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Factorization of moving-average spectral densities by state-space representations and stacking. (English) Zbl 1077.62074
Summary: To factorize a spectral density matrix of a vector moving average process, we propose a state space representation. Although this state space is not necessarily of minimal dimension, its associated system matrices are simple and most matrix multiplications involved are nothing but index shifting. This greatly reduces the complexity of computation. Moreover, we stack every \(q\) consecutive observations of the original process MA(\(q\)) and generate a vector MA(1) process. We consider a similar state space representation for the stacked process. Consequently, the solution hinges on a surprisingly compact discrete algebraic Riccati equation (DARE), which involves only one Toeplitz and one Hankel block matrix composed of autocovariance functions. One solution to this equation is given by the so-called iterative projection algorithm. Each iteration of the stacked version is equivalent to \(q\) iterations of the unstacked one.
We show that the convergence behavior of the iterative projection algorithm is characterized by the decreasing rate of the partial correlation coefficients for the stacked process. In fact, the calculation of the partial correlation coefficients via the Whittle algorithm, which takes a very simple form in this case, offers another solution to the problem. To achieve computational efficiency, we apply the general Newton procedure given by P. Lancaster and L. Rodman [Algebraic Riccati equations. (1995; Zbl 0836.15005)] to the DARE and obtain an algorithm of quadratic convergence rate. One immediate application of the new algorithms is polynomial stabilization. We also discuss various issues such as check of positivity and numerical implementation.

MSC:
62M15 Inference from stochastic processes and spectral analysis
47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
65C60 Computational problems in statistics (MSC2010)
93B20 Minimal systems representations
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[1] Anderson, B.D.O., An algebraic solution to the spectral factorization problem, IEEE trans. automatic control, 12, 410-414, (1967)
[2] Arnold, W.F.; Laub, A.J., Generalized eigenproblem algorithms and software for algebraic Riccati equations, Proc. IEEE, 72, 1746-1754, (1984)
[3] Barraud, A., A new numerical solution of \(\dot{X} = A_1 X + X A_2 + D\), \(X(0) = C\), IEEE trans. automat. control, 22, 976-977, (1977) · Zbl 0383.65049
[4] Bartels, R.H.; Stewart, G.W., Solution of the matrix equation \(\mathit{AX} + \mathit{XB} = C\), Comm. ACM, 15, 820-826, (1972) · Zbl 1372.65121
[5] Bauer, F.L., Ein direktes iterations verfahren zur Hurwitz-zerlegung eines polynoms, Arch. elelktr. uebertragung, 9, 285-290, (1955)
[6] F.L. Bauer, Beiträge zur entwicklung numerischer verfahren für programmgesteuerte rechenanlagen, ii. direktes faktorisierung eines polynoms, Sitz. Ber. Bayer. Adad. Wiss. (1956), 163-203.
[7] Brockwell, P.J.; Davis, R.A., Time seriestheory and models, (1991), Springer Berlin
[8] Caines, P.E., Linear stochastic systems, (1988), Wiley New York · Zbl 0781.93093
[9] Cleveland, W.S., The inverse autocorrelations of a time series and their applications, Technometrics, 14, 277-293, (1972) · Zbl 0276.62079
[10] Faurre, P.L., Stochastic realization algorithms, (), 1-26
[11] Gajić, Z.; Qureshi, M.T.J., Lyapunov matrix equation in system stability and control, (1995), Academic Press New York
[12] Golub, G.H.; Van Loan, C.F., Matrix computations, (1996), The John Hopkins University Press Baltimore and London · Zbl 0865.65009
[13] Goodman, T.N.T.; Micchelli, C.A.; Rodriguez, G.; Seatzu, S., Spectral factorization of Laurent polynomials, Adv. computat. math., 7, 429-445, (1997)
[14] Hannan, E.J., Multiple time series, (1970), Wiley New York · Zbl 0211.49804
[15] Ježek, J.; Kučera, V., Efficient algorithm for matrix spectral factorization, Automatica, 21, (1985) · Zbl 0585.93015
[16] Lancaster, P.; Rodman, L., Algebraic Riccati equations, (1995), Clarendon Press Oxford · Zbl 0836.15005
[17] L.M. Li, The state space model of a stationary process and the stochastic realization algorithm, Master’s Thesis, Peking University, 1991.
[18] Li, L.M., A method to factorize the spectral density of multiple moving average processes, J. systems sci. math., 7, 169-176, (1994) · Zbl 0804.60031
[19] Reinsel, G., Elements of multivariate time series analysis, (1997), Springer Berlin · Zbl 0873.62086
[20] Rozanov, Y.A., Stationary random processes, (1967), Holden-Day San Francisco · Zbl 0152.16302
[21] Sayed, A.H.; Kailath, T., A survey of spectral factorization methods, Numer. linear algebra appl., 8, 467-496, (2001) · Zbl 1053.47013
[22] Tuel, W.G., Computer algorithm for spectral factorization of rational matrices, IBM J. res. development, 12, 163-170, (1968) · Zbl 0155.48704
[23] Whittle, P., On the Fitting of multivariate autoregressions, Biometrika, 5, 129-134, (1963) · Zbl 0129.11304
[24] Wilson, G., Factorization of covariance generating function, SIAM J. numer. anal., 6, (1969) · Zbl 0176.46401
[25] Wilson, G., The factorization of matrical spectral densities, SIAM J. appl. math., 23, 420-426, (1972) · Zbl 0227.65042
[26] Wilson, G.T., A convergence theorem for spectral factorization, J. multivariate anal., 8, 222-232, (1978) · Zbl 0384.60027
[27] Youla, D.C., On the factorization of rational matrices, IRE trans. inform. theory, 7, 172-189, (1961) · Zbl 0103.25201
[28] Youla, D.C.; Kazanjian, N.N., Bauer-type factorization of positive matrices and the theory of matrix polynomials orthogonal on the circle, IEEE trans. circuits systems, 25, 57-69, (1978) · Zbl 0417.93018
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