where?is a corrected forecast of the ith model through the SPPM procedure. Due to the fact that every dynamical model has a systematic bias, its predictive skill can be improved by reducing the systematic error. The SPPM is a statistical model which is designed to correct a systematic bias of dynamical prediction. After applying the SPPM to individual model separately, the MME prediction is obtained by the equal-weighting simple composite of the corrected individual predictions.

The SPPM is a kind of pointwise regression model. The predictor of the model is a pattern of predicted precipitation in certain domain and the predictand is precipitation at each grid point over the global domain. The main idea is to predict the predictand at each grid by projecting the predictor field onto the covariance pattern between the large-scale predictor field and the one-point predictand. The model equation is as follows:

where x and t denote spatial and temporal grid, respectively. P indicates a time series projected by the covariance pattern between predictand, PRCP(t), and predictor field, Ψ(x,t), in certain domain (D). The parameter, α, is a regression coefficient of the projected time series, P, on the predictand during a training period, T.

In this statistical prediction, selection of the predictor domain (D) plays a crucial role on the predictive skill. In general, traditional pattern projection models use a fixed geographical domain whose location and size are fixed for predictor during the whole forecast period. This method is somewhat appropriate to the regional climate prediction, where the number of predictands is limited. However, when the prediction target covers a wide region so that the number of predictands is large, it is difficult to subjectively choose the predictor domain. Therefore, a method is required in order to objectively select prediction domain. In the SPPM process, the optimal predictor domain is automatically selected with an objective criterion.

The SPPM consists of two steps to obtain final prediction. The first step is a selection of predictor domain, and the second step is a prediction by the pattern projection of Eq. (2.6.2). In the first step, to select predictor domain correlation coefficients between predictand and precipitation (predictor variable) at each grid are calculated to search possible predictor domain. Among them, some grids, having significantly higher correlation, are selected as a predictor grid. In this case, the selected grids may be split into several regions. In the traditional statistical model, only one geographical domain, including significant grids, is selected as a predictor domain, so other grids outside the selected domain are not used. However, in the SPPM, all significant grid points are gathered and a reconstructed domain is constructed by lining up the selected grid points. There constructed domain is regarded as a predictor domain (D). Using the selected predictors, the SPPM produces a corrected forecast based on Eq. (2.6.2). More detail procedure on the SPPM procedure is referred to Kug et al. (2007b).

The MME-SPPM2 is an improved version of the MME-SPPM. In the MME-SPPM, all participant models are used for simple composite at each grid after individual model is corrected by the SPPM. However, in the MME-SPPM2 some model predictions that have destructive poor skills are excluded from simple composite. That is, only qualified model predictions that have predictable skill at each grid point are used for the multi-model ensemble. The poor models may degrade the skill of the multi-model ensemble, therefore, the MME-SPPM skill can be improved by removing skill-less predictions of the poor model. In order to know whether each model prediction has predictable skill or not, double-cross-validation method (Kaas et al. 1996) is used in the SPPM procedure. In the double-cross-validation process, hindcast skill of individual model at each grid is calculated during the training period. If the correlation skill during the training period is higher than a threshold value, the model is selected.