A Bayesian Network (BN) is a directed acyclic graph in which nodes represent random variables and edges indicate conditional dependencies among them. The graph defines a factorization of the Joint Probability Distribution (JPD) which allows to efficiently perform probabilistic inferencer. Static BNs have been already applied as statistical downscaling methods; in this case, the nodes represent stations (local observations) and the links the spatial dependencies among them. Moreover, an extra node representing the atmospheric state (weather types) is included as an evidential variable (the future value is known). Therefore, the posterior distribution of the stations can be directly computed from a predicted weather type using the BN. The standard analogue method is the simplest case where no spatial dependencies are considered (it corresponds to a Naive Bayes inverse classifier). In this paper we study daily rainfall occurrence over the Iberian Peninsula and ERA-40 reanalysis.
Different statistical learning methods allow to automatically obtain a graph and the corresponding JPD from data. We have applied different standard learning algorithms (K2 and B) to infer BNs from data and to downscale rainfall occurrence obtaining a probabilistic forecast. In this case, no significant improvement over a standard analog method is achieved (the stations tend to be independent given the atmospheric state variable). We show that this result is a consequence of the static character of BNs. Thus, we analyze the different extensions of BNs to include temporal dependencies.
The most popular dynamic extension of BNs considers a sequence of time slices corresponding to different snapshots in time of the same static BN, connected by temporal links (Dynamical Bayesian Networks, DBN). Both spatial intra-slice and dynamic inter-slice connections can be imposed or learnt from data according to different strategies. We have compared static and dynamic models considering three different approaches. The first method is a two step algorithm where we first learn a common structure for the time slices (static model) and then the inter-slice connections. In the second approach, both intra and inter-slice connections are learnt simultaneously. In these two cases only a few inter-slice connections are established, since spatial relationships tend to be stronger than the temporal ones at daily basis. In order to favour dynamic relationships, in the last approach, we learn inter-slice connections before the intra-slice ones. This third approach obtained the best results in terms of ROC skill area. As a conclusion we show that the analog downscaling approach can be naturally extended to include both spatial and temporal relationships using the framework of DBNs. The validation results show a small average improvement of the skill; however, for certain stations (those with poorest analog skill), a significant improvement has been obtained.
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