Anthony Fiche
email: anthony.fiche@ensta-bretagne.fr

Jean-Christophe Cexus
email: jean-christophe.cexus@ensta-bretagne.fr

Arnaud Martin
email: arnaud.martin@univ-rennes1.fr

Ali Khenchaf
email: ali.khenchaf@ensta-bretagne.fr

Features modeling with an α-stable distribution: application to pattern recognition based on continuous belief functions

Keywords: Gaussian and α-stable model, unimodal features, continuous belief functions, supervised pattern recognition, Kolmogorov-Smirnov test, Bayesian approach

Features modeling with an α-stable distribution: Application to pattern recognition based on continuous belief functions

Introduction

The choice of a model has an important role in the problem of estimation. For example, the Gaussian model is a very efficient model which fits data in many applications as it is very simple to use and saves computation time. However, as is the case for all distribution models, Gaussian laws have some weaknesses and results can end-up being skewed. Indeed, as the Gaussian probability density function (pdf) is symmetrical, it is not valid when the pdf is not symmetrical. It is therefore difficult to choose the right model which fits data for each application. The Gaussian distribution belongs to a family of distributions called stable distributions. This family of distributions allows the representation of heavy tails and skewness. A distribution is said to have a heavy tail if the tail decays slower than the tail of the Gaussian distribution. Therefore, the property of skewness means that it is impossible to find a mode where probability density function is symmetrical. The main property of stable laws introduced by Lévy [START_REF] Lévy | Théorie des erreurs : La loi de Gauss et les lois exceptionnelles[END_REF] is that the sum of two independent stable random variables gives a stable random variable. α-stable distributions are used in different fields of research such as radar [START_REF] Achim | SAR image denoising via Bayesian wavelet shrinkage based on heavy-tailed modeling[END_REF][START_REF] Banerjee | Adaptive target detection in foliage-penetrating SAR images using alpha-stable models[END_REF], image processing [START_REF] Kuruoglu | Skewed α-stable distributions for modelling textures[END_REF] or finance [START_REF] Mcculloch | Financial applications of stable distributions, Handbook of statistics[END_REF][START_REF] Fama | The behavior of stock-market prices[END_REF], ... The aim of this study is to show the interest in fitting data with an α-stable distribution. In [START_REF] Williams | Bayesian Data Fusion of Multi-View Synthetic Aperture Sonar Imagery for Seabed Classification[END_REF], the author proposes to characterize the sea floor from a vector of features modeled by Gaussian mixture models (GMMs) using an Autonomous Underwater Vehicle (AUV). However, the problem with the Bayesian approach is the difficulty in considering the uncertainty of data. We therefore favored the use of an approach based on the theory of belief functions [START_REF] Dempster | Upper and Lower probabilities induced by a multivalued mapping[END_REF][START_REF] Shafer | A mathematical theory of evidence[END_REF]. In [START_REF] Fiche | Bayesian approach and continuous belief functions for classification[END_REF], the authors compared a Bayesian and belief classifier where data from sensors are modeled using GMMs estimated via an Expectation-Maximization (EM) algorithm [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF]. This work has been extended to data modeled by α-stable mixture models [START_REF] Fiche | Influence de l'estimation des paramètres de texture pour la classification de données complexes[END_REF]. However, it is difficult to choose between these two models because the results are roughly the same. This paper raises two problems. Firstly, it is necessary to work on a data set where features are modeled by α-stable pdfs. The second problem is to know how to apply the theory of belief functions when data from sensors are modeled by α-stable distributions. This point has been dealt with in [START_REF] Fiche | Continuous belief functions and α-stable distributions[END_REF].

The recent characterization of uncertain environments has taken an important place in several fields of application such as the SOund Navigation And Ranging (SONAR). SONAR has therefore been used for the detection of underwater mines [START_REF] Quidu | Mine classification based on raw sonar data: an approach combining Fourier descriptors, statistical models and genetic algorithms[END_REF]. In [START_REF] Laanaya | Multi-view fusion based on belief functions for seabed recognition[END_REF], the author developed techniques to perform automatic classification of sediments on the seabed from sonar images. In [START_REF] Fiche | Bayesian approach and continuous belief functions for classification[END_REF], the authors classified sonar images by extracting features and modeling them with GMMs. In [START_REF] Pouliquen | Sea-bed identification using echo-sounder signals[END_REF], the authors characterized the sea floor using data from a mono-beam echo-sounder. A data set represents an echo signal amplitude according to time. They compare the time envelope of echo signal amplitude with a set of theoretical reference curves. In this paper, we build a classifier based on the theory of belief functions where a vector of features extracted from a mono-beam echo-sounder is modeled by a Gaussian and an α-stable distribution. The feature pdfs have only one mode. We finally show that it would be interesting to fit features with an α-stable model.

This paper is organized as follows: we first introduce the definition of stability, the method to construct α-stable probability density functions and the methods of estimation in Section 2. The definitions in the multivariate case are also presented. We introduce the notion of belief functions in discrete and continuous cases in Section 3. Belief functions are then calculated in the particular case of α-stable distributions and a belief classifier is constructed in Section 4. We finally classify data by modeling data with Gaussian and α-stable distributions before the generated data are classified by comparing the theory of belief functions with Gaussian and α-stable distributions in Section 5. We compare the results obtained with the theory of belief functions using a Bayesian approach.

The α-stable distributions

Gauss introduced the probability density function called "Gaussian distribution" (in honor of Gauss) in his study of astronomy. Laplace and Poisson developed the theory of characteristic function by calculating the analytical expression of Fourier transform of a probability density function. Laplace found that the Fourier transform of a Gaussian law is also a Gaussian law. Cauchy tried to calculate the Fourier transform of a "generalized Gaussian" function with the expression f n (x) = 1 π +∞ 0 exp (-ct n ) cos (tx)dt, but he did not solve the problem. When the integer n is a real α, we define the family of α-stable distributions. However, Cauchy did not know at that time if he had defined a probability density function. With the results of Polya and Bernstein, Janicki and Weron [START_REF] Janicki | Simulation and chaotic behavior of α-stable stochastic processes[END_REF] demonstrated that the family of α-stable laws are probability density functions. The mathematician Paul Lévy studied the central limit theorem and showed, with the constraint of infinite variance, that the limit law is a stable law [START_REF] Lévy | Théorie des erreurs : La loi de Gauss et les lois exceptionnelles[END_REF]. Motivated by this property, Lévy calculated the Fourier transform for all α-stable distributions. These stable laws which satisfy the generalized central limit theorem are interesting as they allow the modeling of data as impulsive noise when the Gaussian model is not valid. Another property of these laws is the ability to model heavy tails and skewness. In the following, we define the notions of stability, characteristic function, the way to build a pdf from characteristic function, the different estimators of α-stable distributions and the extension of α-stable distribution to the multivariate case.

Definition of stability

Stability is when the sum of two independent random variables which follow α laws, gives an α law. Mathematically, this definition means the following: a random variable X is stable, denoted X ∼ S α (β, γ, δ), if for all (a,b) ∈ R + × R + , there are c ∈ R + and d ∈ R such that:

aX 1 + bX 2 = cX + d, (1)
with X 1 and X 2 two independent α-stable random variables which follow the same distribution as X. If Equation (1) defines the notion of stability, it does not give any indication as to how to parameterize an α-stable distribution. We therefore prefer to use the definition given by characteristic function to refer to an α-stable distribution.

Characteristic function

Several equivalent definitions have been suggested in the literature to parameterize an α-stable distribution from its characteristic function [START_REF] Taqqu | Stable non-gaussian random processes[END_REF][START_REF] Zolotarev | One-dimensional stable distributions[END_REF]. Zolotarev [START_REF] Zolotarev | One-dimensional stable distributions[END_REF] proposed the following:

φ S α (β,γ,δ) (t) =            exp( jtδ -|γt| α [1 + jβ tan( πα 2 )sign(t)(|t| 1-α -1)]) if α 1, exp( jtδ -|γt|[1 + jβ 2 π sign(t) log |t|]) if α = 1, (2)
where each feature has specific values: [START_REF] Lévy | Théorie des erreurs : La loi de Gauss et les lois exceptionnelles[END_REF] is the skewness parameter.

• α ∈]0, 2] is the characteristic exponent. • β ∈ [-1,
• γ ∈ R + * represents the scale parameter.

• δ ∈ R is the location parameter.

The advantage of this parameterization compared to [START_REF] Taqqu | Stable non-gaussian random processes[END_REF] is that the values of the characterization and probability density functions are continuous for all parameters. In fact, the parameterization defined by [START_REF] Taqqu | Stable non-gaussian random processes[END_REF] is discontinuous when α = 1 and β = 0.

The probability density function

The representation of an α-stable pdf, denoted f S α (β,γ,δ) , is obtained by calculating the Fourier transform of its characteristic function:

f S α (β,γ,δ) (x) = ∞ -∞ φ S α (β,γ,δ) (t) exp(-jtx)dt. (3)
However, this definition is problematic for two reasons: while the integrand function is complex, its bounds are infinite. Nolan [START_REF] Nolan | Numerical calculation of stable densities and distribution functions[END_REF] therefore proposed a way to represent normalized α-stable distributions (i.e. γ = 1 and δ = 0). The main idea of Nolan [START_REF] Nolan | Numerical calculation of stable densities and distribution functions[END_REF] is to use variable modifications so that the integral has finite bounds. Each parameter has an influence on the shape of the f S α (β,γ,δ) . The curve has a large peak when α is near 0 and a Gaussian shape when α = 2 (Figure 1(a)). The shape of the distribution skews to the left if β = 1, to the right if β = -1 (Figure 1(b)) while the distribution is symmetrical when β = 0. Finally, the scale parameter enlarges or compresses the shape of the distribution (Figure 1(c)) and the location parameter leads to the translation of the mode of the f S α (β,γ,δ) (Figure 1(d)).

An overview of α-stable estimators

The estimators of α-stable distributions are decomposed into three families:

• the sample quantile methods [START_REF] Fama | Parameter estimates for symmetric stable distributions[END_REF][START_REF] Mcculloch | Simple consistent estimators of the parameters of stable laws[END_REF] • the sample characteristic function methods [START_REF] Press | Applied multivariate analysis[END_REF][START_REF] Press | Estimation in univariate and multivariate stable distributions[END_REF][START_REF] Koutrouvelis | Regression-type estimation of the parameters of stable laws[END_REF][START_REF] Koutrouvelis | An iterative procedure for the estimation of the parameters of stable laws[END_REF] • the Maximum Likelihood Estimation [START_REF] Brorsen | Maximum likelihood estimates of symmetric stable distribution parameters[END_REF][START_REF] Mcculloch | Linear regression with stable disturbances[END_REF][START_REF] Dumouchel | Stable distributions in statistical inference[END_REF][START_REF] Nolan | Maximum likelihood estimation and diagnostics for stable distributions[END_REF].

Fama and Roll [START_REF] Fama | Parameter estimates for symmetric stable distributions[END_REF] developed a method based on quantiles. However, the algorithm proposed by Fama and Roll suffers from a small asymptotic bias in α and γ and restrictions on α ∈]0.6, 2] and β = 0. McCulloch [START_REF] Mcculloch | Simple consistent estimators of the parameters of stable laws[END_REF] extended the quantile method to the asymmetric case (i.e. β = 0). The McCulloch estimator is valid for α ∈]0.6, 2].

Press [START_REF] Press | Applied multivariate analysis[END_REF][START_REF] Press | Estimation in univariate and multivariate stable distributions[END_REF] proposed a method based on transformations of characteristic function. In [START_REF] Weron | Performance of the estimators of stable law parameters[END_REF], the author compared the performances of several estimators. For example, the Press method is efficient for specific values. Koutrouvelis [START_REF] Koutrouvelis | Regression-type estimation of the parameters of stable laws[END_REF] extended the Press method and proposed a regression method to estimate the parameters of α-stable distributions. He proposed a second version of his algorithm which is distinct that it is iterative [START_REF] Koutrouvelis | An iterative procedure for the estimation of the parameters of stable laws[END_REF]. In [START_REF] Akgiray | Estimation of stable-law parameters: A comparative study[END_REF], the authors proved that the method proposed by Koutrouvelis is better than both the quantile method and the Press method because it gives consistent and asymptotically unbiased estimates.

The Maximum Likelihood Estimation (MLE) was first studied in the symmetric case [START_REF] Brorsen | Maximum likelihood estimates of symmetric stable distribution parameters[END_REF][START_REF] Mcculloch | Linear regression with stable disturbances[END_REF]. Dumouchel [START_REF] Dumouchel | Stable distributions in statistical inference[END_REF] developed an approximate maximum likelihood method. The MLE was also developed in the asymmetric case.

Nolan [START_REF] Nolan | Maximum likelihood estimation and diagnostics for stable distributions[END_REF] extended the MLE in general case. The problem with the MLE is the calculation of the α-stable probability density function because there is no closed-form expression. Moreover, the computational algorithm is timeconsuming.

Consequently, we estimate a univariate α-stable distribution using the Koutrouvelis method [START_REF] Koutrouvelis | An iterative procedure for the estimation of the parameters of stable laws[END_REF].

Multivariate stable distributions

It is possible to extend the α-stable distributions to the multivariate case. A random vector X ∈ R is stable if for all a, b ∈ R + , there are c ∈ R + and D ∈ R d such that:

aX 1 + bX 2 = cX + D, (4)
where X 1 and X 2 are two independent and identically distributed random vectors which follow the same distribution as X.

The characteristic function of an multivariate α-stable distribution, denoted X ∼ S α,d (σ, δ), has the form:

φ S α,d (σ,δ) (t) =                exp - S d | < t, s > | α (1 -jsgn(< t, s >) tan( πα 2 )σ(ds) + j < δ, t > if α 1, exp - S d | < t, s > |(1 + j π 2 sgn(< t, s >) ln(< t, s >))σ(ds) + j < δ, t > if α = 1, (5)
with

• S d = {x ∈ R d |||x|| = 1} the d-dimensional unit sphere.
• σ(.) a finite Borel measure on S d .

• δ, t ∈ R d .
The expression for the characteristic function involves an integration over the unit sphere S d . The measure σ is called the spectral measure and δ is called the location parameter. The problem with a multivariate α-stable distribution is that characteristic function forms a non-parametric set. To avoid this problem, it is possible to consider a discrete spectral measure [START_REF] Modarres | A method for simulating stable random vectors[END_REF] which has the form:

σ(.) = K i=1 γ i δ s i (.), (6)
with γ i corresponding to weight and δ s i the Dirac measure in s i . For an α-stable distribution in R 2 with K mass points, the quantity s i = {cos(θ i ), sin(θ i )} with θ i = 2π(i-1) K . There are two methods to estimate an α-stable random vector:

• the PROJection method [START_REF] Mcculloch | Estimation of the bivariate stable spectral representation by the projection method[END_REF] (PROJ)

• the Empirical Characteristic Function method [START_REF] Nolan | Estimation of stable spectral measures[END_REF] (ECF) These two algorithms have the same performances in terms of estimation and computation time.

At this point, it is important to underline that the features extracted from mono-beam echo-sounders have the properties of heavy-tails and skewness. Consequently, we decided to estimate the data with an α-stable model. These data are however imprecise and uncertain: the imprecision and uncertainty of data can be linked to poor quality estimation of these data. To take these constraints into consideration, we used an uncertain theory called the theory of belief functions. Consequently, the goal of the next section is to present the theory of belief functions.

The belief functions

The final objective of this paper is to classify synthetic and real data using the theory of belief functions. Data obtained from sensors are generally imprecise and uncertain as noises can disrupt their acquisition. It is possible to consider these constraints by using the theory of belief functions. We will first develop the theory of belief functions within a discrete framework before characterizing it in real numbers.

Discrete belief functions

This section outlines basic tools in relation to the theory of belief functions.

Definitions

Discrete belief functions were introduced by Dempster [START_REF] Dempster | Upper and Lower probabilities induced by a multivalued mapping[END_REF], and formalized by Shafer [START_REF] Shafer | A mathematical theory of evidence[END_REF] where he considers a discrete set of n exclusive events C i called the frame of discernment:

Θ = {C 1 , . . . , C n }. (7)
Θ can be interpreted as all the assumptions to a problem. Belief functions are defined as 2 Θ onto [0,1]. The objective of discrete belief functions is to attribute a weight of belief to each element A ∈ 2 Θ . Belief functions must follow the normalization:

A⊆Θ m Θ (A) = 1, (8)
where m Θ is called the basic belief assignment (bba).

A focal element is a subset of A where m Θ (A) > 0 and several functions in one-to-one correspondence are built from bba:

bel

Θ (A) = B⊆A,B ∅ m Θ (B), ( 9
)
pl Θ (A) = A∩B ∅ m Θ (B), (10)
q Θ (A) = B⊂Θ,B⊇A m Θ (B). (11)
The credibility function of A called bel Θ (A) is all the elements B ⊆ A which believe partially in A. This function can be interpreted as a minimum of belief in A. On the contrary, a plausibility function pl Θ (A) illustrates the maximum belief in A while commonality function q Θ (A) represents the sum of bba allocated to the superset of A. This function is very useful as shown below.

Combination rule

We consider M different experts who give mass m Θ i (i = 1, . . . , M) on each element A ⊆ Θ. Alternatively, it is possible to combine them using combination rules. There are several combination rules [START_REF] Sentz | Combination of evidence in Dempster-Shafer theory[END_REF] in the literature which differently address conflicts between sources. The most common rule is the conjunctive combination [START_REF] Ph | The combination of evidence in the transferable belief model[END_REF] where the resultant mass of A is obtained by:

m Θ (A) = B 1 ∩...B n =A ∅ M i=1 m Θ i (B i ) , ∀A ∈ 2 Θ . ( 12
)
The mass of the empty set is given by:

m Θ (∅) = B 1 ∩...B n =∅ M i=1 m Θ i (B i ) , ∀A ∈ 2 Θ . ( 13
)
This rule allows us to stay in the open world. However, this is not practical as calculations are difficult. It is possible to calculate resultant mass with communality functions where each mass m Θ i (i = 1, . . . , M) must be converted into its communality function to calculate the resultant communality function as follows:

q Θ (A) = M i=1 q Θ i (A). ( 14
)
The final mass is obtained by carrying out the inverse operation of Equation ( 11) [START_REF] Shafer | A mathematical theory of evidence[END_REF].

Pignistic probability

To make a decision on Θ, several operators exist such that maximum credibility or maximum plausibility with pignistic probability being the most commonly used operator [START_REF] Ph | Constructing the pignistic probability function in a context of uncertainty[END_REF]. This name comes from pignus, a bet, in Latin. This operator approaches the pair (bel,pl) by uniformly sharing a mass of focal elements on each singleton C i . This operator is defined by:

betP(C i ) = A⊂Θ,C i ∈A m Θ (A) |A|(1 -m Θ (∅)) , (15)
where |A| represents the cardinality of A. We choose the decision C i by evaluating max 1≤k≤n betP(C k ).

Continuous belief functions

The basic description of continuous belief functions was accomplished by Shafer [START_REF] Shafer | A mathematical theory of evidence[END_REF], then by Nguyen [START_REF] Nguyen | On random sets and belief functions[END_REF] and Strat [START_REF] Strat | Continuous belief functions for evidential reasoning[END_REF]. Recently, Smets [START_REF] Ph | Belief functions on real numbers[END_REF] extended the definition of belief functions to the set of reals R = R ∪ {-∞, +∞} and masses are only attributed to intervals of R.

Definitions

Let us consider I = {[x, y], (x, y], [x, y), (x, y); x, y ∈ R} as a set of closed, half-opened and opened intervals of R. Focal elements are closed intervals of R. The quantity m I (x, y) are basic belief densities linked to a specific pdf. If x > y, then m I (x, y) = 0. With these definitions, it is possible to define the same functions as in the discrete case. The interval [a, b] being a set of R with a ≤ b, the previous functions can be defined as follows:

bel R ([a, b]) = x=b x=a y=b y=x m I (x, y)dydx, ( 16
) pl R ([a, b]) = x=b x=-∞ y=+∞ y=max(a,x) m I (x, y)dydx, (17)
q R ([a, b]) = x=a x=-∞ y=+∞ y=b m I (x, y)dydx. (18)

Pignistic probability

The definition of pignistic probability for a < b is:

Bet f ([a, b]) = x=+∞ x=-∞ y=+∞ y=x |[a, b] ∩ [x, y]| |[x, y]| m I (x, y)dxdy. (19)
It is possible to calculate pignistic probabilities to have basic belief densities. However, many basic belief densities exist for the same pignistic probability. To resolve this issue, we can use the consonant basic belief density. A basic belief density is said to be "consonant" when focal elements are nested. Focal elements I u can be labeled as an index u such that I u ⊆ I u with u > u. This definition is used to apply the least commitment principle, which consists in choosing the least informative belief function when a belief function is not totally defined and is only known to belong to a family of functions. The least commitment principle relies on an order relation between belief functions in order to determine if a belief function is more or less committed than another. For example, it is possible to define on order based on the commonality function:

(∀A ⊆ Θ, q Θ 1 (A) ≤ q Θ 2 (A)) ⇔ (m Θ 1 ⊆ q m Θ 2 ). ( 20
)
The mass function m Θ 2 is less committed than m Θ 1 according to the commonality function. The function Bet f can be induced by a set of isopignistic belief functions Biso(Bet f ). Many papers [START_REF] Ph | Belief functions on real numbers[END_REF][START_REF] Ristic | Target classification approach based on the belief function theory[END_REF][START_REF] Caron | Least Committed basic belief density induced by a multivariate Gaussian pdf[END_REF] deal with the particular case of continuous belief functions with nested focal elements. For example, Smets [START_REF] Ph | Belief functions on real numbers[END_REF] proved that the least committed basic belief assignment m R for the commonality ordering attributed to an interval I = [x, y] with y > x related to a bell-shaped1 pignistic probability function is determined by2 :

m R ([x, y]) = θ(y)δ d (x -ζ(y)), (21)
θ(y) = (ζ(y) -y) dBet f (y) dy . ( 22
)
The resultant basic belief assignment m R is consonant and belongs to the set Biso(Bet f ). However, it is difficult to build belief functions in the particular case of multimodal pdfs because the frame of discernment has connected sets.

In [START_REF] Doré | Constructing of a consonant belief function induced by a multimodal probability density function[END_REF][START_REF] Vannobel | Continuous belief functions: singletons plausibility function in conjunctive and disjunctive combination operations of consonant bbds[END_REF], the authors propose a way to build belief functions with connected sets by using a credal measure and an index function.

Credal measure and index function

In [START_REF] Doré | Constructing of a consonant belief function induced by a multimodal probability density function[END_REF], the authors propose a way to calculate belief functions from any probability density function. They use an index function f and a specific index space I to scan the set of focal elements F :

f I : I → F , (23)
y -→ f I (y). ( 24
)
The authors introduce a positive measure µ Ω such that I dµ Ω (y) ≤ 1 describes unconnected sets. The pair ( f I , µ Ω ) defines a belief function. For all A ∈ Ω, they define subsets which belong to the Borel set:

F ⊆A = {y ∈ I| f I (y) ⊆ A}, (25)
F ∩A = {y ∈ I| f I (y) ∩ A ∅}, (26)
F ⊇A = {y ∈ I| f I (y) ⊇ A}. ( 27
)
From these definitions, they compute belief functions:

bel

Ω (A) = F ⊆A dµ Ω (y), (28)
pl

Ω (A) = F ∩A dµ Ω (y), (29)
q Ω (A) =

F ⊇A dµ Ω (y). ( 30
)
They continue their study by considering consonant belief functions. The set of focal elements F must be ordered from the operator ⊆. They define an index function f from R + to F such that:

y ≥ x =⇒ f (y) ⊆ f (x). ( 31
)
They generate consonant sets by using a continuous function g from R d to I = [0, α max [. The α-cuts are the set:

f I cs = {x ∈ R d |g(x) ≥ α}. ( 32
)
They finally define the index function:

f I cs : I = [0, α max ] → { f I cs (α)|α ∈ I}, (33) α -→ f I cs (α). ( 34
)
The information available is the conditional pignistic density Bet f . However, many basic belief densities exist for the same pignistic probability Bet f . The least commitment principle allows the least informative basic belief density to be chosen, where the focal elements are the α-cuts of Bet f such that3 :

dµ Ω (y)(α) = λ( f I cs (α))dλ(α). ( 35
)

Continuous belief functions and α-stable distribution

In this section, we model data distributions as a single α-stable distribution. We must however introduce the notion of plausibility function for an α-stable distribution. We first describe how to calculate the plausibility function knowing the pdf in R before we extend to in R d . We finally explain how we construct our belief classifier.

Link between pignistic probability function and plausibility function in R

The information available is the conditional pignistic density Bet f

[C i ] with C i ∈ θ. The function Bet f [C i
] is supposed to be bell-shaped for all α-stable distributions (proved by Yamazato [START_REF] Yamazato | Unimodality of infinitely divisible distribution functions of class L[END_REF]).

The plausibility function from a mass m R is obtained by an integral of Equation ( 22) between [x, +∞[:

pl R [C i ](I) = +∞ x (ζ(t) -t) dBet f (t) dt dt. (36)
By assuming that Bet f is symmetrical, an integration by parts can simplify Equation ( 36) :

pl R [C i ](I) = 2(x -µ)Bet f (x) + 2 +∞ x Bet f (t)dt. (37)
Now let us consider a particular case where symmetrical Bet f is an α-stable distribution (the parameter β = 0). We already know that Bet f (x) = f S α (β,γ,δ) (x). We can calculate

+∞ x
Bet f (t)dt by using the Chasles' theorem:

+∞ -∞ f S α (β,γ,δ) (t)dt = x -∞ f S α (β,γ,δ) (t)dt + +∞ x f S α (β,γ,δ) (t)dt. ( 38
)
By definition, a pd f has the quantity

+∞ -∞ f S α (β,γ,δ) (t)dt = 1 and
x -∞ f S α (β,γ,δ) (t)dt represents the definition of the α-stable cumulative density function F S α (β,γ,δ) .

Consequently, Equation (37) can be simplified:

pl R [C i ](I) = 2(x -µ) f S α (β,γ,δ) (x) + 2(1 -F S α (β,γ,δ) (x)). ( 39
)
The plausibility function related to an interval I = [x, y] can also be seen as the area defined under the α cut -cut such that α cut = Bet f (x). The notation pl[C i ](x) is equivalent to pl[C i ](I). Now us let consider an asymmetric α-stable probability density function. We must proceed numerically to calculate plausibility function at point x 1 > µ, with µ the mode of the probability density function (Figure 2). The plausibility function related to an interval I 1 = [x 1 , y 1 ] is defined by the area defined under the α cut -cut such that

α cut = Bet f (x 1 ): pl R [C i ](I 1 ) = x 1 -∞ Bet f (t)dt + (y 1 -x 1 )Bet f (x 1 ) + +∞ y 1 Bet f (t)dt. ( 40
)
By definition, a pd f has the quantity

+∞ y 1 f S α (β,γ,δ) (t)dt = 1 -F S α (β,γ,δ) (y 1 ) and x 1 -∞ f S α (β,γ,δ) (t)dt = F S α (β,γ,δ) (x 1 )
. In general, we know only one point y 1 . We estimate numerically x 1 such that f S α (β,γ,δ) (y 1 ) = f S α (β,γ,δ) (x 1 ). Finally, the plausibility function related to the interval I 1 is:

pl R [C i ](I 1 ) = 1 + F S α (β,γ,δ) (x 1 ) -F(y 1 ) + (y 1 -x 1 ) f (x 1 ). ( 41
)
In practice, we base the classification on several features defined for different classes Θ = {C 1 , .., C n }. For example, the Haralick parameters "contrast" and "homogeneity" have different values for the classes rock, silt and sand. The features are modeled by a probability density function because the features have continuous values. We can calculate a plausibility function related to its probability density functions by using the least commitment principle. Several plausibility functions associated to the same feature can be combined by using the generalized Bayes theorem [START_REF] Ph | Belief functions: The disjunctive rule of combination and the generalized Bayesian theorem[END_REF][START_REF] Delmotte | Target identification based on the transferable belief model interpretation of Dempster-Shafer model[END_REF] to calculate mass functions allocated to A of an interval I:

m R [x](A) = C j ∈A pl j (x) C j ∈A c (1 -pl j (x)). ( 42
)
For several features, it is possible to combine the mass functions with combination rules.

To validate our approach, we classify planes using kinematic data as in [START_REF] Ristic | Target classification approach based on the belief function theory[END_REF][START_REF] Caron | Least Committed basic belief density induced by a multivariate Gaussian pdf[END_REF] and compare the decision with the approach of Caron et al. [START_REF] Caron | Least Committed basic belief density induced by a multivariate Gaussian pdf[END_REF]. We associate a Gaussian probability density function for each speed's target (Figure 3):

• Commercial defined by the probability density function of speed S 2 (0, 8, 722.5).

• Bomber defined by the probability density function of speed S 2 (0, 7, 690).

• Fighter defined by the probability density function of speed S 2 (0, 10, 730).

We can observe that the decision is the same with the both approaches in the particular case of Gaussian probability density functions.

Link between pignistic probability function and plausibility function to

R d It is possible to extend plausibility function in R d .
In [START_REF] Caron | Least Committed basic belief density induced by a multivariate Gaussian pdf[END_REF], the authors calculate plausibility function in the Gaussian pdf situation of mode δ and matrix of covariance Σ. Mass function is built in such a way that isoprobability surfaces S i with 1 ≤ i ≤ n are focal elements. In R 2 , isoprobability points of a multivariate Gaussian pdf of mean δ and covariance matrix Σ are ellipses. In dimension d, the authors defined focal elements as the nested sets HV α enclosed by the isoprobability hyperconics HC α = {x ∈ R d |(x -δ) t Σ -1 (x -δ)}. They obtain bbd by applying the least commitment principle:

m R d (HV α ) = α d+2 2 -1 2 d+2 2 Γ( d+2 2 ) exp (- 1 2 α) with α ≥ 0. ( 43
)
Equation ( 43) defines a χ 2 distribution with d + 2 degrees of freedom. The plausibility function at point x belonging to a surface S i corresponds to the volume delimited by S i (we give an example in Figure 4 for an α-stable pdf). Consequently, the plausibility function at point x is defined by:

pl R d (x ∈ R d ) = α=+∞ α=(x-δ) t Σ -1 (x-δ) α d+2 2 -1 2 d+2 2 Γ( d+2 2 ) exp (- 1 2 α)dα. (44)
Equation ( 44) can be simplified by:

pl R d (x ∈ R d ) = 1 -F d+2 ((x -δ)(Σ) -1 (x -δ)). (45)
The function F d+2 is a cumulative density function of the χ 2 distribution with d + 2 degrees of liberty (d is the dimension of vector x).

The authors also calculate plausibility functions in the particular case of GMMs with their pdfs denoted by p GMM :

p GMM (x) = k=n k=1 w k N(x, δ k , Σ k ), (46)
where N(x, δ k , Σ k ) is the normal distribution of the k components with mean δ k and matrix of covariance Σ k and w k is the weight of each mixture. They assign belief to nested sets belonging to a component. Consequently, the plausibility function of the GMMs can be seen as a weighted sum of plausibility functions defined by each component of the mixture:

pl R d (x ∈ R d ) = 1 - k=n k=1 w k F d+2 ((x -δ k )(Σ k ) -1 (x -δ k )). ( 47
)
We want to extend the calculation of plausibility function for any α-stable pdf. However, there is no closed-form expression for α-stable pdf. We use the approach developed by Doré et al. [START_REF] Doré | Constructing of a consonant belief function induced by a multimodal probability density function[END_REF] to build belief functions.

Belief classifier

Let us consider a data set with N samples from d sensors. For example, each feature of a vector x ∈ R d can be seen as a piece of information from a sensor. The classification is divided into two steps. N × p (with p ∈]0, 1[) samples are first picked out randomly from the data set for the learning base, noted X, such that:

X =              x 1 1 . . . x N×p 1 . . . . . . . . . x 1 d . . . x N×p d             
All columns of X belong to a class C i with 1 ≤ i ≤ n. Probability density functions (Gaussian or α-stable models) are estimated from samples belonging to classes. The rest of the samples is used for the test base (N × (1p) vectors). We use a validation test to determine if samples belong to an estimated model. If the test is not valid, we stop the classification, otherwise we continue the classification (Figure 5). The classification step diagram in one and d dimensions is shown in Figure 6.

Plausibility functions knowing classes for x are calculated either from their probability density functions by using Equation ( 41) for an unsymmetric α-stable probability density function or by using Equation ( 45) for a Gaussian probability density function. We obtain d mass functions (one for each feature) at point x with the generalized Bayes theorem [START_REF] Ph | Belief functions: The disjunctive rule of combination and the generalized Bayesian theorem[END_REF][START_REF] Delmotte | Target identification based on the transferable belief model interpretation of Dempster-Shafer model[END_REF] (Equation ( 42)). These d mass functions are combined by a combination rule to obtain a single mass function (section 3.1.2). Finally, the mass function is transformed into pignistic probability (Equation ( 19)) to make the decision.

It is also possible to work with a vector x of d dimensions. For an α-stable probability density function, plausibility functions are calculated for each feature by using the approach of Doré et al [START_REF] Doré | Constructing of a consonant belief function induced by a multimodal probability density function[END_REF] (section 3.3). For a Gaussian probability density function, we use Equation [START_REF] Vannobel | Continuous belief functions: singletons plausibility function in conjunctive and disjunctive combination operations of consonant bbds[END_REF] to calculate plausibilty functions. We calculate one mass function at point x by using the generalized Bayes theorem (Equation ( 42)). There is no combination step because we are calculating a single mass function. We use Equation [START_REF] Zolotarev | One-dimensional stable distributions[END_REF] to transform the mass function into pignistic probabilities. The decision is chosen by using the maximum number of pignistic probabilities.

Application to pattern recognition

The aim is to build a belief classifier and to perform a classification of synthetic and real data by estimating features using a Gaussian and α-stable pdf. In [START_REF] Fiche | Influence de l'estimation des paramètres de texture pour la classification de données complexes[END_REF], synthetic data are classified by modeling features using Gaussian and α-stable mixture models. By observing confidence intervals, the hypothesis of α-stable mixture models is significantly better than the hypothesis of Gaussian mixture models. However, when the number of Gaussian distributions increases, classification accuracies are significantly the same. Images from a side-scan sonar are automatically classified by extracting Haralick features [START_REF] Haralick | Textural features for image classification[END_REF][START_REF] Haralick | Statistical and structural approaches to texture[END_REF]. However, classification accuracies are roughly the same.

In this section, we limit algorithms with a vector of features in dimension d ≤ 2. Indeed, the generalization of α-stable pdf in dimension d > 2 increases CPU time because there is no closed-form expression. Consequently, we distinguish two cases during the classification step:

• The one dimension case: each dimension is considered as a feature.

• The two case: we considered a vector of two features.

We also use a statistical test called Kolmogorov-Smirnov test (K-S test) to evaluate the quality of each model. A Kolmogorov-Smirnov test (K-S test) with a significance level of 5 % is also used to determine if two datasets differ significantly. Let us assume two null hypotheses:

• H 1 : "Data follow a Gaussian distribution".

• H 2 : "Data follow an α-stable distribution".

The notation H 1 and H 2 are used for the rest of the paper. If the test rejects a null hypothesis H i , we stop the classification. Otherwise, if the test is valid, we then classify the samples belonging to the test base by using the belief classifier.

The results obtained with the belief functions are compared using a Bayesian approach. The prior probability p(C i ) is first calculated corresponding to the proportion of each class in the learning set. For each class, the application of Bayes theorem gives posterior probabilities:

p(C i /x) = p(x/C i )p(C i ) n u=1 p(x/C u )p(C u ) . ( 48
)
Finally, the decision is chosen by using the maximum nimber of posterior probabilities. We first present a classification of synthetic data, belonging to Gaussian distributions, by modeling features using a Gaussian distribution and α-stable pdf. We then compare the α-stable and Gaussian model with synthetic data belonging to α-stable distributions. The same approach is finally applied to real data from a mono-beam echo-sounder.

Application to synthetic data generated by Gaussian distributions

In this subsection, we show that the α-stable and Gaussian model have the same behavior during the classification when synthetic data are generated by Gaussian distributions. We simulated three Gaussian pdfs in R × R (c.f. Table 1). Each distribution holds 3000 samples. A mesh between [-4, 4] × [-4, 4] enables the level curves of pdf to be plotted (Figure 7).

Results

We generated data and split samples randomly into two sets: one third for the learning base and the rest for the test base. The learning base was used to estimate each mean and variance for the Gaussian distributions and each parameter α, β, γ and δ for the α-stable distributions [START_REF] Fama | The behavior of stock-market prices[END_REF][START_REF] Koutrouvelis | Regression-type estimation of the parameters of stable laws[END_REF][START_REF] Mcculloch | Simple consistent estimators of stable distribution parameters[END_REF]. We needed to make a mesh between [-4, 4] × [-4, 4] to calculate plausibility functions in R 2 with the α-stable model. Consequently, samples were chosen possessing features with dimensions within [-4, 4].

H

The one dimension case.

For each class C i , we consider each feature as a source of information. The estimated probability density functions can be seen as pignistic probability functions. For each pignistic probability function, we associate a mass function which will subsequently be combined with other mass functions.

The K-S test is valid for the two null hypotheses H 1 and H 2 (Table 2). We can assert that samples are drawn from the same distribution. The proposed method performs well in the particular case of synthetic data belonging to Gaussian distributions. However, the classification rates (Table 3) are very low because there is confusion between classes (Figure 8). The classification accuracy obtained with the Bayesian approach is significantly the same as the belief functions. Indeed, the Bayesian performs well provided that the probability density functions are wellestimated. The classification results depend on the prior probabilitiy estimation: if a prior probability of a class is underestimated, the confusion between classes increases.

We extended our study to a vector of two dimensions to try to improve the classification rate.

The two dimension case.

We generalized the study by considering a vector of two features. The learning base was used to estimate each mean and matrix of covariance for the Gaussian distributions (Figure 9) and each parameter α, σ, θ and δ for the α-stable distributions [START_REF] Nolan | Estimation of stable spectral measures[END_REF] (Figure 10). As in one dimension, the two null hypotheses H 1 and H 2 were verified (Table 4). The classification results (Table 3) are somewhat less than impressive because there is confusion between classes (between C 1 and C 2 cf. Figure 7). Furthermore, the work in two dimensions enables better results to be obtained than the combination. The Bayesian approach gives significantly the same results as the theory of belief functions. Indeed, the estimations of model and the estimation of prior probabilities are well-estimated.

We extended our study by considering synthetic data generated by α-stable distributions.

Bayesian approach Belief approach

Classification accuracies (%) 95 % confidence intervals Classification accuracies (%) 95 % confidence intervals

H

Application to synthetic data generated by α-stable distributions

In this subsection, we show the interest in modeling features in one and two dimensions with a single α-stable distribution. 4: Statistical values for a Kolmogorov-Smirnov test with a significance level of 5 % (p-value: the critical value to reject the null hypothesis, ksstat: the greatest discrepancy between the observed and expected cumulative frequencies).

α σ θ δ First α-stable pdf (C 1 ) 1.5 1 1 0 π 2 0 0 Second α-stable pdf (C 2 ) 1.5 1 1 0 π 2 2 1.4 Third α-stable pdf (C 3 ) 1.5 1 1 0 π 2 1 0.5
Table 5: Numerical values for the α-stable pdf.

Presentation of the data

Three classes of artificial data sets were generated from 2D α-stable distributions [START_REF] Modarres | A method for simulating stable random vectors[END_REF] (c.f. Table 5 and [START_REF] Nolan | Calculation of multidimensional stable densities[END_REF] for the significance of σ and θ) with 3000 samples. A mesh between [-4, 4] × [-4, 4] enables the level curves of pdf to be plotted (Figure 11).

Results

The one dimension case. We can assert that the estimation with the α-stable hypothesis is better than the Gaussian hypothesis (Figure 12). There is a difference between the mode of the Gaussian model and the mode of α-stable model. Indeed, the mean, calculated from samples, for the Gaussian model does not correspond to the mode when samples do not belong to a Gaussian law. Moreover, the values of pdfs are not the same compared to the real pdfs. 6: Statistical values for a Kolmogorov-Smirnov test with a significance level of 5 % (p-value: the critical value to reject the null hypothesis, ksstat: the greatest discrepancy between the observed and expected cumulative frequencies).

p-value ksstat

The K-S test is not satisfied for the null hypothesis H 1 for each class and each component. Consequently, we stopped the classification step for the Gaussian model. However, the K-S test is valid for the null hypothesis H 2 (Table 6). We can observe that the results obtained with the theory of belief functions are significantly the same as the results obtained using the Bayesian approach. This phenomenon can be explained by the fact that prior probabilities were well-estimated. Consequently, we changed voluntarily the values of prior probabilities. We fixed arbitrarily the prior probabilities: p(C 1 ) = 1/6, p(C 2 ) = 2/3 and p(C 3 ) = 1/6. The classification accuracy decreased to 53.10 % (Table 7). Indeed, the learning base and the test base can sometimes be significantly different. For example, the prior probability of class sand can be greater than the prior probabilities of classes rock and silt, although it does not represent the truth, and the classification rate of class sand can be favoured. This phenomenon introduces confusion between classes and the classification rate can decrease. Consequently, the learning step is an important step for the Bayesian approach and can have a bad effect on the classification rate whereas the theory of belief functions is less sensitive to the learning step. The two dimension case.

Bayesian approach

We considered a vector of two features. In Figure 13, we draw the iso-contour lines of data estimated with the Gaussian model and in Figure 14 the iso-contour lines of data estimated with the α-stable model As in one dimension, the null hypothesis H 1 is rejected at significance level 5 %, but the null hypothesis H 2 is valid (Table 8).

The classification results (Table 9) are somewhat less than impressive because there is confusion between classes. Furthermore, we obtain the same classification accuracies with the vector of two dimensions compared to the combination of two features. However, the classification decreases when the prior probabilities are not well-estimated (Table 7). We also compared these results with a GMM. It is well-known that the GMM can accurately model an arbitrary continuous distribution. This model estimation can be performed efficiently using the Expectation-Maximization (EM) algorithm. We compared classification accuracies by increasing the number of components. We drew the classification accuracy (Figure 15) in relation to the number of components. Let us assume the null hypothesis:

• G n : "Data follow a Gaussian mixture model with n components".

The K-S test rejects the null hypothesis G n for a number of components n < 4. The representations of iso-contour lines (Figure 16, Figure 17 and Figure 18) confirm the K-S test. Furthermore, the classification accuracies are lower than classification rates with a single α-stable pdf.

However, we can not increase indefinitely the number of components because a phenomenon of overfitting can appear (for example the representation of iso-contour by using a GMM with 5 components). Consequently, the number of 4 components seems to be a good compromise between the overfitting and good estimation. The classification accuracies are roughly the same between the single α-stable model and the GMM with 4 components. p-value ksstat C 1 0.0683 0.1003 C 1 0.3694 0.0688 C 3 0.1154 0.0932 Table 8: Statistical values for a Kolmogorov-Smirnov test with a significance level of 5 % (p-value: the critical value to reject the null hypothesis, ksstat: the greatest discrepancy between the observed and expected cumulative frequencies).

Application to real data from a mono-beam echo-sounder 5.3.1. Presentation of data

We used a data set from a mono-beam echo-sounder given by the Service Hydrographique et Océanique de la Marine (SHOM). Raw data represents an echo signal amplitude according to time. In [START_REF] Pouliquen | Sea-bed identification using echo-sounder signals[END_REF], the authors classify seven types of sea floor by comparing the time envelope of echo signal amplitude with a set of theoretical references curves. In our application, raw data (Figure 19) were processed using the Quester Tangent Corporation (QTC) software [START_REF] Caughey | Sea bottom classification from echo sounding data[END_REF] to

Conclusions

In this paper, we have shown the advantages in using the α-stable distributions to model data with heavy-tails. We considered the imprecision and uncertainty of data for modeling pdf and the theory of belief functions offers a way to model these constraints. We have examined the fundamental definitions of this theory and have shown a way to calculate plausibility functions when the knowledge of sensors is an α-stable pdf. We finally validated our model by making a comparison with the model examined by Caron et al. [START_REF] Caron | Least Committed basic belief density induced by a multivariate Gaussian pdf[END_REF] where the pdf is a Gaussian.

Synthetic and real data were finally classified using the theory of belief functions. We estimated the learning base, calculated plausibility functions, combined plausibility functions and calculated maximum pignistic probability to obtain the classification accuracy. The K-S test shows that synthetic data can be modeled by an α-stable model. The proposed method performs well for synthetic data. The consideration of two features improves the classification rate compared to the combination of two features for the Gaussian data. This study was then extended to a real application by using features extracted from a mono-beam echo-sounder. The Gaussian model is limited when modeling features compared to the α-stable model because the α-stable model has more degrees of freedom. To sum up, the interest in using an α-stable model is that data, in both one or two dimensions, can be modeled. The problem with the α-stable model is the computation time. Indeed, we proceed numerically to calculate plausibility functions (discretization for the bivariate α-stable pdf). Furthermore, it is difficult to generalize in dimension d > 2 without increasing CPU time.

We present preliminary results about classification problem by modeling features with α-stable probability density functions in dimension d ≤ 2. In future works, we plan to make experiments in the high-dimensional case with d ≥ 3. One solution in perspective to improve the computation time is to optimize the code in another language such that C/C++. Another perspective to this work is the use of other features from sensors to improve the classification accuracy, images from side-scan sonar.

  with x = ζ(y) satisfying Bet f (ζ(y)) = Bet f (y) and θ(y):

5. 1 . 1 .First

 11 Presentation of the data µ V

  ksstat p-value ksstat C 1 0.1509 0.0873 0.1509 0.0873 C 1 0.0490 0.0650 0.1997 0.0836 C 3 0.1399 0.0870 0.1665 0.0841 Table

  Influence of parameter α with β = 0, γ = 1 and δ = 0. Influence of parameter β with α = 1.5, γ = 1 and δ = 0. Influence of parameter γ with α = 1.5, β = 0 and δ = 0. Influence of parameter δ with α = 1.5, β = 0 and γ = 1.

Figure 1 :Figure 2 :

 12 Figure 1: Different α-stable pdf.

Figure 3 :

 3 Figure 3: Representation of probability density function of speed and the pignistic probabilities calculated from the Fiche et al. and Caron et al. [43] approaches.

Figure 4 :Figure 5 :

 45 Figure 4: Isoprobability surface of f .

Figure 6 :

 6 Figure 6: Scheme of classification step in one and d dimensions.

Figure 7 :

 7 Figure 7: Iso-contour lines of generated Gaussian pdf.

Figure 8 :

 8 Figure 8: Empirical pdf and its estimations (The first row corresponds to the first feature and the second row corresponds to the second feature).

Figure 9 :

 9 Figure 9: Iso-contour lines of data estimated with a Gaussian model.

Figure 10 :

 10 Figure 10: Iso-contour lines of data estimated with an α-stable model.

Figure 11 :

 11 Figure 11: Iso-contour lines of generated α-stable pdf.

3 Figure 12 :

 312 Figure 12: Empirical pdf and its estimations (The first row corresponds to the first feature and the second row corresponds to the second feature).

Figure 13 :

 13 Figure 13: Iso-contour lines of data estimated with a Gaussian model.

Figure 14 :

 14 Figure 14: Iso-contour lines of data estimated with an α-stable model.

  interval for the α-stable model Confidence intervals for the Gaussian model

Figure 15 :

 15 Figure 15: Classification rates according the number of components.

Figure 16 :

 16 Figure 16: Representation of iso-contour by modeling data with a mixture of Gaussian for the class C 1 (from left to right: 2, 3, 4 and 5 components).

Figure 17 :

 17 Figure 17: Representation of iso-contour by modeling data with a mixture of Gaussian for the class C 2 (from left to right: 2, 3, 4 and 5 components).

Figure 18 :Figure 19 :

 1819 Figure 18: Representation of iso-contour by modeling data with a mixture of Gaussian for the class C 3 (from left to right: 2, 3, 4 and 5 components).

Figure 20 :

 20 Figure 20: Running sample variances.

Figure 21 :

 21 Figure21: Empirical pdf and its estimations (The first row corresponds to the feature called "third quantile calculated on echo signal amplitude" and the second row corresponds to the feature called "25th quantile calculated on cumulative energy").

Figure 22 :

 22 Figure 22: Level curves of three empirical α-stable distributions with 10000 samples (The first column refers to the class rock, the second column refers to the class sand and the third column refers to the class silt).

Figure 23 :

 23 Figure 23: Level curves with an α-stable model estimation (The first column refers to the class rock, the second column refers to the class sand and the third column refers to the class silt).

Figure 24 :

 24 Figure 24: Level curves with a Gaussian model estimation (The first column refers to the class rock, the second column refers to the class sand and the third column refers to the class silt).

Table 1 :

 1 Numerical values for the Gaussian pdf.

Table 2 :

 2 Statistical values for a Kolmogorov-Smirnov test with a significance level of 5 % (p-value: the critical value to reject the null hypothesis, ksstat: the greatest discrepancy between the observed and expected cumulative frequencies).

			1	H 2
			p-value ksstat p-value ksstat
	C 1	first feature	0.7021 0.0542 0.9241 0.0422
		second feature 0.8799 0.0452 0.9241 0.0422
	C 2	first feature	0.8693 0.0464 0.3736 0.0712
		second feature 0.6855 0.0557 0.3736 0.0712
	C 3	first feature	0.8437 0.0464 0.4150 0.0667
		second feature 0.6602 0.0551 0.7866 0.0493

Table 3 :

 3 Classification accuracies and 95 % confidence intervals.

Table

	C 1	first feature	0.1758 0.0851
		second feature 0.7614 0.0517
	C 2	first feature	0.8487 0.0458
		second feature 0.4794 0.0630
	C 3	first feature	0.5505 0.0621
		second feature 0.2753 0.0776

Table 7 :

 7 Classification accuracies and 95 % confidence intervals with the prior probabilities p(C 1 ) = 1/6, p(C 2 ) = 2/3 and p(C 3 ) = 1/6.

i.e. the pdf is unimodal with a mode µ, continuous and strictly monotonous increasing (decreasing) at left (right) of the mode.

δ d refers to the Dirac's measure.

λ refers to the Lebesgue's measure.

Acknowledgements

The authors would like to thank the Service Hydrographique et Océanique de la Marine (SHOM) for the data and G. Le Chenadec for his advice concerning the data.

Belief approach

Classification accuracies (%) 95 % confidence intervals Classification accuracies (%) 95 % confidence intervals obtain some features, which were normalized between [0,1]. These data had already been used for the navigation of an Autonomous Underwater Vehicle (AUV) [START_REF] Martin | Autonomous Underwater Vehicle sensors fusion by the theory of belief functions for Rapid Environment Assessment[END_REF]. The frame of discernment is Θ = {silt, rock, sand}, with 4853 samples from silt, 6017 from rock and 7338 from sand. The selection of features plays an important role in classification because the classification accuracy can be different according to the features. In [START_REF] Karoui | Seabed segmentation using optimized statistics of sonar textures[END_REF], the author gave an overview of methods based on features to classify data and choose features which discriminate all classes. Some features calculated by the QTC software can be modeled by an α-stable pdf. There is a graphical test called converging variance test [START_REF] Granger | Infinite Variance" and Research Strategy in Time Series Analysis[END_REF] which enables us to determine if samples belong to an α-stable distribution. This test calculates the estimation of variances by varying the number of samples. If the estimation of variance diverges, the samples can belong to an α-stable distribution. We give an example of this test for the feature called "third quantile calculated on echo signal amplitude" of the class rock (Figure 20). We chose the features called the "third quantile calculated on echo signal amplitude" and the "25th quantile calculated on cumulative energy".

Results

We randomly select 5000 samples for the data set. Half the samples were used for the learning base and the rest for the test base. We first consider features in one dimension, i.e. two mass functions were calculated and combined to obtain a single mass function.

The one dimension case. We can observe that the assumption of the α-stable model can easily accommodate the data compared to the Gaussian model (Figure 21).

The α-stable hypothesis is valid with the K-S test whereas the K-S test rejects the Gaussian hypothesis (Table 10). The approach using the theory of belief functions (classification accuracy of 82.68 %) give better results than the Bayesian approach (classification accuracy of 80.64 %) but not significantly. This phenomenon can be explained by the fact that the estimated models and prior probabilities are well-estimated.

The two dimensions case. We extended the study by using a vector of two features. The level curves of empirical data (Figure 22) show there is confusion between the class silt and the class sand. The α-stable model (Figure 23) considers the confusion between classes compared to the Gaussian model (Figure 24


Louis Bertrand

Vincent Caradec

Jean-Sébastien Eideliman

Disability and Employability: Professional Categorisations and Individual Experiences at the Boundaries of Disability

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

At the edges of disability

If language is everywhere and always an instrument of action and power [START_REF] Bourdieu | Ce que parler veut dire[END_REF], the vocabulary used in the domain of disability has particular importance. As a result, there is an abundance of conceptual definitions of and approaches to disability produced by individual researchers [START_REF] Thomas | How is Disability Understood? An Examination of Sociological Approaches[END_REF][START_REF] Ville | Introduction à la sociologie du handicap: histoire, politiques et expérience[END_REF] and international institutions (UN 1 , WHO 2 , OECD 3 , INDCP 4 ) aiming to clarify the variety of notions in use and draft a common framework. This is one of the roles of the WHO"s international disability classifications, which are continuously updated and subject to endless debate (Alter, European Journal of Disability Research, 2013). Since the International Statistical Classification of Diseases (ICD) granted a specific branch for the field of disability with the creation of the International Classification of Impairments, Disabilities and Handicaps (ICIDH) in 1980, disability has become accepted as a complex and multidimensional notion integrating, at the minimum, organic, functional, and social aspects. The main disagreements have concerned questions of form (how to build a field that is not entirely defined by negative terms?) as well as fundamental questions, concerning in particular the inclusion of the surrounding environment in the very definition of disability. As others have aptly summed it 1 The United Nations 2 World Health Organization 3 Organisation for Economic Co-operation and Development 4 International Network on the Disability Creation Process up [START_REF] Barnes | Exploring Disability: ASociological Introduction[END_REF], the individual model of disability (disability is fundamentally the consequence of a malfunction internal to the individual) and the social model (disability is fundamentally the consequence of a society that builds obstacles to social participation for certain categories of individuals) are in constant tension, although the former long seemed to be in a hegemonic position until it gave way to the accumulating assaults of the Quebecois classification (Fougeyrollas et alii., 1998) and international disabled persons associations [START_REF] Davis | The Disability Studies Reader[END_REF].

These definition difficulties did nothing to facilitate the 20th century production of the field of disability, in both the social (through new social policies) and sociological [START_REF] Bourdieu | The Genesis of the Concepts of Habitus and Field[END_REF] senses of the term. Although there is a core where the variety ofcriteria applicable todisability overlaps, more surprisingly there are also a numerous situations withstrikinginconsistencies, and this among all the applied criteria. In France, the vast national study Handicaps-Incapacités-Dépendance (Handicaps-Incapacities-Dependency), conducted in the late 1990s, was partly conceived with the goal of tallying the number of disabled persons, but it only multipliedinterrogations. As J.-F. Ravaud, A. Letourmy and I. [START_REF] Ravaud | Identifying the Population with Disability: the Approach of an INSEE Survey on Daily Life and Health[END_REF] have shown, the percentage of disabled people in the French population ranges from 4% to 40%, depending on whether a narrow definition is applied, based on the official recognition of a disability, or a wide definition, based on the declaration of at least one impairment, with a variety of intermediary positions such as self-declaration of a disability (10% of the French population responds to this criteria, and this figure is the one most often chosen and cited in the press and scientific reports). As the rosettes of Jean-François Ravaud, Alain Letourmy and Isabelle Ville (ibid.,p. 537 sqq.) suggest, disability may be represented by the form of a flower with arelatively tight centre (where all the defining criteria come together) and many petals, each representing conformity with a given criteria.A significant proportion of those declaring themselves disabled thus do not correspond to any objective criteria allowing them to be counted in this category, while inversely, an equally significant proportion of people with official"handicapped" recognition do not declare themselves as disabled.

This special issue is devoted to the idea that approaching disability from the periphery has a heuristic value that allows an improved understanding of just what the field of disability is. By paying attention to borderline situations, by examining cases where disability merges into other qualifications (validity, illness, poverty…), by positioning ourselves on the boundaries shaping the field of disability, we learn as much, if not more, about its core as we would by addressing the most typical figures of disability head-on. Using Ravaud, Letourmy and Ville"s image of the rosette, one might say that our approach is focused not on the centre of the flower where the overlapping criteria (official disability recognition, presence of everyday limitations, self-designation as disabled person, etc.) make situations conceptually obvious, but instead on the petals, where the various dimensions of disability disconnect.

From this angle, the domain of employment and the issue of the recognition of "handicapped workers" provide particularly interesting material for analysis.

The five contributions gathered in this issue thus address various borders of disability while sharing a common theme, the relationship to employment. The first two deal with the boundary between disability and illness; the first, (Jacques Rodriguez, "A course of treatment: Putting people with tuberculosis to work in England and France in the 1920s"), questions the place of work in the treatment of people suffering from tuberculosisin the early 20 th century, when the category of disability was still only in its infancy but ripe for development in these early assemblies of sick people with irreversible physical effects [START_REF] Ville | From Inaptitude for Work to Trial of the Self: The Vicissitudes of Meanings of Disability[END_REF]; the second (Audrey Parron, "Autonomy issues for young adults dealing with psychic disorders") focuses on mentally ill youths" transition to adulthood, at the moment when the issues of their autonomy and a potential recognition of psychic disability, possible under French law since 2005, take urgency. The third contribution (Samuel Neuberg, "Poverty as a situation of disability: Social workers" reticence to back Active Solidarity Income (RSA) beneficiaries" requests for Disabled Adults Allowance (AAH)") also deals with programs for gaining autonomy and support for finding employment, but in this case focusing on people receiving social assistance, on the border between disability and social marginality. Lastly, the final two contributions address disabled persons more explicitly, questioning the boundary between disabled workers and disabled persons. The first (Louis Bertrand, Vincent Caradec and Jean-Sébastien Eideliman, "Situating disability. The recognition of "handicapped workers" in France") deals with the relative weight of social situations and individual characteristics in the recognition of "handicapped workers" in the French setting, while the second (TrudieKnijn and Frits van Wel, "Better at work: Activation of partially disabled workers in the Netherlands") addresses the effects of new policies for the recognition of partial occupational disability on individuals in the Netherlands.

Between the right to work and the right to not work

The origins of assuming political and social responsibility for disabled people are intimately connected to employment issues. In the history of disability in France, Henri-Jacques [START_REF] Stiker | A History of Disability[END_REF] singles out historical efforts by the State, economically and socially liberal at the time, to assure a form of national solidarity with people injured at work (starting with the 1898 law on work accidents), disabled veterans (after World War I), and people ill with tuberculosis (following the epidemics of the late 19 th and early 20 th centuries). A pair of issues was raised from the very first measures addressing these populations: the occupational reclassification of individuals who fell victim to political and social processes (industrialization, war, urbanization) and the establishment of pensions and benefits. Put in other terms, the right to work and the right to not work are jointly affirmed, the charge of determining who will benefit from which right being left to the institutions charged with administering these populations.

The term "right" used here containsa fundamental ambiguity, though.Is it really a matter of rights that individuals may assert, or is it instead a matter of injunctions that individuals might have trouble resisting? As Jacques Rodriguez shows in his commentary on two texts written by doctors, work combines some very different values in the first rehabilitation centres in France and England. Sometimes vaunted for its therapeutic values, sometimes advocated for giving social and economic utility back to disadvantaged individuals, the doctors behind these pioneering initiatives handle work as aninstrument for governing the population. Although the English case gives greater emphasis to the economic value of work, a blend of moral, economic, and therapeutic values is found on both sides of the Channel.

There are as many histories of taking charge ofdisability as there are national histories [START_REF] Woodill | From Charity and Exclusion to Emerging Independence: An Introduction to the History of Disabilities[END_REF]. Specific systems were gradually put in place for the employment of people who were not yet called disabled. Studies based on international comparisons [START_REF] Velche | L'emploi des personnes handicapées en Europe : où les politiques adoptées traduisent les différences culturelles[END_REF][START_REF] Oakes | Perspectives on Disability, Discrimination, Accomodations and Law. A Comparison of the Canadian and American Experience[END_REF] put countries with established forms of affirmative action into opposition, primarilybased on those with disabled-worker quotas for large businesses (such as Germany, Austria, Spain, France, Italy) and others that borrowed a page from the civil rights movement and targetedtheir legislation on fighting discrimination against disabled workers (including Canada, the United States, the United Kingdom, Sweden). The Netherlands is in an intermediate position, since it recently became possible forlarge companies to negotiate the establishment of quotas with social partners without any national obligation having been voted. These systems differ in many other ways, from the generosity of benefits to the relative focus of measures toward the most disabled people [START_REF] Velche | L'emploi des personnes handicapées en Europe : où les politiques adoptées traduisent les différences culturelles[END_REF], showing the diverse ways the right to work and the right not to work may be arbitrated in specific national contexts.

Beyond this national diversity, there are similarities to be found in how these arbitrations have developed. Once again, the Netherlands provides an interesting example, as TrudieKnijn and Frits van Wel show: while policies for disabled persons had long favoured a high rate of inactive people living off rather generous and widely granted disability pensions, there was an about-face in the early 2000s. In most cases disability pensions were replaced with income supplements or unemployment benefits, which vary according to the rate of disability recognized by the administration and are conditional on active efforts to find work.

If the Netherlands"s case is extreme, it dramatically demonstrates the trend that the right to work is gaining the upper hand over the right not to work. In the French case, for example, after the 1970s-1980s period when the establishment of the Allocation AdulteHandicapé (Handicapped Adult Benefit; AAH; created in 1975) allowed a certain number of people to develop a feeling of social utility outside of the sphere of the labour market while not compounding the already high unemployment figures [START_REF] Ville | Le handicap comme épreuve de soi. Politiques sociales, pratiques institutionnelles et expérience[END_REF], a variety of reforms in the field of disability sapped the justifiability of not making an effort to rejoin the workforce.

In consequence, the opposition between AAH beneficiaries, who since 1975 had been considered at least temporarily exempt from looking for work, and beneficiaries of the Reconnaissance de la qualité de travailleurhandicapé(Recognition of the Quality of Handicapped Workers; RQTH), who to the contrary were considered to be workers disabled in strength or in actions, faded away in the 2000s [START_REF] Bertrand | Politiques sociales du handicap et politiques d"insertion : continuités, innovations, convergences[END_REF]. Not only could AAH and RQTH henceforth be combined, but disability professionals had to examine the working capacities of all AAH claimants. In the name of disabled persons" right to work, an injunction to be as integrated as possible into the job market is gradually established, to the detriment of other forms of non-market occupational engagements [START_REF] Ville | To Work or not to Work?The Occupational Trajectories of Wheelchair Users[END_REF].

Employability and autonomy

Concomitant with these developments, the notion of employability experienced rapid success on both sides of the Atlantic in the latter part of the 20 th century [START_REF] Gazier | L'employabilité: brève radiographie d'un concept en mutation[END_REF]. It was also used in the field of disability, which is unsurprising since disabled persons have a higher likelihood than others of being both less productive and more discriminated against on the employment market. But in addition to its descriptive character, the notion of employability is also a template orienting how one might view relationships with employment: thinking in terms of employability makes employment an unambiguously desirable good and transforms the opposition employable/unemployable (which in the field of disability could refer to the opposition between pensioned people unfit for work and disabled workers) into a continuum of degrees of employability. One could draw parallels with the notion of educatability, whose success in the sub-field of disabled childhood came with the idea that no child was uneducatable (as was thought and officially affirmed until the latter half of the 20 th century) and that it was rather a question of finding the suitable pedagogy for each child (Chauvière and Plaisance, 2000) so that all might receive an education, perceived as an inalienable collective good. In this special journal issue, although employment appearsto bea more or less distant possibility depending on the article concerned, it at least remains a horizon that orients support professionals" practices, and to a lesser extent their publics" as well. In the first articles this horizon seems unattainable (particularly for tuberculosis sufferers in the early 20 th century, and to a lesser degree for youth suffering from psychic problems and beneficiaries of social assistance), while it seems distinctly closer, although challenging to attain, in the latter two contributions (for disabled workers in France and partially unfit workers in the Netherlands).

Considering this variety of situations through the lens of employment prompts us to look into the professional worlds that have developed around these publics in search of, or at least perceived as lacking, employability. There are many such worlds and they do not necessarily communicate easily amongst themselves. Naturally, there are some significant differences in employment support professionals" practices from one country to the next, as shown by the examples developed here of France (representing a hybrid case) and the Netherlands (more activationist) (Barbieret Ludwig-Mayerhofer, 2004), and from one sector to another as well. Yet people likely to be considered disabled frequently navigate between several sectors, especially the medical, social service, and medico-social. Employment is a concern in all these sectors, but it assumes different connotations for the various concerned professionals: doctors see itmore as a vector of healing and moralization (see contributions by Jacques Rodriguez and Audrey Parron), social workers in terms of social integration and autonomy(Audrey Parron and Samuel Neuberg), and disability professionals as a path to normalization and social integration (Bertrand, Caradec and Eideliman; Knijn and van Wel).

For many of these professionals, doing one"s work well consists of supporting their publics on their paths to employmentnot necessarily bringing them to the destination, but getting them to move forward in what they think is the right direction. The quest for improved employability is thus often confused with the search for greater autonomy, which has become a cardinal value in contemporary societies [START_REF] Ehrenberg | The weariness of the self: Diagnosing the History of Depression in the Contemporary Age[END_REF]. In the case of young adults with psychic problems, Audrey Parronclearly demonstrates that the professions supporting them (doctors and social workers) do not aim for just any form of autonomy, but pursue only what they consider "good autonomy," which may in fact be strictly supervised by institutions. What is important is not that young adults be able to get by at any cost (which could lead to bad autonomy resulting in forms of social marginalisation), but that they advance toward a responsible adult life in society, evaluated according to a given number of criteria for social integration (housing, employment, forming romantic partnerships…) This perspective and these criteria are also found among the social workers that Samuel Neuberg studied. They also see the path to employment as a long and winding road, punctuated with way-posts and "impediments" that they work to remove one by one. These impediments often concern the same kinds of material elements (housing, cleanliness, transportation, child care…), which in this case serve as so many points for professionals to addresshence giving themselves and their publics the feeling that things are moving in the right direction. In this context, disability is thought of as aparticular sort of impediment that prevents employment and progress, to the point that disability benefitclaims are perceived as a last resort that in a way indicate support"s failure.

Shifting now to the disability professionals" side, as the last two articles do, we see that support in moving toward employment does not end with the establishment of a claim to benefits, since it is possible to combine benefits and employment according to different modalities in different countries. But here again, a significant proportion of professional practices are guided by a rationale of moving claimants toward the goal of increasing employability.

Administrative categories and experiences of disability

These articles prompt us to think about categorization in the domains of handicap and employmentconjointly, and to wonder what is at issue in these categories for professionals and individuals alike. As one might expect, the issues differ depending on which side of the counter you are on: articles addressing the professionals" perspective (those by Jacques Rodriguez and Samuel Neuberg) reveal rationales quite different from those addressing their publics" viewpoints (Knijn and van Wel). These differences are even more obvious in articles contrasting the two points of view (those of Audrey Parron and Louis Bertrand, Vincent Caradecand Jean-Sébastien Eideliman): professionals are situated in relation to the meaningstheir specific fields givethe categories,so that a claim for disability recognition will have quite distinct meaningsfor those working in the domain of social assistance and those working in the field of disability. But for the individuals whose claims they process, these categorizations are part of a much broader universe of meaning that encompasses various spheres of their social lives: depending on their age, gender, and academic and occupational careers, the possibility of new job training, recognition of a degree of disability, or the evaluation of a degree of employability may have widely divergent meanings and consequences.

And yet one of the objectives of themovement toward the individualisation of social policy, which also touches the domain of disability [START_REF] Bertrand | Saisir les personnes. Le fonctionnement des commissions dans l"aide au logement et aux personnes handicapées[END_REF], is to reduce these gaps by making professionals study claimants" situations case by case, in relation to their needs and expectations. In the domain of disability, this is why professionals evaluating claims applications are required to take account of "Life Plans" that claimants may have formulatedto back up their request (although it is optional). The article by Louis Bertrand, Vincent Caradec and Jean-Sébastien Eideliman shows that this desire to account for the particularities of claimants" situations does not stop professionals from comprehending these situations very differently than those who are experiencing them, who,when they call upon the disability sector, are often seeking some kind of protection from the pressures they are subjected to at a given moment in their life course.

In addition to highlighting the great diversity of motives behind recourse to support, this article also stresses that disability-related categorizations are not only the bases of rights and statuses, but are also elements of identification, in the sense of qualifications that individuals may or may not integrate into their self-perception. Although there are a variety of forms of resistance to designation of the self or a loved one as a disabled person, expressions like "disabled worker," "disability," and "unfitness" hold strong meanings for people. In this vein,TrudieKnijn and Frits van Welshow that the new disability policy in the Netherlands produces discernable effects on its beneficiaries, who react in varying ways during the conducted sociological interviews. Contrary to the authors" expectations, individuals do little to challenge the system"s tendency to be stingy in attributing high degrees of disability and instead signal repeatedly that their occupational capacities are under-estimated, as manifest in what they find to be excessive degrees of disability. The evaluation is therefore an incident where an individual"s declassing is brutally objectified, even if it does open access to new rights. This is at the heart of the complexity of administrative instruments, which are simultaneously somewhat arbitrary quantifications caught up in politico-economic imperatives and qualifications that could have important subjective consequences. Disability in general is particular in how it is placed at the intersection of public health issues, social policy, and individual experience, making it fruitful for analysing contemporary social transformations.

Nota bene: The articles in this issue that were originally in French were translated by Juliette Rogers, to whom we are deeply grateful.

References:

Alter, European Journal of Disability Research.(2013). International Classification of Functioning, Disability and Health (ICF, WHO): Uses in the Field of Education. Special issue.7(1).

  Handicap et employabilité. Catégorisations professionnelles et expériences individuelles aux frontières du handicap Louis Bertrand, Vincent Caradec and Jean-Sébastien Eideliman -Université Lille 3 / CeRIES