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Additional info for Measure Theory Applications to Stochastic Analysis

Example text

Xsn . Cette prédiction coïncide avec X0 si s0 est l’un des sites d’observation. Si X est gaussien, X0 n’est autre que l’espérance conditionnelle E(X0 | Xs1 , . . , Xsn ) ; inconditionnellement, la loi de cette prédiction est gaussienne et l’erreur est X0 − X0 ∼ N (0, τ 2 (s0 )). 34). Si Σ n’est pas connue, elle sera estimée préalablement (cf. 3). 10. Krigeage universel : la meilleure prédiction linéaire de X0 sans biais est X0 = {t cΣ −1 + t (z0 − t ZΣ −1 c)(t ZΣ −1 Z)−1t ZΣ −1 }X. 36) La variance de l’erreur de prédiction est τ 2 (s0 ) = σ02 − t cΣ −1 c + t (z0 − t ZΣ −1 c)(t ZΣ −1 Z)−1 (z0 − t ZΣ −1 c).

N) un résidu centré spatialement corrélé. 40 1 Modèle spatial du second ordre et géostatistique Notant X = t (Xs1 , . . , Xsn ), ε = t (εs1 , . . , εsn ), Z = t (zs1 , . . 32) s’écrit matriciellement : X = Zδ + ε, avec E(ε) = 0 et Cov(ε) = Σ. La deuxième étape consiste à modéliser Σ à partir d’une fonction de covariance, d’un variogramme ou encore d’un modèle AR spatial. 11. Pluies dans l’Etat du Parana (données parana du package geoR [181] de R ) Ces données donnent la hauteur de pluie moyenne sur différentes années durant la période mai–juin pour 143 stations du réseau météorologique de l’Etat de Parana, Brésil.

7-b) de demi-support L+ = {(1, 0), (2, 0), (−1, 1), (0, 1), (0, 2), (1, 1), (0, 2)}, et de coefficients : c1,0 = 2aκ2 , c0,1 = 2bκ2 , c2,0 = 2a2 κ2 , c0,2 = 2b2 κ2 c−1,1 = −2abκ2, σe2 = σε2 κ2 où κ2 = (1 + 2a2 + 2b2 )−1 . 34 1 Modèle spatial du second ordre et géostatistique L+ + L L L R R (a) (b) Fig. 7. (a) Support R = {(1, 0), (0, 1)} du modèle SAR causal et support L du CAR associé ; (b) support R = {(1, 0), (0, 1), (−1, 0), (0, −1)} du modèle SAR non-causal et support L du CAR associé. 3. Le SAR factorisant : Xs,t = αXs−1,t + βXs,t−1 − αβXs−1,t−1 + εs,t , |α| et |β| < 1, est un CAR aux 8-ppv, de coefficients c1,0 = α(1 + α2 )−1 , c0,1 = β(1 + β 2 )−1 , c1,1 = c−1,1 = −c1,0 × c0,1 σe2 = σε2 κ2 où κ2 = (1 + α2 )−1 (1 + β 2 )−1 Dans ces trois exemples, κ2 < 1 est le gain en variance de la prédiction CAR de X comparée à la prédiction SAR.

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