[Sur la structure des modèles bilinéaires sur ]
Les modèles bilinéaires (BL) classiques sont largement utilisés pour la modélisation des données non gaussiennes. Cependant, lʼextension de ces modèles au cas spatial (SBL) donne une nouvelle classe de modèles susceptibles de prendre en considération la non gaussianité et le comportement spatial. Le but principal de cette Note consiste à étudier la structure de certains modèles SBL qui jouent un rôle très important dans lʼanalyse statistique spatiale. Nous établissons des conditions nécessaires et suffisantes pour lʼexistence de solutions stationnaires aux seconds ordres, réguliers et ergodiques basées sur les fonctions de transferts. En utilisant la représentation ARMA spatiale, on montre que la variance de lʼerreur de prédiction linéaire est toujours plus grande que celle obtenue par SBL.
One-dimensionally indexed bilinear models are widely used for modeling non-Gaussian dataset. Extending BL models to multidimensionally indexed (spatial) one yields a novel class of models which are capable of taking into account the non-Gaussianity character and spatiality behavior. Hence, the main contribution here is to study the -structure of some SBL models which play an important role in spatial statistical analysis. So, we establish necessary and sufficient conditions for the existence of regular second order stationary and ergodic solutions in terms of its transfer functions. As a consequence, we observe that the second order structure is similar to a weak ARMA field, and that the variance of the best linear prediction error is always greater than the one obtained from an SBL model.
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@article{CRMATH_2012__350_7-8_427_0,
author = {Bibi, Abdelouahab and Kimouche, Karima},
title = {On $ {\mathbb{L}}_{2}$-structure of bilinear models on $ {\mathbb{Z}}^{d}$},
journal = {Comptes Rendus. Math\'ematique},
pages = {427--432},
publisher = {Elsevier},
volume = {350},
number = {7-8},
year = {2012},
doi = {10.1016/j.crma.2012.04.002},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2012.04.002/}
}
TY - JOUR
AU - Bibi, Abdelouahab
AU - Kimouche, Karima
TI - On $ {\mathbb{L}}_{2}$-structure of bilinear models on $ {\mathbb{Z}}^{d}$
JO - Comptes Rendus. Mathématique
PY - 2012
SP - 427
EP - 432
VL - 350
IS - 7-8
PB - Elsevier
UR - http://www.numdam.org/articles/10.1016/j.crma.2012.04.002/
DO - 10.1016/j.crma.2012.04.002
LA - en
ID - CRMATH_2012__350_7-8_427_0
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%A Kimouche, Karima
%T On $ {\mathbb{L}}_{2}$-structure of bilinear models on $ {\mathbb{Z}}^{d}$
%J Comptes Rendus. Mathématique
%D 2012
%P 427-432
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%N 7-8
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2012.04.002/
%R 10.1016/j.crma.2012.04.002
%G en
%F CRMATH_2012__350_7-8_427_0
Bibi, Abdelouahab; Kimouche, Karima. On $ {\mathbb{L}}_{2}$-structure of bilinear models on $ {\mathbb{Z}}^{d}$. Comptes Rendus. Mathématique, Tome 350 (2012) no. 7-8, pp. 427-432. doi : 10.1016/j.crma.2012.04.002. http://www.numdam.org/articles/10.1016/j.crma.2012.04.002/
[1] M. Amirmazlaghani, H. Amindavar, Image denoising using two-dimensional GARCH model, in: Systems, Signals and Image Processing, 2007, pp. 397–400.
[2] On the covariance structure of time-varying bilinear models, Stoch. Anal. Appl., Volume 21 (2003), pp. 25-60
[3] A fixed point approach to model random fields, ALEA, Volume 3 (2007), pp. 111-132
[4] Spatial Statistics and Modeling, Springer, 2010
[5] ARCH-type bilinear models with double long memory, Stochastic Process. Appl., Volume 100 (2002), pp. 275-300
[6] Multiple Viener–Itô Integrals, Lecture Notes in Mathematics, vol. 849, Springer, 1981
[7] Sur un modèle autorégressif non linéaire, ergodicité et ergodicité géométrique, JSTA, Volume 8 (1987), pp. 195-204
[8] On Wiener–Itô representation and the best linear predictions for bilinear time series, J. Appl. Probab., Volume 26 (1989), pp. 274-286
[9] Separable lower triangular bilinear model, J. Appl. Probab., Volume 41 (2004), pp. 221-235
[10] Gaussian maximum likelihood estimation for ARMA models II: Spatial processes, Bernoulli, Volume 12 (2006), pp. 403-429
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