Le rôle du numérique dans les pratiques des enseignantes

Dima Hanna, dima.hanna@live.com Christiana Charalampopoulou, christiana.charalampopoulou@univ-tlse2.fr Chercheures associées UMR EFTS Université Toulouse -Jean Jaurès Les enseignantes sont ainsi invitées à adapter leurs pratiques professionnelles à ces injonctions institutionnelles. L'Environnement Numérique de Travail (ENT), implanté en collège depuis maintenant plus de quinze ans, apparaît pour les instances décisionnelles comme un soutien à ces transformations. Il désigne un ensemble intégré de services numériques mis à disposition de toustes les acteurrices de la communauté éducative d'un ou plusieurs établissements scolaires.

«

  Mobiliser fortement les potentialités du numérique » tel est l'objectif que le Ministère de l'Éducation nationale (MEN) français présente dans son plan : le numérique au service de l'école de la confiance (2019). L'intégration des Technologies de l'Information et de la Communication pour l'Éducation (TICE) engage l'ensemble du système éducatif dans des transformations pédagogiques et organisationnelles profondes, de l'école maternelle à l'enseignement supérieur.

  Géographie, même s'ils assument individuellement leurs tâches face à leurs élèves. Dans ce cadre, la mise en partage, la distribution et la coordination des tâches réalisées par les enseignantes est plébiscitée. Il ne s'agit pas d'une simple juxtaposition mais d'une complémentarité et d'interactions en permanence entre elleseux.Des recherches effectuées dans différentes régions en France observent une faible implication des enseignantes dans le travail collaboratif via l'ENT. À titre indicatif, Bruillard (2011) démontre que les apports originaux en classe issus de l'ENT sont rares et relèvent peu de déclarations de travail collaboratif entre les professionnelles. De même, seul un petit nombre d'enseignantes saisit le potentiel collaboratif lié à l'outil de mise en ligne de documents proposé par l'ENT(Voulgre, 2011).Les réponses à notre enquête renforcent leurs observations et démontrent que le travail collaboratif est peu plébiscité par les répondantes. En effet, seulement 20% préparent des contenus pédagogiques avec d'autres enseignantes et à peine 8% réalisent une séquence à plusieurs. De plus, 76% des enseignantes déclarent ne pas constituer une base commune d'apprentissage sur l'ENT.Bruillard É., 2011, « Le déploiement des ENT dans l'enseignement secondaire : entre acteurs multiples, dénis et illusions », Revue française de pédagogie, n°177, 4, 101-130.Voulgre É., 2011, « Une approche systémique des TICE dans le système scolaire français : entre finalités prescrites, ressources et usages par les enseignants », Thèse de doctorat inédite, Rouen, 356p.Par innovation, nous entendons « un processus qui a pour intention une action de changement et pour moyen l'introduction d'un élément ou d'un système dans un contexte déjà structuré »(Cros, 1998). Des études antérieures ont montré que les enseignantes n'estiment pas l'ENT capable de favoriser l'émergence de nouvelles pratiques pédagogiques. Il apparaît pourtant que, dans certains cas, elles et ils s'appuient sur l'ENT pour développer des innovations(Genevois et Poyet, 2010). Nos résultats renforcent ce dernier point En effet, parmi celles et ceux qui ont répondu au questionnaire, 60% indiquent que l'ENT soutient leur capacité d'innovation pédagogique. L'analyse des entretiens montre que la perception de l'évolution des pratiques professionnelles grâce à l'ENT fait l'unanimité : « Ça a complètement changé mes pratiques professionnelles », « L'ENT reste un plus. Je ne saurais pas revenir en arrière ». Les enseignantes déclarent que l'outil leur permet d'innover dans leurs pratiques pédagogiques notamment dans la communication avec leurs élèves. Elles et ils réalisent un appui personnalisé aux élèves en situation de handicap ou avec des problèmes d'absentéisme grâce à l'échange et au dépôt des cours en ligne. Au vu de nos résultats, les pratiques collaboratives qui en découlent ne sont ni intuitives, ni spontanées. L'ENT est plus utilisé comme instrument de communication que comme outil de collaboration. Ce manque d'usages collaboratifs est dû, entre autres, à la disparité des compétences numériques dans le corps éducatif et à des problématiques purement techniques. Néanmoins, les enseignantes reconnaissent l'ENT comme une innovation à la fois technologique et pédagogique prenant tout son sens dans la relation enseignant-élève (conception de séquence, suivi individuel, évaluation de l'élève, classe inversée). Dynamiques du changement en éducation et en formation. Considérations plurielles sur l'innovation (7-16), Versailles : Institut National de Recherche Pédagogique.

	Selon les enseignantes interrogées, les deux activités les plus fréquemment utilisées sont la messagerie et le cahier de textes. Elles et ils indiquent n'avoir recours aux blogs que de manière hebdomadaire ou mensuelle. Quant au reste des outils que propose l'ENT, les enseignantes affirment les mobiliser de manière beaucoup plus sporadique. Cette recherche propose donc d'étudier les usages de l'ENT des enseignantes de collège. Pour quoi est-il utilisé ? Permet-t-il de développer un travail collaboratif ? Est-il un levier pour l'innovation pédagogique ? Pour répondre à ces questions, nous avons effectué une enquête par questionnaire et par entretien semi-directif auprès des enseignants des collèges de l'Académie de Toulouse. Hanna D., Charalampopoulou C., 2019, « Travail collaboratif sur l'ENT et innovation dans les pratiques professionnelles des enseignants », Spirale, n°63, 23-35. Par ailleurs, les enseignantes entretenues indiquent que leurs pratiques collaboratives consistent à : -partager entre collègues des ressources audio-visuelles collectées sur Internet à des fins pédagogiques ; -préparer des fiches à destination des élèves en pratiquant l'écriture collaborative via le module « documents partagés ». Plusieurs tentatives de travail collaboratif à partir de l'ENT ont vu le jour durant la réforme du collège de 2016, notamment en lien avec l'instauration des Enseignements Pratiques Interdisciplinaires (EPI : création de blog, partage de documents). La prescription a joué un rôle important dans l'intégration de cet outil d'un point de vue interdisciplinaire. « Il s'agissait des documents à remplir pour donner les axes des projets, les compétences qui seraient travaillées, les notions visées. Chaque discipline se saisissait de ces documents et les remplissait collaborativement, parce qu'il y avait une pression institutionnelle », avoue un enseignant. En outre, les entretiens ont révélé que les enseignantes cherchant à travailler ensemble se retrouvent face à la réticence de certaines de leurs collègues ; d'autres disent préférer Marcel J.-F., Dupriez V., Périsset Bagnoud D., 2007, « Le métier d'enseignant : nouvelles pratiques, nouvelles recherches » in J.-F. Marcel, V. Dupriez, D. Périsset Bagnoud et M. Tardif (dir.) Coordonner, collaborer, coopérer. De nouvelles pratiques enseignantes (7-17), Bruxelles : De Boeck Université. L'ENT est source d'innovation pédagogique Ces professionnelles estiment que l'ENT favorise également l'innovation dans leurs pratiques en classe : « J'ai pu mettre en place des pédagogies actives de façon plus efficiente depuis que l'ENT existe : les travaux collaboratifs avec les élèves, des éléments de classe inversée », indique un enseignant. De plus, quelques enseignantes disent avoir modifié leurs façons de communiquer avec les familles d'élèves. Ainsi, 71,5% considèrent l'ENT comme une innovation à la fois technologique et pédagogique. Les raisons de ce positionnement sont prioritairement liées aux potentiels que le dispositif offre pour l'accompagnement des élèves dans leur apprentissage. Les enseignantes réalisent également que leur rôle est primordial dans l'usage pédagogique de l'ENT : « On va pouvoir détourner l'ENT à des fins pédagogiques ». Toutefois, elles et ils mettent en avant une limite de l'outil : « C'est une innovation pédagogique mais, pour moi, il ne va pas encore assez loin finalement » ; et sa remplaçabilité : « On aurait pu faire la classe inversée avec un autre support, pas forcément avec l'ENT ». L'ENT apparaît donc comme outil mobilisé pour inciter les enseignantes à travailler Distances et savoirs, n°8, 4, 565-583. ensemble. Genevois S., Poyet F., 2010, « Espaces numériques de travail (ENT) et "école étendue" », Cros F. (dir.), 1998,

Accessible par Internet, il est destiné aux enseignantes, aux élèves, aux parents et aux personnels de direction. Parmi les modules de l'ENT proposés, nous distinguons principalement les services de gestion (cahier de textes), les services de mises à disposition des ressources (classe virtuelle), les services d'administration (gestion des usagers) et les services de communication et de collaboration (messagerie, forum).

L'ENT n'encourage pas au travail collaboratif des enseignantes

Le travail collaboratif entre professionnelles correspond à la poursuite ensemble d'un objectif commun

(Marcel et al., 2017)

. Dans le cas de l'enseignement secondaire, l'enseignante des Mathématiques prépare par exemple des séquences avec l'enseignante d'Histoire-L'outil est principalement perçu comme un moyen facilitant les échanges entre professionnelles ; ainsi 84% des répondantes l'utilisent pour communiquer avec leurs collègues afin d'organiser des projets pédagogiques dans le cadre de la classe ou de l'établissement. Cependant, l'ENT reste peu investi dans le cadre de projets pédagogiques entre établissements ; seules 8% des interrogées le mobilisent dans ce cas. Pour 5 des 8 enseignantes interviewées, « ce n'est pas un espace de travail collaboratif. C'est un espace de communication dans les usages ». développer des projets communs en présentiel. La culture de l'établissement apparaît comme décisive pour développer et perpétuer des pratiques collaboratives. Enfin, d'autres freins proviennent des aspects techniques de l'ENT, tels que des problèmes de logistique ou d'espace de stockage trop limité.


Benoît Cadre
email: cadre@bretagne.ens-cachan.fr

Bruno Pelletier
email: pelletier@math.univ-montp2.fr

Pierre Pudlo
email: pudlo@math.univ-montp2.fr

Clustering by estimation of density level sets at a fixed probability

In density-based clustering methods, the clusters are defined as the connected components of the upper level sets of the underlying density f . In this setting, the practitioner fixes a probability p, and associates with it a threshold t (p) such that the level set { f ≥ t (p) } has a probability p with respect to the distribution induced by f . This paper is devoted to the estimation of the threshold t (p) , of the level set { f ≥ t (p) }, as well as of the number k(t (p) ) of connected components of this level set. Given a nonparametric density estimate fn of f based on an i.i.d. n-sample drawn from f , we first propose a computationally simple estimate t

(p) n of t (p) , and we establish a concentration inequality for this estimate. Next, we consider the plug-in level set estimate { fn ≥ t (p) n }, and we establish the exact convergence rate of the Lebesgue measure of the symmetric difference between { f ≥ t (p) } and { fn ≥ t (p) n }. Finally, we propose a computationally simple graph-based estimate of k(t (p) ), which is shown to be consistent. Thus, the methodology yields a complete procedure for analyzing the grouping structure of the data, as p varies over (0; 1). Index Terms -Kernel estimate; Density level sets; Nonparametric statistics; Clustering. AMS 2000 Classification -62H30, 62H12.

Introduction

Cluster analysis encompasses a number of popular statistical techniques aiming at classifying the observations into different groups, called clusters, of similar items; see, e.g., Chapter 10 in [START_REF] Duda | Pattern Classification. Second Edition[END_REF], and Chapter 14 in [START_REF] Hastie | The Elements of Statistical Learning: Data Mining, Inference, and Prediction[END_REF], for a general exposition on the subject. In general, no prior knowledge on the groups and their number is available, in which case clustering is an unsupervised learning problem. According to [START_REF] Hastie | The Elements of Statistical Learning: Data Mining, Inference, and Prediction[END_REF], clustering methods may be categorized in three ensembles, namely combinatorial algorithms, mixture modeling, and mode seekers. The methods proposed and studied in this paper pertain to the third class and rely on the ideas of density-based clustering; see [START_REF] Hartigan | Clustering Algorithms[END_REF].

Let us recall the nonparametric definition of a cluster given by [START_REF] Hartigan | Clustering Algorithms[END_REF]. Let X be a R d -valued random variable with density f . For any t ≥ 0, denote by L (t) the t-upper-level set of f , i.e.,

L (t) = { f ≥ t} = {x ∈ R d : f (x) ≥ t}.
(1.1)

According to [START_REF] Hartigan | Clustering Algorithms[END_REF], the clusters are the connected components of L (t), whence relating population clusters with domains of mass concentration.

Density level sets are therefore the basic objects of Hartigan's approach to the clustering problem. They also play a prominent role in various scientific applications, including anomaly or novelty detection, medical imaging, and computer vision. The theory behind their estimation has developed significantly in the recent years. Excess-mass level set estimates are studied in [START_REF] Hartigan | Estimation of a convex density contour in two dimensions[END_REF], [START_REF] Muller | Excess mass estimates and tests of multimodality[END_REF], [START_REF] Nolan | The excess-mass elipsoid[END_REF], [START_REF] Polonik | Measuring mass concentration and estimating density contour clusters-an excess mass approach[END_REF][START_REF] Polonik | Minimum volume sets and generalized quantile processes[END_REF], [START_REF] Tsybakov | On nonparametric estimation of density level sets[END_REF]. Other popular level set estimates are the plug-in level set estimates, formed by replacing the density f with a density estimate fn in (1.1). Under some assumptions, consistency and rates of convergence (for the volume of the symmetric difference) have been established in [START_REF] Baillo | Set estimation and nonparametric detection[END_REF][START_REF] Baillo | Convergence rates in nonparametric estimation of level sets[END_REF], [START_REF] Baillo | Total error in a plug-in estimator of level sets[END_REF], and an exact convergence rate is obtained in [START_REF] Cadre | Kernel estimation of density level sets[END_REF]. Recently, [START_REF] Mason | Asymptotic normality of plug-in level set estimates[END_REF] derive the asymptotic normality of the volume of the symmetric difference for kernel plug-in level set estimates; see also related works in [START_REF] Molchanov | A limit theorem for solutions of inequalities[END_REF], [START_REF] Cuevas | Plug-in estimation of general level sets[END_REF].

In the context of clustering, algorithms relying on the definition of [START_REF] Hartigan | Clustering Algorithms[END_REF] are typically composed of two main operations. First, observations falling into an estimation of L (t) are extracted, and next, these extracted observations are partitioned into groups; see, e.g., [START_REF] Cuevas | Estimating the number of clusters[END_REF], [START_REF] Biau | A graph-based estimator of the number of clusters[END_REF], and the references therein. However, to interpret the cluster analysis, the extracted set of observations must be related to a probability instead of a threshold of the level set. Such an objective may be reached as follows: given a probability p ∈ (0; 1), define t (p) as the largest threshold such that the probability of L (t (p) ) is greater than p, i.e.,

t (p) = sup t ≥ 0 : P X ∈ L (t) ≥ p . (1.2) Note that P(X ∈ L (t (p) )) = p whenever P( f (X) = t (p) ) = 0.
The parameter p has to be understood as a resolution level fixed by the practitioner: if p is close to 1, almost all the sample is in the level set, while if p is small, L (t (p) ) is a small domain concentrated around the largest mode of f .

Hence, in a cluster analysis, the practitioner fixes a probability p, depending on the objectives of his study. For a complete study, he needs to estimate, from a set of observations, the threshold t (p) , the level set L (t (p) ), as well as the number of clusters, i.e. the number of connected components of L (t (p) ). Assessing the number of clusters is also a major challenge in cluster analysis, due to its interpretation in terms of population diversity. When a hierarchical cluster analysis is needed, a dendrogram (see, e.g., Hastie et al., 2009, p. 521) may be produced by varying the value of p over (0, 1). The aim of this paper is to address these estimation problems, given a set of i.i.d. observations X 1 , • • • , X n drawn from f . Estimation of t (p) and L (t (p) ). In [START_REF] Cadre | Kernel estimation of density level sets[END_REF], a consistent estimate of t (p) is defined as a solution in t of the equation

{ fn ≥t} fn (x)dx = p, (1.3)
where fn is a nonparametric density estimate of f based on the observations

X 1 , • • • , X n .
In practice, though, computing such an estimate would require multiple evaluations of integrals, yielding a time-consuming procedure. Following an idea that goes back to [START_REF] Hyndman | Computing and graphing highest density regions[END_REF], we propose to consider the estimate t (p) n defined as the (1p)-quantile of the empirical distribution of fn (X 1 ), . . . , fn (X n ).

Such an estimate may be easily computed using an order statistic. We first establish a concentration inequality for t

(p)
n , depending on the supremum norm of fnf (Theorem 2.1). Next we specialize to the case where fn is a nonparametric kernel density estimate, and we consider the plug-in level set estimate L n (t

(p) n ) defined by L n (t (p) n ) = { fn ≥ t (p)
n }. The distance between two Borel sets in R d is defined as the Lebesgue measure λ of the symmetric difference denoted ∆ (i.e., A∆B = (A ∩ B c ) ∪ (A c ∩ B) for all sets A, B). Our second result (Theorem 2.3) states that, under suitable conditions, L n (t (p) n ) is consistent in the following sense:

nh d n λ L n (t (p) n )∆L (t (p) ) P -→ C (p) f .
Here,

C (p)
f is an explicit constant, depending on f and p, that can be consistently estimated.

Estimation of the number of clusters. Then, we consider the estimation of the number of clusters of L (t (p) ). A theoretical estimator could be defined as the number of connected components of the plug-in level set estimate L n (t (p) n ), for any estimate t (p) n of t (p) . However, heavy numerical computations are required to evaluate this number in practice, especially when the dimension d is large. For this reason, stability criterions with respect to resampling, or small perturbations of the data set, are frequently employed in practice, despite the negative results of [START_REF] Ben-David | A sober look at clustering stablity[END_REF] and [START_REF] Ben-David | Stability of k-means clustering[END_REF]. The approach developed in [START_REF] Biau | A graph-based estimator of the number of clusters[END_REF] and summarized below is based on a graph and leads to a dramatic decrease of the computational burden; see also [START_REF] Ben-David | A sober look at clustering stablity[END_REF]. In [START_REF] Biau | A graph-based estimator of the number of clusters[END_REF], the threshold t > 0 is fixed. Set (r n ) n a sequence of positive numbers, and define the graph G n (t) whose vertices are the observations X i for which fn (X i ) ≥ t, and where two vertices are connected by an edge whenever they are at a distance no more than r n . [START_REF] Biau | A graph-based estimator of the number of clusters[END_REF] prove that, with probability one, the graph G n (t) and the set L (t) have the same number of connected components, provided n is large enough. Hence the number of connected components, say k n (t), of G n (t) is a strongly consistent estimate of the number of connected components k(t) of L (t). In practice, however, only the probability p is fixed, hence the threshold defined by (1.2) is unknown. Moreover, in the abovementioned paper, the behavior of k n (t) depends on the behavior of the gradient of fn ; when fn is a kernel density estimate for instance, this leads to restrictive conditions on the bandwidth sequence. In comparison with [START_REF] Biau | A graph-based estimator of the number of clusters[END_REF], one can sum up our contribution (Theorem 3.1) as follows: only the probability p is fixed, and the associated threshold is estimated, leading to an efficient and tractable method for clustering. Moreover, the concentration inequality for the estimator is obtained whatever the behavior of the gradient of fn , hence a better inequality.

The paper is organized as follows. Section 2 is devoted to the estimation of the threshold t (p) and the level set L (t (p) ). In Section 3, we study the estimator of the number of clusters of L (t (p) ). Section 4, Section 5, and Section 6 are devoted to the proofs. Finally, several auxiliary results for the proofs are postponed in the Appendices, at the end of the paper.

2 Level set and threshold estimation

Notations

Let fn be an arbitrary nonparametric density estimate of f . For t ≥ 0, the t-upper level sets of f and fn will be denoted by L (t) and L n (t) respectively, i.e.,

L (t) = f ≥ t , and L n (t) = fn ≥ t .
Given a real number p in (0; 1), our first objective is to estimate a level t (p) ∈ R such that L (t (p) ) has µ-coverage equal to p, where µ is the law of X. To this aim, let H and H n be the functions defined for all t ≥ 0 respectively by

H(t) = P( f (X) ≤ t), H n (t) = 1 n n ∑ i=1 1 fn (X i ) ≤ t .
Next, for all p ∈ (0; 1), we define the (1p)-quantile of the law of f (X), i.e.

t (p) = inf{t ∈ R : H(t) ≥ 1 -p}, (2.1)
and its estimate based on the sample fn

(X 1 ), • • • , f (X n ): t (p) n = inf{t ∈ R : H n (t) ≥ 1 -p}. (2.2)
In comparison with the estimator of t (p) defined as a solution of (1.3), the estimate t

(p)
n is easily computed, by considering the order statistic induced by the sample fn (X 1 ),. . . , fn (X n ). Moreover, note that the set of discontinuities for H is at most countable, and that whenever H is continuous at t (p) , the two definitions (1.2) and

(2.1) coincide. In this case, we have µ(L (t (p) )) = p. We shall consider L n (t

(p) n ) as an estimate of L (t (p) ).
Whenever f is of class C 1 , we let T 0 be the subset of the range of f defined by

T 0 = t ∈ (0; sup R d f ) : inf { f =t} ∇ f = 0 .
This set naturally arises when considering the distribution of f (X). Indeed, the Implicit Function Theorem implies that T 0 contains the set of points in (0; sup R d f ) which charges the distribution of f (X). We shall assume throughout that the density f satisfies the following conditions.

Assumption 1 [on f ]

(i) The density f is of class C 2 with a bounded hessian matrix, and f (x) → 0 as x → ∞.

(ii) T 0 has Lebesgue content 0.

(iii) λ({ f = t}) = 0 for all t > 0.
Assumptions 1-(ii) and 1-(iii) are essentially imposed for the sake of the simplicity of the exposition, allowing the main results to be stated for almost all p ∈ (0; 1). By Assumption 1-(i), the upper t-level set L (t) is compact for all t > 0, as well as its boundary { f = t}. Assumption 1-(iii), which ensures the continuity of H, roughly means that each flat part of f has a null volume. Moreover, it is proved in Lemma A.1 that under Assumption 1-(i), we have T

0 = f (X ) \ {0; sup R d f }, where X = {∇ f = 0} is the set of critical points of f . Suppose in addition that f is of class C k , with k ≥ d.
Then, Sard's Theorem (see, e.g., [START_REF] Aubin | A course in Differential Geometry[END_REF] ensures that the Lebesgue measure of f (X ) is 0, hence implying Assumption 1-(ii).

Let us introduce some additional notations. We let . 2 and . ∞ be the L 2 (λ)and L ∞ (λ)-norms on functions respectively, and . be the the usual Euclidean norm. At last, H stands for the (d -1)-dimensional Hausdorff measure (see, e.g., [START_REF] Evans | Measure Theory and Fine Properties of Functions[END_REF]. Recall that H agrees with ordinary (d -1)dimensional surface area on nice sets.

The next subsection is devoted to the study of the asymptotic behavior of t

(p) n and L n (t (p) n ), when t (p)
n is defined by (2.2). The case of an arbitrary density estimate fn is considered first. Next, we specialize the result in the case where fn is a kernel density estimator.

Asymptotic behavior of t (p) n

Our first result provides a concentration inequality for t (p) n defined by (2.2) when fn is an arbitrary density estimate.

Theorem 2.1. Suppose that f satisfies Assumption 1. Then, for almost all p ∈ (0; 1) and for all η > 0, we have

P |t (p) n -t (p) | ≥ η ≤ P fn -f ∞ ≥ C 1 η +C 2 n 2 exp -nC 1 η 2 ,
where C 1 and C 2 are positive constants.

We now specialize the above result in the case where fn is a nonparametric kernel density estimate of f with kernel K and bandwidth sequence

(h n ) n , namely fn (x) = 1 nh d n n ∑ i=1 K x -X i h n . (2.
3)

The following assumptions on h n and K will be needed in the sequel.

Assumption 2a [on

h n ] nh d n log n → ∞, and nh d+4 n → 0. Assumption 3 [on K]
The kernel K is a density on R d with radial symmetry:

K(x) = Φ ( x ) ,
where Φ : R + → R + is a decreasing function with compact support.

Under Assumption 3 the class of functions

K x -. h : h > 0 ; x ∈ R d
has a polynomial discrimination (see, e.g., Pollard, 1984, Problem II.28, p. 42).

Then, sharp almost-sure convergence rates on fnf can be established (see, e.g., [START_REF] Giné | Rates of strong uniform consistency for multivariate kernel density estimators[END_REF]Guillou, 2002, Einmahl and[START_REF] Einmahl | Uniform in bandwidth consistency of kerneltype function estimators[END_REF]. More precisely, since E fn -

f ∞ = O(h 2 n ) under Assumption 1-(i)
, one deduces from the above-mentioned papers that, if Assumptions 2a and 3 also hold, then for all η > 0,

∑ n P v n fn -f ∞ ≥ η < ∞, (2.4)
where

(v n ) n is any sequence satisfying v n = o( nh d n / log n).
Combined with the concentration inequality in Theorem 2.1, we obtain the following corollary.

Corollary 2.2. Suppose that f satisfies Assumption 1. Let fn be the nonparametric kernel density estimate (2.3) satisfying Assumptions 2a and 3. Then, for almost all p ∈ (0; 1), we have

nh d n log n t (p) n -t (p) a.s -→ 0.
Even if the above result is non-optimal, it turns out to be enough for a cluster analysis, as showed in the next section.

Asymptotic behavior of

L n (t (p) n )
We shall need a slightly stronger assumption than Assumption 2a on the bandwidth sequence (h n ) n .

Assumption 2b [on

h n ] nh d n (log n) 16 → ∞, and nh d+4 n (log n) 2 → 0.
Under this set of conditions on the bandwidth sequence, one may apply the main result in [START_REF] Cadre | Kernel estimation of density level sets[END_REF].

The next result is an equivalent to Corollary 2.1 of [START_REF] Cadre | Kernel estimation of density level sets[END_REF], in which the estimate of t (p) is defined as a solution of (1.3). It shows that L n (t (p) n ) is consistent for the volume of the symmetric difference. Hence, this estimate can be used as a reliable basis for performing a cluster analysis in practice.

Theorem 2.3. Suppose that f satisfies Assumption 1 and that d ≥ 2. Let fn be the nonparametric kernel density estimate (2.3) satisfying Assumptions 2b and 3. Then, for almost all p ∈ (0; 1), we have

nh d n λ L n (t (p) n )∆L (t (p) ) P -→ 2 π K 2 t (p) { f =t (p) } 1 ∇ f dH .
The deterministic limit in the above theorem depends on the unknown density f . However, one can prove that if (α n ) n is a sequence of positive numbers tending to 0 and such that α 2 n nh d n /(log n) 2 → ∞, then, for almost all p ∈ (0; 1),

t (p) n α n λ L n (t (p) n ) \ L n (t (p) n + α n ) P -→ t (p) { f =t (p) } 1 ∇ f dH .
The proof of the above result is similar to the one of Lemma 4.6 in Cadre ( 2006), using our Corollary 2.2. Combined with Theorem 2.3, we then have, for almost all p ∈ (0; 1),

α n nh d n t (p) n λ L n (t (p) n ) \ L n (t (p) n + α n ) λ L n (t (p) n )∆L (t (p) ) P -→ 2 π K 2 ,
which yields a feasible way to estimate λ(L n (t (p) n )∆L (t (p) )). Remark 2.4. According to Proposition A.2 in Appendix A, on any interval I ⊂ (0; sup R d f ) with I ∩ T 0 = / 0, the random variable f (X) has a density on I, which is given by

g(t) = t { f =t} 1 ∇ f dH , t ∈ I.
Thus the normalized distance between L n (t (p) n ) and L (t (p) ) in Theorem 2.3 corresponds to the density g at point t (p) , up to a multiplicative constant.

3 Estimation of the number of clusters

Notations

As in the previous section, let us start with an arbitrary nonparametric density estimate fn of f . We first recall the methodology developed by [START_REF] Biau | A graph-based estimator of the number of clusters[END_REF] to estimate the number of clusters k(t) of L (t). Set J n (t) = j ≤ n : fn (X j ) ≥ t , i.e., {X j : j ∈ J n (t)} is the part of the n-sample lying in the t-level set of fn . Let (r n ) n be a sequence of positive numbers vanishing as n → ∞. Define the graph G n (t) with vertices {X j : j ∈ J n (t)} and where, for i, j ∈ J n (t), X i and X j are joined by an edge if and only if X i -X j ≤ r n . Then we set k n (t) as the number of connected components of the graph G n (t).

Under suitable assumptions, [START_REF] Biau | A graph-based estimator of the number of clusters[END_REF] prove that, with probability one, k n (t) = k(t) provided n is large enough. In our setting however, the threshold t (p) is unknown and has to be estimated. Hence, the main result in [START_REF] Biau | A graph-based estimator of the number of clusters[END_REF] may not be applied in order to estimate the number of clusters k(t (p) ).

Let t (p)

n be an arbitrary estimator of t (p) . In the next subsection, we state a concentration inequality for k n (t (p) n ). Then, we specialize this result to the case where fn is the kernel estimate (2.3) and t (p) n is given by (2.2).

Asymptotic behavior of k n (t (p) n )

In what follows, ω d denotes the volume of the Euclidean unit ball in R d .

Theorem 3.1. Suppose that f satisfies Assumption 1. Let (ε n ) n and (ε ′ n ) n be two sequences of positive numbers such that ε n + ε ′ n = o(r n ). For almost all p ∈ (0; 1), there exists a positive constant C, depending only on f and p, such that, if n is large enough,

P k n (t (p) n ) = k(t (p) ) ≤ 2 P fn -f ∞ > ε n + 2 P |t (p) n -t (p) | > ε ′ n +Cr -d n exp -t (p) ω d 4 d+1 nr d n .
In comparison with the result in [START_REF] Biau | A graph-based estimator of the number of clusters[END_REF] for a fixed threshold, the above concentration inequality does not require any assumption on the gradient of fn . As a consequence, when fn is a nonparametric kernel estimate for instance, the conditions imposed on the bandwidth are less restrictive. Now consider the particular case where fn is defined by (2.3) and t (p) n is defined by (2.2). Letting (v n ) n be a sequence such that v n = o( nh d n / log n), and choosing the sequences (r n ) n , (ε ′ n ) n and (ε n ) n in Theorem 3.1 so that ε n = ε ′ n = 1/v n and v n r n → ∞, we deduce the following from Theorem 3.1, Theorem 2.1 and (2.4).

Corollary 3.2. Suppose that f satisfies Assumption 1. Let fn be the kernel density estimate (2.3) satisfying Assumptions 2a and 3, and let t (p) n be the estimate of t (p) defined by (2.2). Then, for almost all p ∈ (0; 1), we have almost surely

k n (t (p) n ) = k(t (p) ),
provided n is large enough.

4 Proof of Theorem 2.1: convergence of t (p) n

Auxiliary results

We shall assume throughout this subsection that Assumptions 1, 2a and 3 hold.

First note that under Assumption 1, H is a bijection from (0; sup R d f ) to (0; 1). Indeed, Assumption 1-(iii) implies that H is a continuous function. Moreover, under Assumption 1-(i), H is increasing: for suppose it were not, then for some t ≥ 0 and some ε > 0,

0 = H(t + ε) -H(t) = {t< f ≤t+ε} f dλ,
which is impossible, because λ({t < f < t + ε}) > 0. Then we denote by G the inverse of H restricted to (0; sup R d f ).

Lemma 4.1. The function G is almost everywhere differentiable.

Proof. As stated above, H is increasing. Hence, by the Lebesgue derivation Theorem, for almost all t, H is differentiable with derivative H ′ (t) > 0. Thus, G is almost everywhere differentiable.

The Levy metric d L between any real-valued functions ϕ 1 , ϕ 2 on R is defined by

d L (ϕ 1 , ϕ 2 ) = inf θ > 0 : ∀x ∈ R, ϕ 1 (x -θ) -θ ≤ ϕ 2 (x) ≤ ϕ 1 (x + θ) + θ ,
(see, e.g., Billingsley, 1995, 14.5). Recall that convergence in distribution is equivalent to convergence of the underlying distribution functions for the metric d L .

Lemma 4.2. Let x 0 be a real number, and let ϕ 1 be an increasing function with a derivative at point x 0 . There exists C > 0 such that, for any increasing function

ϕ 2 with d L (ϕ 1 , ϕ 2 ) ≤ 1, |ϕ 1 (x 0 ) -ϕ 2 (x 0 )| ≤ Cd L (ϕ 1 , ϕ 2 ).
Proof. Let θ be any positive number such that, for all x ∈ R,

ϕ 1 (x -θ) -θ ≤ ϕ 2 (x) ≤ ϕ 1 (x + θ) + θ. (4.1)
Since ϕ 1 is differentiable at x 0 ,

ϕ 1 (x 0 ± θ) = ϕ 1 (x 0 ) ± θϕ ′ 1 (x 0 ) + θψ ± (θ) (4.2)
where each function ψ ± satisfies ψ ± (θ) → 0 when θ → 0 + . Using (4.1) and ( 4.2), we obtain

-θ(ϕ ′ 1 (x 0 ) + 1) + θψ -(θ) ≤ ϕ 2 (x 0 ) -ϕ 1 (x 0 ) ≤ θ(ϕ ′ 1 (x 0 ) + 1) + θψ + (θ).
Taking the infimum over θ satisfying (4.1) gives the announced result with any C such that, for all δ ≤ 1,

ϕ ′ 1 (x 0 ) + 1 + max |ψ -(δ)|, |ψ + (δ)| ≤ C.
Let L ℓ (t) denote the lower t-level set of the unknown density f , i.e., L ℓ (t) = {x ∈ R d : f (x) ≤ t}. Moreover, we set

V n = sup t≥0 µ n L ℓ (t) -µ L ℓ (t) , (4.3)
where µ n = 1 n ∑ n i=1 δ X i is the empirical measure indexed by the sample, δ x denoting the Dirac measure at point x. The next lemma borrows elements from the Vapnik-Chervonenkis theory; we refer the reader to [START_REF] Devroye | A Probabilistic Theory of Pattern Recognition[END_REF] for materials on the subject. Lemma 4.3. There exists a constant C such that, for all η > 0, we have

P (V n ≥ η) ≤ Cn 2 exp -nη 2 /32 .
Proof. Let A be the collection of lower level sets, namely

A = {L ℓ (t),t ≥ 0}.
Observe that the Vapnik-Chervonenkis dimension of A is 2. Then, by the Vapnik-Chervonenkis inequality (see, e.g., Devroye et al., 1996, Theorem 12.5), we obtain the stated result.

Proof of Theorem 2.1

We first proceed to bound d L (H, H n ). We have H n (t) = µ n L ℓ n (t) , and

H(t) = µ L ℓ (t) where L ℓ n (t) = {x ∈ R d : fn (x) ≤ t} and L ℓ (t) = {x ∈ R d : f (x) ≤ t}.

The triangular inequality gives

L ℓ t -fn -f ∞ ⊂ L ℓ n (t) ⊂ L ℓ t + fn -f ∞ ,
which, applying µ n , yields

µ n L ℓ t -fn -f ∞ ≤ H n (t) ≤ µ n L ℓ t + fn -f ∞ .
Moreover, by definition of V n in (4.3), we have

H(s) -V n ≤ µ n L ℓ (s) ≤ H(s) +V n ,
for all real number s. The two last inequalities give

H(t -fn -f ∞ ) -V n ≤ H n (t) ≤ H(t + fn -f ∞ ) +V n .
Using the fact that H is non-decreasing, we obtain

d L (H, H n ) ≤ max fn -f ∞ ,V n . (4.4)
By Lemma 4.1, G is almost everywhere differentiable. Let us fix p ∈ (0; 1) such that G is differentiable at 1p, and observe that G(1p) = t (p) . Denote by G n the pseudo-inverse of H n , i.e.

G n (s) = inf{t ≥ 0 : H n (t) ≥ s}, and remark that G n (1 -p) = t (p) n . Moreover, we always have d L (H, H n ) ≤ 1 because 0 ≤ H(t) ≤ 1 and 0 ≤ H n (t) ≤ 1 for all t ∈ R. Hence, since d L (H, H n ) = d L (G, G n ), we obtain from Lemma 4.2 that for some constant C, t (p) n -t (p) = |G n (1 -p) -G(1 -p)| ≤ Cd L (H, H n ) .
Theorem 2.1 is now a straightforward consequence of (4.4) and Lemma 4.3.

5 Proof of Theorem 2.3: convergence of L n (t (p) n )

Auxiliary results

We shall assume throughout this subsection that Assumptions 1, 2b and 3 hold.

Lemma 5.1. For almost all p ∈ (0; 1), we have

(i) (log n) × λ L n (t (p) )∆L (t (p) ) P -→ 0 and (ii) (log n) × λ L n (t (p) n )∆L (t (p) ) P -→ 0.
Proof. We only prove (ii). Set ε n = log n/ nh d n , which vanishes under Assumption 2b. Moreover, let N 1 , N 2 be defined as

N c 1 = p ∈ (0; 1) : lim ε→0 1 ε λ t (p) -ε ≤ f ≤ t (p) + ε exists ; N c 2 = p ∈ (0; 1) : 1 ε n |t (p) n -t (p) | a.s.
-→ 0 .

Both N 1 and N 2 have a null Lebesgue measure: the first property is a consequence of the Lebesgue derivation Theorem and the fact that H is a bijection from (0; sup R d f ) onto (0; 1). The second one is a direct consequence of Theorem 2.1.

Hence, one only needs to prove the lemma for all p ∈ N c 1 ∩ N c 2 . We now fix p in this set, and we denote by Ω n the event

Ω n = { fn -f ∞ ≤ ε n } ∩ {|t (p) n -t (p) | ≤ ε n }.
Since P(Ω n ) → 1 by (2.4), it suffices to show that the stated convergence holds on the event Ω n . Simple calculations yields

λ L n (t (p) n )∆L (t (p) ) = λ fn ≥ t (p) n ; f < t (p) + λ fn < t (p) n ; f ≥ t (p) .
But, on the event Ω n , we have fn

+ε n ≥ f ≥ fn -ε n and t (p) n -ε n ≤ t (p) ≤ t (p) n +ε n . Consequently, if n is large enough, λ L n (t (p) n )∆L (t (p) ) ≤ λ t (p) -2ε n ≤ f < t (p) + λ t (p) ≤ f ≤ t (p) + 2ε n = λ t (p) -2ε n ≤ f ≤ t (p) + 2ε n ≤ Cε n , for some constant C, because p ∈ N c
1 and (ε n ) n vanishes. The last inequality proves the lemma, since by Assumption 2b, ε n log n → 0.

In the sequel, μn denotes the smoothed empirical measure, which is the random measure with density fn , defined for all Borel set A ⊂ R d by μn (A) = A fn dλ.

Lemma 5.2. For almost all p ∈ (0; 1),

(i) nh d n μn (L n (t (p) )) -µ(L n (t (p) )) P -→ 0 and (ii) nh d n μn (L n (t (p) n )) -µ(L n (t (p) n )) P -→ 0.
Proof. We only prove (ii). Fix p ∈ (0; 1) such that the result in Lemma 5.1 holds. Observe that

μn (L n (t (p) n )) -µ(L n (t (p) n )) ≤ L n (t (p) n )∆L (t (p) ) | fn -f |dλ + L (t (p) ) ( fn -f )dλ ≤ λ L n (t (p) n )∆L (t (p) ) fn -f ∞ + L (t (p) ) ( fn -f )dλ . (5.1)
Recall that K is a radial function with compact support. Since nh d+4 n → 0 and L (t (p) ) is compact for all p ∈ (0; 1), it is a classical exercise to prove that for all p ∈ (0; 1),

nh d n L (t (p) ) ( fn -f )dλ P -→ 0.
(5.2) (see, e.g., Cadre, 2006, Lemma 4.2). Moreover, by (2.4) and Lemma 5.1,

nh d n λ L n (t (p) n )∆L (t (p) ) fn -f ∞ P -→ 0.
(5.

3)

The inequalities (5.1), (5.2) and (5.3) prove the assertion of the lemma.

Lemma 5.3. For almost all p ∈ (0; 1),

nh d n µ L n (t (p) n ) -µ L (t (p) ) P -→ 0.
Proof. Let ε n = log n/ nh d n and N be the set defined by

N c = p ∈ (0; 1) : 1 ε n |t (p) n -t (p) | a.s.
-→ 0 .

By Corollary 2.2, N has a null Lebesgue measure. If p ∈ N c , then almost surely, we have

t (p) -ε n ≤ t (p) n ≤ t (p) + ε n for large enough n. Hence, L n (t (p) + ε n ) ⊂ L n (t (p) n ) ⊂ L n (t (p) -ε n ).
Consequently, one only needs to prove that for almost all p ∈ N c , the two results above hold:

nh d n µ L n (t (p) ± ε n ) -µ L (t (p) ) P -→ 0.
(5.4)

For the sake of simplicity, we only prove the "+" part of (5.4).

One can follow the arguments of the proofs of Propositions 3.1 and 3.2 in [START_REF] Cadre | Kernel estimation of density level sets[END_REF], to obtain that for almost all p ∈ N c , there exists J = J(p) with

nh d n µ L n (t (p) + ε n ) ∩ V n P -→ J and nh d n µ L n (t (p) + ε n ) c ∩ Vn P -→ J,
where we set

V n = t (p) -ε n ≤ f < t (p) and Vn = t (p) ≤ f < t (p) + 3ε n . Thus, for almost all p ∈ N c nh d n µ L n (t (p) + ε n ) ∩ V n -µ L n (t (p) + ε n ) c ∩ Vn P -→ 0.
(5.5)

Now let p ∈ N c satisfying the above result, and set Ω n = { fnf ∞ ≤ 2ε n }. By (2.4), P(Ω n ) → 1 hence one only needs to prove that the result holds on the event

Ω n . But, on Ω n , µ L n (t (p) + ε n ) -µ L (t (p) ) = µ fn ≥ t (p) + ε n ; f < t (p) -µ fn < t (p) + ε n ; f ≥ t (p) = µ L n (t (p) + ε n ) ∩ V n -µ L n (t (p) + ε n ) c ∩ Vn .
Consequently, by (5.5), we have on

Ω n nh d n µ L n (t (p) + ε n ) -µ L (t (p) ) P -→ 0.
This proves the "+" part of (5.4). The "-" part is obtained with similar arguments.

Proof of Theorem 2.3

Let t 0 ∈ T c 0 . Since f is of class C 2 , there exists an open set I(t 0 ) containing t 0 such that inf

{ f ∈I(t 0 )} ∇ f > 0.
Thus, by Theorem 2.1 in [START_REF] Cadre | Kernel estimation of density level sets[END_REF], we have, for almost all t ∈ I(t 0 ),

nh d n λ (L n (t)∆L (t)) P -→ 2 π K 2 t { f =t} 1 ∇ f dH .
Recalling now that the Lebesgue measure of T 0 is 0, and that H is a bijection from (0; sup R d f ) onto (0; 1), it follows that the above result remains true for almost all p ∈ (0; 1), with t (p) instead of t. As a consequence, one only needs to prove that for almost all p ∈ (0; 1), nh d n D n (p) → 0 in probability, where p) ) .

D n (p) = λ L n (t (p) n )∆L (t (p) ) -λ L n (t (p) )∆L (t (
After some calculations, D n (p) may be expressed as

D n (p) = R d 1{t (p) n ≤ fn < t (p) }g dλ - R d 1{t (p) ≤ fn < t (p)
n }g dλ, where g = 1 -21{ f ≥ t (p) }. For simplicity, we assume that 0 < t (p) n ≤ t (p) . Recall that μn is the random measure with density fn . Thus,

D n (p) ≤ λ t (p) n ≤ fn < t (p) ≤ 1 t (p) n μn t (p)
n ≤ fn < t (p) .

The factor 1/t

(p)
n in the right-hand side of the last inequality might be asymptotically bounded by some constant C, using Corollary 2.2. Hence, for all n large enough, and for almost all p ∈ (0; 1), p) ) .

D n (p) ≤ C μn L n (t (p) n ) -μn L n (t (
(5.6)

The right-hand term in (5.6) may be bounded from above by μn L n (t

(p) n ) -μn L n (t (p) ) ≤ μn L n (t (p) n ) -µ L n (t (p) n ) + µ L n (t (p) n ) -µ L (t (p) ) + µ L (t (p) ) -μn L n (t (p) ) .
By Lemma 5.2 and Lemma 5.3, we obtain, for almost all p ∈ (0; 1),

nh d n μn L n (t (p) n ) -μn L n (t (p) ) P -→ 0,
which, according to (5.6), gives the stated result.

6 Proof of Theorem 3.1: convergence of k n (t (p) n )

Preliminaries

We assume in the whole section that Assumption 1 holds. Since H is a bijection from (0; sup R d f ) onto (0; 1) and since the Lebesgue measure of T 0 is 0, one only needs to prove Theorem 3.1 for each probability p ∈ (0; 1) such that t (p) / ∈ T 0 . Now fix such a probability p. Because f is of class C 2 , there exists a closed interval I ⊂ (0; +∞) such that t (p) is in the interior of I, and inf { f ∈I} ∇ f > 0. For ease of notation, we now set

k (p) n = k n (t (p) n ), k (p) = k(t (p) ), J n = J n (t (p) n ), and G n = G n (t (p) n ).
Proof. In the proof, a graph is denoted as (set of vertices, set of edges).

On

V ′ n = {X j : j ∈ J ′ n }, the graph G n = (V n , E n ) induces the subgraph G ′ n = (V ′ n , E ′ n ).
We first proceed to prove that G ′ n has exactly k (p) connected components on Γ ′ n . To this aim, observe first that

J ′ n ⊂ { j ≤ n : f (X j ) ≥ t (p) + ε n + ε ′ n }, (6.2) by definition of J ′ n , provided fn -f ∞ ≤ ε n and |t (p) n -t (p) | ≤ ε ′ n . Conversely, j ≤ n is such that (X j ) ≥ t (p) + ε n + ε ′ n and if |t (p) n -t (p) | ≤ ε ′ n , then fn (X j ) ≥ f (X j ) -fn -f ∞ ≥ f (X j ) -ε n ≥ t (p) + ε ′ n ≥ t (p) n . Hence, if |t (p) n -t (p) | ≤ ε ′ n J ′ n ⊃ { j ≤ n : f (X j ) ≥ t (p) + ε n + ε ′ n }. (6.3)
This shows that the two sets in (6.2) and ( 6.3) are in fact equal as soon as fn -

f ∞ ≤ ε n and |t (p) n -t (p) | ≤ ε ′ n , i.e., J ′ n = { j ≤ n : f (X j ) ≥ t (p) + ε n + ε ′ n }. (6.4)
In particular, on Γ ′ n , we have

V ′ n = V n ∩ L (t (p) + ε n + ε ′ n ). (6.5)
We are now ready to prove that k

′ n = k (p) . Since n is large enough, L (t (p) + ε n + ε ′ n ) has exactly k (p)
+ ε n + ε ′ n ). First, if X i and X j belong to different connected components of L (t (p) + ε n + ε ′ n ), then necessarily, X i -X j ≥ D min .
Since n is large enough, we have r n < D min , and so no edge of G ′ n connects X i to X j . Second, if X i and X j belong to the same connected component of L (t (p) + ε n + ε ′ n ), then on Γ ′ n , they are contained in some balls of P n . If they happen to lie in the same ball, then X i -X j ≤ r n /2 and so they are connected by an edge in G ′ n . Otherwise, there exists a path of edges in G ′ n connecting X i to X j , and so they belong to the same connected component of G ′ n . This follows from the fact that, whenever n is large enough, the union of the balls of P n has the same topology as L (t (p)

+ ε n + ε ′ n ); in particular, their Lemma A.1. We have f (X ) \ {0; sup R d f } = T 0 . Proof. Let x ∈ X . If f (x) = 0 or f (x) = sup R d f
, then obviously f (x) ∈ T 0 and hence, f (X ) \ {0; sup R d f } ⊂ T 0 . Conversely, T 0 ⊂ f (X ) by continuity of ∇ f and because the set { f = t} is compact whenever t = 0.

The next proposition describes the absolutely continuous part of the random variable f (X).

Proposition A.2. Let I be a compact interval of R ⋆ + such that I ∩ T 0 = / 0. Then, the random variable f (X) has a density g on I, which is given by

g(t) = t { f =t} 1 ∇ f dH , t ∈ I.
Proof. Since { f ∈ I} is compact and { f ∈ I} ∩ {∇ f = 0} = / 0, we have inf

{ f ∈I} ∇ f > 0.
Now, let J be any interval included in I. Observe that f is a locally Lipschitz function and that 1{ f ∈ J} is integrable. According to Theorem 2, Chapter 3 in [START_REF] Evans | Measure Theory and Fine Properties of Functions[END_REF],

P ( f (X) ∈ J) = { f ∈J} f dλ = J { f =s} f ∇ f dH ds,
hence the lemma.

B Auxiliary results on L (t)

In this Appendix, we only presume that Assumption 1-(i) holds. We denote by I a closed and non-empty interval such that inf

{ f ∈I} ∇ f > 0.
The following lemma, stated without proof, is a consequence of Theorem 3.1 in [START_REF] Milnor | Morse Theory[END_REF] p.12 and Theorem 5.2.1 in [START_REF] Jost | Riemannian Geometry and Geometric Analysis[END_REF] p.176; see also Lemma A.1 in [START_REF] Pelletier | Strong consistency of spectral clustering on level sets[END_REF]. Recall that for all t ≥ 0, (t), where the C ℓ (t)'s denote the connected components of L (t).

L (t) = C 1 (t) ∪ • • • ∪ C k(t)
Lemma B.1. There exists a one-parameter group of diffeomorphisms (ϕ u ) u∈R such that for all s,t in I, ϕ t-s is a diffeomorphism from L (s) onto L (t). Consequently, for all s,t in I, (i) L (s) and L (t) have the same number of connected components;

(ii) C ℓ (t) ⊂ C ℓ (s) whenever s < t and 1 ≤ ℓ ≤ k(t);

(iii) C ℓ (t) = ϕ t-s (C ℓ (s)) whenever 1 ≤ ℓ ≤ k(t). Proof. (i) If x is in the interior of L (t), then x is in the interior of L (t + δ 0 ) for some δ 0 . Thus, for some r 0 > 0, B(x, r 0 ) ⊂ L (t + δ 0 ). We can assume that r 0 < 1. Then, if δ < δ 0 and r < r 0 , B(x, r) ⊂ B(x, r 0 ) ⊂ L (t + δ 0 ) ⊂ L (t + rδ).

Hence, for such a pair (r, δ), B(x, r) ∩ L (t + rδ) = B(x, r), which gives the result.

(ii) Let x be an element of the boundary of L (t), and denote by H 1/r the homothety with center x and similitude ratio 1/r. For r, δ > 0, we have where A r,δ = H 1/r B(x, r) ∩ L (t + rδ) . We claim that as (r, δ) → (0, 0), the indicator function of the set A r,δ converges toward the indicator function of the set

A 0,0 = ξ ∈ R d : ξ -x ≤ 1, ∇ f (x) • (ξ -x) > 0 .
Observe that H 1/r B(x, r) = B(x, 1), and fix ξ ∈ B(x, 1). Then, ξ is in H 1/r L (t + rδ) if and only if f x + r(ξx) ≥ t + rδ. Moreover, f x + r(ξx) = t +

Lemma B. 2 .

 2 Let t ∈ I, and fix x ∈ L (t).

  (i) If x is in the interior of L (t), then lim (δ,r)→(0,0) r -d λ (B(x, r) ∩ L (t + rδ)) = ω d . (ii) If x is on the boundary of (t), then lim (δ,r)→(0,0) r -d λ (B(x, r) ∩ L (t + rδ)) = ω d 2 .

1

  r d λ (B(x, r) ∩ L (t + rδ)) = λ A r,δ (B.1)

  connected components by Lemma B.1. Hence, one only needs to prove that any pair of vertices X i and X j of G ′ n is linked by an edge of E ′

n if and only if both vertices lie in the same connected components of L (t

(p)

In what follows, B(x, r) stands for the Euclidean closed ball centered at x ∈ R d with radius r.

Let P n be a finite covering of L (t (p) + ε n + ε ′ n ) by closed balls B(x, r n /4) with centers at x ∈ L (t (p) + ε n + ε ′ n ), constructed in such a way that

for some positive constant C 1 . Let J ′ n be the subset of J n defined by

Define the event Γ n on which every ball of the covering P n contains at least one point X j with j ∈ J ′ n , i.e.,

Finally, we set

In the sequel, the statement "n is large enough" means that n satisfies the three following conditions:

, where α and β are given by Proposition B.3 and Proposition B.4 respectively,

In condition (iii) above, D min denotes the smallest distance between two different connected components of L (min I). By Lemma B.1, each level set L (t) has exactly k (p) connected components, provided t ∈ I. Hence,

where for all t, the C ℓ (t)'s denote the connected components of L (t).

Lemma 6.1. Assume that n is large enough. Then, on Γ ′ n , k

number of connected components are equal. As a consequence, since on Γ n , each ball of

And, if we decompose G ′ n into its connected components

we have also obtained that

0 by Proposition B.4, so that B(X j , r n /4) intersects some ball of the covering P n . This proves that any vertex of V ′′ n is connected by an edge of G n to at least one vertex in

n is smaller than the number of connected components of G ′ n , which is equal to k (p) by (6.6). But since n is large enough, so that r n < D min , each vertex in V ′′ n cannot be connected simultaneously to different components of G ′ n by (6.7) and (6.8). Therefore k

6.2 Proof of Theorem 3.1

By Lemma 6.1, provided n is large enough,

We assume in this section that n is large enough, so that the set of assumptions on n of the preliminaries holds. If we set

(6.9)

21

We now proceed to bound P(Γ c n ). First observe that

, and recall that by (6.4), J ′ n and Jn coincide on Γ ′′ n . Then, for all A ∈ P n ,

By Proposition B.3, for any closed ball A centered at x in L (t (p) ) with radius r n /4, we have

(6.12) With (6.1), (6.10), (6.11) and (6.12), we deduce that

According to (6.9) and the inequality 1u ≤ exp(-u) for all u ∈ R, we obtain

This concludes the proof.

A Auxiliary results on f and H

In this Appendix, we only presume that Assumption 1-(i) holds. Recall that X is the subset of R d composed of the critical points of f , i.e.,

The following lemma characterizes the set T 0 .

r∇ f (x) • (ξx) + o(r) when r → 0. Recalling that ∇ f (x) = 0, this gives, for any ξ ∈ R d , lim

By Lemma B.2, ψ is bounded from below by some constant C < ω d /2 on V ∩ U, where V is some open neighborhood of L (t) × (0, 0). Since dist(L (t) × (0, 0), R d+3 \V ) > 0, as a distance between two disjoint closed sets, one of them being compact, the result is proved.

The proof of the next result is left to the reader, since it can be obtained by adapting the proofs of Lemma B.2 and Proposition B.3. Proposition B.4. There exists β > 0 such that, if r 2 + δ 2 < β 2 , then, for all (t, x) such that t ∈ I and x ∈ L (trδ), the closed ball B(x, r/4) intersects L (t + rδ).


Yaroub Elloumi
email: yaroub.elloumi@esiee.fr

Cataract Grading Method Based on Deep Convolutional Neural Networks and Stacking Ensemble Learning

Keywords: Fundus images, Cataract, Deep learning, Ensemble learning

Purpose The cataract is the most common cause of severe vision impairment or blindness worldwide. It is essential to periodically diagnose the retina in order to prevent cataract severity, and so to enhance the life quality of cataract-affected patients. Cataract grading through a fundus image is feasible with higher accuracy. However, a delay of early cataract screening is registered caused by deficiency of ophthalmologists and imaging devices. The challenge is to propose a CAD system to grade the cataract from retinal images.

Method In this paper, an ensemble learning framework for cataract grading is put forward, where three convolutional deep neural networks are stacked in order to provide higher performance grading. The main contributions of this work are given as follows: (1) Preprocessing and data augmentation of fundus images are performed to ensure the robustness of the cataract grading; (2) The well-known DL architectures (Inception-V3, MobileNet-V2 and NasNet-Mobile) are fine-tuned and learned as base classifiers; (3) A stacking method is propounded to combine the features of base classifiers.

Results

The evaluation is conducted using a dataset of 590 fundus images selected from two public databases. The suggested framework achieves 93.97% accuracy, 95.59% sensitivity, 91.67% specificity, 94.20% precision and 94.89% F-measure for cataract grading.

Conclusion

The proposed framework successfully grades fundus images into cataract severity. Moreover, stacking ensemble learning allows achieving a performance that significantly surpasses the ones realized by each DL architecture, applied separately.

Introduction

The eye lens is a circular and transparent component, located beyond the iris, which allows refracting the visual information into the retina [1]. The cataract is an ocular disease that affects the eye lens, hence becoming cloudy. As reported by the world health organization in 2020, the cataract caused 65.2 million people suffering from moderate or severe vision impairment or blindness, which exceeds widely the persons having the same severity caused by glaucoma, diabetic retinopathy, corneal opacities and trachoma all together. More than 50% of worldwide blindness cases are caused by the cataract where the number of blind people will exceed 40 million in 2025 [START_REF] Zhou | Automatic Cataract Classification Using Deep Neural Network With Discrete State Transition[END_REF][START_REF] Cao | Hierarchical method for cataract grading based on retinal images using improved Haar wavelet[END_REF]. The cataract is involved due to smoking, genetics, alcohol, nutritional or metabolism disorders, medications or a long exposure to sunlight [START_REF] Agarwal | Mobile Application Based Cataract Detection System[END_REF][START_REF] Song | An Improved Semi-Supervised Learning Method on Cataract Fundus Image Classification[END_REF][START_REF] Sigit | Classification of Cataract Slit-Lamp Image Based on Machine Learning[END_REF]. In addition, it is correlated with several other pathologies [1,[START_REF] Agarwal | Mobile Application Based Cataract Detection System[END_REF].

The cataract can be diagnosed through the slit-lamp photography, which consists in reflecting light inside the lens structure and deducing the cataract disease from the non-uniform illumination of the refracted intensity. The Lens Opacities Classification System III, the American Cooperative Cataract Research Group protocol and the Oxford Clinical Cataract Classification are the wellknown protocols for classifying the cataract disease, which are based on convoluted procedures and require well-experienced ophthalmologists [START_REF] Zhou | Automatic Cataract Classification Using Deep Neural Network With Discrete State Transition[END_REF][START_REF] Cao | Hierarchical method for cataract grading based on retinal images using improved Haar wavelet[END_REF]. The ultrasound backscattering signal, the optical coherence tomography and the ultrasound biomicroscope are also used for cataract screening. However, their diagnosis processes are costly and based on complex operations. The cataract is easily diagnosed through the ophthalmoscopy where the blurriness of the retina components is similar to the quality of the visual acuity. Cataract screening and grading through a retinal fundus image is feasible with higher accuracy, even with inexperienced graders [START_REF] Zhou | Automatic Cataract Classification Using Deep Neural Network With Discrete State Transition[END_REF][START_REF] Cao | Hierarchical method for cataract grading based on retinal images using improved Haar wavelet[END_REF], which promotes accessing to eye diagnosis.

While the cataract develops slowly, it is hard to recognize it since its symptoms are similar to a poor visual acuity [START_REF] Cao | Hierarchical method for cataract grading based on retinal images using improved Haar wavelet[END_REF][START_REF] Agarwal | Mobile Application Based Cataract Detection System[END_REF]. Indeed, it is always diagnosed through a visual acuity test. This pathology imposes limiting activities and affects the life quality [1,[START_REF] Agarwal | Mobile Application Based Cataract Detection System[END_REF]. Therefore, it is recommended to make a periodical diagnosis in order to enhance the life quality of cataract-affected patients [START_REF] Cao | Hierarchical method for cataract grading based on retinal images using improved Haar wavelet[END_REF]. However, the process of cataract diagnosis is time-consuming and leads to an important workload, especially with the higher rate of requested diagnosis. In addition, an alarming worldwide deficiency of ophthalmologists is registered [START_REF] Wong | Global prevalence of age-related macular degeneration and disease burden projection for 2020 and 2040: a systematic review and meta-analysis[END_REF]. The lack of ophthalmologists will be aggravated in future years and associated to an augmentation of elderly people. Consequently, a delay is registered to ensures early cataract diagnosis, and will persist in the years ahead.

Therefore, several studies have focused on proposing a Computer-Aided-Diagnosis (CAD) system for cataract detection and grading. The common challenge is to succeed in detecting and grading the cataract in terms of opacities. For this purpose, several methods have been suggested which have realized optimal performances of cataract detection. However, they have failed to achieve a higher performance of cataract grading, even resorting to Ensemble Learning (EL). Elsewhere, those methods have been always evaluated using private datasets where a lack a public database containing cataract-affected fundus images has been noticed, in contrast to the other ocular pathologies such as diabetic retinopathy and glaucoma. The evaluation processes have been always performed using few hundreds of fundus images, inadequately partitioned in terms of grading, which might not guarantee a reliable evaluation. Therefore, our main objective is to provide a CAD system for cataract grading. The first challenge consists in ensuring higher accuracy when classifying images into cataract stages. In fact, several Deep Learning (DL) architectures are dedicated for the classification problem, which are varied in terms of processing principles, and so in terms of classification results. Our main idea consists in stacking features of DL architectures in order to provide higher performance grading. The second challenge is to ensure such grading even using a small dataset. To address this problem, we firstly select three DL architectures composed by lightweight convolutional blocks that allow converging the trained model without resorting to a large number of images. Secondly, we fine-tune those DL architectures initially trained using the ImageNet dataset. In addition, we perform a well-off data augmentation to diversify retina modelling.

The remainder of this paper is organized as follows. Section 2 presents a literature review about the cataractaffected retinal images and the existing methods for grading. Section 3 details the learning framework of cataract grading. Section 4 describes the experimentation and the discussion of the provided results. Section 5 presents the conclusion and some future work.

Literature review

The cataract occurs when proteins are accumulated in different locations of the lens [START_REF] Zhou | Automatic Cataract Classification Using Deep Neural Network With Discrete State Transition[END_REF][START_REF] Sigit | Classification of Cataract Slit-Lamp Image Based on Machine Learning[END_REF] which becomes cloudy. In terms of severity, vision becomes blurred with distressed colors and troubled with bright light [START_REF] Agarwal | Mobile Application Based Cataract Detection System[END_REF][START_REF] Sigit | Classification of Cataract Slit-Lamp Image Based on Machine Learning[END_REF], which corresponds to vision impairment. If the lens is totally clouded, the light is disrupted when projected on the retina, which avoids reflecting real images, and hence leading to blindness [START_REF] Guo | A computer-aided healthcare system for cataract classification and grading based on fundus image analysis[END_REF][START_REF] Agarwal | Mobile Application Based Cataract Detection System[END_REF]. Respectively, fundus images are graded into non-cataract, mild, moderate or severe cataract, as shown in Fig. 1. Before the cataract disease, all retinal components, even the micro-vascular structures, are explicitly illustrated. In the mild stage, only choroid and capillary vessels cannot be distinguished among the other components. The moderate stage is deduced when only the optic disk and the main vessels are visible. It becomes difficult to observe any retinal structures in the severe stage. Some work has been focused on proposing a CAD system for cataract detection and grading. Recently, the proposed methods have been based on machine learning, while varied in terms of types and principles. Some methods have employed single-machine learning for grading. In the work suggested in [START_REF] Guo | A computer-aided healthcare system for cataract classification and grading based on fundus image analysis[END_REF], features were extracted, whose processing was based on the sketch method with discrete cosine transforms and discrete wavelet transforms. After that, a multiclass discriminant analysis algorithm was used for cataract classification. The dataset was composed by 445 fundus images where 199 are non-cataract and 148, 71 and 27 are in mild, moderate and severe cataract stages. The classification performance of cataract detection and grading was 90.9% and 77.1%, respectively. In [START_REF] Dong | Classification of cataract fundus image based on deep learning[END_REF], the suggested method consisted in performing a DL architecture to provide a feature map that was input to an SVM classifier so as to grade images. The DL architecture was composed successively by a convolution layer with a filter kernel size of 11, an activation layer with a ReLU function, a pooling layer, and a normalization layer. A dataset of 7,851 fundus images was used for training, where an accuracy rate of 90.82% was achieved. The method put forward in [START_REF] Xiong | An Approach to Evaluate Blurriness in Retinal Images with Vitreous Opacity for Cataract Diagnosis[END_REF] extracted features reflecting visible structures, local standard deviation and contrast of vessels against background. The extracted features allowed to train a decision tree that ensured a four-class grading with 83.8% of accuracy. The method suggested in [START_REF] Zhou | Automatic Cataract Classification Using Deep Neural Network With Discrete State Transition[END_REF] figured out the cataract grading using a deep neural network. Firstly, improved Haar wavelet features and retinal structure features were extracted from the retinal image. Then, the feature vector was input to a Multi-Layer Perceptron (MLP) classifier with an exponential discrete state transition. The accuracy of cataract detection and grading is respectively 92.85% and 89.23% using a dataset of 1,355 retinal images. In [START_REF] Imran | Cataract Detection and Grading with Retinal Images Using SOM-RBF Neural Network[END_REF], green channels of retinal images were preprocessed by applying the histogram equalization and the top-bottom hat transformation. Then, the wavelet and texture features were extracted. The classification was done using Self Organizing Maps (SOM) and a Radial Basis Function (RBF) neural network to process clustering, where 91.7% of accuracy for cataract grading was achieved.

Several methods have exploited multiple machine learning algorithms to perform a higher performance of cataract detection and grading. In [START_REF] Song | An Improved Semi-Supervised Learning Method on Cataract Fundus Image Classification[END_REF], The eigenvalues were calculated from the gray-level and gray-gradiant cooccurrence matrices to be used as texture features. In addition, the discrete wavelet transform and the discrete cosine transform were performed to provide wavelet and sketch features. Thereafter, The Support Vector Machine (SVM) and the Logistic Regression (LR) were used for classification. Each machine learning algorithm was performed several times as a binary classifier following the one-Vs-rest principle. Then, binary classifiers of each machine learning algorithm were combined to perform the four types of grading classification. A dataset of 2,000 retinal images was used, which was split equitably between training and testing steps. The ensemble classifier based on the SVM outperformed the LR one where the accuracy of the four-stage cataract grading was about 88.60%. The work described in [START_REF] Yang | Exploiting ensemble learning for automatic cataract detection and grading[END_REF] consisted in extracting independent wavelet, sketch and texture based features from preprocessed retinal images. The SVM and Back Propagation Neural Network (BPNN) were used as based classifiers, where their results were provided to EL to perform the final classification. The method was evaluated using 1,239 fundus images, where their numbers in terms of increased severity were 767, 246, 128, and 98. The experimentation showed that EL achieved a better performance than each single learning model, where the realized accuracy was 93.2% and 84.5 % for cataract detection and grading, respectively. In [START_REF] Cao | Hierarchical method for cataract grading based on retinal images using improved Haar wavelet[END_REF], the Haar wavelet decomposition was used to extract features that reflected how the retinal components were modelled in the fundus image. The optimal features were selected through a BPNN classifier. Thereafter, three types of adjacent two-class classification based on majority voting were performed to ensure the four-class classification problem of the cataract. The experimentation was performed using 1,355 retinal images, where the accuracy of the two-class and four-class classification was respectively 94.83% and 85.98%. The method exposed in [START_REF] Hnoohom | Comparison of Ensemble Learning Algorithms for Cataract Detection from Fundus Images[END_REF] was dedicated to cataract detection. It started by extracting texture and sketch features from preprocessed images. Then, an EL experimentation was carried out utilizing the decision tree, BPNN and sequential minimal optimization classifiers and using a dataset of 374 fundus images. Each classifier was trained first by texture features, second by sketch features, and third by combining both of them. It was deduced that merging features would allow realizing better accuracy than using each kind of features separately. Thereafter, the classifiers trained using the same feature set were stacked. It was synthetized that the stacked learning method achieved the detection accuracy of 95.47%, which exceeded the one provided by each single classifier.

Ensemble learning framework for

cataract severity grading

Preprocessing & data augmentation

The fundus images are captured through varied devices with different technologies, hence provided with multiple resolutions. For this purpose, all fundus images are resized in order to be normalized. The new sizes are determined with respect to the input of each employed DL architecture. In addition, the fundus images have varied and unbalanced contrasts caused either by the lamp used in the capture task or by the retina morphology, as presented in the first and second images in Fig. 2. To avoid this limitation, the histogram equalization is applied to enhance the whole contrast ratio per images and to normalize the contrast of all images of the dataset [START_REF] Li | Automatic Detection of Diabetic Retinopathy in Retinal Fundus Photographs Based on Deep Learning Algorithm[END_REF], as shown in the third and fourth images in Fig. 2.

The retinas are illustrated through different Field-Of-Views (FOVs) that depend on the capture devices. Moreover, the FOV gap leads to a similar difference of masks used to encompass the retinas. For this purpose, a data augmentation process is suggested, which consists in creating new images by modifying the initial ones. This step allows raising the dataset size to augment the method robustness. As regards the FOV and mask differences, each fundus image is zoomed, sheared to a random corner, and shifted and flipped in the horizontal and vertical directions, as respectively depicted in the fundus images in Fig. 3. Furthermore, the fundus image is flipped respectively in horizontal, vertical and both horizontal and vertical directions.

DL model for cataract grading 3.2.1. Inception-V3

Inception-V3 is used for image classification, which is extended from the GoogLeNet network suggested by Google in 2014 [START_REF] Kristiani | iSEC: An Optimized Deep Learning Model for Image Classification on Edge Computing[END_REF]. Inception-V3 consists in performing several small convolutions on the same level instead of using large convolutions [START_REF] Hagos | Transfer Learning based Detection of Diabetic Retinopathy from Small Dataset[END_REF]. The first level of the Inception module is composed of three convolutions and one max-pooling. In the last level, the provided channels are non-linearly merged [START_REF] Dong | Inception v3 based cervical cell classification combined with artificially extracted features[END_REF]. Therefore, the network parameters are reduced, which implies accelerating either the training or test steps. In addition, the features are efficiently extracted, which increases the classification accuracy [START_REF] Zhao | Intelligent Recognition of Fatigue and Sleepiness Based on InceptionV3-LSTM via Multi-Feature Fusion[END_REF]. Inception-V3 contains dense layers to increase the network depth, which decreases again the computational complexity. It is trained on the ImageNet dataset, to identify 1,000 classes [START_REF] Li | Automatic Detection of Diabetic Retinopathy in Retinal Fundus Photographs Based on Deep Learning Algorithm[END_REF]. Inception-V3 proves higher accuracy than the original GoogLeNet, Resnet-50 and AlexNet [START_REF] Zhao | Intelligent Recognition of Fatigue and Sleepiness Based on InceptionV3-LSTM via Multi-Feature Fusion[END_REF]. Due to its flexibility and accuracy, Inception-V3 achieves a higher performance when doing Transfer Learning (TL) even with small datasets [START_REF] Dong | Inception v3 based cervical cell classification combined with artificially extracted features[END_REF].

MobileNet-V2

The MobileNet-V2 architecture contains convolutional layers, where each one is followed by a batch normalization and a ReLU6 nonlinear activation function, except the output layer [START_REF] Zhao | Intelligent Recognition of Fatigue and Sleepiness Based on InceptionV3-LSTM via Multi-Feature Fusion[END_REF][START_REF] Wang | A MobileNets Convolutional Neural Network for GIS Partial Discharge Pattern Recognition in the Ubiquitous Power Internet of Things Context: Optimization, Comparison, and Application[END_REF]. In MobileNet-V2, expensive convolutional layers are factorized into lightweight separable convolution blocks. In each block, the input is filtered through a (3x3) depthwise convolutional layer [START_REF] Aguiar | Visual Trunk Detection Using Transfer Learning and a Deep Learning-Based Coprocessor[END_REF]. The provided output channels are projected via a (1x1) pointwise convolutional layer. The MobileNet-V2 blocks are built with residual connections that assist converging the network weights [START_REF] Michele | MobileNet Convolutional Neural Networks and Support Vector Machines for Palmprint Recognition[END_REF]. The use of (3x3) depthwise separable convolutions and the resolution modification through layers imply decreasing the computational complexity with respect to the standard convolutions despite a slight accuracy reduction [START_REF] Aguiar | Visual Trunk Detection Using Transfer Learning and a Deep Learning-Based Coprocessor[END_REF][START_REF] Toğaçar | COVID-19 detection using deep learning models to exploit Social Mimic Optimization and structured chest X-ray images using fuzzy color and stacking approaches[END_REF].

NasNet-Mobile

The Neural architecture search (Nas) framework is a research algorithm based on a neural architecture [START_REF] Gupta | InstaCovNet-19: A deep learning classification model for the detection of COVID-19 patients using Chest X-ray[END_REF]. The objective is to provide an optimal Convolutional Neural Network (CNN) architecture for a given dataset, called Nas Network (NasNet). It automatically proposes a sequence of blocks that present a whole architecture. Thereupon, the provided NasNet architecture is unkown before the training process [START_REF] Cogan | MAPGI: Accurate identification of anatomical landmarks and diseased tissue in gastrointestinal tract using deep learning[END_REF]. The NasNet research algorithm is based on a controller recurrent neural network that samples convolutional neural blocks and arranges them into different architectures. The controller uses reinforcement learning where the proposed architectures are trained and their accuracy is utilized to enhance the following ones [START_REF] Cogan | MAPGI: Accurate identification of anatomical landmarks and diseased tissue in gastrointestinal tract using deep learning[END_REF][START_REF] Bahri | Remote Sensing Image Classification via Improved Cross-Entropy Loss and Transfer Learning Strategy Based on Deep Convolutional Neural Networks[END_REF]. To classify whatever image size, the generated architectures are built using two types of convolutional cells [START_REF] Zoph | Learning Transferable Architectures for Scalable Image Recognition[END_REF]: (1) normal cells that provide a feature map having the same resolution than the input, and (2) reduction cells that provide a feature map where the height and the width are reduced by half. Following the same architecture building, the NasNet framework might provide two different architectures [START_REF] Bahri | Remote Sensing Image Classification via Improved Cross-Entropy Loss and Transfer Learning Strategy Based on Deep Convolutional Neural Networks[END_REF]. The first one is called NasNet-Large, which is composed of 84 million parameters. The second one is NASNet-Mobile made up of only four million parameters. This architecture is trained on ImageNet and performs acceptable classification. Despite the lower size, the TL of the NasNet-Mobile architecture maintains similar accuracy even with a smaller dataset [START_REF] Winoto | Small and Slim Deep Convolutional Neural Network for Mobile Device[END_REF].

Transfer learning and fine-tuning

TL and Fine-Tuning (FT) are well-known machine learning methods where the knowledge obtained through a trained model dedicated to a problem is used to resolve another one [START_REF] Mateen | Exudate Detection for Diabetic Retinopathy Using Pretrained Convolutional Neural Networks[END_REF]. Generally, a great dataset is required to train a CNN, which is not guaranteed in several problems. For this purpose, both methods are used to avoid the problem of the dataset size, to enhance the performance, to reduce the redundancy and to accelerate the training [START_REF] Michele | MobileNet Convolutional Neural Networks and Support Vector Machines for Palmprint Recognition[END_REF]. The TL consists in training the model in order to adjust the objective of the related dataset to the target of the current one [START_REF] Mateen | Exudate Detection for Diabetic Retinopathy Using Pretrained Convolutional Neural Networks[END_REF]. In the FT, the weight values are retrieved from the trained model and used as the initialization to be updated through the training process [START_REF] Michele | MobileNet Convolutional Neural Networks and Support Vector Machines for Palmprint Recognition[END_REF]. Due to the size of the dataset used in this study, our framework leads to extract the features of the pre-trained CNN model of Inception-V3, MobileNet-V2 and Nasnet-Mobile to be used for TL and FT. The selected CNN models are initially trained with the publicly "ImageNet" dataset that contains 1,000 categories [START_REF] Li | Automatic Detection of Diabetic Retinopathy in Retinal Fundus Photographs Based on Deep Learning Algorithm[END_REF]. The main goal of the targeted models is to classify the fundus image into: healthy, cataract mild stage, cataract moderate stage or cataract severe stage. Therefore, the last three layers are substituted to adapt the result into the aimed four classes. After the last global average pooling, a flatten layer is processed to merge the channel of the provided feature map. Thereafter, a first fully connected dense layer is applied with the ReLU activation function in order to reduce the feature map resolution. The last softmax fully connected dense layer is altered which is composed of four neurons to generate four probability distributions that correspond to the four output categories [START_REF] Mateen | Exudate Detection for Diabetic Retinopathy Using Pretrained Convolutional Neural Networks[END_REF]. Furthermore, FT is preformed where the layer weights are unfreezing during training. Hence, their values are able to be updated based on the back propagation principle with a learning rate of 0.00001 [START_REF] Li | Automatic Detection of Diabetic Retinopathy in Retinal Fundus Photographs Based on Deep Learning Algorithm[END_REF]. A callback is activated to save the better weight after each epoch to improve the performance and accelerate model training. For the three DL architectures, the train process is performed using the "Adam" optimizer and the " categorical_crossentropy " loss function through 150 epochs.

Ensemble learning framework for cataract grading

An EL system is a learning model that clusters several classifiers, called base classifiers, to merge their decisions into a single one, called a meta classifier [START_REF] Książek | Development of novel ensemble model using stacking learning and evolutionary computation techniques for automated hepatocellular carcinoma detection[END_REF]. The main purpose is to provide a higher accurate decision than the ones separately given by each base classifier [START_REF] Książek | Development of novel ensemble model using stacking learning and evolutionary computation techniques for automated hepatocellular carcinoma detection[END_REF][START_REF] Zhou | The ensemble deep learning model for novel COVID-19 on CT images[END_REF]. These base classifiers are called homogeneous if they belong to the same algorithm trained to various datasets. In contrast, heterogeneous base classifiers are different methods learned from the same dataset. Two categories of EL systems are extensively used in the work appeared in the past few years [START_REF] Gupta | Improving malware detection using big data and ensemble learning[END_REF]. The first one was the data level EL, where single classifiers were resampled to select the base classifier ones, such as the bagging method. The combination level was a second category of EL where individual classifiers were combined, including voting, ensemble selection and stacking. In the voting method, each base classifier would generate a label for each dataset instance, which were retrieved by the meta model that decided a unique label through a voting principle. Knowing that some base classifiers on an EL system might decrease the performance, the ensemble selection was a heuristic algorithm that allowed selecting an optimal subset of base classifiers that would guarantee a better performance. The base classifiers in the stacking technique provided their result as an input of the meta model. For the stacking EL, the training principles of classifiers were considerably different, which led to varied expertise. Stacking consists in combining the predictions of all base classifiers into another learning model [START_REF] Winoto | Small and Slim Deep Convolutional Neural Network for Mobile Device[END_REF][START_REF] Gupta | Improving malware detection using big data and ensemble learning[END_REF][START_REF] An | Deep ensemble learning for Alzheimer's disease classification[END_REF].

Our main objective is to ensure cataract grading with a higher performance. The MobileNet-V2, Inception-V3 and Nasnet-Mobile models are selected as heterogeneous classifiers since they have different architectures and hence multiple prediction principles. Those models are known by their higher performance, even though their lightweight criteria prevent them from achieving optimal classification. Consequently, we design a stacked ensemble classifier where the three CNN models are considered as base-classifiers in order to combine the predictions. The classification capability and the greater difference of the base classifier leads to a better classification performance of the whole EL. The metaclassifier rectifies the prediction errors, reduces the bias carried out by the base classifiers and prevents the overfitting effect [START_REF] Mateen | Exudate Detection for Diabetic Retinopathy Using Pretrained Convolutional Neural Networks[END_REF][START_REF] Agarwal | A-Stacking and A-Bagging: Adaptive versions of ensemble learning algorithms for spoof fingerprint detection[END_REF].Moreover, their difference allows benefiting from the advantages of each model to provide a complementary prediction. As a result, the metaclassifier ensures achieving high prediction accuracy. The choice of the meta-classifier is a critical issue in a way that the performance of the whole framework must be better than each base classifier, tested separately. For this purpose, several classifiers are experimentally evaluated, as detailed in section 4. kernel provides the optimal grading accuracy. The main steps of the EL framework, illustrated in Fig. 4, are as follows:

1) Extract the feature from each base classifier separately into a vector having a size of (1*4); 2) Concatenate the feature probability provided into three vectors provided from the base classifiers into a single meta-feature vector with a size of (1*12); 3) Provide the meta-feature vector to the meta classifier to classify the input image into cataract stages.

Experimental results

Dataset

Based on the literature review detailed in section 2, the evaluation of the cataract detection and grading methods is always conducted using private datasets. In our case, we construct a dataset of retinal images selected from two public databases recently published in the Kaggle platform. The "Cataract Dataset" [START_REF]Cataract Dataset[END_REF] is composed of 600 fundus images having a size of 2592*1728, where 100 are affected by the cataract disease. The second database is called "Ocular Disease Recognition (ODiR)" [START_REF]Ocular Disease Recognition[END_REF] containing 8,000 fundus images with different image sizes, where 293 are reached by the cataract. The cataract-affected images of both databases are labeled through the intervention of two ophthalmologists belonging to different hospital centers.

The used dataset contains 590 fundus images, where 220 are confirmed as non-cataract, while 65, 145 and 160 images are diagnosed as mild, moderate and severe cataracts, respectively. Thereafter, each grading set is randomly partitioned into five subsets. three subsets, representing 60% of the whole dataset, are dedicated for training; each of the last two subsets are respectively used for validation and testing. To ensure a reliable evaluation, we conduct the evaluation through a 5-fold cross validation approach, where the five subsets are affected differently to the training, validation and testing process, as depicted in Fig. 5.

Software environment and evaluation metrics

The framework is coded with python language and using the "Keras" API dedicated for Deep learning processing. The training and testing phase are run on the "google Colab" cloud service. To evaluate the performance of our method, five metrics are computed, which are Sensitivity (Sens), Specificity (Spec), Accuracy (Acc), Precision (Prec) and F-Measure (F-M) given in equations ( 1) -( 5), respectively.

Sens =

(1)

Spec = (2)
Acc = (3)
Prec = (4)
F-M = × × (5)
where TP, TN, FP and FN are respectively True Positive, True Negative, False Positive and False Negative detected images.

Meta classifier algorithm choice

The aim of this sub-section is to identify the machine learning algorithm and its configuration, to be used as a meta classifier that allows achieving the highest performance of cataract grading. Within this objective, different machine learning algorithms are used having as an input the extracted features of all the three base classifiers. For instance, the Multi-Layer Perceptron (MLP) is employed, where the hidden layer size is varied among 8, 10 and 12, and where their results are shown in the second column of Table 1. For the Random Forest (RF), the tree number is fixed to 200, while the tree depth is varied following the above values {8, 10, 12}, where the performance metrics are illustrated in the third column. The SVMs are trained with the linear, RBF and polynomial kernels separately, where the achieved performance metrics are depicted in the fourth column. We deduce that using the SVM classifier with the RBF kernel allows achieving the higher performance of cataract grading, thus chosen as a meta classifier of the proposed framework. In this sub-section, we evaluate the cataract grading using the 5-fold cross validation approach based on the five metrics as shown in Table 2. For a better analysis of the achieved results, the box plots illustration is propounded in Fig. 6 to present the performance of these validations.

We deduce that all validations achieved high performances in terms of the five metrics. The accuracy, sensitivity and F-measure values are very close to their average. Moreover, reduced gaps are deduced between values for specificity and precision with variation values of ±1.04% and ±0.72%, respectively. Accordingly, a higher performance is guaranteed whatever the fundus image dataset used for testing or training. This result approves of the robustness for the suggested framework when used in clinical context with new fundus images.

Ensemble learning evaluation with respect to base classifiers

The four-grading performance of the suggested EL framework is evaluated and compared to the ones realized by each base classifier, separately. For this purpose, the experimentation consists in extracting features from each base classifier and given to an algorithm classifier to label images. To ensure a credible evaluation, the same SVM classifier with the RBF kernel is associated in the output of each DL architecture. In addition, the same subset of fundus images is used either for base classifiers or a metaclassifier, where their cataract-grading results is detailed in the confusion matrices of Fig. 7.

The EL framework succeeds in properly classifying a larger number of fundus images, as deduced when adding and comparing blue cells for each matrix. Moreover, we deduce that it registers less numbers either of overestimated or under-estimated fundus images, as noticed by half-matrices limited by the diagonal lines. The performance metrics are illustrated in Fig. 8, where the EL framework significantly surpasses the base classifiers in all the five metrics.

Subsequently, we analyse the achieved result for each grade separately. For the non-cataract grade, the Inception-V3 architecture falls into four fundus images, in contrast to the two other DL architectures. Thereby, the EL framework settles for the features extracted by the Mobilenet-V2 and Nasnet-Mobile architectures. For the mild grade, all single classifiers fall into more than 33% of images. The proposed framework takes benefit from the diversity of all extracted features to enhance the final classification. In addition, the EL classification ensures the maximal labelling of moderate cataract affected images, like the Mobilenet-V2 architecture even with the lower accuracy achieved by Inception-V3 and Nasnet-Mobile. Finally, the framework takes the average results of the three base classifiers for the severe grade images, where it is overpassed by only one wrong image grading with respect to two base classifiers. We compare the grading performance of our method to the existing ones. The study is performed in terms of overall accuracy, where values are represented in the last column of Table 3. We deduce that our framework outperforms the methods based on a single CNN architecture [START_REF] Zhou | Automatic Cataract Classification Using Deep Neural Network With Discrete State Transition[END_REF][START_REF] Dong | Classification of cataract fundus image based on deep learning[END_REF][START_REF] Imran | Cataract Detection and Grading with Retinal Images Using SOM-RBF Neural Network[END_REF]. By the way, those methods achieve accuracies similar the ones realized by any base-classifier, separately. Moreover, our framework surpasses the methods that employed an EL principles [START_REF] Cao | Hierarchical method for cataract grading based on retinal images using improved Haar wavelet[END_REF][START_REF] Yang | Exploiting ensemble learning for automatic cataract detection and grading[END_REF]. The better grading result is due to the adequate choice of base classifiers. In addition, it is caused by the ensemble learning principle with consists at take benefit from all base classifiers rather than chosen the best ones.

Conclusion

The cataract is the first worldwide cause of vision impairment and blindness. The challenge is to propose a CAD system to grade the cataract from retinal images. For such a need, we have put forward in this paper an ensemble learning framework of stacking three DL architectures, where higher-performance cataract grading has been achieved. The realized performance has outperformed the ones realized by each single DL architecture. Accordingly, this framework will promote accessing to eye diagnosis, hence avoiding an advanced cataract state.

In our future work, we aim to extend our framework to consider other ocular pathologies, such as diabetic retinopathy, glaucoma or aged macular degeneration. Moreover, the employed lightweight learning machines, either as base-classifiers or as meta-classifiers, have low computational complexity. Consequently, the framework may be implemented in smartphone devices to be provided as a mobile-aided screening system for cataract grading, which will widely increase the accessibility to eye diagnosis.

Figure 1 .

 1 Figure 1. Retinal images of all cataract grades. (a) Non-cataract. (b) Mild cataract. (c) Moderate cataract. (d) Severe cataract

Figure 2 .

 2 Figure 2. (a), (b): Original fundus images; (c), (d): Fundus images after histogram equalization.

Figure 3 .

 3 Figure 3. Different data augmentation processing.

Figure 4 .

 4 Figure 4. Processing pipeline of ensemble learning framework for cataract grading

Figure 5 .

 5 Figure 5. Dispatching of subsets for 5-fold cross validation

4. 4 .

 4 Ensemble learning framework evaluation 4.4.1. Five-cross validation of the proposed framework

Figure 6 .Figure 7 .

 67 Figure 6. Performance visualization of cataract grading using box plots

Table 1 .

 1 Performance comparison between SVM and RF classifiers

	Classifier					MLP					RF				SVM
	Parameters	Hidden	layer	size	8	10	12	Max	depth	8	10	12	Kernel	Linear	Poly	RBF
	Sensitivity				88.57	94.29	94.29			94.29	92.86	94.29		94.29	94.37	95.59
	Specificity				91.67	87.50	85.42			85.42	87.50	87.50		85.42	89.36	91.67
	Accuracy				89.83	91.53	90.68			90.68	90.68	91.53		90.68	92.37	93.97
	Precision				93.94	91.67	90.41			90.41	91.55	91.67		90.41	93.06	94.20
	F-Measure				91.18	92.96	92.31			92.31	92.20	92.96		92.31	93.71	94.89

Table 2 .

 2 Cataract grading performance for 5-flod cross validation

	Validations				Metrics
		Sens	Spec	Acc	Prec	F-M
	1	90,00% 91,67% 90,68% 94,03% 91,97%
	2	94,29% 89,58% 92,37% 92,96% 93,62%
	3	94,20% 89,80% 92,37% 92,86% 93,53%
	4	94,12% 89,80% 92,31% 92,75% 93,43%
	5	95,59% 91,67% 93,97% 94,20% 94,89%

Table 3 .

 3 Performance comparison between existing methods of cataract grading

	Methods of cataract grading	Classification algorithm	Acc (%)
	Guo et al. [9] (2015)	Discriminant analysis algorithm	77.1
	Xiong et al. [40] (2017)	Decision tree	83.84
	Yang et al. [14] (2016)	Stacking EL of SVM and BPNN	84.5
	Cao et al. [3] (2020)	Voting EL of two-class BPNN	85.98
	Song et al. [4] (2019)	Eigenvalue computing + SVM	88.6
	Zhou et al. [2] (2019)	CNN	89.23
	Dong et al. [11] (2017)	CNN + SVM	90.82
	Imran et al. [13] (2019)	SOM-RBF NN	91.7
	Our method	Stacking EL of 3 CNN	93.97

Acknowledgements The author would like to acknowledge the help of Dr. Aymen Daldoul and Dr. Nesrine Abroug, ophthalmologists respectively in the hospital center of Nevers agglomeration (France) and in Department of Ophthalmology of Fattouma Bourguiba University Hospital (Tunisia), to provide the groundtruth of retinal fundus images.

Funding this work was supported by the PHC-UTIQUE 19G1408 Research program.

Conflict of interest

The authors declare that they have no confict of interest to this work.