Fitting ecological process models to spatial patterns using scalewise variances and moment equations

Ecological spatial patterns are structured by a multiplicity of processes acting over a wide range of scales. We propose a new method, based on the scalewise variance--that is, the variance as a function of spatial scale, calculated here with wavelet kernel functions--to disentangle the signature of processes that act at different and similar scales on observed spatial patterns. We derive exact and approximate analytical solutions for the expected scalewise variance under different individual-based, spatially explicit models for sessile organisms (e.g., plants), using moment equations. We further determine the probability distribution of independently observed scalewise variances for a given expectation, including complete spatial randomness. Thus, we provide a new analytical test of the null model of spatial randomness to understand at which scales, if any, the variance departs significantly from randomness. We also derive the likelihood function that is needed to estimate parameters of spatial models and their uncertainties from observed patterns. The methods are demonstrated through numerical examples and case studies of four tropical tree species on Barro Colorado Island, Panama. The methods developed here constitute powerful new tools for investigating effects of ecological processes on spatial point patterns and for statistical inference of process models from spatial patterns.