Analysis of Italian Rainfall Data with a Hierarchical Bayesian Space-Time Model

Climate and meteorological data are characterised by many different scales of spatial and temporal variability often in conjunction with non stationarity, anisotropy and quite complicated space-time interactions. Furthermore climate and meteorological studies must be carried out on large amount of data, often coming from different sources, in order to capture long and short term dependencies, large and small scale spatial effects. All this leads to severe computational problems and the need for the development of complex ad hoc models. Furthermore, for this reason, meteorologists are often constrained to apply potentially unrealistic simplifying assumptions in order to adopt standard statistical models. This kind of models, generally, assume spatial data to be temporally independent and spatial structure not varying over time (separable covariance structure) to estimate the spatial correlation structure and it do not consider the temporal dynamic of the process and the temporal correlation as a function of the spatial domain (Royle, 2000). These limitations can severely affect the estimates quality and the efficiency of traditional space-time statistical models and methods. Alternative models are relatively easy to formulate in the traditional LMM or GLMM frameworks, but a lack of understanding of the underlying processes and the “curse of dimensionality” make the implementation of these models challenging (Wikle et al., 2002). The Bayesian framework represents a natural way to analyse spatio-temporal data and it gives the concrete possibility to overcome the afore mentioned limits (Berliner, Levine and Shea, 2000). In particular, the hierarchical Bayesian space-time modelling approach allows to deal with space-time dependence and interactions by modelling all the relevant process component in several stages. Such models become feasible to implement in high dimensions. Several recent example of Bayesian hierarchical models are present in the literature: for an extensive review see Huang et al., 2007, Benerjee, Carlin and Gelfand, 2004, Wikle et al. 2003 and Wikle, 2000. In this paper we consider a hierarchical Bayesian space-time model, proposed in Wikle, Berliner and Cressie, 1998, to treat monthly rainfall data related to the Italian area and collected between January 2003 and December 2006. The choice of such model is strictly related to the own features of the precipitation process. It’s fairly well known that precipitation process involves complicated spatial structure, temporal structure and spatio-temporal interactions and that the interest of meteorologist are properly in the understanding the behaviour of this process features in order to build prediction maps or hydrological balance equations and so on. These considerations combined with the further necessity of working with a large dataset don’t allow the use of standard statistical approaches and can be more effectively treated in the hierarchical Bayesian space-time modelling approach. Indeed the chosen model allows us to provide a mechanism for combining data from very different sources; to incorporate physical knowledge and background science in the model development and in the specifications of priors on model parameters; to provide posterior distributions of quantities of interest which can be used for scientific inference strategy and to work with very large datasets. These advantages are reached by the model specification through the following five hierarchical levels: 1) the measurement process, as the precipitation process plus an error term; 2) the large and small scales features, incorporated as a linear combination of three sources of variation: time, space and space-time interaction; 3) model parameters: each of these sources are then represented according to physical knowledge; 4 and 5) priors on parameters and hyperpriors are specified respectively in the fourth and fifth stage to complete the model specification. In Particular in the second stage one can decompose the precipitation process into three meaningful components letting the meteorologist to be able to understand and measure how the rainfall is determined by the spatial effect, by the temporal seasonality and by the space-time interactions too. In this stage, the pure spatial and temporal effect describe the well known climate effect whereas the dynamical short time and small spatial scale effect can be easily interpreted as the weather contribution. In this way the rainfall amount in a given site depends on its spatial location and on which period it has been observed as a consequence of the climate effect but it surely depends also on what had happened in the neighbouring sites and previously in time, in other words on the weather contribution. The estimation of such flexible model is obtain through a complex and computer intensive MCMC procedure. Moreover many of the advances in hierarchical Bayesian spatio-temporal modelling have been properly due to the application of the recent MCMC techniques to the Bayesian theory (Wikle et al., 2002). The aim of the present work is to estimate and to understand the spatial and the temporal large scale features (climate effect) of the precipitation process and to isolate them from the spatio-temporal ones (weather effect) for the Italian area. The obtained information are, in a further step, used to obtain predictions maps. The computations are developed by the authors using the R software environment (Development Core Team, 2007).