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A recursive approach for determining matrix inverses as applied to causal time series processes

Author

Listed:
  • Serge B. Provost

    (The University of Western Ontario)

  • John N. Haddad

    (Notre Dame University-Louaize)

Abstract

A decomposition of a certain type of positive definite quadratic forms in correlated normal random variables is obtained from successive applications of blockwise inversion to the leading submatrices of a symmetric positive definite matrix. This result can be utilized to determine Mahalanobis-type distances and allows for the calculation of the full likelihood functions in instances where the observations secured from certain causal processes are irregularly spaced or incomplete. Applications to some autoregressive moving-average models are pointed out and an illustrative numerical example is presented.

Suggested Citation

  • Serge B. Provost & John N. Haddad, 2019. "A recursive approach for determining matrix inverses as applied to causal time series processes," METRON, Springer;Sapienza Università di Roma, vol. 77(1), pages 53-62, April.
    • Handle: RePEc:spr:metron:v:77:y:2019:i:1:d:10.1007_s40300-019-00147-4
    DOI: 10.1007/s40300-019-00147-4
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    References listed on IDEAS

    as
    1. Jianqing Fan & Yuan Liao & Martina Mincheva, 2013. "Large covariance estimation by thresholding principal orthogonal complements," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(4), pages 603-680, September.
    2. Tommaso Proietti & Alessandro Giovannelli, 2018. "A Durbin–Levinson regularized estimator of high-dimensional autocovariance matrices," Biometrika, Biometrika Trust, vol. 105(4), pages 783-795.
    3. John N. Haddad, 1995. "The Recursive Property Of The Inverse Of The Covariance Matrix Of A Moving‐Average Process Of General Order," Journal of Time Series Analysis, Wiley Blackwell, vol. 16(6), pages 551-554, November.
    4. John N. Haddad, 2004. "On the closed form of the covariance matrix and its inverse of the causal ARMA process," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(4), pages 443-448, July.
    5. Cosme, Iria C.S. & Fernandes, Isaac F. & de Carvalho, João L. & Xavier-de-Souza, Samuel, 2018. "Memory-usage advantageous block recursive matrix inverse," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 125-136.
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