Empirical Orthogonal Function (EOF) Analysis for gappy data

[Updates]: The following approach has serious shortcomings, which I have recently become aware of. In a comparison of gappy EOF approaches Taylor et al. (2013) [pdf] show that this traditional approach is not as accurate as others. Specifically, the approach of DINEOF (Data Interpolating Empirical Orthogonal Functions) proved to be the most accurate. I have outlined the DINEOF algorithm in another post [link]. and show a comparison of gappoy EOF methods here: http://menugget.blogspot.de/2014/09/pca-eof-for-data-with-missing-values.html. The R package "sinkr" now contains a version of the function ("eof") for easy installation: https://github.com/menugget/sinkr

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The following is a function for the calculation of Empirical Orthogonal Functions (EOF). For those coming from a more biologically-oriented background and are familiar with Principal Component Analysis (PCA), the methods are similar. In the climate sciences the method is usually used for the decomposition of a data field into dominant spatial-temporal modes. 

Clarke and Ainsworth's BIOENV and BVSTEP (and BIO-BIO etc...)

Nonmetric Multidimensional Scaling (NMDS) plot of vegetation sample dissimilarities with best correlating environmental variables (left) and species (right) plotted as vectors (datasets "varespec" and "varechem" from the package vegan)

The R package "vegan" contains a version of Clarke and Ainsworth's (1993) BIOENV analysis allowing for the comparison of distance/similarity matrices between two sets of data having either samples or variables in common. The typical setup is in the exploration of environmental variables that best correlate to sample similarities of the biological community (e.g. species biomass or abundance). In this case, the similarity matrix of the community is fixed, while subsets of the environmental variables are used in the calculation of the environmental similarity matrix. A correlation coefficient (typically Spearman rank correlation coefficient) is then calculated between the two matrices and the best subset of environmental variables can then be identified and further subjected to a permutation test to determine significance.

This can be a very helpful analysis in the exploration of the often highly dimensional space of community samples. The method is also widely accepted by the scientific community due to its flexibility across a wide variety of data and is completely non-parametric - Clarke and Ainsworth's (1993) paper describing the method has 674 citations on Google Scholar at the time of this posting. 

The R package "vegan" incorporates this routine in the function bioenv(). An example of a BIOENV exploration between vegetation community data (dataset "varespec" in the vegan package) and the environmental data (dataset "varechem" in the vegan package) :