Développements récents dans la modélisation de la persistance à long terme Developments in modelling long term persistence

Résumé

Afin de modéliser efficacement la persistance dans les séries chronologiques rencontrées en hydrologie, des développements récents autour du modèle fractionnaire auto-régressif à moyenne mobile (FARMA) (fractional autoregressive-moving average model) sont présentés. On s'intéresse particulièrement ici à de nouvelles procédures permettant d'estimer les paramètres du modèle FARMA d'une manière efficace au point de vue calcul. Pour obtenir les distributions d'échantillons des estimateurs des paramètres à partir de petits échantillons, une technique faisant appel au bootstrap peut être utilisée. Des applications pratiques à des séries de débits en rivière, de précipitations et de températures, montrent l'utilité des modèles FARMA.

Mots-clés:

• Modélisation,
• persistance à long terme,
• modèles FARMA,
• hydrologie

Abstract

In order to model effectively persistence in hydrologic tune series, recent developments in fractional autoregressive-moving average (FARMA) models are presented. A time series possesses persistence or long memory if it has an autocorrelation structure that attenuates slowly to zero with increasing lags. Based on the controversy surrounding the Hurst phenomenon, some hydrologists claim that it is important to employ stochastic models which have the ability to model long memory when it is present in a given time series. Fractional Gaussian noise models and approximations thereof were developed within the field of hydrology in order to be able to model long memory. However, a particularly flexible set of models having the capability to describe long memory is the FARMA family of models, which constitutes a direct generalization of autoregressive integrated moving average (ARIMA) models.

In particular, like an ARIMA model, a FARMA model contains autoregressive and moving average parameters. Whereas the differencing operator d is restricted to be zero or take on positive integer values in an ARIMA model, the parameter d in a FARMA model can have real values and is estimated along with the other model parameters. For a specified range of values for the d parameter, a FARMA model has long memory. Besides reviewing the background and main theoretical properties of FARMA models, simulation and forecasting techniques are presented. Additionally, procedures for estimating the parameters of a FARMA model are given and a bootstrapping technique is described to obtain the small sample distributions of the estimated parameters.

To explain how to apply FARMA models in practice and demonstrate their usefulness, they are fitted to riverflow, precipitation and temperature time series.

Keywords:

• Modelling,
• long term persistence,
• FARMA models,
• hydrology