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

In real life spatial patterns often evolve through time, and so to understa nd the relationship between a generating mechanism and the resulting spatia l pattern we need to consider the full space-time structure. We therefore e xtend the linear lattice-based spatial growth-interaction process of Rensha w (Renshaw E 1994a J. R. Stat. Sec. B 56 75-91), and explore the spectral p aradigm between it and the purely spatial construction of Jefferson and And erson (Jefferson J H and Anderson J D 1987 Proc. Conf. of the Advisory Grou p 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 t he general inverse space-time problem, namely how to construct spatial inte raction parameters which will produce a process with given spectral structu re. We then show that although the construction of fractal-type processes w ith pure power-law spectra requires global interaction, approximating power -sine-law spectra require spatial interaction across only a finite number o f sites. Simulated one- and two-dimensional examples illustrate the princip les involved.