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Multivariate spectral analysis using Cholesky decomposition

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  • Ming Dai

Abstract

We propose to smooth the Cholesky decomposition of a raw estimate of a multivariate spectrum, allowing different degrees of smoothness for different elements. The final spectral estimate is reconstructed from the smoothed Cholesky elements, and is consistent and positive definite. More importantly, the Cholesky decomposition matrix of the spectrum can be used as a transfer function in generating time series whose spectrum is identical to the given spectrum at the Fourier frequencies. This not only provides us with much flexibility in simulations, but also allows us to construct bootstrap confidence intervals for the multivariate spectrum by generating bootstrap samples using the Cholesky decomposition of the spectral estimate. A numerical example and an application to electroencephalogram data are used as illustrations. Copyright Biometrika Trust 2004, Oxford University Press.

Suggested Citation

  • Ming Dai, 2004. "Multivariate spectral analysis using Cholesky decomposition," Biometrika, Biometrika Trust, vol. 91(3), pages 629-643, September.
    • Handle: RePEc:oup:biomet:v:91:y:2004:i:3:p:629-643
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    Cited by:

    1. von Sachs, Rainer, 2019. "Spectral Analysis of Multivariate Time Series," LIDAM Discussion Papers ISBA 2019008, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Chau, Van Vinh & von Sachs, Rainer, 2017. "Positive-Definite Multivariate Spectral Estimation: A Geometric Wavelet Approach," LIDAM Discussion Papers ISBA 2017002, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. Hu, Zhixiong & Prado, Raquel, 2023. "Fast Bayesian inference on spectral analysis of multivariate stationary time series," Computational Statistics & Data Analysis, Elsevier, vol. 178(C).
    4. Robert T. Krafty & Ori Rosen & David S. Stoffer & Daniel J. Buysse & Martica H. Hall, 2017. "Conditional Spectral Analysis of Replicated Multiple Time Series With Application to Nocturnal Physiology," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(520), pages 1405-1416, October.
    5. Fiecas, Mark & von Sachs, Rainer, 2012. "Spectral density shrinkage for high-dimensional time series," LIDAM Discussion Papers ISBA 2012037, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. Chau, Van Vinh & Ombao, Hernando & von Sachs, Rainer, 2017. "Data depth and rank-based tests for covariance and spectral density matrices," LIDAM Discussion Papers ISBA 2017019, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    7. Fiecas, Mark & von Sachs, Rainer, 2013. "Data-driven Shrinkage of the Spectral Density Matrix of a High-dimensional Time Series," LIDAM Discussion Papers ISBA 2013044, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    8. Jentsch, Carsten & Kreiss, Jens-Peter, 2010. "The multiple hybrid bootstrap -- Resampling multivariate linear processes," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2320-2345, November.
    9. Meier, Alexander & Kirch, Claudia & Meyer, Renate, 2020. "Bayesian nonparametric analysis of multivariate time series: A matrix Gamma Process approach," Journal of Multivariate Analysis, Elsevier, vol. 175(C).
    10. Addisu H. Addis & Hugh T. Blair & Paul R. Kenyon & Stephen T. Morris & Nicola M. Schreurs & Dorian J. Garrick, 2023. "Agent-Based Modelling to Improve Beef Production from Dairy Cattle: Young Beef Production," Agriculture, MDPI, vol. 13(4), pages 1-10, April.
    11. Zhang, Shibin, 2016. "Adaptive spectral estimation for nonstationary multivariate time series," Computational Statistics & Data Analysis, Elsevier, vol. 103(C), pages 330-349.
    12. Evangelos E. Ioannidis, 2022. "A new non‐parametric cross‐spectrum estimator," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(5), pages 808-827, September.

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