Connectionists: "Similarity Covariance" pre-print

[Roberto D. Pascual-Marqui]

Dear Colleagues,
The following pre-print on "similarity covariance" might be of interest:

http://arxiv.org/abs/1301.4291

A measure of association between vectors based on "similarity covariance"

RD Pascual-Marqui, D Lehmann, K Kochi, T Kinoshita, N Yamada

The "similarity covariance" definition introduced in this study is motivated
by the seminal work of Szekely et al on "distance covariance" (Ann. Statist.
2007, 35: 2769-2794; Ann. Appl. Stat. 2009, 3: 1236-1265). Instead of using
Euclidean distances "d" as in Szekely et al, we use "similarity", which can be
defined as "exp(-d/s)", where the scaling parameter s>0 controls how rapidly
the similarity falls off with distance. Scale parameters are chosen by
maximizing the similarity correlation. The motivation for using "similarity"
originates in spectral clustering theory (see e.g. Ng et al 2001, Advances in
Neural Information Processing Systems 14: 849-856). We show that a particular
form of similarity correlation is asymptotically equivalent to distance
correlation for large values of the scale parameter. Furthermore, we extend
similarity correlation to coherence between complex valued vectors, including
its partitioning into real and imaginary contributions. Several toy examples
are used for comparing distance and similarity correlations. For instance,
points on a noiseless straight line give distance and similarity correlation
values equal to 1; but points on a noiseless circle produces near zero distance
correlation (dCorr=0.02) while the similarity correlation is distinctly non
zero (sCorr=0.36). In distinction to the distance approach, similarity gives
more importance to small distances, which emphasizes the local properties of
functional relations. This paper represents a preliminary empirical study,
showing that the novel similarity association has some distinct practical
advantages over distance based association.For the sake of reproducible
research, the software code implementing all methods here (using lazarus
free-pascal "www.lazarus.freepascal.org"), including all test data, are freely
available at: "sites.google.com/site/pascualmarqui/home/similaritycovariance".

Sincerely,
Roberto

...
Roberto D. Pascual-Marqui, PhD, PD
The KEY Institute for Brain-Mind Research, University Hospital of
Psychiatry Zurich (Switzerland)
Department of Community Psychiatric Medicine, Shiga University of
Medical Science (Japan) (pascualm at belle.shiga-med.ac.jp)
[www.keyinst.uzh.ch/loreta] [www.researcherid.com/rid/A-2012-2008]