Block Circulant Matrices and the Spectra of Multivariate Stationary

Spec. Matrices 2021; 9:36–51 Research Article Open Access Special Issue: Combinatorial Matrices Marianna Bolla*, Tamás Szabados, Máté Baranyi, and Fatma Abdelkhalek Block circulant matrices and the spectra of multivariate stationary sequences https://doi.org/10.1515/spma-2020-0121 Received October 30, 2020; accepted December 16, 2020 Abstract: Given a weakly stationary, multivariate time series with absolutely summable autocovariances, asymptotic relation is proved between the eigenvalues of the block Toeplitz matrix of the rst n autocovariances and the union of spectra of the spectral density matrices at the n Fourier frequencies, as n ! ∞. For the proof, eigenvalues and eigenvectors of block circulant matrices are used. The proved theorem has important consequences as for the analogies between the time and frequency domain calculations. In particular, the complex principal components are used for low-rank approximation of the process; whereas, the block Cholesky decomposition of the block Toeplitz matrix gives rise to dimension reduction within the innovation subspaces. The results are illustrated on a nancial time series. Keywords: weakly stationary time series, autocovariance and spectral density matrix, nite Fourier transform, complex principal components, block Cholesky decomposition MSC: 42B05, 62M15, 65F30 1 Introduction 2 Let fXtg be a weakly stationary, real-valued time series (t 2 Z). Assume that E(Xt) = 0, c(0) = E(Xt ) > 0, and the sequence of autocovariances c(h), (h = 1, 2, ... ) is absolutely summable; obviously, c(h) = c(−h). n Then it is known that the Toeplitz matrix Cn = [c(i − j)]i,j=1, that is the covariance matrix of the random T T vector (X1, ... , Xn) , is positive denite for all n 2 N (the vectors are column vectors and denotes the transposition). Under the absolute summability condition, the spectral density function f of the process exists and it is continuous. We can as well use [−π, π] or [0, 2π] for a complete period of f , and in view of f(ω) = f(−ω) = f(2π − ω), it suces to conne ourselves to the [0, π] interval. In [1], an asymptotic relation between 2π the eigenvalues of Cn and the values f (ωj) is proved as n ! ∞, where ωj = j n is the jth Fourier frequency, j = 0, 1, ... , n − 1. In the present paper we prove a similar, albeit more complicated, result in the multivariate case. Let fXtg d be a weakly stationary d-dimensional time series, t 2 Z and the state space is R . Again, E(Xt) = 0 and the sequence of the d × d autocovariance matrices C(0), C(1), C(2), ... is absolutely summable (entrywise); also T C(−h) = C (h) and C(0) is the usual covariance matrix of Xt. Now the nd × nd covariance matrix Cn of the T T T compounded random vector (X1 , ... , Xn ) is the following symmetric, positive semidenite block-Toeplitz *Corresponding Author: Marianna Bolla: Department of Stochastics, Institute of Mathematics, Budapest University of Tech- nology and Economics Muegyetem rkp. 3. Bldg H/5. H-1111 Budapest, Hungary, E-mail: [email protected] Tamás Szabados: Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics Muegyetem rkp. 3. Bldg H/5. H-1111 Budapest, Hungary, E-mail: [email protected] Máté Baranyi: Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics Mue- gyetem rkp. 3. Bldg H/5. H-1111 Budapest, Hungary, E-mail: [email protected] Fatma Abdelkhalek: Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics Muegyetem rkp. 3. Bldg H/5. H-1111 Budapest, Hungary, E-mail: [email protected] Open Access. © 2021 Marianna Bolla et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. Block Circulant Matrices and Spectra Ë 37 matrix: 2 C(0) C(1) C(2) ··· C(n − 1) 3 6 CT(1) C(0) C(1) ··· C(n − 2) 7 6 7 6 CT(2) CT(1) C(0) ··· C(n − 3) 7 Cn := 6 7 , (1) 6 7 6 . .. 7 4 . 5 CT(n − 1) CT(n − 2) CT(n − 3) ··· C(0) the (i, j) block of which is C(j − i). The symmetry comes from the fact, that the (j, i) block is C(i − j) = CT(j − i). At the same time, we consider the (under our conditions, existing) spectral density matrix f of the process. This is a self-adjoint, positive semidenite, d × d matrix, which is, in fact, the Fourier transform of the autocovariance matrices: ∞ 1 X f (ω) = C(h)e−ihω , ω 2 [0, 2π]. 2π h=−∞ It has real diagonal, but usually complex o-diagonal entries. In view of f (−ω) = f (ω), it suces to conne ourselves to the [0, π] interval (here · denotes the complex, entrywise conjugation). In our main theorem, we state that for ‘large’ n, the eigenvalues of Cn asymptotically comprise the union of the spectra of the spectral density matrices f (ωj), j = 0, 1, ... , n − 1. (We use spectra in two dierent meanings here: for the eigenvalues and for the spectral representation of weakly stationary processes.) The theorem also has the computational benet that instead of nding the eigenvalues of an nd × nd matrix (that needs O(n3d3) operations) we can nd the eigenvalues of number n of d×d matrices (that means only O(nd3) operations). Under our conditions, the rank r of f (ω) is almost surely constant on [0, π], but it can be less than d. It is true for sliding summations (two-sided moving average processes), in particular, for regular ones (one- sided moving average processes). Processes of rational spectral density, e.g., VARMA processes, are such. At the same time, the dimension of the innovation subspaces, which equals to r, can be concluded from the diagonal blocks of the block Cholesky decomposition of Cn, and it can also be reduced in the time domain with usual principal component (PC) analysis techniques. In parallel, in the frequency domain, an m-rank approximation with m ≤ r is also possible by using PC transformation with m complex PCs, where m depends on the gap of the spectrum of f , see also [2]. The organization of the paper is as follows. In Section 2, we introduce block circulant matrices and show how existing results for their spectra are applicable in our situation. It is important, that because of the du- plication of the eigenvalues, eigenvectors of both real and complex coordinates are available. Both kinds of eigenvectors are intensively studied. After this preparation, we prove our main theorem in Section 3. Section 4 discusses consequences of this theorem, like bounds for the spectra, complex PC transformation, low rank approximation of the process, illustrated on real-life nancial data. Eventually, in Section 5, conclusions are drawn. 2 Preliminaries To characterize the eigenvalues of the block-Toeplitz matrix Cn, we need the symmetric block circulant (s) matrix Cn that we consider for odd n, say n = 2k+1 here (for even n, the calculations are similar); for the def- inition, see [3]. In fact, the rows of a circulant matrix are cyclic permutations of the preceding ones; whereas, in the block circulant case, when permuting, we take transposes of the blocks if those are not symmetric themselves, see the example of Equation (2). Spectra of block circulant matrices are well characterized, but (s) Cn is not block circulant, in general; this is why Cn is constructed, by disregarding the autocovariances of n order greater than 2 . This can be done only on the assumption that the sequences of autocovariances are (en- n trywise) absolutely summable. It is crucial that k is the integer part of 2 , as the constructed block circulant n matrix is symmetric only with this choice of k. (In case of even n, the choice 2 is perfect, and it causes only 38 Ë Marianna Bolla et al. minor changes in the spectrum, as will be discussed later.) This choice also has benets when the autocovariance matrices are estimated based on an n-length long time series X1, ... , Xn: n−h X n Cˆ(h) = Cˆ∗(−h) = X X∗, h = 0, 1, ... , b c t+h t 2 t=1 n ˆ if EXt = 0, otherwise the estimated mean is subtracted. Here at least 2 sample entries are used for each C(h) n with h ≤ 2 that makes the estimate more accurate. (Note that the above estimate is asymptotically unbiased n even though its ergodicity is guaranteed only if h = o(n) as n ! ∞. Though 2 is obviously not o(n), as the n proof of the main theorem will show, disregarding the last 2 autocovariances will guarantee the convergence of the spectrum.) (s) The (i, j) block of Cn for 1 ≤ i ≤ j ≤ n is ( C(j − i), j − i ≤ k C(s)(block , block ) = n i j C(n − (j − i)), j − i > k; whereas, for i > j, it is ( CT(i − j), i − j ≤ k C(s)(block , block ) = n i j CT(n − (i − j)), i − j > k. (s) In this way, Cn is a symmetric block Toeplitz matrix, like Cn, and it is the same as Cn within the blocks (i, j)s (s) for which jj − ij ≤ k holds. However, Cn is also a block circulant matrix that ts our purposes. For example, if n = 7 and k = 3, then we have 2 3 C(0) C(1) C(2) C(3) C(3) C(2) C(1) 6 T 7 6C (1) C(0) C(1) C(2) C(3) C(3) C(2)7 6 T T 7 6C (2) C (1) C(0) C(1) C(2) C(3) C(3)7 (s) 6 7 C := 6CT(3) CT(2) CT(1) C(0) C(1) C(2) C(3)7 .

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