# Computing the Best Index PDFs combining Index PDFs from two SFSs This function implements the computation to obtain the index Probability Density Functions (PDFs) (e.g. NAO index) obtained to combining the Index PDFs for two Seasonal Forecast Systems (SFSs), the Best Index estimation (see Sanchez-Garcia, E. et al (2019), tools:::Rd_expr_doi("10.5194/asr-16-165-2019") for more details about the methodology applied to estimate the Best Index). ```r BEI_PDFBest( index_obs, index_hind1, index_hind2, index_fcst1 = NULL, index_fcst2 = NULL, method_BC = "none", time_dim_name = "time", na.rm = FALSE ) ``` ## Arguments - `index_obs`: Index (e.g. NAO index) array from an observational database or reanalysis with at least a temporal dimension (by default 'time'), which must be greater than 2. - `index_hind1`: Index (e.g. NAO index) array from a SFS (named SFS1) with at least two dimensions (time , member) or (time, statistic). The temporal dimension, by default 'time', must be greater than 2. The dimension 'member' must be greater than 1. The dimension 'statistic' must be equal to 2, for containing the two paramenters of a normal distribution (mean and sd) representing the ensemble of a SFS. It is not possible to have the dimension 'member' and 'statistic' at the same time. - `index_hind2`: Index (e.g. NAO index) array from a SFS (named SFS2) with at least two dimensions (time , member) or (time, statistic). The temporal dimension, by default 'time', must be greater than 2. The dimension 'member' must be greater than 1. The dimension 'statistic' must be equal to 2, for containing the two paramenters of a normal distribution (mean and sd) representing the ensemble of a SFS. It is not possible to have the dimension 'member' and 'statistic' together. - `index_fcst1`: (optional, default = NULL) Index (e.g. NAO index) array from forescating of SFS1 with at least two dimensions (time , member) or (time, statistic). The temporal dimension, by default 'time', must be equal to 1, the forecast year target. The dimension 'member' must be greater than 1. The dimension 'statistic' must be equal to 2, for containing the two paramenters of a normal distribution (mean and sd) representing the ensemble of a SFS. It is not possible to have the dimension 'member' and 'statistic' together. - `index_fcst2`: (optional, default = NULL) Index (e.g. NAO index) array from forescating of SFS2 with at least two dimensions (time , member) or (time, statistic). The temporal dimension, by default 'time', must be equal to 1, the forecast year target. The dimension 'member' must be greater than 1. The dimension 'statistic' must be equal to 2, for containing the two paramenters of a normal distribution (mean and sd) representing the ensemble of a SFS. It is not possible to have the dimension 'member' and 'statistic' together. - `method_BC`: A character vector of maximun length 2 indicating the bias correction methodology to be applied on each SFS. If it is 'none' or any of its elements is 'none', the bias correction won't be applied. Available methods developped are "ME" (a bias correction scheme based on the mean error or bias between observation and predictions to correct the predicted values), and "LMEV" (a bias correction scheme based on a linear model using ensemble variance of index as predictor). (see Sanchez-Garcia, E. et al (2019), tools:::Rd_expr_doi("10.5194/asr-16-165-2019") for more details). - `time_dim_name`: A character string indicating the name of the temporal dimension, by default 'time'. - `na.rm`: Logical (default = FALSE). Should missing values be removed? ## Returns BEI_PDFBest() returns an array with the parameters that caracterize the PDFs, with at least a temporal dimension, by default 'time' and dimension 'statistic' equal to 2. The firt statistic is the parameter 'mean' of the PDF for the best estimation combining the two SFSs PDFs. The second statistic is the parameter 'standard deviation' of the PDF for the best estimation combining the two SFSs PDFs. If index_fcst1 and/or index_fcst2 are null, returns the values for hindcast period. Otherwise, it returns the values for a forecast year. ## Examples ```r # Example 1 for the BEI_PDFBest function index_obs<- rnorm(10, sd = 3) dim(index_obs) <- c(time = 5, season = 2) index_hind1 <- rnorm(40, mean = 0.2, sd = 3) dim(index_hind1) <- c(time = 5, member = 4, season = 2) index_hind2 <- rnorm(60, mean = -0.5, sd = 4) dim(index_hind2) <- c(time = 5, member = 6, season = 2) index_fcst1 <- rnorm(16, mean = 0.2, sd = 3) dim(index_fcst1) <- c(time = 1, member = 8, season = 2) index_fcst2 <- rnorm(18, mean = -0.5, sd = 4) dim(index_fcst2) <- c(time = 1, member = 9, season = 2) method_BC <- 'ME' res <- BEI_PDFBest(index_obs, index_hind1, index_hind2, index_fcst1, index_fcst2, method_BC) # Example 2 for the BEI_PDFBest function index_obs<- rnorm(10, sd = 3) dim(index_obs) <- c(time = 5, season = 2) index_hind1 <- rnorm(40, mean = 0.2, sd = 3) dim(index_hind1) <- c(time = 5, member = 4, season = 2) index_hind2 <- rnorm(60, mean = -0.5, sd = 4) dim(index_hind2) <- c(time = 5, member = 6, season = 2) index_fcst1 <- rnorm(16, mean = 0.2, sd = 3) dim(index_fcst1) <- c(time = 1, member = 8, season = 2) index_fcst2 <- rnorm(18, mean = -0.5, sd = 4) dim(index_fcst2) <- c(time = 1, member = 9, season = 2) method_BC <- c('LMEV', 'ME') res <- BEI_PDFBest(index_obs, index_hind1, index_hind2, index_fcst1, index_fcst2, method_BC) ``` ## References Regionally improved seasonal forecast of precipitation through Best estimation of winter NAO, Sanchez-Garcia, E. et al., Adv. Sci. Res., 16, 165174, 2019, tools:::Rd_expr_doi("10.5194/asr-16-165-2019") ## Author(s) Eroteida Sanchez-Garcia - AEMET, esanchezg@aemet.es