dataesr/scientific-paragraphs-categorization huggingface dataset:
• Adjust for spatial biases using the CycleGAN model from Step 2. • Rescale data by applying the inverse of the pointwise min-max normalization. Step 3: Application of CycleGAN to correct spatial dependence • Reorder QQ data with the Schaake Shuffle technique for period 1 such that the rank structure of the data obtained at the end of Step 3 is reproduced. Step 4: Reordering of the QQ data Raw-CycleGAN outputs for period 1. MBC-CycleGAN outputs for period 1. where i denotes the realizations of at the time step i across the k dimensions (and similarly for m with ). The energy statistic goes to zero when the two multivariate samples and are drawn from the same distribution. During training, computations of energy distances are performed every 10 epochs, i. e. each time that the Cycle-GAN has worked 10 times through the entire training dataset.
Design of experiments For evaluation purposes, the proposed MBC-CycleGAN method is applied to adjust climate simulations outputs with SAFRAN data as references. Bias correction is performed on separate seasons in order to preserve seasonal properties. In the following, for sake of clarity, only the winter results are presented. Data are available for the 1979-2016 period (i. e, 3420 winter days), and need to be divided into a calibration period and a projection period to train and evaluate our algorithm. In accordance with common practices in machine learning, the 1979-2016 period is split as follows: 70% (2394 days) as training dataset and 30% (1026 days) as evaluation dataset. In this study, two different crossvalidation methods-that differ in how calibration and projection periods are constructed-are used to evaluate our methodology.
Implementing and evaluating both the PP and MOS approaches in such a validation context permits to determine which approach is better suited in our context of bias correction of climate simulations. For the MOS approach, MBC-CycleGAN is applied directly to IPSL data according to the 4 steps already described in Sect. 3. 3. Concerning the implementation of the PP approach, the same procedure is applied but the CycleGAN model is trained in a slightly different way. Indeed, as already explained in Sect. 1, a PP approach consists in establishing the statistical relationships between large-cale predictors and localscale predictands from observational or reanalysis data (including for the predictors) before applying them to climate model data. Hence, large-scale predictors temporally matching the SAFRAN dataset are needed to a PP approach.
For this purpose, a new climate dataset is constructed for both temperature and precipitation as follows: initial localscale SAFRAN data with 8 km × 8 km spatial resolution are upscaled using conservative interpolation on a large-scale grid of 32 km × 32 km spatial resolution. Then, the obtained large-scale data are regridded using bilinear interpolation to the initial grid of SAFRAN, allowing to train CycleGAN. It results in "biased" daily maps of temperature and precipitation (large-scale predictors) of the initial SAFRAN data (local-scale predictands), temporally matching the chronology of the SAFRAN time series. Using these new datahereafter referred to as "low-resolution (LR) SAFRAN"-a CycleGAN model is trained for the implementation of the PP approach by learning the transfer of maps from 1d-BC large-scale predictors (QQ(LR SAFRAN)) to maps from local-scale predictands (SAFRAN).
Hence, to provide a global picture of the performances of the MBC-CycleGAN method in the nonstationary context, three bias correction exercises of climate data with different statistical changes are performed with respect to SAFRAN references: • the correction of IPSL simulations that present different marginal and spatial properties from SAFRAN, and with potentially different changes than those from SAFRAN. • the correction of LR SAFRAN dataset (presented above), whose marginal and spatial properties as well as their changes are in line with those from SAFRAN. • the correction of a third dataset called IPSLbis (presented below) that presents different marginal and spatial properties from SAFRAN, but for which their changes are in line with those from SAFRAN. For the sake of clarity, a summary of the different attributes of the three datasets to correct is presented in Table  1.
LR SAFRAN dataset already presented above has, by construction, little bias with SAFRAN references: its biases are only due to the interpolation technique used to obtain data with a lower resolution. Hence, statistical changes between the calibration and projection periods for LR SAFRAN are in line with those from the SAFRAN dataset. Adjusting LR SAFRAN data for the projection period permits to assess if the MBC-CycleGAN method is able to reproduce the changes from the reference in the correction. Also, the LR SAFRAN dataset presents the particularity of being synchronous in time with references. Hence, in addition to evaluate the proposed method in terms of distributional properties, which is not considered as sufficient to identify successful bias correction techniques  (Maraun 2016) , this pairwise correspondence between predictors and predictands offers the possibility to directly compare corrected daily maps with those from the references using classic forecast verification statistics.
As IPSL simulations compute a different combination of variability and warming than those from the SAFRAN reanalysis, IPSL model and SAFRAN references are likely to present disagreeing changes in their statistical (marginal and dependence) properties between calibration and projection periods. To evaluate the influence of these potential disagreeing changes on the performance of correction of the proposed method, we constructed the third dataset, referred to as "IPSLbis", for the projection period only. IPSLbis is specifically constructed so that its marginal and dependence changes between calibration and projection periods are in line with those from the reference. In order to ease the comparison of results with the first bias correction exercise, we forced IPSLbis to have the same changes as LR SAFRAN.
This is reached by using a two-step procedure that takes advantage of a nonstationary quantile mapping technique for marginal changes (CDF-t,  Vrac et al. 2012 ) and a matrixrecorrelation technique for dependence changes  (Bárdossy and Pegram 2012). More details about the generation of the IPSLbis data can be found in Appendix 3 and a detailed evaluation of the evolution of statistical properties of the different dataset between the calibration and projection period is provided in Appendix 4. In particular, results presented in Appendix 4 indicate that, as expected, changes in spatial structures from SAFRAN references are (globally) in agreement with those from LR SAFRAN for both temperature and precipitation. However, concerning changes in spatial structures for IPSL simulations, conclusions are not the same depending on the physical variable.
Comparisons to existing MBCs: R 2 D 2 and dOTC Although evaluating the performance of correction for IPSL simulations is of primary interest, applying our method on these three datasets (IPSL, IPSLbis, LR SAFRAN) permits to assess gradually how well our method is performing depending on the biases present in the dataset to correct. Note that, as IPSL and IPSLbis data during calibration are identical, there is no need to train for a second time the CycleGAN model for IPSLbis data: the CycleGAN model trained with IPSL data can be used directly to adjust IPSLbis simulations for the projection period. In addition, two MBCs with different assumptions about nonstationarity are applied for comparison using the second cross-validation method: the "Rank Resampling For Distributions And Dependences" (R 2 D 2 , Vrac and Thao 2020) and the "Dynamical Optimal Transport Correction" (dOTC,  Robin et al.
2019 ) methods. R 2 D 2 , developed in the context of marginal/dependence category, relies on an analogue-based method that allows to resample ranks from a reference dataset according to some conditioning information and reconstructs dependence structure of the simulated time series. The information to condition the analogues can be multivariate by considering, for example, a set of variables to be corrected at a given time t. Conditioning for the ranks resampling can also be extended to ranks sequences, i. e. conditioning by not only one but several lagged time steps. Please note that, for the different implementations of R 2 D 2 in this study, the multivariate con- ditioning used includes 4 grid points that cover uniformly the region of interest. In addition, 5 lagged time steps are used for the conditioning, as it has been found to stabilize the R 2 D 2 method (not shown).
Results In this section, analyses are presented for the winter season (December, January and February) only. CycleGAN models are trained during the calibration period and selected such that energy distances on ranks are minimized. All evaluations are performed on the projection period for the corrected outputs obtained from the two cross-validation methods and results are compared to those from the reference dataset. For bias-corrected precipitation time series, thresholding of 1 mm is applied before evaluation to replace values lower than 1 mm by 0. Bias correction outputs from the first and second cross-validation methods are evaluated in terms of both marginal and spatial properties. Analyses of temporal properties are only provided for outputs from the second cross-validation method, in which calibration and projection periods are divided chronologically and hence do not distort temporal properties, contrary to the first crossvalidation method that randomly defines these periods.
Training of MBC-CycleGANs Figure  5  shows energy distances with respect to SAFRAN references for temperature computed on physical values (Fig. 5a, b ) and ranks (Fig. 5c, d ) for LR SAFRAN, plain IPSL simulations, 1d-QQ, and MBC-CycleGAN (MBC-CG) outputs during the training on the calibration period. In addition, results for Raw-CycleGAN (Raw-CG) are presented. Differences between Raw-CG and MBC-CG only lie in their marginal properties: while Raw-CG corresponds to the outputs obtained from the CycleGAN after denormalization at the end of Step 3, MBC-CG is the combination of the spatial structure from Raw-CG and univariate properties from QQ outputs (see the flowchart provided in Fig. 3 ). The results for precipitation are presented in Fig. S1  of the Supplement. Clearly, Fig. 5a, b  show large energy distances computed on physical values of temperature for LR SAFRAN and IPSL datasets, indicating some biases on spatial structures for those dataset with respect to SAFRAN references.
Univariate distribution properties Once the CycleGAN models have been selected for both the PP and MOS approaches, the corrections of IPSL simulations can be performed for the projection period. First, biascorrected data are evaluated in terms of univariate statistics. For temperature and precipitation, differences of mean values between the bias corrected data and the SAFRAN references are computed at each grid cell. For temperature mean, absolute differences are computed, while for precipitation variables having absolute zeros, relative mean differences are more appropriate. Maps of differences with respect to the reference-for IPSL simulations and the bias-corrected data-are displayed in Fig. 6  for both temperature and precipitation. The mean absolute error (MAE) with respect to the reference dataset is also reported on each map.
Spatial correlations Quality of the corrections in terms of spatial correlations is now assessed. For each grid cell, spatial dependencies are evaluated for temperature and precipitation by computing Pearson pairwise correlations between the cell of interest and each of the remaining 783 grid cells over the region of Paris for the different climate datasets. The biases of these 783 spatial Pearson correlations are then summarized by computing the Mean Squared Error (MSE) with the corresponding 783 correlations computed for the references. By computing the MSE values for each grid cell, 784 MSE values are obtained for each climate dataset and can be intercompared from one dataset to another. Figure  7  shows the boxplots of the MSE values obtained for both temperature and precipitation for the plain IPSL simulations and BC outputs.
Univariate distribution properties Similarly to the first cross-validation method, univariate properties are evaluated using mean differences computed at each grid cell. Figure  8  shows, for the bias-corrected outputs from the three bias correction exercises, the maps of temperature mean differences with respect to SAFRAN references. Maps for precipitation relative mean differences are presented in Fig. S6  of the Supplement. For information purposes only, standard deviation relative mean differences for temperature and precipitation are also displayed in Figs. S7 and S8, respectively. For temperature, values of IPSL and IPSLbis mean differences (Fig. 8b, c ) are high, indicating strong biases of temperature mean with respect to the SAFRAN reference dataset (Fig. 8a ), although less pronounced for IPSLbis.
Spatial correlations We now evaluate the ability of MBC-CycleGAN to adjust spatial dependence. First, as for the Sect. 5. 1, we compute MSE of spatial Pearson correlations for both temperature and precipitation. Figure  9  displays the results with boxplots for the different datasets to correct and their adjusted outputs. Scatterplots of MSE values with respect to QQ outputs are presented in Fig. S9  to better assess the potential benefits of using MBC methods relative to univariate ones. For temperature (Fig. 9a ), the positive values of MSE for IPSL suggest biases with respect to the SAFRAN references, illustrating the necessity to correct spatial properties of the model before using it in subsequent analyses. For IPSLbis, MSE values are slightly smaller, but still indicates strong differences of spatial correlations with respect to the references.
The difference of results between IPSL and IPSLbis highlights that discrepancies of changes with the references can potentially have a non-negligible effect on spatial properties; in fact, reducing those discrepancies as it is done with the generation of IPSLbis leads here to reduce biases in spatial correlations. Concerning LR SAFRAN, MSE values are small, suggesting that upscaling the reference dataset deteriorates only slightly its spatial structure. By simply correcting univariate distributions, the three QQ outputs do not present a particular improvement of temperature MSE values. Clear improvements of the spatial correlation structures are provided by the MBC-CycleGAN method for the adjustment of IPSL, IPSLbis and LR SAFRAN, although some differences of performances are observed between the three corrected outputs.
Temperature MSE values are indeed closer to 0 for the correction of LR SAFRAN than for the correction of IPSLbis and IPSL, for which similar results are obtained. Concerning Spatial-R 2 D 2 , the corrections of IPSL and IPSLbis provide major improvements in adjusting the spatial correlations. In particular, better results are obtained for the correction of IPSLbis. However, with regard to the Spatial-R 2 D 2 outputs with LR SAFRAN, the benefits provided by R 2 D 2 are smaller, as not all of the spatial correlations are improved. This result can better be seen in Fig. S9e. This contrasted performance for the R 2 D 2 method appears in the context of the correction of LR SAFRAN that already presents small spatial biases with respect to SAFRAN references. The correction obtained for LR SAFRAN suggests that the R 2 D 2 method is too constrained by the selected con- ditioning to find an appropriate collection of analogues for the projection period of this specific dataset.
To better assess spatial structure adjustments brought by MBCs, the calculation of energy distances between the biascorrected time series and the references are performed for each physical variable according to two different multivariate distributions: • on values of the physical variable directly over the whole region of Paris to assess differences of spatial properties (i. e. , including both the marginals and their dependence); • on ranks of the physical variable over the whole region of Paris to assess differences of spatial dependence structures (i. e. , without the influence of marginal properties). Values of energy distances are estimated using a bootstrap method. It consists for each dataset in (i) sampling (with replacement) daily fields, (ii) computing the energy distance on the bootstrapped dataset, and (iii) repeating the previous two steps 1000 times to construct the bootstrap sampling distribution.
From this bootstrap sampling, distribution is deduced by the bootstrap estimator (mean of the 1000 energy distances obtained) and a 90% bootstrap sampling interval to provide uncertainty bands of the estimated distance. Results for temperature and precipitation are displayed in Fig. 10. The closer the values of the energy distances are to 0, the closer the spatial properties of the outputs are to the one of the reference data. For temperature, the two estimators of energy distances on physical values (Fig. 10a ) and ranks (Fig. 10b ) for IPSL and IPSLbis data are quite high compared to those for LR SAFRAN, which is in agreement with the differences of spatial properties already observed between these datasets and the references in Fig. 9. For the three QQ outputs, while energy distances on physical values are lower (Fig.
10a ), similar energy distances on ranks as those from the dataset to correct are obtained (Fig. 10b ). It highlights again that, although the QQ method adjusts the univariate distributions, it is not supposed to modify rank sequence of time series, and therefore spatial dependence structures, during the correction procedure. With regard to the three MBC methods for the correction of IPSL, dOTC performs slightly better on raw values (Fig. 10a ) than MBC-CycleGAN and R 2 D 2 , for which comparable results are obtained. For energy distances computed on ranks (Fig. 10b ), dOTC and R 2 D 2 produce sim- ilar results. Slightly poorer performances of MBC-Cycle-GAN are obtained compared to the two other MBC methods, although strongly improving the spatial dependence structures of IPSL simulations.
Temporal structure In this section, bias-corrected data are evaluated relative to temporal properties. As a reminder, MBC-CycleGAN and dOTC methods have been specifically implemented to only adjust marginal and spatial properties of climate simulations. Similarly, the R 2 D 2 algorithm is applied to adjust marginal and spatial features but, contrary to the two other methods, it also takes into account (part of) the temporal dependence properties through the multivariate conditioning chosen for its implementation, as previously explained in Sect. 4. In theory, this choice of conditioning dimensions allows R 2 D 2 to partially recover temporal properties of the reference dataset  (Vrac and Thao 2020). Adjusting spatial coherence necessarily modifies the rank sequences of the initial time series during the correction procedure (e.
11c ). In fact, IPSLbis temporal properties are inherited from IPSL outputs: even in a high-dimensional context, the two-step procedureand in particular, the matrix-recorrelation technique-used to construct IPSLbis from IPSL does not lead to a strong modification of temporal properties. This result on temporal properties of data preprocessed with this matrix-recorrelation technique is consistent with the conclusions obtained in  François et al. (2020)  for a MBC method (MRec) using the same matrix-recorrelation. For LR SAFRAN outputs (Fig. 11d ), values of AR1 differences are very close to 0, highlighting that the upscaling step used to construct LR SAFRAN data does not strongly modify the temporal properties of the initial SAFRAN reference dataset, which was expected by construction. Difference maps for temperature from QQ outputs (Fig.
11e-g ) are relatively similar to those from the three datasets to adjust, respectively. However, for the three MBC methods used to adjust spatial dependence structure, modifications of temporal properties for temperature are not equivalent. With regard to MBC-CycleGAN and dOTC outputs (Fig. 11h, i, j, n, o  and p ), temporal statistics are close to that from the QQ outputs. It hence suggests that both MBC-CycleGAN and dOTC algorithms, although correcting the spatial features, perform little changes of the (n) (o) (p) (g) (h) (i) (j) (k) (l) (m) temporal sequencing of the time series to correct. For MBC-CycleGAN, this is partly explained by the fact that, within the CycleGAN procedure, input maps from QQ outputs are transformed to outputs with improved spatial features, whilst not modifying too much the initial input image.
It hence results in partially preserving the temporal properties of the QQ outputs used as inputs of the CycleGAN while providing improvements of the spatial representation. This particular point is thereafter discussed in greater details. Concerning R 2 D 2 outputs, different results are obtained depending on the dataset to correct. For the correction of both IPSL and IPSLbis (Fig. 11k , l), R 2 D 2 provides small improvements of temporal properties of temperature, which illustrates that, by including lags in the conditional dimensions, R 2 D 2 is able to improve-in addition to spatial properties-temporal structure of climate datasets. However, for the correction of LR SAFRAN (Fig. 11m ), a deterioration of AR1 temperature differences is obtained with respect to initial LR SAFRAN data (Fig.
11d ). This result can be linked with the previously mentioned contrasted performances of the R 2 D 2 method to adjust LR SAFRAN dataset in Subsect. 5. 2. 2. For precipitation (Fig. S10 ), same conclusions hold for IPSL, IPSLbis and LR SAFRAN outputs. However, contrary to temperature, 1d-QQ corrections of IPSL and IPSLbis (Figs. S10e, f ) show a pronounced improvement of temporal properties for precipitation, highlighting the potential influence of marginal properties of precipitation time series on its autocorrelation values. Moreover, the improvements of temporal properties of temperature provided by R 2 D 2 for the corrections of IPSL and IPSLbis are no longer observed for precipitation (Fig. S10k, l ). Instead, temporal properties with unexpected behaviors are obtained, potentially due to the difficulty of R 2 D 2 to correct physical variables with events occuring at local scale, such as precipitation  (Vrac and Thao 2020).
As the LR SAFRAN dataset is temporally matching the SAFRAN dataset by construction, classic forecast statistics such as Root Mean Square Error (RMSE) can indeed be interesting to assess the performances of MBC methods. Table  2  shows, for temperature and precipitation, the RMSE values with respect to SAFRAN references for the different BC outputs of LR SAFRAN. For temperature, the RMSE value between daily maps of the reference and the LR SAFRAN dataset is around 0. 36. Slight improvement in terms of RMSE is provided by the QQ method (RMSE = 0. 31). As the QQ method preserves the temporal sequencing of the times series to correct, this improvement is only due to the correction of marginal properties. The MBC-CycleGAN presents better results (RMSE = 0. 23), permitting to state with more confidence that, while adjustment of spatial dependence structure are performed, it modifies only slightly the temporal sequencing of the times series to correct.