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- W938679934 abstract "Complex systems in nature fluctuate exhibit very rich spatio– temporal structure. We study the dynamical properties of real complex systems such as e.g. economy and/or financial market by looking at their spectral properties eg. density of eigenvalues under umbrella of Free Random Variables Calculus back-of-an -envelope calculation of complicated problems in RandomMatrix Theory. Classical Methodology General idea: In the absence of information on the phenomenon take large number of possible explanatory variables and large number of output variables, look for correlations between pairs hoping to find some signal. Problem:Standard tools for identifying hidden spatio–temporal structures like • Factor Component Analysis • Principal Component Analysis are rapidly marred by measurement noise, quantified by r = N/T and caught into dimensionality curse trap. This usually leads to biased estimates and spurious correlations. Solution: Random Matrix Theory with reacher structures can be difficult to get the result • Canwe have amore user-friendly version of RMT, to easily incorporate dynamical parameter ? Models considered we will assume, that cross–correlations of N variables can be described by the two–point covariance (correlation) function, Cia,jb ≡ 〈XiaXjb〉 . (1) For Xia ≡ xia − 〈xia〉, which describe the fluctuations (with zeromean) of the returns around the trend, and collect them into a rectangular N × T matrix X. The average 〈. . .〉 is understood as taken according to some probability distribution whose functional shape is stable over time, but whose parameters may be time– dependent. With cross–covariances and auto– covariances factorized, non–random, and decoupled the temporal dependence of the distribution of variable is the same, and the structure of cross–correlations does not evolve in time Cia,jb = CijAab (2) we will consider these distinct cases: • C = A = I • C 6= I A = I • C 6= A 6= I • Cross–correlations Free Random Variables A generalization of probability theory to noncommutative random variables, such as infinite (Hermitian) randommatricesH. It relies on the concept of freeness, which is noncommutative independence. Classical Probability Noncommutative probability (FRV) x random variable, p(x) H random matrix, P (H) pdf spectral density (λ)dλ characteristic function gx(z) ≡ 〈 e 〉 Green’s function GH(z) = 1 N 〈" @default.
- W938679934 created "2016-06-24" @default.
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- W938679934 date "2010-10-15" @default.
- W938679934 modified "2023-09-24" @default.
- W938679934 title "Applying free random variables to the analysis of temporal correlations in real complex systems" @default.
- W938679934 hasPublicationYear "2010" @default.
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