From: McCune, B. & J. B. Grace Analysis of Ecological Communities. MjM Software Design, Gleneden Beach, Oregon Tables, Figures, and Equations

Figure Timeline of development of some ordination methods and their application in ecology. BC = Bray-Curtis ordination, CA(RA) = correspondence analysis (reciprocal averaging), CCA = canonical correspondence analysis, DCA = detrended correspondence analysis, ISOMAP = isometric feature mapping, LLE = locally linear embedding, NMS = nonmetric multidimensional scaling, and PCA = principal components analysis.

Figure Reduction of a p-dimensional matrix to a k-dimensional ordination.

Figure A two-dimensional ordination of sample units in species space. Distances between sample units approximate dissimilarity in species composition.

Fig Overlay with different symbols for different topographic classes of sample units (stands).

Fig Overlay with related points enclosed by loops or arcs.

Figure Overlay with sizes of symbols proportional to the magnitude of a variable, in this case abundance of the species Festuca idahoensis.

Figure Scatterplot of abundance, measured as cover, of one species in relation to two ordination axes. Each point represents a plot. Symbol size is proportional to species abundance. Upper right: 2-D ordination of species composition. Lower right: scatterplot of abundance of “ALSA” (Alectoria sarmentosa) against score on Axis 1 (horizontal axis). Upper left: scatterplot of abundance of ALSA against Axis 2. Superimposed on the two abundance scatterplots are the least squares regression lines and a smoothed envelope. See text for interpretation of r and tau.

Figure Example of a joint plot. The angles and lengths of the radiating lines indicate the direction and strength of relationships of the variables with the ordination scores.

Figure Method of calculating a joint plot vector in PC-ORD.

Figure Finding the angle of the vector using vector fitting as described by Dargie (1984); r 2 = 0.66 between moisture and three ordination axes. Assume we have a k-dimensional ordination to which we are fitting a vector for variable V. Regress V against the ordination scores X: Dargie’s method

Figure Scaling the length of the vector using vector fitting. Vectors were scaled to unit length in three dimensions then projected onto two dimensions. The numbers are r 2 values in three dimensions.

Figure Grid overlay on an artificial data set. The grid represents a 2-D environmental gradient underlying noiseless species responses. Inset: the ideal result is a regular grid.

Figure Successional vectors. Lines connect sequential measurements on permanent sample units.

Table Coefficients of determination for the correlations between ordination distances and distances in the original 40-dimensional space. For this example, the data matrix was 100 sample units  40 species. The number of entity pairs used in the correlation = 4950.

Table Suggested methods for calculating proportion of variance represented by ordination axes. The after-the-fact method is the r 2 between Euclidean distances in the ordination space and distances in the original space. The distances in the original space can be calculated with various distance measures, but this choice of distance measure is important. “Built-in” means the method is a part of the ordination method itself.