Recursive Estimation of The Covariance Matrix and Its Convergence for Multivariate Normal Hidden Markov Models
DOI:
https://doi.org/10.11113/mjfas.v21n3.3458Keywords:
Multivariate hidden Markov model, Covariance matrix, Expectation maximization algorithm, Monotone convergence.Abstract
This work discusses the covariance matrix estimates and convergence analysis for multivariate normal hidden Markov models. This study findings a series of covariance matrix estimators converges monotonically increasing to a stationary point of the likelihood function through the application of the expectation maximization algorithm.
References
Cappé, O. (2005). Inference in hidden Markov models. Springer.
Ross, S. M. (2019). Introduction to probability models. Academic Press. https://doi.org/10.1016/b978-0-12-814346-9.00006-8
Trichilli, Y., Boujelbène Abbes, M., & Masmoudi, A. (2020). Predicting the effect of Googling investor sentiment on Islamic stock market returns: A five-state hidden Markov model. International Journal of Islamic and Middle Eastern Finance and Management, 13(2), 165–193. https://doi.org/10.1108/IMEFM-07-2018-0218
Zhang, M., Jiang, X., Fang, Z., Zeng, Y., & Xu, K. (2019). High-order hidden Markov model for trend prediction in financial time series. Physica A: Statistical Mechanics and Its Applications, 517, 1–12. https://doi.org/10.1016/j.physa.2018.10.053
Nguyen, N. (2018). Hidden Markov model for stock trading. International Journal of Financial Studies, 6(36), 1–17. https://doi.org/10.3390/ijfs6020036
Somani, P., Talele, S., & Sawant, S. (2014). Stock market prediction using hidden Markov model. 2014 IEEE 7th Joint International Information Technology and Artificial Intelligence Conference (pp. 89–92). IEEE.
Gupta, A., & Dhingra, B. (2012). Stock market prediction using hidden Markov models. Students Conference on Engineering Systems (pp. 1–4).
Luck, A., Giehr, P., Nordstrom, K., Walter, J., & Wolf, V. (2019). Hidden Markov modelling reveals neighborhood dependence of DNMT3a and 3b activity. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 16(5), 1598–1609. https://doi.org/10.1109/TCBB.2019.2910814
Zarrabi, N., Schluesche, P., Meisterernst, M., Börsch, M., & Lamb, D. C. (2018). Analyzing the dynamics of single TBP-DNA-NC2 complexes using hidden Markov models. Biophysical Journal, 115(12), 2310–2326. https://doi.org/10.1016/j.bpj.2018.11.015
Tao, H., & Lu, X. (2019). Smoke vehicle detection based on multi-feature fusion and hidden Markov model. Journal of Real-Time Image Processing, 17(3), 745–758. https://doi.org/10.1007/s11554-019-00856-z
Paroli, R., & Spezia, L. (1999). Gaussian hidden Markov models: Parameters estimation and applications to air pollution data [Technical Report No. 94]. Università Cattolica del Sacro Cuore.
Rabiner, L. R. (1989). A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE, 77(2), 257–286. https://doi.org/10.25300/MISQ/2017/41.3.08
Cutajar, M., Gatt, E., Grech, I., Casha, O., & Micallef, J. (2013). Comparative study of automatic speech recognition techniques. IET Signal Processing, 7(1), 25–46. https://doi.org/10.1049/iet-spr.2012.0151
Mouaz, B., Abderrahim, B. H., & Abdelmajid, E. (2019). Speech recognition of Moroccan dialect using hidden Markov models. IAES International Journal of Artificial Intelligence, 8(1), 7–13. https://doi.org/10.11591/ijai.v8.i1.pp7-13
Khiatani, D., & Ghose, U. (2017). Weather forecasting using hidden Markov model. 2017 International Conference on Computing, Communication and Automation (pp. 220–225). https://doi.org/10.1109/IC3TSN.2017.8284480
Fikri, M., Samsurizal, Christiono, & Mauriraya, K. T. (2020). Pemodelan cuaca menggunakan model hidden Markov untuk pemanfaatan energi surya. Kiilat, 9(2), 217–224.
Fikri, M., & Abdul-Malek, Z. (2023). Partial discharge diagnosis and remaining useful lifetime in XLPE extruded power cables under DC voltage: A review. Electrical Engineering. https://doi.org/10.1007/s00202-023-01935-y
Fikri, M., et al. (2023). Clustering suara corona discharge berdasarkan tegangan menggunakan metode fuzzy c-mean. Elkomika: Jurnal Teknik Energi Elektrik, Teknik Telekomunikasi, & Teknik Elektronika, 11(3), 609–624.
Pasra, N., Fikri, M., Mauriraya, K. T., Rijanto, T., & Buditjahjanto, I. G. P. A. (2023). Deteksi suara corona discharge berdasarkan noise menggunakan metode LPC dan Euclidean distance. Elkomika: Jurnal Teknik Energi Elektrik, Teknik Telekomunikasi, & Teknik Elektronika, 11(1), 72–85.
Fikri, M., Christiono, & Mulyadi, I. G. K. (2022). Clustering fenomena corona discharge berdasarkan suara menggunakan metode LPC dan Euclidean distance. Elkomika: Jurnal Teknik Energi Elektrik, Teknik Telekomunikasi, & Teknik Elektronika, 10(3), 689–701.
Fikri, M., et al. (2024). Comparison of corona discharge identification in 20 kV cubicles based on voltage and noise using ED, HMM, and FCM. Jurnal Teknologi, 86(5), 11–22.
Barbu, V. S., & Limnios, N. (2009). Semi-Markov chains and hidden semi-Markov models toward applications. Springer.
Spezia, L. (2010). Bayesian analysis of multivariate Gaussian hidden Markov models with an unknown number of regimes. Journal of Time Series Analysis, 31, 1–11. https://doi.org/10.1111/j.1467-9892.2009.00635.x
Spezia, L., Futter, M. N., & Brewer, M. J. (2011). Periodic multivariate normal hidden Markov models for the analysis of water quality time series. Environmetrics, 22(3), 304–317. https://doi.org/10.1002/env.1051
Fikri, M., Abdul-Malek, Z., & Atmadja, K. (2023). Recursive parameter estimation and its convergence for multivariate normal hidden Markov model. Thai Statistician.
Fikri, M., Abdul-Malek, Z., Esa, M. R. M., & Supriyanto, E. (2023). Recursive parameter estimation and its convergence for multivariate normal hidden Markov inhomogeneous models. Malaysian Journal of Fundamental and Applied Sciences, 19(5), 840–854.
Pourahmadi, M. (1999). Joint mean-covariance models with applications to longitudinal data: Unconstrained parameterisation. Biometrika, 86(3), 677–690.
Ledoit, O., & Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis, 88, 365–411. https://doi.org/10.1016/S0047-259X(03)00096-4
Rothman, A. J., Levina, E., & Zhu, J. (2010). A new approach to Cholesky-based covariance regularization in high dimensions. Biometrika, 97(3), 539–550. https://doi.org/10.1093/biomet/asq022
Pourahmadi, M. (2011). Covariance estimation: The GLM and regularization perspectives. Statistical Science, 26(3), 369–387. https://doi.org/10.1214/11-STS358
Lam, C. (2016). Nonparametric eigenvalue-regularized precision or covariance matrix estimator. Annals of Statistics, 44(3), 928–953. https://doi.org/10.1214/15-AOS1393
Ledoit, O., & Wolf, M. (2020). Analytical nonlinear shrinkage of large-dimensional covariance matrices. Annals of Statistics, 48(5), 3043–3065.
Hamilton, J. D. (1990). Analysis of time series subject to changes in regime. Journal of Econometrics, 45, 39–70.
Setiawaty, B., Santoso, D. H., & Ardana, N. K. K. (2007). Pemodelan nilai tukar rupiah terhadap $US menggunakan deret waktu hidden Markov satu waktu sebelumnya. Jurnal Matematika dan Aplikasinya.
Baum, L. E. (1972). An inequality and associated maximization technique in statistical estimation of probabilistic functions of Markov processes. Proceedings of the Third Symposium on Inequalities (pp. 1–8).
MacDonald, I. L., & Zucchini, W. (1997). Hidden Markov and other models for discrete-valued time series. Chapman and Hall.
Paroli, R., Redaelli, G., & Spezia, L. (2000). Poisson hidden Markov models for time series of overdispersed insurance counts. ASTIN Colloquium (pp. 461–474).
Spezia, L. (2010). Bayesian analysis of multivariate Gaussian hidden Markov models with an unknown number of regimes (Vol. 31, pp. 1–11). https://doi.org/10.1111/j.1467-9892.2009.00635.x
Grimmett, G. R., & Stirzaker, D. R. (2001). Probability and random processes. Oxford University Press.
Baum, L. E., Petrie, T., Soules, G., & Weiss, N. (1970). A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Annals of Mathematical Statistics, 41(1), 164–171.
Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society: Series B, 39(1), 1–22. https://doi.org/10.1111/j.2517-6161.1977.tb01600.x
Wu, J. (1983). On the convergence properties of the EM algorithm. Annals of Statistics, 11(1), 95–103.
Zangwill, W. I. (1969). Nonlinear programming: A unified approach. Prentice-Hall.
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