Bayesian Analysis

Markov switching Dirichlet process mixture regression

Athanasios Kottas and Matthew A. Taddy

Full-text: Open access

Abstract

Markov switching models can be used to study heterogeneous populations that are observed over time. This paper explores modeling the group characteristics nonparametrically, under both homogeneous and nonhomogeneous Markov switching for group probabilities. The model formulation involves a finite mixture of conditionally independent Dirichlet process mixtures, with a Markov chain defining the mixing distribution. The proposed methodology focuses on settings where the number of subpopulations is small and can be assumed to be known, and flexible modeling is required for group regressions. We develop Dirichlet process mixture prior probability models for the joint distribution of individual group responses and covariates. The implied conditional distribution of the response given the covariates is then used for inference. The modeling framework allows for both non-linearities in the resulting regression functions and non-standard shapes in the response distributions. We design a simulation-based model fitting method for full posterior inference. Furthermore, we propose a general approach for inclusion of external covariates dependent on the Markov chain but conditionally independent from the response. The methodology is applied to a problem from fisheries research involving analysis of stock-recruitment data under shifts in the ecosystem state.

Article information

Source
Bayesian Anal. Volume 4, Number 4 (2009), 793-816.

Dates
First available in Project Euclid: 22 June 2012

Permanent link to this document
http://projecteuclid.org/euclid.ba/1340369825

Digital Object Identifier
doi:10.1214/09-BA430

Mathematical Reviews number (MathSciNet)
MR2570089

Zentralblatt MATH identifier
1330.62141

Keywords
Dirichlet process prior hidden Markov model Markov chain Monte Carlo multivariate normal mixture stock-recruitment relationship

Citation

Taddy, Matthew A.; Kottas, Athanasios. Markov switching Dirichlet process mixture regression. Bayesian Anal. 4 (2009), no. 4, 793--816. doi:10.1214/09-BA430. http://projecteuclid.org/euclid.ba/1340369825.


Export citation

References

  • Antoniak, C. (1974). "Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems." Annals of Statistics, 2: 1152 – 1174.
  • Berliner, L. M. and Lu, Z.-Q. (1999). "Markov switching time series models with application to a daily runoff series." Water Resources Research, 35: 523–534.
  • Billio, M., Monfort, A., and Robert, C. P. (1999). "Bayesian estimation of switching ARMA" models. Journal of Econometrics, 93: 229–255.
  • Blackwell, D. and MacQueen, J. (1973). "Ferguson distributions via Pólya urn schemes." Annals of Statistics, 1: 353–355.
  • Chib, S. (1996). "Calculating posterior distributions and modal estimates in Markov mixture models." Journal of Econometrics, 75: 79–97.
  • Connor, R. and Mosimann, J. (1969). "Concepts of independence for proportions with a generalization of the Dirichlet distribution." Journal of the American Statistical Association, 64: 194–206.
  • Ferguson, T. (1973). "A Bayesian analysis of some nonparametric problems." Annals of Statistics, 1: 209–230.
  • Frühwirth-Schnatter, S. (2006). Finite Mixture and Markov Switching Models. Springer.
  • Gelfand, A. E. and Kottas, A. (2002). "A Computational Approach for Full Nonparametric Bayesian Inference under Dirichlet Process Mixture Models." Journal of Computational and Graphical Statistics, 11: 289–305.
  • Goldfeld, S. M. and Quandt, R. E. (1973). "A Markov model for switching regression." Journal of Econometrics, 1: 3–16.
  • Hamilton, J. D. (1989). "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle." Econometrica, 57: 357–384.
  • Hare, S. R. and Mantua, N. J. (2000). "Empirical evidence for North Pacific regime shifts in 1977 and 1989." Progress in Oceanography, 47: 103–145.
  • Hughes, J. P. and Guttorp, P. (1994). "A class of stochastic model for relating synoptic atmospheric patters to regional hydrologic phenomena." Water Resources Research, 30: 1535–1546.
  • Hurn, M., Justel, A., and Robert, C. P. (2003). "Estimating mixtures of regressions." Journal of Computational and Graphical Statistics, 12: 55–79.
  • Ishwaran, H. and James, L. (2001). "Gibbs sampling methods for stick-breaking priors." Journal of the American Statistical Association, 96: 161–173.
  • Ishwaran, H. and Zarepour, M. (2000). "Markov chain Monte Carlo in approximate Dirichlet and beta two-parameter process hierarchical models." Biometrika, 87: 371–390.
  • Jacobson, L. D., Bograd, S. J., Parrish, R. H., Mendelssohn, R., and Schwing, F. B. (2005). "An ecosystem-based hypothesis for climatic effects on surplus production in California sardine and environmentally dependent surplus production models." Canadian Journal of Fisheries and Aquatic Sciences, 62: 1782–1796.
  • MacCall, A. D. (2002). "An hypothesis explaining biological regimes in sardine-producing Pacific boundary current systems." In Climate and fisheries: interacting paradigms, scales, and policy approaches: the IRI-IPRC Pacific Climate-Fisheries Workshop, 39–42. International Research Institute for Climate Prediction, Columbia University.
  • McCulloch, R. E. and Tsay, R. S. (1994). "Statistical Analysis of Economic Time Series via Markov Switching Models." Journal of Time Series Analysis, 15: 523–539.
  • McGowan, J. A., Cayan, D. R., and Dorman, L. M. (1998). "Climate-ocean variability and ecosystem response in the Northeast Pacific." Science, 281: 210–217.
  • Müller, P., Erkanli, A., and West, M. (1996). "Bayesian curve fitting using multivariate Normal mixtures." Biometrika, 83: 67–79.
  • Munch, S. B., Kottas, A., and Mangel, M. (2005). "Bayesian nonparametric analysis of stock-recruitment relationships." Canadian Journal of Fisheries and Aquatic Sciences, 62: 1808–1821.
  • Neal, R. (2000). "Markov Chain Sampling Methods for Dirichlet Process Mixture Models." Journal of Computational and Graphical Statistics, 9: 249–265.
  • Quandt, R. E. and Ramsey, J. B. (1978). "Estimating Mixtures of Normal Distributions and switching regressions (with discussion)." Journal of the American Statistical Association, 73: 730–752.
  • Quinn, T. J. I. and Derisio, R. B. (1999). Quantitative Fish Dynamics. Oxford University Press.
  • Robert, C., Celeux, G., and Diebolt, J. (1993). "Bayesian estimation of hidden Markov chains: a stochastic implementation." Statistics & Probability Letters, 16: 77–83.
  • Scott, S. (2002). "Bayesian methods for hidden Markov models: recursive computing for the 21st century." Journal of the American Statistical Association, 97: 337–351.
  • Sethuraman, J. (1994). "A constructive definition of Dirichlet priors." Statistica Sinica, 4: 639–6650.
  • Shi, J. Q., Murray-Smith, R., and Titterington, D. M. (2005). "Hierarchical Gaussian process mixtures for regression." Statistics and Computing, 15: 31–41.
  • Shumway, R. H. and Stoffer, D. S. (1991). "Dynamic linear models with switching." Journal of the American Statistical Association, 86: 763–769.
  • Taddy, M. and Kottas, A. (2009). "A Bayesian Nonparametric Approach to Inference for Quantile Regression." Journal of Business and Economic Statistics, to appear.
  • Wada, T. and Jacobson, L. D. (1998). "Regimes and stock-recruitment relationships in Japanese sardine, 1951-1995." Canadian Journal of Fisheries and Aquatic Sciences, 55: 2455–2463.