A measure of association between vectors based on “similarity covariance”
{Cite as: RD Pascual-Marqui, D Lehmann, K Kochi, T Kinoshita, N Yamada: A measure of association between vectors based on “similarity covariance”, 2013-01-21, arXiv:1301.4291 [stat.ME], http://arxiv.org/abs/1301.4291}
RD Pascual-Marqui1,2, D Lehmann2, K Kochi2, T Kinoshita3, N Yamada1
1Department of Psychiatry, Shiga University of Medical Science, Japan; 2The KEY Institute for Brain-Mind Research, University of Zurich, Switzerland; 3Department of Neuropsychiatry, Kansai Medical University, Japan.
Corresponding author:
RD Pascual-Marqui; pascualm at key.uzh.ch; pascualm at belle.shiga-med.ac.jp; www.uzh.ch/keyinst/loreta.htm
The KEY Institute for Brain-Mind Research, University Hospital of Psychiatry, Zurich, Switzerland
Department of Psychiatry, Shiga University of Medical Sciences, Shiga, Japan
Abstract
The “maximum similarity correlation” definition introduced in this study is motivated by the seminal work of Székely 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 Székely 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 discussed here (using lazarus free-pascal “www.lazarus.freepascal.org”), including the test data as text files are freely available, under a Creative Commons Attribution-Noncommercial-ShareAlike license (creativecommons.org/licenses/by-nc-sa/3.0).