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Revista Colombiana de Estadística

Print version ISSN 0120-1751

Rev.Colomb.Estad. vol.45 no.1 Bogotá Jan./June 2022  Epub Jan 17, 2023

https://doi.org/10.15446/rce.v45n1.85385 

Artículos originales de investigación

Variable Selection in Switching Dynamic Regression Models

Selección de variables en modelos de regresión dinámicos de cambios de régimen

DAYNA P. SALDANA-ZEPEDA1  a 

CIRO VELASCO-CRUZ2  b 

VICTOR H. TORRES-PRECIADO3  c 

1 ESCUELA DE MERCADOTECNIA, UNIVERSIDAD DE COLIMA, COLIMA, MEXICO

2 DEPARTAMENTO DE ESTADISTICA, SOCIOECONOMIA ESTADISTICA E INFOMATICA, COLEGIO DE POSTGRADUADOS, TEXCOCO, MEXICO

3 FACULTAD DE ECONOMIA, UNIVERSIDAD DE COLIMA, COLIMA, MEXICO


Abstract

Complex dynamic phenomena in which dynamics is related to events (modes) that cause structural changes over time, are well described by the switching linear dynamical system (SLDS). We extend the SLDS by allowing the measurement noise to be mode-specific, a flexible way to model non stationary data. Additionally, for models that are functions of explanatory variables, we adapt a variable selection method to identify which of them are significant in each mode. Our proposed model is a flexible Bayesian nonparametric model that allows to learn about the number of modes and their location, and within each mode, it identifies the significant variables and estimates the regression coefficients. The model performance is evaluated by simulation and two application examples from a dataset of meteorological time series of Barranquilla, Colombia are presented.

Key words: bayesian filtering and smoothing; dirichlet process; hierarchical model; state-space model

Resumen

Fenómenos dinámicos complejos en los que la dinámica está relacionada con eventos (modos) que provocan cambios estructurales a lo largo del tiempo, se aproximan mediante un sistema dinámico lineal de cambio de régimen (SDLR). Extendemos el SDLR al permitir que el error de medición sea específico del modo, una forma flexible de modelar datos no estacionarios. Además, para los modelos que son funciones de variables explicativas, adaptamos un método de selección de variables para identificar cuáles de ellas son significativas en cada modo. El modelo propuesto es un modelo bayesiano no paramétrico flexible que permite conocer el número de modos y su ubicación, y dentro de cada modo, identifica las variables significativas y estima los coeficientes de regresión. El desempeño del modelo se evalúa mediante simulación y se presentan dos ejemplos de aplicación de un conjunto de datos de series de tiempo meteorológicas de Barranquilla, Colombia.

Palabras clave: filtrado y suavizamiento bayesianos; modelos de espacio-estado; modelos jerárquicos; procesos Dirichlet

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References

Antoniak, C. (1974), 'Mixtures of dirichlet processes with applications to bayesian nonparametric problems', The Annals of Statistics 2(6), 1152-1174. [ Links ]

Barber, D. (2012), Bayesian Reasoning and Machine Learning, Cambridge University Press. [ Links ]

Bishop, C. (2006), Pattern Recognition and Machine Learning, Springer. [ Links ]

Blackwell, D. & MacQueen, J. (1973), 'Ferguson distributions via Polya urn schemes', The Annals of Statistics 1(2), 353-355. [ Links ]

Bregler, C. (1997, June), Learning and recognizing human dynamics in video sequences, in 'Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition', pp. 568-574. [ Links ]

Carvalho, C. & Lopes, H. (2007), 'Simulation based sequential analysis of markov switching stochastic volatility models', Computational Statistics and Data Analysis 51, 4526-4542. [ Links ]

Du, K., Mu, C., Deng, J. & Yuan, F. (2013), 'Study on atmospheric visibility variations and the impacts of meteorological parameters using high temporal resolution data: an application of environmental internet of things in china', International Journal of Sustainable Development and World Ecology 20(3), 238-247. [ Links ]

Escobar, M. (1988), Estimating the Means of Several Normal Populations by Nonparametric Estimation of the Distribution of the Means, PhD thesis, Yale University. [ Links ]

Escobar, M. & West, M. (1995), 'Bayesian density estimation and inference using mixtures', Journal of the American Statistical Association 90(430), 577-588. [ Links ]

Ferguson, T. (1973), 'A bayesian analysis of some nonparametric problems', The Annals of Statistics 1(2), 209-230. [ Links ]

Fox, E., Sudderth, E., Jordan, M. & Willsky, A. (2011 a), 'Bayesian nonparametric inference of switching dynamic linear models', IEEE Transactions on signal processing 59(4), 1569-1585. [ Links ]

Fox, E., Sudderth, E., Jordan, M. & Willsky, A. (2011b), 'A sticky hdp-hmm with application to speaker diarization', The Annals of Applied Statistics 5(2A), 1020-1056. [ Links ]

Han, M., Ren, W. & Liu, X. (2015), 'Joint mutual information-based input variable selection for multivariate time series modeling', Engineering Applications of Artificial Intelligence 37, 250-257. [ Links ]

Huang, W., Tan, J., Kan, H., Zhao, N., Song, W., Song, G., Chen, G., Jiang, L., Jiang, C., Chen, R. & Chen, B. (2009), 'Visibility, air quality and daily mortality in shanghai, china', Science of The Total Environment 407(10), 3295-3300. [ Links ]

Huerta, G., Sansó, B. & Stroud, J. R. (2004), 'A spatiotemporal model for mexico city ozone levels', Journal of the Royal Statistical Society 53(2), 231-248. [ Links ]

Ishwaran, H. & James, L. (2001), 'Gibbs sampling methods for stick-breaking priors', Journal of the American Statistical Association 96(453), 161-173. [ Links ]

Ishwaran, H. & James, L. (2002), 'Approximate dirichlet process computing in finite normal mixtures: Smoothing and prior information', Journal of Computational and Graphical Statistics 11(3), 1-26. [ Links ]

Ishwaran, H. & Zarepour, M. (2000), 'Markov chain monte carlo in approximate dirichlet and beta two-parameter process hierarchical models', Biometrika 87(2), 371-390. [ Links ]

Ishwaran, H. & Zarepour, M. (2002a), 'Dirichlet prior sieves in finite normal mixtures', Statistica Sinica 12(3), 941-963. [ Links ]

Ishwaran, H. & Zarepour, M. (2002), 'Exact and approximate sum representations for the dirichlet process', The Canadian Journal of Statistics 30(2), 269-283. [ Links ]

Kalman, R. (1960), 'A new approach to linear filtering and prediction problems', Journal of Basic Engineering 82, 35-45. [ Links ]

Kalman, R. (1963), 'Mathematical description of linear dynamical systems', Journal of the Society for Industrial and Applied Mathematics 1(2), 152-192. [ Links ]

Kim, C. (1994), 'Dynamic linear models with markov switching', Journal of Econometrics 60(1-2), 1-22. [ Links ]

Kuo, L. & Mallick, B. (1998), 'Variable selection for regression models', The Indian Journal of Statistics. Special Issue on Bayesian Analysis 60(1), 65-81. [ Links ]

Lamon III, E., Carpenter, S. & Stow, C. (1998), 'Forecasting PCB concentrations in Lake Michigan salmonids: a dynamic linear model approach', Ecological Applications 8(3), 659-668. [ Links ]

MacEachern, S. N. (1994), 'Estimating normal means with a conjugate style dirichlet process prior', Communications in Statistics-Simulation and Computation 23(3), 727-741. [ Links ]

Majewski, G., Kleniewska, M. & Brandyk, A. (2011), 'Seasonal variation of particulate matter mass concentration and content of metals', Polish Journal of Environmental Studies 20(2), 417-427. [ Links ]

Majewski, G., Rogula-Kozlowska, W., Czechowski, P. O., Badyda, A. & Brandyk, A. (2015), 'The impact of selected parameters on visibility: First results from a long-term campaign in warsaw, poland', Atmosphere 6, 1154-1174. [ Links ]

McAlinn, K. & West, M. (2016), Dynamic bayesian predictive synthesis in time series forecasting, Technical report, Duke University. [ Links ]

Meinhold, R. & Singpurwalla, N. (1983), 'Understanding the kalman filter', The American Statistician 37(2), 123-127. [ Links ]

National Centers for Environmental Information (2021), 'Local climatological data'. https://www.ncei.noaa.gov/data/local-climatological-data/Links ]

Pavlovic, V., Rehg, J. & MacCormick, J. (2001), Learning switching linear models of human motion., in 'Advances in Neural Information Processing Systems , Vol. 13, Neural Information Processing Systems (NIPS) 2000. [ Links ]

Petris, G., Petrone, S. & Campagnoli, P. (2009), Dynamic Linear Models with R, Springer-Verlag. [ Links ]

Rauch, H., Striebel, C. & Tung, F. (1965), 'Maximum likelihood estimates of linear dynamic systems , AIAA Journal 3(8), 1445-1450. [ Links ]

Redner, R. & Walker, H. (1984), 'Mixture densities, maximum likelihood and the em algorithm', SIAM Review 26(2), 195-239. [ Links ]

Rodríguez, A. (2007), Some Advances in Bayesian Nonparametric Modeling, PhD thesis, Duke University. [ Links ]

Sethuraman, J. (1994), 'A constructive definition of dirichlet priors', Statistica Sinica 4, 639-650. [ Links ]

Stephens, M. (2000), 'Dealing with label switching in mixture models , Journal of the Royal Statistical Society 62(4), 795-809. [ Links ]

Teh, Y. W., Jordan, M. I., Beal, M. J. & Blei, D. M. (2006), 'Hierarchical dirichlet processes , Journal of the American Statistical Association 101, 1566-1581. [ Links ]

Thach, T.-Q., Wong, C.-M., Chan, K.-P., Chau, Y.-K., Chung, Y.-N., Ou, C.-Q., Yang, L. & Hedley, A. J. (2010), 'Daily visibility and mortality: Assessment of health benefits from improved visibility in hong kong', Environmental Research 110(6), 617-623. [ Links ]

Tsai, Y., Kuo, S.-C., Lee, W.-J., Chen, C.-L. & Chen, P.-T. (2007), 'Long-term visibility trends in one highly urbanized, one highly industrialized, and two rural areas of taiwan , Science of The Total Environment 382(2-3), 324-341. [ Links ]

Velasco-Cruz, C., Leman, S. C., Hudy, M. & Smith, E. P. (2012), 'Assessing the risk of rising temperature on brook trout: a spatial dynamic linear risk model , Journal of Agricultural, Biological, and Environmental Statistics 17(2), 246-264. [ Links ]

Wang, L. & Wang, X. (2013), 'Hierarchical dirichlet process model for gene expression clustering', EURASIP Journal on Bioinformatics and Systems Biology 1(5). [ Links ]

Watson, A., Ramirez, C. & Salud, E. (2009), 'Predicting visibility of aircraft , PLOS ONE 5(7), 1-16. [ Links ]

West, M. (2013), Bayesian Dynamic Modelling, Oxford University Press, chapter 8. [ Links ]

West, M. & Harrison, J. (1997), Bayesian Forecasting and Dynamic Models, 2 edn, Springer. [ Links ]

Zeng, Y. & Wu, S., eds (2013), State-space models. Applications in Economics and Finance, Springer. [ Links ]

Received: February 2020; Accepted: December 2021

aPh.D. E-mail: daynasz@ucol.mx

bPh.D. E-mail: cvelasco@vt.edu

cPh.D. E-mail: tpreciado04@gmail.com

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