Numéro spécial : Génération aléatoire de conditions météorologiques
Modeling of air temperatures: preprocessing and trends, reduced stationary process, extremes, simulation
Journal de la société française de statistique, Volume 156 (2015) no. 1, pp. 138-168.

Our first goal is to give a complete methodology to get a simulation model valid for a very large number of air temperature series. Existing simulators have quite good properties for the bulk of the distribution but they are unable to reproduce extreme values as well as cold or hot waves. Thus we focus on this aspect in order to cover both the bulk of the data and extreme values.

First we give some new results on preprocessing (the phase during which trends and seasonalities are removed). In this context, the choice of non parametric trends requires some new results on cross validation. Once additive (mean) and multiplicative (variance) trends and seasonality are suppressed, we test the cyclo-stationarity of the reduced series. In almost all cases, there does not remain any trend even for extremes values.

To model the observations, we start from non parametric estimates of the expectation and variance from a day conditional to the previous one and from studies of correlations. The best class of model seems the seasonal FARCH class. Then for mathematical improvements supported by physical ones, FARCH process is interpreted as the Euler scheme of continuous-time diffusions and thus as misspecified models and modified by adaptive truncation of innovations using new results of extreme theory on diffusions. These results are plugged in the likelihood of FARCH processes to get a convenient estimation.

We detail some results of the application to about 200 stations from Eurasia and United States, including the validation procedures for the model. The comparison to some existing models is also considered

Notre premier objectif est de donner une méthodologie complète pour obtenir un modèle de simulation valable pour un très grand nombre de séries de température de l’air . Simulateurs existants ont très bonnes propriétés pour l’essentiel de la distribution, mais ils sont incapables de se reproduire valeurs extrêmes ainsi que les vagues de froid ou de chaud. Ainsi, nous nous concentrons sur cet aspect afin de couvrir à la fois l’essentiel des données et les valeurs extrêmes.

D’abord, nous donnons quelques résultats nouveaux sur prétraitement ( la phase au cours de laquelle les tendances et saisonnalités sont supprimés ) . Dans ce contexte, le choix des tendances non paramétriques nécessite de nouveaux résultats sur la validation croisée . Une fois additifs (moyenne) et multiplicatif ( variance) Tendances et saisonnalité sont supprimés, nous testons la cyclo- stationnarité de la série réduite . Dans presque tous les cas , il ne reste aucune tendance , même pour les valeurs extrêmes .

Pour modéliser les observations , nous commençons à partir des estimations non paramétriques de l’ espérance et la variance d’une journée conditionnel à la précédente et à partir des études de corrélations . La meilleure classe de modèle semble la classe Farch saison . Puis, des améliorations mathématiques supportées par les physiques , procédé Farch est interprété comme le schéma d’Euler des diffusions en temps continu , et donc que les modèles mal spécifiés et modifiés par la troncature adaptative des innovations en utilisant de nouveaux résultats de la théorie sur les diffusions extrême . Ces résultats sont branchés sur la probabilité des processus Farch pour obtenir une estimation pratique.

Nous détaillons certains résultats de l’application d’ environ 200 stations d’Eurasie et aux états-Unis , y compris les procédures de validation du modèle. La comparaison de certains modèles existants est également envisagée.

Keywords: temperature, time series, preprocessing, cyclo-stationarity, extremes, diffusion, simulation model
Mot clés : température, série temporelle, pretraitement, cyclo-stationnarité, extrêmes, diffusion, modèle de simulation
@article{JSFS_2015__156_1_138_0,
     author = {Dacunha-Castelle, Didier and Hoang, Thi Thu Huong and Parey, Sylvie},
     title = {Modeling of air temperatures: preprocessing and trends, reduced stationary process, extremes, simulation},
     journal = {Journal de la soci\'et\'e fran\c{c}aise de statistique},
     pages = {138--168},
     publisher = {Soci\'et\'e fran\c{c}aise de statistique},
     volume = {156},
     number = {1},
     year = {2015},
     zbl = {1316.62167},
     language = {en},
     url = {http://archive.numdam.org/item/JSFS_2015__156_1_138_0/}
}
TY  - JOUR
AU  - Dacunha-Castelle, Didier
AU  - Hoang, Thi Thu Huong
AU  - Parey, Sylvie
TI  - Modeling of air temperatures: preprocessing and trends, reduced stationary process, extremes, simulation
JO  - Journal de la société française de statistique
PY  - 2015
SP  - 138
EP  - 168
VL  - 156
IS  - 1
PB  - Société française de statistique
UR  - http://archive.numdam.org/item/JSFS_2015__156_1_138_0/
LA  - en
ID  - JSFS_2015__156_1_138_0
ER  - 
%0 Journal Article
%A Dacunha-Castelle, Didier
%A Hoang, Thi Thu Huong
%A Parey, Sylvie
%T Modeling of air temperatures: preprocessing and trends, reduced stationary process, extremes, simulation
%J Journal de la société française de statistique
%D 2015
%P 138-168
%V 156
%N 1
%I Société française de statistique
%U http://archive.numdam.org/item/JSFS_2015__156_1_138_0/
%G en
%F JSFS_2015__156_1_138_0
Dacunha-Castelle, Didier; Hoang, Thi Thu Huong; Parey, Sylvie. Modeling of air temperatures: preprocessing and trends, reduced stationary process, extremes, simulation. Journal de la société française de statistique, Volume 156 (2015) no. 1, pp. 138-168. http://archive.numdam.org/item/JSFS_2015__156_1_138_0/

[1] A.M.G., Klein Tank et al Daily datasets of 20th-century surface air temperature and precipitation series for the European climate assessment, Int. J. of Clim., Volume 22 (2002), pp. 1441-1453

[2] Benth, J.S.; Benth, F.E. A critical view on temperature modelling for application in weather derivatives markets, Energy Economics, Volume 34 (2011), pp. 592-602

[3] Berman, S.M. Maxima and large excursions of stationary gaussian processes, Trans.Amer.Math.Soc., Volume 160 (1973) no. 67-85 | Zbl

[4] Chen, M.; An, H. The probabilistic properties of the nonlinear autoregressive model with conditional heteroskedasticity, Acta Mathematicae applicatae sinica, Volume 15 (1999) | Zbl

[5] Cleveland, R.B; Cleveland, W.S; Mcrae, J.E; Terpenning, I. STL: a seasonal-trend decompostion procedure based on Loess (with discussion), Journal of Official Statistics, Volume 6 (1990), pp. 3-73

[6] Campell, S.D.; Diebold, F.X. Weather Forecasting for Weather Derivatives, Journal of the American Statistical Association, Volume 100 (2005), pp. 6-16 | Zbl

[7] Cleveland, W.S. Robust locally weighted regression and smoothing scatterplots, Journal of the American Statistical Association, Volume 74 (1979), pp. 829-836 | Zbl

[8] Chu, C.K.; Marron, J.S. Comparison of two bandwidth selectors with dependent errors, Ann. Stat., Volume 19 (1991), pp. 1906-1918 | Zbl

[9] Cline, D. B.H.; Pu, H.H. Geometric ergodicity of on linear time series, Statistica Sinica, Volume 9 (1999), pp. 1103-1118 | Zbl

[10] Davis, R.A. Maximum and minimum of one-dimensional diffusions, Stochastic Processes and their applications, Volume 13 (1982), pp. 1-9 | Zbl

[11] Dacunha- Castelle, D.; Florens-Zmirou, D. Estimation of the coefficients of a diffusion from discrete observations, Stochastics, Volume 19 (1986), pp. 263-284 | Zbl

[12] Doukhan, P. Mixing : properties and examples, Springer-Verlag, 1994 | Zbl

[13] Francisco-Fernández, M.; Vilar-Fernández, J.M. Bandwidth selection for the local polynomial estimator under dependence: a simulation study, Computational Statistics, Volume 20 (2005), pp. 539-558 | Zbl

[14] Furrer, E.M.; Katz, R.W. Improving the simulation of extreme precipitation events by stochastic weather generators, Water Resources Research, Volume 44 (2008)

[15] Florens-Zmirou, D. Estimation de la variance d’une diffusion à partir d’une observation discrétisée, C.R.A.S., t.309, Série I (1989), pp. 195-200 | Zbl

[16] Gobet, E.; Hoffmann, M.; Reib, M. Nonparametric estimation of scalar diffusions based on low frequency data, Ann. Stat., Volume 32 (2004), pp. 2223-2253 | Zbl

[17] Golfarb, D.; Idnani, A. Dual and Primal-Dual Methods for Solving Strictly Convex Quadratic Programs, J. P. Hennart (ed.), Numerical Analysis, Springer-Verlag, Berlin, 1982, pp. 226-239 | Zbl

[18] Godfarb, D.; Idnani, A. A numerically stable dual method for solving strictly convex quadratic programs, Mathematical Programming, Volume 27 (1983), pp. 1-33 | Zbl

[19] Hall, P.; Lahiri, S.N.; Polzehl, J. On bandwidth choice in nonparametric regression with both short and long-range dependent errors, Ann. Stat., Volume 23 (1995), pp. 1921-1936 | Zbl

[20] Hoang, T.T.H Séries chronologiques non stationnaires non linéaires. Le cas des séries de températures en Europe., Université Paris Sud 11 (2010) (Ph. D. Thesis)

[21] Hansen, L.P.; Scheinkman, J.A.; Touzi, N. Indentification of scalar diffusions using eigenvectors, Journal of econometrics, Volume 86 (1998), pp. 1-32 | Zbl

[22] Katz, R. Overview of extreme value analysis under climate change., US CLIVAR/NCAR ASP Researcher Colloquium on statistical assessment of extreme weather phenomena under climate change, NCAR, Boulder, CO (2011)

[23] Kessler, M.; Sorensen, M. Estimating equations based on eigenfunctions for a discretely observed diffusion process, Bernoulli, Volume 5 (1999) no. 299-314 | Zbl

[24] Marron, J.S. Partiontioned cross-validation, Econometric Rev., Volume 6 (1987), pp. 271-284 | Zbl

[25] Mraoua, M.; Bari, D. Temperature stochastic modeling and weather derivatives pricing: empirical study with Moroccan data, Afrika Statistika, Volume 2 (2007) no. 1, pp. 22-43 | Zbl

[26] Menne, M.J.; Durre, I.; Vose, R.S.; Gleason, B.E.; Houston, T.G. An overview of the global historical climatology network- Daily database, Journal of Atmospheric and Oceanic Technology, Volume 29 (2012), pp. 897-910

[27] Nelder, J.A.; Mead, R.A. A simplex algorithm for function minimization, Computer Journal, Volume 7 (1965), pp. 308-313 | Zbl

[28] Parey, S.; Dacunha-Castelle, D.; Hoang, T.T.H. Mean and variance evolutions of the hot and cold temperatures in Europe, Climate Dynamics (2009)

[29] Parey, S.; Hoang, T.T.T; Dacunha-Castelle, D. The role of variance in the evolution of observed temperature extremes in Eurasia and in the United States (2013) (accepted by JPR)

[30] Resnick, S.I. Heavy-Tail Phenomena: Probabilistic and Statistical Modeling, Springer- Verlag, 2007 | Zbl

[31] Resnick, S.I. Extreme values, regular variation and point processes, Springer- Verlag, 1987 | Zbl

[32] Richardson, C.W. Stochastic simulation of daily precipitation, temperature, and solar radiation, Water Resources Research, Volume 17 (1981), pp. 182-190

[33] Ruppert, D.; Wand, M.P. Multivariate locally weighted least squares regression, Ann. Stat., Volume 22 (1994), pp. 1346-1370 | Zbl

[34] Sura, P. Stochastic models of climate extremes: Theory and observations, Springer-Verlag, 2012

[35] Wang, H. Nonlinear ARMA models with functional MA coefficients, Journal of Time series analysis, Volume 29 (2008), pp. 1032-1056 | Zbl

[36] Wilks, D.S.; Wilby, R.L. The weather generation game: a review of stochastic weather models, Progress in Physical Geography, Volume 23 (1999), pp. 329-357