Stability of a spatial structure and compromise in a multiway matrices statistical analysis: application to physicai and chemical properties in a reservoir lake
This paper is based on the observation that three-dimensional data matrices (sites x times x variables) often used in limnological investigations require statistical analyses fitted to experimental objectives. Many apparently different statistical tools (3-mode PCA of TUCKER, 1964; KROONENBERG, 1983; projection of variables WILLIAMS and STEPHENSON, 1973; DOLEDEC and CHESSEL, 1987) may be useful to clarify limnological problems such as : 1) the temporal variability of a pattern (elimination of spatial heterogenity) : 2) the spatial structure of a pattern (elimination of temporal effects, mapping of an average situation) : 3) the temporal variability of Lake stratification (stability, modification or inversion) : 4) the spatial structure of temporal variability, and 5) the between variables typology of a spatial and temporal structure. Our methodological approach allowed us to assess the temporal stability of the spatial structure of the Lake waters (question 3) using a multitable analysis known as triadic analysis (THIOULOUSE and CHESSEL, 1987). As part of the limnological study of a reservoir Lake (Sorme reservoir Lake, Saône-et-Loire, France) 10 commonly used physical and chemical variables were studied from July 1980 to October 1931. During this period, 12 water samples were taken near the surface at each of the 10 stations scattered along the Sorme Lake (see figure 1). Main morphometric features of the Sorme Lake are : 1) a surface area equal to 230 ha, 2) a 25 km long perimeter and 3) a volume of 9.5 106 m3 with a maximum depth of 13 meters upstream of the dam and an average depth of about 4 meters. Seasonal tidal range was only a few meters. Only 2 of the 3 concepts of triadic analysis stated by THIOULOUSE and CHESSEL, 1987 are developed here : 1) for each of the 12 tables (stations x variables) coming from the 12 sampling dates, data are first centered (elimination of mean) and standardized (division by standard deviation) (see figure 2). The resulting table Y called interstructure matrix, i.e. interstructure between each of the sampling dates matrix, is organized to have sampling dates as columns and the ten physical and chemical variables at each station successively as fines. Principal Component Analysis (PCA on the variance-covariance matrix) is then applied to the interstructure matrix. In our case it is a one-dimensional matrix, i.e. according to physical and chemical variables, there is only one spatial structure common to each sampling date (figure 3 and 2) compromise matrix are associated with the successive PCA factors of the interstructure (figure 4). According to the previous remark, only the first factor is considered. Data are reorganized to have physical and chemical variables as columns, and stations as fines. This last table defines a compromise matrix labelled Z. The mapping of the numerical values of matrix Z renders a ten-dimensional description of the permanent spatial structure (figure 5). To summarize the multivariate description, matrix Z is processed with a PCA on the variance-covariance matrix producing a three-dimensional compromise (figure 6). The interpretation of the compromise table by mapping the factorial scores of the PCA leads to a functional scheme of the reservoir Lake waters distinguishing five sectors (see figure 7) as a function of water depth, influence of tidal range, influence of tributaries and of the Sorme River. 3 stations are periodically isolated from the reservoir and produce 3 sectors with lower pH and temperature values and higher concentrations in ammonia and sulphate according to the influence of tributaries. The 4th sector is associated with the former submerged valley, i.e. main channel of the Sorme River prior to the dam closure, and demonstrated an ionic gradient concerning mainly nitrate and chloride-concentrations. The 5th sector, opposed to the latter, consists in the deeper area of the Sorme Lake which reveals rather homogeneous waters near the surface.