Show Summary Details

Page of

Printed from Oxford Research Encyclopedias, Climate Science. Under the terms of the licence agreement, an individual user may print out a single article for personal use (for details see Privacy Policy and Legal Notice).

date: 27 June 2022

Empirical-Statistical Downscaling: Nonlinear Statistical Downscalingfree

Empirical-Statistical Downscaling: Nonlinear Statistical Downscalingfree

  • Aristita BusuiocAristita BusuiocRomanian National Meteorological Administration


Empirical-statistical downscaling (ESD) models use statistical relationships to infer local climate information from large-scale climate information produced by global climate models (GCMs), as an alternative to the dynamical downscaling provided by regional climate models (RCMs). Among various statistical downscaling approaches, the nonlinear methods are mainly used to construct downscaling models for local variables that strongly deviate from linearity and normality, such as daily precipitation. These approaches are also appropriate to handle downscaling of extreme rainfall.

There are nonlinear downscaling techniques of various complexities. The simplest one is represented by the analog method that originated in the late 1960s from the need to obtain local details of short-term weather forecasting for various variables (air temperature, precipitation, wind, etc.). Its first application as a statistical downscaling approach in climate science was carried out in the late 1990s. More sophisticated statistical downscaling models have been developed based on a wide range of nonlinear functions. Among them, the artificial neural network (ANN) was the first nonlinear regression–type method used as a statistical downscaling technique in climate science in the late 1990s. The ANN was inspired by the human brain, and it was used early in artificial intelligence and robotics. The impressive development of machine learning algorithms that can automatically extract information from a vast amount of data, usually through nonlinear multivariate models, contributed to improvements of ANN downscaling models and the development of other new, machine learning-based downscaling models to overcome some ANN drawbacks, such as support vector machine and random forest techniques. The mixed models combining various machine learning downscaling approaches maximize the downscaling skill in local climate change applications, especially for extreme rainfall indices.

Other nonlinear statistical downscaling approaches refer to conditional weather generators, combining a standard weather generator (WG) with a separate statistical downscaling model by conditioning the WG parameters on large-scale predictors via a nonlinear approach. The most popular ways to condition the WG parameters are the weather-type approach and generalized linear models.

This article discusses various aspects of nonlinear statistical downscaling approaches, their strengths and weaknesses, as well as comparison with linear statistical downscaling models. A proper validation of the nonlinear statistical downscaling models is an important issue, allowing selection of an appropriate model to obtain credible information on local climate change. Selection of large-scale predictors, the model’s ability to reproduce historical trends, extreme events, and the uncertainty related to future downscaled changes are important issues to be addressed.

A better estimation of the uncertainty related to downscaled climate change projections can be achieved by using ensembles of more GCMs as drivers, including their ability to simulate the input in downscaling models. Comparison between more future statistical downscaled climate change signals and those derived from dynamical downscaling driven by the same global model, including a complex validation of the RCMs, gives a measure of the reliability of downscaled regional climate changes.


  • Downscaling

Introduction to the Field

Impact applications, such as hydrological and crop models, need high-resolution information on various climate variables on the scale of a river basin or a farm, scales that are not available from the usual global climate models (GCMs). Downscaling techniques produce regional climate information on a finer scale from global climate change scenarios, based on the assumption that there is a systematic link between the large-scale and local climates (Benestad, 2016; Christensen et al., 2007a).

There are two types of downscaling approaches: Dynamical downscaling is based on regional climate models (RCMs) nested in a GCM (Christensen et al., 2007b; Giorgi & Mearns, 1999), and empirical-statistical downscaling (ESD) is based on developing statistical relationships between large-scale atmospheric variables (predictors), available from GCMs, and observed local-scale variables of interest (predictands) (Hewitson & Crane, 1996; von Storch et al., 1993). The two downscaling approaches are complementary, each having different strengths and weaknesses (Benestad, 2016), with the ESDs usually being preferred due to their low computational cost.

Various ESD approaches could be placed in three groups (see Christensen et al., 2007a; von Storch, 2000; Wilby et al., 2004): weather classification schemes (see Bardossy & Plate, 1992, Corte-Real et al., 1999; Zorita & von Storch, 1999), stochastic WGs (see Busuioc & von Storch, 2003; Semenov & Barrow, 1997; Wilks, 1998), and regression models (see Huth et al., 2008; von Storch et al., 1993; Weichert & Bürger, 1998). Each of them presents advantages and disadvantages (Christensen et al., 2007a; Maraun et al., 2010; Wilby et al., 2004). In terms of the source of chosen predictors under calibration, ESDs may involve two fundamentally different strategies, such as perfect prognosis (PP) or model output statistics (MOS), whether the predictors are chosen from either observations (or reanalysis as a surrogate of observations) or model outputs (see Benestad, 2016; Maraun et al., 2010; Maraun & Widmann, 2018). An additional hybrid strategy utilizes a common data space (i.e., a combination of both reanalysis and model output) by using common empirical orthogonal functions (EOFs; Benestad, 2001).

Regression models are the most popular statistical downscaling techniques, representing linear or nonlinear relationships between large-scale predictors and local-scale predictands. The linear ESDs are widely used in climate change applications due to their simplicity and straightforward application covering techniques, such as simple or multiple linear regressions (Hellström et al., 2001; Huth, 2004) and canonical correlation analysis (Busuioc et al., 1999; von Storch et al., 1993). However, they are limited by some assumptions like well-defined linear predictor–predictand mathematical relationships and normality of the data used in their calibration, which are not always satisfied in reality. This is the case for precipitation, one of the most important variables used in hydrological modeling, for which there is an increasing demand for high-spatial-resolution (up to 1 km or less) and high-temporal-resolution (up to 1 hr or less) data. At this temporal scale, precipitation intensities are more and more skewed, and the linear ESDs are not appropriate, nonlinear approaches being needed. Therefore, nonlinear ESDs are demanded more in cases where the weather variables are downscaled (e.g., daily data), for which the Gaussian assumption is not feasible, as opposed to climate data (e.g., monthly statistics). They are also more appropriate for dealing with downscaling of extreme rainfall and often involve concepts like those used in more general machine learning and artificial intelligence.

The artificial neural network (ANN) has been among the first nonlinear regression–type methods used in statistical downscaling in climate science (Hewitson & Crane, 1996). The explosion of big data sets used in hydrological science that derived from various sources of measurements (radar and satellite, traditional gauge networks), paleoclimate proxies, reanalysis products, and GCM or RCM large-scale simulations needed special algorithms to extract useful information. Machine learning-based methods are very popular ways to automatically extract information from large predictor and predictand data sets, usually through nonlinear multivariate models, without the need to construct explicit physical or statistical models (He et al., 2016; Vandal et al., 2019). The main strength of machine learning algorithms, as opposed to traditional ANNs, is their ability to tackle different types of problems, from classification to prediction and parameter selection. Achieving this goal is feasible by using high-performance computing to combine machine learning-based algorithms with big data to develop automated algorithms for various applications. Based on machine learning algorithms, improved ANN (Pan et al., 2019) as well as new statistical downscaling techniques, such as support vector machine (SVM; Anandhi et al., 2008) and random forest (RF; He et al., 2016), have been developed. Hybrid models combining various versions of the machine learning-based models increase the performance of the downscaling process, especially for extreme rainfall (Pham et al., 2019).

The machine learning-based downscaling models are purely statistical approaches in which physical or conceptual ideas only play a minor role. An alternative approach is weather typing, yielding physically interpretable linkage to surface climate (Vrac & Naveau, 2007; Wilby et al., 2004). Analog method, as a particular case of weather typing (Dayon et al., 2015; Zorita & von Storch, 1999), is the simplest nonlinear statistical downscaling technique to infer weather variables at the local scale. Weather types are mostly applied to condition WGs to create physically motivated precipitation generators, as a hybrid statistical downscaling model (nonlinear conditional WG) combining an unconditional WG with a separate statistical downscaling model based on weather typing (Bardossy & Plate, 1992; Maraun et al., 2010; Maraun & Widman, 2018). In these approaches, WGs take as their starting point a distribution of precipitation in each time interval.

Other alternative approaches, explicitly considering the mechanisms of precipitation generation in a simplified stochastic framework, are given by full-field WGs, including a filtered autoregressive Gaussian process (D’Onofrio et al., 2014; Rebora et al., 2006), Poisson cluster models (Kilsby et al., 2007; Rodriguez-Iturbe et al., 1987), and multifractal cascade models (Lovejoi & Mandelbrox, 1985). These techniques use coarse-scale precipitation as a predictor and can disaggregate daily rainfall for any subhourly time scale (Kossieres et al., 2016).

This article discusses various aspects of pure regression-type nonlinear models (i.e., ANN, SVM, and RF approaches) and analogs, as well as hybrid downscaling models given by nonlinear conditional WGs, highlighting their strengths and weaknesses and their comparisons, including to linear approaches. A comparison with dynamical downscaling is also considered.

Overview of Nonlinear Statistical Downscaling

The statistical downscaling techniques have their origins in developing operational numerical weather forecasting and in trying to get local details on weather variables through the first ESD technique, known as the analog method, since 1951 (Benestad, 2016). More sophisticated nonlinear statistical downscaling techniques, such as ANNs, SVM, and RFs, are among the newest ESD methods, and they involve concepts similar to those used in more general machine learning and artificial intelligence.

The historical development of nonlinear statistical downscaling techniques, including their strengths and weaknesses, as well as their ability to reproduce extreme events and various statistical parameters are presented. In climate change studies, to maximize the skill of the downscaled results, combinations of various downscaling techniques, including RCMs, are used. Various overviews of all ESD types have been presented (Christensen et al., 2007a; Maraun et al., 2010; Wilby et al., 2004).

Analog Method

The analog method is the oldest statistical downscaling approach, developed for the first time by Lorenz (1969) for short-term weather forecasting, and then applied for short-term climate prediction (Barnett & Preisendorfer, 1978; van den Dool, 1994) and even for decadal prediction (Hawkings et al., 2011). Its first application as a statistical downscaling approach was presented by Zorita et al. (1995). The analog method is based on the idea that, for a large-scale circulation pattern simulated by GCM (or a global numerical forecasting model), its analog is defined as the most similar large-scale pattern selected from historical observations (training period), according to a defined similarity criterion. The observed local weather state (represented by air temperature, precipitation, wind, etc.) corresponding to the chosen analog is then associated with the simulated (forecasted) large-scale pattern (Zorita & von Storch, 1999). A random selection among more analogs (e.g., the k most similar situations) has been proposed as an alternative to the selection of only one analog (Lall & Sharma, 1996). Various similarity criteria are used, the most popular being the Euclidian distance, but other similarity measures have been proposed, such as the Teweles–Wobus score (TWS; Dayon et al., 2015; Guilbaud & Obled, 1998; Obled, 2002). Referring to prediction on a global scale, van del Dool (1994) concluded that it is almost impossible to find good multilevel multivariate natural analogs, because thousands of observation years are needed.

However, for downscaling purposes, the degree of freedom can be reduced by previously filtering out the background noise of atmospheric field through EOF analysis as well as by reducing the area of interest from a global scale to a continental or smaller scale (Zorita & von Storch, 1999). Due to its simplicity and quite good performance compared to other more complex models, the analog method continued to be applied in many regional climate change scenarios (Ben Daoud et al., 2011; Dayon et al., 2015; Imberd & Benestad, 2005; Schmidli et al., 2007; Timbal & McAvaney, 2001; Yates et al., 2003) to project future changes in local/regional predictand variables from GCM climate change scenarios. Besides the simplicity of calculation, the main advantages of the analog method are the physical coherence between predictand variables and their spatial coherence, which is a very important issue for hydrological models.

However, some limitations can be revealed: the necessity for long observations to find a reasonable analog, the inability to produce a local weather state that has never been observed in the training period, the inability to preserve the time structure, and underestimation of the daily standard deviation of predictands (Beersma & Buishand, 2003; Benestad, 2010; Dayon et al., 2015; Maraun et al., 2010; Young, 1994). Imbert and Benestad (2005) suggested a method for dealing with new values outside the range of the historical sample by shifting the whole probability density function (PDF) according to a trend predicted by linear ESD models. However, this technique does not resolve the problem of the distorted upper tail that is produced by using the extended EOF (Benestad, 2010).


ANNs have their origins in models of human brain function (Hsieh & Tang, 1998). The human brain is one of the great wonders of nature, being much more performant than the most advanced artificial intelligence program running on a supercomputer. This performance is given by the brain’s massively parallel computing structure, achieved by neurons that are interconnected in a network. Inspired by the human brain model, many scientists tried to develop similar models—ANNs—for data analysis and prediction. At the beginning, the ANN techniques were very popular in artificial intelligence, robotics, and many other fields (Crick, 1989), the most well-known application being pattern recognition, such as cloud classification in satellite imagery analysis (Lee et al., 1990). The historical development of the ANN models and the early difficulties in their adaptation to meteorology and oceanography, including proposed techniques to overcome them, have been presented by Hsieh and Tang (1998). Details on the ANN algorithm were presented by Bishop (1995). In meteorology, ANNs were first used in improving weather prediction (Eccel et al., 2007) and seasonal forecasts (Tangang et al., 1998). They have since evolved into powerful nonlinear statistical downscaling models used in climate science.

ANNs are regression-type approaches used in climate science to develop nonlinear relationships between large-scale predictors and local predictands. ANN provides an alternative to linear statistical models, which are often limited by assumptions of linearity, normality, non-multicollinearity, and so on (Cavazos, 1999). The most widely used neural network is the multilayer perceptron (MLP; Rumelhart et al., 1986), composed of layers of individual processing units usually called neurons. Each neuron takes its input from all elements in the previous layer, evaluates a nonlinear function of the inputs, and forwards this result to the next layer. Each connection between two neurons is given a relative weight, which is adjusted during a training phase in order to minimize some measure of error. More complex models have one or more “hidden” layers of neurons between the input and output layers. In general, most ANN applications have only one or two hidden layers (Hertz et al., 1991).

The ANN parameters are estimated in a training process using observed data. In comparison with multiple linear regression, theoretically the ANN can represent any arbitrary nonlinear function, but in traditional ANNs, the type of function should be manually selected. Neural networks are extremely flexible nonlinear regressors, but they are prone to overfitting due to the large number of parameters. The model output may fit the data very well during the training period but produce poor forecasts during the test period, since an ANN model is usually capable of learning the signals in the data, but as training progresses, it often starts learning the noise as well.

An early use of the ANN in statistical downscaling was presented by Hewitson and Crane (1996). The method has been mainly applied for daily precipitation downscaling that exhibits a significant nonlinear relationship (Cavazos et al., 2002; Haylock et al., 2006; Osslon et al., 2001; Trigo & Palutikof, 2001; Vu et al., 2015; Weichert & Bürger, 1998). Only a few ANN models have been applied to monthly precipitation (Okkan & Kirdemir, 2016; Schoof & Prior, 2001). Some studies applied the method for daily temperature downscaling and compared the results with those obtained from other ESDs (Gaitan et al., 2014; Huth et al., 2008; Schoof & Prior, 2001). ANN has also been used as an MOS technique to improve numerical weather prediction, such as minimum temperature (Eccel et al., 2007). It has been found that, in comparison with multivariate linear regression, the ANN advantage is small.

The various types of neural networks that have internal memory structures that can store the past values of input variables through time are more suitable for complex nonlinear system modeling. Time-lagged feed-forward networks (TLFNs) and recurrent networks (RNNs) are the two most-used ways of introducing “memory” in a neural network in order to develop a temporal neural network. Dibike and Coulibaly (2006) used the TLFN approach for downscaling daily precipitation and temperature.

Improved versions of the ANN models have been developed based on more efficient learning algorithms, such as extreme learning machine (ELM) and stepwise extreme learning machine (SWELM), which combines the stepwise feature selection method into ELM, since model complexity and computational time consumption by a traditional ANN impede application of stepwise feature selection (MoradiKhaneghahi et al., 2019). It was found that the SWELM outperformed the ANN algorithm for temperature downscaling in the United States.

Novel developments in machine learning techniques, such as deep neural networks (DNNs), have been carried out and used to reduce the bias in the satellite precipitation products (Tao et al., 2016). The difference between DNNs and traditional ANNs is that DNNs aim to automatically extract information at multiple levels of abstraction to allow a system to learn complicated functional relationships between the input and the output directly from the data, while traditional neural networks tend to use manually designed features. A particular form of DNN, the convolutional neural network (CNN), has recently been used to improve precipitation estimation from reanalysis precipitation products (Pan et al., 2019). This model is an alternative to the existing precipitation-related parameterization schemes for numerical precipitation estimation, the model being trained in such a way as to learn precipitation-related dynamic features from the surrounding dynamic fields by optimizing a hierarchical set of spatial convolution kernels. Testing the model at 14 geogrid points across the contiguous United States showed that the estimated values from the CNN model outperform the reanalysis products and other ESD products using linear regression, nearest neighbor, RF, or fully connected DNN (Pan et al., 2019). This novel approach can be used for improving precipitation prediction.

A hybrid model combining a linear stochastic model and a nonlinear ANN has been used for drought forecasting in India (Mishra et al., 2007). The drought was represented by the Standardized Precipitation Index. Results showed the hybrid model forecasted drought with greater skill than the individual stochastic and ANN models. These promising results suggest this method can be applied in other areas with different climates.


SVM, with components for classification (SVC) and regression (SVR), is a machine learning technique based on statistical learning theory proposed by Vapnik (1995). SVM models are used to capture nonlinear regression relationships between variables, but, in comparison with traditional ANNs, they are more efficient in getting a global optimum solution leading to an optimum network structure. The SVM learning algorithm automatically decides the model architecture (number of hidden units), in contrast to standard ANNs, which use a subjective selection of the hidden units (Anandhi et al., 2008; Sahardi et al., 2017). The nonlinear regression functions in SVRs are based on kernel functions, and their associated weights are estimated during the training process from the data. The SVM approach has been used for statistical downscaling of monthly (Sahardi et al., 2017; Tripathi et al., 2006) and daily precipitation (Chen et al., 2010a; Raje & Mujumdar, 2011; Vandal et al., 2019).

However, the SVR presents some drawbacks, such as complexity in computation and overfitting, which are addressed in improved algorithms, such as the sparse Bayesian learning (SBL) algorithm known as relevance vector machine (RVM) and least-squares SVM (LS-SVM). The RVM proposes a fully probabilistic framework and introduces prior information on the model weights, leading to identification of a few training vectors associated with non-zero weights, which are called relevant vectors. In comparison to SVM, the most compelling feature of RVM is that it utilizes fewer kernel functions, avoiding overfitting. The RVM approach has been applied in downscaling of monthly precipitation (Sahardi et al., 2017; Tipping, 2001). The LS-SVM algorithm provides a computational advantage over standard SVM by converting the quadratic optimization problem into a system of linear equations. The LS-SVM can be used for both classification and regression problems (Anandhi et al., 2008; Okkan & Kirdemir, 2016; Okkan & Serbes, 2012).

The SVM technique also has some drawbacks when dealing with large data samples. Lee et al. (2005) proposed an improved algorithm based on SVM, a new smoothing strategy for solving the regression of large-scale training data called the smooth SVM (SSVM). Chen et al. (2010b) applied this technique in downscaling of daily precipitation and compared the results with those achieved through the ANN method. It has been found that the SSVM algorithm is capable of producing satisfactory results in terms of daily and monthly mean precipitation that are better than ANN, but it is less skillful at reproducing extreme daily precipitation. For improving downscaling of extreme rainfall, a hybrid model has been proposed by combination of the SVM techniques with the RF techniques (Pham et al., 2019).


The RF method was proposed by Breiman in 2001, and it became a popular machine learning technique due to its ability to deal with complex nonlinear relationships between variables while minimizing problems with overfitting. The RF model is an ensemble machine learning technique based on a combination of classification and regression tree methods, so that it is composed of an ensemble of decision trees generated through bootstrap samples of the training data and random-variable subset selection. These models have only two calibration parameters: the number of variables and the number of trees. There are other important advantages of the RF models (He et al., 2016; Pang et al., 2017): They provide a full PDF of the model outputs (forecast or classification label); the decision trees are nonparametric and robust in the presence of outliers, noise, and overfitting; and the relative influence of the predictors can be ranked.

RF has been used in various hydrological applications, such as seasonal streamflow forecasting (Zhao et al., 2012), improving rainfall estimation from satellite information related to cloud physical properties (Kuhnlein et al., 2014), and statistical downscaling of daily temperature (Eccel et al., 2007; Pang et al., 2017) and wind (Davy et al., 2010). The RF potential for spatial statistical downscaling of precipitation has been proved by He et al. (2016), who showed the advantage of using two independent RF models in a simulation of extreme precipitation. They highlighted that the method can be adapted to include any type of covariate (e.g., discrete or continuous) and can be adjusted to different types of precipitation (e.g., stratiform, convective, and orographic).

Some studies showed that hybrid models combining the robustness of RF in classification and the superiority of SVM to fit highly nonlinear data are more skillful in developing statistical downscaling models of daily rainfall (Pour et al., 2016). To improve the statistical downscaling of extreme rainfall, a combination of RF and least-squares SVR has been proposed by Pham et al. (2019). In this case, RF is used for classification of three rainfall states (dry day, non-extreme-rainfall day, and extreme-rainfall day), and least-squares SVR is used for rainfall-amount prediction. They found that RF outperforms other rainfall classification methods, including linear discriminant analysis (LDA) and support vector classification (SVC).

Hybrid Statistical Downscaling Models: WGs

The standard stochastic models or WGs have been developed to generate an ensemble of statistically equivalent random time series of weather variables (e.g., synthetic weather series), theoretically of unlimited length, reproducing their long-term observed statistical properties (Richardson, 1981; Wilks, 1998). This approach is based on the assumption that the meteorological processes exhibit a stationary stochastic process, meaning that the PDF is constant over time but with random fluctuations. Most WGs have been developed as a precipitation generator, the other variables being simulated conditional on the generated precipitation. Various types of WGs have been described in several reviews (e.g., Maraun et al., 2010; Wilks, 2010, 2012; Wilks & Wilby, 1999). Yin and Chen (2020) carried out a comprehensive review of various WG approaches published before the end of the 2010s.

The parameters of the basic WGs are calibrated on observations only, with these models being known as unconditional generators, and they cannot be directly used to build local climate change scenarios since their parameters are not linked by the predictors simulated by the GCMs. To overcome this issue, the unconditional models are extended to so-called conditional versions by conditioning their parameters on large-scale conditions (predictors) simulated by GCMs to obtain future changes in the PDF of weather variables of interest that is carried out through a separate statistical downscaling model. Therefore, the conditional WGs are hybrid statistical downscaling models combining unconditional WGs with a statistical downscaling model, which is mainly considered a PP ESD. In terms of the ESD type, the conditional WGs can be roughly grouped into linear and nonlinear approaches (the classification adopted in this article; a short overview of nonlinear WGs is presented). In comparison to pure PP approaches that do not explicitly model either spatial or temporal correlations, the WGs can explicitly aim to generate time series or spatial fields with the observed temporal or spatial structure or temporally disaggregate a monthly/daily precipitation amount to an hourly/subhourly scale (Kossieres et al., 2016; Maraun et al., 2010; Soerup et al., 2016). One of the main advantages of the conditional WGs is their ability to capture the tail behavior of climate variables, in comparison to other ESDs (i.e., regression models and weather classification schemes) that tend to underestimate the variance of local/regional climate, resulting in a poor simulation of extremes (Yin & Chen, 2020).

The most popular nonlinear WGs use the weather-type approach to condition the WG parameters (Maraun & Widmann, 2018; Yin & Chen, 2020). In this approach, the WG parameters are calibrated separately for each class of weather types (Bardossy & Plate, 1992; Corte-Real et al., 1999; Fowler et al., 2005). Weather types can be defined subjectively, such as Lamb weather types (Lamb, 1972) and the German Weather Service classification scheme (e.g., Hess & Brezowsky, 1969), or objectively by using various automated statistical tools (e.g., Bardossy et al., 2005; Corte-Real et al., 1999; Fowler et al., 2000). Dynamic variables (e.g., sea-level pressure) are usually considered in the classification of weather types representing circulation patterns. However, these nonlinear WGs are useful for representing internal climate variability, and they could fail to correctly reproduce forced future changes (Maraun & Widmann, 2018; Yin & Chen, 2020). Therefore, it is crucial that thermodynamic variables (e.g., temperature and humidity) also be included in weather-type classification (e.g., Huth et al., 2008; Murawski et al., 2016; Philipp et al., 2007) when they are used to condition WGs. Nonlinear WGs using the weather types for the conditioning process have mainly been developed for daily time steps. For example, Corte-Real et al. (1999) conditioned the parameters of single-site rainfall WG (occurrence probability and distribution function) on four circulation patterns based on daily sea-level pressure fields. They found that the distributions of wet and dry spells have been best simulated using a first-order two-state Markov model conditioned on daily circulation patterns of both the current day and the previous day. This approach has been extended to a multisite WG of Wilks type (Wilks, 1998) by Qian et al. (2002). The classification scheme of the German Weather Service has been used to condition a multidimensional stochastic model developed for the space–time distribution of daily precipitation (Bardossy & Plate, 1992). The rainfall is linked to the atmospheric circulation patterns using conditional distributions and conditional spatial covariance functions.

For nonparametric WGs, Beersma and Buishand (2003) extended the unconditional version of a multisite stochastic model based on the K-NN (nearest neighbors) resampling approach proposed by Buishand and Brandsma (2001), for conditional resampling on three daily indices of the atmospheric circulation. In this approach, one searches for days in the historical record that have atmospheric circulation characteristics similar to those of the conditioning day. One of these nearest neighbors is randomly selected, and the observed values of that nearest neighbor are adopted as the simulated values for the conditioning day. However, this approach tends to underestimate the lengths of wet/dry spells. A new method for generation of multisite precipitation occurrence has been proposed by coupling a discrete version of the K-NN resampling approach with a genetic algorithm, along with model adaptation to climate change (Lee & Sing, 2019).

Some studies attempted to develop subdaily-scale nonlinear conditional WGs. For example, Fowler et al. (2000) used three groups of the Lamb weather types based on daily sea-level pressure to condition the parameters of a stochastic model based on the Neyman–Scott Rectangular Pulses (NSRP) process for generating single-site precipitation series at temporal resolutions down to the hourly scale. An extended multisite version has been developed by Fowler et al. (2005).

Other approaches refer to the nonhomogeneous hidden Markov models (NHMMs) that simulate a series of hidden weather states representing characteristic spatial weather patterns, the transition probabilities between them being simulated by a Markov chain model conditioned on large-scale predictors, and finally the multisite daily precipitation patterns are generated conditional on these hidden weather states only (Hughes & Guttorp, 1994; Vrac et al., 2007). In this case, a weather state is defined as discrete and scalar. A K-NN-based nonparametric nonhomogeneous hidden Markov model (NNHMM) has been developed by Mehrotra and Sharma (2005), allowing the weather state to be a continuous random variable and the multisite daily rainfall vector to be conditional on the previous day’s rainfall. Their results concluded that using continuous weather states is more successful in capturing the day-to-day rainfall characteristics and more reliable for representation of extreme events, in comparison to discrete-state NHMM. A further improvement compared to the NHMM and NNHMM approaches has been proposed by considering rainfall occurrences directly conditional upon the atmospheric variables, avoiding the use of discrete or continuous weather states (Mehrotra & Sharma, 2010). This approach downscales rainfall occurrences at multiple stations using a parametric modified Markov model to compute transition probabilities conditional on large-scale predictors and an aggregated wetness state over a previous period designed to provide longer term persistence. The rainfall amounts on wet days are generated through a nonparametric kernel density simulator conditional on previous time-step rainfall and large-scale predictors. The proposed technique provides a better representation of the observed low-frequency variability and spell extremes by incorporating the behavior of the climate over recent past periods, making it more adaptable for future changes.

Generalized linear models (GLMs) are also used to condition the WG parameters using a link function so that the conditional mean of the predictand is modeled as a linear function of a set of predictors (Kim et al., 2012; Verdin et al., 2015). For example, logistic link function (logistic regression) is used for precipitation occurrence, and logarithmic link function for precipitation intensity (Maraun et al., 2010). To reduce the overdispersion phenomenon, Kim et al. (2012) introduced smoothed seasonally aggregated climate statistics as additional covariates in the GLM. If an additional interest is the dependence of the extreme tail on a set of predictors, vector generalized linear models (VGLMs) have been developed to estimate a vector of parameters of a distribution, instead of the conditional mean of a distribution only. For example, Ben Alaya et al. (2015) developed a Bernoulli-generalized Pareto multivariate autoregressive (BMAR) model for multisite statistical downscaling of daily precipitation by using a VGLM to condition the parameters of the Bernoulli-generalized Pareto distribution (precipitation occurrences and precipitation amounts) at the same time.

Methodological Issues of Nonlinear Statistical Downscaling Models

Model Validation

Validation of all downscaling products is one of the most important issues in supplying reliable information to end users on a regional scale. Several end users are interested in having any downscaling approach correctly reproduce some specific characteristics of the climate variables of interest, at a specific space and time scale. An open European network developing a systematic validation framework of the downscaling approaches (RCMs and ESDs) has been carried through the COST Action VALUE (Maraun et al., 2015). Generally, the validation techniques are similar for all ESDs, with a few particularities for some nonlinear techniques. For example, the machine learning-based methods (DNN, SVM, and RF) automatically extract the optimum model.

There are some common assumptions for all ESDs that should be evaluated (Benestad, 2016; Takayabu et al., 2016; Vu et al., 2015):


There is a strong and stable relationship (stationarity) between predictors and predictands that also should be relevant in the future perturbed climate.


The GCM driver is skillful in reproducing the variability of large-scale predictors.


The large-scale predictors capture the climate change signal.

All ESD validation procedures follow similar steps:


The quality and quantity of observational data used in ESD development are evaluated. The main quality issues are inhomogeneities and outliers that may induce errors in estimation of real trends and extreme events (Maraun et al., 2010). Outliers are especially important in nonlinear models related to daily precipitation, but some machine learning-based techniques, such as RFs, are robust in the presence of outliers (He et al., 2016). Long enough observational time series are needed to estimate trends, extremes, and natural variability. High-resolution gridded data sets have been compiled for some regions (Haylock et al., 2008), but their quality is affected by sparse data over some areas.


Any validation procedure should be applied to an independent data set, with the models calibrated on a separate interval, but with the required observations that are long enough (Busuioc & von Storch, 2003; Wilby et al., 2004; Zorita & von Storch, 1999). For short observational time series, the cross-validation method is suggested as an alternative approach (Benestad, 2016; Gutierez et al., 2019; Maraun et al., 2015). A spatial validation can be also used, especially in the case of full-field WGs of the autoregressive type, when their output statistics are validated for independent stations not used in the model fitting (Burton et al., 2010a). In machine learning-based ESDs, the cross-validation is automatically included in the model development.


The validation procedure itself consists in quantifying the mismatch between various statistics, called indices, derived from the ESD outputs and observations using various performance measures. A comprehensive list of various climate indices and performance measures was presented by Maraun et al. (2015). Early attempts have been made to propose standard indices to be used in validation of the downscaling approaches, such as those presented by the Expert Team on Climate Change Detection and Indices, and later in the STARDEX project focusing mainly on extreme indices (Goodes et al., 2012), as well as in the ENSEMBLE project (van der Linden & Mitchell, 2009). The simplest climate indices are mean and variance, and the simplest performance measures refer to bias, mean absolute error (MAE), correlation, and root mean square errors (RMSE). The outputs of the nonlinear statistical downscaling models are mainly daily and subdaily time series, including binary events, such as wet/dry days. In this case, indices quantifying extremes represented by quantile or distributions rather than individual values are interesting, and special performance measures are used. The significance of differences between long-term means derived from observations and estimated by the downscaling procedure is estimated by Student’s t-test, while the variance is evaluated with the F-test. The validation of the probability distributions of precipitation is carried out using the χ2 test or Kolmogorov–Smirnov test (Semenov et al., 1998; Soerup et al., 2016).


Plausible ESD climate change projections require skillful GCMs with respect to the predictors used as inputs. This issue is addressed by making use of common EOFs (Benestad, 2001) and by comparing the statistics of the principal components derived from the GCMs and reanalysis (Benestad, 2016; Busuioc et al., 2006). In the case of conditional WGs on large-scale weather types, it is important to assess the GCM’s ability to reproduce the frequency, seasonality, and persistence of the weather types and the ability of weather types to predict local climate variability (Yin & Chen, 2020). For example, Murawsky et al. (2016) showed that most CMIP5 GCMs are able to capture characteristics of weather types well, while the relationships between weather type and climate variables over the Rhine catchment are quite diverse.


Evaluation of ESD credibility under nonstationary climates is needed. It is important to distinguish between stationarity of the statistical relations and nonstationarity under global warming. Sarhadi et al. (2017) proposed a strategy to examine the credibility of an SVM-based statistical downscaling under nonstationary conditions. This is based on an implicit assumption that climate conditions in the current time or future will have “signatures” over time and space. For any specific region, future climate states and empirical relations may be assumed to have signatures in the past climate.

Intercomparison of Downscaling Methods

Studies on ESD models proved that it is difficult to find a single model that works best for all climates—that is, to find a magic solution for everything (Takayabu et al., 2016)—and this conclusion is also true for nonlinear models. A comprehensive overview of this issue was presented by Christensen et al. (2007a) and of Europe was presented by Gutierez et al. (2019). Other comparisons have been made over some limited areas. For example, analyzing the performance of six ESDs, including some nonlinear models, in their ability to reproduce the precipitation statistics for a European region of complex topography (Alps), Schmidli et al. (2007) concluded that the results were dependent on the season, downscaling method, and region. Hellström et al. (2001) found that both statistical and dynamic downscaling models show nearly equal ability to reproduce the seasonal precipitation cycle in Sweden. A similar conclusion with respect to performance of the dynamic and statistical downscaling techniques in reproducing extreme events over various European regions has been drawn from the STARDEX project (Goodess et al., 2012) and ENSEMBLES project (van der Linden & Mitchel, 2009).

Even for the same class of ESD models, their performance could be different for various regions. For example, Vu et al. (2018) compared five single-site precipitation WGs for four diverse climate regions (Mediterranean climate, western United States, temperate climate of eastern Australia, and tropical monsoon region in northern Vietnam) and found that all the WGs performed differently with respect to climate regime and various precipitation statistics. However, Semenov et al. (1998) found that the semi-empirical distributions used by the LARS-WG model are more appropriate than WGEN in reproducing some observed data characteristics for various sites in the United States, Europe, and Asia. Mehrotra et al. (2006) found that the Wilks model has an overall advantage over two other multisite WGs (K-NN and the parametric hidden Markov model) for simulation of point rainfall occurrence over a homogeneous region in Australia.

Studies regarding ANN performance in comparison to other ESDs have been carried out over small areas and found that the result is dependent on the predictand type and ANN version. For example, in the case of daily precipitation amount, the model’s performance seems to be poor (Weichert & Bürger, 1998; Wilby et al., 1998), while in the case of temporal smoothing precipitation (5-day moving average filter), the performance is improved (Hewitson & Crane, 1996). The ANNs were found to be the best (among five other ESDs) at modeling the interannual variability of the extreme indices in the United Kingdom (Haylock et al., 2006). Gaitan et al. (2014) found that the nonlinear Bayesian neural network (BNN) outperforms the linear regression in downscaling of maximum and minimum temperatures in Quebec, Canada. However, Huth et al. (2008) compared several linear and nonlinear statistical downscaling methods for winter daily temperature at eight European stations according to four performance criteria, and they found that the pointwise linear regression appears to be the best method when considering all the criteria together, while the deviations of statistical distributions from normality are only captured by the neural networks, and the classification methods yield the best spatial correlations.

Only a few studies have carried out ESD intercomparisons with new machine learning methods. Pang et al. (2017) compared three machine learning methods (RF, SVM, and ANN) and multiple linear regression (MLR) to downscale daily mean temperature, and they found that RF performance is higher than that of the other models, according to five selected performance criteria. Hybrid models combining the RF and SVM approaches improved the daily precipitation downscaling, including extremes (Pham et al., 2019; Pour et al., 2016). However, in other cases, the linear models were more performant. For example, Vandal et al. (2019) found that linear models based on Bias Correction Spatial Disaggregation (BCSD) outperform various nonlinear approaches based on machine learning methods in downscaling of daily precipitation. In other cases, comparing BCSD with other quantile-mapping-based techniques used in downscaling of daily temperature and precipitation, Yang et al. (2019) found that, even if overall BCSD outperforms other methods, there are differences in terms of some extreme indices. Furthermore, multiple linear regression has been proven to have better performance than the ANN and RF models in downscaling of minimum temperature predicted by meteorological models (Eccel et al., 2007).

On the other hand, Zorita and von Storch (1999) found that the simple analog method performs better than other, more complicated methods. This conclusion could be supported also by the ability of this statistical downscaling technique to be transferable to a future climate (Dayon et al., 2015).

Therefore, it appears that more downscaling approaches should be tested over a homogeneous region to cover as much as possible the uncertainties related to statistical downscaling methods. It is an advantage to include as many different types of downscaling models, global models, and emission scenarios as possible when developing local climate change scenarios (Haylock et al., 2006). Also, it’s good to keep in mind that it is not always the most complicated method that gives a more certain result.

Predictor Selection

Appropriate selection of the large-scale predictors is a crucial issue in developing any skillful and credible statistical downscaling model. The main aspects taken into account for predictor selection in nonlinear approaches are generally similar to those for all ESDs, with some differences. The first aspect is related to the size of the predictor domain. The domain of large-scale atmospheric variables should be large enough to capture the synoptic forcing features represented by the GCM (Hewitson & Crane, 1996). The second aspect is related to predictor selection itself.

Any ideal predictor for any ESD should have the following characteristics (Benestad, 2016; Dayon et al., 2015; Maraun et al., 2010; Maraun & Widmann, 2018):


It should explain a large part of the local predictand variability, including long-term trends.


It should be realistically reproduced by a GCM.


It should make physical sense and should capture well the future climate change signal.

Comprsehensive overviews of predictor selection in developing various ESDs have been presented in many studies (Goodess et al., 2012; Gutierrez et al., 2019; Maraun et al., 2010; Wilby et al., 2004). It is recommended that the candidate predictor suite contain variables describing temperature, sea-level pressure, atmospheric circulation, thickness, stability, and moisture content (Vu et al., 2015). A literature review of predictor selection for various statistical downscaling techniques (linear and nonlinear) was presented by Anandhi et al. (2008). It has been found that a combination of predictors leads to a statistical downscaling model with better performance than a downscaling model built with a single predictor, and the best choice is a combination of dynamic and thermodynamic variables. The raw large-scale predictors are used in nonlinear ESDs based on analogs for downscaling daily variables (Dayon et al., 2015; Schmidli et al., 2007). For regression-type nonlinear models, the raw predictors are usually standardized (by subtraction of mean and division by standard deviation) when downscaling either daily variables (Chen et al., 2010b; Pham et al., 2019; Vu et al., 2015) or monthly variables (Anandhi et al., 2008). The case is similar for linear ESDs (Busuioc et al., 2006; Huth, 2004; Schmidli et al., 2007).

The various approaches to identifying significant predictors are mainly dependent on the type of predictor–predictand relationship and the stability of model performance on the calibration/validation interval. For example, correlation coefficient and stepwise regression are more appropriate for predictor selection in linear downscaling techniques, while mutual information and decision tree are more appropriate for nonlinear techniques (Nourani et al., 2019; Teegavarapu & Goly, 2018). Machine learning techniques have been proven effective in capturing highly nonlinear relationships between predictors and predictands, and RF-based techniques have the ability to automatically rank the relative influence of the predictors (He et al., 2016). However, most machine learning techniques (ANNs, SVM, and RF) are considered black boxes, where the relationships between predictors and predictands remain hidden. Sachindra and Kanae (2019) proposed a novel challenging approach, based on the use of parallel multiple populations in genetic programming, to identify an optimum set of predictors and to filter out irrelevant and redundant information in the set of predictors. This approach produces explicit predictor–predictand relationships, and it has been tested for downscaling of large-scale climate data to daily minimum and maximum temperatures.

Usually, the raw large-scale predictors are high-dimensional grid-point fields. Various predictor transformation methods are used for dimensionality reduction and to eliminate noise. The EOF technique is widely used for this objective, being applied either to each predictor variable (Anandhi et al., 2008; Busuioc & von Storch, 2003; Hewiston & Crane, 1996; Zorita & von Storch, 1999) or to combined predictors (Benestad, 2001; Busuioc et al., 2006; Gaitan et al., 2014). This technique is unsupervised, ignoring the dependency on the response variable, and is also inappropriate for dimension reduction purposes for nonlinear predictors (Sarhadi et al., 2017). Barshan et al. (2011) proposed Supervised Principal Component Analysis (S-PCA), which is a generalization of PCA, extracting the principal components of predictors that have maximal dependency on the predictand. A kernel version of the S-PCA (K-S-PCA) to capture the nonlinear variability between predictors and predictand has been proposed by Sarhadi et al. (2017).

Another physically motivated transformation is given by weather types, which are straightforward ways of allowing for nonlinear relations between large-scale predictors and predictands (Maraun et al., 2010). Weather types are mostly used to condition WGs (Bardossy & Plate, 1992; Corte-Real et al., 1999; Fowler et al., 2005; Qian et al., 2002). In this case, weather types should include both dynamic and thermodynamic properties (Maraun & Widmann, 2018; Yin & Chen, 2020).

Issues in Statistical Downscaling of Climate Change Projections

Downscaling of future climate change scenarios based on nonlinear statistical downscaling models developed on observations requires that the selected models, proved to be valid for the current climate, be also valid in the future under uncertain climate conditions. Additionally, taking into account the performance of various GCMs used as drivers for statistical downscaling models, some issues need to be addressed to supply credible regional climate change projections for various user needs. These issues are the same for all ESDs and are the stationarity or transferability of statistical relationships in the future and various uncertainties related to GCM drivers.


The transferability or credibility of the established statistical relationship in the future is a challenging issue, since future climate conditions could be very different from the current conditions used to develop the statistical relationship. Some studies tried to solve this problem by calibrating a model on the coldest (driest) years and then validating it on the warmest (wettest) years, and vice versa (Liu et al., 2015; Sarhadi et al., 2017). Other contrasting climate conditions could be El Niño–Southern Oscillation (ENSO) versus non-ENSO years, and vice versa. A more complex technique has been proposed (Busuioc et al., 1999) based on the assumption that, if there is a similarity between future ESD climate change signals and those derived directly from the GCM grid-point data, and if GCM is able to reproduce the empirically observed ESD link, it can be concluded that the established downscaling technique is also valid for the future changed climate.

Other methods to evaluate the statistical downscaling model’s transferability to the future climate refer to a perfect model framework (Erlandsen et al., 2020), by comparing the future predictand changes simulated directly from an RCM with statistical downscaled changes resulting from the development of the predictand–predictor relationship using the RCM simulations (driven by reanalysis), the comparison being made at the RCM native resolution. A similarity between the two climate change signals can be considered evidence of statistical downscaling transferability. Additionally, if RCMs are statistically downscaled using the relationship built with reanalyses (predictors) and observed predictands (real case), a similarity between the future predictand changes derived through the statistical downscaling model and directly from the RCM simulations, greatly increases the confidence in the realism of the RCM and statistical downscaling results (Dayon et al., 2015; Vrac et al., 2007). Dayon et al. (2015) also used this technique to select the optimum combination of large-scale predictors for analog methods (i.e., the combination giving the best statistical downscaling performance for the current climate as well as for similar future climate change signals).

The credibility of statistical downscaling models under nonstationary climate has also been addressed by Sarhadi et al. (2017) in the context of using two machine learning methods (support vector regression and relevance vector machine) in downscaling of monthly precipitation. They argued that if the model is calibrated under nonstationary conditions, it will be qualified for future generalization under approximate nonstationarity. The main idea is based on an implicit assumption that climate conditions in the current time or the future will have “signatures” over time and space.


Quantifying uncertainty in climate change projections is crucial. Uncertainty derives from anthropogenic climate forcing (represented by so-called emissions scenarios), GCM response (determined by model physics and subgrid parameterizations), and natural climate variability induced by internal variability of the chaotic climate system and natural forcings, such as solar activity (Benestad, 2016; Deser et al., 2012; Maraun et al., 2010). The additional uncertainty derived from the downscaling technique (RCMs and ESDs) is also important in regional climate change projections. The uncertainty related to the first two sources and downscaling techniques can be addressed by using various future trajectories of anthropogenic emissions of greenhouse gases and GCM/RCM ensembles (multimodel ensembles), respectively, and they can be potentially reduced (Benestad, 2016; Deser et al., 2012). Important initiatives, such as the PRUDENCE, ENSEMBLES, CMIP5, and CORDEX projects, as well as the CMIP6 project (in progress), are examples.

However, uncertainty due to natural variability is unlikely to be reduced, given the inherently unpredictable nature of unforced climate fluctuations beyond a few years to decades (Deser et al., 2012). The relative roles of various sources of uncertainty were summarized by Maraun et al. (2010). The concept of the “uncertainty cascade” was also challenged by Benestad et al. (2017), who highlighted that the different downscaling approaches introduce both errors as well as information/constraints, and if the constraints are stronger than the errors, then the consequence is reduced uncertainty rather than increased.

Uncertainty due to ESD has been only partially addressed in the ENSEMBLES project (van der Linden & Mitchel, 2009), but the European COST VALUE initiative is designed to validate and compare RCMs and ESDs, with the aim of enhancing multidisciplinary collaboration among impact modelers, climatologists, statisticians, and stakeholders. Synthesis of this work has been presented by Maraun et al. (2015), and intercomparison of a large ESD ensemble was presented by Gutierrez et al. (2019). The main conclusion was that, even if most ESDs improve the raw model (reanalysis of RCM) biases, no approach seems to be superior. A similar conclusion has been also drawn in other studies by intercomparison of various downscaling methods over limited areas, such as comparison of RCMs and ESDs (Busuioc et al., 2006; Haylock et al., 2006; Hellström et al., 2001; Schmidli et al., 2007; Soerup et al., 2016; Tang et al., 2016); of various types of ESDs, including linear and nonlinear approaches (Ben Alaya et al., 2015; Gaitan et al., 2014; Huth et al., 2008; Pang et al., 2017; Vandal et al., 2019; Weichert & Bürger, 1998; Wilby et al., 1998; Zorita & von Storch, 1999); or of various types of nonlinear models (Evin et al., 2018; Mehrotra et al., 2006; Semenov et al., 1998; Vu et al., 2018). An interesting method for evaluating the uncertainty of the ESDs was presented by Rajendran and Cheung (2015).

Impressive development of nonlinear statistical downscaling models for many places in the world occurred in the first decades of the 21st century, especially in the category of WGs of various complexities and machine learning-based approaches. In comparison to the high computational cost of RCMs, nonlinear statistical downscaling models can be used to build large ensembles of local climate change scenarios. However, a limited number of nonlinear statistical downscaling models have addressed the uncertainty issue by using large ensembles of GCM drivers (Burton et al., 2010b; Dayon et al., 2015; Dobler et al., 2013; Liu et al., 2015; Okkan & Kirdemir, 2016). Most climate scenarios based on WGs (mainly single-site WGs) used a change factor computed from the RCM simulations to perturb their parameters (Blekinsop et al., 2013; Burton et al., 2010b; Chen et al., 2012; Dobler et al., 2013; Soerup et al., 2016). However, machine learning-based models, such as SVMs and RFs, are still in an early phase of development, and they have not been used yet to construct local climate change scenarios.

The credibility of statistically downscaled results is also influenced by the performance of the GCM (RCM) in reproducing the variability of the predictors used as drivers in the statistical downscaling model (Benestad, 2016; Busuioc et al., 2006; Takayabu et al., 2016; Trigo & Palutokof, 2001). Therefore, GCM/RCM validation is an essential issue in any downscaling technique (RCM or ESDs). Mehrotra and Sharma (2010) adjusted the GCM predictors for current and future climate before their application in a stochastic downscaling procedure. However, there is no proof that this adjustment is valid for the future.

Added Value and Future Challenges

Generally, the main advantage of any ESD is its ability to supply information at point scale, as needed in many impact studies, based on physically documented large-scale information derived from GCMs. In particular, nonlinear statistical downscaling techniques are able to provide information on various statistics for main climate variables (temperature and precipitation) at high spatial (up to 1 km) and temporal scales (up to 1 hr or less), as are needed in many hydrological and agricultural applications.

Research studies carried out since the end of the 20th century regarding the development of complex nonlinear models, especially those related to multisite and full-field WGs, have provided substantial added value in reproducing various precipitation statistics (mean wet day frequency and intensity, and frequency of dry and wet spell lengths) in complex topography, such as an Alpine river catchment (see Keller et al., 2015; Terzago et al., 2018) or the Spanish Pyrenees (Burton et al., 2010a); extreme value distributions at the individual and spatial scales (Hundecha et al., 2009); statistical features of extreme rare events (see Evin et al., 2018; Furrer & Katz, 2008); and spatial patterns of extreme precipitation at an urban scale of 1-hr temporal resolution and a 2-km grid (Soerup et al., 2016). In this way, the low performance of statistical downscaling in simulations of temperature and precipitation extremes has been much improved. The added value of some spatiotemporal WGs (in comparison to RCMs) in capturing the natural variability and extremes of precipitation has been also highlighted (Burton et al., 2010b).

There has also been important progress in developing new nonlinear statistical downscaling models based on machine learning approaches (ANNs, SVMs, and RFs), which became very popular due to their ability to automatically extract information from very large predictor and predictand data sets, without the need to construct explicit physical or statistical models (see He et al., 2016; Pan et al., 2019; Pang et al., 2017; Sahardi et al., 2017; Schoof & Prior, 2001). These models are used in a variety of applications, such as improving weather predictions (Eccel et al., 2007), seasonal prediction (Mishra et al., 2007), better parameterization of subgrid-scale processes (Pan et al., 2019), and statistical downscaling at various time scales (daily or monthly; Pang et al., 2017; Pham et al., 2019; Sahardi et al., 2017; Vandal et al., 2019). However, only a few SVM- or RF-based downscaling models have been used to construct local climate change scenarios, which remain a challenge for the near future.

As an alternative to most machine learning-based downscaling methods (ANN, SVM, and RF), which are considered black boxes, a novel challenging approach based on the use of parallel multiple populations in genetic programming has been proposed (Sachindra & Kanae, 2019). It aims to produce explicit predictor–predictand relationships, to identify an optimum set of predictors, and to filter out irrelevant and redundant information in the predictor data set.

Due to their low computational cost compared to RCMs, nonlinear ESDs are suitable for downscaling of large GCM/RCM ensembles (Burton et al., 2010b; Dobler et al., 2013; Liu et al., 2015) to assess the uncertainties on projection of future climate changes on regional scale. An important future challenge for the nonlinear models is their combination with RCMs, with a priori RCM validation (see Dayon et al., 2015; D’Onofrio et al., 2014; Takayabu et al., 2016).

Hybrid statistical downscaling approaches combining various nonlinear models and using the advantages of each method are a good way to approach seasonal predictions of indices describing extreme events, such as drought (Mishra et al., 2007), or improving model performance in downscaling of daily rainfall, including extremes (Pham et al., 2019). Hybrid statistical models based on nonlinear conditional WGs also offer a good way to improve the downscaling of multisite precipitation on daily and subdaily scales, including distribution of wet and dry spell length, extreme events, and low-frequency variability (Fowler et al., 2005; Lee & Sing, 2019; Mehrotra & Sharma, 2005, 2010). Improvements of these approaches should be explored.

Statistical downscaling techniques can be also used as diagnostic tools for RCM validation via their ability to reproduce the spatial characteristics of predictors and the strength of the connection between predictor and predictands (see Busuioc et al., 2006; Takayabu et al., 2016). Inclusion of the RCM in a downscaling chain from GCMs to high-spatial-resolution daily precipitation via a stochastic downscaling technique (e.g., RainFARM) has been proven to increase the stochastic downscaling performance (D’Onofrio et al., 2014). A similar technique has also been applied by Dayon et al. (2015) using analog statistical downscaling.

The main disadvantages of all ESDs are their need for long-enough, high-quality, and high-resolution observational data to be fitted and validated. However, these data are also needed for RCM validation. Extension of the successful research results presented in this article by providing high-resolution spatial and temporal climate information is dependent on the availability of these data, which is a big challenge for many places. New observation points would be very useful. Using satellite information to obtain high-resolution spatial precipitation data could be one solution, in light of improved techniques like statistical downscaling based on ANN (see Pan et al., 2019; Tao et al., 2016).

However, even if great progress has been made in developing proficient nonlinear models for reproducing various precipitation and temperature statistics on high spatial and temporal scales, there are still few large ensembles of regional climate change scenarios based on these techniques (Burton et al., 2010b; Dayon et al., 2015; Dobler et al., 2013; Liu et al., 2015; Okkan & Kirdemir, 2016). The main reasons could be the complexity of the models, which assume high standard statistical knowledge, as well as the difficulty of adapting some of them to the climate change context (i.e., to perturb their parameters for large-scale variability). To overcome this problem, “change factor” and pattern-scaling techniques have been extensively used until now. For example, Soerup et al. (2016) used the change factor derived from 13 RCMs to perturb the parameters of a full-field WG based on a spatiotemporal NSRP model and obtained more realistic spatial patterns of extremes than those observed in RCM precipitation. Similar results have been obtained by Burton et al. (2010b) using the same model.

The current development of higher spatial resolution GCMs and RCMs raises questions about the future utility of RCMs in general, and of ESDs in particular. Taking into account the weaknesses of state-of-the-art GCM and RCM performance in reproducing regional/local climate variability that are not related to spatial resolution, both RCMs and ESDs should be still regarded as alternative research tools for addressing some scientific questions in the future, and their use together adds supplementary confidence to the local climate change signal. Important other questions are described in the World Climate Research Programme’s “Grand Challenges,” one of which is weather and climate extremes. Addressing this question in the context of downscaling requires much effort from climate scientists, as well as their robust theoretical knowledge of meteorology, climatology, and statistics. For example, Yin and Chen (2020) suggested using the progress in the extreme theories (Papalexiou et al., 2018) and development of statistical methods in nonlinear and high-dimensional dependence modeling, to improve the simulation of extreme values (Hao & Singh, 2016).

Further Reading

  • Biesbroek, R., Badloe, S., & Athanasiadis, I. N. (2020). Machine learning for research on climate change adaptation policy integration: an exploratory UK case study. Regional Environmental Change, 20, 85.
  • Cowpertwait, P., Ocio D., Collazos G., De Cos, O., & Stocker, C. (2013). Regionalised spatiotemporal rainfall and temperature models for flood studies in the Basque Country, Spain. Hydrology and Earth System Science, 17, 479–494.
  • Furrer, E. M., & Katz, R. W. (2007). Generalized linear modeling approach to stochastic weather generators. Climate Research, 34, 129–144.
  • Hashmi, M. Z., Shamseldin, A. Y., & Melville, B. W. (2011). Statistical downscaling of watershed precipitation using Gene Expression Programming (GEP). Environmental Modelling & Software, 26(12), 1639–1646.
  • Kannan, S., & Ghosh, S. (2010). Prediction of daily rainfall state in a river basin using statistical downscaling from GCM output. Stochastic Environmental Research and Risk Assessment, 25(4), 457–474.
  • Maraun, D. (2019). Statistical downscaling for climate science. Oxford research encyclopedia of climate science. Oxford University Press.
  • Mezghani, A., & Hingray, B. (2009). A combined downscaling-disaggregation weather generator for stochastic generation of multisite hourly weather variables over complex terrain: Development and multi-scale validation for the Upper Rhone River basin. Journal of Hydrology, 377, 245–260.
  • Parding, K. M., Dobler, A., McSweeney, C. F., Landgren, O. A., Benestad, R., Erlandsen, H. B., Mezghani, A., Gregow, H., Räty O., Viktor, E., El Zohbi, J., Christensen, O. B., & Loukos, H. (2020). GCMeval—An interactive tool for evaluation and selection of climate model ensembles. Climate Services, 18, 100167.
  • Park, J., Onof, C., & Kim, D. (2019). A hybrid stochastic rainfall model that reproduces some important rainfall characteristics at hourly to yearly timescales, Hydrology and Earth System Science, 23, 989–1014.
  • Sachindra, D. A., Ahmed, K., Rashid, Md. M., Shahid, S., & Perera, B. J. C. (2018). Statistical downscaling of precipitation using machine learning techniques. Atmospheric Research, 212, 240–258.


  • Anandhi, A., Srinivas, V. V., Nanjundiah, R. S., & Kumar, D. N. (2008). Downscaling precipitation to river basin in India for IPCC SRES scenarios using support vector machine. International Journal of Climatology, 28, 401–420.
  • Bardossy, A., Bogardi, I., & Matyasovszky, I. (2005). Fuzzy rule-based downscaling of precipitation. Theoretical and Applied Climatology, 82(1–2), 119–129.
  • Bardossy, A., & Plate, E. J. (1992). Space–time model for daily rainfall using atmospheric circulation patterns. Water Resources Research, 28, 1247–1259.
  • Barnett, T., & Preisendorfer, R. (1978). Multifield analog prediction of short-term climate fluctuations using a climate state vector. Journal of Atmospheric Science, 35, 1771–1787.
  • Barshan, E., Ghodsi, A., Azimifar, Z., & Zolghadri Jahromi, M. (2011). Supervised principal component analysis: Visualization, classification and regression on subspaces and submanifolds. Pattern Recognition, 44, 1357–1371.
  • Beersma, J. J., & Buishand, T. A. (2003). Multi-site simulation of daily precipitation and temperature conditional on the atmospheric circulation. Climate Research, 25, 121–133.
  • Ben Alaya, M. A., Hebana, C., & Ouarda, T. B. M. J. (2015). Probabilistic multisite statistical downscaling for daily precipitation using a Bernoulli-generalized Pareto multivariate autoregressive model. Journal of Climate, 28, 2349–2364.
  • Ben Daoud, A., Sauquet, E., Lang, M., Bontron, G., & Obled, C. (2011). Precipitation forecasting through an analog sorting technique: A comparative study. Advances in Geosciences, 29, 103–107.
  • Benestad, R. E. (2001). A comparison between two empirical downscaling strategies. International Journal of Climatology, 21, 1645–1668.
  • Benestad, R. E. (2010). Downscaling precipitation extremes: Correction of analog models through PDF predictions. Theoretical and Applied Climatology, 100, 1–21.
  • Benestad, R. E. (2016). Downscaling climate information. Oxford research encyclopedia of climate science. Oxford University Press.
  • Benestad, R. E., Sillmann, J., Thorarinsdottir, T. L., Guttorp, P., Mesquita, M. D. S., Tye, M. R., Uotila, P., Maule, C. F., Thejll, P., Drews, M., & Parding, K. M. (2017). New vigour involving statisticians to overcome ensemble fatigue. Nature Climate Change, 7(10), 697–703.
  • Bishop, C. M. (1995). Neural networks for pattern recognition. Oxford University Press.
  • Blenkinsop, S., Harpham C., Burton, A., Goderniaux, P., Brouyère, S., & Fowler, H. J. (2013). Downscaling transient climate change with a stochastic weather generator for the Geer catchment, Belgium. Climate Research, 57(2), 95–109.
  • Breiman, L. (2001). Random forests. Machine Learning, 45(1), 5–32.
  • Buishand, T. A., & Brandsma, T. (2001). Multisite simulation of daily precipitation and temperature in the Rhine basin by nearest-neighbour resampling. Water Resources Research, 37, 2761–2776.
  • Burton, A., Fowler, H. J., Blenkinsop, S., & Kilsby, C. G. (2010b). Downscaling transient climate change using a Neyman-Scott Rectangular Pulses stochastic rainfall model. Journal of Hydrology, 381(1–2), 18–32.
  • Burton, A., Fowler, H. J., Kilsby, C. G., & O’Connell, P. E. (2010a). A stochastic model for the spatial-temporal simulation of nonhomogeneous rainfall occurrence and amounts. Water Resources Research, 46, W11501.
  • Busuioc, A., Giorgi, F., Bi, X., & Ionita, M. (2006). Comparison of regional climate model and statistical downscaling simulations of different winter precipitation change scenarios over Romania. Theoretical and Applied Climatology, 86, 101–124.
  • Busuioc, A., & von Storch, H. (2003). Conditional stochastic model for generating daily precipitation time series. Climate Research, 24, 181–195.
  • Busuioc, A., von Storch, H., & Schnur, R. (1999). Verification of GCM generated regional precipitation and of statistical downscaling estimates. Journal of Climate, 12(1), 258–272.
  • Cavazos, T. (1999). Large-scale circulation anomalies conducive to extreme events and simulation of daily precipitation in northeastern Mexico and southeastern Texas. Journal of Climate, 12, 1516–1523.
  • Cavazos, T., Comrie, A. C., & Liverman, D. M. (2002). Intraseasonal variability associated with wet monsoons in southeast Arizona. Journal of Climate, 15, 2477–2490.
  • Chen, J., Brissette, F. P., & Leconte, R. (2012). Downscaling of weather generator parameters to quantify hydrological impacts of climate change. Climate Research, 51, 185–200.
  • Chen, H., Guo, J., Xiong, W., Guo, S., & Xu, C-Y. (2010b). Downscaling GCMs using the Smooth Support Vector Machine method to predict daily precipitation in the Hanjiang Basin. Advances in Atmospheric Sciences, 27(2), 274–284.
  • Chen, S.-T., Yu, P.-S., & Tang, Y.-H. (2010a). Statistical downscaling of daily precipitation using support vector machines and multivariate analysis. Journal of Hydrology, 385, 13–22.
  • Christensen, J. H., Carter, T. R., Rummukainen, M., & Amanatidis, G. (2007b). Evaluating the performance and utility of regional climate models: The PRUDENCE project. Climatic Change, 81(1), 1–6.
  • Christensen, J. H., Hewitson, B., Busuioc, A., Chen, A., Gao, X., Held, I. R., Jones, R., Kolli, R. K., Kwon, W. T., Laprise, R., Rueda, V., Mearns, L. O., Menéndez, C. G., Räisänen, J., Rinke, A., Sarr, A., & Whetton, P. (2007a). Regional climate projections. In Climate change 2007: The physical science basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press.
  • Corte-Real, J., Quian, B., & Xu, H. (1999). A weather generator for obtaining daily precipitation scenarios based on circulation patterns. Climate Dynamics, 15(12), 921–935.
  • Crick, F. (1989). The recent excitement about neural networks. Nature, 337, 129–132.
  • Davy, R. J., Woods, M. J., Russell, C. J., & Coppin, P. A. (2010). Statistical downscaling of wind variability from meteorological fields. Boundary-Layer Meteorology, 135(1), 161–175.
  • Dayon, G., Boé, J., & Martin, E. (2015). Transferability in the future climate of a statistical downscaling method for precipitation in France. Journal of Geophysical Research: Atmospheres, 120, 1023–1043.
  • Deser, C., Knutti, R., Solomon, S., & Phillips, A. S. (2012). Communication of the role of natural variability in future North American climate. Nature Climate Change, 2, 775–779.
  • Dibike, Y. B., & Coulibaly, P. (2006). Temporal neural networks for downscaling climate variability and extremes. Neural Networks, 19, 135–144.
  • Dobler, C., Bürger, G., & Stötter, J. (2013). Simulating future precipitation extremes in a complex Alpine catchment. Natural Hazards and Earth System Science, 13, 263–277.
  • D’Onofrio, D., Palazzi, E., von Handerberg, J., Provenzale, A., & Calmanti, S. (2014). Stochastic rainfall downscaling of climate models. Journal of Hydrometeorology, 15, 830–843.
  • Eccel, E., Ghielmi, L., Granitto, P., Barbiero, R., Grazzini, F., & Cesari, D. (2007). Prediction of minimum temperatures in an alpine region by linear and non-linear post-processing of meteorological models. Nonlinear Processes in Geophysics, 14(3), 211–222.
  • Erlandsen, H. B., Parding, K. M., Benestad, R., & Mezghani, A. (2020). A hybrid downscaling approach for future temperature and precipitation change. Journal of Applied Meteorology and Climatology, 59(11), 1793–1807.
  • Evin, G., Favre, A. C., & Hingray, B. (2018). Stochastic generation of multi-site daily precipitation focusing on extreme events. Hydrology and Earth System Science, 22, 655–672.
  • Fowler, H. J., Kilsby, C. G., & O’Connell, P. E. (2000). A stochastic rainfall model for the assessment of regional water resource systems under changed climatic condition. Hydrology and Earth System Sciences, 4(2), 263–281.
  • Fowler, H. J., Kilsby, C. G., O’Connell, P. E., & Burton, A. (2005). A weather‐type conditioned multi‐site stochastic rainfall model for the generation of scenarios of climate variability and change. Journal of Hydrology, 308, 50–66.
  • Furrer, E. M., & Katz, R. W. (2008). Improving the simulation of extreme precipitation events by stochastic weather generators. Water Resources Research, 44, W12439.
  • Gaitan, C. F., Hsieh, W. W., Cannon, A. J., & Gachon, P. (2014). Evaluation of linear and non-linear downscaling methods in terms of daily variability and climate indices: Surface temperature in southern Ontario and Quebec, Canada. Atmosphere-Ocean, 52(3), 211–221.
  • Giorgi, F., & Mearns, L. O. (1999). Introduction to special section: Regional climate modeling revisited. Journal of Geophysical Research, 104, 6335–6352.
  • Goodess, C. M., Anagnostopoulou, C., Bárdossy, A., Frei, C., Harpham, C., Haylock, M. R., Hundecha, Y., Maheras, P., Ribalaygua, J., Schmidli, J., Schmith, T., Tolika, K., Tomozeiu, R., & Wilby, R. L. (2012). An intercomparison of statistical downscaling methods for Europe and European regions: Assessing their performance with respect to extreme weather events and implications for climate change applications (Climate Research Unit Research Report). Climate Research Unit, University of East Anglia.
  • Guilbaud, S., & Obled, C. (1998). Prévision quantitative des précipitations journalières par une technique de recherche de journées antérieures analogues: Optimisation du critère d’analogie[Daily quantitative precipitation forecast by an analogue technique: Optimisation of the analogy criterion]. Comptes Redus de l’Académie des Science—Series IIA—Earth and Planetary Science, 327(3), 181–188.
  • Gutierrez, J. M., Maraun, D., Widman, M., Huth, R., Hertig, E., Benestad, R., Roessler, O., Wibig, J., Wilcke, R., Kotlarski, S., San Martín, D., Herra, S., Bedia, J., Casanueva, A., Manzanas, R., Iturbide, M., Vrac, M., Dubrovsky, M., Ribalaygua, J. . . . Pagé. C. (2019). An intercomparison of a large ensemble of statistical downscaling methods over Europe—Results from the VALUE perfect predictor cross-validation experiment. International Journal of Climatology, 39(9), 3750–3785.
  • Hao, Z., & Singh, V. P. (2016). Review of dependence modeling in hydrology and water resources. Progress in Physical Geography: Earth and Environment, 40(4), 549–578.
  • Hawkins, E., Robson, J., Sutton, R., Smith, D., & Keenlyside, N. (2011). Evaluating the potential for statistical decadal predictions of sea surface temperatures with a perfect model approach. Climate Dynamics, 37(11–12), 2495–2509.
  • Haylock, M. R., Cawley, G. C., Harpham, C., Wilby, R. L., & Goodess, C. M. (2006). Downscaling heavy precipitation over the United Kingdom: A comparison of dynamical and statistical methods and their future scenarios. International Journal of Climatology, 26(10), 1397–1415.
  • Haylock, M. R., Hofstra, N., Klein Tank, A. M. G., Klok, E. J., Jones, P. D., & New, M. (2008). A European daily high-resolution gridded data set of surface temperature and precipitation for 1950–2006. Journal of Geophysical Research, 113, D20119.
  • He, X., Chaney, N. W., Schleiss, M., & Sheffield, J. (2016). Spatial downscaling of precipitation using adaptable random forests. Water Resources Research, 52, 8217–8237.
  • Hellström, C., Chen, D., Achberger, C., & Räisänen, J. (2001). Comparison of climate change scenarios for Sweden based on statistical and dynamical downscaling of monthly precipitation. Climate Research, 19, 45–55.
  • Hertz, J., Krogh, A., & Palmer, R. G. (1991). Introduction to the theory of neural computation. Addison-Wesley.
  • Hess, P., & Brezowsky, H. (1969). Katalog der Groflwetterlagen Europas. Berichte des Deutschen Wetterdienstes, 15 .
  • Hewitson, B. C., & Crane, R. G. (1996). Climate downscaling techniques and applications. Climate Research, 7, 85–95.
  • Hsieh, W. W., & Tang, B. (1998). Applying neural network models to prediction and data analysis in meteorology and oceanography. Bulletin of the American Meteorological Society, 79(9), 1855–1870.
  • Hughes, J. P., & Guttorp, P. (1994). A class of stochastic models for relating synoptic atmospheric patterns to regional hydrologic phenomena. Water Resources Research, 30, 1535–1546.
  • Hundecha, Y., Pahlow, M., & Schumann, A. (2009). Modeling of daily precipitation at multiple locations using a mixture of distributions to characterize the extremes. Water Resources Research, 45, W12412.
  • Huth, R. (2004). Sensitivity of local daily temperature change estimates to the selection of downscaling models and predictors. Journal of Climate, 17, 640–652.
  • Huth, R., Kliegrová, S., & Metelka, L. (2008). Non-linearity in statistical downscaling: Does it bring an improvement for daily temperature in Europe? International Journal of Climatology, 28(4), 465–477.
  • Imbert, A., & Benestad, R. E. (2005). An improvement of analog model strategy for more reliable local climate change scenarios. Theoretical and Applied Climatology, 82, 245–255.
  • Keller, D. E., Fischer, A. M., Frei, C., Liniger, M., Appenzeller, C., & Knutti, R. (2015). Implementation and validation of a Wilks-type multi-site daily precipitation generator over a typical Alpine river catchment. Hydrology and Earth System Science, 19, 2163–2177.
  • Kilsby, C. G., Jones, P. D., Burton, A., Ford, A. C., Fowler, H. J., Harpham, C., James, P., Smith, A., & Wilby, R. L. (2007). A daily weather generator for use in climate change studies. Environmental Modelling & Software, 22(12), 1705–1719.
  • Kim, Y., Katz, R. W., Rajagopalan, B., Podestá, G. P., & Furrer, E. M. (2012). Reducing overdispersion in stochastic weather generators using a generalized linear modeling approach. Climate Research, 53, 13–24.
  • Kossieris, P., Makropoulos, C., Onof, C., & Koutsoyiannis, D. (2016). A rainfall disaggregation scheme for sub-hourly time scales: Coupling a Bartlett-Lewis based model with adjusting procedures. Journal of Hydrology, 556, 980-992.
  • Kuhnlein, M., Appelhans, T., Thies, B., & Nauß, T. (2014). Precipitation estimates from MSG SEVIRI daytime, nighttime, and Twilight data with random forests. Journal of Applied Meteorology and Climatology, 53(11), 2457–2480.
  • Lall, U., & Sharma, A. (1996). A nearest neighbor bootstrap for resampling hydrologic time series. Water Resources Research, 32(3), 679–693.
  • Lamb, H. H. (1972). British Isles weather types and a register of daily sequence of circulation patterns, 1861–1971. In Geophysical Memoir 116. HMSO.
  • Lee, J., Weger, R. C., Sengupta, S. K., & Welch, R. M. (1990). A neural network approach to cloud classification. IEEE Transaction on Geoscience and Remote Sensing, 28, 846–855.
  • Lee, T., & Sing, V. (2019). Discrete k-nearest neighbor resampling for simulating multisite precipitation occurrence and model adaption to climate change. Geoscientific Model Development, 12, 1189–1207.
  • Lee, Y., Hsieh, W., & Huang, C. (2005). SSVR: A smooth support vector machine for insensitive regression. IEEE Transactions on Knowledge and Data Engineering, 17, 678–685.
  • Liu, W., Zhang, A., Wang, L., Fu, G., Chen, D., Liu, C., & Cai, T. (2015). Projecting the future climate impacts on streamflow in Tangwang River basin (China) using a rainfall generator and two hydrological models. Climate Research, 62, 79–97.
  • Lorenz, E. N. (1969). Atmospheric predictability as revealed by naturally occurring analogs. Journal of Atmospheric Science, 26, 639–646.
  • Lovejoy, S., & Mandelbrot, B. (1985). Fractal properties of rain and a fractal model. Tellus A, 37, 209–232.
  • Maraun, D., Wetterhall, F., Ireson, A. M., Chandler, R. E., Kendon, E. J., Widmann, M., Brienen, S., Rust, H. W., Sauter, T., Themeßl, M., Venema, V. K. C., Chun, K. P., Goodess, C. M., Jones, R. G., Onof, C., Vrac, M., & Thiele-Eich, I. (2010). Precipitation downscaling under climate change: Recent developments to bridge the gap between dynamical models and the end user. Review of Geophysics, 48, RG3003.
  • Maraun, D., & Widmann, M. (2018). Statistical downscaling and bias correction for climate research. Cambridge University Press.
  • Maraun, D., Widmann, M., Gutiérrez, J. M., Kotlarski, S., Chandler, R. E., Hertig, E., Wibig, J., Huth, R., & Wilcke, R. A. I. (2015). VALUE: A framework to validate downscaling approaches for climate change studies. Earth’s Future, 3, 1–14.
  • Mehrotra, R., & Sharma, A. (2005). A nonparametric nonhomogeneous hidden Markov model for downscaling of multi‐site daily rainfall occurrences. Journal of Geophysical Research: Atmospheres, 110, D16108.
  • Mehrotra, R., Srikanthan, R., & Sharma, A. (2006). A comparison of three stochastic multi‐site precipitation occurrence generators. Journal of Hydrology, 331, 280–292.
  • Mishra, A. K., Desai, V. R., & Singh, V. P. (2007). Drought forecasting using a hybrid stochastic and neural network model. Journal of Hydrological Engineering, 12(6), 626–638.
  • MoradiKhaneghahi, M., Lee, T., & Sing, V. (2019). Stepwise extreme learning machine for statistical downscaling of daily maximum and minimum temperature. Stochastic Environmental Research and Risk Assessment, 33, 1035–1056.
  • Murawski, A., Zimmer, J., & Merz, B. (2016). High spatial and temporal organization of changes in precipitation over Germany for 1951–2006. International Journal of Climatology, 36(6), 2582–2597.
  • Nourani, V., Razzaghzadeh, Z., Baghanam, A. H., & Molajou, A. (2019). ANN-based statistical downscaling of climatic parameters using decision tree predictor screening method. Theoretical and Applied Climatology, 137(3–4), 1729–1746.
  • Obled, C. (2002). Quantitative precipitation forecasts: A statistical adaptation of model outputs through an analogues sorting approach. Atmospheric Research, 63(3–4), 303–324.
  • Okkan, U., & Kirdemir, U. (2016). Downscaling of monthly precipitation using CMIP5 climate models operated under RCPs. Meteorological Applications, 23, 514–528.
  • Okkan, U., & Serbes, Z. A. (2012). Rainfall–runoff modeling using least squares support vector machines. Environmetrics, 23(6), 549–564.
  • Olsson, J., Uvo, C., & Jinno, K. (2001). Statistical atmospheric downscaling of short-term extreme rainfall by neural networks. Physics and Chemistry of the Earth, Part B, 26(9), 695–700.
  • Pan, B., Hsu, K., AghaKouchak, A., & Sorooshian, S. (2019). Improving precipitation estimation using convolutional neural network. Water Resources Research, 55(3), 2301–2321.
  • Pang, B., Yue, J., Zhao, G., & Xu, Z. (2017). Statistical downscaling of temperature with the random forest model. Advances in Meteorology, 2017(7-8), 1–11.
  • Papalexiou, S. M., AghaKouchak, A., & Foufoula-Georgiou, E. (2018). A diagnostic framework for understanding climatology of tails of hourly precipitation extremes in the United States. Water Resources Research, 54, 6725–6738.
  • Pham, Q. B., Yang, T.-C., Kuo, C.-M., Tseng, H.-W., & Yu, P.-S. (2019). Combining random forest and least square support vector regression for improving extreme rainfall downscaling. Water, 11, 451.
  • Philipp, A., Della-Marta, P. M., Jacobeit, J., Fereday, D. R., Jones, P. D., Moberg, A., & Wanner, H. (2007). Long-term variability of daily North Atlantic-European pressure patterns since 1850 classified by simulated annealing clustering. Journal of Climate, 20, 4065–4095.
  • Pour, S. H., Shahid, S., & Chung, E. S. (2016). A hybrid model for statistical downscaling of daily rainfall. Procedia Engineering, 154, 1424–1430.
  • Qian, B., Corte-Real, J. O., & Xu, H. (2002). Multisite stochastic weather models for impact studies International Journal of Climatology, 22(11), 1377–1397.
  • Raje, D., & Mujumdar, P. P. (2011). A comparison of three methods for downscaling daily precipitation in the Punjab region. Hydrological Processes, 25(23), 3575–3589.
  • Rajendran, Q. S., & Cheung, S. H. (2015). BUASCSDSEC—Uncertainty assessment of coupled classification and statistical downscaling using Gaussian process error coupling. International Journal of Environmental Science and Development, 6(3), 211–216.
  • Rebora, N., Ferraris, L., von Hardenberg, J., & Provenzale, A. (2006). The RainFARM: Rainfall downscaling by a filtered autoregressive model. Journal of Hydrometeorology, 7, 724–738.
  • Richardson, C. W. (1981). Stochastic simulation of daily precipitation, temperature, and solar radiation. Water Resources Research, 17, 182–190.
  • Rodriguez-Iturbe, I., Cox, D. R., & Isham, V. (1987). Some models for rainfall based on stochastic point processes. Proceedings of the Royal Society of London. Series A, 410, 269–288.
  • Rumelhart, D. E., Hinton, E., & Williams, J. (1986). Learning internal representation by error propagation. In D. E. Rumelhart, J. L. McClelland, & P. R. Group, Eds. (Eds.), Parallel distributed processing (Vol. 1, pp. 318–362). MIT Press.
  • Sachindra, D. A., & Kanae, S. (2019). Machine learning for downscaling: The use of parallel multiple populations in genetic programming. Stochastic Environmental Research and Risk Assessment, 33, 1497–1533.
  • Sarhadi, A., Burn, D. H., Yang, G., & Ghodsi, A. (2017). Advances in projection of climate change impacts using supervised nonlinear dimensionality reduction techniques. Climate Dynamics, 48, 1329–1351.
  • Schmidli, J., Goodess, C. M., Frei, C., Haylock, M. R., Hundecha, Y., Ribalaygua, J., & Schmith, T. (2007). Statistical and dynamical downscaling of precipitation: An evaluation and comparison of scenarios for the European Alps. Journal of Geophysical Research, 112, D04105.
  • Schoof, J. T., & Pryor, S. (2001). Downscaling temperature and precipitation: A comparison of regression-based methods and artificial neural networks. International Journal of Climatology, 21(7), 773–790.
  • Semenov, M. A., & Barrow, E. M. (1997). Use of a stochastic weather generator in the development of climate change scenarios. Climatic Change, 35, 397–414.
  • Semenov, M. A., Brooks, R. J., Barrow, E. M., & Richardson, C. W. (1998). Comparison of the WGEN and LARS-WG stochastic weather generators in diverse climates. Climate Research, 10, 95–107.
  • Sørup, H. J. D., Christensen, O. B., Arnbjerg-Nielsen, K., & Mikkelsen, P. S. (2016). Downscaling future precipitation extremes to urban hydrology scales using a spatio-temporal Neyman–Scott weather generator. Hydrology and Earth System Science, 20, 1387–1403.
  • Takayabu, I., Kanamaru, H., Dairaku, K., Benestad, R., von Storch, H., & Christensen, J. H. (2016). Reconsidering the quality and utility of downscaling. Journal of Meteorological Society of Japan, 94A, 31–45.
  • Tang, J., Niu, X., Wang, S., Gao, H., Wang, X., & Wu, J. (2016). Statistical downscaling and dynamical downscaling of regional climate in China: Present climate evaluations and future climate projections. Journal of Geophysical Research: Atmospheres, 121, 2110–2129.
  • Tangang, F. T., Hsieh, W. W., & Tang, B. (1998). Forecasting the regional sea surface temperatures of the tropical Pacific by neural network models, with wind stress and sea level pressure as predictors. Journal of Geophysical Research, 103, 7511–7522.
  • Tao, Y., Gao, X., Hsu, K., & Sorooshan, S. (2016). A deep neural network modeling framework to reduce bias in satellite precipitation products. Journal of Hydrometeorology, 17, 931–945.
  • Teegavarapu, R. S. V., & Goly, A. (2018). Optimal selection of predictor variables in statistical downscaling models of precipitation. Water Resources Management, 32(6), 1969–1992.
  • Terzago, S., Palazzi, E., & von Hardenber, J. (2018). Stochastic downscaling of precipitation in complex orography: A simple method to reproduce a realistic fine-scale climatology. Natural Hazards and Earth System Science, 18, 2825–2840.
  • Timbal, B., & McAvaney, B. J. (2001). An analogue-based method to downscale surface air temperature: Application for Australia. Climate Dynamics, 17(12), 947–963.
  • Tipping, M. E. (2001). Sparse Bayesian learning and the relevance vector. Journal of Machine Learning Research, 1, 211–244.
  • Trigo, R. M., & Palutikof, J. P. (2001). Precipitation scenarios over Iberia: A comparison between direct GCM output and different downscaling techniques. Journal of Climate, 14, 4422–4446.
  • Tripathi, S., Srinivas, V. V., & Nanjundiah, R. S. (2006). Downscaling of precipitation for climate change scenarios: A support vector machine approach. Journal of Hydrology, 330(3–4), 621–640.
  • Vandal, T., Kodra, E., & Ganguly, A. (2019). Intercomparison of machine learning methods for statistical downscaling: The case of daily and extreme precipitation. Theoretical and Applied Climatology, 137, 557–570.
  • van den Dool, H. (1994). Searching for analogs, how long must we wait? Tellus, 46A, 314–324.
  • van der Linden, P., & Mitchell, J. F. B. (Eds.). (2009). ENSEMBLES: Climate change and its impacts: Summary of research and results from the ENSEMBLES project. Hadley Centre.
  • Vapnik, V. N. (1995). The nature of statistical learning theory. Springer Verlag.
  • Verdin, A., Rajagopalan, B., Kleiber, W., & Katz, R. W. (2015). Coupled stochastic weather generation using spatial and generalized linear models. Stochastic Environmental Research and Risk Assessment, 29, 347–356.
  • von Storch, H. (2000). Review of empirical downscaling techniques. In Regional climate development under global warming. General technical report no. 4. Conference proceedings REGCLIM spring meeting Jevnaker, Torbjornrud, Norway, 2000 (pp. 29–46).
  • von Storch, H., Zorita, E., & Cubasch, U. (1993). Downscaling of global climate change estimates to regional scales: An application to Iberian rainfall in wintertime. Journal of Climate, 6, 1161–1171.
  • Vrac, M., & Naveau, P. (2007). Stochastic downscaling of precipitation: From dry events to heavy rainfalls. Water Resources Research, 43, W07402.
  • Vrac, M., Stein, M. L., Hayhoe, K., & Liang, X. Z. (2007). A general method for validating statistical downscaling methods under future climate change. Geophysical Research Letters, 34, L18701.
  • Vu, M. T., Aribarg, T., Supratid, S., Raghavan, S. V., & Liong, S-Y. (2015). Statistical downscaling rainfall using artificial neural network: Significantly wetter Bangkok? Theoretical and Applied Climatology, 126(3–4), 453–467.
  • Vu, T. M., Mishra, A. K., Konapala, G., & Liu, D. (2018). Evaluation of multiple stochastic rainfall generators in diverse climatic region. Stochastic Environmental Research and Risk Assessment, 32, 1337–1353.
  • Weichert, A., & Bürger, G. (1998). Linear versus nonlinear techniques in downscaling. Climate Research, 10, 83–93.
  • Wilby, R. L., Charles, S. P., Zorita, E., Timbal, B., Whetton, P., & Mearns, L. O. (2004). Guidelines for use of climate scenarios developed from statistical downscaling methods (Supporting material of the Intergovernmental Panel on Climate Change).
  • Wilby, R. L., Wigley, T. M. L., Conway, D., Jones, P. D., Hewitson, B. C., Main, J., & Wilks, D. S. (1998). Statistical downscaling of general circulation model output: A comparison of methods. Water Resources Research, 34, 2995–3008.
  • Wilks, D. (1998). Multisite generalization of a daily stochastic precipitation generation model. Journal of Hydrology, 210, 178–191.
  • Wilks, D. S. (2010). Use of stochastic weather generator for precipitation downscaling. Wiley Interdisciplinary Reviews: Climate Change, 1(6), 898–907.
  • Wilks, D. S. (2012). Stochastic weather generators for climate-change downscaling, part II: Multivariable and spatially coherent multisite downscaling. Wiley Interdisciplinary Reviews: Climate Change, 3(3), 267–278.
  • Wilks, D. S., & Wilby, R. L. (1999). The weather generator game: A review of stochastic weather models. Progress in Physical Geography, 23, 329–357.
  • Yang, Y., Tang, J., Xiong, Z., Wang, S., & Yuan, J. (2019). An intercomparison of multiple statistical downscaling methods for daily precipitation and temperature over China: Present climate evaluations. Climate Dynamics, 53, 4629–4649. .
  • Yates, D., Gangopadhyay, S., Rajagopalan, B., & Strzepek, K. (2003). A technique for generating regional climate scenarios using a nearest‐neighbor algorithm. Water Resources Research, 39(7), 1199.
  • Yin, S., & Chen, D. (2020). Weather generator. Oxford research encyclopedia of climate science. Oxford University Press.
  • Young, K. (1994). A multivariate chain model for simulating climatic parameters from daily data. Journal of Applied Meteorology, 33(6), 661–671.
  • Zhao, T., Yang, D., Cai, X., & Cao, Y. (2012). Predict seasonal low flows in the upper Yangtze River using random forests model. Journal of Hydroelectric Engineering, 31(3), 18–38.
  • Zorita, E., Hughes, J., Lettenmaier, D., & von Storch, H. (1995). Stochastic downscaling of regional circulation patterns for climate model diagnosis and estimation of local precipitation. Journal of Climate, 8, 1023–1042.
  • Zorita, E., & von Storch, H. (1999). The analog method as a simple statistical downscaling technique: Comparison with more complicated methods. Journal of Climate, 12, 2474–2489.