dataesr/scientific-paragraphs-categorization huggingface dataset:
To select the most appropriate kernel model (given the training dataset), we investigate whether the considered GP model is capable of: Predictability. This concept is related to whether the GP model is capable of predicting "yet-unseen" input configurations, i. e. , samples that have not been used for training, which can be assessed by using independent test samples, bootstrap or cross-validation approaches (e. g. ,  Hastie et al. , 2009). Two indicators are estimated. The first indicator, denoted Q 2 , measures the deviation from the true output value. Given a test set T, Q 2 is defined as follows: Q 2 = 1 - ∑ i∈T (y i -ŷi ) 2 ∑ i∈T (y i -ŷi ) 2 (10) where ŷi is the ith GP-based prediction of the model output y i , and y = 1 |T| ∑ i∈T y i is the average value for the test set. A coefficient Q 2 close to 1.
However, adding more model parameters results in increasing the explainability. To counterbalance this tendency (related to overfitting), a penalty term is generally introduced (see, e. g. ,  Höge et al. , 2018)  to select a simpler model. Here, simplicity refers to the number of GP model parameters, but alternative definitions exist (see, for instance,  Rougier and Priebe (2020) , who introduce the concept of flexibility). By assuming that the true GP model exists and that it is among the set of candidate GP models, we propose relying on the Bayesian information criterion BIC  (Schwarz, 1978)  described as follows: BIC = 2 log(l) + k. log(n) ( 12 ) where k is the number of parameters and n is the number of observations. Stability. We explore to what extent the kernel correlation matrix (derived from the covariance matrix) is stable to the perturbations in the training dataset.
We evaluate an error measure between the correlation matrices R0 , estimated using the GP model fitted with the whole training dataset and Rj , estimated at the jth iteration of the n cv -fold cross validation procedure. For instance, when n cv = 5, the training dataset corresponds to the whole training dataset, from which 20% of the observations have been randomly removed. The error measure is defined by: err = 1 n cv ∑ ncv k=1 ⃦ ⃦R 0 -Rk 2 F ⃦ ⃦ ( 13 ) where ‖. ‖ F is the Frobenius matrix norm. By restricting to the nondiagonal elements in the upper triangular part of R, Eq. (  13 ) is calcu- lated as 1 ncv ∑ ncv k=1 ( 1 np ∑ np i=1 (r i,0 -ri,k ) 2 ) , where ri,0 is the ith coefficient of R0 (read in column order for instance), ri,k is the ith coefficient of Rk , and n p is the number of terms of R.
e. , (u 1 ,u 3 ), and (u 2 ,u 4 ,u 5 )); -Low rank approximation. A matrix with rank of q = 2 is tested (denoted LR2); -Ordinal variable. The levels are ordered by following the level index. For the expert-based grouping, we assume that another expert does not know the correct ordering and assumes an unrealistic order (denoted OW), i. e. , u 4 <u 2 <u 5 <u 3 <u 1. The training dataset is defined through random sampling by considering m x-points per u-level with m = 4, 5 and 6, i. e. , with different training dataset sizes of 20, 25 and 30.
Implementation procedure In this section, we apply the GP-based procedure described in Sect. 2 to the cases described in Sect. 3. To ensure identifiability of the model defined with k(. ,. ), we treat k cont as a covariance kernel and each k j cat as a correlation kernel (j = 1 …,J). For all GP models, we consider a Matérn 5/2 covariance matrix for the continuous variables. For the ordinal kernel k O cat , a Matérn 5/2 continuous kernel is combined with a quadratic nondecreasing spline warping function F (see Eq. (  9 )). We consider GP models with constant trends and fit them using the R package kergp  (Deville et al. , 2018)  by applying a prescaling and centering of the predictability measured by 1-Q 2 (log 10 scale); (c) coverage error measured by 1-CA. The horizontal dashed line corresponds to 5%, i.
Application to the synthetic case Considering the synthetic test case, Fig. 4  depicts the GP-derived correlation matrices for each of the kernel assumptions described in Sect. 3. 1 using the training dataset of intermediate size (m = 5). The application of the expert-based and ordinal kernel assumptions (Fig. 4c, e ) shows some consistent structures among the levels, namely, the highly correlated group for u 1 to u 4 and the particular behavior of u 5. The magnitude of the interdependencies between the identified group and u 5 differs slightly: assumption E indicates a low-to-moderate correlation (~10%), whereas assumption O indicates a decreasing correlation from 25 to 15% for u 4-1. The structure of interlevel dependencies is richer for the general (Fig. 4a ) and LR2-based GP models (Fig.
4b ). For these correlation matrices, the interpretation is also less straightforward than for E or O, for which some group structures are more easily identified. The visual inspection of both matrices suggests, however, some interesting features, i. e. , the increasing correlation between u 1-4 (for Gen) and the particular behavior of u 5 , as well as a possible grouping of u 2-4 (for LR2). Capturing the correlation structure can be hard for these cases given the size of the training dataset (of 25); this is further discussed below when analyzing the stability criterion. In this case, the exchangeable assumption (i. e. , compound symmetry CS) leads to a lowto-moderate intercorrelation coefficient of ~40% (not shown). The four criteria for kernel model selection are examined in Fig. 5. For the sake of comparability between the different assumptions for m, we preferably plot the difference of BIC with the minimum value over the whole experiment (named ΔBIC).
Several observations can be made: -Explainability measured by BIC (Fig. 5a ) appears efficient here to exclude the unrealistic assumptions OW and EW: they present very large BIC differences regardless of m. For instance,  Burnham and Anderson (2002)  suggested a difference of the considered information criterion (relative to the minimum value) of at least 10 to support the ranking between model candidates with confidence; -BIC appears to be very informative to discriminate the kernel assumptions and clearly selects the ordinal assumption as the most appropriate; -The Q 2 criterion is commonly used in the computer experiment community to rank different models with respect to their predictive capability. In our case, basing the analysis on this unique criterion is difficult: at low size of the training dataset (m = 4), models E, CS and O all minimize 1-Q 2 , and show similar performance (see median values in Fig.
5b ): selecting one of them is thus hardly achievable. Furthermore, the model associated with the unrealistic expert-based grouping EW has a satisfactory predictive capability at low m values (although the width of the confidence interval is larger than the others); -For larger m value -here m = 6 -(i. e. , with the largest size of the training dataset), the Q 2 criterion allows us to identify model O as the most appropriate model with respect to the predictive capability (log 10 (1-Q 2 ) is the lowest in Fig. 5b ); -The other facet of predictability related to the coverage of the prediction intervals suggests discarding kernel assumptions Gen and LR2 (because CA is here far larger than the level of the prediction interval) but hardly allows differentiating the other assumptions regardless of m (Fig.
5c ); -The stability criterion (Fig. 5d ) tends to suffer from the same sensitivity as Q 2 to the size of the training dataset but has a higher discriminative power: at low m values (m = 4), both kernel models CS and O are selected as very stable (because of low log 10 (err) values in Fig. 5d ). However, a high stability is also reached for EW. For a large m value, the ordinal assumption is then clearly selected as the assumption leading to the most stable correlation matrix. From this analysis, we can conclude that the ordinal assumption allows us to successfully fulfill three of the criteria (explainability, predictability, and stability), especially at sufficiently high m values (m ≥ 5). For coverage, the ordinal assumption is not ranked first, but CA appears of a reasonable order of magnitude (median value of ~85%), i.
e. , with moderate deviation from the level of the 95% predictive interval. For the small size of the training dataset (m = 4), unambiguously rejecting selecting EW and CS becomes difficult if the explainability criterion (BIC criterion) is not considered, which can partly be explained by the analysis of the EW-based correlation matrix (Fig. 4d ), which reveals a (quasi)homogeneous structure with correlation coefficients ranging from 44 to 56%, i. e. , of the same order of magnitude as the CS assumption (of ~40%), hence indicating that both GP models should perform similarly, suggesting that at a low number of training points, the CS assumption remains the most reasonable assumption (when the goal is the joint maximization of predictability, stability and explainability).
For instance, in the cyclone real case application, Fig. 11a  gives the tree structure derived from the analysis of the cyclone real case. However, from the analysis of Fig. 11 , we note several differences between the structure constructed using the DT trained using the whole dataset (Fig. 11a ) and the ones at each iteration of the cross-validation procedure (i. e. , using three DT model setups with a "perturbed" training database, Fig. 11b-d ); in particular, for the leftmost part related to the track variable (outlined by a green rectangle), the grouping of tracks is similar for Figs. 11a  and c  but differs for Fig. 11b  and is even absent for Fig. 11d. This high sensitivity of the derived structure (to the changes in the training dataset) has already been identified in the literature  (Breiman, 2001) : in addition to bringing some confusion regarding the dependencies between levels, the drawback is also a poorer predictability: this is shown by the low Q 2 values for each test case in Table  2.
Concluding remarks, recommendations and further work Model uncertainties (related to the structure/form of the model or to the unambiguous choice of "appropriate" physical laws) are generally analyzed in computer-based studies by defining a categorical variable, i. e. , a multilevel indicator. To obtain deeper insights into how the simulator output values computed for two levels of the categorical variable correlate, the interlevel dependence structure is learned from a series of computer experiments (100-200) by means of a GP-based correlation matrix. Table  3  summarizes the key aspects of the GP approach for three test cases together with a comparison to tree-based methods, showing that the GP-based approach can be seen as a satisfactory compromise because: (1) it clearly achieves higher predictability with a Q 2 value > 90% given a moderate size of the training dataset (typically 100-200), whatever the considered case (Table  2 ), while reaching high performance with respect to different criteria (explainability, simplicity, stability); and (2) the correlation matrices provide a concise and graphical way of interlevel dependence structure (using the interpretation provided in Sect.
Software and data availability Software name: kergp Developers: Yves Deville, David Ginsbourger, Olivier Roustant. Contributors: Nicolas Durrande Maintainer: Olivier Roustant roustant@insa-toulouse. fr. System requirements: Windows, Linux, Mac. Program language: R. Availability: https://cran. r-project. org/web/packages/kergp/index. html. License: GPL-3. 0. Documentation: https://cran. r-project. org/web/packages/kergp/ kergp. pdf. Reproducible code: a Jupyter Notebook is available on Zenodo at  (Rohmer, 2022). It showcases the implementation of Gaussian process models using mixed continuous/categorical inputs variables with an application on the real case in the domain of marine flooding (application case 1 described in Sect. 3. 2, and results discussed in Sect. 4. 3). The design of numerical experiments and the simulation results for application case 1 are provided; to re-use these numerical simulation results, please contact Jeremy Rohmer, BRGM (j.
Acknowledgments This research was conducted within the frame of the Chair in Applied Mathematics OQUAIDO, gathering partners in technological research ( BRGM, CEA, IFPEN, IRSN, Safran, Storengy ) and academia ( CNRS, Ecole Centrale de Lyon, Mines Saint-Etienne, University of Grenoble, University of Nice, University of Toulouse ) around advanced methods for computer experiments. Data used for real case 1 are based on work funded by the  French National Research Agency  within the  SPICy  project ( ANR -14 -CE03 -0013 ). Data used for real case 2 are based on ULTIMATE- CO2  Project, funded by the  European Commission 's  Seventh Framework Program  [FP7/2007-2013] under grant agreement n • 281196. We are grateful to  Prof. J. Rougier  as well as to two anonymous reviewers for their comments that led to improvements of the article.
Study cohort All adult patients that underwent PRLA or TLA between January 2015 and December 2020, registered in EUROCRINE®, were included in the analysis. Patients without histopathology results were excluded. Robot-assisted cases and bilateral adrenalectomies were excluded as they present subgroups with possible different advantages and morbidity, and as they were unequally divided amongst both study groups  [12]. EUROCRINE® is an online, endocrine surgical quality registry that initially focused on rare endocrine tumors but has evolved into a pan-European database collecting data on all endocrine surgical procedures. It has already led to several papers in the eld of adrenal surgery  [12, 13]. Data entry is not mandatory, but EUROCRINE® board members are responsible for assessment of compliance on national level.
Postoperative characteristics included length of hospital stay (de ned as the number of days in hospital after surgery, dichotomized at a threshold of 2 days), 30-day morbidity, hospital readmission, and 30-day all-cause mortality. In view of the information provided from prede ned and free text data elds, all complications were categorized according to the Clavien-Dindo classi cation  [14]. Also, histopathological outcomes and completeness of resection were collected. The primary outcome was morbidity (Clavien-Dindo score of 2 or more). The secondary outcomes were length of hospital stay and conversion to open surgery. Sensitivity analyses were performed for patients with tumor sizes > 50 mm, patients with BMI > 30 kg/m 2 , patients with pheochromocytoma, and according to side of the tumor as they present additional surgical and anatomical challenges.
Propensity score matching was used to account for possible confounders on outcome  [18]. Propensity scores (ranging from 0 to 1, the probability of a patient assigned to PRLA or LTA) were derived using a logistic regression model including age, sex, BMI, tumour size, center experience, and hormonal excess status. One-to-one propensity score matching was performed by nearest neighbour matching within a calliper of 0. 2. The balance of confounding variables between both groups of the matched data was assessed with standardized mean differences (SMD). SMD < 10% is a recommended threshold for reporting balance  [18]. All statistical analyses were conducted using IBM SPSS Statistics for Windows (Version 25. 0, Armonk, NY) and R 3. 6. 3 open-source software with 'MatchIt' package (http://www. Rproject.
Data from high-and low-volume centres could be evaluated. Different subgroup analyses have con rmed the overall results. Some limitations of this study are acknowledged. Data on speci c surgeon experience within the highand low-volume experienced centres is lacking. Length of hospital stay might be in uenced by differences of health care reimbursement amongst the participating European countries. As other registries, EUROCRINE registry is prone to typing and coding errors, and missing data. No information on estimated blood loss, operative times, and healthcare costs was available. The type of complication data had been registered in the prede ned and free text data elds for only 108 patients. Both surgical techniques are feasible and safe and have a clear advantage on open surgery. The shorter length of hospital stay after PRLA is important as it can in uence future healthcare and hospital ward planning, and it may subsequently reduce healthcare costs.
ACKNOWLEDGMENTS We would like to thank all participating clinics in the EUROCRINE database (https://eurocrine. eu) and all members of the EUROCRINE ® council:  David Scott-Coombes ,  Anders Bergenfelz ,  Thomas Clerici ,  Philipp Riss ,  Thomas Musholt ,  Erik Nordenström ,  Sam Van Slycke ,  Maurizio Iacobone ,  Laurent Brunaud ,  Marcin Barczynski ,  Konstantinos Nastos ,  Marco Raffaelli ,  Marit Helene Hansen ,  Christina Martinez Santos ,  Olov Norlén ,  Menno Vriens ,  Özer Makay ,  Fausto Palazzo , and  Arnold Trupka.
Mass distributions are used illustratively, and the measured grain diameters are instead used in any further analyses. As we use the semimajor axis, particle diameters are nominally maxima, notwithstanding their extension in 3D. All particle size data shown in Fig. 2d , h, l, p were obtained at 4000-4300 magnification, and all data in Fig. 6d  at 3700-4300 magnification: annotated SEM images are provided as Supplementary Fig. 1. The number and size of particles that can be classified in this way is dependent on image size and resolution; accordingly, in order to ensure the total particle population was appropriately and representatively sampled, we also measured particles using a wider range of magnifications. Decreasing and increasing the magnification to 500 × and 10,000 × , respectively, did not appreciably affect the fraction of largest or smallest particles observed.
Acknowledgements H. T. and J. F. were supported by a  Royal Society University Research Fellowship   UF140716  and fieldwork by F. W. was financially supported by the  EC-FP7   Vuelco  project. H. T. , J. C. , and C. I. S. thank  Alex Barria  and  Miguel Obando  for logistical assistance in the field. H. U. was supported through an  Envision PhD studentship. Carmel Pinnington  and  Sara Baldock  assisted with SEM imaging. We thank  Lionel Wilson  and  Duncan Woodcock  for discussion of particle transfer through pathways.
Code availability Python code used to analyse and plot data are provided in a Jupyter Notebook at the following repository: https://github. com/ jifarquharson/cordon-caulle-ash-vents/, archived via Zenodo: Jamie Farquharson. (2022). jifarquharson/cordon-caulle-ash-vents: (v1. 0). Zenodo. https://doi. org/10. 5281/zenodo. 6815907.
Feature Selection A recursive feature elimination was then applied to reduce redundant features. The Random Forest classifier ranks the importance of features through their impact on the model accuracy. Recursive feature elimination can be used to rank the feature by looping through Random Forest classifiers and removing the least important feature until the accuracy remains stable. Here we generated collections of features and a 4-fold cross-validation was utilized to test the performance of the various feature collections on the MLP model. Cross-validation is a universal method for assessing the accuracy of a classifier based on a limited data sample. The dataset was generated from the field-observed data in combination with the Sentinel-2 imagery, and the whole dataset was split into 5 individual sets.
A test set was held out for final evaluation. Then, three of the remaining datasets were used as training datasets, initially used to fit the model. The last dataset is used as a validation dataset, which provides an unbiased evaluation of a model fit on the training dataset while tuning the parameters. The process is repeated 4 times. The feature collection including the fewest number of features achieving the same performance as using all features was selected for the final model. The performance metrics of recall and precision were used to evaluate models using different feature collections  [63]. Precision is defined as the fraction of relevant instances among the retrieved instances, i. e. , the number of true positives divided by the total number of elements labelled as belonging to the positive class.
Feature Selection A recursive feature elimination was then applied to reduce redundant features. The Random Forest classifier ranks the importance of features through their impact on the model accuracy. Recursive feature elimination can be used to rank the feature by looping through Random Forest classifiers and removing the least important feature until the accuracy remains stable. Here we generated collections of features and a 4-fold crossvalidation was utilized to test the performance of the various feature collections on the MLP model. Cross-validation is a universal method for assessing the accuracy of a classifier based on a limited data sample. The dataset was generated from the field-observed data in combination with the Sentinel-2 imagery, and the whole dataset was split into 5 individual sets.
Mapping Faidherbia albida Trees Using Artificial Neural Networks The Multi-Layer Perception (MLP) model was adopted for the binary classification of Faidherbia albida trees. MLP is an artificial neural network with multiple densely connected layers between the input and output layers. The densely connected layers in MLP models combine all the features of the previous layer, which is suitable for complex classification tasks. We used the artificial neural network instead of other non-parametric classifiers, such as Random Forests or Support Vector Machines, which have been applied in previous tree species classifications, because of the irregular spectral signatures of the dataset and the uneven composition of species in the data structure. Considering the image resolution and the size of tree canopy, any given Faidherbia albida pixel will usually be a mix of reflectance from Faidherbia albida canopies and background vegetation or bare soil.
To explore the agreement between our remote sensing based Faidherbia albida distribution map and the potential occurrence based on ecological modelling, we obtained a Faidherbia albida potential occurrence map that was provided by Kindt  [64]  at a 5-km resolution. The occurrence map was generated from 9 bioclimatic layers, 2 topographic layers and 3 soil maps by the R package BiodiversityR. Details about the data sources and resolution of the layers can be found in Table  S4. This map consists of a unitless Faidherbia albida habitat suitability index varying from 0 (lowest) to 910 (highest). The absence-presence threshold for suitability of Faidherbia albida was provided by the SDM model as 534, meaning that suitability values above 534 correspond to the likely presence of the species with increasing values associated with higher likelihood of species presence.
The final selected features for the MLP model (Table  1 ) show that the seasonal dynamic features account for more than 50% of all selected features, and the typical span of the two months forming the difference feature is around 5 months, usually representing one month in the start of the dry season and the other month at the end of the dry season. The acquired times of the selected data by recursive feature elimination are February (start of Faidherbia albida senescence), July (leaf-off) and October (leafout). In terms of feature type (vegetation index or single band reflectance), the NDVI and NDI54 dominate the selected features.
The final selected features for the MLP model (Table  1 ) show that the seasonal dynamic features account for more than 50% of all selected features, and the typical span of the two months forming the difference feature is around 5 months, usually representing one month in the start of the dry season and the other month at the end of the dry season. The acquired times of the selected data by recursive feature elimination are February (start of Faidherbia albida senescence), July (leaf-off) and October (leaf-out). In terms of feature type (vegetation index or single band reflectance), the NDVI and NDI54 dominate the selected features.