Parametrizations, fixed and random effects

Azzouz Dermoune (1) , Cristian Preda (2)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 MODAL - MOdel for Data Analysis and Learning

We consider the problem of estimating the random element s of a finite dimensional vector space S from the continuous data corrupted by noise with unknown variance σ 2 w. The mean E(s) (the fixed effect) of s belongs to a known vector subspace F of S, and the likelihood of the centred component s − E(s) (the random effect) belongs to an unknown supplementary space E of F relative to S and has the PDF proportional to exp{−q(s)/2σ 2 s }, where σ 2 s is some unknown positive parameter. We introduce the notion of bases separating the fixed and random effects and define comparison criteria between two separating bases using the partition functions and the maximum likelihood method. We illustrate our results for climate change detection using the set S of cubic splines. We show the influence of the choice of separating basis on the estimation of the linear tendency of the temperature and the signal-to-noise ratio σ 2 w /σ 2 s .

Mots clés: General linear model; fixed effect; random effect; cubic spline; smoothing parameter; likelihood; climate change detection

Domaines: Applications [stat.AP]; Méthodologie [stat.ME]

Azzouz Dermoune, Cristian Preda. Parametrizations, fixed and random effects. Journal of Multivariate Analysis, 2016, 154, pp.162 - 176. ⟨10.1016/j.jmva.2016.11.001⟩. ⟨hal-01424782⟩