A methodology to predict monthly rainfall process is proposed. The prediction strategy is developed under the framework of dynamic probability models and empirical functions. The parameters of the dynamic probability model are changing with time while the structure of the probability model will remain unchanged. The parameters of the dynamic probability model are estimated at every point in time by using empirical functions. An empirical function is a time difference equation that establishes the relationship between a random vector that belongs to the probability model and a set of time series, which are observations of climatological phenomena. In time series literature the empirical functions are known as multivariate transfer function models. A mathematical relationship between the dynamic probability model and the empirical functions was derived after taking the first moment of both the theoretical and the empirical models. Thus, the parameters of the dynamic probability model become a set of empirical functions.

The maximum likelihood method was used to estimate the parameters of the dynamic probability model because this method is invariant under linear transformations and because most of the time their estimators are consistent. Typically, the resulting likelihood function is a highly nonlinear function with some constraints. The suggested optimization method consists of two steps: The first one is dedicated to estimate the initial point, which is obtained after estimating the parameters of the empirical functions. The second step consists of obtaining the final estimates of the parameters of the dynamic probability model. The sequential quadratic programming (SQP) algorithm was selected to solve the constrained nonlinear optimization problem. Thus, if an initial point is carefully selected, then the nonlinear algorithm will converge to a satisfactory local maximum, and consequently, the optimal parameters of the dynamic probability model will be available. The dynamic probability model and the empirical functions will be used to compute the probability that in a particular station and time the rainfall level will exceed the normal behavior, or the rainfall level would be less than the normal level, or the rainfall level would be in the normal range. Once, the probability is known for one of the previous three stages, the conditional expected rainfall will be predicted.

The proposed algorithm was successfully applied to predict the monthly rainfall process of five rainfall stations located in Puerto Rico (PR) with the longest rainfall records. PR is a small Caribbean island. Nevertheless there have been 91 rainfall stations installed since 1899 and 65 of them are currently recording rainfall data on a daily basis. One hundred and one years (1901-2001) of monthly rainfall records were studied. The empirical functions include the following observations: the SST in the North Atlantic (5-20N, 60-30W), SST in the South Atlantic (0-20S, 30W-10E), SST in Tropical Equatorial (10S-10N, 0-360). The data set also includes the SST in the equatorial Pacific: el Niņo 1-2 (0-10S, 90-80W), el Niņo 3 (5N-5S, 150-90W), el Niņo 3-4, (5N-5S, 160E-150W), and el Niņo 3-4 (5N-5S, 170-120W), and the North Atlantic Oscillation index.