No abstract

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 AppleTM 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 HyperCardTM 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’.

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.

This paper addresses the question of studying the joint structure of three data tables R, L and Q. In our motivating ecological example, the central table L 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 of L are the sites-by-environment data table R 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 called RLQ analysis (R-mode linked to Qmode), which consists in the general singular value decomposition of the triplet (RtDILDjQ,Dq,Dp) where DI, D j, Dq, ])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 of R and Q. An example in bird ecology illustrates the potential of this method for community ecologists.

1. 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 spedes-environment relationships by considering each data set equally. Different analytical techniques are appropriate for fulfilling different objectives. 2. 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. 3. The method was demonstrated using faunistic and environmental data from Friday (Freshzvater 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. 4. 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.

Ecological studies often require studying the common structure of a pair of data tables. Co‐inertia analysis is a multivariate method for coupling two tables. It is often neglected by ecologists who prefer the widely used methods of redundancy analysis and canonical correspondence analysis. We present the co‐inertia criterion for measuring the adequacy between two data sets. Co‐inertia analysis is based on this criterion as are canonical correspondence analysis or canonical correlation analysis, but the latter two have additional constraints. Co‐inertia analysis is very flexible and allows many possibilities for coupling. Co‐inertia analysis is suitable for quantitative and/or qualitative or fuzzy environmental variables. Moreover, various weighting of sites and various transformations and/or centering of species data are available for this method. Hence, more ecological considerations can be taken into account in the statistical procedures. Moreover, the principle of this method is very general and can be easily extended to the case of distance matrices or to the case of more than two tables. Simulated ecological data are used to compare the co‐inertia approach with other available methods.

No abstract

The design and objective of a community study imply the selection of the appropriate ordination technique in terms of species response models and weighting options. In this paper, we start from the observation that existing two‐table ordination techniques and related measures of niche breadth inevitably weight a sample in proportion to its abundance. We introduce a new multivariate method, which gives a more even weight to all sampling units, including those which are species poor or individual poor. We use this new method of analysis which we call OMI (for Outlying Mean Index) to address the question of niche separation and niche breadth. The Outlying Mean Index, or species marginality, measures the distance between the mean habitat conditions used by species (species centroid), and the mean habitat conditions of the sampling area (origin of the niche hyperspace), and OMI analysis places species along habitat conditions using a maximization of their mean OMI. Therefore, the position of the species depends on their niche deviation from a reference, which represents neither the mean nor the most abundant species, but a theoretical ubiquitous species that tolerates the most general habitat conditions (i.e., a hypothetical species uniformly distributed among habitat conditions). We demonstrate that OMI analysis is well suited for the investigation of multidimensional niche breadths in the case of strong limiting factors (e.g., meteorological conditions) or strong driving forces (e.g., longitudinal stream gradient). Furthermore, the analysis helps in finding which ecological factors are most important for community structure and organization and provides a separation of species based on their niche characteristics.