Multivariate analysis of spatial patterns: a unified approach to local and global structures

We propose a new approach to the multivariate analysis of data sets with known sampling site spatial positions. A between-sites neighbouring relationship must be derived from site positions and this relationship is introduced into the multivariate analyses through neighbouring weights (number of neighbours at each site) and through the matrix of the neighbouring graph. Eigenvector analysis methods (e.g. principal component analysis, correspondence analysis) can then be used to detect total, local and global structures. The introduction of the D-centring (centring with respect to the neighbouring weights) allows us to write a total variance decomposition into local and global components, and to propose a unified view of several methods. After a brief review of the matrix approach to this problem, we present the results obtained on both simulated and real data sets, showing how spatial structure can be detected and analysed. Freely available computer programs to perform computations and graphical displays are proposed.     

A new technique for ordering asymmetrical three-dimensional data sets in ecology

The aim of this paper is to tackle the problem that arises from asymmetrical data cubes formed by two crossed factors fixed by the experimenter (factor A and factor B, e.g., sites and dates) and a factor which is not controlled for (the species). The entries of this cube are densities in species. We approach this kind of data by the comparison of patterns, that is to say by analyzing first the effect of factor B on the species-factor A pattern, and second the effect of factor A on the species-factor B pattern. The analysis of patterns instead of individual responses requires a correspondence analysis. We use a method we call Foucart's correspondence analysis to coordinate the correspondence analyses of several independent matrices of species x factor A (respectively B) type, corresponding to each modality of factor B (respectively A). Such coordination makes it possible to evaluate the effect of factor B (respectively A) on the species-factor A (respectively B) pattern. The results obtained by such a procedure are much more insightful than those resulting from a classical single correspondence analysis applied to the global matrix that is obtained by simply unrolling the data cube, juxtaposing for example the individual species x factor A matrices through modalities of factor B. This is because a single global correspondence analysis combines three effects of factors in a way that cannot be determined from factorial maps (factor A, factor B, and factor A x factor B interaction) whereas the applications of Foucart's correspondence analysis clearly discriminate two different issues. Using two data sets, we illustrate that this technique proves to be particularly powerful in the analyses of ecological convergence which include several distinct data sets and in the analyses of spatiotemporal variations of species distributions.     

Testing sexual segregation and aggregation: old ways are best

The study of sexual segregation has received increasing attention over the last two decades. Several hypotheses have been proposed to explain the existence of sexual segregation, such as the "predation risk hypothesis," the "forage selection hypothesis," and the "activity budget hypothesis." Testing which hypothesis drives sexual segregation is hampered, however, by the lack of consensus regarding a formal measurement of sexual segregation. By using a derivation of the well-known chi-square (here called the sexual segregation and aggregation statistic [SSAS]) instead of existent segregation coefficients, we offer a reliable way to test for temporal variation in the occurrence of sexual segregation and aggregation, even in cases where a large proportion of animals are observed alone. A randomization procedure provides a test for the null hypothesis of independence of the distributions of males and females among the groups. The usefulness of SSAS in the study of sexual segregation is demonstrated with three case studies on ungulate populations belonging to species with contrasting life histories and annual grouping patterns (isard, red deer, and roe deer). The existent segregation coefficients were unreliable since, for a given value, sexual segregation could or could not occur. Similarly, the existent segregation coefficients performed badly when males and females aggregated. The new SSAS was not prone to such limitations and allowed clear conclusions regarding whether males and females segregate, aggregate, or simply mix at random applicable to all species.     

Measuring biological diversity using Euclidean metrics

We study the complementary use of Rao's theory of diversity (1986) and Euclidean metrics. The first outcome is a Euclidean diversity coefficient. This index allows to measure the diversity in a set of species beyond their relative abundances using biological information about the dissimilarity between the species. It also involves geometrical interpretations and graphical representations. Moreover, several populations (e.g., different sites) can be compared using a Euclidean dissimilarity coefficient derived from the Euclidean diversity coefficient. These proposals are used to compare breeding bird communities living in comparable habitat gradients in different parts of the world.     

Matching species traits to environmental variables: a new three-table ordination method

This paper addresses the question of studying the joint structure of three data tablesR,L andQ. In our motivating ecological example, the central tableL is a sites-by-species table that contains the number of organisms of a set of species that occurs at a set of sites. At the margins ofL are the sites-by-environment data tableR and the species-by-trait data table Q. For relating the biological traits of organisms to the characteristics of the environment in which they live, we propose a statistical technique calledRLQ analysis (R-mode linked toQ-mode), which consists in the general singular value decomposition of the triplet (R ^t D _I LD _J Q,D _q ,D _p ) whereD _I ,D _J ,D _q ,D _p are diagonal weight matrices, which are chosen in relation to the type of data that is being analyzed (quantitative, qualitative, etc.). In the special case where the central table is analysed by correspondence analysis,RLQ maximizes the covariance between linear combinations of columns ofR andQ. An example in bird ecology illustrates the potential of this method for community ecologists.