Improved statistical downscaling of daily precipitation using SDSM platform and data-mining methods

Transcription

1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 33: (2013) Published online 1 November 2012 in Wiley Online Library (wileyonlinelibrary.com) DOI: /joc.3611 Improved statistical downscaling of daily precipitation using SDSM platform and data-mining methods H. Tavakol-Davani, a M. Nasseri b * and B. Zahraie c a School of Civil Engineering, University of Tehran, Tehran, Iran b School of Civil Engineering, University of Tehran, Tehran, Iran c Center of Excellence for Engineering and Management of Civil Infrastructures, School of Civil Engineering, University of Tehran, Tehran, Iran ABSTRACT: In this paper, an extension of the statistical downscaling model (SDSM), namely data-mining downscaling model (DMDM), has been developed. DMDM has the same platform as the most cited statistical downscaling models, namely SDSM and ASD. Multiple linear regression (MLR), ridge regression (RR), multivariate adaptive regression splines (MARS) and model tree (MT) constitute the mathematical core of DMDM. DMDM uses linear basis functions in MARS and linear regression rules in MT to keep the linear structure of SDSM; therefore, all of the SDSM assumptions are also valid in DMDM. These methods highlight the effect of data partitioning for meteorological predictors in the downscaling procedure. Inputs and output of the presented approaches are the same as SDSM and ASD. In the case study of this research, NCEP/NCAR databases have been used for calibration and validation. According to the inherent linearity of the methods, suitable predictor selection has been done with stepwise regression as a preprocessing stage. The results of DMDM have been compared with observed precipitation in 12 rain gauge stations that are scattered in different basins in Iran and represent different climate regimes. Comparison between the results of SDSM and DMDM has indicated that the presented approach can highly improve downscaling efficiency in terms of reproducing monthly standard deviation and skewness for both calibration and validation datasets. Among the proposed methods in DMDM, the results of the case study have shown that MT has provided better performances both in modelling occurrence and amount of precipitation. Also, MT is potentially recognized as a powerful diagnostic tool that could extract information in key atmospheric drivers affecting local weather. It also has fewer parameters during dry seasons, in which the number of historical precipitation events might not be enough for calibrating SDSM model in many arid and semi-arid regions. KEY WORDS statistical downscaling; data-mining methods; climate change; MARS; model tree Received 20 January 2012; Revised 14 August 2012; Accepted 25 September Introduction Future scenarios of climate change are achieved from global circulation models (GCMs) which are the premier meteorological sources for approximating plausible future climate. This information is spatially too coarse to determine regional effects of climate change, and it must be transformed to finer resolutions to be applicable in local analysis. Downscaling approaches depict suitable methods to extract regional-scale meteorological variables from GCM outputs. Two general types of mathematical approaches for downscaling of GCM simulations can be listed as follows: Dynamical approaches or regional climate models (Mearns et al., 2003) Statistical approaches: * Correspondence to: M. Nasseri, School of Civil Engineering, University of Tehran, Tehran, Iran. mm_nasseri@yahoo.com, mnasseri@ut.ac.ir Regression-based (empirical) methods (Enke and Spekat, 1997; Faucher et al., 1999; Wilby et al., 1999; Li and Sailor, 2000; Hessami et al., 2008; Raje and Mujumdar, 2011), Weather pattern approaches (Bárdossy and Plate, 1992; Yarnal et al., 2001; Anandhi et al., 2011), Stochastic weather generators (Semenov and Barrow, 1997; Bates et al., 1998). Because of less preprocessing requirements and computational costs, statistical approaches in general and regression-based methods in specific have received more attention among the aforementioned downscaling approaches. Among different statistical approaches, regression-based methods are famous for simplicity in implementation. Different tools have been developed for statistical downscaling using artificial neural networks (ANNs) (Pasini, 2009; Tomassetti et al., 2009; Mendes and Marengo, 2010; Fistikoglu and Okkan, 2011), k- nearest neighbors (Yates, et al., 2003; Gangopadhyay et al., 2005; Raje and Mujumdar, 2011), support vector 2012 Royal Meteorological Society

2 2562 H. TAVAKOL-DAVANI ET AL. machines (SVMs) (Tripathi et al., 2006; Chen et al., 2010) and linear regression (Wilby et al., 2002; Hessami et al., 2008). Finding empirical relationships between large- and regional-scale climates is the aim of statistical downscaling methods. Correlation between GCM-scale meteorological variables (predictors) and local meteorological variables such as precipitation and temperature (predictands) is the master point of statistical downscaling procedures. Advantages and disadvantages of statistical downscaling methods have been discussed by Hessami et al. (2008). Statistical downscaling model (SDSM) and automated statistical downscaling (ASD) are among the most cited statistical/empirical downscaling tools. These packages implement linear regression to estimate the amount and/or the occurrence of local meteorological predictands; simple linear and ridge regression (RR) methods are the mathematical kernels of SDSM and ASD, respectively. SDSM and ASD behave well in keeping mean of predictands, but they are weak in simulating standard deviation and extreme values (Wilby et al., 2004; Hessami et al., 2008). In this paper, SDSM (Wilby et al., 2002) and ASD (Hessami et al., 2008) models have been extended using two linear data-mining (DM) methods of model tree (MT) and multivariate adaptive regression splines (MARS). The main reason of using these DM methods is the linearity of their mathematical kernels, which make them Figure 1. Location map of rain gauges of interest in the five catchments (Jazmoorian, Sefidrood, Mordab-anzali, Shapoor-dalky and Mond basin).

3 SDSM, DATA-MINING METHODS, CLIMATE CHANGE 2563 consistent with SDSM and ASD. DM methods perform well in the regression field, and therefore it is anticipated that MT and MARS might outperform multiple linear regression (MLR) and RR in statistical downscaling of GCM simulations. In the next section (Section 2), the study areas have been described. In Section 3, the platform of SDSM software, MT, MARS and the proposed approach are presented. Later, in Section 4, the downscaling results are presented. In the last section (Section 5), concluding remarks are presented. 2. Case study 2.1. Local data The selected rain gauge stations in this research are scattered in five basins namely Hamoon-Jazmoorian, Sefidrood, Mordab-Anzali, Shapoor-dalky and Mond with areas of , , 3 224, and km 2, respectively. Hamoon-Jazmoorian is located in an arid region in southeast of Iran near Iran Pakistan border. Sefidrood and Mordab-Anzali basins are located in a wet region in north of Iran near Caspian Sea and Shapoor-dalky and Mond basins are located in a relatively arid region in southwest of Iran near Persian Gulf. These basins are selected based on availability of observed meteorological data and also they represent various climatic and hydrologic conditions. Twelve rain gauge stations in these basins have been selected (Figure 1). Approximately in all stations, dry and wet seasons are in the periods of May through October and November through April, respectively. According to Table 1, years of daily precipitation records are available for these stations. In this study, the first 75% of recorded datasets is used for calibration and the last 25% is used for validation in each station. Daily records of precipitation are collected from the national database of the Iranian Ministry of Energy Large-scale datasets In this research, the outputs of Hadley Center s GCM (HadCM3) have been utilized. The special report on emission scenarios (SRES), A2 and B2 scenarios have been used to project the future climate conditions. Large-scale NCEP reanalysis atmospheric data (Table 1) have been used as the model predictors. Spatial resolution (dimensions of grid box) of HadCM3 outputs is 3.75 (long.) 2.5 (lat.), whereas it is 2.5 (long.) 2.5 (lat.) for NCEP data. Therefore, projected large-scale predictors of NCEP on HadCM3 computational grid box have been used. These data and HadCM3 daily simulations are supported and distributed by the Canadian Climate Change Scenarios Network (CCCSN) ( and also the Canadian Climate Impacts Scenarios (CCIS) website ( There are 26 different atmospheric variables for each grid box in this database. For each rain gauge station, nine boxes covering the study areas have been selected. Figure 1 depicts centre of each meteorological grid box and location of the selected rain gauge stations. As it can be seen in this figure, the grid boxes cover a large area over the selected basins and around them. In addition, 1- to 3-d lagged series of the predictors have also been considered as new predictors in this research. For each station, 828 (9 (no. of boxes) 23 (atmospheric variables) 4 (number of lags)) predictors have been analyzed, which are explained in more details in the next sections of the paper. 3. Methodology 3.1. Statistical downscaling model SDSM software is developed based on multiple linear regression downscaling model (MLRDM). SDSM presents several weather ensembles and its outputs are the average of these ensembles. These ensembles are the results of linear regression models with stochastic terms No. Station code Station name Table 1. Basic information about 12 raingauges. Abbr. Basin Length of dataset (year) Longitude ( E) Latitude ( N) Statistical characteristics of observed daily rainfall (mm) Mean Max. Std Dehrood Deh. Jazmoorian Delfard Del. Jazmoorian Khoramshahi Kho. Jazmoorian Kharposht Kha. Jazmoorian Farshekan Far. Sefidrood Rasht Ras. Sefidrood Kasma Kas. Mordab-anzali Shanderman Shan. Mordab-anzali Shapoor Shap. Shapoor-dalky Shoorjareh Shoo. Shapoor-dalky Arsanjan Ars. Shapoor-dalky Khanzanian Khan. Mond Max., maximum; Std., standard deviation.

4 2564 H. TAVAKOL-DAVANI ET AL. d m p s p d m g p p p g Figure 2. SDSM structure (after Wilby et al., 2002). of bias correction. The general platform of SDSM has been shown in Figure 2 (Wilby et al., 2002). Because of the linear concepts of SDSM, selection of the predictors is based on the correlation or partial correlation analysis between the interested predictand and the predictors and weights of the predictors which are estimated via ordinary least-square method. In addition of ordinary least-square method, dual simplex method has been also provided in SDSM for calibration because of instability of regression coefficients for nonorthogonal predictor vectors. Hessami et al. (2008) added a new option of using RR (Hoerl and Kennard, 1970) in their downscaling model, namely ASD as a remedy of the nonorthogonality impact of the predictor vectors. SDSM model contains two separate sub-models to determine occurrence and amount of conditional meteorological variables (or discrete variables) such as precipitation and amount model for unconditional variables (or continues variables) such as temperature or evaporation. Therefore, SDSM can be classified as a conditional weather generator in which regression equations are used to estimate the parameters of daily precipitation occurrence and amount, separately, so it is slightly more sophisticated than a straightforward regression model. Statistical downscaling using SDSM consists of the following steps: 1. In first step, suitable predictors should be selected. SDSM provides the ability of some statistical analysis for users to select the best predictors. In SDSM, predictors should have acceptable unconditional and conditional correlations with the predictand. Also, partial correlation, P-value and explained variance of the predictors can be checked while using SDSM. Figure 3. DMDM procedure.

5 SDSM, DATA-MINING METHODS, CLIMATE CHANGE 2565 Acceptable ranges for these statistics are proposed by Wilby et al. (2004).The scatter plot is another tool provided in SDSM in order to select the appropriate predictors. 2. A MLR model is calibrated to simulate the precipitation occurrence which is called unconditional model. This model can be calibrated by two different methods namely ordinary least-square and dual simplex methods. An autoregressive term can be added to this model. For each month, one MLR model must be calibrated for occurrence estimation. The days with and without events (precipitation) are represented with 1 and 0, respectively. For each day and ensemble, a uniformly distributed random number between 0 and 1 is generated. If the random number is less than the output of the occurrence model in that day, precipitation occurs. Otherwise, precipitation does not occur. 3. Another MLR model, namely conditional model, is calibrated to simulate the precipitation amount. This model is calibrated using the available rainy days dataset. Like the unconditional model, SDSM calibrate different conditional models for 12 months of year. For a day which is identified as a rainy day in the previous step, output of the amount model is calculated. Then, a random number is added to the output to consider the modelling error. This random number is generated using a normal distribution function with zero mean and standard deviation equal to standard error. 4. The result of the previous step is compared with a predefined threshold. If the result is less than the threshold, the precipitation will not occur. Otherwise, the result is considered as the rainfall amount in that day and in that ensemble. (a) (b) (c) (d) Figure 4. Comparison of SDSM and MLRDM in Del. rain gauge. (a) Monthly mean, (b) monthly standard deviation, (c) monthly skewness and (d) q q plot (Left column: calibration period, right column: validation period).

6 2566 H. TAVAKOL-DAVANI ET AL. Table 2. Large-scale Predictors from NCEP database. No. Predictor Abbreviation 1 Mean sea level pressure mslp 2 Surface airflow strength p f 3 Surface zonal velocity p u 4 Surface meridional velocity p v 5 Surface vorticity p z 6 Surface divergence p_zh hpa airflow strength p5_f hpa zonal velocity p5_u hpa meridional velocity p5_v hpa vorticity p5_z hpa bivergence p5zh hpa airflow strength p8_f hpa zonal velocity p8_u hpa meridional velocity p8_v hpa vorticity p8_z hpa divergence p8zh hpa geopotential height p hpa geopotential height p Relative humidity at 500 hpa r Relative humidity at 850 hpa r Near surface relative humidity rhum 22 Near surface specific humidity shum 23 Mean temperature at 2 m temp 5. Furthermore, in SDSM variance inflation (VIF) and bias correction (b) factors are set to 12 and 1, respectively in the calibration period. Then they are calculated for scenario generation using Equations (1) and (2) for each monthly model (Hessami et al., 2008): b = Mean obs Mean mod + 1 (1) VIF = 12 (Var obs Var mod ) Ste 2 (2) where Mean obs and Mean mod are the mean values of the observed and modelled precipitation, respectively. Var obs and Var mod are the variances of observed and modelled precipitation for the calibration period and Ste is the standard error in the same period. b 1 is added to the amount of precipitation in each day and VIF/12 is multiplied to the standard deviation of modelling error. While the downscaling model is calibrated using NCEP dataset, in estimating VIF and bias correction, variables with the subscript mod are estimated using downscaling model outputs based on GCM simulations. This approach allows the modeller to take into account the bias of GCM in the downscaling process. 6. Finally, in order to achieve a single downscaled time series from all projected ensembles, their arithmetic means are calculated. SDSM package allows monthly (12 separate models), seasonal (4 models) or annual (1 model) calibration. Seasonal or annual models are recommended in data sparse situations to increase the sample of wet-days available in the calibration period. Different transformation methods, such as fourth or square root transformations, are available in SDSM to reduce data variance in semi-arid areas. In this study, SDSM has been rewritten in MATLAB environment. Accuracy and compatibility of the MAT- LAB code with SDSM package has been tested using several datasets. Then the SDSM MATLAB code has been extended to include the selected DM methods. The selected DM methods have been described in the next section Implemented methods and methodology In this paper, MARS and MT have been used to extend SDSM model at single-site level. The details of these methods are presented in the following sections. Table 3. Selected predictors for occurrence model in Deh., Kas. and Khan. stations. Station Predictor Lag Longitude Latitude Coefficient of determination Deh. p5_v p5_z r r r r r p8_v Kas. p f p u p5_v r r r Khan. p5_v p8_z r r shum rhum

7 SDSM, DATA-MINING METHODS, CLIMATE CHANGE 2567 (a) (b) (c) (d) Figure 5. Deh. rain gauge in Jazmoorian basin. (a) Monthly mean, (b) monthly standard deviation, (c) monthly skewness and (d) q q plot (Left column: calibration period, right column: validation period) Multivariate adaptive regression splines Friedman and Stuetzle (1981) initiated MARS method, and then Friedman (1991) developed this idea as a new multivariate regression based on linear estimation and continuous data partitioning. This method automatically finds nonlinearity and suitable data structure for model parameters using weighted summation of some conditional linear or polynomial basis functions. General formulation of MARS function (f ) is as follows: f (x) = n C i B i (x) i=1 B i (x) = max (0, d i x) or B i (x) = max (0, x d i ) (3) where n is the number of basis functions, C i is the weight of basis function i and d is the basis function node or knot and i refers to basis function counter. Application of MARS technique consists of the following two steps: 1. Forward procedure: In this procedure, by adding basis functions iteratively, the parameters are estimated using various least-square methods. The forward phase is executed until one of the following conditions is met: a. Reached maximum number of basis functions (This value is optional and can be defined by the user), b. The modelling error does not improve in several iterations, c. The modelling error is less than the desirable error, d. The number of model coefficients in the current iteration is equal to or larger than the maximum allowable number. 2. Backward procedure: In this procedure, leave one out evaluation of basis functions is done iteratively to

8 2568 H. TAVAKOL-DAVANI ET AL. (a) (b) (c) (d) Figure 6. Kas. rain gauge in Mordab-anzali basin. (a) Monthly mean, (b) monthly standard deviation, (c) monthly skewness and (d) q q plot (Left column: calibration period, right column: validation period). avoid over fitting and helping in generalization and pruning MARS constructed structures. In the backward procedure, the model is simplified in each iteration by deleting one least-important basis function. The importance of basis function is determined using the generalized cross validation (GCV) index which is calculated as follows: MSE train. GCV = ( 1 enp ) 2 (4) n where MSE train. is mean squared error (MSE) of the training data, n is the number of data cases in the training dataset and enp is the effective number of parameters which is calculated as follows: enp = k + c k 1 (5) 2 where k is the number of basis functions in the model and c is the number of knots. At the end of the backward phase, from those best models of each size, one model with lowest GCV value is selected (Jekabsons, 2010a). Different hydrological applications of MARS in hydrological sciences can be found in the literature (Coulibaly and Baldwin, 2005; Buccola and Wood, 2010; Herrera et al., 2010). The ARESLab package provided by Jekabsons (2010) has been utilized in this study for developing MARS module of data-mining downscaling model (DMDM). In this paper, no constraint for constructing MARS is considered. Different revisions of MARS such as FAST MARS (Friedman, 1993.), ARESLab (Jekabsons, 2010) have been developed. In this study, linear basis functions of MARS have been used to achieve consistency with SDSM Model tree MT has a tree-based data structure and modular splitting rules in nonterminal nodes of the tree structure; also the linear regression functions at the leaves of the conceptual mathematical tree. MT is similar to a piecewise linear estimation. It learns efficiently and can successfully tackle physical tasks with high dimensionality up to hundreds of variables. Compared with other DM techniques, model-tree learning is fast and the results

9 SDSM, DATA-MINING METHODS, CLIMATE CHANGE 2569 (a) (b) (c) (d) Figure 7. Khan. rain gauge in Mond basin. (a) Monthly mean, (b) monthly standard deviation, (c) monthly skewness and (d) q q plot (Left column: calibration period, right column: validation period). (a) (b) Figure 8. Daily precipitation of Deh. station in the first year of validation period: (a) dry season and (b) wet season.

10 2570 H. TAVAKOL-DAVANI ET AL. Table 4. Selected predictors for amount model in Deh., Kas. and Khan. stations. Station Predictor Lag Longitude Latitude Coefficient of determination Deh. p5zh r rhum p8_f Kas. p f p u p5_z p8_v r Khan. p8_v r r p8_f Table 5. Number of parameters for occurrence model in Deh. station. Model Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. MLR RR MARS MT No. of Calibration data are interpretable. MT combines a conventional decision tree with possibility of generating linear regression functions at its leaves. M5 is a well-known model-tree paradigm originally developed by Quinlan (1986, 1993). This method is a piecewise linear model and more flexible than pure linear models. So it takes an intermediate position between the linear concepts such as MLR and truly nonlinear framework such as ANNs. In this DM algorithm, the parameter space splits the dataset into subspaces and a linear regression is built for each set. The construction of a MT is similar to that of decision tree. The resulting model can be classified as a modular model containing linear models each being utilized on a particular subset of the input space. This idea is not new. One of the first applications of multiple linear models for describing dynamical system behaviour was in the 1970s by Becker (1976) and Becker and Kundzewicz (1987) in rainfall runoff modelling. This approach has recently found more supporters in climatological and hydrological sciences (Faucher et al., 1999; Li and Sailor, 2000; Xiong et al., 2001; Solomatine and Xue, 2004). It is worthwhile to mention that inherent ability of MT in identifying conditional forms may play an important role in structure recognition of data-sparse natural phenomena such as rainfall events in arid and semi-arid area. Details of the algorithm can be found in Quinlan (1993) and Solomatine and Dulal (2003). A new version of M5 developed by Wang and Witten (1997) namely M5 has been used in this study. M5 algorithm has the following two phases: 1. Growing phase: In this step, the tree is created with one node, then M5 recursively tries to develop the computational tree branches in order to minimize variations of output variables of intra-subset in each branch. This goal is achieved through maximization of standard deviation reduction (SDR) index for each subset. This index is calculated using the following equation: SDR = Std (T ) i T i T Std (T i ) (6) where T is the set of data that reaches to the supposed node, T 1, T 2,... are the subsets results of division of T and Std indicates the standard deviation. Then, a linear regression model is fitted to each subset in the leaf. 2. Pruning phase: At the end of previous phase, a tree with lots of leaves is created which typically over fits the data. In pruning phase, the tree is pruned back from each leaf until an estimate of the expected error that will be experienced at each node cannot be reduced any further. First, the absolute error in each leaf is calculated, then, the following factor is multiplied to the error: F = n + v (7) n v where n is the number of training samples reaching the supposed node and v is the number of parameters of the node. Then, M5 starts to simplify its structure through omitting the terms of the regression models recursively in order to minimize the product of F and the absolute error (Jekabsons, 2010b). M5 algorithm and its various details are thoroughly described by Wang and Witten (1997).

11 SDSM, DATA-MINING METHODS, CLIMATE CHANGE 2571 The M5 lab package in MATLAB environment provided by Jekabsons (2010) is used for MT modelling in this research. In the next section, the proposed downscaling approaches have been described in brief Linear downscaling using DMDM Linear DMDM is a fully automated MATLAB package developed in the current study based on SDSM procedure of downscaling. In DMDM, four linear-based regression models including MLR (which is also used in SDSM package), linear MARS, MT and RR are available for calibrating both occurrence and amount models of meteorological predictands. Auto calibration capability is also available for all five models as well. According to the implemented procedure in SDSM and ASD, DMDM includes two separate subroutines for precipitation occurrence and amount ensemble simulation. It also includes VIF and bias correlation similar to SDSM and ASD. The following steps should be taken to use DMDM model for downscaling precipitation: 1. A uniformly distributed random number in [0, 1] is generated to determine whether precipitation occurs. Similar to SDSM, for each day and in each ensemble, a wet day occurs when the random number is less than or equal to the output of the calibrated occurrence model which can be any of the five MLR, MARS, MT and RR models. 2. Another model (from the set of four available models in DMDM) is calibrated to simulate the precipitation amount using the rainy days data. Similar to SDSM, DMDM can calibrate different conditional models for 12 months of the year. For a day which is identified as a rainy day in the previous step, output of the amount model is calculated. Then, similar to SDSM, a normally distributed number is added to the output to consider the modelling error. This random number is generated using a normal distribution function with zero mean and standard deviation equal to standard error. 3. In the last step, the results from the previous step are compared with a user-defined threshold to avoid generation of irrational results (such as negative values or too small positive values which can interrupt the dry spell analysis). These three steps are also shown in Figure 3. These steps are similar to SDSM and the only major difference between DMDM and SDSM is the four linear models which are available in DMDM. Also in DMDM, possibility of considering different sets of predictors for precipitation occurrence and amount modelling has been provided. The following five steps have been performed in this study for evaluation of the performance of the models: Singularity analysis is carried out in DMDM. In this study, in model calibration phase, NCEP variables are used while in scenario generation phase, GCM variables are exploited. The calibrated models in DMDM must be checked for possible over fitting, extrapolation and singular response modes. In this paper, computed precipitation values which are greater than ten times of the maximum observed, are considered as singular results. Consequently, the model combinations which produce such results are rejected. This threshold is selected based on engineering judgment and can be different for other basins. The absolute relative errors are calculated for mean, standard deviation and skewness in the dry and wet seasons for the model which passes the previous step. Mean of the absolute errors for the two wet and dry seasons estimated in the previous step are calculated which reduces the six error terms to three. Final error (FER) is calculated using Equation (8) assuming 3, 2 and 1 as the relative weights of three error terms of mean error (Error mean ), standard deviation error (Error std. ) and skewness error (Error skw. ), respectively. It must be noted that the error weights in this formula are selected based on engineering judgment and can be changed based on the modeller s judgment. FER = 3Error mean + 2Error std. + Error skw. 6 (8) Figure 9. Amount model for January in Deh. station using MT model (x 1 = p5zh, x 2 = r850, x 3 = rhum and x 4 = p8_f).

12 2572 H. TAVAKOL-DAVANI ET AL. Table 6. Number of parameters for amount model in Deh. station. Model Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. MLR RR MARS MT No. of Calibration data Finally, the model with the least-fer value is selected as the best one. In this study, no transformation has been used as preprocessing for application of DMDM, MLRDM and SDSM. In the next section, modelling results and the advantages and disadvantages of the proposed methodology are described Issue of parsimony As mentioned in the previous sections of the paper, the proposed methodology consists of two distinct modules. Both in occurrence and amount simulations, number of observations versus number of model parameters are one of the most important factors when comparing results of different models. Both MLR and RR methods use fixed number of parameters for a specific number of predictors. But in DM methods, the number of parameters depends on the dataset structure and can be different for the same number of predictors. In this study, in order to address issue of parsimony, Akaike information criterion (AIC) has been used. The AIC is a measure of goodness of fit involving not only the weighted errors, but the number of observations, n, and the number of model parameters, p (Tong, 1983, p. 135): ( ) Rm AIC = n ln + 2 p (9) n The smaller the AIC, the better is the fit. R m is loss function. It should be noted that, in the literature, different loss functions have been used for estimating AIC. Examples can be found in Ljung (1999) and Olea (2006). In this study, FER (Equation (8)) is considered as loss function in Equation 10. In the next section, results and discussion of results are presented. 4. Results and discussion As mentioned earlier, since the proposed occurrence and amount models are linear, stepwise regression has been adopted to select the best combination of predictors. SDSM package uses the same predictors for occurrence and amount of precipitation, but in this study, separate sets of predictors have been considered for modelling occurrence and amount of precipitation. For example, the differences between provided degree of freedom in using different dataset for occurrence and amount simulation have been presented in Figure 4 in Deh. station. As shown in this figure, the results of MLRDM (possibility of using different datasets for occurrence and amount modelling), both in the calibration and validation periods is slightly better than SDSM (same datasets for occurrence and amount modelling). MLRDM performance has been slightly better in producing monthly mean, standard deviation and skewness both in the calibration and validation periods. Also, q q plots of observed versus computed precipitation have confirmed the better consistency of MLRDM results distribution with the observed records particularly in extreme values of the calibration period. But in validation period, this superiority is not significant. It should be noted that same comparison for all 12 stations has been carried out and the results have shown minor improvements of SDSM results when separate sets of predictors are used for occurrence and amount modelling. Although the improvement of SDSM results can be considered minor, since the procedure of predictor selection is automated in MLRDM, it can be considered as a more reliable downscaling tool compared with SDSM. In the rest of this section, MLRDM results refer to the results of the modified SDSM model which is reproduced in DMDM package, so that it can work with different sets of predictors for occurrence and amount. Tables 2 and 3 show the selected predictors for occurrence and amount models in Deh., Kas. and Khan. stations. In these tables, coefficients of determination for the selected predictands and predictors have been presented. It should be noted that all listed predictors are significant at p < Selected predictors are mostly relative humidity and zonal velocity in different geopotential heights without any time lags. Generally, they are scattered in all of the nine boxes shown in Figure 1. In the current research, 100 ensembles of precipitation have been produced in each station. The mean results of ensembles in the three selected stations, namely Deh., Kas. and Khan., are shown in Figures 5 7 for calibration and validation periods. Both MRLDM and DMDM performances have been acceptable in simulating the monthly mean for the calibration and validation periods while for the selected three stations shown in Figures 5 7, DMDM does not show significant superiority over MLRDM. Figures 5 7 also show that the performance of DMDM has been better than MLRDM in simulating the monthly standard deviations. Skewness has been simulated better by DMDM in the calibration and validation periods as well.

13 SDSM, DATA-MINING METHODS, CLIMATE CHANGE 2573 Table 7. DMDM selection for Deh. (based on the validation period). Model combination Dry season Wet season Selection No Occ. model Amo. model Mean Std. Skw. Mean Std. Skw. Singularity evaluation FER 1 a MLR MLR Pass b MLR RR Pass MLR MARS Pass MLR MT Pass b RR MLR Pass b RR RR Pass RR MARS Pass RR MT Pass MARS MLR Pass MARS RR Pass MARS MARS Pass MARS MT Pass MT MLR Pass MT RR Pass MT MARS Pass MT MT Pass Observation Selected combination is marked by grey colour. a This model is equal to MLRDM or SDSM model. b This model is equal to ASD model. Table 8. DMDM best combination selection results in 12 stations (base on the validation period). No. Station Occurrence model Amount model 1 Deh. MT MLR 2 Del. MT MT 3 Kho. MT MARS 4 Kha. MARS MLR 5 Far. MT MT 6 Ras. MT MT 7 Kas. MT MT 8 Shan. MT MT 9 Shap. MT MARS 10 Shoo. MARS MT 11 Ars. MT MT 12 Khan. MT MLR Figure 8 shows daily downscaled rainfalls of Deh. station in dry and wet seasons. In the dry season (Figure 8(a)), the performance of both models has been almost the same. As can be seen in Figure 8(b), DMDM has performed better than MLRDM in modelling and presenting the high daily values in the wet season. DMDM performance in the validation period may be sensitive to the over-fitting effect, so Tables 4 and 5 have been presented to show the number of parameters for occurrence and amount model in Deh. station. For the occurrence models, the number of parameters has been significantly higher than MLR and RR models while there have been sufficient samples for calibration. In other words, the over-fitting has not happened. In the case of the amount model, the number of parameters has not been increased significantly compared with MLR and RR models. It is also worth mentioning that MLR and RR models have been over-fitted for the months of June, August and September while MARS and MT showed more flexibility in these months by using a single parameter. In other words, the proposed models develop intelligently based on the available calibration data. Amount model structures for January in Deh. station are demonstrated in Equations (10) (12) and Figure 9. MLR: R = x x x x 4 (10) RR : R = 0.23x x x x 4 (11) MARS:B 1 = max(0, x 2 ) B 2 = max(0, x ) B 3 = B 2 max(0, x 4 ) R = B B B 3 (12) where R is the rainfall (mm) and x 1, x 2, x 3 and x 4 are p5zh, r850, rhum and p8_f as shown in Table 3, respectively. BF i (i = 1, 2, 3) are the calibrated basic functions of MARS method. Due to the large number of parameters in MARS and MT for occurrence model, they are not shown in this paper. In Table 6, statistical evaluation of the combined different models (proposed in this study and used in SDSM and ASD packages for occurrence and amount) for Deh. station has been presented as an example. All models have passed the singularity test. The model which has had the least values of the errors (estimated FER) is shown in grey in the Table 7. This model uses MT and MLR in modelling occurrence and amount, respectively. It can be seen in Table 6, choosing MT for modelling amount and occurrence also leads to relatively good results.

14 2574 H. TAVAKOL-DAVANI ET AL. Table 9. Comparison of MLRDM and DMDM performance in all studied station. Calibration Validation Dry season Wet season Dry season Wet season Station Model Mean Std. Skw. Mean Std. Skw. Mean Std. Skw. Mean Std. Skw. FER AIC Deh. MLRDM DMDM Obs Del. MLRDM DMDM Obs Kho. MLRDM DMDM Obs Kha. MLRDM DMDM Obs Far. MLRDM DMDM Obs Ras. MLRDM DMDM Obs Kas. MLRDM DMDM Obs Shan. MLRDM DMDM Obs Shap. MLRDM DMDM Obs Shoo. MLRDM DMDM Obs Ars. MLRDM DMDM Obs Khan. MLRDM DMDM Obs

15 SDSM, DATA-MINING METHODS, CLIMATE CHANGE 2575 (a) (b) (c) Figure 10. Five-year moving average for A2 and B2 scenarios in (a) Deh., (b) Kas. and (c) Khan. stations. Table 8 presents DMDM best combinations in 12 studied stations. According to this table, MT was the most selected model both for occurrence and amount modelling. Then, MARS and MLR achieved the second and third ranks, respectively. It is worthwhile to mention that in Kas. station best combination, according to FER criteria, is MT-MARS (MT for occurrence and MARS for amount modelling). But this combination produces singular rainfall values more than the threshold defined. So, based on singularity analysis MT-MT has been reported as the best combination in this table. Overall, the results have shown that MT can provide a useful diagnostic tool for identifying the key determinants of precipitation amount at a given site, including threshold states in the atmosphere, or combinations of conditions that generate extreme events. Table 9 shows the comparison between MLRDM and DMDM performances in all studied station using the evaluation procedure in Equation (3). In this table, mean values, standard deviation (std.), skewness (skw.) of the simulated and observed rainfall for dry and wet seasons for both calibration and validation datasets are presented. FER and AIC values for comparing the results of DMDM and MLRDM are shown in this table for the validation period as well. Both FER and AIC values have been shown in this table show superiority of DMDM over MLRDM. To evaluate the climate change impacts on the studied stations, SRES A2 and B2 scenarios have been considered for the period. In Figure 10, 5-year moving average of precipitation is presented for A2 and B2 scenarios in Deh., Kas. and Khan. stations. The results of downscaling using MLRDM and DMDM are generally different because of the different values of bias correction and VIF factors calculated in the calibration period by these models. As it can be seen in Figure 10, except for Khan. station, the proposed models produced significantly different results compared with MLRDM model which shows high level of uncertainty associated

16 2576 (a) 2000 Precipitation (mm) H. TAVAKOL-DAVANI ET AL th Percentile (MLRDM) 95th Percentile(MLRDM) 5th Percentile (DMDM) 95th Percentile (DMDM) Maximum (Calibration period) Minimum (Calibration period) Year (b) Precipitation (mm) th Percentile (MLRDM) 95th Percentile (MLRDM) 5th Percentile (DMDM) 95th Percentile (DMDM) Maximum (Calibration period) Minimum (Calibration period) Year (c) Precipitation (mm) th Percentile (MLRDM) 95th Percentile (MLRDM) 5th Percentile (DMDM) 95th Percentile (DMDM) Maximum (Calibration period) Minimum (Calibration period) Year Figure 11. Annual precipitation for A2 and B2 scenarios and [5, 95]th ensemble percentiles for DMDM and MLRDM in Deh., Kas. and Khan. Stations. (a) Deh. station (A2), (b) Deh. station (B2), (c) Kas. station (A2), (d) Kas. station (B2), (e) Khan. station (A2) and (f) Khan. station (B2). with downscaling modelling procedure in future climate change projections. To assess output uncertainty of the downscaled scenarios using DMDM and MLRDM, annual precipitation of 5th and 95th percentiles of ensemble results for the both scenarios in Deh., Kas. and Khan. stations and their minimum and maximum in their calibration periods are presented. As it can be seen in this Figure 11, the uncertainty bound of DMDM results is narrower than SDSM which shows less difficulty for the decision makers. 5. Conclusions Presented results in this paper depict better performance of the proposed model in preserving historical monthly 2012 Royal Meteorological Society mean precipitation and closer estimation of historical monthly mean standard deviation and skewness compared with SDSM and ASD. Stepwise regression based on quantile evaluation is used here for meteorological feature (input) selection. DMDM has proved to be a suitable alternative for daily precipitation downscaling and it can be also further developed for downscaling of other conditional and unconditional meteorological variables. Two proposed statistical norms, FER and AIC, have been efficiently used to evaluate the performances of the proposed downscaling models. Using FER and AIC allows modeller to perform evaluations just based on model errors with and without incorporating issue of parsimony. Results of precipitation downscaling in 12 rain gauge stations scattered over three different basins in Iran Int. J. Climatol. 33: (2013)

17 2577 SDSM, DATA-MINING METHODS, CLIMATE CHANGE (d) Precipitation (mm) th Percentile (MLRDM) 95th Percentile (MLRDM) 5th Percentile (DMDM) 95th Percentile (DMDM) Maximum (Calibration period) Minimum (Calibration period) Year (e) Precipitation (mm) th Percentile (MLRDM) 95th Percentile(MLRDM) 5th Percentile (DMDM) 95th Percentile (DMDM) Maximum (Calibration period) Minimum (Calibration period) Year (f) Precipitation (mm) th Percentile (MLRDM) 95th Percentile (MLRDM) 5th Percentile(DMDM) 95th Percentile (DMDM) Maximum (Calibration period) Minimum (Calibration period) Year Figure 11. Continued. have shown that MT has been selected 11 and 8 times (from 12 models) for occurrence and amount modelling, respectively. While the total number of MT parameters is higher than SDSM and ASD, no over-fitting has occurred. On basis of the FER values estimated in this study, DMDM has outperformed MLRDM in the northern basins with wet climate (Sefidrood and Mordabanzali). The improvement in the performance of DMDM compared with MLRDM in the semi-arid basins in the case study of this research has not been as significant as its performance in the aforementioned wet basins. It should be noted that both DMDM and SDSM can be used for semi-arid/data sparse regions while application of SDSM might be limited to seasonal or annual time scales because of possibility of over fitting or ill-condition occurrence. Analysis of daily time series of downscaled precipitation values in the 12 rain gauge stations used in the case 2012 Royal Meteorological Society study of this research shows that except for Del. and Khan. stations, the differences between the results of DMDM and MLRDM have been higher than the differences between the values estimated for A2 and B2 scenarios by these models. This finding shows high uncertainty associated with downscaling techniques utilized in the climate change related studies. While the need for careful selection of proper climate change scenarios has been emphasized in the previous studies, the results of this study have shown that uncertainty associated with downscaling technique is at least as important as scenario itself. Future work might consider the SDSM platform coupled with nonlinear methods such as SVM, adaptive neuro-fuzzy inference systems (ANFIS) both in regression (=amount) and classification (=occurrence) mode. It must be noted that feature selection in this type of modelling must be based on nonlinear evaluation, because a linear correlation is not valid. This issue Int. J. Climatol. 33: (2013)