Daniel Chessel

We propose a multivariate approach to the study of geographic species distribution which does not require absence data. Building on Hutchinson's concept of the ecological niche, this factor analysis compares, in the multidimensional space of ecological variables, the distribution of the localities where the focal species was observed to a reference set describing the whole study area. The first factor extracted maximizes the marginality of the focal species, defined as the ecological distance between the species optimum and the mean habitat within the reference area. The other factors maximize the specialization of this focal species, defined as the ratio of the ecological variance in mean habitat to that observed for the focal species. Eigenvectors and eigenvalues are readily interpreted and can be used to build habitat-suitability maps. This approach is recommended in situations where absence data are not available (many data banks), unreliable (most cryptic or rare species), or meaningless (invaders). We provide an illustration and validation of the method for the alpine ibex, a species reintroduced in Switzerland which presumably has not yet recolonized its entire range.

SUMMARY We present an unconventional procedure (fuzzy coding) to structure biological and environmental information, which uses positive scores to describe the affinity of a species for different modalities (i.e. categories) of a given variable. Fuzzy coding is essential for the synthesis of long‐term ecological data because it enables analysis of diverse kinds of biological information derived from a variety of sources (e.g. samples, literature). A fuzzy coded table can be processed by correspondence analysis. An example using aquatic beetles illustrates the properties of such a fuzzy correspondence analysis. Fuzzy coded tables were used in all articles of this issue to examine relationships between spatial‐temporal habitat variability and species traits, which were obtained from a long‐term study of the Upper Rhône River, France. Fuzzy correspondence analysis can be programmed with the equations given in this paper or can be performed using ADE (Environmental Data Analysis) software that has been adapted to analyse such long‐term ecological data. On Macintosh Apple TM computers, ADE performs simple linear ordination, more recently developed methods (e.g. principal component analysis with respect to instrumental variables, canonical correspondence analysis, co‐inertia analysis, local and spatial analyses), and provides a graphical display of results of these and other types of analysis (e.g. biplot, mapping, modelling curves). ADE consists of a program library that exploits the potential of the HyperCard TM interface. ADE in an open system, which offers the user a variety of facilities to create a specific sequence of programs. The mathematical background of ADE is supported by the algebraic model known as ‘duality diagram’.

SUMMARY Methods used for the study of species–environment relationships can be grouped into: (i) simple indirect and direct gradient analysis and multivariate direct gradient analysis (e.g. canonical correspondence analysis), all of which search for non‐symmetric patterns between environmental data sets and species data sets; and (ii) analysis of juxtaposed tables, canonical correlation analysis, and intertable ordination, which examine species–environment relationships by considering each data set equally. Different analytical techniques are appropriate for fulfilling different objectives. We propose a method, co‐inertia analysis, that can synthesize various approaches encountered in the ecological literature. Co‐inertia analysis is based on the mathematically coherent Euclidean model and can be universally reproduced (i.e. independently of software) because of its numerical stability. The method performs simultaneous analysis of two tables. The optimizing criterion in co‐inertia analysis is that the resulting sample scores (environmental scores and faunistic scores) are the most covariant. Such analysis is particularly suitable for the simultaneous detection of faunistic and environmental features in studies of ecosystem structure. The method was demonstrated using faunistic and environmental data from Friday ( Freshwater Biology 18, 87‐104, 1987). In this example, non‐symmetric analyses is inappropriate because of the large number of variables (species and environmental variables) compared with the small number of samples. Co‐inertia analysis is an extension of the analysis of cross tables previously attempted by others. It serves as a general method to relate any kinds of data set, using any kinds of standard analysis (e.g. principal components analysis, correspondence analysis, multiple correspondence analysis) or between‐class and within‐class analyses.