling using verbal, graphical and mathematical approaches.

Empirical dynamic modelling (EDM) is becoming an increasingly popular method for understanding the dynamics of ecosystems. It has been applied to laboratory, terrestrial, freshwater and marine systems, used to forecast natural populations and has addressed fundamental ecological questions. Despite its increasing use, we have not found full explanations of EDM in the ecological literature, limiting understanding and reproducibility. Here we expand upon existing work by providing a detailed introduction to EDM. We use three progressively more complex approaches. A short verbal explanation of EDM is then explicitly demonstrated by graphically working through a simple example. We then introduce a full mathematical description of the steps involved. Conceptually, EDM translates a time series of data into a path through a multi-dimensional space, whose axes are lagged values of the time series. A time step is chosen from which to make a prediction. The state of the system at that time step corresponds to a 'focal point' in the multi-dimensional space. The set (called the library) of candidate nearest neighbours to the focal point is constructed, to determine the nearest neighbours that are then used to make the prediction. Our mathematical explanation explicitly documents which points in the multi-dimensional space should not be considered as focal points. We suggest a new option for excluding points from the library that may be useful for short-term time series that are often found in ecology. We focus on the core simplex and S-map algorithms of EDM. Our new R package, pbsEDM, enhances understanding (by outputting intermediate calculations), reproduces our results and can be applied to new data. Our work improves the clarity of the inner workings of EDM, a prerequisite for EDM to reach its full potential in ecology and have wide uptake in the provision of advice to managers of natural resources.