Statistical analysis of multidimensional fuzzy set ordinations.

A protocol for the creation and statistical analysis of multidimensional fuzzy set ordinations (MFSO) is presented, including a simple forward stepwise variable selection algorithm and goodness-of-fit statistics. Mapping sample point distributions from the fuzzy topological space of fuzzy set ordination (FSO) to a Euclidean space enables analysis by a broad range of parametric statistical methods. The results obtained exhibit immediate interpretability, as each axis in an MFSO represents a single environmental variable and is orthogonal to all other axes by design. MFSO is tested on five data sets: three simulated coenospaces, and vegetation data from Bryce Canyon National Park, Utah, USA, and the Shoshone National Forest, Wyoming, USA. The resulting ordinations on the test data sets achieve (1) high efficiency in representing the underlying dissimilarity matrix in low dimensionality (> 90% of the efficiency of principal coordinates analysis); (2) a high level of fidelity in reconstructing the location and configuration of samples in the simulated coenospaces; (3) a high level of resistance to noise in vegetation or environment data; and (4) a low level of sensitivity to sample size or placement along simulated environmental gradients. Environmental variable effect sizes are easily estimated, and probabilities of observing the results obtained are easily calculated by parametric distributions or permutation statistics.