Modelling directional spatial processes in ecological data
Introduction
It is well known that spatial distributions of species are influenced by environmental gradients (Huston, 1996). Since the article of Legendre and Fortin (1989), the importance of spatial structures has been well understood by ecologists. This has led to a number of methodological developments to study spatial patterns in ecology. Methods devised in other domains have also been applied to ecology. For example, geostatistical tools have been, and still are, used to investigate spatial relationships in an ecological perspective; Peterson et al. (2007) is a recent example of the use of geostatistics in river modelling. Legendre (1990) proposed to use polynomials of the geographic coordinates of the sites to represent spatial relationships in models aimed at explaining species variation. More recently, the development of principal coordinates of neighbour matrices (PCNM) (Borcard and Legendre, 2002, Borcard et al., 2004, Legendre and Borcard, 2006) has provided a new and more powerful way for studying spatial variation. It has also significantly enhanced the proportion of variation explained by spatial models. Dray et al. (2006) developed the framework of Moran's eigenvector maps (MEM), which is a generalization of the PCNM approach. Griffith and Peres-Neto (2006) unified the PCNM, MEM, and spatial filtering methods (Griffith, 2000) into a family called eigenfunction-based spatial analysis. Borcard et al. (1992) showed through variation partitioning that spatial relationships and environment can explain both separate and common variation of the distributions of species. To this day, however, no methodological development has shown how to model the influence of asymmetric, directional process on species distributions or other response variables of interest.
At broad or fine scales, the spatial distribution of species is often structured by abiotic and/or biotic gradient(s). We propose that gradients influencing species spatial distributions can be studied via spatial variables (eigenfunctions) that represent directional spatial processes. Dray et al. (2006) deplored the absence of methods capable of modelling asymmetric directional spatial processes; the present paper fills that gap. Here, a new framework is presented, which is also part of the eigenfunction-based spatial filtering framework, with the added feature that it considers space in an asymmetric way. Variables created via this framework will be called asymmetric eigenvector maps (AEM). This method was created for situations where a hypothesized asymmetric, directional spatial process influences the species distribution at scales ranging from fine to broad (e.g. the directional effects of a river network, or of currents in a sea, river, stream, or fluvial lake, on species distributions). Since the AEM framework creates spatial variables corresponding to an asymmetric, directional process, these variables can be used to model the spatial structure of any set of response variables, e.g. single-species population data, multi-species community, meta-population, or meta-community, influenced by an asymmetric spatial process. A process is a phenomenon organized along space and/or time. We define a directional spatial process as a process that can be represented by directional arrows in geographic space. The structures resulting from directional processes are asymmetric. To test the functioning and limits of the new AEM method, simulations have been carried out in a two-dimensional spatial context.
Section snippets
Method
The Dray et al. (2006) MEM method consists in the diagonalization of a spatial weighting matrix (W). Matrix W is a resemblance matrix that can be constructed through the Hadamard product between two previously computed resemblance matrices: a connectivity matrix showing which sites are linked to one another by connexions, and a weighting matrix which gives the weight associated to each pair of sites. As developed by Dray et al. (2006), no direction can be imposed on the created MEM spatial
Simulation study
We carried out a range of simulations to better understand the behaviour of AEM eigenfunctions in different situations. AEM eigenfunctions were first tested for type I error. For power evaluation, they were compared to MEM eigenfunctions in the presence of asymmetric generating processes, for different types of spatial structures, using the proportions of variance explained as criterion.
Simulations were first used to estimate the type I error of AEM analysis. Two sets of simulations with a
Ecological illustration
To illustrate the application of AEM analysis to real ecological situations, we used data collected on 42 lakes of the Mastigouche Reserve, Québec, Canada (46°40′N, 73°20′W) and analyzed by Magnan et al. (1994). The dependent data matrix describes brook trout (Salvelinus fontinalis) diet composition in those lakes. In each lake, 20 stomachs were sampled during daytime by anglers in June 1989. Mean percent wet mass was recorded for nine functional prey categories: zoobenthos, amphipods,
Discussion
The objective of spatial modelling using geographic eigenfunctions differs from that of standard canonical modelling using only environmental variables as the explanatory table. Magnan et al. (1994) did both types of modelling, acknowledging the fact that the presence of spatial structures in communities is of great interest: it indicates that some process has been at work to create these structures. Ecologists now understand that spatial structures can be produced by two different mechanisms (
Supplements
An R package called “AEM” is available online. It contains all the functions used to perform the analyses presented in this paper.
Acknowledgements
We are grateful to Prof. P. Magnan, Université du Québec à Trois-Rivières, who allowed us to use the brook trout diet data for illustration of the AEM method. This research was supported by NSERC grant no. OGP0007738 to P. Legendre.
References (36)
- et al.
A modelling study on hydrodynamics and pollutant dispersion in the Suez Canal
Ecological Modelling
(2000) - et al.
All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices
Ecological Modelling
(2002) - et al.
Spatial modelling: a comprehensive framework for principal coordinate analysis of neighbour matrices (PCNM)
Ecological Modelling
(2006) - et al.
An empirical comparison of permutation methods for tests of partial regression coefficients in a linear model
Journal of Statistical Computation and Simulation
(1999) - et al.
Les arbres et les représentations des proximités
(1988) Théorie des graphes et ses applications
(1958)- Blanchet, F.G., Legendre, P., Borcard, D., in press. Forward selection of explanatory variables....
- et al.
Dissecting the spatial structure of ecological data at multiple scales
Ecology
(2004) - et al.
Partialling out the spatial component of ecological variation
Ecology
(1992) - et al.
Individual variations in habitat use and morphology in brook charr
Journal of Fish Biology
(1997)