Critical Windows: A Method For Detecting Lagged Variables In Ecological Time Series

Many developmental processes in the life sciences, ecology and even in economics depend strongly on the environmental conditions occurring in a bounded time interval, the results occurring often far later. Examples are as diverse as plant phenology, grapewine maturation, diapause induction and so on. The method proposed here, aims at detecting quickly such effects. The basic idea is to regress the recorded results of a series of replications of the process against a function of an independent time series. This variable is defined on a set of periods of time, systematically scanned by varying their lower and upper bounds. In simple cases when this function is the integral and the effect strictly limited in a time window, the response model, under the form of correlation coefficients, is tractable and its shape is predictable. It is the same when the window is a bell-shaped function and can be fitted with other weighting functions as the Beta and the polynomials. The null hypothesis of absence of influence of any past interval is tested by Monte-Carlo simulation. The most likely window of influence is determined by the maximum correlation coefficient, and the bivariate confidence interval is estimated by bootstrap. The period found with a rectangular shaped window can be used as a starting point for more specific windows. This technique has the advantage of avoiding to split the climatic series into arbitrary slices, thus multiplying the predictors and complicating the models selection. It is closely linked to continuous lag distributed models with the simplification that the variable of interest is not explicitly time dependent. Examples are given for the prediction of aphids population dynamics, male morphs induction in aphids, and the phenology of the ash tree.