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Fitting hidden semi-Markov models to breakpoint rainfall data

Part of:Geophysics Geophysics (general)Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

John Sansom and
Peter Thomson
John Sansom*
Affiliation:
National Institute of Water and Atmospheric Research, New Zealand
Peter Thomson*
Affiliation:
Statistics Research Associates Ltd
*
1 Postal address: PO Box 14–901, National Institute of Water and Atmospheric Research, Wellington, New Zealand.
2 Postal address: PO Box 12–649, Statistics Research Associates Ltd, Wellington, New Zealand. Email: peter@statsresearch.co.nz
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Abstract

The paper proposes a hidden semi-Markov model for breakpoint rainfall data that consist of both the times at which rain-rate changes and the steady rates between such changes. The model builds on and extends the seminal work of Ferguson (1980) on variable duration models for speech. For the rainfall data the observations are modelled as mixtures of log-normal distributions within unobserved states where the states evolve in time according to a semi-Markov process. For the latter, parametric forms need to be specified for the state transition probabilities and dwell-time distributions.

Recursions for constructing the likelihood are developed and the EM algorithm used to fit the parameters of the model. The choice of dwell-time distribution is discussed with a mixture of distributions over disjoint domains providing a flexible alternative. The methods are also extended to deal with censored data. An application of the model to a large-scale bivariate dataset of breakpoint rainfall measurements at Wellington, New Zealand, is discussed.

Keywords

Hidden semi-Markov modelshigh-resolution rainfall datadwell-time distributionsEM algorithm

MSC classification

Primary:62M05: Markov processes
Secondary:86A10: Meteorology and atmospheric physics

Information

Type
Estimation problems
Information
Journal of Applied Probability , Volume 38 , Issue A: Probability, Statistics and Seismology , 2001, pp. 142 - 157
Copyright
Copyright © Applied Probability Trust 2001 

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References

Akaike, H. (1977). An entropy maximisation principle. In Applications of Statistics , ed. Krishnaiah, P. R., North Holland, Amsterdam, 27–41.Google Scholar
Baum, L. E. and Petrie, T. (1966). Statistical inference for probabilistic functions of finite state Markov chains. Ann. Math. Statist. 37, 1554–1563.Google Scholar
Biondini, R. (1976). Cloud motion and rainfall statistics. J. Appl. Meteorology 15, 205–224.Google Scholar
Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 39, 1–38.Google Scholar
Elliott, R. J., Aggoun, L. and Moore, J. B. (1995). Hidden Markov Models. Springer, New York.Google Scholar
Ferguson, J. D. (1980). Variable duration models for speech. In Symposium on the Application of Hidden Markov Models to Text and Speech , ed. Ferguson, J. D., Institute for Defense Analyses, Princeton, NJ, 143–179.Google Scholar
Guedon, Y. and Cocozza-Thivent, C. (1990). Explicit state occupancy modelling by hidden semi-Markov models: application of Derin's scheme. Comput. Speech Language 4, 167–192.Google Scholar
Hughes, J. P. (1997). Computing the observed information in the hidden Markov model using the EM algorithm. Statist. Probab. Lett. 32, 107–114.CrossRefGoogle Scholar
Hughes, J. P., Guttorp, P. and Charles, S. P. (1999). A non-homogeneous hidden Markov model for precipitation occurrence. J. Roy. Statist. Soc. Ser. C 48, 15–30.Google Scholar
Kedem, B. H., Pavlopoulos, H., Guan, X. and Short, D. A. (1994). A probability distribution model for rain rate. J. Appl. Meteorology 33, 1486–1493.Google Scholar
Levinson, S. E., Rabiner, L. R. and Sondhi, M. M. (1983). An introduction to the application of the theory of probabilistic functions of a Markov process to automatic speech recognition. Bell Systems Tech. J. 62, 1035–1074.Google Scholar
Macdonald, Ll. and Zucchini, W. (1997). Hidden Markov and other Models for Discrete-valued Time Series. Chapman and Hall, London.Google Scholar
Meilijson, I. (1989). A fast improvement to the EM algorithm in its own terms. J. Roy. Statist. Soc. Ser. B 51, 127–138.Google Scholar
Rabiner, L. R. (1989). A tutorial on hidden Markov models and selected applications in speech recognition. Proc. IEEE 77, 257–285.Google Scholar
Rogers, R. R. (1979). A Short Course in Cloud Physics. Pergamon, New York.Google Scholar
Sansom, J. (1995). Rainfall discrimination and spatial variation using breakpoint data. J. Climate 8, 624–636.2.0.CO;2>CrossRefGoogle Scholar
Sansom, J. (1998). A hidden Markov model for rainfall using breakpoint data. J. Climate 11, 42–53.Google Scholar
Sansom, J. (1999). Large-scale spatial variability of rainfall through hidden semi-Markov models of breakpoint data. J. Geophysical Res. 104 (D24), 31631–31643.Google Scholar
Sansom, J. and Thomson, P. J. (1992). Rainfall classification using breakpoint pluviograph data. J. Climate 5, 755–764.Google Scholar
Sansom, J. and Thomson, P. J. (1998). Detecting components in censored and truncated meteorological data. Environmetrics 9, 673–688.Google Scholar
Sansom, J. and Thomson, P. J. (2000). Fitting hidden semi-Markov models. Tech. Rep. 77, National Institute of Water and Atmospheric Research, New Zealand.Google Scholar
Zucchini, W. and Guttorp, P. (1991). A hidden Markov model for space-time precipitation. Water Resources Res. 27, 1917–1923.CrossRefGoogle Scholar