PP-ESD modeling requires (1) predictand data from weather stations or other observational systems, (2) reanalysis datasets for the construction of predictors, and (3) GCM or RCM output for the construction of simulated predictors if the PP-ESD models are used for downscaling simulated climates. To understand the workflow demonstrated in later sections, the reader needs to be aware of few important package design choices related to data structure and preprocessing.

The package adopts the OOP paradigm and treats every predictand data archive (e.g., weather station or glacier) as an object. Since the current version of the package focuses only on station-based downscaling, we will henceforth describe it only as the weather station object. The package accepts the (typical for weather stations) comma-separated value (CSV) file format. These files contain the predictand time series, such as a temperature record, as well as weather station attributes like the weather station's name, ID, and location. The read_station_csv from the pyESD.weatherstation module initiates each weather station as a separate object using the StationOperator that features all the other functionalities. The weather station object is associated with at least one predictand dataset (i.e., the values of at least one climate variable recorded at that particular station). Furthermore, the initialized object includes all attributes and methods required for the complete downscaling cycle. For instance, the package adopts the fit and predict framework of the scikit-learn Python package (Pedregosa et al., 2011) that can be directly applied to the weather station object.

The PP-ESD approach is highly sensitive to the choice of predictors and learning models (Maraun et al., 2019a; Gutiérrez et al., 2019). Moreover, since PP-ESD models are empirical in nature, the predictors serve as proxies for all the relevant physical processes and must be informative enough to account for the local predictand variability (Huth, 1999, 2004; Maraun and Widmann, 2018). Therefore, the selection of potential predictors should be informed by our knowledge of the atmospheric dynamics that control the climate variability of the study area. For example, synoptic-scale climate features, such as atmospheric teleconnection patterns, control much of the regional-scale climate variability. It is therefore recommended to consider these as potential predictors. Statistical techniques, such as methods for feature selection or dimension reduction, may then be applied to reduce the list of physically relevant potential predictors to a smaller selection of predictors that have a robust statistical relationship with the predictand. These steps contribute to the performance of the models and also resolve some of the issues related to multicollinearity and overfitting (e.g., Mutz et al., 2016). The pyESD package adopts three different wrapper feature selection techniques that can be explored for different models: (1) recursive feature elimination (Chen and Jeong, 2007), (2) tree-based feature selection (Zhou et al., 2021), and (3) sequential feature selection (Ferri et al., 1994). The methods are included in pyESD.feature_selection as RecursiveFeatureElimination, TreeBasedSelection, and SequetialFeatureSelection, respectively. Furthermore, classical filter feature selection techniques, such as correlation analyses, are also included as a method of the weather station object.

Predictors are typically constructed by (1) computing the regional means of a physically relevant climate variable or (2) constructing index time series for relevant synoptic-scale climate phenomena. The package allows the user to consider a few important aspects for each type of predictor.

The area over which the climate variable is averaged can significantly affect model performance. In complex terrain with high-frequency topography, for example, choosing a smaller spatial extent may result in the predictor having a higher explanatory power. Therefore, a radius (with a default value of 200 km) around the weather station may be defined by the user to determine the size of the area used for the computation of the regional means.

The empirical relationship between local predictand and large-scale predictors is often complicated due to the complex dynamics in the climate system. However, ML algorithms have been demonstrated to perform well in extracting hidden patterns in climate data that are relevant for building more complex transfer functions (e.g., Raissi and Karniadakis, 2018). Specifically, neural networks have been explored for downscaling climate information due to their ability to establish a complex and nonlinear relationship between predictands and predictors (e.g., Nourani et al., 2019; Gardner and Dorling, 1998; Vu et al., 2016). Moreover, support vector machine (SVM) models have been used to capture the links between predictors and predictands by mapping the low-dimensional data into a high-dimensional feature space with the use of kernel functions (e.g., Anandhi et al., 2008; Tripathi et al., 2006). Previous studies have also applied multi-model ensembles due to their ability to reduce model variance and capture the distribution of the training data (e.g., Xu et al., 2020; Massaoudi et al., 2021; Gu et al., 2022).

Selecting the most appropriate model or algorithm for a specific location or predictand can be challenging because one needs to consider many case-specific factors like data dimensionality, distribution, temporal resolution, and explainability. This problem is exacerbated by the lack of well-established frameworks for climate information downscaling (Gutiérrez et al., 2019). The pyESD package addresses this challenge with the implementation of many ML models that are different with regard to their theoretical paradigms, assumptions, and model structure. The implementation of commonly used models in the same package allows researchers to experiment with different learning models and to replicate and update their research based on emerging recommendations for specific predictands and geographical locations. The implementation of statistical and ML models in pyESD mainly relies on the open-source scientific framework scikit-learn tool (Pedregosa et al., 2011). In the following subsections, we briefly explain the principles behind the ML methods that are included in the pyESD package.

The process of training and testing the PP-ESD models is the most critical stage in the downscaling procedure, since it determines much of the robustness of the final models, as well as the accuracy of the predictions they generate. The process typically involves the following steps: (1) the observational records are separated into training and testing datasets. (2) The training datasets are used to establish the transfer functions that make up the PP-ESD models. (3) The trained models are then evaluated on the independent testing datasets (Sect. 2.5). In the model training process, hyperparameter optimization techniques (e.g., GridSearchCV) are used to fine-tune the transfer function parameters, such as regression coefficients, to optimize model performance. Cross-validation (CV) techniques are applied to split the whole training dataset into smaller training and validation data sections and allow the assessment and iterative improvement of the model parameters during training while also preventing overfitting (Moore, 2001; Santos et al., 2018). In this category of techniques, the k-fold framework is the most used for climate information downscaling models. It partitions the training data into k equally sized and mutually exclusive subsamples, which are also referred to as folds (Stone, 1976; Markatou et al., 2005). More specifically, for each iteration step, one fold is used for model validation, and the remaining k-1 folds are used for model training. The leave-one-out CV technique (Lachenbruch and Mickey, 1968) is an alternative and has been used for the development of ESD models (e.g., Gutiérrez et al., 2013). Cross-validation techniques rely on the fundamental assumption of independent and identically distributed (i.i.d) data. They, therefore, treat the data as a result of a generative process that has no memory of previously generated samples (Arlot and Celisse, 2010). The assumption of i.i.d might not be valid for time series data (e.g., Bergmeir and Benítez, 2012) due to seasonal effects, for example. To circumvent this problem, monthly bootstrapped resampling and time series splitters are included in the pyESD package. The pyESD.splitter module contains all CV frameworks available for model training, including the k-fold, leave-one-out, and other CV schemes. The validation metrics used for optimizing the model parameters include the coefficient of determination (R2) (Eq. 11), root mean squared error (RMSE) (Eq. 13), mean absolute error (MAE) (Eq. 14), and others that are summarized in Sect. 2.5. The final values for the validation metrics, which reflect the model performance during training, are arithmetic means of the individual values for each iteration. In this paper, we refer to them as CV performance metrics (i.e., CV R2, CV RMSE, and CV MAE).

In the process of downscaling climate information, best practice involves the use of stringent model evaluation schemes with independent data outside the training data range (Wilby et al., 2004). Retaining a section of the data as a testing dataset (Sect. 2.4) is recommended if longer records (e.g., ≥30 years) are available. It allows (a) a completely independent evaluation of the trained model's performance and (b) an assessment of the sensitivity of the model to the chosen training dataset. In the case of time series, the latter can provide insights into the model's sensitivity to the calibration period and the temporal stationarity of the model's transfer functions. If the records are short (e.g., <30 years), the CV metrics (Sect. 2.4) can be used, albeit with caveats, as nonideal estimates for the model's performance (e.g., Mutz et al., 2021). For the remainder of this section, however, we will assume that longer records and completely independent testing datasets are available.

The PP-ESD model is evaluated on the basis of the model's predictions y^ and the observed values y. In pyESD, the following performance metrics are implemented.

The coefficient of determination (R2) represents the fraction of the predictand's observed variance that can be explained by the predictors. It can be seen as a measure of how well the model predicts the unseen data (Wilks, 2011). The R2 for the predicted values y^i in relation to the observed data yi for i=1, …, n samples is defined as11R2y,y^=1-∑i=1nyi-y^i2∑i=1nyi-y¯2,where y¯ is the mean of the observed data, ∑i=1nyi-y^i2is the sum of squared residuals (SSR), and ∑i=1nyi-y¯2 is the total sum of squares (SST). R2 can range from -∞ to 1, where 1 is the best possible score and negative values are indicative of an arbitrary, worse model. An R2 value of 0 is indicative of a model that would always predict the y¯. In this case, the model represents no improvement over simply using the mean y¯ as a model.

Pearson's correlation coefficient (PCC) evaluates the linear correlation between the model predictions yi and observed data xi. The PCC of 1 indicates a perfect positive correlation, -1 indicates a perfect anticorrelation, and 0 indicates no correlation between the predicted and observed values. The PCC for n samples is defined as12PCCxy=∑i=1nxi-x¯yi-y¯∑i=1nxi-x¯2∑i=1nyi-y¯2,where the x¯ and y¯ are the means of the xi and xi values, respectively.

The root mean squared error (RMSE) estimates the mean magnitude of error between the predictions and observations. The RMSE is given in the physical units of the observed data and not standardized. Smaller values indicate better model performance. The RMSE for predictions y^i and observations yi of n samples is calculated as13RMSEy,y^=1n∑i=1ny^i-yi2.The mean absolute error (MAE) is a scale-dependent accuracy measure that also provides information on the errors between the predictions and observations. The MAE is estimated as the sum of absolute errors normalized by the sample size (n). The MAE is calculated as14MAEy,y^=1n∑i=1ny^i-yi.

The developed and tested PP-ESD model can finally be coupled to coarse-scale climate information. If the PP-ESD model was developed with the intention to downscale predictions of future climate change, the next logical step is to couple it to GCM simulations forced with different greenhouse gas concentration scenarios. Since PP-ESD is the bias-free downscaling alternative to MOS-ESD, PP-ESD models may be coupled to all GCMs, provided that the predictors are adequately represented by the GCMs. This condition may be alleviated to an extent by standardizing the simulated predictor (Bedia et al., 2020). An analysis of the distribution similarity between the observed and simulated predictors can be conducted to test the assumption of representation. For example, the Kolmogorov–Smirnov (KS) test, which is implemented as part of the pyESD package utilities, is a nonparametric statistical hypothesis test that can be used to evaluate the null hypothesis (H0) that the observation-based predictors and simulated predictors are of the same theoretical distribution.

The first step in ESD–GCM coupling is to utilize the GCM output to recreate the predictors used in the training of the ESD model. This may involve anything from constructing simple temperature regional means to reconstructing multivariate indices for more complex climate phenomena. In the case of index-based predictors such as NAO, EA, SCAN, and others, the simulated indices are reconstructed by projecting the pressure anomalies of the GCM onto the EOF loading patterns of the predictors (e.g., Mutz et al., 2016). This ensures that the physical meaning of the index values is maintained. The ESD model then takes these simulated predictors as input and generates local-scale predictions according to the model's transfer functions. The added value of the resulting downscaling product can be evaluated by comparing the downscaled values to the raw outputs of different GCMs and RCMs. Finally, the high-resolution local-scale predictions can be used to drive climate change impact assessment models to predict flood frequency (e.g., Padulano et al., 2021; Hodgkins et al., 2017), agricultural changes (e.g., Mearns et al., 1996), changes in water resources (e.g., Dau et al., 2021), and more.

All implemented predictor selection methods demonstrated merit, and the correlation analyses revealed strong linear dependencies between the predictand variables and potential predictors (Figs. A1 and A2). For example, precipitation records are highly correlated (PCC ≥0.5) with large-scale total precipitation (tp), atmospheric relative humidity (r), and zonal wind velocity (u) up to the mid-tropospheric level (i.e., 500–1000 hPa) (Fig. A1). The temperature records are highly correlated (PPC ≥0.7) with near-surface temperature (t2m), atmospheric temperature (t on all levels), and dew-point temperature depression (dtd) up to the mid-troposphere (Fig. A2). Both predictands also show a good correlation (PCC ≥0.25) with the indices of the atmospheric teleconnection patterns (i.e., NAO, EA, EAWR, and SCAN). The predictor selection methods (i.e., recursive, tree-based, and sequential) perform similarly for all the precipitation and temperature stations (Fig. 4). More specifically, the three methods yield CV R2 values of 0.5 to 0.75 (Fig. 4a), CV RMSE values of ≤25 mm per month (Fig. 4c) for precipitation, CV R2 values of ≥0.95 (Fig. 4b), and CV RMSE values of 0.3 to 0.6 ∘C (Fig. 4d) for temperature stations. Since the methods did not show a significant difference in performance, the recursive method was applied for the refinement of the set of predictors, since it allows more flexibility and a stepwise iteration of several combinations of potential predictors (e.g., Mutz et al., 2021; Hammami et al., 2012; Li et al., 2020). The frequencies with which specific predictors were selected using the recursive method are listed in Table 2.

The predictors tp and t2m were included for most of the precipitation and temperature station records, respectively. This indicates that variations in the larger-scale precipitation and temperature fields already explain much of the local-scale predictand variability in the vicinity of the weather stations. Many of the refined predictor sets also included indices of the NAO (9 of 18 precipitation stations, 5 of 9 temperature stations), SCAN (11 of 18 precipitation stations, 5 of 9 temperature stations), EA (10 of 18 precipitation stations, 4 of 9 temperature stations), and EAWR (11 of 18 precipitation stations, 3 of 9 temperature stations). This confirms the strong manifestation of Northern Hemisphere atmospheric teleconnection patterns in the local-scale precipitation and temperature in the catchment (e.g., Bárdossy, 2010; Ludwig et al., 2003). Their exclusion from the other stations is likely due to the fact that their variability might already be captured by zonal and meridional wind speeds and synoptic pressure variables like geopotential height (z) and mean sea level pressure (slp) (Hurrell and Van Loon, 1997; Hurrell, 1995; Barnston and Livezey, 1987; Maraun and Widmann, 2018). Relative humidity was selected as a predictor for most of the precipitation stations. This is consistent with the results of many other studies (e.g., Gutiérrez et al., 2019; Hammami et al., 2012) and our physical understanding of it as a measure of humidity that takes saturation vapor pressure into consideration.

We experimented with seven different regressors before deciding on the regressor that would be used to establish the final ESD models (see Sect.3.2.2). A total of 126 precipitation and 63 temperature experimental models were generated with the seven regressors. Overall, most of the experimental models performed reasonably well with a mean CV R2 of ≥0.5 for precipitation and ≥0.9 for temperature stations (Fig. 5). The MLP models, on the other hand, performed relatively poorly with CV R2 values of ≤0.4 for precipitation and ≤0.9 for temperature. This is due to the fact that MLP model calibration requires longer records and a more complex architecture to capture most of the informative patterns in the training data. This study, however, uses a simplified architecture to make the results reproducible without higher computational requirements. The result can likely be improved with more data (e.g., by using daily values) and an increase in hidden layers (Sect. 2.2.3). The overall performance of the experimental models underlines the methods' suitability for downscaling.

Among the better-performing precipitation models, the LassoLarsCV and ARD methods yielded the best results (CV R2=0.55–0.75, CV RSME = 20–23 mm per month), followed by the RandomForest and bagging ensembles (CV R2=0.48–0.70, CV RSME = 21 to 25 mm per month), as well as the XGBoost ensemble regressor (CV R2=0.39–0.65, CV RMSE = 22–27 mm per month). Stacking all experimental models into a meta-regressor also yields good results (CV R2=0.45–0.7, CV RMSE = 20–26 mm per month) despite the poor performance of the MLP regressors. Based on these results, the LassoLarsCV, ARD, RandomForest, and bagging regressors were selected as the final base learner for the stacking model. ExtaTree was chosen as the final meta-learner to prevent overfitting issues by placing an additional discriminative threshold on all the base regressor's predictions (Geurts et al., 2006).

The experimental temperature models showed similar patterns in performance but performed better overall. LassoLarCV and ARD emerge as the best-performing models (CV R2=0.85–0.98, CV RMSE = 0.2–0.6 ∘C), followed by the RandomForest and bagging regressors (CV R2=0.8–0.96, CV RMSE = 0.3–0.7 ∘C), as well as the XGBoost and stacking ensemble regressors (CV R2=0.75–0.96, CV RMSE = 0.3–0.8 ∘C). Therefore, we also selected stacking (with LassoLarsCV, ARD, RandomForest, bagging) for the final temperature models, too.

Following the analysis of the seven experimental models (Sect. 4.2), the recursive predictor selection method and stacking learning model (with LassoLarsCV, ARD, RandomForest, and bagging) were selected for the generation of the final ESD models. The models were trained on the 1958–2010 data in a CV setting and evaluated on the retained data in the 2011–2020 period. R2, RMSE, and MAE were used as performance metrics for the CV setting and the final evaluation (Tables 3 and 4). The models' performance was good overall but varied notably between different stations. The prediction skill estimates were higher for temperature than for precipitation. For temperature (Table 4), the explained variance estimates (“Fit R2”) are in the range of 0.81–0.98 (μ=0.94), and CV R2 values are in the range of 0.84 to 0.98 (μ=0.93), whereas for precipitation (Table 3), the explained variance estimates are in the range of 0.58–0.84 (μ=0.71), and CV R2 values are in the range of 0.54–0.72 (0.65). The accuracy measures display a similar discrepancy with CV RMSE of 0.3–0.6 ∘C (μ=0.42 ∘C) and CV MAE of 0.2–0.50 ∘C (μ=0.34 ∘C) for temperature, as well as CV RMSE of 20–24 mm per month (μ=21 mm per month) and CV MAE of 14–18 mm per month (μ=16 mm per month) for precipitation.

The final model evaluation using independent, retained data from 2011–2020 yielded R2 values of up to 0.95 as well as average RMSE and MAE of ∼1.0 ∘C for temperature and R2 values of up to 0.74, average RMSE of 22 mm per month, and MAE of 17 mm per month for precipitation. The discrepancy in temperature and precipitation model performance is unsurprising, since the thermodynamics and atmospheric dynamics controlling precipitation variability are more difficult to represent (e.g., Shepherd, 2014). Regardless, the overall performance speaks in favor of applying the study's approach to downscale midlatitude climate in complex terrain. Moreover, the models' similar performance during CV and the final evaluation indicates that the models were not overfitted and that the predictand–predictor relationships hold outside the observed period. Finally, it is worth noting that the stacking regressor performed better than the individual base models, even when all the potential regressors of the initial experiments (Sect. 4.2) were stacked into a meta-regressor. Such improvements demonstrate the advantage of the ease of experimentation through a package like pyESD.

We visualize a prediction example (Fig. 6) to (a) provide a less abstract presentation of these results and (b) demonstrate the type of figure generated by the plotting utility functions in the pyESD.plot module. The figure depicts the predictions generated by the final ESD model for the Hechingen station, a station that records precipitation and temperature (station ID 7 and 1, respectively). The observed and predicted values for 2011–2020 are highly correlated, with PCCs of 0.85 (Fig. 6a) for precipitation and 0.97 (Fig. 6b) for temperature. The time series comparisons also demonstrate the models' abilities to predict the variability of the observed values in both the training and testing period (Fig. 6a and b). Prior to this study, PP-ESD models had not been directly applied to the weather stations in the catchment. However, our models are among the best performing for temperature and precipitation when we compare them to models from other studies across Europe (e.g., Gutiérrez et al., 2019; Hertig et al., 2019; Schmidli et al., 2007). For instance, Gutiérrez et al. (2019) performed an intercomparison of statistical downscaling model performance for 86 stations across Europe using the MOS, PP, and WG methods. The Spearman correlation of the downscaled and observed values yielded R values in the range of ∼0.0–0.7 (with many stations ≤0.5) for precipitation and 0.3–0.95 for temperature. These comparisons also underline the suitability of the pyESD methods for downscaling climate information even in complex mountainous regions.

The predictions of local precipitation and temperature responses to 21st century climate change were generated by coupling the final ESD models to MPI-ESM simulations forced with greenhouse gas concentration scenarios RCP2.6, RCP4.5, and RCP8.5 (Sect. 3.2.3). The results are presented as deviations from the monthly long-term means of the training period (1958–2010) and referred to as “anomalies” hereafter. The annual mean anomaly time series were computed with a 1-year moving average with a centered mean (Figs. 7 and 9).

The precipitation predictions (Fig. 7) for RCP8.5 (RCP4.5) show a strong (weak) positive trend towards the end of the century. This trend is even more pronounced for the predicted temperatures (Fig. 9) in the catchment. The predicted precipitation changes vary greatly between weather stations. Furthermore, the RCPs change the magnitude but not the pattern of the predictions for each station. For instance, stations that show an increase (decrease) in precipitation for the RCP2.6 predict a greater increase (decrease) in response to RCP4.5 and RCP8.5. The annual and seasonal 30-year end-of-century climatologies show an overall increase in precipitation in response to both RCP2.6 and RCP4.5 (Fig. 8) for most of the stations. The annual end-of-century climatologies deviate from the present day (1958–2010) by ca. -5 to 20 mm per month for RCP8.5 and ca. ≤5 mm per month for RCP2.6. Overall, the ESD models predict a precipitation increase of ca. 10 %–20 % until the end of the century. Furthermore, the seasonal climatologies reveal a shift of maximum precipitation away from the summer season for some stations. Such shifts in seasonality and an overall decrease in summer precipitation have previously been predicted (e.g., Gobiet et al., 2014; Paparrizos et al., 2017; Feldmann et al., 2013). Prior to this study, no ESD–GCM-based predictions of the 21st century precipitation changes had been developed for the weather stations of the catchment. However, the models' predictions of the precipitation response to higher greenhouse gas concentration scenarios are comparable to coarser predictions by other studies using RCMs or ESD models (Feldmann et al., 2013; Kunstmann et al., 2004; Paparrizos et al., 2017; Lau et al., 2013). The precipitation predictions generated in this case study can be used further for climate impact assessments, such as assessments of the probability of flooding and drought across the hydrological catchment. The projected shifts in seasonality across the catchment represents potentially valuable information for agricultural planning.

The predicted temperature anomalies (Fig. 9) reveal a strong (weak) positive trend for RCP8.5 (RCP4.5). The end-of-century climatologies reveal only moderate warming of ca. -0.5 to 1 ∘C for RCP2.6 and significant warming (ca. 2–4 ∘C) for all seasons in response to RCP8.5 (Fig. 10). More specifically, the investigated region is predicted to experience the most warming (≥3 ∘C) in the summer season. There are few differences in predicted warming between the stations of the catchment. Generally, the estimated magnitude of warming towards the end of the century is in agreement with the IPCC report (IPCC, 2021) and other downscaled estimates (e.g., Kunstmann et al., 2004; Gutiérrez et al., 2019). The predicted warming would likely implicate societal and ecological systems and stresses the need for efficient adaptation and mitigation strategies.

The case study highlights the efficiency and robustness of the downscaling steps implemented in the pyESD package. However, as noted in previous sections, the accuracy of the predictions generated by a GCM–ESD model coupling relies on the predictors being adequately represented by the GCMs. KS tests were performed to evaluate this for the temporal overlap (1979–2000) between the ERA5 reanalysis product and the MPI-ESM GCM output (Sect. 3.2.3). Results from these tests show significant differences in the distribution of ERA5 and MPI-ESM when the raw monthly time series are considered, thus violating the assumptions of the PP-ESD approach. However, this issue does not persist for monthly standardized anomalies of precipitation and temperature (Fig. 11). Previous studies yielded similar results when using seasonal standardizers (Bedia et al., 2020) and principal component transformations (Benestad et al., 2015a), both of which are included in the pyESD package.

A comparison of the ESD-generated annual 20-year climatologies for the mid-century (2040–2060) and the end of the century (2080–2100) to the model output of GCMs and RCMs (i.e., EURO-CORDEX) reveals several differences. The GCMs (MPI-ESM and HadGEM2) predict ∼20 mm per month (∼30 %) higher precipitation rates than the ESD models and RCMs. The ESD-based precipitation predictions of this study are closest to the RCM estimates but ∼≥5 mm per month higher in magnitude for most of the stations (Fig. 12). The closeness of the ESD-based and RCM-based estimates underlines the added value of our ESD approach for downscaling precipitation. However, there are significant (∼4 ∘C) differences between the ESD-based and RCM-based temperature estimates (Fig. 13). The ESD-based temperature predictions were higher than those of the RCM but lower than those of the GCM. Both the RCM and ESD models used boundary conditions from the same GCM (MPI-ESM). The RCM reduced the GCM temperatures by more (∼8 ∘C) than the ESD models (∼4 ∘C or less). This may be a reflection of both (a) the selection of GCM near-surface temperatures as predictors in the ESD models and (b) the shrinking of regression coefficients when the ESD transfer functions are determined.