where S(t) is a multi-model ensemble prediction for day t, a time mean of the observed state, a weight for model i, i the model index, N the number of models, a time mean of the prediction by model i, and Fi(t) a prediction by model i.

where O(t) denotes a observed state, t time, and t-train the length of the training period.

In most of the applications of multi-model ensemble method, the Gauss-Jordan elimination algorithm can be efficiently used in order to minimize the above function J. However, there are singular value problems in this algorithm especially for the application to precipitation forecasts because of many of zero rain events. This problem can satisfactorily be solved by an algorithm, known as singular value decomposition (SVD). Instead of using the Gauss-Jordan elimination algorithm, the SVD is, therefore, the method of choice for solving the equation (2.2.1). Further details on the SVD can be found below.