Friday, April 19, 2013

1304.4982 (Vinayak et al.)

Emerging spectra of singular correlation matrices under small power-map

Vinayak, Rudi Schäfer, Thomas H. Seligman
A spectral study of power-map deformed singular random correlation matrices is presented in this paper. The study is inspired from the multivariate analysis of large dimensional datasets where stationarity or quasi-stationarity could often be a good approximation for a non-stationary time series only over a short time horizon. Correlation matrices estimated over such short time horizons are singular, with a significant portion of the spectrum falling exactly on zero. The power map has been proposed to remove this spectral degeneracy, where the correlation matrix is deformed as soon as the exponent is varied from one. Importance of the power map has been illustrated for the singular matrices encountered in financial application, viz. the portfolio optimization. To get a clearer picture of deformed matrices, the exponent of the power map has been chosen to be very close to one such that the emerging spectrum corresponding to the zeros of the original spectrum falls into a gap between zero and nonzero part of the entire spectrum. Analytical approximate estimate of the first and the second moments for the emerging spectra have been calculated for the deformed Wishart matrices of random matrix theory. Furthermore, the study has been extended to the correlated Wishart ensembles with two examples, viz. when the true correlation matrix has block-diagonal structure while in the second example it has a smooth band profile off the diagonal. In both cases the spectra have been analyzed numerically using the analytical results as a benchmark situation. It is shown that the emerging spectra are sensitive to the true correlation.
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