Modeling of air temperatures: preprocessing and trends, reduced stationary process, extremes, simulation

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.