Measuring information-based complexity across scales using cluster analysis

Abstract

Scaling of ecological data can present a challenge firstly because of the large amount of information contained in an ecological data set, and secondly because of the problem of fitting data to models that we want to use to capture structure. We present a measure of similarity between data collected at several scales using the same set of attributes. The measure is based on the concept of Kolmogorov complexity and implemented through minimal message length estimates of information content and cluster analysis (the models). The similarity represents common patterns across scales, within the model class. We thus provide a novel solution to the problem of simultaneously considering data structure, model fit and scale. The methods are illustrated in application to an ecological data set.

Introduction

Scale is a concept central to all ecological studies, whether relating to space or time. Sayama et al. (2003) demonstrated that there are powerful linkages between scales, contradicting the erroneous, though commonly held, assumption that it is possible to neatly partition evolutionary effects at different spatial scales, to study a molecule, individual, population, metapopulation, species or ecosystem. It is these dynamic linkages among the levels, rather than the number of levels themselves, that should probably be the focus of attention. Hogeweg (2002) argues that ‘processes do not, in biotic systems, operate in isolation and the existence of entanglement at different time and space scales does not need explanation, being there by default’. Ignoring it by segregating time and space scales is simply a modelling artefact.

There are several ways in which scale appears as a feature in ecological studies. An omnipresent problem is the modifiable areal unit problem (MUAP; see Openshaw, 1984, Fotheringham and Wong, 1991, Nakaya, 2000, Brunsdon, 2002, Holt et al., 1996, Jelinski and Wu, 1996). Ecological units do not come in convenient packets and the size, shape and distribution of samples will all have effects on any study; this aspect of scale has already received considerable research (Brunsdon, 2002, Pavlov et al., 2001; see also methods developed by Juhász-Nagy and Podani, 1983). The effects of scale can, however, be mitigated by employing fuzzy concepts. This allows any individual sample to partake of several component structures and leads to consistent estimates of cluster parameters. Bar-Yam (2002) proposed that agglomerative clustering indicates the mechanism by which information is lost as the level of uncertainty increases across scales, but a quantitative measure will depend on the similarity coefficient and the particular algorithm used for clustering. Pavlov et al. (2001) and Puzicha and Buhman (1998) use similar ideas to obtain segmentation of images based on texture variation and using fractal concepts; specifically Pavlov et al. (2001) suggest using wavelet decompositions. The number and distribution of samples also links to the part-whole problem (cf. Szabo, 1996) and also to the relationship between habitat heterogeneity and spontaneous pattern production (Sayama et al., 2003).

Here we shall consider a different problem that concerns the estimation of common structure between levels. We ask, does common structure exist and, if so, how strong is it? And how does it decay as differences in scale increase? Some argue that scale invariance, or the presence of similar structure across different spatial or temporal scales, should be expected for complex systems (e.g., Brown et al., 2002). However, Wolpert and Macready (2000) recently put forward another view that over different space and time scales, the patterns exhibited by such a complex system should vary greatly, and in ways that are unexpected given the patterns on the other scales. The degree of dissimilarity plotted against scale would therefore provide a profile, which can be used as a system descriptor, and compared with other system profiles irrespective of the subject matter. Binder and Plazas (2001) recommend a similar procedure. This obviously requires a suitable measure of dissimilarity or similarity between data at two or more scales. Several authors have suggested ordination methods for multiscale analyses (Noy-Meir and Anderson, 1971, Borcard and Legendre, 2002); however, these do not in general provide a similarity measure between scales. Another approach makes use of fractal and multifractal analysis (e.g., based on Rényi's generalized entropy functions; Borda-de-Agua et al., 2002); however, again the degree of self-similarity cannot easily be determined. In this paper, we present a clustering approach to determine similarity between scales. The problems caused MUAP can be overcome by using fuzzy clustering. This allows us to identify common structure at different scales using the minimum message length principle. The method was applied to ecological data to test its efficacy at detecting changes in community structure in terms of the composition and relative abundances of species in the community.