The Emergence of Pattern in Random Processes
We consider both time series as well as spatial distributions (in 1-4 dimensions). In the first, we observe that time series for individual and independently deviating random variables can manifest pattern through the emergence of peak-to-peak sequences that are visible to the eye yet fail all Fourier analysis schemes and reveal a seeming periodicity of 3-events per cycle. We note that this can explain observations of apparent cycles in mammalian animal populations. We consider models, as well, based on the Langevin equation of kinetic theory and the Smolouchowski relation that present circumstances where the apparent period can vary from 3-4 and, for a special subclass of problems, to periods between 2 and 3. We explore how cataloged observational data from global earthquake catalogues, magnetospheric AL index observations, Old Faithful Geyser eruption data, and the performance of the Standard & Poor's 500 index (percent daily variation) manifest different degrees of statistical agreement with the theory we derived. We present a simple model for many mammalian population cycles whose underlying phenomenological basis has strong biological implications. We then employ directed graphs to explore nearest-neighbor relationships and isolate the character of spatial clustering in 1-4 dimension. We observe that the one-dimensional problem is formally equivalent to that presented by peak-to-peak sequences in time series and also demonstrates a mean number of points per cluster of 3 in one dimension. We then take the first moment of each of the clusters formed, and observed that they too form clusters. We observe the emergence of a hierarchy of clusters and the emergence of universal cluster numbers, analogous to branching ratios and, possibly, Feigenbaum numbers. These, in turn, are related to fractals as well as succularity and lacunarity, although the exact nature of this connection has not been identified. Finally, we show how hierarchical clustering emerging from random distributions may help provide an explanation for observations of hierarchical clustering in cosmology via the virial theorem and simulation results relating to the gravitational stabilization in a self-similar way of very large self-gravitating ensembles.