Spectra of Sample Covariance Matrices for Multiple Time Series

Transcription

1 Spectra of Sample Covariance Matrices for Multiple Time Series Reimer Kühn, Peter Sollich Disordered System Group, Department of Mathematics, King s College London VIIIth Brunel-Bielefeld Workshop on Random Matrices, Dec 14-15, 212

2 Sample Covariance Matrices of Multiple Time Series Covariance matrix of stationary stochastic process x t = (x a t ), t Z, 1 a p: C ab ij = 1 M M t=1 x a i+t xb j+t = 1 M (XXT ) ab ij. Here X = (x it ) is pn M matrix with entries x it = x i+t. Expect finite sample fluctuation around mean C ab ij = xa i xb j ±O(1/ M) = C ab (i j)±o(1/ M) C is randomly perturbed block Toeplitz matrix. Spectrum of C as N, fixed p and α = N/M? Known result as α : Szegö s Theorem ρ (λ) = 1 p p s=1 2π dq 2π δ(λ Ĉ s (q))

3 Compare with Wishart Laguerre Ensemble Empirical covariances for N data, evaluated on the basis of M measurements for each variable. Use N M matrices X = (x it ) with i.i.d. entries x it to compute: C ij = 1 M (XXT ) ij = 1 M M t=1 x it x jt. Expect finite sample fluctuation around mean. C ij = x i x j ±O(1/ M) = δ ij ±O(1/ M) Spectrum of C as N, fixed α = N/M? Marčenko Pastur-Law ρ α (λ) = 1 4α (λ (1+α)) 2 2παλ

4 Principal Differences Rows of X for the multi-time series covariance problem groups of shifted sections of a set of p time series (x t ) t Z, x t R p. X = x 1 x 2 x 3... x M x 2 x 3 x 4... x 1+M..... x N x N+1 x N+2... x N+M Number of random variables in the problem is O(N), rather than O(N 2 ) as in the Wishart Laguerre ensemble. Extensive body of knowledge about the Wishart-Laguerre ensemble and its variants (applications in multivariate statistics, signal-processing, finance,...) Very little is known. Existence proofs (Basak, Bose, Sen 211). p = 1-case solved only recently. (RK, P Sollich, EPL 212)

5 Spectral Density and Resolvent Spectral density of sample covariance matrix from resolvent 1 [λε ρ(λ) = lim Im Tr 1I C ] 1, λ ε = λ iε N πnp Express as (S F Edwards & R C Jones, JPA, 1976) 1 ρ α (λ) = lim Im ln N πn p λ Tr [ λ ε 1I C ] = lim 2 Im lnz Np, N πn p λ where N p = Np and Z Np is a Gaussian integral: Z Np = k,a du ka 2π/i exp { i 2 ka,lb } u ka (λ ε δ ab δ kl Ckl ab )u lb.

6 Performing the Average Standard Approach Replica Method 1 lnz Np = lim Z n n ln N n p For integer n, Z n N p is partition function of n identical copies of the system (n-th power of Gaussian integral) Experience: final result has structure of replica-symmetric high-temperature solution annealed calculation (n = 1). lnz Np ln Z Np Do annealed calculation from the start Z Np = k,a du ka 2π/i exp { i 2 ka,lb } u ka (λ ε δ ab δ kl Ckl ab )u lb

7 Performing the Average (contd.) Insert definition of C, and α p = αp, Z Np = ka du ka 2π/i exp i 2 λ ε ka u 2 ka + i 2 α p M zi 2 i=1 with disorder dependence of Z Np only through the M variables z i = 1 Np ka x a k+i u ka, 1 i M. By CLT (for weakly dependent rv s) normally distributed for large M with z i =, z i z j = 1 N p ka,lb x a k+i xb l+j u kau lb Q ij and Q given in terms of true process auto-covariance Q ij = z i z j = 1 N p ka,lb C ab (i j +k l)u ka u lb

8 Exploiting Szegö s Theorem for Spectral Sums {z i } average is Gaussian; with α p = αp: Z Np = ka du ka 2π/i exp i 2 λ ε ka u 2 ka 1 2 lndet(1i iα pq) Q is a Toeplitz matrix. evaluate ln det (1I iα p Q) using Szegö s theorem: where Q µ = 1 N p lndet(1i iα p Q) ka,lb (M 1)/2 µ= (M 1)/2 Ĉ ab (q µ )e iq µ(k l) uka u lb = 1 p ) ln (1 iα p Q µ with û a (q µ ) = 1 N Nk=1 e iq µk u ka and q µ = 2π M µ. ab Ĉ ab (q µ )û a(q µ )u b (q µ )

9 Closed Form Approximation & Scaling Get closed form expression of Z Np = ν I ν with I ν = i p sĉs(p ν ) p s=1 e i s x sλ ε /Ĉ s (p ν ) dx s ( 1 i α 2 sx s ) 2/α where Ĉ s (p ν ), s = 1,...,p are eigenvalues of Ĉ(p ν ) = (Ĉ ab (p ν )) Gives ρ α (λ) = 1 p π dq π p s=1 1 Ĉ s (q) ρ s { λ Ĉ s (q) } For uncorrelated data Ĉ s (q) 1, and ρ s is independent of s identify ρ s with the spectral density for covariance matrices of p time series of i.i.d. (uncorrelated) data.

10 Numerical Tests Spectral density for x n N(,1) α = ρ(λ).8.6 ρ(ln(λ)) λ ln(λ) (Left) p = 1: simulation results (green); analytic approximation for ρ () α (λ) (red), Marčenko-Pastur law (blue-dashed) (. From RK, P Sollich, EPL 212.) (Right) logarithmic spectral density; simulation results for p = 1,...,4. In all α =.1

11 AR-1 α =.1, p = 1 x n = a 1 x n a 2 1 ξ n (Logarithmic) Spectral density for AR-1 α = ρ(ln(λ)).6 ρ(ln(λ)) ln(λ) ln(λ) Left: i.i.d. data, simulation (green) and analytic result (red). Right a 1 =.8. Comparing scaling based on the empirical scaling function (black) with that based on the analytic result (red) and simulations (green).

12 AR-1 α =.1, p = 2 and p = 3 (Logarithmic) Spectral density for AR-1 α =.1 x n = Ax n 1 +σξ n ρ(ln(λ)) ρ(ln(λ)) ln(λ) ln(λ) (Left) Spectra for p = 2, uncorrelated data and A = [.8.1;.1.8], (Right) p = 3, uncorrelated data and A = [.8.2.1;.2.6.1;.1.1.5]. In both cases σ =.2, and simulation results are compared with scaling using an approximate evaluation of scaling integrals.

13 Summary Computed DOS of sample covariance matrices for multiple time-series using annealed calculation. Key ingredient: Szegö s theorem for (block) Toeplitz matrices Rectangular window and decorrelation approximation Closed form approximation. Use of Szegös theorem suggests a scaling form for DOS. scaling is requires knowledge of a function on R p! DOS for i.i.d. data is insufficient. currently working on effective methods to evaluate scaling function for p > 1. Lots of possible applications.