NXG Logic Explorer - Marčenko-Pastur and Tracy-Widom Distributions
The Marčenko-Pastur (MP) Law is one of the most fundamental elements of random matrix theory. The NXG Logic Explorer computer program employs the MP Law in a variety of ways, including component subtraction and super-resolution root MUSIC dimension reduction.
Component subtraction is a quantitative finance technique which attempts to remove the "market" effect due to overall market-wide sentiment. This is accomplished by decorrelating asset return prices, which can make a portfolio less biased from sentiment, and more adaptive to real market changes. Denoising is a noise removal approach which is commonly used to reduce uncertainty in asset prices over a time horizon. Decorrelation of assets requires use of principal components analysis (PCA) and multivariate regression to remove various levels of correlation. Noise reduction for asset price return histories requires fitting the limit distribution of eigenvalue density (MP Law) of the asset correlation matrix, determining the noise cutoff, and then applying multivariate regression to remove noise from price returns.
In super-resolution root MUSIC, the Explorer package exploits the MP Law to estimate the noise cuttoff for eigenvalues extracted from a covariance or correlation matrix. Using a leave-one-out approach, the noise eigenvectors when each object is left out of covariance matrix determination are cross-multiplied with each left out object's vector element to estimate a mode vector, which inverts the cross-product. As such, when the noise is lower for an object (for which the truth table is really in a given class, but left out during eigendecomposition), its inverse is greater, resulting in greater mode vector element values for the class that the left out object is really a member of.
The Tracy-Widom Law can also be used for determining the mean and standard deviation of lambda+ (noise cuttof level); however, we have chosen use of MP fitting. Another game-changing feature in Explorer is that derivative-free function approximation of MP is performed using particle swarm optimization exclusively.
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