Improving community analysis with the Beals smoothing function

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Abstract
The Beals smoothing function, also known as the sociological favorability index, is a data transformation for heterogeneous ecological community data. After Beals smoothing, virtually any ordination method can successfully extract the dominant patterns in the data. Presence–absence (1,0) values in the raw data matrix are replaced by continuous probabilities. The transformed value in a given cell represents the probability of a particular species occurring in a particular sample unit, based on the other species that were present in that sample unit. These probabilities are constructed from a matrix of joint occurrences between species. Beals smoothing is appropriate for heterogeneous community data sets with a large number of zeros in the data matrix (i.e. most samples contain a fairly small proportion of the species). The method may also be useful for smoothing out differences due to varying sampling intensity or where sample quality is uneven. Beals smoothing is also desirable when the data set consists of many small sample units such that the data are inherently noisy. Smoothing such data sets in effect blends information from the small sample units. If the sample size is large, the blending is likely to be robust.