Spatial modelling for mixed-state observations

Abstract

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.Comment: Published in at http://dx.doi.org/10.1214/08-EJS173 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Mixed-state spatio-temporal auto-models

“…The concept has been extended to the multivariate case by [12]. Then it made possible to define auto-models on a lattice for mixed-state observations, see [11]. We consider now the spatio-temporal framework, adding a temporal dynamics over the spatial scheme.…”

“…We consider now the spatio-temporal framework, adding a temporal dynamics over the spatial scheme. We consider a Markov chain in time, of Markov fields in space, those fields being defined in an analogous way to the auto-models for mixed-state observations of [11].…”

“…Random variables with mixed states have been first introduced in [12] and developed in [11]. We present the results in the context of the mixed-state space E = {0} ∪ (0, ∞), but they hold equally for a general state space E = F ∪ G, where F = {e 1 , ..., e M } and G is a subset of R p (see [11]). Of course, E can be written E = [0, ∞[ but the previous writing points out the dual feature of the measurements.…”

“…The main idea is to consider a temporal Markov chain of spatial Markov fields, the latter accounting for the mixed feature of the data. More precisely, we take for those random fields mixed-state auto-models presented in [11].…”

“…In section 2 we define mixed-state variables and then turn out to the temporal dynamics on mixed-state Markovian fields; we crucially use some results given in [12] and [11], that concern the building of spatial multi-parameter auto-models. The general Markov Chain of Markov Fields dynamics is thoroughly described in [10].…”