Electronic Journal of Statistics

Spatial modelling for mixed-state observations

Cécile Hardouin and Jian-Feng Yao

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In several application fields like daily pluviometry data modelling, or motion analysis from image sequences, observations contain two components of different nature. A first part is made with discrete values accounting for some symbolic information and a second part records a continuous (real-valued) measurement. We call such type of observations “mixed-state observations".

This paper introduces spatial models suited for the analysis of these kinds of data. We consider multi-parameter auto-models whose local conditional distributions belong to a mixed state exponential family. Specific examples with exponential distributions are detailed, and we present some experimental results for modelling motion measurements from video sequences.

Article information

Electron. J. Statist., Volume 2 (2008), 213-233.

First available in Project Euclid: 27 March 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H05: Characterization and structure theory 62E10: Characterization and structure theory
Secondary: 62M40: Random fields; image analysis

Multivariate analysis Distribution theory Mixed-state variables Auto-models Spatial cooperation Markov random fields


Hardouin, Cécile; Yao, Jian-Feng. Spatial modelling for mixed-state observations. Electron. J. Statist. 2 (2008), 213--233. doi:10.1214/08-EJS173. https://projecteuclid.org/euclid.ejs/1206641967

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  • [1] Pierre Ailliot, Craig Thompson, and Peter Thompson. Space time modeling of precipitation using a hidden markov model and censored gaussian distributions. Technical report, Victoria University of Wellington, http://pagesperso.univ-brest.fr/, ailliot/doc/rainailliot.pdf, 2006.
  • [2] David J. Allcroft and Chris A. Glasbey. A latent Gaussian Markov random-field model for spatiotemporal rainfall disaggregation., J. Roy. Statist. Soc. Ser. C, 52(4):487–498, 2003. ISSN 0035-9254.
  • [3] Barry C. Arnold, Enrique Castillo, and José María Sarabia., Conditional specification of statistical models. Springer Series in Statistics. Springer-Verlag, New York, 1999. ISBN 0-387-98761-4.
  • [4] Julian Besag. Spatial interaction and the statistical analysis of lattice systems., J. Roy. Statist. Soc. Ser. B, 36:192–236, 1974. ISSN 0035-9246. With discussion by D. R. Cox, A. G. Hawkes, P. Clifford, P. Whittle, K. Ord, R. Mead, J. M. Hammersley, and M. S. Bartlett and with a reply by the author.
  • [5] Patrick Bouthemy, Cécile Hardouin, Gwënaelle Piriou, and Jian-Feng Yao. Mixed-state auto-models and motion texture modeling., Journal of Mathematical Imaging and Vision, 25(3):387–402, 2006.
  • [6] Noel Cressie and Subhash Lele. New models for Markov random fields., J. Appl. Probab., 29(4):877–884, 1992. ISSN 0021-9002.
  • [7] Ronan Fablet and Patrick Bouthemy. Motion recognition using non parametric image motion models estimated from temporal and multiscale cooccurrence statistics., IEEE Trans. on Pattern Analysis and Machine Intelligence, 25(12) :1619–1624, December 2003.
  • [8] Xavier Guyon., Random fields on a network: Modeling, Statistics and Applications. Probability and its Applications (New York). Springer-Verlag, New York, 1995. ISBN 0-387-94428-1.
  • [9] Cécile Hardouin and Jian-Feng Yao. Multi-parameter auto-models and their application. Technical report, IRMAR/Université de Rennes 1, http://hal.archives-ouvertes.fr/hal -00154382/fr/, 2006. Forthcoming in, Biometrika.
  • [10] Mark S. Kaiser and Noel Cressie. The construction of multivariate distributions from Markov random fields., J. Multivariate Anal., 73(2):199–220, 2000. ISSN 0047-259X.
  • [11] Mark S. Kaiser, Noel Cressie, and Jaehyung Lee. Spatial mixture models based on exponential family conditional distributions., Statist. Sinica, 12(2):449–474, 2002. ISSN 1017-0405.
  • [12] Fabien Salzenstein and Wojciech Pieczynski. Parameter estimation in hidden fuzzy markov random fields and image segmentation., CVGIP: Graph. Models Image Process., 59:205–220, 1997.