Asma Trabelsi

Zied Elouedi

Eric Lefevre

An ensemble classifier through rough set reducts for handling data with evidential attributes

Keywords: Ensemble classifier, evidential data, rough set theory, uncertain data, reducts

Ensemble classifier is a well-known method that has been used to solve several machine learning problems. To have reliable results, one should ensure the build of a good ensemble. In order to do so, researchers have proposed some heuristics like Random Subspace Ensemble (RSM), Rough set based ensemble, etc. The drawback of these mentioned approaches is their disability to handle uncertain data especially when uncertainty is represented by the evidence theory. The aim of this paper is to adapt both RSM and Rough set based ensemble in order to let them working in the context of evidential data. Three ensemble classifier approaches based on the rough set theory have been proposed and have been compared with each other. For the comparison purpose, we have relied on Ensemble Enhanced Evidential k Nearest Neighbor (EEk-NN) classifier, real world datasets from the UCI repository as well as synthetic databases.

Introduction

Recent years have witnessed the revival of artificial intelligence (AI) in academic and industry circles. One of the popular applications of AI is Machine Learning (ML), which has seen spectacular developments, and continues to find applicability in a wide range of domains, such as Information search, Information retrieval [START_REF] Chen | Machine learning for information retrieval: Neural networks, symbolic learning, and genetic algorithms[END_REF], speech recognition [START_REF] Padmanabhan | Machine learning in automatic speech recognition: A survey[END_REF], and personal assistants on mobile phones [START_REF] Canbek | On the track of artificial intelligence: Learning with intelligent personal assistants[END_REF], etc.

The ever-growing availability of data imposes an increasing demand for new and more powerful learning techniques. Therefore, in the last decade, we have seen the growth of several learning techniques, such as the ensemble learning [START_REF] Reformat | Building ensemble classifiers using belief functions and owa operators[END_REF][START_REF] Quost | Classifier fusion in the Dempster-Shafer framework using optimized t-norm based combination rules[END_REF]. This approach allows a significant improvement in the machine learning results by combining several models. Thus, this approach offers better predictive/classification performance compared to a single model [START_REF] Quost | Classifier fusion in the Dempster-Shafer framework using optimized t-norm based combination rules[END_REF]. That is why ensemble methods ranked first in many prestigious machine learning competitions, such as the Netflix Competition, KDD 2009, and Kaggle.

The construction of an ensemble system requires two substantial steps. The first one is to select a set of individual classifiers while the second one consists of combining the output predictions of these classifiers. The choice of the set of individual classifiers as well as the fusion operator could influence the ensemble performance.

The diversity between the base classifiers has been defended as a successful mean to construct a performing ensemble system and could be ensured in different manners. One among the diversity techniques that has been proven to be efficient and effective [START_REF] Bryll | Attribute bagging: improving accuracy of classifier ensembles by using random feature subsets[END_REF] is to vary the input feature space.

The process of generating diverse feature subsets with good predicting potential is still undergoing research study. The Random Subspace Method (RSM) is often used in the literature [START_REF] Kotsiantis | Combining bagging, boosting and random subspace ensembles for regression problems[END_REF], but its major limitation is to randomly partition the original input set. So, random selection may potentially increase the risk of selecting irrelevant and redundant features.

Despite the promising results of RSM based ensemble classifier, several other techniques and frameworks have been developed to enhance the prediction performance [START_REF] Saha | Rough set based ensemble classifier forweb page classification[END_REF]. Among them, we highlight the rough set theory that has been successfully used to reduce the set of features of any dataset [START_REF] Pawlak | Rough sets[END_REF]. This theory has been successfully applied in machine learning, particularly for feature reduction. It consists of finding the minimal feature subsets allowing the same discrimination as the initial set. The reduced set is called a reduct.

Ensemble classifiers through rough set reducts have been applied in a wide range of practical problems, such as text classification [START_REF] Shi | Rough set and ensemble learning based semisupervised algorithm for text classification[END_REF], biomedical classification [START_REF] Shi | A novel ensemble algorithm for biomedical classification based on ant colony optimization[END_REF], tumor classification [START_REF] Wang | Tumor classification by combining PNN classifier ensemble with neighborhood rough set based gene reduction[END_REF], web services classification [START_REF] Saha | Classification of web services using tensor space model and rough ensemble classifier[END_REF], etc.

It is important to note that real world data are almost imperfect (i.e. incomplete or/ and uncertain). This imperfection is due to multiple external factors such as obstacles, interference, missing information, etc. Several studies have been made to handle data imperfection. From these lines of research, we mention evidential theory [START_REF] Lee | Imprecise and uncertain information in databases: An evidential approach[END_REF], the probability theory [START_REF] Alizadehsani | Handling of uncertainty in medical data using machine learning and probability theory techniques: A review of 30 years[END_REF] and the fuzzy set theory [START_REF] Arunraj | Modeling uncertainty in risk assessment: An integrated approach with fuzzy set theory and monte carlo simulation[END_REF].

The evidence theory is commonly used to represent data imperfection thanks to its flexibility and capability to represent all kinds of imperfection including total and partial ignorance [START_REF] Song | Classifying vaguely labeled data based on evidential fusion[END_REF]. The data represented with the evidence framework are called evidential data [START_REF] Samet | Evidential database: a new generalization of databases?[END_REF].

In spite of their great importance, rough set ensemble classifiers has not been applied on data with evidential features [START_REF] Trabelsi | Learning decision rules from uncertain data using rough sets[END_REF][START_REF] Campagner | Belief functions and rough sets: Survey and new insights[END_REF]. To unlock this research field, we propose a rough set based ensemble for processing such kind of data. More particularly, we treat the case of imperfect features represented within the evidence framework. Our approach consists of three main levels: reduct generation, reduct selection for training individual classifiers and classifier fusion.

To sum up, the idea of this paper is to propose new ensemble classifier approaches to cope with uncertain data, more precisely when the uncertainty affects the values of attributes and is represented with the evidence theory. To the best of our knowledge, we are among the first to develop ensemble rough set classifiers in the context of evidential data. Three new approaches based on the rough set theory have been proposed and compared with each other. They will also be compared with individual EEk-NN classifier and the RSM technique. The choice of the rough set theory is justified by the fact that it has given promising results in a which context [START_REF] Shi | Rough set and ensemble learning based semisupervised algorithm for text classification[END_REF][START_REF] Shi | A novel ensemble algorithm for biomedical classification based on ant colony optimization[END_REF][START_REF] Wang | Tumor classification by combining PNN classifier ensemble with neighborhood rough set based gene reduction[END_REF][START_REF] Saha | Classification of web services using tensor space model and rough ensemble classifier[END_REF].

The remaining of this paper is organized as follows. In Section 2, we provide a brief description of the main concepts underlying the evidence theory, we define the concept of data with evidential features and we highlight the fundamental concepts of the rough set theory. We present, in Section 3, our ensemble classifier through rough set reducts to cope with evidential data, particularly data with evidential features. The experimentation settings and results are given in Section 4. We draw conclusions and some future work directions in Section 5.

Preliminary knowledge

In this section, we present the preliminary knowledge of two well known mathematical theories, namely the evidence theory and the rough set theory.

The evidence theory

The evidence theory, also referred to as belief function theory, is regarded as a very effective and efficient way for representing and managing uncertainty [START_REF] Dempster | Upper and Lower probabilities induced by a multivalued mapping[END_REF][START_REF] Shafer | A mathematical theory of evidence[END_REF]. This theory is extensively used for handling several real-world applications, including image processing [START_REF] Huang | Application of belief functions to medical image segmentation: A review[END_REF], business decision [START_REF] Srivastava | Belief Functions in Business Decisions[END_REF], multi-sensor fusion [START_REF] Ullah | Multisensor data fusion based on modified belief entropy in Dempster-Shafer theory for smart environment[END_REF][START_REF] Liang | A networked method for multi-evidence-based information fusion[END_REF], pattern recognition [START_REF] Denoeux | Handling possibilistic labels in pattern classification using evidential reasoning[END_REF], medical diagnosis [START_REF] Straszecka | Combining uncertainty and imprecision in models of medical diagnosis[END_REF][START_REF] Ellouze | Artificial intelligencebased diabetes diagnosis with belief functions theory[END_REF], classification [START_REF] Elouedi | Belief decision trees: theoretical foundations[END_REF][START_REF] Trabelsi | Classification systems based on rough sets under the belief function framework[END_REF][START_REF] Zhu | A fuzzy preference-based Dempster-Shafer evidence theory for decision fusion[END_REF], clustering [START_REF] Denoeux | NN-EVCLUS: Neural network-based evidential clustering[END_REF] and target tracking [START_REF] Daum | Multitarget-multisensor tracking: principles and techniques[END_REF]. In what follows, we provide a brief overview of its fundamental concepts as interpreted by the Transferable Belief Model (TBM) [START_REF] Smets | The transferable belief model[END_REF]. We also point out other basic concepts including the special belief functions, the belief function bel and the decision making process. We at last present the Dempster rule for combining distinct information sources.

Frame of discernment

Let Θ = {θ 1 , θ 2 , . . . , θ c } denotes the frame of discernment including a finite non empty set of c elementary hypotheses that are assumed to be exhaustive and mutually exclusive. The power set of Θ, denoted by 2 Θ , is made up of all the subsets of Θ:

2 Θ = { / 0, {θ 1 }, {θ 2 }, {θ 1 , θ 2 }, . . . , Θ} (1)
where each element of 2 Θ is called a proposition or an event.

Basic belief assignment

An expert's belief over the subsets of the frame of discernment Θ are represented by the socalled basic belief assignment (bba) denoted by m. It is carried out in the following manner:

∑ A⊆Θ m(A) = 1 (2)
The basic belief mass (bbm), denoted by m(A), implies the degree of belief exactly assigned to the event A. Because of a lack of information, this quantity cannot be distributed to any strict subset of A. It is worth noting that every subset A of 2 Θ having fulfilled m(A) > 0 is called a focal element.

With the aim of expressing some particular uncertainty situations, some special bbas have been proposed, including the vacuous, the certain bbas and the simple support mass function. The vacuous bba represents the total ignorance (i.e. m(Θ) = 1). The certain bba is a bba with a singleton as its unique focal element. A bba is called simple support function (ssf) if it has at most two focal elements: the frame of discernment Θ and a strict subset of Θ called the focus of the ssf.

Belief function bel

A belief function bel, relative to a bba m, assigns to any subset A of Θ the sum of beliefs exactly committed to every subset of A by m [START_REF] Shafer | A mathematical theory of evidence[END_REF]. In other words, it implies the total belief that one commits to A without being also committed to A. It has to be noted that m( / 0) is not included in bel(A), since / 0 is a subset of both A and A. The belief function bel is defined as follows:

bel : 2 Θ → [0, 1] bel(A) = ∑ / 0̸ =B⊆A m(B) (3)

The Dempster combination rule

The combination of a set of imperfect data is a crucial task owing to its ability to achieve a piece of more accurate information and improve decision making. The evidence theory is considered as a powerful tool to merge imperfect data including uncertain, imprecise and incomplete data. Indeed, several fusion rules have been proposed to aggregate a set of data induced from distinct information sources. The Dempster rule is one among the commonly used rules. Let m 1 and m 2 be two bbas defined in the same frame of discernment Θ. The Dempster rule is set to:

(m 1 ⊕ m 2 )(A) = k(m 1 ∩ ⃝m 2 )(A) (4) (m 1 ⊕ m 2 )( / 0) = 0
where

k -1 = 1 -(m 1 ∩ ⃝m 2 )( / 0) (5)
and

(m 1 ∩ ⃝m 2 )(A) = ∑ B,C⊆Θ:B∩C=A m 1 (B).m 2 (C) (6)
where ∩ ⃝ represents the conjunctive fusion rule also proposed within the evidence framework.

Data with evidential features

A dataset with evidential features is a dataset composed by M objects O j (i.e. j ∈ {1, . . . , M}) where each of them is described by N features A = {A 1 , . . . , A N } that are expressed within the evidence theory and a class label d j ∈ d. Each feature A k (i.e. k ∈ {1, . . . , N}) has a domain of discrete values denoted by Θ A k . The advantage of evidential data over the other theories is the ability to represent all kinds of data uncertainty. In fact, it allows us to manage total certainty, partial ignorance and total ignorance. An example of data characterized by evidential attributes is given below. Example: Overall, a bank loan officer has to predict the customer profitability levels d={Good, Bad} on the basis of some parameters (features). To put it simply, in this example, Table 1 describes the data knowledge for training, where we relied on three characteristics:

• Income with possible values Θ Income ={No, Low, Average, High}.

• Property: This feature reflects whether the loan requested by the client is greater or less than its property value and consequently it takes values into Θ Property ={Greater, Less}.

• Unpaid Credit: This feature provides information about client's unpaid Credit with possible values Θ UnpaidCredit ={Yes, No}. The example of

(Θ Property ) = 0.1 m UnpaidCredit 1 ({Yes}) = 1 Good O m Income 2 ({Average}) = 1 m Property 2 ({Greater}) = 1 m UnpaidCredit 2 ({No}) = 1 Good O m Income

Basic concepts for rough set theory

The rough set theory, proposed by Pawlak [START_REF] Pawlak | Rough sets[END_REF], is an efficient way for dealing with various machine learning problems such as clustering [START_REF] Parmar | MMR: an algorithm for clustering categorical data using rough set theory[END_REF], classification [START_REF] Khoo | A rough-set-based approach for classification and rule induction[END_REF], feature selection [START_REF] Bhatt | On fuzzy-rough sets approach to feature selection[END_REF][START_REF] Kumar | Fuzzy-rough discriminative feature selection and classification algorithm, with application to microarray and image datasets[END_REF], etc. One of our ultimate goals throughout this paper is to extract the most suitable feature subsets for classifier ensemble. Thus, we mainly relied on rough sets for generating the smallest subsets of relevant features (i.e. called reducts) and selecting the appropriate ones. In practical terms, a data set has to be represented through a Decision Table (DT ) which is defined as a pair DT = (U, A ∪ {d}). The universe

U = {O 1 , . . . , O M } reflects a non-empty finite set of M objects, A = {A 1 , . . . , A N } is a non-empty finite set of N features with values V (O i ) = {V 1 (O i ), . . . ,V N (O i )} for each object O i
DM(O i , O j ) = {A k ∈ A|V k (O i ) ̸ = V k (O j ) and d i ̸ = d j } (7) ∀ i, j = {1, . . .

, M}

Once the discernibility matrix DM is computed, the discernibility function can be defined as follows:

f (DM) = ∧{∨(DM(O i , O j ))|∀O i , O j ∈ U, DM(O i , O j ) ̸ = / 0} (8)
The discernibility function has to be converted from a conjunctive normal form into a disjunctive normal form for picking out all possible reducts.

A rough set based ensemble for data with evidential features

We propose to develop a rough set based ensemble classifier for dealing with data with evidential features. It aims to construct an ensemble of evidential classifiers (i.e. classifiers trained from data with evidential features) through rough set reducts. The general structure of our proposed framework is depicted in Figure 1 where EC i (i.e. i ∈ {1, . . . , L}) corresponds to the i th selected Evidential Classifer. As depicted in this figure, our proposed framework consists of three main steps: reduct generation from data with evidential features, reduct selection to ameliorate the prediction performance and classifier combination. In the following, we describe in further detail each of these steps.

A framework for generating reducts from data with evidential features

Reduct computation has been proven as an NP-hard problem which has led to the introduction of several heuristics. The Rosetta software is one among the most effective approaches for reducts generation. It includes a set of algorithms for extracting multiple reducts such as the SAVGenetic Reducer [START_REF] El-Monsef | Classification of sand samples according to radioactivity content by the use of euclidean and rough sets techniques[END_REF], a genetic algorithm for picking out approximate reducts. As our main purpose is to address data with evidential features, we propose to extend the SAVGenetic algorithm for processing such kind of data. By analogy to the standard version, our evidential SAVGenetic reducer starts by calculating a discernibility matrix Λ ′ [START_REF] Trabelsi | Feature selection from partially uncertain data within the belief function framework[END_REF]. Assume that U={O 1 ,. . .,O M } is a given data set with M objects. Each object O i (i ∈ {1, . . . , M}) is described by N evidential features A = {A 1 , . . . , A N } and a certain class label d i ∈ d. Note that each attribute A k (i.e. k ∈ {1,. . . ,N}) has a domain of discrete values denoted by Θ A k . The entries of the discernibility matrix Λ ′ are set to:

Λ ′ (O i , O j ) = {A k ∈ A|bel(O i ̸ = O j |A k ) > T and d i ̸ = d j } (9)
where T refers to a tolerance threshold (i.e. T is set to 0.1 with the aim of maximizing the search space) and:

bel(O i ̸ = O j |A k ) = ∑ E i ,E j ,∀a i ∈E i ,a j ∈E j ,a i ̸ =a j m k i (E i )m k j (E j ) (10)
with E i and E j are subsets of Θ A k . The idea behind Equation 10 is to link between the events: attribute A k of objects O i and O j having different values as well as the distance between two mass functions. In particular, two mass functions may well be identical, while we would have absolutely no idea of whether their attribute values are equal or not. Even worse, we may have strong evidence of equal values that would give a positive distance (say, two probabilities slightly different, but with an important weight on the same value), and no evidence at all of equality that would give a null distance (say, two vacuous mass functions). This is obviously an undesirable property.

The process of extracting reducts through a discernibility matrix is regarded as a set cover problem. It consists of finding the minimal hitting sets from the non empty sets of the obtained discernability matrix. Since the minimal hitting set is an NP-hard problem, we relied on the genetic algorithm for picking out approximate hitting sets, meaning approximate reducts. Let us denote by A the set containing the elements of the discernibility matrix Λ ′ and let us denote by ζ ′ the set containing the non empty sets of Λ ′ , the fitness function corresponds to our genetic algorithm for each candidate solution B ∈ 2 N is set to [START_REF] Jin | Multi-objective machine learning[END_REF]:

f (B) = (1 -α) × |A| -|B| |B| + α × min{ε, |[𭟋 ∈ ζ ′ |𭟋 ∩ B ̸ = / 0]| |ζ ′ | } (11)
The fitness function f (B) rewards not only subsets that are hitting sets (i.e. meaning subsets having a non empty intersection with all elements of the discernability matrix) but also subsets with shortest size. Herein, α ∈ [0, 1] refers to the adaptive weighting between the two parts and ε reflects the minimal hitting set fraction. In this paper, we set ε with the aim of extracting minimal sets and we set α to 0.5 to balance the two parts.

Reduct selection for ensemble learning

We could very well wind up with hundreds or even thousands of reducts. The process of constructing ensemble systems with all generated classifiers is extremely costly, especially for high dimensional databases. An alternative solution consists of selecting the most appropriate reducts for ensemble learning. We present, in what follows, three approaches enabling us to pick out the most suitable reducts for an ensemble of evidential classifiers.

Select Diverse Reducts (DR)

One of the main keys for constructing a successful rough set ensemble is to ensure a good diversity between the chosen reducts. Getting inspiration from [START_REF] Debie | Reduct based ensemble of learning classifier system for real-valued classification problems[END_REF], we propose a new heuristic for selecting diverse reducts from the pool of generated ones. Our algorithm (see Algorithm 1) starts by picking out the reduct with the minimum cost, meaning the smallest one. Then, it computes the diversity between the chosen reduct and the remaining ones using Algorithm 2. The reduct which has to be chosen is the one with the highest diversity degree. As in [START_REF] Debie | Reduct based ensemble of learning classifier system for real-valued classification problems[END_REF], the diversity measure is the inverse of the average similarity between a candidate reduct R j and the L ′ chosen ones (i.e RED_Chosen) as follows:

Div j = 1 - 1 L ′ ∑ R i ∈RED_Chosen |R j ∩ R i | |R j ∪ R i | (12)
The most diverse reduct R j will then be chosen for constructing the ensemble system and it will be removed from the current reduct set RED. This process will be repeated until at most L reducts are selected or the reduct pool RED is empty.

Accuracy-Diversity Assessment Function for reduct selection (AD-AF)

The study conducted by Opitz [START_REF] Opitz | Popular ensemble methods: An empirical study[END_REF] has demonstrated that both the accuracy of individual classifiers and the diversity of base classifiers may improve the performance of an ensemble system. So, a good classifier ensemble has to be constructed on the basis of accurate individual classifiers that are diverse as much as possible. As already stated above, an ensemble system with rough set reducts is considered as a valid alternative for ensuring good diversity between the base classifiers. Herein, we propose to construct a good rough set based ensemble classifier by making a trade-off between the diversity and the accuracy of each individual classifier. More precisely, we relied on the assessment function, proposed by Opitz [START_REF] Opitz | Popular ensemble methods: An empirical study[END_REF], to extract the reducts enabling the construction

Fitness( f , Ens_Cls) = Acc( f , Ens_Cls) + ω × Div( f , Ens_Cls) (13)
where Ens_Cls states the current ensemble of classifiers, Acc( f , Ens_Cls) reflects the average accuracy of the base classifiers, Div( f , Ens_Cls) represents the diversity between base classifiers and ω corresponds to the parameter that balances Accuracy and Diversity.

Several classifier diversity measures exist in the literature. Authors in [START_REF] Kuncheva | Measures of diversity in classifier ensembles and their relationship with the ensemble accuracy[END_REF] have distinguished pairwise and non-pairwise diversity measures. The choice of the most convenient one remains a tricky problem. In this paper, we relied on the disagreement measure, which is a pairwise one, for computing classifier diversity. The parameter ω has to be adjusted automatically for maximizing the fitness function value. More concretely, we keep the value of ω when Fitness is increasing, while we increase it if Accuracy is stable and Diversity is decreasing and we decrease it if Accuracy is decreasing and Diversity is stable. The value of ω will be set to 1 as the initial value and the changing amount of ω will be set to 10% based on its current value. Our method differs from Opitz's approach in the extent to which it takes into consideration the accuracy between reducts in addition to the diversity and the accuracy of individual classifiers. The diversity between reducts is used to reduce the search space of reducts to improve search efficiency. It will be computed at each iteration, using Equation 12, and the candidate reducts with a diversity measure smaller than a threshold S (i.e. in the experimentation parts, we have set S to 0.7 to enable the selection of the most diverse reducts) will then be removed from the search space of reducts. Our proposed framework is detailed in Figure 2 and Algorithm 3.

We start by retrieving the reduct R 1 with the lowest cost and constructing the first evidential Classifier EC 1 (i.e. RED_Choosen = {R 1 } and Ens_Cls= {EC 1 }).

The diversity Div j between the current selected reducts RED_Choosen and each reduct R j ∈ RED is the calculated. The reducts R j with a diversity measure smaller then a threshold S will be removed from the reduct pool RED. Each candidate reduct R j ∈ RED will be evaluated using Equation 13. The reduct R k enabling the highest fitness function will be selected for constructing our ensemble learning (i.e. Ens_Cls = {Ens_Cls, EC k }). This process has to be repeated until at most a number L of reducts is reached or the current reduct pool RED is empty.

Ensemble Accuracy Assessment Function for reduct selection (EA-AF)

The wrapper approach, using the classifier accuracy as feature selection criterion, has been successfully used for solving several pattern recognition problems. In fact, it allows us to pick out the feature subset that achieves the greatest classification accuracy. Herein, we follow the same process as the previously presented approach but we relied on the ensemble accuracy as a fitness function for extracting the most appropriate reducts for an ensemble of evidential classifiers. The fitness function is set to:

Fitness( f j , Ens_Cls) = max R k ∈RED (EnsAcc( f k , Ens_Cls)) (14)
where EnsAcc( f k , EnsCls) reflects the ensemble accuracy of the already chosen classifiers RED_Chosen and the candidate classifier f k . One important element which has to be highlighted is the maximum number L of selected reducts,

Fitness( f j , Ens_Cls) = max R k ∈RED (Fitness( f k , Ens_Cls)) {% See Equation 20} 16: Ens_Cls ← {Ens_Cls, f j } 17: RED_Chosen ← {RED_Chosen, R j } 18: RED=RED-R j 19: L ′ =L ′ +1 20: until L ′ = L or isEmpty(RED) = true
meaning selected classifiers. According to a study conducted in [START_REF] Opitz | Popular ensemble methods: An empirical study[END_REF], ensembles of 25 classifiers are sufficient to improve the ensemble performance. In the remaining of this paper, we set L to 25.

Time complexity

It is important to point out the complexity of our different proposed approaches. Let us denote by N the total number of reducts, N ′ the number of candidate reducts, L the maximum number of chosen reducts and L ′ the real number of chosen reducts, the time complexity of each approach is given computed as:

• DR: For the approach Select Diverse Reducts, we repeat the process L times and for each time, we compare reducts with the L ′ chosen ones. So the process takes O(L * L ′ * N).

• AD-AF: A part of the DR approach is included in the AD-AF one. We only need to compute the accuracy induced by all candidate reducts. The time complexity will be computed as

O(L * (L ′ * (N + N ′ )))
• EA-AF: This third approach is the more complex one in terms of time complexity which is computed as

O(L * (L ′ * (N + ( N ′ * (N ′ +1) 2
))))

Multiple classifier integration

As previously mentioned, each individual evidential classifier yields decisions in terms of belief functions. Numerous combination operators have been proposed within the evidence theory to aggregate the classifiers outputs. In this paper, we relied on the Dempster combination rule presented in subsection 2.1.4. This choice is justified by the fact that we combine individual classifiers trained with diverse reducts.

Experimentation settings and results

This section is devoted to examining the performance of the three proposed reduct selection approaches for ensemble learning. In what follows, we detail our experimentation settings and results.

Experimentation settings

We have relied on some numerical and mixed real world databases acquired from the UCI machine learning databases [START_REF] Bache | UCI repository databases[END_REF], where some of them contain missing values. We have also generated some Synthetic datasets using Python and Scikit-Learn, a Machine Learning Library. The advantage of this library is that it accepts various parameters allowing to control the looks and feels of datasets. One Artificial dataset has been generated using Python. Then, we have added noise to the attributes of this generated data in order to create newest datasets. Different noise intensities have been explored like 0.15, 0.30 and 0.5. These values represent the fraction of samples whose attributes have been assigned randomly. Consequently, we have obtained four synthetic databases, the first one with no noise is called SD, the second one with a 0.15 as noise intensity is called SD_0.15, the third one with 0.30 as noise intensity is called SD_0.30 and the last one with 0.50 as noise intensity is called SD_0.50. Table 2 provides a description of the used databases.

Once data are collected and generated, we have modeled data thanks to the evidence theory. In a practical point of view, missing values have to be imputed and continuous variables have usually to be discretized into bins. However, the uncertainty introduced by missing values imputation and continuous variables discretization have to be addressed. Herein, we propose to generate databases with evidential features from the mentioned ones. That is, the missing values will be represented by vacuous bbas and symbolic attributes have to be expressed through certain bbas. With regards to continuous variables, they have been transformed into beliefs using the Evidential c-Means approach (EcM) [START_REF] Masson | ECM: An evidential version of the fuzzy c-means algorithm[END_REF]. The EcM approach enables to associate for each instance feature a bba representing its membership to each cluster. In our case, we have set c to 3 for reducing the time complexity. The EcM algorithm starts by creating the user requested number of clusters for each feature. It then estimates, for each feature, the distance between each instance and each cluster's center and generates a bba using the distance value. Afterwards, it tries to minimize an objective function and computes recursively the cluster's center until no more minimization is possible [START_REF] Masson | ECM: An evidential version of the fuzzy c-means algorithm[END_REF][START_REF] Samet | Evidential data mining: precise support and confidence[END_REF].

The choice of the base classifier is also a crucial task. To the best of our knowledge, machine learning classifiers that handle data with evidential features are limited. Note that we have introduced, in previous work, a novel classifier for handing such a kind of data called Enhanced Evidential k Nearest Neighbor (EEk-NN) [START_REF] Trabelsi | A novel k-nn approach for data with uncertain attribute values[END_REF]. For the evaluation process, we relied on the (EEk-NN) as a base classifier. A key issue that has to be addressed is the parameter k which reflects the number of neighbors. For our tests, we evaluate three k values which correspond to 3, 5 and 7 respectively. To do so, we were based on some standard information retrieval measures, notably the Percentage of Correctly Classification (PCC), the recall and the precision.

Experimentation results

We conduct some experiments to compare our proposed approach for reduct selection and to pick out the best one among them. Our comparison is based on the ensemble size, the ensemble reduct diversity and the performance measure. We also make comparisons using an hypothesis test.

Comparison in terms of the ensemble size

The comparison results between our three heuristics for reduct selection in terms of the ensemble size are given in Table 3 where GR represents the total number of generated reducts. From these tables, we can see that a huge number of reducts may be produced for a given database (e.g. we have 8191 reducts for the Hepatitis database and 975 reducts for the Thoracic Surger database). So, it is crucial to select the most appropriate reducts for an ensemble of classifiers. As previously mentioned, we have proposed three approaches for reduct selection, namely the DR, the AD-AF and EA-AF methods. The experimentation results have proven that both AD-AF and EA-AF methods have yielded smallest ensembles compared with the DR approach. Let's take the Hepatitis database with k equals 3 as an example, the ensemble size achieved by the DR, the AD-AF and the EA-AF are equal to 25, 5 and 3 respectively. From this point of view, we can deduce the efficiency of the AD-AF and EA-AF approaches for generating ensemble EEk-NN with reduced size.

Comparison in terms of the Reduct diversity

We have mainly relied on the Jaccard distance J δ for measuring the reduct diversity. This measure highly depends on the number of reducts. In fact, the maximum diversity is yielded when there is an empty intersection of the generated reducts. It is set to:

J δ = |R 1 ∪ R 2 ∪ . . . ∪ R M ′ | -|R 1 ∩ R 2 ∩ . . . ∩ R M ′ | |R 1 ∪ R 2 ∪ . . . ∪ R M ′ | (15)
The obtained results are given in Table 4 where we can remark that the DR method has achieved in most cases the most diverse reducts comparatively with the AD-AF and EA-AF approaches. This can be explained by the specific feature of the Jaccard measure. In fact, it promotes the ensemble constructed with the largest number of reducts. The results still show that the AD-AF and EA-AF methods are able to provide sets of reducts with higher diversity compared to DR on some datasets (e.g., Hepatitis).

Classification performance Comparison

The evidence theory has not only the advantage to manage and represent uncertainty but it also proposes a set of combination rules to merge evidence acquired from several information sources, notably the evidential outputs of an ensemble of classifiers. The Dempster operator is a well used rule in the context of classifier fusion within the evidence theory [START_REF] Quost | Classifier fusion in the Dempster-Shafer framework using optimized t-norm based combination rules[END_REF]. From this, we use the Dempster rule for combining the selected individual classifiers obtained by the DR, the AD-AF and the EA-AF techniques. Following a 5-folds cross-validation process, we carry out a comparative study between these methods when based on the PCC, the Recall and the Precision. This approach has also been compared with individual EEk-NN classifier and the Random Selected Reduct (RSR) that consists of selecting a defined number of reducts to build the ensemble system [START_REF] Bar | Multiple reducts computation in rough sets with applications to ensemble classification[END_REF]. The experimentation results for Heart, Japanese, SD, SD_0.15, SD_0.30 and SD_0.50 databases are given from Table 5 to Table 10. From the obtained results, we can remark that the performance of an ensemble system is greatly influenced by the selected reduct approach.

From those tables, we can also remark that both individual EEk-NN classifier and ensemble classifiers built from the RSR method provide the most poorly performance comparatively with the DR, the AD-AF and the EA-AF approaches for all the tested databases and for all values of k but RSR still a little bit better than individual EEk-NN classifier. We can conclude that ensemble systems stay anyway better than individual classifier and we can attribute the failure of RSR with respect to DR, the AD-AF and the EA-AF to the existence of redundant features as part of the RSR selected reducts. To cope with a redundancy problem, we have firstly proposed the DR approach allowing the selection of diverse reducts from the original pool. The experimentation results have proven the impact of diversity when constructing an ensemble system. In fact, the PCC, the Recall and the Precision results achieved by the DR approach are greater than those obtained by the RSR method for almost all cases. Since the diversity between reducts is not sufficient for achieving the best performance, we have proposed the AD-AF method that takes into consideration in addition to the diversity between the selected reducts the diversity and the accuracy of base classifiers. The obtained results have proven the effectiveness of this method over the DR approach for almost all databases. Thus, we can deduce the effectiveness of the AD-AF technique over the DR in terms of the ensemble size as well as the classification performance. At last, we showed that the EA-AF has yielded the best classification performance for these databases and for the different values of k. We have obtained the same interpretation for all databases.

In this paper, we have also studied the effect of noise on our proposed approach. Different noise levels have been considered 0, 0.15, 0.30 and 0.50. The experimentation results are given from Table 7 to Table 10 and from Figure 3 to Figure 6. From these tables, we can remark that the more uncertainty we have, the lower accuracy we obtain but the EA-AF approach still yield the best results compared with the other approaches. For that reason, we have shown the precisionrecall curves only for EA-AF results. These curves show the tradeoff between the Precision and the Recall for different thresholds. We can remark that in case of no and low uncertainty, we have a high area under the curves representing high recall and high precision and consequently low false positive and low false negative rate. For high uncertainty (Figure 5 and Figure 6), we can remark that curves approach the less accurate area.

Results based on hypothesis test

As presented earlier, a big part of machine learning is to select the best machine learning approach. That means that we have to select the model maximizing the accuracy. Researchers today ask this key question "Is the difference in skill between two machine learning models real, or due to a statistical chance?". In this part of the paper, we use statistical hypothesis testing to address this question. There exist several statistical tests. To cite a few, we mention the McNemar's test and 5×2 Cross-Validation, Paired t-test, etc. These statistical tests are widely used for comparing machine learning approaches. The idea is to compute a pvalue and to compare it with a variable al pha to decide if we have to reject null hypothesis or not (i.e. null hypothesis means that there is no difference between two models).To sum up:

• if pvalue >= α, we fail to reject null hypothesis • if pvalue < α, we reject null hypothesis In this paper, we have chosen to compare the different machine learning approaches based on the Paired t-test Test Statistic. We have ten pairs of models to check: {EEk-NN, RSR}, {EEk-NN, AD -AF}, {EEk-NN, EA -AF}, {RSR, DR}, {RSR, AD -AF}, {RSR, EA -AF}, {EEk-NN, RSR}, {DR,AD -AF}, {DR, EA -AF},{AD -AF, EA -AF}. For the statistical test, we set α to 0.05, the default value used in the majority of scientific articles.

The achieved results have mentioned that the pvalue for this present case study is between 0.001 and 0.023 for all pairs of models, all databases and all values of k, As the pvalue is smaller than the considered significance level (i.e., 0.05), we can reject the null hypothesis. Therefore, the results statistically provide convincing evidence that the EEk-NN, the DR, the AD -AF and the EA -AF approaches perform differently. On average, the accuracy for the EA -AF approaches is better than the DR and the AD -AF approaches.

Conclusion and future works

In this paper, we have proposed a novel framework for selecting a successfully ensemble of classifiers for addressing data with evidential features. Our framework consists firstly of generating all possible reducts and then selecting the most suitable ones for training individual classifiers. Three approaches have been proposed for selecting the best reducts, namely the Diversity Reduct method (DR), the Accuracy-Diversity Assessment Function method (AD-AF) and the Ensemble Accuracy Assessment Function method (EA-AF). These mentioned approaches have been compared in terms of ensemble size, reduct diversity, classification performance and the difference between the approaches using a statistical test. The achieved results have shown that the EA-AF approach has yielded the best results in terms of the performance criterion using the Dempster combination rule to merge the outputs of the classifiers. Unfortunately, we cannot demonstrate that our method brings better or similar results to further other methods as we are almost the first authors providing this framework. The comparison could not be limited to ensemble classifiers, but generally to classification methods. One limitation of the proposed approach is that bba calculation for big databases requires too many computation resources, as clustering can be numerically hard. As a future work, we look forward to reducing the approach's complexity. We also aim to study the impact of some combination rules in the ensemble performance as well as to use other interesting evaluation criteria like the statistical test.

3 ( 6 (

 36 Θ Property ) = 0.1 m UnpaidCredit 6 ({Yes}) = 1 Bad

  and d corresponds to the decision value (i.e. the decision class). The discernibility function is one among the efficient solutions for reduct extraction. It consists firstly of computing a discernibility matrix DM from a given decision table DT . Each element DM(O i , O j ) represents the set of all features discerning objects O i and O j with different class d. Entries of the discernibility matrix DM are computed as follows:

Table 3 :

 3 Comparison in terms of ensemble size (a) Ensemble size for Heart database GR DR AD-AF EA-AF k=3 127 Ensemble size for Vote Records database GR DR AD-AF EA-AF k=3 Ensemble size for Hepatitis database GR DR AD-AF EA-AF k=3 8191

Figure 3 :

 3 Figure 3: The precision-recall curves for the SD dataset

Figure 4 :Figure 5 :Figure 6 :

 456 Figure 4: The precision-recall curves for the SD_0.15 dataset

Table 1 :

 1 Example of data with evidential attributes

	O	Income		Property	UnpaidCredit	d
	O	m Income 1	({High}) = 1	m Property 1	({Greater}) = 0.6 m Property 1	({Less}) = 0.3 m Property 1

  Table 1 presents different kinds of bbas. The objects of both Income and UnpaidCredit are represented by certain mass functions. For the feature Property, objects O 2 and O 4 are also represented by certain mass functions, objects O 3 and O 5 are represented by simple support mass functions and objects O 1 and O 6 are represented by general mass functions.

  Algorithm 1 Select diverse reducts 1: input: A pool of reducts RED, L is the maximum Number of chosen reducts 2: output: L ′ diverse reducts 3: RED_Chosen ← / 0 4: R 1 =min R∈RED cost(R)5: RED_Chosen ← {RED_Chosen, R 1 } 6: L ′ =1 7: RED= RED -R 1 ; 8: While L ′ < L or isEmpty(RED) = false Do 9: Div ←-ReductDiversity(RED_Chosen, RED) {%Computed through Algorithm 2} 10: R_best= arg max R j ∈RED Div j 11: RED_Chosen ← {RED_Chosen, R_best} 12: RED= RED -R_best; 13: L ′ = L ′ + 1 14: end while Algorithm 2 ReductDiversity(RED_Chosen,RED) 1: input: Candidate reducts RED and selected reducts RED_Chosen 2: output: Diversity between reducts Div 3: for j= 1 to |RED| 4: Sim j =0; 5: for each R i ∈ RED_Chosen 6: Sim j = Sim j + of accurate individual classifiers with the greatest possible level of diversity. Opitz's assessment function for a classifier f is set to:

		|RED j ∩R i |
		|RED j ∪R i |
	7: end for 8: Div j = 1-	Sim j |RED_Chosen|
	9: end for

  Repeat 11: Div ←-ReductDiversity(RED_Chosen, RED) {% Computed through Algorithm 2} 12: RED_New ←-R j ∈ RED with Div j > S 13: RED = RED_New 14: Choose a new reduct R j from RED satisfying: 15:

	Algorithm 3 Accuracy-Diversity assessment function for reduct selection Generated 1: input: A pool of reducts RED, L is the maximum Number of chosen reducts reducts RED 2: output: Chosen L ′ diverse reducts
	3: RED_Chosen ← / 0
	4: Ens_Cls ← / 0
	5: R 1 =min R∈RED cost(R) 6: RED_Chosen ← {RED_Chosen, R 1 } 7: L ′ =1	Set a number L of maximum chosen reducts
	8: RED= RED -R 1 ;
	9: Ens_Cls ← {Ens_Cls, f 1 }
	10:	with the minimum cost Choose Reduct R i	R i
			Add the chosen
			reduct into
			RED_Chosen
			Remove all
			R j in RED
			where Div j < S
			Select best reduct R j
		If |RED_Chosen| = L or
			isEmpty(RED)
			No
	Yes
			Compute the
			fitness function
			Construct
			Ensemble
			Classifier
		Figure 2: Reduct selection for ensemble learning

Table 2 :

 2 Description of databases DatabasesHeart Japanese Vote records Hepatitis Wine Thoracic Surgery SD SD_0.15 SD_0.30 SD_0.50

	Total instances	270	690	435	155	178	470	700	700	700	700
	Total attributes	13	15	16	19	13	17	9 9 No 9 9 No No No
	Number of classes	3	2	2	2	3	2	4	4	4	4

Table 4 :

 4 Comparison in terms of reduct diversity (a) Reduct diversity for Heart database

				(b) Reduct diversity for Japanese
	DR AD-AF EA-AF k=3 0.6 0.5 0.5 k=5 0.6 0.5 0.54 k=7 0.6 0.42 0.86	database k=3 0.69 DR AD-AF EA-AF 0.67 0.69 k=5 0.69 0.65 0.63 k=7 0.69 0.69 0.64
	(c) Reduct diversity for Vote Records	(d) Reduct diversity for Hepatitis
	database			database
	DR AD-AF EA-AF	DR AD-AF EA-AF
	k=3 0.5	0.49	0.47	k=3 0.54	0.58	0.58
	k=5 0.5	0.5	0.48	k=5 0.54	0.58	0.46
	k=7 0.5	0.48	0.50	k=7 0.54	0.58	0.54
	(e) Reduct diversity for Thoracic Surger	(f) Reduct diversity for Wine Data
	database			database
	DR AD-AF EA-AF	DR AD-AF EA-AF
	k=3 0.94	0.93	0.94	k=3 0.82	0.75	0.81
	k=5 0.94	0.93	0.92	k=5 0.82	0.79	0.76
	k=7 0.94	0.92	0.90	k=7 0.82	0.81	0.74
	(g) Reduct diversity for SD database DR AD-AF EA-AF k=3 0.79 0.77 0.74 k=5 0.79 0.77 0.74 k=7 0.79 0.75 0.73	(h) Reduct diversity for SD_0.15 database DR AD-AF EA-AF k=3 0.79 0.78 0.79 k=5 0.79 0.77 0.78 k=7 0.79 0.78 0.77
	(i) Reduct diversity for SD_0.30 database	(j) Reduct diversity for SD_0.50 database
	DR AD-AF EA-AF	DR AD-AF EA-AF
	k=3 0.79	0.77	0.73	k=3 0.79	0.76	0.75
	k=5 0.79	0.76	0.73	k=5 0.79	0.76	0.74
	k=7 0.79	0.77	0.74	k=7 0.79	0.76	0.75

Table 7 :

 7 Results

				for SD database
			EEk-NN RSR	DR	AD-AF AE-AF
		PCC	65.83	67.90 77.95 82.34	85.39
	k=3	Recall	55.15	56.93 73.87 77.23	85.83
		Precision 56.20	58.43 75.36 81.19	86.66
		PCC	67.43	69.03 78.17 84.52	85.39
	k=5	Recall	57.46	59.89 75.78 79.38	85.83
		Precision 60.12	60.67 76.13 83.24	86.74
		PCC	70.21	71.80 79.43 84. 21	85.9
	k=7	Recall	60.13	60.07 76.18 80.33	86.43
		Precision 61.72	62.86 77.15 84.81	86.94


Médéric Jeanne

Marie-Laure Vuillaume

Dévina C Ung

Valerie E Vancollie

Christel Wagner

Stephan C Collins

Sandrine Vonwill

Damien Haye

Nora Chelloug

Rolph Pfundt

Joost Kummeling

Marie-Pierre Moizard

Sylviane Marouillat

Tjitske Kleefstra

Binnaz Yalcin

Dr Frédéric Laumonnier
email: frederic.laumonnier@inserm.fr

Annick Toutain

Haploinsufficiency of the HIRA gene located in the 22q11 deletion syndrome region is associated with abnormal neurodevelopment and impaired dendritic outgrowth

Keywords: HIRA haploinsufficiency, 22q11, functional neurogenomics, dendritic outgrowth, animal model, neurodevelopmental disorder

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The 22q11.2 deletion syndrome (22q11DS), also known as DiGeorge syndrome or velocardiofacial syndrome, is the most frequent chromosomal microdeletion syndrome. The phenotypic expression of the 22q11DS is highly heterogeneous and includes congenital heart defects, thymus and parathyroides hypoplasia or agenesis leading respectively to immune deficiency and hypocalcemia. The phenotypic spectrum also includes a large variety of neurodevelopmental disorders (NDD). Indeed, intellectual disability (ID) is present in about 50% of the patients with a mean total IQ of approximately 70 [START_REF] Swillen | Intelligence and psychosocial adjustment in velocardiofacial syndrome: a study of 37 children and adolescents with VCFS[END_REF][START_REF] Smedt | Deciphering Developmental Disorders Study (2017) Prevalence and architecture of de novo mutations in developmental disorders[END_REF]). The frequency of Autism Spectrum Disorders (ASD) is estimated to range between 15% to 50% [START_REF] Fine | Autism spectrum disorders and symptoms in children with molecularly confirmed 22q11.2 deletion syndrome[END_REF][START_REF] Vorstman | The 22q11.2 deletion in children: high rate of autistic disorders and early onset of psychotic symptoms[END_REF][START_REF] Bruining | Dissecting the clinical heterogeneity of autism spectrum disorders through defined genotypes[END_REF]. Psychiatric disorders are also very frequent as Attention Deficit/Hyperactivity Disorder is present in 37% of affected children and psychotic disorders in 41% of the adult patients (Schneider et al. 2014). This makes the 22q11DS the highest known genetic risk factor for developing schizophrenia and the second most common genetic cause of developmental delay [START_REF] Rauch | Diagnostic yield of various genetic approaches in patients with unexplained developmental delay or mental retardation[END_REF]. Considerable work has been done over the past 20 years to address the complex pathophysiology of the neurodevelopmental phenotype of the 22q11DS. Nevertheless, until now, the etiology of the cognitive and psychiatric phenotype remains speculative.

Among the 32 protein-coding genes located in the 1.5 Mb proximal 22q11.2 deletion, HIRA (Histone cell cycle regulator [MIM: 600237]) encodes a histone chaperone and plays an important role in the epigenetic regulation of gene expression since its tissue-specific knock-out affects many basic cellular processes including DNA damage, limited de novo methylation, and global aberrant transcription [START_REF] Nashun | Continuous Histone Replacement by Hira Is Essential for Normal Transcriptional Regulation and De Novo DNA Methylation during Mouse Oogenesis[END_REF][START_REF] Dilg | HIRA Is Required for Heart Development and Directly Regulates Tnni2 and Tnnt3[END_REF][START_REF] Valenzuela | HIRA deficiency in muscle fibers causes hypertrophy and susceptibility to oxidative stress[END_REF]. The HIRA protein contains a WD40 repeat (WDR) domain and belongs to the WDR family proteins, recently pointed out for their putative role in brain development and neuronal connectivity [START_REF] Kannan | WD40-repeat 47, a microtubule-associated protein, is essential for brain development and autophagy[END_REF]). Finally, it has been recently shown that HIRA regulates neural progenitor cell proliferation and neurogenesis particularly by controlling the Wnt signalling pathway [START_REF] Li | Histone chaperone HIRA regulates neural progenitor cell proliferation and neurogenesis via β-catenin[END_REF].

Here we report the identification of de novo truncating variants in the HIRA gene in two unrelated patients referred to the genetic consultation for diagnostic purpose of a neurodevelopmental disorder. To provide evidence for the pathogenicity of these variants, we performed functional analyses combining in vitro mouse neuronal models in which Hira gene expression was decreased, with the neuroanatomical analysis of Hira heterozygous knock-out mice, and the neurodevelopmental expression profile of Hira mRNA. Our findings revealed that HIRA haploinsufficiency leads to abnormal defects in both in vitro and in vivo models, suggesting that this gene would be involved in the regulation of neuronal differentiation and maturation.

Methods and materials

Patients

The study was approved by the local institutional review boards, and written informed consent was obtained from the patients' parents, including explicit permission to share clinical and identifying information.

Genomic DNA and RNA were obtained from the University Hospital of Tours (France) and from the Radboud University Medical Center in Nijmegen (The Netherlands).

Hira mRNA neurodevelopmental expression analyses

Total RNAs were extracted from neuronal cultures or from mouse brains using Trizol (Invitrogen, USA) and purified by the DirectZol kit (Ozyme, France). Complementary DNAs were obtained from 2 µg of total RNAs using the Prime Script RT Reagent commercial kit (Takara Bio, Nojihigashi, Japan). Primers were designed with Primer3 software: Forward primer (5'CGTACTGCTGCTGTGCTGTT3') was located in exon 9 of Hira and the reverse primer (5'CTGGGAGAAGTCGAGGAATG3') was located in exon 11. qPCR experiments were run in duplicates on the LightCycler 480 (Roche Diagnostics, Meylan, France) using the SYBR Green Master Mix commercial kit (Applied Biosystem, Foster City, CA, USA). Relative expression was assessed using the advanced E-method from LightCycler software (Roche Diagnostics, Meylan, France). Hypoxanthin phosphoribosyltransferase 1 (HPRT1) for human samples, Cyclin C1 (Cyc1) and Beta-2-microglobulin (B2m) for mouse samples were used as reference genes for normalization.

Hira knockdown expression analysis

Plasmids containing a small hairpin RNA (shRNA) targeting HIRA human and murine transcripts, as well as plasmids containing a scramble shRNA were used (Origene, TF500962; Rockville, MD, USA). The target sequence of the effective shRNA (5'GACAGTATGAATGCTACCTCTACTCCTGC3') overlapped exons 3 and 4 of Hira mRNA. The target sequence of the scramble shRNA is as follows 5'GCACTACCAGAGCTAACTCAGATAGTACT3'. The expression of the shRNA was driven by a U6 promoter. These constructions contained the chloramphenicol resistance gene and a CMV driven RFP gene, which expresses RFP protein constitutively in mammalian cells. The plasmidic DNAs were extracted using the Endo Free Plasmid Maxi Kit (Qiagen, Venlo, Netherlands). Both effective shRNA and non-effective shRNA were transfected in HEK293T cells in order to evaluate their efficiency to repress the expression of endogenous HIRA. HEK293T cultures were maintained in Dulbecco's modified Eagle's medium (DMEM)-Glutamax (Invitrogen, USA) supplemented with 10% Fetal Bovine Serum (Eurobio, France) and were transfected with the appropriate plasmid using Lipofectamine 2000 (Invitrogen, USA). In brief, 2,5 µg of DNA were combined with 6 µL of Lipofectamine 2000 in 300 µL of DMEM. The DNA/Lipofectamine 2000 mixture was incubated for 20 min at room temperature. Complexes were added dropwise onto cells growing after half of the medium was removed. Next, cells were incubated for 4 hours at 37°C and 5% CO2.

The transfection medium was then replaced by DMEM/10% Fetal Bovine Serum, and total RNA lysates were obtained 72 hours post-transfection. In order to measure the expression levels of HIRA in each condition, RT-qPCR analyses were performed on 5 independent transfections.

Primary hippocampal cultures

All mouse experiments were performed according to protocols approved by the University Francois-Rabelais of Tours and INSERM (Project authorization number 01456.03, Ministry of Research).

Hippocampi or cortices were dissected from embryonic day 17.5 C57BL/6J mouse embryos (Janvier Labs, Le Genest Saint-Isle, France), manually dispersed in cold PBS (Fisher Scientific, Waltham, MA, USA), and triturated with papain (10 U/ml; Worthington Biochemical, Lakewood, NJ, USA) for 22 min at 37°C. Cells were resuspended in DMEM/F12 (Fisher Scientific) with 10% fetal bovine serum (Eurobio, Courtaboeuf, France) and centrifuged at 250 g for 3 min, and the final pellet was resuspended in primary neuron growth medium (PNGM; Lonza, Basel, Switzerland). Dissociated cells were then plated onto glass coverslips coated in poly-D-lysine (Sigma Aldrich, Saint-Louis, MO, USA) and laminin at a density of 400 cells per mm 2 . The cultures were kept in Primary Neuron Growth Medium, and half of the medium was changed twice a week.

Transfections and cellular analyses

Hippocampal neurons were transfected 11 days after plating (Day in vitro 11, DIV11) with the appropriate plasmids (non-effective shRNA and effective shRNA) with the DNAin Neuro transfection Kit (Amsbio, Abingdon, UK). In brief, 500 ng of plasmid DNA was combined with 1 µL of DNAin Neuro in 100 µL of PNBM medium. The DNA and DNAin Neuro mixture was incubated for 15 min at room temperature.

Complexes were added dropwise onto cells growing after half of the medium was removed and stored. Next, cells were incubated for 4 hours at 37°C and 5% CO2. After incubation, the medium was changed. Cells were fixed at DIV14 using 4% paraformaldehyde and coverslips were mounted on a glass slide using ProLong Diamond Antifade Mountant with DAPI (Invitrogen). Individual neurons were directly imaged under fluorescence and confocal microscopy (Leica SP8, Leica, Wetzlar, Germany) using RFP (562 nm) labeling as a tracer of morphology. Image analysis was done using Imaris software (Bitplane Scientific Software, Zurich, Switzerland) allowing the evaluation of the number of neurites per neurons, the number of neurites by branching level and the mean neurite length. The neuronal branching was evaluated using Sholl analysis on ImageJ software (Wayne Rasband, NIH, Bethesda, MD, USA). Statistical analysis was performed using the GraphPad Prism 6.0 software (La Jolla, CA, USA). For the neuronal morphological study, the data were first analyzed using the D'Agostino-Pearson normality test. When the result was positive, the data were then analyzed using an Ordinary one-way ANOVA followed by Tukey's multiple comparisons. P<0.05 was assumed significant.

Generation of Hira +/-knockout mice

The construction of the Hira allele was based on the "Knockout-first allele" method [START_REF] Skarnes | A conditional knockout resource for the genome-wide study of mouse gene function[END_REF].

A "critical" exon (exon 4) present on all transcripts was first identified. A promotor-less targeting cassette was inserted in intron 3 in a C57BL/6N blastocyst, which created a frameshift mutation, thus generating the KO-first allele (also known as the tm1a allele). A heterozygous-by-heterozygous ("het-by-het") breeding scheme was used to propagate the line. After weaning, animals were housed three to four mice per cage with WT controls housed separately, in specific-pathogen-free environment in individually ventilated cages under 12/12 light/dark cycle with temperature-controlled conditions and free access to food and water with hardwood bedding. All animals were regularly monitored for health and welfare concerns.

Mouse Hira +/-neuroanatomical study

The neuroanatomical study was carried out using a cohort of 10 male mice aged 16 weeks (n=5 per genotype). Mouse brain samples were fixed in 4% buffered formalin for exactly 48 hours. A total of 40 brain parameters, consisting of surface area and length measurements, were taken blind to the genotype across one sagittal section at Lateral +0.60 mm. These parameters were distributed across 22 developmentally distinct brain regions related to the cerebrum, the cortex, the pons, the cerebellum, the ventricle, the corpus callosum, the thalamus, the caudate putamen, the hippocampus, the fimbria, the anterior commissure, the stria medullaris, the fornix, the optic chiasm, the hypothalamus, the pontine nuclei, the substantia nigra, the transverse fibers of the pons, the cingulate gyrus, the dorsal subiculum, the inferior colliculus and the superior colliculus. These brain regions were further delineated into six main brain categories (brain size, commissures, ventricles, cortex, subcortex and cerebellum). Data were analyzed using a two-tailed Student t-test assuming equal variance to determine whether a brain region is associated with neuroanatomical defects or not.

Ethical considerations in animal use

The care and use of Hira +/-mice in the Wellcome Sanger Institute study was carried out in accordance with UK Home Office regulations, UK Animals (Scientific Procedures) Act of 1986 under two UK Home Office licences (80/2485 and P77453634) that approved this work, which were reviewed regularly by the Wellcome Sanger Institute Animal Welfare and Ethical Review Body.

Results

We identified two cases with neurodevelopmental disorder and de novo genetic variants of HIRA. The patient 1 was a 4 year-old female born to non-consanguineous Algerian parents. She was born at 38 weeks.

Birth parameters were unknown except for the weight which was at the 25 th percentile. Neonatal period was marked by hypotonia and feeding difficulties with velopharyngeal insufficiency. She had psychomotor retardation, moderate ID and motor stereotypies. She also had growth retardation and mild microcephaly (OFC: -2.8 SD). At physical examination, facial features were suggestive of 22q11DS as she had narrow and slightly upslanted palpebral fissures, bilateral ptosis, tubular nose with hypoplastic alae nasi. The mouth was small, with a high palate and a short uvula. Ears were posteriorly rotated with small lobes. Brain MRI showed diffuse atrophy of the white matter. Metabolic investigation, blood count and calcemia were normal.

Array-CGH (cytoshure ISCA 60k, Agilent, Santa-Clara, CA, USA) identified a de novo 28 kb intragenic deletion encompassing exons 2 to 13 of HIRA (Fig. 1). This deletion resulted in a disruption of the open reading frame of HIRA leading to a premature stop codon in exon 14 (Fig. 1).

The Patient 2 was a 6 year-old female who presented behavioural problems mainly confined to symptoms of ASD. She had average head circumference for age. She also had diabetes mellitus type 1 and recurrent infections since 4 years of age. Array-CGH did not show any pathogenic CNV but whole-exome sequencing revealed a de novo heterozygous splice site variant in the HIRA gene, (hg19, chr22:19394706C>G; NM_003325.4: c.302+1G>C). The cDNA analysis confirmed the predicted deletion of exon 4 leading to a frameshift with a premature stop codon in exon 6, p.(Thr101Thrfs*20) (Fig. 1).

We also looked in public databases (DECIPHER, ClinVar, denovoDB) for additional de novo heterozygous HIRA variants potentially associated with pathogenicity. Interestingly, the denovo-db database (release Following these observations, we hypothesized that haploinsufficiency of HIRA might cause the phenotype observed in our patients and play a role in the developmental phenotype of the 22q11DS. We therefore studied the cerebral pattern of Hira expression and assessed the impact of haploinsufficiency of Hira in the neuronal plasticity and brain neuroanatomy in mice.

Expression analysis of Hira mRNA by RT-qPCR in mouse total brain extracts revealed a maximal level at birth with an approximately 2-fold increased expression compared to embryonic and postnatal expression (Fig. 2). Furthermore, RT-qPCR analyses performed in developing primary hippocampal cultures from mouse embryonic brains showed that Hira mRNA expression was maximal at 4 days of culture (Day in Vitro, DIV4) and significantly decreased when neurons become fully mature with functional synapses (DIV10 to DIV21). The pattern of Hira mRNA expression observed in total brain and in primary hippocampal neurons would thus correlate with the neuritogenesis and early dendritogenesis stages suggesting that Hira might modulate neuronal differentiation and early maturation processes.

To address the impact of Hira haploinsufficiency in developing primary hippocampal neuronal cultures, we used a shRNA approach to knock-down endogenous Hira gene expression. We first sought to validate the efficiency of our approach by transfecting of a shRNA expression plasmid targeting HIRA in HEK293T cell lines. Our experiments resulted in a 56% (P=0.0079) reduction of HIRA mRNA expression level, in accordance with a haploinsufficiency model. As the shRNA targets a sequence of HIRA gene that is identical between human and mouse orthologs, we then assessed the impact of Hira haploinsufficiency in developing mouse neuronal cultures. Primary hippocampal neurons transfected at DIV11 with the effective shRNA displayed morphological impairments compared to controls. Particularly, they displayed a 46% (P=0.0051) decrease in the number of neuritic intersections per neuron, a 28% (P=0.0177) decrease in the mean total neurite length per neuron and a 23% decrease in the number of neurites per neuron compared to neurons transfected with the non-effective shRNA (i.e. scramble shRNA). The decrease in the number of neurites per neurons was not statistically significant (P=0.1490) but the assessment of the number of neurites by branching level showed a 33% (P=0.0241) decrease in the number of neurites in the third branching level compared to neurons transfected with the non-effective shRNA. These results suggested that a 2-fold reduction of Hira gene expression dysregulates dendritic arborization in primary neuronal cultures, suggesting that the HIRA protein is essential for dendritic development growth and stability.

Since no previous mouse studies reported the implication of Hira in the context of brain development, and in order to study the role of Hira haploinsufficiency in vivo, we developed a heterozygous mouse model using the knock-out first allele method [START_REF] Skarnes | A conditional knockout resource for the genome-wide study of mouse gene function[END_REF]). The strategy relies on the identification of an exon common to all transcript variants (exon 4), upstream of which a LacZ cassette was inserted. Exon 4 of the Hira allele was flanked by loxP sequences bilaterally. The resulting Hira tm1a(EUCOMM)Wtsi mice were then phenotyped. At weaning age, mouse survival was assessed from 187 successfully genotyped mice originating from several litters and derived from a het-by-het breeding scheme. We obtained no homozygous mice suggesting that Hira is essential for viability. To determine the window of death, we carried out a recessive lethality screen at mouse embryonic day 14.5 (E14.5). We looked at 41 embryos, none of which were homozygous, suggesting that homozygous mice die before E14.5. Using a recently developed and highly robust approach for the assessment of 40 brain parameters distributed across 22 developmentally distinct brain regions [START_REF] Collins | A Method for Parasagittal Sectioning for Neuroanatomical Quantification of Brain Structures in the Adult Mouse[END_REF], we analyzed neuroanatomical defects in Hira +/-transgenic mice (data available upon request). These parameters encompass six main brain categories: brain size, commissures (callosal, anterior and stria medullaris), ventricles (lateral and fourth), cortex (motor and cingulate), subcortex (hippocampus, caudate putamen, fimbria, thalamus, hypothalamus, substantia nigra, subiculum and colliculus), and cerebellum (granule layer, cerebellar nuclei, number of folia, pons, nerves, and pontine nuclei). This consisted of a systematic quantification on a single sagittal brain section at Lateral +0.60 mm. To minimize environmental and genetic variations, mice of each genotype were analyzed at 16 weeks of age and bred on the same genetic background (Bl/6N). Minor anomalies were identified in Hira +/- mice when compared to wild type, with suggestive evidence of possible convergence with microcephaly in some brain regions. For example, the hippocampus showed reduced height in the molecular layer (-20%, P=0.04). The corpus callosum and the fornix showed a marginally reduced area (-10%, P=0.04 and -9%, P=0.016, respectively) (Fig. 3).

Discussion

We report here de novo truncating variants in the HIRA gene, which is located in the commonly deleted region of the 22q11DS, identified in patients presenting with ID or ASD symptoms. These two variants were not found in control databases (GnomAD) and were predicted to be pathogenic and to lead to a lossof-function, knowing that HIRA is predicted intolerant to loss-of-function (pLI=1). This observation and the further identification of two additional cases with NDD and a de novo HIRA truncating mutation led us to consider HIRA as a candidate gene for aspects of the neurodevelopmental phenotype of 22q11DS.

Several genes have already been proposed as candidate genes for the neuropsychiatric phenotype of the 22q11DS, such as COMT, PRODH, RANBP1 or even ZDHHC8 [START_REF] Jacquet | PRODH mutations and hyperprolinemia in a subset of schizophrenic patients[END_REF][START_REF] Raux | Involvement of hyperprolinemia in cognitive and psychiatric features of the 22q11 deletion syndrome[END_REF][START_REF] Mukai | Palmitoylation-dependent neurodevelopmental deficits in a mouse model of 22q11 microdeletion[END_REF][START_REF] Mukai | Molecular substrates of altered axonal growth and brain connectivity in a mouse model of schizophrenia[END_REF][START_REF] Vorstman | Proline affects brain function in 22q11DS children with the low activity COMT 158 allele[END_REF][START_REF] Philip | Cognitive, behavioural and psychiatric phenotype in 22q11.2 deletion syndrome[END_REF][START_REF] Gothelf | Risk factors and the evolution of psychosis in 22q11.2 deletion syndrome: a longitudinal 2-site study[END_REF][START_REF] Meechan | Modeling a model: Mouse genetics, 22q11.2 Deletion Syndrome, and disorders of cortical circuit development[END_REF][START_REF] Paronett | Ranbp1, Deleted in DiGeorge/22q11.2 Deletion Syndrome, is a Microcephaly Gene That Selectively Disrupts Layer 2/3 Cortical Projection Neuron Generation[END_REF][START_REF] Moutin | Palmitoylation of cdc42 Promotes Spine Stabilization and Rescues Spine Density Deficit in a Mouse Model of 22q11.2 Deletion Syndrome[END_REF]. Nevertheless, the pathophysiology of the neurodevelopmental phenotype remains unexplained. The role of HIRA in development has already been suggested as the homozygous KO is lethal in mice, which we also show in this study, and the heterozygous KO results in gastrulation defects [START_REF] Roberts | Targeted Mutagenesis of the Hira Gene Results in Gastrulation Defects and Patterning Abnormalities of Mesoendodermal Derivatives Prior to Early Embryonic Lethality[END_REF]. Moreover, as a chromatin organization regulator HIRA is supposed to be necessary during brain development. Indeed, the chromatin remodelling process is frequently impaired in neurodevelopmental disorders and genes involved in this process are known to be responsible for multiple conditions associated with ID or ASD. A recent study described that genes located in the 22q11.2 region are significantly associated with networks involved in brain development and associated with pathogenesis of schizophrenia and ASD [START_REF] Forsyth | Synaptic and Gene Regulatory Mechanisms in Schizophrenia, Autism, and 22q11.2 CNV Mediated Risk for Neuropsychiatric Disorders[END_REF]. In particular, HIRA was highlighted as a candidate driver of neurodevelopmental disorders such as schizophrenia and ASD.

Here, we bring new evidence of the implication of HIRA in brain development and disorder. We observed that mouse Hira was significantly expressed in both total brain and primary hippocampal cultures during times corresponding to neuritogenesis and early dendritogenesis stages. Furthermore, we demonstrated that a 2-fold reduction in Hira gene expression induced alteration of neuronal morphology in primary hippocampal cultures by impairing both dendritic growth and branching. Interestingly, primary hippocampal neurons from mouse models of 22q11.2 deletion also have impaired dendritic growth [START_REF] Mukai | Palmitoylation-dependent neurodevelopmental deficits in a mouse model of 22q11 microdeletion[END_REF][START_REF] Fénelon | The pattern of cortical dysfunction in a mouse model of a schizophrenia-related microdeletion[END_REF][START_REF] Moutin | Palmitoylation of cdc42 Promotes Spine Stabilization and Rescues Spine Density Deficit in a Mouse Model of 22q11.2 Deletion Syndrome[END_REF]. Although it has been demonstrated that ZDHHC8 is involved in this neuronal phenotype, our results suggest that HIRA is also involved in the impaired dendritic growth observed in the 22q11DS mouse model. Additionally, it has been demonstrated that HIRA regulates neurogenesis by controlling the Wnt signalling pathway [START_REF] Li | Histone chaperone HIRA regulates neural progenitor cell proliferation and neurogenesis via β-catenin[END_REF]. Indeed, HIRA regulates βcatenin levels by recruiting the H3K4 trimethyltransferase SETD1A to the β-catenin promoter. In humans, loss-of-function variants in SETD1A have been associated with schizophrenia and developmental disorders [START_REF] Singh | Rare loss-of-function variants in SETD1A are associated with schizophrenia and developmental disorders[END_REF][START_REF] Kummeling | Characterization of SETD1A haploinsufficiency in humans and Drosophila defines a novel neurodevelopmental syndrome[END_REF]. Moreover, HIRA defects inhibit β-catenin expression and result in altered neurogenesis. Years of studies has revealed that the Wnt signalling pathway is necessary for the guidance and branching of the axon and dendrites, as well as synapses formation and their structural remodeling. Therefore, we could hypothesize that the impaired dendritic growth and branching observed in our neuronal model results from the alteration of the Wnt pathway caused by the haploinsufficiency of

HIRA.

As shown in our study, it has been previously demonstrated that homozygous KO of Hira is lethal in mice [START_REF] Roberts | Targeted Mutagenesis of the Hira Gene Results in Gastrulation Defects and Patterning Abnormalities of Mesoendodermal Derivatives Prior to Early Embryonic Lethality[END_REF]. Although conditional KO models in heart or muscle tissues have been described, there is no mouse model studying the impact of Hira haploinsufficiency on neurological phenotype [START_REF] Dilg | HIRA Is Required for Heart Development and Directly Regulates Tnni2 and Tnnt3[END_REF][START_REF] Valenzuela | HIRA deficiency in muscle fibers causes hypertrophy and susceptibility to oxidative stress[END_REF]. Therefore, we developed and analyzed a heterozygous Hira +/-mouse model to assess the impact of Hira haploinsufficiency on brain anatomy. We showed that heterozygous Hira knockout mice resulted in subtle neuroanatomical defects with some evidence of possible microcephaly in some brain regions. In particular, the corpus callosum and the molecular layer of the hippocampus displayed a statistically significant reduction in surface area and height respectively. It is likely that neuroanatomical anomalies exhibited in mice at the tissue level might be partially caused by abnormal neuritogenesis seen in primary neurons. It is noteworthy that nearly 50% of 22q11DS patients have microcephaly. Furthermore, MRI studies in 22q11DS patients have noted a decrease in total brain volume by 9 to 11% [START_REF] Kates | Frontal and caudate alterations in velocardiofacial syndrome (deletion at chromosome 22q11.2)[END_REF][START_REF] Simon | Volumetric, connective, and morphologic changes in the brains of children with chromosome 22q11.2 deletion syndrome: an integrative study[END_REF]. Hypoplastic corpus callosum, abnormalities of the septum pellucidi and hippocampal abnormalities have also been reported [START_REF] Ryan | Spectrum of clinical features associated with interstitial chromosome 22q11 deletions: a European collaborative study[END_REF][START_REF] Andrade | Hippocampal malrotation is associated with chromosome 22q11.2 microdeletion[END_REF][START_REF] Bohm | Neuroradiographic findings in 22q11.2 deletion syndrome[END_REF]. Although mild, the neuroanatomical findings of the Hira +/-mouse model should be put in the context of the neuroradiographic abnormalities reported in 22q11DS patients, potentially sharing common features such as corpus callosum hypoplasia and hippocampal malformations.

In conclusion, neurobiological and genetic analyses have provided important insights into the genetic basis of psychiatric and cognitive symptoms observed in patients carrying 22q11.2 microdeletions. Nevertheless, the neurodevelopmental pathogenesis of this syndrome seems to be extremely complex and may probably require the effects of reduced dosage for multiple genes within the 22q11.2 deletion, possibly interacting with permissive variants in modifier genes elsewhere in the genome. Our results demonstrate that the haploinsufficiency of HIRA impairs dendritic growth, neuronal branching and neuroanatomical development suggesting that HIRA should be considered as a major gene in the landscape of the complex pathophysiology of the neurodevelopmental phenotype of 22q11DS.

Figures titles and legends

  v.1.6.1, May 2020 accessed) included two additional cases. This includes a case from the Deciphering Developmental Disorders (DDD) study (individual DDD4K.03620) (Deciphering Developmental Disorders Study 2017), with NDD associated with a de novo truncating variant in HIRA consisting in a stop-gained mutation, hg19 chr22:19341607G>A, NM_003325.4:c.2596C>T and leading to a truncated HIRA protein, p.(Gln866*). We also found one variant of unknown significance, chr22:19341564A>G NM_003325.4:c.269T>C, p.(Met880Thr), carried by a patient for which the phenotype was not clearly described but included in a large autism spectrum disorder cohort (de Rubeis et al. 2014).

Figure 1 .

 1 Figure 1. Molecular genetic investigations in the two patients with HIRA genetic alteration. (A)

Figure 2 .

 2 Figure 2. Neuronal expression study and functional analyses of Hira in mouse brain tissue and primary hippocampal cultures. (A) Relative Hira mRNA expression levels at different stages of embryonic and postnatal development. E : embryonic day, P : postnatal day. RT-qPCR analysis of Hira mRNA expression level is shown relative to the Gapdh reference gene. Data are presented as dot plots with

Figure 3 .

 3 Figure 3. Mouse Hira +/-displays minor reductions in neuroanatomical phenotypes.(A) Histogram comparing male Hira +/-to WT and showing variation (decreased-minus scale or increased-

Figure 3

 3 Figure 3 Click here to access/download;Figure;New Figure 3 revisions BY.pdf

Acknowledgements

We would like to thank the patients and their families. We also thank the « IBiSA Electron Microscopy Facility » of the University of Tours for management and access to the confocal microscopy platform. This work was funded by the Association pour le Développement de la Neurogénétique (ADN) and The Fondation de France (to F.L.), and the Institut National de la Santé et de la Recherche Médicale (Inserm) for providing material support. The authors thank staff at the Research Support Facility (Sanger Institute)

for their excellent care of the mice and the members of the genome engineering, genotyping, phenotyping and database teams for their contribution to this work.

Declarations:

Funding : This work was funded by the Association pour le Développement de la Neurogénétique (ADN) and The Fondation de France (to F.L.), and the Institut National de la Santé et de la Recherche Médicale (Inserm) for providing material support.

Conflicts of interest/Competing interests :

On behalf of all authors, the corresponding author states that there is no conflict of interest. Consent to participate : The study was approved by the local institutional review boards, and written informed consent was obtained from the patients' parents, including explicit permission to share clinical and identifying information.

Consent for publication : The study was approved by the local institutional review boards, and written informed consent was obtained from the patients' parents, including explicit permission to share clinical and identifying information.

Disclosures

On behalf of all authors, the corresponding author states that there is no conflict of interest.


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