The development of space-time interaction processes with given spectral structure

In real life spatial patterns often evolve through time, and so to understand the relationship between a generating mechanism and the resulting spatial pattern we need to consider the full space-time structure. We therefore extend the linear lattice-based spatial growth-interaction process of Renshaw (Renshaw E 1994a J. R. Stat. Soc. B 56 75-91), and explore the spectral paradigm between it and the purely spatial construction of Jefferson and Anderson (Jefferson J H and Anderson J D 1987 Proc. Conf. of the Advisory Group for Aerospace Research and Development vol 419 14.1-14.19) which is based on fractionally integrated white noise. This insight enables us to solve the general inverse space-time problem, namely how to construct spatial interaction parameters which will produce a process with given spectral structure. We then show that although the construction of fractal-type processes with pure power-law spectra requires global interaction, approximating power-sine-law spectra require spatial interaction across only a finite number of sites. Simulated one- and two-dimensional examples illustrate the principles involved.