Nawel Zemmal

Nacer Eddine Benzebouchi

Nabiha Azizi

Didier Schwab

Brahim Belhaouari

Unbalanced Learning for Early Automatic Diagnosis of Diabetes Based on Enhanced Resampling Technique and Stacking Classifier

Keywords: Unbalanced learning, multimodal classification, ensemble learning, meta-learning, diabetes diagnosis

Diabetes is characterized by an abnormally enhanced concentration of glucose in the blood serum. It has a damaging impact on several noble body systems, mainly on the cardiovascular, renal, and visual systems. Automated screening allows early diagnosis of certain illness (such as diabetes), which generally increases the chances for successful treatment. Today, machine learning has developed considerably in the domain of medical diagnosis, especially with regard to diabetes diagnosis, and as such, thanks to the integration of the concept of unbalanced learning, which considerably reduces the generation of erroneous classification results. This general concept is dealt with from two different perspectives, i.e. at the data level through modification/balancing of the learning data set as well as at the algorithm level. The present paper takes a hybrid approach towards imbalanced learning in proposing an enhanced multimodal meta-learning method called IRESAMPLE+St to distinguish between normal and diabetic patients. This approach relies on the Stacking paradigm by utilizing the complementarity that may exist between classifiers. In the same focus of this study, a modified RESAMPLE-based technique referred to as IRESAMPLE+ and the SMOTE method is integrated as a preliminary resampling step to overcome and resolve the problem of unbalanced data. The imbalanced Pima Indian Diabetes (PID) data set is optimized through the proposed IRESAMPLE+ method, successfully operating as both an oversampling and undersampling technique, thereby reinforcing the diagnostic accuracy established by the Stacking classifier. The suggested IRESAMPLE+St provides a computerized diabetes diagnostic system with impressive results, Accuracy of 99.87%, Sensitivity of 100%, Specificity of 99.70% and AUROC of 99.90%, comparing them to the principal related studies. The over-performing results reflect the design and engineering successes achieved with the IRESAMPLE+St system for the classification of diabetes.

Introduction

Diabetes is considered one of the major diseases and greatest challenges facing health systems. Due to the modern lifestyle, the incidence of diabetes in the world is in constant increasing, particularly among children. In 2015, diabetes, known as a silent killer, was the direct cause of 1.6 million deaths, and in 2012, hyperglycemia caused an additional 2.2 million deaths. More than 400 million persons in the world are living with diabetes. The number of cases of this chronic disease has quadrupled since 1990, from 108 million in 1980 to 422 million in 2014. The World Health Organization (WHO) warns that if the current tendency persists, its prevalence will increase. 1Diabetes is a chronic metabolic illness that cannot be cured but can be treated and controlled. It is caused by a lack/default of use of a hormone called insulin. Insulin is produced by the pancreas; it allows glucose (sugar) to enter the body's cells to be used as an energy source. In a non-diabetic person, insulin fulfills its role well and cells have the energy they need to function. In the case of insulin insufficiency (insulinopenia) or when it does not effectively perform its function, as is the case in a person with diabetes, glucose cannot be used as fuel for cells. It then accumulates in the blood and causes an increase in the sugar level (hyperglycemia). In the long term, hyperglycemia causes certain complications affecting many noble systems of the body, such as ocular, renal, nerve, heart, and blood vessel complications. Automated screening allows an early diagnosis of certain illness [START_REF] Lamari | SMOTE-ENN-Based Data Sampling and Improved Dynamic Ensemble Selection for Imbalanced Medical Data Classification[END_REF][START_REF] Zemmal | An Enhanced Feature Selection Approach based on Mutual Information for Breast Cancer Diagnosis[END_REF], before the appearance of symptoms/complications, as well as better management and a reduction in social costs. Nowadays, Machine Learning (ML) techniques are playing a fundamental role in the evolution of the domain of medical diagnosis, particularly with respect to diagnosing diabetes. The principal goal of diagnostic aid systems is to enhance diagnostic accuracy. Actually, they are used as a second opinion by physicians to obtain the final diagnosis, which may reduce human mistakes. For this reason, several studies have been suggested concerning the automated classification/diagnosis of diabetes [START_REF] Barakat | Intelligible Support Vector Machines for Diagnosis of Diabetes Mellitus[END_REF][START_REF] Cao | Diabetic plantar pressure analysis using image fusion[END_REF][START_REF] Choudhury | A Survey on Medical Diagnosis of Diabetes Using Machine Learning Techniques[END_REF][START_REF] Pradhan | Phenotype Prediction from Metagenomic Data Using Clustering and Assembly with Multiple Instance Learning (CAMIL)[END_REF]Zou et al., 2019). The majority of medical diagnoses are based on a binary decision (for example the patient is diabetic or non-diabetic), hence the interest of classification into two categories. The classifiers generally used in the classification phase are Support Vector Machine (SVM) [START_REF] Barakat | Intelligible Support Vector Machines for Diagnosis of Diabetes Mellitus[END_REF][START_REF] Pradhan | Phenotype Prediction from Metagenomic Data Using Clustering and Assembly with Multiple Instance Learning (CAMIL)[END_REF][START_REF] Abdillah | Diagnosis of Diabetes using Support Vector Machines with Radial Basis Function Kernels[END_REF], Gaussian Process Classification (GPC) [START_REF] Maniruzzaman | Comparative approaches for classification of diabetes mellitus data: Machine learning paradigm[END_REF], Random Forest (RF) [START_REF] Nai-Arun | Comparison of Classifiers for the Risk of Diabetes Prediction[END_REF]Zou el al., 2018), and Convolutional Neural Network (CNN) [START_REF] Li | Convolutional Neural Network Based Clustering and Manifold Learning Method for Diabetic Plantar Pressure Imaging Dataset[END_REF]. The aim of any classification problem is to find the best features and a linear/non-linear separation boundary in order to well segregate between categories. We note that each classifier has its own manner of producing the margin that divides the classes and the resulting model varies from one classifier to another. That's why the ML Community has claimed that there is not a better (single) classifier in all cases. In other words, there are no dominant classifiers (concerning the error rate) all the others for all problems; consequently, the choice of the appropriate classifier for a given problem is not obvious and remains a challenge in the domain of ML. Likewise, unbalanced learning or learning through imbalanced data is a common problem associated with the often unbalanced medical databases and is seen as another challenge in the area of ML, in particular with supervised learning. Class-imbalance has recently appeared in many areas of applications, including disease screening [START_REF] Nnamoko | Efficient Treatment of Outliers and Class Imbalance for Diabetes Prediction[END_REF], rare event prediction [START_REF] Li | Convolutional Neural Network Based Clustering and Manifold Learning Method for Diabetic Plantar Pressure Imaging Dataset[END_REF], and spam filtering [START_REF] Ratadiya | Spam filtering on forums: A synthetic oversampling based approach for imbalanced data classification[END_REF] , where there are more samples available for some categories than for others. In particular, in a binary classification problem, the learning data of the minority class is much less represented than those of the majority class, which disrupts the ML algorithms [START_REF] Chawla | SMOTE: Synthetic minority over-sampling technique[END_REF][START_REF] Japkowicz | The class imbalance problem: A systematic study[END_REF]. Figure 1 is an illustrative example of imbalanced data. In fact, all ML techniques always tend to neglect the minority class, when in most cases it is most interesting. And this is due to the fact that the separation boundary or decision boundary is converging towards the majority class, i.e. ML algorithms only learn to predict recurrent classes and thus may lead to misclassification results on the part of the classifiers, especially on the validation set. For this purpose, it would be interesting to apply a data set balancing module so as to avoid learning errors and skewed classification. Two aspects are addressed by means of this module, that is, the processing at the data level as well as at the algorithm level. However, most approaches are addressing this issue primarily at the data level in terms of changing/balancing the learning data set through resampling techniques (oversampling or undersampling), such as the RESAMPLE and Synthetic Minority Over-sampling Technique (SMOTE) strategies, which are considered among the widely adopted methods to circumvent problems related to unbalanced data [START_REF] Nnamoko | Efficient Treatment of Outliers and Class Imbalance for Diabetes Prediction[END_REF][START_REF] Li | Convolutional Neural Network Based Clustering and Manifold Learning Method for Diabetic Plantar Pressure Imaging Dataset[END_REF][START_REF] Chawla | SMOTE: Synthetic minority over-sampling technique[END_REF][START_REF] Japkowicz | The class imbalance problem: A systematic study[END_REF]. Oversampling is a way to rebalance the data set; it consists of increasing the number of instances belonging to the minority class by replicating them in a random way. Undersampling consists of randomly removing from the learning base instances belonging to the majority class, so as to rebalance the class distribution. Data resampling aims to prevent the problem of data imbalance to restore a more correct/normal situation, and can also be used to improve learning in order to obtain a more robust model. Relatively few methods treat the class imbalance problem at the algorithm level that consists of adapting traditional learning models so as to attenuate the bias against majority classes and to fit them to mine-related data having biased distributions [START_REF] Krawczyk | Learning from imbalanced data: open challenges and future directions[END_REF]. Such a branch is not as widely taken up among the researchers because of its delicacy in terms of design that is directly dependent upon the dataset being used. On the other hand, Ensemble Learning (EL) is becoming increasingly widespread as a means of dealing with class imbalance [START_REF] Krawczyk | Learning from imbalanced data: open challenges and future directions[END_REF]. For this reason, ensemble learning (combination of classifiers) has become the widely adopted tendency in medical diagnostic support systems. Currently, EL occupies an important place in the field of Machine Learning (ML) and Artificial Intelligence (AI) because of these honorable results in various applications [START_REF] Lamari | SMOTE-ENN-Based Data Sampling and Improved Dynamic Ensemble Selection for Imbalanced Medical Data Classification[END_REF][START_REF] Abdillah | Diagnosis of Diabetes using Support Vector Machines with Radial Basis Function Kernels[END_REF][START_REF] Benzebouchi | Multi-classifier system for authorship verification task using word embeddings[END_REF][START_REF] Ashraf | Using Ensemble StackingC Method and Base Classifiers to Ameliorate Prediction Accuracy of Pedagogical Data[END_REF]. The main idea of EL contributes to improving the performance of unbalanced learning by merging various classifiers so as to reduce variance, bias, and/or otherwise enhance predictions. Ensemble Learning methods in ML and AI consist of seeking to benefit from the varied expertise of different classifiers to obtain by combination a final decision model that is expected to be better than any classifier when it's considered separately. Figure 2 The success of EL methods lies in the fact that they guarantee a lower error than its best classifier. Stacking (also called meta-classification) is an ensemble learning paradigm that allows combining various basic classifiers using a high-level meta-classifier, unlike other algorithms such as bagging and boosting based on voting methods. On the one hand, combining the characteristics of several classification models to create a new is more powerful model. The variability and diversity between different classifiers of the model are an important condition in order to take advantage of the complementarity that may exist between them. In other words, generate a set of complementary (multimodal) classifiers that can be combined (meta-learned) to arrive at an optimal solution. Diversity/Complementarity is considered a fundamental property of the concept of multimodality. This article proposes a hybrid approach-based improved multimodal meta-classifier method called IRESAMPLE+St towards unbalanced learning for diagnosing diabetes on the basis of the Stacking algorithm by using the diversity that exists between various classifiers. With the same purpose of the study is also to obtain a balanced data set involving a data pre-processing phase through widespread resampling techniques such as the modified RESAMPLE-based method also known as IRESAMPLE+ and the SMOTE technique, in order to form the basic classifiers of the pool as well as to get a more efficient model. The basic classifiers pool consists of five heterogeneous classifiers, namely multilayer perceptron (MLP), K-Nearest Neighbors (K-NN), Support Vector Machine (SVM), Random Forest (RF), and Naive Bayes (NB). The classification model is generated from different basic classifiers that will be used as new input data for a meta-classifier as meta-characteristics. The main task of metaclassifier or stacking aggregator, referred to as SVM, is to learn the best way to combine the different models of classification to obtain the most accurate/best classifier.

In this study, the following search questions will be exanimated closely:

 Does data preprocessing play a significant role in the performance of the proposed system?  Which resampling technique is better adapted for balancing data?  Which aggregation method is better suited for the combination of classifiers?  Does the combination of classifiers have an impact on performance?

The following points highlight the principal contributions of this article:

 Hybridizing between two unbalanced data classification approaches, i.e. the data level-based and the algorithm level-based.  Proposal for a modified resampling strategy called RESAMPLE-based IRESAMPLE+ to enhance the classification performance on the Pima Indians Diabetes (PID) unbalanced medical data set that operates successfully as both an oversampling and undersampling technique.  Analyzing how a meta-classification-based ensemble learning known as stacking aggregator behaves while integrating a cross-training module as well as generating a pool of complementary basic classifiers.

This article is structured as follows: an overview of the related work introduced in Section 2. Section 3 represents the meta-classification architecture as well as explains the different stages of the proposed approach. Section 4 illustrates the experimental results of this study. Finally, the conclusion of this work presented in section 5.

Related studies

Various studies have been led to develop computerized systems for the detection of diabetes. Automated diagnosis of diabetes was performed using different machine learning (ML) paradigms, some of which include techniques dealing with the unbalanced data problem and ensemble methods. [START_REF] Barakat | Intelligible Support Vector Machines for Diagnosis of Diabetes Mellitus[END_REF] proposed a diabetes diagnosis system using the SVM Classifier with RBF kernel (gamma of 0.0005 and C of 5), as well as two rules extraction techniques, SQRex-SVM and eclectic, constituting the last unit of the proposed approach, thus transforming the SVM black-box into a more comprehensible diagnostic model. An under-sampling strategy is adopted in this approach to solve the class imbalance problem, which is the K-means clustering algorithm. The results obtained showed Accuracy (Acc) of 94%, Sensitivity (Sen) of 93% and Specificity (Spe) of 94% using the leave-one-out cross-validation method. [START_REF] Pradhan | Phenotype Prediction from Metagenomic Data Using Clustering and Assembly with Multiple Instance Learning (CAMIL)[END_REF] suggested a diabetes screening approach employing several data preprocessing techniques, such as normalization, discretization, and feature selection. This latter concept is applied by using the correlation-based function selector (CFS) algorithm. The SVM classifier is adopted for the classification/detection phase of diabetes using the k-fold cross-validation (CV) method with k=2. The results of the model showed that the rate of properly classified instances was 86.46% and Area Under Receiver Operating Characteristic (AUROC) curve was 83.00%. [START_REF] Nai-Arun | Comparison of Classifiers for the Risk of Diabetes Prediction[END_REF] contributed to an analytical study of the performance of different ML techniques (decision tree, artificial neural network, logistic regression, and naïve bayes) to predict diabetes, including ensemble learning methods (bagging, boosting, and random forest). A comparison of 13 classification models is carried out in order to choose the best one. Experimental results showed that the Random Forest (RF)-based ensemble method outperforms all other models with an Acc of 85.558% and an AUROC of 91.20% applying 10-fold CV. [START_REF] Abdillah | Diagnosis of Diabetes using Support Vector Machines with Radial Basis Function Kernels[END_REF] presented a tool for early diagnosis of diabetes by combining two ML techniques, namely: SVM with Radial Basis Function (RBF) kernel. This model is trained using the kfold cross-validation technique with k=10, and its performance is assessed on the basis of performance measures derived from the confusion matrix, defined as follows: Acc = 80.22%, Sen = 82.56%, Spe = 79.12%, and AUROC = 80.84%. [START_REF] Perveen | Performance Analysis of Data Mining Classification Techniques to Predict Diabetes[END_REF] presented an ensemble learning approach based on the AdaBoost algorithm for diagnosing diabetes, through three age groups of the Canadian population. The analytic study showed that the performance of AdaBoost using J48 as a basic classifier outperforms the bagging using J48 as a basic classifier and the J48 decision tree algorithm as a single classifier. 2020) presented an ensemble method based on voting using fifteen ML algorithms to diagnose type-2 diabetes disease. Only the four paradigms such as K-NN, SVM, ANN, and NB were considered in this study, thus representing the basic classifier pool and majority voting was adopted to reach a final decision with an accuracy of 98.60%. [START_REF] Maniruzzaman | Classification and prediction of diabetes disease using machine learning paradigm[END_REF] opted for an analysis of different classifiers such as NB, DT, RF, and Adaboost (AB) in order to maintain the most efficient classifier for the distinction between normal and diabetes patients. The logistic regression (LR) algorithm is considered in this work as a feature selection technique to determine diabetes risk factors according to p-value and odds ratio (OR). The RF classifierbased LR paradigm performed best with an Acc of 94.25%, a Sen of 99.57%, and an AUROC of 95.00% by applying the 10-fold CV protocol. [START_REF] Ramani | MapReduce-based big data framework using modified artificial neural network classifier for diabetic chronic disease prediction[END_REF] suggested a modified ANN classifier approach to diabetes prediction that uses a MapReduce scheme in order to give a realizable frame within predictive programming paradigms for the map and reduce functions. The min-max normalization technique is performed for the pre-processing phase which consists of resizing the outputs of a value range to a new value range (e.g. 0 to 1 or -1 to 1). The proposed study reached an accuracy level of approximately 99.60%. Tama and Rhea (2019) presented an exploratory study concerning diabetes early detection on the basis of five various ensemble learning strategies including the bagging, random subspace, rotation forest, boosting, and DECORATE approaches. Each of them is made up of eight tree-based basic classifiers, i.e. CART, C4.5, REPT, RT, NBT, FT, BFT, and LMT. This study's performance is evaluated in terms of AUROC using a replicate cross-validation strategy (10 × 5𝐶𝑉). [START_REF] Mahabub | A robust voting approach for diabetes prediction using traditional machine learning techniques[END_REF] proposed a diabetes early detection system through an analytical study of several ML paradigms like K-NN, AdaBoost (AB), DT, RF, SVM, GradientBoosting (GB), LR, MLP, MultinomialNB, ExtremeGB, GaussianNB in an aim to select three most successful classifiers (K-NN, SVM and MLP) and subsequently used to construct the base classifier pool adopting a voting approach (hard and soft voting). For more information concerning computerized diabetes systems, the authors can be referred to recent reviews [START_REF] Choudhury | A Survey on Medical Diagnosis of Diabetes Using Machine Learning Techniques[END_REF], Sankar [START_REF] Ganesh | Diagnosis of diabetes type-II using hybrid machine learning based ensemble model[END_REF]. Most of the studies cited in the literature focus on a single classifier for the diagnostic/prediction phase of diabetes. In addition, few works use EL algorithms that are based primarily on homogeneous basic classifiers as well as voting techniques involving aggregation at the class-level, which supplies less information upon the merger. While, very few studies deal with the concept of unbalanced data at the data level through the SMOTE technique.

To the best of our knowledge, this is the first study to propose an unbalanced multimodal learning approach from an ensemble of a wide variety of heterogeneous classifiers that will be merged via a stacking aggregator or meta-classifier that handles fusion at the score-level through the rich information available at this level by applying the suggested IRESAMPLE+ technique.

Proposed IRESAMPLE+St System

The methodology of the proposed IRESAMPLE+St system is presented through a schema illustrated in Figure 3. The principal preoccupation expressed in this study consists of data balancing with the use of the enhanced RESAMPLE technique which is called IRESAMPLE+. Furthermore, this paper also seeks to generate an ensemble of complementary classifiers as well as the analysis of different fusion approaches using a new fusion strategy known as meta-classifier.

Data preprocessing

Data preprocessing is seen as a necessary phase in the domain of Machine Learning (ML) and data mining in order to eliminate noisy data while training classifiers on better quality data. Data resampling is a frequently used notion in an attempt to lessen the impact of data imbalance during learning. In general, this concept of resampling involves reducing the sample size of the majority class (undersampling) or increasing the sample size of the minority class (over-sampling). Thus, for a two-class classification problem, the minority class has a relatively smaller number of samples relative to the majority class. This latter class then has a proportionally larger number of samples. In this study, the preliminary step for data pre-processing is the make use of appropriate resampling techniques; it is about constituting a balanced data set permitting an optimal representation/distribution of the data so as to train the basic classifiers pool (promotes balanced learning). This is done via the SMOTE technique as well as a modified method based on the RESAMPLE technique, named IRESAMPLE+, commonly used owing to its better performance in the field of ML [START_REF] Borges | Ensemble of machine learning algorithms for cryptocurrency investment with different data resampling methods[END_REF][START_REF] Elreedy | A Comprehensive Analysis of Synthetic Minority Oversampling Technique (SMOTE) for Handling Class Imbalance[END_REF].

The general principle of the SMOTE technique (supervised oversampling strategy) is to produce artificial instances to extend the boundaries of the minority class; these instances are randomly generated along line segments of a number of k-nearest neighbors that belong to the same class. Thus, this strategy makes the minority class area larger and general. The SMOTE strategy uses four parameters to determine the number of instances to create:

𝑆𝑀𝑂𝑇𝐸 ← 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 (𝑆, 𝑃, 𝐾, 𝐶)

(1) 𝑤ℎ𝑒𝑟𝑒, -𝑆 indicates the random number seed, -𝑃 indicates the percentage of SMOTE instances to establish, -𝐾 is the number of nearest neighbors to utilize, and -𝐶 define the index of the nominal class value to SMOTE (minority class). Figure 4 shows an example of a synthetic instance generation. This principle is based on the difference between a minority instance and its nearest neighbor; this difference is multiplied by a random number between 0 and 1, and then added to the characteristic vector. The following pseudo-code illustrates this concept: Pseudo-code: SMOTE (oversampling strategy) Input: unbalanced data set (diabetes) Output: balanced data set Process: 1. For a sufficient number of synthetic instances do 2. Select a minority instance 𝐼 3. Select one of the nearest neighboring instances 𝑁 4. Select a random weight between 0 and 1 𝑊 5. Create the new synthetic instance 𝑆 6. For each attribute do 7.

Compute: 𝑣𝑎𝑙𝑢𝑒𝑆 = 𝑣𝑎𝑙𝑢𝑒𝐼 + (𝑣𝑎𝑙𝑢𝑒𝑁 -𝑉𝑎𝑙𝑢𝑒𝐼) × 𝑊 8. End for 9. End for The RESAMPLE technique (supervised subsampling strategy) involves creating a randomized subsample of a data set employing sampling with or without replacement, in that every instance of the sub-ensemble has the same probability of being chosen. RESAMPLE is supposed to be an impartial representation of a cluster. The following terms refer to the number of instances to be selected as part of the RESAMPLE strategy: 𝑅𝑒𝑠𝑎𝑚𝑝𝑙𝑒 ← 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 (𝑆, 𝑍, 𝐵, 𝑛𝑜 -𝑟𝑒𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡, 𝑉)

(2) 𝑤ℎ𝑒𝑟𝑒, -𝑆 is the random number seed, -𝑍 indicates the size of the output data set as a percentage of the input data set, -𝐵 is the bias factor to a uniform distribution of classes, -𝑛𝑜 -𝑟𝑒𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 disables replacement of instances, and -𝑉 reverses the choice -just available with -no-replacement-. Figure 5 shows an example of RESAMPLE technique. The goal is indeed to analyze each of these two methods (SMOTE, RESAMPLE, and IRESAMPLE+ based on the RESAMPLE approach) in order to keep the most efficient strategy.

Ensemble Classifiers approach: Basic classifiers & Aggregation strategy

Since there is no uniform classification model that can simultaneously and exhaustively solve any type of machine learning problem, it is also impossible to have an optimal classifier that can learn any distribution of learning data. This explains why it is difficult to emphasize the excellence of one ML paradigm to the detriment of another; therefore, emphasis should be placed on the use of a combination of classifiers. In effect, the aim of the combination of classifiers lies in the possibility of simultaneously constructing a set of diverse/complementary and efficient classifiers in order to achieve better decisionmaking through various aggregation strategies.

The IRESAMPLE + St multimodal learning method is in fact based on the early fusion concept [START_REF] Lamari | SMOTE-ENN-Based Data Sampling and Improved Dynamic Ensemble Selection for Imbalanced Medical Data Classification[END_REF], which concretely means that the different outputs/characteristics issued from the basic classifiers will be combined between them "meta-features", thus constituting entries relative to a second classification model "meta-classifier" to generate the final result. Thus, no decision regarding the diagnosis of diabetes can be made once the characteristics/outputs of various classifiers are merged.

In the present case, however, the role of the basic classifiers essentially consists of different modalities, where these different classifiers bring certain types of specific information as well as their own/unique point of view. On the one hand, this is done by taking advantage of the independence between the basic classifiers, and on the other hand, by profiting from the complementarity/diversity likely to be present within the context of these classifiers. This study adopts the parallel ensemble notion that consists of the parallel production of distinct basic classifiers "𝑚". This approach enabled us to create a set of basic classifiers following several experiments, and after a detailed review of the literature on other ML tasks. Figure 6 illustrates the proposed system structure for diabetes diagnosis. With a view to reaching a final decision regarding the proposed ensemble approach, two different classtype merging techniques (i. e. Majority Voting (MV) & Weighted Majority Voting (WMV)) are applied [START_REF] Lamari | SMOTE-ENN-Based Data Sampling and Improved Dynamic Ensemble Selection for Imbalanced Medical Data Classification[END_REF][START_REF] Benzebouchi | Multi-classifier system for authorship verification task using word embeddings[END_REF], as well as the "stacking aggregator" or "meta-classifier" strategy. As for the second fusion strategy utilized during this work, namely the meta-classifier, it involves recourse to classifiers that, in many cases, are heterogeneous; in other words, it makes use of distinct types of learners, thus leading to a heterogeneous ensemble. The general principle of this algorithm aims to learn how to optimally aggregate the predictions of several successful ML models by means of a stacking aggregator.

The architecture of a Stacking scheme involves two or more basic classifiers, commonly referred to as first-level models, and a meta-model that combines the predictions of those basic models, known as the second-level model. Basic models form on the basis of an initial training dataset, and then the metaclassifier (or meta-model) is trained around the outputs/predictions of the basic models in terms of characteristics "meta-features".

In fact, two arguments require to be defined in the context of the construction phase of this proposed ensemble system: the basic learners "𝑚" to be adjusted and the meta-learner responsible for setting up the aggregation phase. Those basic learners ("𝑚 = 5") correspond to the following algorithms, namely Multilayer Perceptron (MLP), K-Nearest Neighbors (K-NN), Support Vector Machine (SVM), Random Forest (RF), and Naive Bayes (NB). At the classification/diagnosis stage, the SVM meta-learner (taking into account the characteristics of the five basic learners) is adopted because of its performance in dealing with popular problems related to binary classification. This is considered in the domain of medical diagnosis as the best binary separator.

The IRESAMPLE+St ensemble approach relying on the Stacking algorithm integrates the k-fold crosstraining strategy with 𝑘 = 10 (which is similar to the k-fold cross-validation technique) in order to construct the meta-model, so that all instances are taken into account to form the meta-classifier as well as the resulting model. This process consists of randomly cutting (without replacement) the original data set into k equivalent samples or 𝑘 parts of approximately equal size. The first part is retained for testing and the model is trained on the 𝑘 -1 parts. This means that the basic learners are then trained on the 𝑘 -1 parts, and validated/tested by means of one of the remaining k parts so that this process is repeated k times, such that each 𝑘 sub-part is utilized exactly once as a validation set. In this way, the concept of the Stacking operation "meta-classification" can be expressed through the following algorithm by using the k-fold cross-training method.

Algorithm

Computational complexity

Algorithm computational complexity is an interesting study topic in the ML domain. In general, an ML model's complexity is assessed through the 𝐵𝑖𝑔 -𝑂 notation. This concept can be split into two categories: time complexity, relating to how long the model/algorithm takes to run (which is always specified in terms relative to a certain input size "𝑛"), and spatial complexity, concerning the amount of memory consumed by it. The proposed meta-classification ensemble approach consists of a basic classifier pool "𝑚", a number of training samples "𝑛", and a features set "𝑓 "generated as an input for the meta-classifier by the basic classifiers. It should be mentioned that the computational cost concept is not involved in the online (realtime) classification process, it only occurs in the training phase (offline) with a low and reasonable computation time estimated at about 15 𝑚𝑖𝑛.

This IRESAMPLE+St approach is defined as follows with respect to its computational complexity: 𝑡𝑟𝑎𝑖𝑛_𝑡𝑖𝑚𝑒_𝑐𝑜𝑚𝑝𝑙𝑒𝑥𝑖𝑡𝑦 = 𝑂(𝑆𝑡(𝑛 2 . 𝑚. 𝑓) and 𝑠𝑝𝑎𝑐𝑒_𝑐𝑜𝑚𝑝𝑙𝑒𝑥𝑖𝑡𝑦 = 𝑂(𝑚. 𝑓). where "𝑚" denotes the sum value with respect to the time complexity (in terms of the 𝐵𝑖𝑔 -𝑂 notation) of all classifiers used (MLP, K-NN, SVM, RF, and NB), in which MLP is 𝑂(𝑛 2 . 𝑓. 𝑙 𝑖 ) , K-NN is 𝑂(𝑛. 𝑓. 𝑘 number of neighbors ), SVM is 𝑂(𝑛 𝑠𝑣 2 . 𝑓), RF is 𝑂(𝑛. 𝑙𝑜𝑔(𝑛). 𝑓. 𝑡 number of trees ), and NB is 𝑂(𝑛. 𝑓). In this case, 𝑠𝑣 represents the number of support vectors, 𝑙 equals the number of hidden layers, and 𝑖 indicates the number of neurons in each layer.

Experimental results evaluation

With a view to evaluating the "meta-classifier" fusion method for diabetes diagnosis, the k-fold crossvalidation technique is applied with 𝑘 = 10. The implementation of the proposed task is performed via the Weka tool, an open-source library dedicated to ML algorithms, written in Java, and also giving access to well-known tools such as deeplearning4j, scikit-learn and R. The following subdivisions describe in detail the experiments carried out in the framework of this work as well as the achieved results.

Used data set

The suggested approach to diagnosing diabetes uses an ensemble of 768 instances and 8 characteristics from the Pima Indians Diabetes (PID) Database2 , of which 500 cases represent normal patients and 268 cases with diabetes. The attributes of the diabetes database are described in Table 1.

Evaluation criteria

In order to assess the performance of the proposed model, we use performance measures commonly used in the medical domain which are given as follows in Eqs. ( 6), ( 7), ( 8), ( 9), ( 10), (11), and (12): Where, True Positives (TP), True Negatives (TN), False Positives (FP), and False negatives (FN) are described in Table 2. The suggested method was also evaluated according to the area under receiver operating characteristic (AUROC) curve [START_REF] Bradley | The use of the area under the roc curve in the evaluation of machine learning algorithms[END_REF]).

𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 (𝑆𝐸𝑁) = 𝑇𝑃 𝑇𝑃 + 𝐹𝑁 (6) 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 (𝑆𝑃𝐸) = 𝑇𝑁 𝑇𝑁 + 𝐹𝑃 (

Results and Discussion

Several experiments concerning the constitution of the basic classifier pool and, more precisely, on the number of selected classifiers are carried out during this study [START_REF] Azizi | From static to dynamic ensemble of classifiers selection: Application to Arabic handwritten recognition[END_REF][START_REF] Zemmal | Adaptative S3VM semi supervised learning with features cooperation for breast cancer classification[END_REF], by adopting five distinct models which are the following: the Multilayer Perceptron (MLP), the K-Nearest Neighbors (K-NN), the Support Vector Machine (SVM), the Random Forest (RF), and the Naive Bayes (NB). Such classifiers are moreover formed from the original data set, i.e. without any prior processing, so as to be able to better evaluate and compare the results obtained through these experiments.

1

Experimental set-up

The set-up parameters as well as each algorithm's performance are described as follows:

As for the MLP classifier, ideal outcomes are attained through adjustment of the following settings: " ℎ𝑖𝑑𝑑𝑒𝑛_𝑙𝑎𝑦𝑒𝑟𝑠: ′𝑎′ = ((𝑎𝑡𝑡𝑟𝑖𝑏𝑠 + 𝑐𝑙𝑎𝑠𝑠𝑒𝑠)/2) = 5 ", " 𝑙𝑒𝑎𝑟𝑛𝑖𝑛𝑔_𝑟𝑎𝑡𝑒 = 0.3 ", " 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = 0.2", "𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑝𝑜𝑐ℎ𝑠 = 500", and "𝑣𝑎𝑙𝑖𝑑𝑎𝑡𝑖𝑜𝑛_𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 = 20". The relevant performances achieved are 75.39% ACC, 60.80% SEN, 83.20% SPE, 63.29% F-1 score, 44.92% MCC and 79.30% AUROC.

Concerning the K-NN classifier, the settings on the basis of 𝐾 -number of neighbors to use, 𝑛𝑒𝑎𝑟𝑒𝑠𝑡_𝑁𝑒𝑖𝑔ℎ𝑏𝑜𝑢𝑟_𝑆𝑒𝑎𝑟𝑐ℎ_𝐴𝑙𝑔𝑜𝑟𝑖𝑡ℎ𝑚, 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒_𝑊𝑒𝑖𝑔ℎ𝑡𝑖𝑛𝑔, and 𝑤𝑖𝑛𝑑𝑜𝑤_𝑆𝑖𝑧𝑒 are set so as to obtain optimized results by using " 𝐾 = 2", "𝐴𝑙𝑔𝑜𝑟𝑖𝑡ℎ𝑚: 𝐸𝑢𝑐𝑙𝑖𝑑𝑒𝑎𝑛_𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 -𝑅 𝑓𝑖𝑟𝑠𝑡 -𝑙𝑎𝑠𝑡", " 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒_𝑊𝑒𝑖𝑔ℎ𝑡𝑖𝑛𝑔: 𝑁𝑜 ", and " 𝑤𝑖𝑛𝑑𝑜𝑤_𝑆𝑖𝑧𝑒 = 0 ". The experiments were conducted with a Windows-7 operating system under the following hardware setup: Intel® Core™ i7-8750H CPU @ 2.20 GHz up to 4.10 GHz processor with RAM 64 GB and DDR4-2666, LPDDR3-2133 Memory.

Balancing of the data set using oversampling and subsampling strategy

The present study examines the way in which the pool of basic classifiers can be used as part of the preprocessed/balanced data of the PID database. In other words, this suggested method is based on results obtained through data balancing techniques (SMOTE and the modified RESAMPLE-based method called IRESAMPLE+). As an illustration of this recourse to resampling techniques concerning the proposed approach, the different phases to follow to balance the data using the SMOTE filter and the suggested IRESAMPLE + method are thus presented below. It should be noted that Figure 7 shows that there is a large non-uniformity in data distribution across the classes of the data set (diabetes_PID) employed in this work. The disparity in final classes such as tested_negative non-diabetic and tested_positive diabetic can result in incorrect diagnostic results. Thus, the classifier may be more biased towards the high concentration classes (non-diabetic) than the low concentration class (diabetic), which may also lead to the inaccurate classification of patients to detect the resulting category.

Taking into consideration the nature of the unbalance of the output class, namely tested_positive (diabetic class), an oversampling strategy, SMOTE, is applied to the diabetes data set. The supervised filtering strategy is applied using the settings of function (1) as follows:

𝑆𝑀𝑂𝑇𝐸 ← 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 (1, 86.6, 5, 2)

The make use of the SMOTE method (in this configuration) permitted to increase the size of the minority class (diabetic) from 268 to 500 in order to dispose of a balanced database and to obtain significant results. Figure 8 presents a comparative diagram across the raw data set vs. the application of the SMOTE technique. On the other hand, the subsampling strategy, RESAMPLE, is also practiced on the majority class (nondiabetic) to reduce its size from 500 to 268, so that both classes possess an identical number of instances. This supervised filtering strategy (RESAMPLE) can be configured in the following way:

𝑅𝑒𝑠𝑎𝑚𝑝𝑙𝑒 ← 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 (1, 69.8, 1, 𝑛𝑜 -𝑟𝑒𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡, 𝐹) (21)
Such a setup has in effect engendered a great loss of data (instances), leading ultimately to unsatisfactory results. The present study aims at modifying/improving the general functioning and principle of this technique so that it can operate at the same time as an oversampling and undersampling strategy; this is achieved through an appropriate adjustment at the level of equation ( 2) parameters, while also introducing an iterative structure enabling a significant balancing of the data. The following pseudocode highlights this aspect:

Pseudo-code: IRESAMPLE+ Procedure (oversampling and subsampling strategy) Input: Original unbalanced data set (diabetes) Output: more efficient balanced data set Process:

1. Begin 2. Initialisation 2. Repeat 3. 𝑅𝑒𝑠𝑎𝑚𝑝𝑙𝑒 ← 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 (1, 100, 0, 𝑛𝑜 -𝑟𝑒𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡, 𝐹) (22) 4. Until (number of class 1non-diabetic instances = number of class 2diabetic instances) 5. End Meanwhile, the use of the enhanced RESAMPLE strategy succeeded, on the one hand, in increasing the size of the minority class (diabetics) from 268 to 384 and, on the other hand, in decreasing the size of the majority class (non-diabetics) from 500 to 384. The purpose of this strategy is to provide a balanced data set with exactly the same number of instances in the two classes while preserving the maximum of instances, with the objective of minimizing bias and thus reinforcing the classification/diagnosis accuracy established by the classifier. Figure 9 graphically illustrates the application of the RESAMPLE technique and the modified RESAMPLE-based method known as IRESAMPLE+. Indeed, the use of the SMOTE (oversampling) method permitted to augment the minority class size by randomly adding artificial instances, which can lead to the overlearning problem. The idea of the RESAMPLE (subsampling) technique consists of the random removal of the majority class instances, which can eventually lead to the suppression of the relevant information. In order to overcome the different problems mentioned above, the present study suggests a new strategy called IRESAMPLE+ as a compromise between the two previously discussed techniques (SMOTE and RESAMPLE), that can operate at the same time as an oversampling and undersampling strategy enabling a meaningful analysis and relevant balancing of the PID data set. The effect of these different resampling strategies regarding the distribution of the classes is illustrated in Figure 10.

Models performance analysis

This article aims to separately analyze the specific performance of each classifier performed, whether before or after the application of data resampling techniques (SMOTE and IRESAMPLE+) and this, according to the performance factors mentioned above: ACC, SEN, SPE, PPV, NPV, F-score, MCC, and AUROC. The following tables (Table 3 andTable 4) summarize the obtained outcomes from five models (MLP, K-NN, SVM, RF, and NB). On the basis of findings described in Tables 3 and4, it is clear that, after the application of resampling techniques, the classifiers' performance is still superior to that observed in the original data. In addition, the proposed IRESAMPLE+ filter provides better results compared to the SMOTE filter, and therefore the performance of the classifiers is considerably optimized. Figure 11 more visibly illustrates this comparative study using only AUROC values. In order to enhance the sensitivity and specificity of the proposed multimodal learning as well as the global performance in terms of diagnosis, on the one hand; moreover, on the other hand, to reach a final decision about this study, three aggregation approaches are applied: the Majority voting (MV), the Weighted Majority Voting (WMV), and the SVM meta-classifier (or stacking aggregator). Thus, the obtained results by using the SMOTE, RESAMPLE and IRESAMPLE+ techniques are indicated in Table 5. Based on the performance described above (Table 4 andTable 5), we observe that the concept of Ensemble Learning (EL) using Resampling techniques (SMOTE, RESAMPLE, and IRESAMPLE+) is clearly preferable over separate use of classifiers. In addition, it should be noted that the WMV method generated a more accurate classification with the use of the SMOTE & RESAMPLE filters, while the SVM meta-classifier aggregation paradigm using the IRESAMPLE+ strategy is much more efficient and robust than the other solutions investigated in this work. This approach has also shown greater reliability compared to current findings using the same data set (PID). Figure 12 illustrates the ROC curves of the three aggregation paradigms employed (MV, WMV, and Stacking_SVM) using the proposed IRESAMPLE+ filter. By noting that all curve points for the three techniques used are positioned in the upper half of the ROC space, resulting in a good ROC curve, especially for the SVM meta-classifier model.

The following diagrams demonstrate that the meta-classification approach considerably reduces the error rate compared to separate learning of classifiers and gives better performance using the ACC (Figure 13), SEN (Figure 14), and SPE (Figure 15) evaluation criteria. An experimental study concerning the SVM meta-classifier performance using different kernel methods such as the polynomial kernel, the Radial Basis Function (RBF) kernel and the PUK kernel with multiple K numbers (5 and 10) of the cross-validation approach accompanied by Table 6 illustrating the obtained findings. 13), ( 14), ( 15), ( 16) and ( 17):

𝐾𝑎𝑝𝑝𝑎 = 𝑃 𝑜 -𝑃 𝑒 1-𝑃 𝑒 (13)
where, 𝑃 𝑜 indicates the relative observed accordance between the classification raters, and 𝑃 𝑒 is the hypothetical probability of chance accordance.

𝑀𝐴𝐸 = ∑ |𝑃 𝑖 -𝑂 𝑖 | 𝑛 𝑖=1 𝑛 (14) 𝑅𝑀𝑆𝐸 = √ ∑ (𝑃 𝑖 -𝑂 𝑖 ) 2 𝑛 𝑖=1 𝑛 (15) 𝑅𝐴𝐸 = ∑ |𝑃 𝑖 -𝑂 𝑖 | 𝑛 𝑖=1 ∑ |𝑂 ̅ -𝑂 𝑖 | 𝑛 𝑖=1 (16) 𝑅𝑅𝑆𝐸 = ∑ (𝑃 𝑖 -𝑂 𝑖 ) 2 𝑛 𝑖=1 ∑ (𝑂 ̅ -𝑂 𝑖 ) 𝑛 𝑖=1 2 (17)
where, 𝑃 𝑖 represent the predicted target, 𝑂 𝑖 represent the observed target, and 𝑛 is the number of observations. Table 8 shows the confusion matrix for this study, which measures the quality of the proposed classification/diagnostic system. This table indicates how many correct predictions for each class and the number of incorrect predictions for each class organized according to the predicted class. Each row corresponds to an actual class while each column corresponds to a predicted class. As shown in Tables 7 and8, 384 instances of diabetes were correctly classified as diabetes cases by the proposed method, 383 healthy instances are properly classified as non-diabetes cases. In summary, 767 instances are accurately labeled and only one is not, resulting in 99.74% Cohen's kappa coefficient (𝑘) with MAE = 0.13%, RMSE = 3.61%, RAE = 0.2604%, and RRSE = 7.2168%, which further proves the reliability of the suggested methodology.

The analysis of the experimental results led to the conclusion that the combination of classifiers is preferable to the separate learning of classifiers (considerably reduces the error rate), and that the combination of multimodal features using the meta-classification method has demonstrated its effectiveness in diagnosing diabetes. The objective of this study is access to the conception of a computer-aided diagnosis (CAD) system for diabetes disease which surpasses other systems of literature; this thanks to the aggregate of the characteristics/outputs of several classifiers in metaclassification architecture as well as to the IRESAMPLE+ technique. It should be noted that the majority of studies do not consider the notion of unbalanced data (especially at the data level) in the diabetes classification, which represents a major problem with regard to the field of ML, leading to erroneous classification results. However, there is relatively limited research [START_REF] Nnamoko | Efficient Treatment of Outliers and Class Imbalance for Diabetes Prediction[END_REF][START_REF] Alghamdi | Predicting diabetes mellitus using SMOTE and ensemble machine learning approach: The Henry Ford ExercIse Testing (FIT) project[END_REF] that has taken into account such an unbalanced data concept in adopting the SMOTE technique. Likewise, most of the approaches mentioned in the literature as approaches to diabetes diagnosis are often based or on a standard approach relying on a single classifier's point of view or on a vote-based ensemble approach (class-level aggregation) that provides less information (available) at the time of fusion.

The proposed IRESAMPLE+St approach by the present study addresses with the unbalanced data concept at both grades, i.e. at the data level as well as at the algorithm level. The former concept is processed through the suggested enhanced RESAMPLE method called IRESAMPLE+ (so as to equilibrate distributions and/or eliminate difficult samples); the second is performed by adopting a stacking aggregator (which includes the k-fold CV technique)-based multimodal ensemble approach. By comparing the performances summarized in Table 8 and with the studies reported previously in the related work section, the proposed IRESAMPLE+St approach clearly surpasses all the state-of-the-art approaches by obtaining the most optimal results in terms of accuracy, sensitivity, specificity, AUROC, and by Cohen's kappa. This confirms as well that the suggested IRESAMPLE+St meta-classification method offers a better and more precise early diagnosis of diabetes disease with 99.87% ACC, 100% SEN, 99.70% SPE, 99.90% AUROC, and 99.74% Cohen's kappa.

With the aim of the proposed model generalization along with further experimental evaluation, different benchmark datasets having unbalanced data are utilized/tested. Pima Indians Diabetes, Parkinson's and Cardiac Catheterization Diagnostic are taken from UCI Machine Learning Repository. Prostate Cancer and VA Lung Cancer datasets from Vanderbilt Biostatistics Wiki [START_REF] Frank | Vanderbilt Biostatistics Wiki[END_REF].

Table 10 presents the results achieved in terms of accuracy (ACC) and data balancing by the proposed approach and this, before and after the application of the IRESAMPLE+ suggested resampling technique on each medical dataset used.

Based on Table 10, the experiments conducted on the various medical datasets validates the meaningful performance of the IRESAMPLE+St proposed approach.

Conclusion

Diabetes is considered one of the gravest diseases in the world threatening human health, also called the silent killer. Screening allows early diagnosis of certain diseases, before the appearance of symptoms/complications, as well as better management and a decrease of the social cost.

The main objective of this paper is to design a robust medical decision support system, more precisely diabetes disease by analyzing several advanced paradigms of the combination of classifiers taking into account the context of the often unbalanced medical bases. This approach uses the meta-classification paradigm that has shown better performance with the addition of the enhanced resampling module known as IRESAMPLE+, which makes our system more robust and reliable. The results demonstrated that ensemble learning with the preprocessed data provides a very low error rate. However, the limitation of the suggested approach is that only the diabetes database is treated/tested and that will be overcome in subsequent work by applying other unbalanced data sets through analysis of other advanced resampling techniques. In addition, there are also plans to build a new data set that includes modern healthcare to predict diabetes comprises a urine test, and the glycated hemoglobin A1c (HbA1c) test.

In conclusion, the suggested IRESAMPLE+St approach delivers a more robust Clinical Decision Aid System (CDAS) that permits diabetologists to rapidly diagnose patients with diabetes at an early stage while also providing a second opinion to the doctors with high accuracy in order to support their therapeutic decisions.

Figure 1 .

 1 Figure 1. An example illustrating the concept of unbalanced data

  illustrates this process in the case of a linearly separable two-class problem. The blue dots describe first class samples and the red triangles represent the second class.

Figure 2 .

 2 Figure 2. The general process of ensemble learning methods

Figure 3 .

 3 Figure 3. Proposed approach architecture for diabetes diagnosis

Figure 4 .

 4 Figure 4. Illustration of the SMOTE principle

Figure 5 .

 5 Figure 5. Illustration of the RESAMPLE principle

Figure 6 .

 6 Figure 6. Proposed network architecture to predict diabetesAn appropriate aggregation strategy should be carried out after the basic learners/classifiers have been trained so as to bring together their outputs into a unique form for subsequent use in the final classification. Various strategies have been suggested allowing the combination of complementary basic classifiers of the pool, such as the linear combiner, the product combiner, and the vote combiner.

Figure 7 .

 7 Figure 7. Class imbalance of the diabetes data set

Figure 8 .

 8 Figure 8. Applying the SMOTE filter to class distribution

Figure 9 .

 9 Figure 9. Practice both RESAMPLE and IRESAMPLE+ methods for class distribution

Figure 10 .

 10 Figure 10. Comparison of different resampling techniques on the distribution of PID classes-diabetics (black circles) and non-diabetics (dark red circles)-

Figure 11 .

 11 Figure 11. Impact of applying SMOTE and IRESAMPLE+ filters on the performance of classifiers

Figure 12 .

 12 Figure 12. ROC curve of three aggregation paradigms used

Figure 13 .

 13 Figure 13. Comparison of Accuracy rate between basic classifiers and the meta-classification method

  On the basis of experimental results conducted as part of this work, the ensemble voting method outperforms all other separate classifiers with 86.00% Acc utilizing the 10-fold CV technique.[START_REF] Maniruzzaman | Classification and prediction of diabetes disease using machine learning paradigm[END_REF] presented a pipeline model known as "CAMIL" based on Multiple Instance Learning (MIL) utilizing Clustering (UCLUST, SUMACLUST, and SWARM) and Assembly (SOAPdenovo2) techniques including a Canopy cluster-based pretreatment phase for phenotype prediction of patients with type-2 diabetes through meta-genomic data (MGWAS). The authors applied a vocabulary-based characteristic extraction approach like the Bag of Words (D-BoW and H-BoW) technique as well as the SVM-Light classifier concerning classification. The CAMIL H-BoW SWARM combination showed better performance with the following results: 84.06% Acc, 85.11% F1-score, and 85.64% AUROC. In a study analogous to[START_REF] Mahabub | A robust voting approach for diabetes prediction using traditional machine learning techniques[END_REF],[START_REF] Hasan | Diabetes Prediction Using Ensembling of Different Machine Learning Classifiers[END_REF] examined several voting-based ensemble methods as well as different combinations of various basic classifiers (such as k-NN, DT, RF, AB, NB, and XGBoost (XB)) with the objective of selecting the most robust pool using soft weighted voting according to AUROC values in order to boost diabetes prediction. Also through this study, the authors

	f1-measure
	from 85,7%, 60,3%, 71,3%, 57,5% to 86,2%, 60,8%, 77,3%, 58,3%, respectively.
	A hybrid model that combines split-vote method and instance duplication was proposed in Kumar et al.
	(2022) to perform the prediction of diabetics in the PIMA imbalanced dataset. The concepts of over-
	sampling and under-sampling in conjunction with model weighting were used to increase the
	classification efficiency of the embedded-based ML algorithms. They obtained an accuracy of 89.32%
	with KNN model, 91.44% with NB model and 95.78% with SVM model.

compare MLP algorithm performance (as a unique classifier) with other suggested schemes. A preliminary step comprising outlier rejection, padding the missing or null values, data standardization (z-score normalization), and feature selection (PCA, ICA, and Correlation-based approach) is adopted within this suggested framework. Experimental results demonstrated that the pool made up of AB & XB basic classifiers achieved the highest performance relative to other possible combinations and MLP algorithm, resulting in an AUROC of 95.00%, a Sen of 78.90%, and a Spe of 93.40% by using the fivefold CV protocol and characteristic selection via the correlation technique. Olisa et al. (

2022

) suggested a novel diabetes prediction approach based on a twice-growth deep neural network (2GDNN) classifier applied to PIMA Indian and LMCH diabetes data sets. In addition, different supervised ML algorithms are evaluated, including RF and SVM models employing repeated stratified k-fold cross-validation. A preliminary step involving feature selection and missing value imputation was performed through Spearman correlation and polynomial regression techniques, respectively. Experimental results proved that the 2GDNN model reached highest performance with a sensitivity of 97.24%, an F1-score of 97.26% using the PIMA data set and a sensitivity of 97.33% and an F1-score of 97.27% when using the LMCH diabetes data set.

[START_REF] Sadeghi | Diabetes mellitus risk prediction in the presence of class imbalance using flexible machine learning methods[END_REF]

compared the performance of DNN, XGBoost and RF algorithms for the diagnosis of diabetes in TGLS cohot data. They investigate the impact of changing threshold, cost-sensitive, over and under-sampling strategies to solve the class imbalance problem. Obtained results in terms of AUROC, F1-score and G-means shows that the DNN outperforms the other algorithms. They found that the RENN under-sampling in DNN boosted ROC and Precision-Recall AUCs, g-mean and

: Stacking (meta-classification) with k-fold cross-training Input

  𝐷 𝐶 = {𝑥′ 𝑖 , 𝑦 𝑖 }, where 𝑥′ 𝑖 = {𝐶 𝑘1 (𝑥 𝑖 ), 𝐶 𝑘2 (𝑥 𝑖 ), 𝐶 𝑘3 (𝑥 𝑖 ), … , 𝐶 𝑘𝑇 (𝑥 𝑖 )}

		: forming dataset Ɗ = {𝑥 𝑖 , 𝑦 𝑖 } 𝑖 ← 1 𝑡𝑜 𝑛
		First-level learning algorithms 𝐶1, . . . , 𝐶𝑇
		Second-level meta-learning algorithm 𝐶′
	Output: an ensemble of classifiers 𝐶
	Process:
	1. Stage1: use cross-validation technique in preparing a formation set for meta-classifier
	2. Randomly split Ɗ into K equal-size subparts:Ɗ = {Ɗ 1 , Ɗ 2 , Ɗ 3 , … , Ɗ 𝑘 }
	3. for 𝑘 ← 1 𝑡𝑜 𝐾 do
	4.	Stage 1.1: learn basic-level classifiers
	5.	for 𝑡 ← 1 𝑡𝑜 𝑇 do
	6.	learn a classifier 𝐶 𝑘𝑡 from Ɗ\Ɗ 𝑘
	7.	end for
	8.	Stage 1.2: build a forming set for meta-classifier
	9.	for 𝑥 𝑖 ∈ Ɗ 𝑘 do
	10.
	11.	end for
	12. end for
	13. Stage2: learn a meta-classifier
	14. Learn a new classifier 𝐶′ based on 𝐷 𝐶
	15. Return 𝐶(𝑥) = 𝐶 ′ (𝐶 1 (𝑥), 𝐶 2 (𝑥), 𝐶 3 (𝑥), … , 𝐶 𝑇 (𝑥))

Table 1 .

 1 Brief description of the attributes of the diabetes data.

	Attribute name	Description	Min	Value Max	Mean
	Pregnancies	Number of times pregnant	0	17	3.845
		Plasma glucose concentration a 2
	Glucose	hours in an oral glucose tolerance	0	199	120.895
		test
	BloodPressure	Diastolic blood pressure (𝑚𝑚 𝐻𝑔)	0	122	69.105
	SkinThickness	Triceps skin fold thickness (𝑚𝑚)	0	99	20.536
	Insulin	2-Hour serum insulin (𝑚𝑢 𝑈/𝑚𝑙)	0	846	79.799
	BMI	Body mass index (𝑤𝑒𝑖𝑔ℎ𝑡 𝑖𝑛 𝑘𝑔/ (ℎ𝑒𝑖𝑔ℎ𝑡 𝑖𝑛 𝑚)^2)	0	67.1	31.993
	DiabetesPedigreeFunction	Diabetes pedigree function	0.08	2.42	0.472
	Age	Age (years)	21	81	33.241
	Outcome	Class variable (negative or positive)	0	1	-

Table 2 .

 2 Binary classification confusion matrix (diabetes)

		Positive test (P)	Negative test (N)
	Patients with diabetes (P)	TP	FN
	Non-diabetic patients (N)	FP	TN

  The corresponding values obtained are 72.66% ACC, 55.20% SEN, 82.00% SPE, 58.49% F-1 score, 38.37% MCC and 74.20% AUROC. The optimal results are reached with respect to SVM classifier by setting up the following main variables: 𝑘𝑒𝑟𝑛𝑒𝑙, 𝑔𝑎𝑚𝑚𝑎, and 𝐶, where "𝑘𝑒𝑟𝑛𝑒𝑙: 𝑃𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙" with a 𝑔𝑎𝑚𝑚𝑎 value of 0.5 and parameter 𝐶 equal to 3. The correspondent results reached are 77.47% ACC, 55.60% SEN, 89.20% SPE, 63.

	27% F-
	1 score, 48.42% MCC and 72.40% AUROC.
	The tuning settings with their optimized outcomes obtained using the RF classifier are determined as
	follows: "𝑚𝑎𝑥𝐷𝑒𝑝𝑡ℎ: 𝑢𝑛𝑙𝑖𝑚𝑖𝑡𝑒𝑑 ", "𝑛𝑢𝑚𝐹𝑒𝑎𝑡𝑢𝑟𝑒𝑠 = (< 0 = 𝑖𝑛𝑡 (𝑙𝑜𝑔_2 (#𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑜𝑟𝑠) + 1))",
	and "𝑛𝑢𝑚𝑇𝑟𝑒𝑒𝑠 = 100". The correspondent results achieved include 74.87% ACC, 59.00% SEN,
	83.40% SPE, 62.10% F-1 score, 43.51% MCC and 81.50% AUROC.

The NB classifier performs best by setting both 𝑑𝑖𝑠𝑝𝑙𝑎𝑦𝑀𝑜𝑑𝑒𝑙𝐼𝑛𝑂𝑙𝑑𝐹𝑜𝑟𝑚𝑎𝑡 and 𝑢𝑠𝑒𝐾𝑒𝑟𝑛𝑒𝑙𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑜𝑟 parameters to 𝑓𝑎𝑙𝑠𝑒 and 𝑢𝑠𝑒𝑆𝑢𝑝𝑒𝑟𝑣𝑖𝑠𝑒𝑑𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑖𝑧𝑎𝑡𝑖𝑜𝑛 parameter to 𝑡𝑟𝑢𝑒 . The correspondent performances represent with an ACC of 76.30%, a SEN of 61.20%, a SPE of 84.40%, an F-1 score of 64.30%, a MCC of 46.78% and an AUROC of 81.90%.

Table 3 .

 3 Obtained results using the basic classifiers separately before resampling the diabetes data set

	Classification				Performance measures (%)
	method	ACC	SEN	SPE	PPV	NPV	F-score	MCC	AUROC
	MLP	75.39	60.80	83.20	66.00	79.80	63.29	44.92	79.30
	K-NN	72.66	55.20	82.00	62.20	77.40	58.49	38.37	74.20
	SVM	77.47	55.60	89.20	73.40	78.90	63.27	48.42	72.40
	RF	74.87	59.00	83.40	65.60	79.10	62.10	43.51	81.50
	NB	76.30	61.20	84.40	67.80	80.20	64.30	46.78	81.90

Table 4 .

 4 Obtained results using the basic classifiers separately after resampling the diabetes data set

	Filtre	Classification method	ACC	SEN	SPE	Performance measures (%) PPV NPV F-score	MCC	AUROC
		MLP	77.40	79.80	75.00	76.10	78.80	77.90	54.86	83.30
		K-NN	79.20	84.60	73.80	76.40	82.70	80.30	58.74	83.00
	SMOTE	SVM	79.60	83.40	75.80	77.50	82.00	80.30	59.37	79.60
		RF	80.80	83.80	77.80	79.10	82.80	81.40	61.71	87.80
		NB	76.50	80.80	72.20	74.40	79.00	77.50	53.20	84.60
		MLP	98.31	99.00	97.70	97.70	98.90	98.30	96.62	97.30
		K-NN	98.83	97.90	98.70	98.70	98.00	97.80	97.67	98.90
	IRESAMPLE+	SVM	99.37	99.00	99.70	99.70	99.00	99.60	99.22	99.60
		RF	98.96	98.40	98.50	98.50	98.50	98.00	98.22	98.90
		NB	95.44	97.70	93.20	93.50	97.50	95.50	90.98	98.30

Table 5 .

 5 Obtained results from three aggregation paradigms used for diabetes data set

	Filtre	Aggregation method	ACC	SEN	SPE	Performance measures (%) PPV NPV	F-score	MCC	AUROC
		MV	81.50	85.40	77.60	79.20	84.20	82.20	63.19	81.50
	SMOTE	WMV SVM meta-classifier	82.41 80.50	84.70 84.60	79.80 78.40	80.40 79.30	84.10 81.80	82.10 80.90	65.43 61.00	82.30 80.50
		MV	76.68	75.00	78.40	77.60	75.80	76.70	53.39	76.70
	RESAMPLE	WMV	78.45	77.60	80.30	79.40	77.20	78.50	55.92	78.00
		SVM meta-classifier	72.02	69.00	75.00	73.40	70.80	72.00	44.11	72.00
		MV	98.83	99.70	97.90	98.00	99.70	98.80	97.67	98.80
	IRESAMPLE+	WMV SVM meta-classifier	99.72 99.87	100 100	99.50 99.70	99.50 99.70	100 100	99.70 99.90	99.46 99.74	99.90 99.90

Table 6 .

 6 Obtained results with various kernel optimizations for two K-fold protocols 90%, which leads to the improvement of the diagnostic performance. The key parameters to tune for the SVM meta-classifier are specified as follows: RBF kernel with a 𝑔𝑎𝑚𝑚𝑎 value of 0.5 and the 𝐶 parameter equal to 3. Table7presents other important measures of performance evaluation of the suggested method provided by the Weka tool, such as the Kappa statistic (classification reliability), the Mean Absolute Error (MAE), the Root Mean Square Error (RMSE), the Relative Absolute Error (RAE), and the Root Relative Square Error (RRSE), which are defined as follows in Eqs. (

	k	Used kernel	ACC	SEN	SPE	Performance measures (%) PPV NPV F-score	MCC	AUROC
		Polynomial	99.73	100	99.50	99.50	100	99.70	99.46	99.70
	5	RBF	99.87	100	99.70	99.70	100	99.90	99.74	99.90
		PUK	99.73	99.50	100		100	99.50	99.70	99.46	99.70
		Polynomial	99.61	99.50	99.70	99.70	99.50	99.60	98.42	99.60
	10	RBF	99.87	100	99.70	99.70	100	99.90	99.74	99.90
		PUK	99.61	99.50	99.70	99.70	99.50	99.60	98.42	99.60
	Comparing the obtained results (Table 6), we find that the SVM meta-classifier using the Radial Basis
	Function (RBF) kernel gave the highest performance and this, regardless of the K-fold value (5-
	times/10-times) of the cross-validation approach. Resulting in 99.87% Accuracy (ACC) with Sensitivity
	(SEN) 100%, Specificity (SPE) 99.70%, Positive Predictive Value (PPV) 100%, Negative Predictive
	Value (NPV) 99.50%, F-score 99.90%, Matthews Correlation Coefficient (MCC) 99.74%, and AUROC
	99.

Table 7 .

 7 illustration of the Kappa statistic, MAE, RMSE, RAE and RRSE rate of the proposed method

Table 8 .

 8 Confusion Matrix of the suggested approach

		Positive test (P)	Negative test (N)
	Patients with diabetes (P)	384	0
	Non-diabetic patients (N)	1	383

Table 9

 9 exemplifies the key approaches proposed in the literature and introduces a comparative study of the suggested method performance against the leading existing approaches of diagnosing diabetes in the literature. This table is organized in terms of accuracy (ACC) measurement.As observed in Table9with respect to performances of related studies available in the literature, it indicates as though all systems operate differently in terms of data pre-processing technique and classification method applied towards diagnosing diabetes using the same data set, i.e. PID. For instance,[START_REF] Nilashi | Accuracy Improvement for Diabetes Disease Classification: A Case on a Public Medical Dataset[END_REF] used the SMO+PCA technique as a preprocessing method and the NN paradigm for the classification stage.[START_REF] Bozkurt | Comparison of different methods for determining diabetes[END_REF] investigated different neural networks (NN)-based classifiers, i.e. distributed time delay networks (DTDN), anticipation networks, learning vector quantization, cascade networks, probabilistic neural networks, and time delay networks using the original data set. Also,[START_REF] Singh | A Stacked Generalization Approach for Diagnosis and Prediction of Type 2 Diabetes Mellitus[END_REF] made use of the original data set, i.e. without any preprocessing by adopting the stacked generalization approach on the basis of different kernels of the SVM classifier such as Linear-SVM, Polynomial-SVM, RBF-SVM and Sigmoid by using the meta-learner SMO.[START_REF] Iyer | Diagnosis of diabetes using classification mining techniques[END_REF] employed data pretreatment methods such as transformation and feature selection (FS) in order to apply the NB classifier to the resulting set.[START_REF] Choubey | GA_MLP NN: A Hybrid Intelligent System for Diabetes Disease Diagnosis[END_REF] opted for the feature selection (FS) method as the data set pre-processing utilizing the genetic algorithm (GA) with the MLP classifier to perform the diagnostic phase.[START_REF] Yilmaz | A New Data Preparation Method Based on Clustering Algorithms for Diagnosis Systems of Heart and Diabetes Diseases[END_REF] proposed a modified Kmeans algorithm for the pre-processing phase and to be used with the SVM classifier.[START_REF] Ramezani | A novel hybrid intelligent system with missing value imputation for diabetes diagnosis[END_REF] reported a logistic adaptive network-based fuzzy inference system (LANFIS) utilizing imputation and OT linear dimension reduction algorithm with regard to the data pretreatment.[START_REF] Varma | A computational intelligence approach for a better diagnosis of diabetic patients[END_REF] removed missing values from the data set employing a modified gini index-gaussian fuzzy decision tree as a classification approach.

			Feature selection using LR algorithm	Random Forest (RF)	94.25	99.57		95.00
	(Nai-Arun & Sittidech, 2014)	SRH	FS using the Gain Ratio Algorithm	Bagging	95.31	-	-	-
	(Yilmaz et al., 2014)	PID	Modified K-means Algorithm	Modified K-Means + SVM	96.71	97.31	95.06	-
				Ensemble model
	(Sarwar et al., 2020)	database created	-	(K-NN, SVM, using majority ANN, and NB)	98.60	-	-	-
				voting
	Olisa et al. (2022)	PID and LMCH diabetes	FS and EMV using Spearman correlation regression techniques and polynomial	a twice-growth deep (2GDNN) neural network	97.25	97.24	-	-

Table 10 .

 10 Comparative results of the proposed approach using other well-known medical data sets (#I: instances, #P: positive instances, #N: negative instances, #ACC: accuracy)

	Approach		Before Pre-processing		After Pre-processing
	Datasets	#I	#P	#N	#ACC(%)	#P	#N	#ACC(%)
	Diabetes	768	268	500	75.65	384	384	99.87
	Diabetic Retinopathy Debrecen	1151	611	540	79.30	575	575	95.48
	Breast Cancer	286	85	201	67.48	143	143	82.87
	Unbalanced dataset	856	12	844	98.59	428	428	100
	Parkinson's	195	147	48	91.70	97	97	95.45
	Cardiac Catheterization Diagnostic	3504	2372	1132	92.40	1752	1752	97.38
	Prostate Cancer Dataset	506	213	293	94.78	253	253	98.70
	VA Lung Cancer	138	40	124	92.32	82	82	96.90

https://www.who.int/diabetes/global-report/en/

https://www.kaggle.com/

Authors


Thibault Oujia
email: thibault.oujia@univ-amu.fr

Keigo Matsuda
email: k.matsuda@jamstec.go.jp

Kai Schneider
email: kai.schneider@univ-amu.fr

EXTREME DIVERGENCE AND ROTATION VALUES OF THE INERTIAL PARTICLE VELOCITY IN HIGH REYNOLDS NUMBER TURBULENCE USING DELAUNAY TESSELLATION

The dynamics of inertial particles in homogeneous isotropic turbulence at high Reynolds number, obtained by three-dimensional direct numerical simulation (DNS), is analyzed considering different Stokes numbers. Divergence and rotation of the particle velocity are determined using Delaunay tessellation of the particle positions at subsequent time instants. For large Stokes numbers heavy tails in the probability density functions (PDFs) are found. Large divergence and rotation values, as large as 50 and 40 times of the corresponding standard deviation, respectively, can be observed. The helicity of the particle velocity, which quantifies the swirling motion of the particle flow, is likewise computed. The PDFs change their shape with the Stokes number and show that for large numbers higher probability for vanishing helicity is found. This confirms that for heavy particles helical motion is apparently suppressed.

INTRODUCTION

Driven by numerous applications of particle laden turbulence, e.g. dust in stars or the rain formation in atmospheric clouds, there are many numerical and theoretical studies in the literature, see e.g. [START_REF] Brownlee | Cosmic dust: Collection and research[END_REF][START_REF] Shaw | Particle-turbulence interactions in atmospheric clouds[END_REF]. Understanding the motion of inertial particles is therefore essential to get insight into these phenomena and to develop sound models. The aim here is to determine the dynamical properties of inertial particles in isotropic turbulence using finite time tessellation-based measures. The particle velocity is crucial to understand particle dynamics, i.e. convergence, divergence of particles and the vortical or swirling motion of the particle clouds. The challenging task is that we only know the velocity of the particles at discrete points in space, i.e. at the particle position.

Voronoi tessellation of the particle position has previously been used, e.g. in [START_REF] Monchaux | Preferential concentration of heavy particles: a voronoi analysis[END_REF], for analyzing preferential concentration of inertial particles. In our recent work, [START_REF] Oujia | Divergence and convergence of inertial particles in highreynolds-number turbulence[END_REF], we proposed a Lagrangian approach to determine the divergence of particle velocity using the volume change rate of the Voronoi cells. Here we propose a further new Lagrangian approach using Delaunay tessellation, to determine the spatial velocity derivatives numerically by considering subsequent time instants. In particular the trace of the velocity gradient tensor, corresponding to the divergence, which yields information about sources and sinks and its antisymmetric part, corresponding to the curl, which characterizes the rotation can thus be computed. The helicity, i.e. the scalar product of vorticity and velocity, can be likewise calculated and swirling motion of particle clouds can consequently be quantified. The dynamical properties of the particle velocity can be assessed and detailed statistics of the divergence and of the rotation can be performed.

Moreover, we will address the presence of extreme values in these quantities which is reflected in heavy tails in the probability distribution functions (PDFs). Extreme events are a generic feature of turbulence, see e.g., [START_REF] Moffatt | Extreme events in turbulent flow[END_REF], and make its prediction particularly difficult, e.g., for tornadoes, large floods and other extreme events with devastating impact. In the present work we will show that the divergence and rotation values of the particle velocity likewise have extreme intensities, much larger than their standard deviation.

The remainder of the manuscript summarizes first the DNS flow data we analyze. Then we describe the method to determine the divergence and rotation based on Delaunay tessellation of the particle positions. Numerical results and extreme divergence and rotation values are presented afterwards and finally some conclusions are drawn.

FLOW DATA AND METHODS

Particle position and velocity data are generated by DNS of particle-laden homogeneous isotropic turbulence, presented in [START_REF] Matsuda | Influence of microscale turbulent droplet clustering on radar cloud observations[END_REF]. The incompressible Navier-Stokes equations are solved in a 2π-periodic cube with a fourth order finite difference scheme. Statistically stationary flow is obtained by forcing at large scales. Uniformly distributed dis-12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12) Osaka, Japan, July [19][20][21][22]2022 crete particles are then injected into the fully developed flow and are tracked in the Lagrangian framework. Maxey's model [START_REF] Maxey | The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields[END_REF] for inertial heavy point particles with Stokes drag is used and the inertial dynamics is controlled by the Stokes number, St = τ p /τ η , where τ p is the particle relaxation time and τ η the Kolmogorov time. The equation of particle velocity v v v p j is given by

d t v v v p j = - v v v p j -u u u p j τ p , (1)
where u u u p j is the fluid velocity at particle position x x x p j . The subscript p denotes the quantity at the position of a particle, and the subscript j denotes the particle identification number.

State of the art high resolution DNS with N 3 g = 512 3 grid points is performed for the Taylor-microscale Reynolds number Re λ = 204, where Re λ ≡ u λ /ν, ν is the kinematic viscosity, and λ is the Taylor microscale. The number of particles N is 1.5 × 10 7 and the considered Stokes numbers are St = 0.5, 1, 2 and 5. Particles with different Stokes numbers, including St = 0 were tracked in an identical turbulent flow.

We apply 3D Delaunay tessellation [START_REF] Delaunay | Sur la sphère vide[END_REF] using the Quickhull algorithm [START_REF] Barber | The quickhull algorithm for convex hulls, acm transactions on mathematical software[END_REF] to the particle positions at two consecutive time instants. To define a cell corresponding to a particle, we use the dual graph of the Delaunay tessellation. For stability reasons, instead of using the circumcenter of the Delaunay cell, as done for the Voronoi tessellation, we use the center of gravity to define the vertices of the cell.

To compute the divergence of the particle velocity D(v v v p ), we consider the local number density averaged over a cell, which is the inverse of the corresponding volume. Using the fact that particles satisfy the conservation equation of the density n p , D t n p = -n p ∇ • v v v p where D t = ∂ t + v v v • ∇ is the Lagrangian derivative and considering two time instants t k and t k+1 = t k + ∆t of the Delaunay tessellation with time step ∆t, we can determine the volume change. Thus we obtain [START_REF] Oujia | Divergence and convergence of inertial particles in highreynolds-number turbulence[END_REF],

D(v v v p ) = - 1 n p D t n p = 2 ∆t V k+1 p -V k p V k+1 p +V k p (2)
The curl of the particle velocity, is defined by computing the circulation of the velocity field of particles over a cell. This can also be expressed as the divergence of the velocity of the fluid which has been rotated in a direction π/2 with respect to the direction of the curl. We define

v v v ⊥ x = A x v v v, v v v ⊥ y = A y v v v and v v v ⊥ z = A z v v v
where A x , A y and A z are rotation matrices around the different axes. We obtain that the curl of the particle velocity

C (v v v p ) is given by C (v v v p ) =   D -v v v ⊥ p,x D -v v v ⊥ p,y D -v v v ⊥ p,z   (3)
The accuracy and reliability of the method to compute the divergence, curl and furthermore the velocity gradient tensor of the particle velocity are studied in [START_REF] Oujia | Differential operators using volume change of tessellation cells[END_REF]. We find that the spatial order of the method is the inverse of the dimension of the space.

RESULTS

Figure 1 shows two-dimensional cuts of the particles colored with the divergence D(v v v p ) (top) and the enstrophy ||C (v v v p )|| 2 2 (bottom) for St = 0 (left) and 1 (right). The particles for St = 0, i.e. fluid particles, are randomly distributed because the flow is incompressible. We can deduce that for St = 0 the divergence values different from zero are numerical errors due to the sampling. Their distribution is homogeneous and we find mostly values close to zero, except in some regions, which are characterized by strong enstrophy values. The enstrophy of the fluid particles (bottom, left) shows a non homogeneous distribution and is spotty, characteristic for intermittent quantities. In contrast to fluid particles, inertial particles (Figure 1, right), here for St = 1, are clustered. We can observe the preferential concentration of particles, corresponding to clusters in regions of low fluid vorticity and voids in regions of large fluid vorticity. This observation is explained by the fact that inertial particles are ejected from high fluid vorticity regions due to the centrifugal force. We also find the high convergence and divergence value are observed close to each other. We conjecture that the particles are crossing. In a sufficiently large region of low fluid vorticity, we can observe that divergence and enstrophy values of inertial particles are close to zero.

PDFs of divergence and rotation are shown in Figure 2. As the turbulence is isotropic, we do not distinguish between the different vector components of C (v v v p ), and we compute the histograms of all these components together. For fluid particles in the continuous setting, the divergence of the fluid velocity vanishes exactly, whereas in the discrete setting D(v v v p ) differs from zero due to the sampling and the computation of the discrete divergence. For inertial particles, the PDFs of the divergence and of the rotation deviate from those for fluid particles. This nicely illustrates that for large Stokes numbers heavy tails and extreme values can be observed in both quantities, going up to more than 50 and 40 times of the standard deviation for the divergence and the rotation, respectively. The PDFs of the divergence and of the rotation have much heavier tails compared to the fluid vorticity, and the extreme values increase significantly with the Stokes number. This is quantified further in Table 1, where each of the variance and flatness of the divergence D(v v v p ) and curl C (v v v p ) is given as a function of the Stokes number. We observe that the variance of the divergence increases as the Stokes number increases. The flatness of the divergence first increases with St with a maximum for St = 1 and then decreases. The large flatness value is a sign of strong non-Gaussianity and the spatially intermittent behavior of the divergence for St = 1. These observations suggest that the global number of particles with large divergence values increases with the Stokes number. Note that for St > 1 tails become shallower with increasing St when the PDFs are normalized by the standard deviation, not shown here. We recall that the divergence of the particles is induced by the vorticity of the fluid flow via the centrifugal force. The decay of the flatness for St ≥ 2 can be explained by the particle's inertia. In fact, heavy particles are less subjected to the small scale fluctuations of the fluid vorticity, so that the particles follow less the fluid flow and thus have less tendency to accelerate suddenly. The variance of the curl decreases from St = 0 to 0.5 and then increases as the Stokes number increases. The flatness decreases from St = 0 to 0.5 and then increases with St with a maximum for St = 2 and then decreases. The reduction of the variance and flatness of the curl between St = 0 and 0.5 can be explained by the fact that the particles in the high velocity regions in the fluid domain are ejected due to the centrifugal force. However, for a small St the particle velocity 12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)

Osaka, Japan, July 19-22, 2022 at a given position is still close to that of the fluid. This explains the reduction of the curl value between St = 0 and 0.5 as well as that of the variance. The increase of the variance of the curl value for larger Stokes numbers could be explained by the higher inertia. Due to the sling effect, the particle trajectories may cross each other, or the particles may enter high fluid vorticity regions and experience strong vortical motion.

Figure 3 shows the PDF of the relative helicity of particle velocity for different Stokes numbers and randomly distributed particles. The relative helicity of the particle velocity is defined as

H (v v v p ) = v v v p • C (v v v p ) ||v v v p || 2 ||C (v v v p )|| 2 (4)
We can observe that for St ≤ 1 there are two maxima at H p = ±1, which means that we have a higher probability that the vorticity is oriented in the same or opposite direction as the particle velocity. This corresponds to helical motion, which is a signature of coherent structures in the particle laden flow. In contrast we find for St = 5 two minima at H p = ±1 and a maximum close to 0 which means that the vorticity has higher probability to be orthogonal to the particle velocity, which indicates incoherent particle flow motion. We can observe a transition from a convex to a concave shape at St = 2. Note that the asymmetry of the distribution can be explain by the presence of helicity asymmetry in the fluid flow due to initial forcing.

CONCLUSIONS

We analyzed high resolution DNS data of particle laden turbulence considering different Stokes numbers. We proposed a semi-Lagrangian method to determine the divergence and rotation of the particle velocity. To this end we considered the time change of the Delaunay tessellation of the particle positions and computed the spatial derivatives in the Lagrangian frame. Consequently the divergence, the curl and likewise the helicity of the particle motion could be quantified. We showed that the divergence and curl PDFs exhibit heavy knowledged for granting access to its high performance computing resources. K.M. acknowledges financial support from JSPS KAK-ENHI Grant Number JP20K04298. The DNS data analysed in this project were obtained using the Earth Simulator supercomputer system of JAMSTEC.

Figure 1 :

 1 Figure 1: Spatial distribution of the particles colored with the divergence D(v v v p ) (top) and the enstrophy ||C (v v v p )|| 2 2 (bottom) at particles positions for St = 0 (left) and 1 (right) for a slice of thickness 4η. Note that the color scale is linear for the divergence and logarithmic for the enstrophy.

Figure 2 :

 2 Figure 2: PDFs of divergence D p (top) and rotation C p (bottom) for the particle velocity for different Stokes numbers including fluid particles.

Figure 3 :

 3 Figure 3: PDFs of the relative helicity (cosine of the angle between velocity and vorticity) H p of the particles for different Stokes numbers including fluid particles (St = 0).

Table 1 :

 1 Variance V and flatness F of divergence D p and curl C p as a function of the Stokes number.

	St	0	0.5	1	2	5
	V(D p ) 0.87 16.9 43.8 88.5 152
	F(D p ) 15.9 55.3 64.0 33.5 15.8
	V(C p ) 68.8 42.7 49.7 60.1 82.3
	F(C p ) 7.83 7.02 12.0 17.7 14.9

tails and we found that for large Stokes numbers extreme values are above 40 times the corresponding standard deviations. In future work, we plan to perform multiscale analysis using multiresolution tessellations.