Development of statistical downscaling model based on Volterra series realization, principal components and ridge regression

Impacts of the global climate change in hydrology and water resources are accessed by downscaling of local daily rainfall from large-scale climate variables. This study developed a statistical downscaling model based on the Volterra series, principal components and ridge regression. This model is known, hereafter as SDCRR. The proposed model is applied at four different stations of the Manawatu River basin, in the North Island of New Zealand to downscale daily rainfall. The large-scale climate variables from the National Centers for Environmental Predictions (NCEP) reanalysis data are used in the present study to obtain with the wide range (WR) and the restricted range (RR) of predictors. The developed SDCRR model incorporated the climate change signals sufficiently by working with WR predictors. Further, principal component analysis (PC) was applied to the set of WR predictors, which were also used as the orthogonal filter in the ridge regression model to deal with the multi-collinearity. The ridge regression coefficients determined were less sensitive to random errors, and were capable of reducing the mean square error between the observed and the simulated daily precipitation data. Thus, the combined application of principal component analysis (PCA) and ridge regression improved the performance of the model. This combination is steady enough to capture appropriate information from predictors of the region. The performance of the SDCRR model is compared with that of the widely used statistical downscaling model (SDSM). The results of the study show the SDCRR model has better performance than the SDSM.


Introduction
The economic development of many countries is mostly dependent on water resources to supply the water demand needs for agriculture, tourism, energy, ecological flow, and domestic (Ghavidelfar et al. 2017). The importance of analyzing the consequences of global climate change on the regional water availability is becoming the greatest challenge for the current and the future world. Particularly, the climate system is related to the hydrological cycle, and therefore any changes in the climate system make differences to hydrological variables such as rainfall (Karamouz et al. 2011;Pervez and Henebry 2014a). New Zealand Ministry for the Environment (MfE) noted that the consequences of climate change increased flooding in New Zealand, which is a subsequent cause of intense rainfall events (Robertson and Sowden 2020;Manning et al. 2015;Fuller and Heerdegen 2005). Significantly, the prediction of the future rainfall under a changing climate combined with hydrologic modelling plays a crucial role in providing the basis for obtaining quantitatively the future extreme event return periods. Global climate models (GCMs) are the ideal tools developed to model the path of future climate change at global scales (Liu et al. 2019). Although GCMs provide a sensible representation of global and continental-scale processes, they are unable to provide local sub-grid-scale features and dynamics because they operate at large spatial scales (> 10 4 km 2 ). Thus, these powerful climate models (GCMs) are used at coarser scale, to obtain future projections due to climate change and could not be used to perform the hydrological study at low resolution (Chim et al. 2020). Furthermore, the coarse resolution and poor representation of hydrological 1 3 variables in global climate models can be improved by downscaling (Ali et al. 2019).
Downscaling is a process of rearranging the large-scale circulation impact to local parameters in a form suitable for different types of climate change studies. Based on their working principles, the downscaling techniques are broadly categorized as dynamical and statistical techniques. A detailed discussion about both techniques can be found in Wilby et al. (2004) and Christensen et al. (2007). Specifically, many scholars applied statistical downscaling methods to examine the climate change impacts on hydrological processes at the regional and the local scales (Chim et al. 2020). The statistical downscaling (SD) techniques have gained significant progress in the establishment of new statistical downscaling classes, namely, (i) weather generator, (ii) weather typing, and (iii) multiple regression. The third classification of statistical downscaling techniques is used to derive statistical relationships between predictors (large-scale variables) and the predictand (small-scale variable). Particularly, many relative studies (Khan et al. 2006;Dibike and Coulibaly 2005;Wilby et al. 1998) noted that the statistical downscaling model (SDSM) is widely applied to downscale rainfall and other hydrological parameters. Additionally, Hashmi et al. (2009b) applied Long Ashton Research Station Weather Generator (LARS-WG) in Auckland, New Zealand to downscale changes in extreme rainfall events. Although, LARS-WG is capable of simulating extreme monthly rainfall events, but the future projections of daily rainfall were not addressed. Researchers noticed that linear or multi-regression-based methods might not perform well in downscaling due to the complex relationship between predictors (climate variables) and predictand (rainfall). Therefore, Hashmi et al. (2011) used Gene Expression Programming (GEP) -a non-linear multiple regression model for rainfall downscaling in the Clutha watershed of New Zealand. The results emphasized that the GEP is an efficient tool to downscale daily rainfall, but its validity for extreme event projections is the limitation of the study. Specifically, Munawar et al. (2022) also compared the performance of the two downscaling techniques, SDSM and LARS-WG. Based on the performance, SDSM was proved to be more effective than LARS-WG in context of temperature downscaling but underperformed for the rainfall downscaling. Tahir et al. (2018) used SDSM to identify the increasing pattern in rainfall in the Limbang river basin, Malaysia. However, Hasan et al. (2018) also used SDSM to identify the decreasing trend of rainfall and an increasing trend of maximum and minimum temperature. On the other hand, researchers also found that the SDSM model underperformed in the case of daily rainfall projection performance (Pervez and Henebry 2014;Chim et al. 2020).
Downscaling of rainfall is more challenging than other variables as rainfall has high spatial and temporal variability and holds complex non-linear relationship with other atmospheric variables (Hu et al. 2019). It is identified by Liu et al. (2019) that many studies used weather generators and stochastic models to downscale daily rainfall but these models could not perform the statistical verification with the daily observations. The researchers further mentioned that many studies compared the stochastic output such as quantiles, monthly means, wet-dry spell lengths, temporal autocorrelation, standard deviation, and spatial dependence on the basis of their statistical characteristics whereas the actual performance on daily variations is addressed by very few researchers. Thus, in this, the focus is mainly on the deterministic part (daily performance) instead of the noisy part (stochastic) of the output. The daily rainfall downscaling performance is affected by the chosen predictor set. Also, in this context, Mandal et al. (2016) noted that the statistical downscaling methods often suffers from the multi-collinearity of the predictors, which could be reduced by the principal component analysis (PCA). Sahriman et al. (2014) applied principal component regression (PCR) to improve the rainfall prediction. Also, Loganathan and Mahindrakar (2021) did climate change assessment over the Cauvery river basin by applying an improvised PCR-based downscaling approach to the dataset of daily rainfall and temperature. The main purpose of PCA is to distinguish between the correlated and uncorrelated patterns among the predictor field anomalies (Salvi et al. 2016). Basically, PCA is the simplest way of conserving variance (the statistical properties), while reducing the dimensionality of the data set. PCA uses the linear transformation of the observed dataset into orthogonal variables, which are linearly uncorrelated and these orthogonal variables are called principal components (PCs). The derived PCs are used as the orthogonal filters in the regression model, which is mainly applied to reduce the computational complexity. The application of orthogonal filters on the input (predictors) is represented by an expansion, which represents a realization of the Volterra series (Watanabe 1986). Similar to the Taylor series, the Volterra series is the polynomial representation of nonlinear systems and researchers often applied the Volterra series to model rainfall-runoff systems (Chou 2007). Sehgal et al. (2018) modeled the climate inputs by applying the Volterra model, which is an alternative to the machine learning approaches. Additionally, Nikolaou and Mantha (2000) demonstrated that the Volterra model can be parameterized into the standard linear regression form. However, severe multi-collinearity is the problem with the standard linear regression because it increases the variance of the coefficient estimates. The increased variance makes the coefficient estimates very sensitive to the minor changes in the data, which produces unstable coefficients. Thus, it is beneficial to use ridge regression to reduce the variance of regression estimates and to shrink the parameter estimation coefficients. Ihwal et al. (2020) used kernel ridge regression to perform statistical downscaling and predict rainfall. Further, Herawati et al. (2018) applied ridge regression to obtain more accurate regression coefficients in comparison to ordinary least square (OLS).
In a previous study, Ghosh and Mujumdar (2006) developed a downscaling methodology to study future droughts (extreme events) under climate change by downscaling monthly rainfall projections with 90% accuracy. The successful application of the combination of techniques, namely, multiple linear regression (MLR), principal component analysis (PCA), c-means clustering, and seasonality will encourage its use in improving the accuracy of daily rainfall downscaling. In the case of MLR, the ordinary least square estimations of regression coefficients may be unstable due to the collinearity among climate predictors. However, the ridge regression is an alternative to the standard MLR in case of the existence of the near collinearity among predictors (Vigneau et al. 1997). Although, there has been extensive application of MLR and PCA in the area of hydrological modelling, there is a very limited application of PCA along with ridge regression in the area of statistical downscaling of daily rainfall. Thus, the present study is to develop a statistical downscaling model based on the application of orthogonal filter in the regression model, which will produce the explanatory variables, which is the realization of the Volterra series expansions. Further, the downscaling model uses the ridge regression on the new set of explanatory variables as a new alternative method to produce stable coefficient estimates, which are capable of delivering more accurate rainfall projections.

Study area and data used
The study area of the present study is Manawatu catchment, which is located in the lower half of the North Island of New Zealand. Figure 1 shows the location of the catchment. The catchment has a warm and temperate climate zone and it observes variability in the seasonal rainfall, and the variation between annual rainfall is from 900 to 1500 mm. This study used the daily rainfall data of four rainfall gauging stations, namely, Palmerston North, Marton, Opiki, and Te Rehunga. These rainfall gauging stations are situated in the vicinity of the Manawatu catchment boundary as shown in Fig. 1. Similar to previous downscaling studies, this study used the National Centers for Environmental Prediction (NCEP) reanalysis data to obtain the large-scale predictor variables used in statistical downscaling (Shashikanth et al. 2014;Yang et al. 2018). The large-scale reanalysis datasets were obtained from the second generation Canadian Earth System Mode (CanESM2) project. The CanESM2 dataset is being widely used in hydrological and climate change studies (Chim et al. 2020); thus, this study also used the particular GCM because of its global coverage. These datasets are freely available and can be obtained using the following link: http:// clima te-scena rios. canada. ca/? page= pred-canes m2. Chim et al. (2020) mentioned that many studies such as Pholkern et al. (2018), Trang et al. (2017), and Hoan et al. (2020), have obtained successful downscaling results with the CanESM2 dataset. Along with NCEP data, which extends from 1961 to 2005, the CanESM2 dataset also has three RCPs. Each RCP scenario consists of 26 large-scale predictor variables, which covers the period from 2006 to 2100. All these 26 predictor variables are given in Table 1.

Tools and techniques
In climate change downscaling studies, the assumption of stationarity in the predictor and predictand relationship requires robust relationship to downscale rainfall at the daily time scale. However, the dependence of predictand over several independent predictors is most widely modelled using multiple linear regression (MLR). As a further consequence of working with MLR, the construction of the downscaling model comprises two steps. The first step involves screening of the most suitable predictor variables. The predictor variables are mostly mutually correlated thus, screening helps in filtering the common information shared by them. The chosen predictor variables should have the highest predictive power with appropriate information to form stable statistical relationships between predictors and predictand. The second step focuses on the development of the statistical relationship between the large-scale variables (predictors) and localscale variable (predictand) (Chim et al. 2020). The general process of predictor variable selection is subjective and the predictor variable types selected vary regionally as well as seasonally (Timbal et al. 2008). Further, in the context of climate change studies, the appropriate predictor variables are those that capture the effect of global warning (Wilby et al. 1998). Hewitson and Crane (1996) stated that for the fulfillment of the successful downscaling, the chosen predictors should meet the following criteria: • GCMs should reproduce large-scale predictors appropriately. • Predictor variables and predictand should have valid relationship even outside the calibration period. • Selected predictor variables should sufficiently incorporate climate change signals. Wetterhall (2005) mentioned that it is most difficult for the chosen set of predictor variables to meet all the three criteria. Furthermore, Wetterhall (2005) noted that there is a choice of working with either the 'Restricted Range' (RR) 1 3 of predictors to meet one or two criteria or with the 'Wide Range' (WR) of predictors to cover all the three aspect of successful downscaling. The study classified the predictors as restricted range (RR) for the case when few predictors are screened from the whole set of predictors, whereas the WR is for the case when the whole set of predictors are chosen to perform the downscaling studies.
Further, several mathematical tools and techniques like principal component analysis (PCA), Volterra series realization, and ridge regression are employed to develop the proposed downscaling models, SDCRR, and compare it's performance with the base model SDSM. The brief introduction of the mentioned techniques is provided below:

Principal component analysis
PCA provides a useful technique for dealing with the problem of multi-collinearity. Also, the inadequacy of model results can be avoided by maintaining the patterns and internal variability of the data (Salvi et al. 2013). Additionally, PCA is a powerful statistical tool, which identifies the pattern of the multi-collinear data and reduces the dimension of problem without losing the internal variability of the original data (Rahman and Rahman 2020;Mandal et al. 2016).

Derivation of Volterra series with principal components
The most widely used application of PCA is to compress the data. PCA produces the set of uncorrelated variables by transforming the correlated variables and reduces the size of the original correlated variables into a smaller number of variables called PCs (Loganathan and Mahindrakar 2021). In the context of regression analysis, the PCA combines the input predictor variables in such a way that it retains the most valuable parts of all of the predictor variables. The added benefit after PCA is that the new variables (PCs) are independent of each another. The present study applied the orthogonal transformation on the original climate predictor variables by utilizing the linear combination of basic functional, which is the realization of the Volterra series expansions. The above mentioned idea is described by Watanabe (1986) as "the sum of powers of outputs from the orthogonal filters is considered as the model, and the coefficients of the terms in the model are determined so that the mean-square-error between the outputs of the system to be measured and the model is minimized". The power series expansion of the continuous functions derives the Volterra series. The three kinds of representation are obtained for the kth power functional by: (i) the linear combination of basic functionals constructed from the product of linear functionals, (ii) the product sum of the non-symmetrical kernels, and (iii) the product sum of symmetrical kernels. The third kind of series is known as the Volterra series. However, the other two are the realization of Volterra series expansion, which is equivalent to the expansion of functionals using Wiener's orthogonal series of functionals. Thus, the realization of expansion is equivalent to Volterra series expansion. Watanabe (1986) mentioned that the functionals derived from the orthogonality plays vital role in reducing the computational complexity. The present study applied the orthogonal filters derived as PCs, to obtain the functionals, which are equivalent to the realization of Volterra series expansion. The matrix of PCs, which are orthogonal to each other is multiplied by the chosen set of climate predictors as given below: where Z represents the transformed matrix, X is the matrix of climate predictor variables The added advantage in this transformation is that the columns of Z matrix are now linearly independent of each other. The internal variability of the original data set is maintained by selecting PCs that explains more than the average amount of variance (Kannan and Ghosh 2013). Hence, the use of (1) Z = ∅X PCA reduces the multi-collinearity and the dimensionality problems by working as the orthogonal filters.

Ridge regression
Ridge regression is a mathematical technique used to deal with the input data multi-collinearity. The multi-collinearity produces a large variance, and as a result, the least unbiased parameter estimates are far from the true value. Thus, in this context, the first step in ridge regression is to do transformation of predictand and predictor variables through centralization and scaling procedures. The centralization and scaling procedures are done through the pre-processing of the datasets to reduce the numerical noise, which occurred as a cause of the variance. The ridge estimator reduces the standard error by introducing bias to the regression estimates.
The ridge regression parameter estimates are obtained by minimizing the sum of square error (SSE) as given by the following equation: where Y is the dependent variable, X is the matrix of independent variables, is the ridge regression coefficient, and is the ridge parameter. There are two important properties of ridge regression over the linear regression. The most important one is that it penalizes the β estimate but not all the feature's estimate. If the estimates of β values are very large, then the first part of Eq. (2) and the SSE is minimized, but the second part which is the penalty term is increased. Alternatively, if value of β estimates is small, the penalty term in the above equation is minimized, and the SSE term will increase due to poor generalization. Accordingly, the ridge regression chooses the feature's estimate β to penalize in such a way that less influential features undergo more penalization.
Development of the downscaling model is dependent on the statistical relationship formed between the predictor variables and the predictand. Although the predictor variables are independent but they are mutually dependent which increases the deviation between observed and simulated data. In ridge regression, a small term λ (ridge parameter) is added along the diagonals of X T X. It makes the X T X + λI matrix (Eq. (3)) invertible (all the columns are linearly independent) (Jadhav et al. 2018). The small and positive values of improve the conditioning of the problem and reduce the variance of the regress parameter estimates. Therefore, the mutually dependent predictors are made independent, and thus increases the precisions of the regression coefficient estimates as given by the following equation:

Statistical downscaling model (SDSM)
The statistical downscaling model (SDSM 4.2) is described as a hybrid of the multiple linear regression methods and the stochastic weather generator, and it has a better ability of capturing the inter-annual variability (Wilby and Dawson 2007). The SDSM is the combination of a stochastic weather generator and a transfer function model (Wilby et al. 2002). The downscaling process needs two types of daily data, the local predictand of interest and the large-scale predictor (NCEP) variables. Initially, the predictand (rainfall) data of the selected station are correlated with the NCEP predictor variables within SDSM. The chosen predictor variable set is used in model calibration and validation. Specifically, the best statistical agreement between predictor variables and predictand is obtained by performing cross-correlation, time shift, and transformation processes.
In the case of downscaling daily rainfall, a direct relationship is developed between the predictand, Y i and selected predictor variables, X ij , as: where k is a transformation (fourth root, inverse normal or logarithmic), j is the regression coefficient, and e i is the model error.
The detailed description of technical information of SDSM is available in Wilby and Dawson (2007). The SDSM is the most widely used model for climate change impact studies; therefore, it is used in the present study as a base model to compare the performance of the proposed SDCRR model. The present study used SDSM to downscale daily rainfall of the Palmerston North, Marton, Opiki, and TeRehunga stations located in and near the Manawatu catchment (Fig. 1). The appropriate predictors were screened to construct the statistical regression model between predictand and predictor variables. The predictors were selected based on the correlation and stepwise regression analysis within the SDSM. Additionally, the statistical relationship in SDSM was developed only with the restricted range of predictors because the wide range of predictors will increase the dimensionality problem. The model used NCEP data to generate the current daily rainfall series. While using SDSM, the parameters such as event threshold was set at 0.3 and skewed nature of rainfall was considered by applying fourth root transformation (Chen et al. 2012;Wilby and Dawson 2007).

Model development
Using the concepts mentioned in the above sections (Tools and Techniques), a statistical downscaling method (SDCRR) (Fig. 2) is developed to downscale future rainfall at daily time interval for the Palmerston North, Marton, Opiki, and TeRehunga rainfall gauging stations. The purpose of this proposed model is to obtain future rainfall projections having perturbations due to the climate change impact. Since the earlier study (Ghosh and Mujumdar 2006) related to a statistical downscaling model to project future rainfall using PCA, fuzzy clustering, and MLR to downscale monthly rainfall projections resulted with very high accuracy (90%), in this study, PCA and ridge regression was proposed to develop the model to minimize error between observed and simulated data of daily rainfall. In the statistical downscaling, an empirical relationship is established between the large-scale atmospheric (NCEP) predictor variables and the regional climate predictand (rainfall) variables, to project future rainfall (Sehgal et al. 2018;Ghosh and Mujumdar 2006). This relationship is assumed to be stable, to provide future projections with optimum accuracy. Thus, establishing a stable relationship is the major challenge of any statistical downscaling model; thus, the overall operation of SDCRR involves three major steps: (i) Screening of the NCEP (large scale) climate predictor variables: The predictor variables are mutually correlated; therefore, screening of the most suitable predictor variables is an important step in the downscaling model. This particular step helps in removing the redundant information from the model. (ii) PCA: The second major step is to work with PCs.
The PCA is applied to the screened predictor variables. This particular step reduces the dimensionality problem. (iii) Ridge regression analysis with Volterra series: The selected PCs are used in developing the orthogonal transformation on the standardized large-scale climate predictor variables. The matrix with orthogonally transformed predictor variables is used to regress the observed rainfall. The ridge regression analysis is done to obtain more stable regression coefficients. In this step, the ridge regression analysis minimizes the variance by introducing bias, which further strengthens the possibility of the accurate estimation of the regression coefficients by addressing the multi-collinearity.
The detailed description of each step used for the generation of daily rainfall projections is explained in the following sections: Step 1: Standardization As shown in Fig. 2, the chosen predictor variables must be pre-processed before they are used in the downscaling model, which includes the standardization and centering of the data. Standardization is performed to reduce the systematic biases in the mean and variance of NCEP data (Wilby et al. 2004). The usual process of obtaining the standardized data is by first subtracting the mean and then dividing by the standard deviation of the predictor variable. Furthermore, centering is the process of performing linear rescaling of a variable (Luckett et al. 2012).
Step 2: Screening of predictors The most crucial step in downscaling is the screening of predictor variable. Generally, the potential predictor variables are mutually correlated and thus contains redundant information. Therefore, it is essential to filter the common information content shared among the predictor variables. The most widely used approach of establishing the relationship between the predictor variables and the predictand is by performing the Pearson correlation analysis. Thus, correlation analysis was performed on the WR of 26 atmospheric variables (reanalysis data) to screen the most suitable or relevant predictors set. The suitable predictor variables were selected by building the relationship between observed rainfall (predictand) data and NCEP (predictors) data of the period 1961-2000: Furthermore, the data were divided into two slices. In the first slice , 30 years of data were used for model calibration whereas in the second slice (1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000), 10 years of data were used for model validation. In particular, the historical data of the period (1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005) obtained from the CanESM2 GCM were also used to affirm the model efficacy and further to substantiate the statistical relationship developed during the calibration process. Moreover, the rainfall data of 2005-2020 (for Palmerston), 2005-2019 (for Opiki and Te Rehunga), and 2005-2016 (for Marton) were used in the present study to represent the current climate. The current climate or baseline data were used to compare the future projections obtained for the 2050s (2031-2060).
Although the predictor variable screening is the crucial step, Wetterhall (2005) mentioned that there is always a difference in opinion that the chosen predictors are capable of performing successful downscaling. Consequently, the efficiency of the statistical downscaling model was evaluated with two different predictor sets. The first set of predictor variables was named WR as mentioned in one of the above sections (Tools and Techniques), which means screening most of the predictor variables from the climate data set to execute successful downscaling. The minimum criteria for the predictors to qualify as WR predictor variables are based on the correlation coefficient value of predictor variables and predictand relationship. The minimum satisfactory correlation coefficient value accepted in the present study is based on correlation coefficient (r ≥ 0.15 ) as used in previous studies (Akhter et al. 2019;Borges et al. 2017). The best correlation between predictor variables and predictand was obtained by lagging NCEP predictors by one day as shown in Fig. 3. The figure shows that the correlation coefficient values for most of the lagged predictors are more than the minimum satisfactory limit. Similarly, the second set of predictor variables was named as RR and the final set of these predictor variables was chosen by carrying out the stepwise regression analysis on the set of WR predictor variables. Furthermore, the two different chosen sets of predictor variables were used to compare their relative performance in the proposed statistical downscaling model, SDCRR.
Step 3: Principal component analysis The goal of the proposed SDCRR downscaling model is to evaluate the effects of climate change on the regional rainfall values at the daily time scale. In this context, the most suitable predictor choice is eventually governed by the availability and the method chosen. The present study evaluated the performance of downscaling model by choosing two different predictor variable sets (section: Screening of predictors). The chosen sets were named as the WR and the RR predictors. The WR predictors had benefit of producing more satisfactory results; therefore, the study executed the downscaling studies with 'Wide Range' predictors as shown in Fig. 2. After that, the WR predictors were chosen by satisfying the minimum correlation requirement (r ≥ 0.15) as mentioned in the above sub-section Screening of Predictors (Tools and techniques). Then PCA was applied to the chosen WR predictor set, PCA is used to convert the predictor variables into uncorrelated PCs. The PCs are classified in decreasing order according to the percentage of the variance explained by each PC.
Significantly, the variation of the original data can be explained by the first few PCs and the number of PCs retained or discarded from the model will depend on the choice of the modeler. Although there is no specific rule for selecting the number of PCs, Kannan and Ghosh (2013) adopted Kaiser's rule for the selection of PCs. The basis of the Kaiser rule is to drop all those components, which explains variance of less than 1% in the total variance in the data set. Therefore, the present study selected all those PCs, which served more than 1% of the variance, which altogether served 96% of total variance.
Step 4: Orthogonal transformation The chosen predictor set was multiplied by the number of orthogonal components, which were derived from the 'c' number of PCs as given by the following equation: where PRED is the matrix of the chosen set of predictors.
The sub-section Derivation of Volterra series with principal components (Tools and techniques) describes that the matrix derived in Eq. (5) is the realization of the Volterra series expansion as it is expanded from the linear combination of the basic functionals derived from the orthogonal set of PCs.
Step 5: Obtained ridge regression coefficients The PCs added to the model with small variances may produce unreasonably high variance regression coefficients (Vigneau et al. 1997). In general, adding a small ridge parameter value has significant influence on the PCs with smaller value of variance, thereby retaining the importance of all PCs. The ridge parameter ( ) does not give the importance only to the PCs with higher variances but also tackles the degeneracy caused by the PCs with smaller variance. Finally, the regression coefficients were obtained by applying the ridge regression as in section Ridge regression and can be expressed by the following equation: The transformed matrix formed in Eq. (5) and the ridge regression coefficients obtained in Eq. (6) are used together to regress the observed data Y i and the final simulated rainfall values are as given below: where R t is the simulated rainfall values given by the regression equation formed using the PCs and the ridge regression coefficients.

Performance indices
The performance indices used to access the efficiency of the downscaling model were the normalized root mean square error (NRMSE), the normalized mean absolute error (NMAE) (Sehgal et al. 2018), and the coefficient of determination ( R 2 ).
(i) The normalized root mean square error (NRMSE) is the normalized root mean square error to the scale of [0,1]. The expression for RMSE is given by the following equation: where O i and P i are the observed and simulated rainfall data, respectively, for 'n' number of data points. The model performance is observed to be better if the model evaluator Fig. 3 Absolute values of the correlation coefficient between observed rainfall and NCEP predictor at all four stations 1 3 value is closer to 0. Normalized root mean square error is adopted to ease the comparison of model performance and is expressed as given below: where range is the difference between the maximum and minimum value of the observed dataset.
(ii) The normalized mean absolute error (NMAE) is the mean absolute error normalized to a scale [0,1]. The expression for MAE can be obtained using the following equation: The model performance is better if the value of MAE is closer to 0, normalized mean absolute error (NMAE) is adopted to ease the comparison of model performance and is expressed as follows: (ii) The coefficient of determination ( R 2 ) is expressed as the Pearson's correlation coefficient which is given by the following equation: where O is the mean of the observed rainfall and P is the mean of simulated rainfall. The R 2 value lies between [0,1], and higher values indicate that the simulated results are more accurate.

Model results and discussion
The developed methodology was applied sequentially to downscale the daily rainfall data of four stations located in the vicinity of the Manawatu catchment (New Zealand). The applicability of the derived results is to improve the closeness between the values of simulated and observed rainfall data to analyse the future performance of rainfall projections due to the impact of climate change.

Performance of SDSM
Tables 1, 2 presents the results obtained with SDSM as discussed in the section of Tools and Techniques. The SDSM was used to perform the correlation analysis between the observed rainfall and the NCEP predictors. Subsequently, the correlation coefficients obtained between the predictand and predictor variables resulted in non-satisfactory results. Thus, to obtain the highest possible correlation, the predictor variables were lagged by 1 day, which was identified as the best option of improving the relationship. The selected predictor variables namely the mean sea level pressure, the divergence near the surface, the geostrophic airflow velocity at 500 hPa, and the meridional velocity component at 500 hPa were used to form the statistical relationship for the future rainfall projections at all four stations. The statistical relationship developed by SDSM for the calibration period showed that the daily rainfall series simulated has obtained the goodness-of-fit (R 2 ) value of 0.28, 0.22, 0.31, and 0.25 for Palmerston, Marton, Opiki, and Te Rehunga, respectively, as shown in Table 2. Furthermore, the SDSM R 2 value obtained is comparable to those found with previous studies and some of these studies have reported lower R 2 values than those obtained in the present study (Pervez and Henebry 2014b;Khan et al. 2006).

Evaluation of SDCRR performance with restricted and wide range predictors
Based on the concept mentioned in Tools and Techniques (Screening of Predictors), the set of predictors selected for two distinctive domains produced many common and few uncommon predictors. As per the methodology discussed (  Tables 5 and 6. These uncommon predictors are mainly total precipitation at surface level, specific humidity both at 500 and 850 pressure levels, and divergence of true wind again at both 500 and 850 pressure levels. The uncommon predictor list of WR also includes surface humidity at surface level, wind speed, wind component at 500 pressure level, and vorticity at 850 pressure level. The PCA was applied to the selected set of WR and RR predictor variables. Thus, the derived uncorrelated components were produced in the form of PCs where each component explained a certain percent of variance given in Table 4. The variance explained by the PCs retains the most valuable parts of all predictor variables. The number of PCs were selected based on the requirement of the percent of variance to be accounted for the regression equation as given in the section of Model development (Step 3: Principal component analysis). Accordingly, the selected number of PCs served more than the average amount of total variance. The study selected PCs from WR and RR predictor variables to explain 96% of the total variance of the original climate predictor variables. The number of PCs selected for WR and RR predictor variables were 12 and 10, respectively, at Palmerston. Similarly, the number of PCs selected at other stations were able to explain 96% of the total variance for both WR and RR predictor variables as shown in Table 4 and were further used to perform the calibration and validation of the SDCRR model at all four stations.
The performance of SDCRR model is evaluated based on the goodness-of-fit ( R 2 ) value. It can be depicted from Table 4 that, when 96% of variance is captured by both the WR and RR predictors, the resulted goodness-of-fit ( R 2 ) value is significantly different for both the cases as given in Table 5. This may be due to the reason of working with those extra predictors of WR, which improved goodness-offit ( 2 ) value up to 0.66, as shown in Table 3.
Furthermore, these extra WR predictors such as divergence of true wind specific humidity and total precipitation along with wind (speed and component) and vorticity are found to have significant impact on rainfall downscaling in the study area. Additionally, the R 2 value derived by the RR predictors was less in comparison to the WR predictors for both the calibration and validation period as given in Table 5. This difference in the goodness-of-fit ( R 2 ) value was found with both MLA and ridge regression analysis. It can be seen that the WR predictors produced R 2 value equal to 0.3 and 0.66, with multiple and ridge regression, respectively, whereas the RR predictors produced inferior R 2 values given as 0.16 and 0.59 as shown in Table 5. Interestingly, there is distinct indication that the WR predictors performed well than that of RR predictors. Based on the results obtained, the study further continued working with WR predictors to improve the performance of model in context of downscaling daily rainfall with SDCRR model. Initially, while working with WR predictors, the study also worked with larger and smaller number of PCs such as 16 and 8 given in Table 5. It is shown in the table that the R 2 value decreases with the decreasing number of PCs, which is a decrease in the percentage of variance of the original climate variables. However, even with different PCs (16, 12, and 8), the R 2 value is not showing much variations, indeed it is maintained at 0.66 and 0.77 for calibration and validation period, respectively. The significant reason for this could be the application of ridge regression along with the PCA.
Specifically, the application of PCA and ridge regression improved the closeness between the daily simulated and the observed rainfall data. The application of PCA helped in filtering the common information content shared by different predictor variables either in higher or lower percentage. Further, based on the concept of PCA, the higher or lower percentage of common information content conveyed by different predictor variables was acquired. Selection of different set of PCs given in column 4 (Table 5) was performed to analyze the effect of data variance on the simulated values of daily rainfall. In comparison to the multiple linear regression, the ridge regression estimates are less sensitive to random errors observed in response of dependent variable and produces less variance as shown in Table 5. This could be due to the small value of ridge parameter (0.01), which was used to overcome the conditioning problem. The improved R 2 value was also due to the small mean square error (as explained in the methodology) obtained as the result of reduced variance in ridge estimates. The results derived during the calibration and validation period also shows that the model performance with the ridge regression evolved R 2 values up to 0.66 and 0.77 for calibration and validation samples, respectively, as shown in Table 5. In contrast, the coefficient estimates of MLR remained unstable with potentially noisy PCs and produced R 2 value only up to 0.3.

Performance of SDCRR with restricted and wide range predictors at other stations
At a daily resolution, the SDCRR model performed well for rainfall downscaling and the model performance was evaluated at different stations of the study area as shown in  The results derived during the calibration and validation period showed that the regression analysis used in establishing the statistical relationship between predictors and predictand has more predictive ability with ridge regression in comparison to MLR. It can be clearly seen that the WR  predictors produced R 2 value equal to 0.3 and 0.75 (Marton), 0.22 and 0.6 (Opiki), and 0.24 and 0.52 (Te Rehunga) with multiple and ridge regression, respectively, as shown in Table 6.
To confirm the superior predictive ability of ridge regression, the calibration and validation were performed with the same data set as shown in Table 6. It appeared that the prediction error obtained from MLR is more than that obtained using ridge regression for both the calibration and the validation periods. Beside the fact that the introduction of large number of components may significantly reduce the performance of the model, the ridge regression evolved superior R 2 values up for both calibration and validation samples as shown in Table 6.
The selected performance indices were further applied to evaluate the superior predictive ability of ridge regression. The SDCRR model performed well with other performance indices (NRMSE and NMAE) for rainfall downscaling. The model performance is consistent at four different stations of the study area as shown in Table 7. These satisfactory estimates of all performance indices during both calibration and validation of the model indicated that the model responded in an effective manner to the daily rainfall data downscaling. In context of testing the performance of SDCRR model using the GCM predictors, the study applied standardized historical (1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005) predictor variables of the CanESM2 to downscale and compare the daily observed data of the same period at Palmerston and Opiki stations. The cumulative distribution function (CDF) of daily rainfall series generated by SDCRR model is compared with the observed rainfall series. Figure 4a, b shows the CDF of daily rainfall obtained from the SDCRR model using NCEP  and historical CanESM2 GCM predictors (1991-2005, respectively. The important observation with the SDCRR model is that the CDF obtained from the simulated data shows admissible deviations from the CDF obtained using the observed rainfall of Palmerston and Opiki stations (Fig. 4a).
Similarly, Fig. 5a compares CDF of daily rainfall data simulated using NCEP predictors' with those obtained with the observed daily rainfall data at Marton and Te Rehunga stations. The SDCRR model performs well in capturing 58% of dry days (rainfall ≤ 1 mm/day) well at all four stations. Further, the percent of dry days calculated from CanESM2 (52%) in comparison to the observed rainfall dry days is obtained accurately at all four stations. Thus, it seems that the SDCRR model fairly simulates all stations daily rainfall using GCM (CanESM2) predictors' data as shown in Figs. 4b and 5b.

Future projections using GCM simulation
To obtain the future rainfall projections, the SDCRR model was applied using different climate change scenarios using the Representative Concentration Pathway (RCP) 2.5, RCP 4.5 and RCP 8.5 of CanESM2. The RCPs are the concentration pathways used in the IPCC AR5. These pathways are prescribed for the greenhouse and aerosol concentration. The identification of these pathways is based on the radioactive forcing, which will be produced by the end of this century. Radioactive forcing is caused by holding that extra heat, retained in the lower atmosphere as a result of additional greenhouse emission (IPCC 2014). The RCP 2.5 represents low emission scenario, RCP 4.5 represents intermediate emission, and RCP 8.5 represents the high emission of carbon. The future period of 2031-2060 is selected to investigate the future impact of climate change under different emission scenarios. The CDF of future rainfall projections at daily time step is obtained at four downscaling stations as shown in Fig. 6.
The CDFs are helpful to detect the increase and decrease in rainfall values. The shift in CDFs above and below the observed curve is used as tool to detect the changes in the frequency of high or low rainfall (Kannan and Ghosh 2013). The CDFs obtained for the three scenarios are almost similar to each other at Palmerston, Marton, Opiki, and Te Rehunga rain stations. The stations observe a downward shift to all three scenarios, which indicates an increased frequency of high rainfall during 2031-2060 compared to the observed or base period rainfall; 2005-2020 for Palmerston, 2005-2019 for Opiki and Te Rehunga, and 2005-2016 for Marton. However, the appearance of downward shift is different at each station therefore, the frequency of highest rainfall is expected at Fig. 4 CDF of basin daily rainfall obtained from SDCRR downscaling model using the (a) reanalysis data  for Palmerston and Opiki stations and (b) CanESM2 data (1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005) for Palmerston and Opiki stations Opiki followed by Palmerston, Marton, and Te Rehunga. Although, we obtained the future projections with a single GCM output, results may vary with the application of multiple GCMs (Kannan and Ghosh 2013).

Conclusion
The statistical downscaling methods are capable of filling the gap between large-scale climate variables and local-scale hydrological variable. Among them, SDSM is widely used in numerous studies for its robust capabilities (Osman and Abdellatif 2016;Hashmi et al. 2009a). Furthermore, the study proposed a statistical downscaling model, SDCRR, and the performance of the proposed model was evaluated in terms of its ability to simulate the daily rainfall data. The daily rainfall data of the Manawatu catchment (Palmerston station) in North Island, New Zealand, were used for the analysis. In this study, SDSM was applied as the base model to evaluate the performance of the proposed model. The results of the study show the SDCRR model has better performance than the SDSM. Along with Palmerston station,   for Marton and Te Rehunga stations (b) CanESM2 data (1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005) for Marton and Te Rehunga stations the SDCRR model was applied at three more stations located in and near the Manawatu catchment. The proposed model SDCRR was able to simulate the daily rainfall with optimum accuracy even at other stations.
Selection of predictors under strong influence of complex climate to perform successful downscaling is rather challenging. In this context, the study worked with WR and RR predictors and found that that the argument of working with WR predictors was evident. It found that 20 predictors selected from the dataset of 26 climate variable improved goodness-of-fit ( R 2 ) value up to 0.66, whereas lesser number of predictors (14 out of 26 climate variables) obtained R 2 value up to 0.59. Similarly, the WR range predictors: 19 at Marton, 21 at Opiki, and 16 at Te Rehunga produced acceptable goodness-of-fit ( R 2 ) value up to 0.75, 0.6, and 0.52 at Maton, Opipki, and Te Rehunga, respectively. Also, the dimensionality and multi-collinearity of WR predictors were effectively handled by PCA and ridge regression. Significantly, the combined application of PCs as the orthogonal filters and ridge regression showed a good applicability to simulate daily rainfall for both the calibration and validation period. Moreover, with the application of PCA, the predictors were linearly transformed to produce all the independent components. With increase in the number of predictors, the content of information shared within variables was increased, which were contributing to the variance of the dependent variable. The MLR coefficient estimates were unstable under increased number of PCs (more than in usual regression analysis) but this inflation in the variance of coefficient estimates caused due to increased number of PCs (even with smaller variance) were controlled by the addition of a ridge constant. The calibration and validation results further reinforce the belief of applying the SDCRR model to conduct the climate change studies. Therefore, the study applied the downscaling model SDCRR to acquire the future rainfall projections at the selected stations under CanESM2 scenarios of RCP 2.5, RCP 4.5, and RCP 8.5. The findings of the future projections under different scenarios are that the rainfall will be increased under all RCP scenarios. Thus, due to change in climate, the future rainfall is expected to increase from the period of 19,311,960 in comparison to the current climate of 2005-2020. Although, in the present study, we used only one GCM for downscaling, the future rainfall estimations could be different with other GCMs. Therefore, the future studies are encouraged to assess the model performance by doing the uncertainty modeling with multiple GCMs.
Funding Open Access funding enabled and organized by CAUL and its Member Institutions.

Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
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