J Cuevas

G Jamest

P G Kevrekidis

K J H Law

Vortex Solutions of the Defocusing Discrete Nonlinear Schrodinger Equation

Keywords: DNLS equation, Vortices, Existence, Stability PACS: 05, 45, Yv, 42, 65, Tg, 03, 75, Lm

We consider the existence, stability and dynamical evolution of dark vortex states in the two-dimensional defocusing DNLS equation, a model of interest both to atomic physics and to nonlinear optics. Our considerations are chiefly based on initializing such vortex configurations at the anti-continuum limit of zero coupling between adjacent sites, and continuing them to finite values of the coupling. Discrete defocusing vortices become unstable past a critical coupling strength and, subsequently feature a cascade of alternating stabilization-destabilization windows for any finite lattice.

INTRODUCTION

The study of vortices and their existence, stability and dynamical properties has been a central theme of study in the area of Bose-Einstein condensates (BECs) [START_REF] Pitaevskii | Bose-Einstein Condensation[END_REF]. In particular, the remarkable experiments illustrating the generation of vortices [START_REF] Matthews | [END_REF] and of very robust lattices thereof [3] have stirred a tremendous amount of activity in this area in the past few years, that has by now been summarized in various reviews and books [4,[START_REF]Emergent Nonlinear Phenomena in Bose-Einstein Condensates[END_REF]. Much of this activity has been centered around the robustness of vortex structures in the context of the mean-field dynamics of the BECs (which are controllably accurately described by a nonlinear SchrOdinger (NLS) equation) in the presence of many of the potentials that are relevant to the trapping of atomic BECs including parabolic traps [START_REF] Pitaevskii | Bose-Einstein Condensation[END_REF] and periodic optical lattice ones [6].

On the other hand, the BECs in the presence of periodic potentials have been argued to be well-approximated by models of the discrete nonlinear SchrOdinger (DNLS) type (i.e., resembling the finite-difference discretization of the continuum equation) [7,[START_REF] Kevrekidis | The Discrete Nonlinear Schrodinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives[END_REF]. In that regard, to understand the existence and stability properties of vortices in the presence of periodic potentials, it would be interesting to analyze the discrete analog of the relevant NLS equation. This is also interesting from a different perspective in this BEC context, namely that if finite-difference schemes are employed to analyze the properties of the continuum equation, it is useful to be aware of features introduced by virtue of the discretization.

However, it should be stressed that this is not a problem of restricted importance in the context of quantum fluids; it is also of particular interest in nonlinear optics where two-dimensional optical waveguide arrays have been recently systematically constructed e.g. in fused silica in the form of square lattices [START_REF] Szameit | [END_REF], whereby discrete solitons can be excited. By analogy to their one-dimensional counterparts of discrete dark solitons, which have been created in defocusing waveguide arrays with the photovoltaic nonlinearity [10], we expect that it should be possible to excite discrete dark vortices in defocusing two-dimensional waveguide arrays.

An especially interesting feature of dark solitons that was observed initially in [11] is that on-site discrete dark solitons are stable for sufficiently coarse lattices, but they become destabilized beyond a certain coupling strength among adjacent lattice sites and remain so until the continuum limit where they are again restabilized (as the point spectrum eigenvalue that contributes to the instability becomes zero due to the restoration of the translational invariance in the continuum problem) [11,12]. It is therefore of interest to examine if the instability mechanisms of discrete defocusing vortices are of this same type or are potentially different and how the relevant stability picture is modified as a function of the inter-site coupling strength.

It is this problem of the existence, stability and continuation of the vortex structures as a function of coupling strength that we examine in the present work. We consider, in particular, a two-dimensional discrete nonlinear Schrodinger where Alfln,m = lfln+1,m + lfln-1,m + lfln,m+1 + lfln,m-1 -4lfln,m is the discrete Laplacian and E is a coupling constant.

We study the defocusing case when E > 0 and look for time-periodic solutions with frequency m. Using the ansatz lfln,m(t) = y05¢n,me-iOJt, we obtain [START_REF] Matthews | [END_REF] where we have set C = E/ m. The coupling parameter C > 0 determines the strength of discreteness effects. The limit C---+ +oo corresponds to the continuum (stationary) Gross-Pitaevskii equation and the case C---+ 0 corresponds to the so-called anti-continuum (AC) limit [13].

When Eq. ( 2) is considered on an infinite lattice Z 2 , we look for solutions satisfying I C/Jn,m I ---+ 1 when (n, m) ---+ oo, for which C/Jn,m vanishes at one lattice site, e.g. at (n,m) = (0,0). Such solutions are denoted as discrete vortices, or "dark" vortex solitons. If one trigonometric tum on any path Max( In I, lml) = p around the vortex center changes the argument of C/Jn,m by 2nS (S E Z), then the vortex is said to have a topological charge (or vorticity) equal to S.

In this presentation we numerically investigate the existence and stability of such solutions on a finite lattice of size N x N, N being large; our analysis is performed as a function of the lattice coupling parameter C and we illustrate how to perform relevant continuations from the AC limit. We mainly focus on numerically computing vortex solutions with vorticity S = 1 and S = 2.

NUMERICAL METHOD

We compute vortex solutions of (2) using the Newton method and a continuation with respect to C. The path-following can be initiated either near the continuum limit (for C large) or at the anti-continuum limit C = 0, since in both cases one is able to construct a suitable initial guess for the Newton method. For relatively high C, a suitable initial condition for a vortex with topological chargeS is obtained with a Pade approximation developed for the continuum limit in [14]. Once a vortex is found for a given C, the solution can be continued by increasing or decreasing C. Although this method was found to be efficient, it remains limited to single vortex solutions having explicit continuum approximations. Moreover, when the Newton method is applied to continue these solutions near C = 0, the Jacobian matrix becomes ill-conditioned (and non-invertible for C = 0) and the iteration does not converge.

In what follows we introduce a different method having a wider applicability, and for which the above mentioned singularity is removed. We consider a finite N for n ?: 1, for n::; -1, for n = 0.

(3)

For C = 0, a single vortex at (n, m) = (0, 0) corresponds to fixing Ro,o = 0 andRn,m = 1 everywhere else. As deduced in [15], angles en,m fulfill sin ( en+1,m-en,m)sin ( en,m-en-1,m) +sin ( en,m+1-en,m)sin ( en,m-en,m-1) = 0 [START_REF]Emergent Nonlinear Phenomena in Bose-Einstein Condensates[END_REF] with (n, m) E r \ { (0, ±1), (±1, 0), (±M, m), (n, ±M)}, supplemented by the four following relations at (n, m) = (0, ±1), (±1, 0) sin ( e1,±1-eo,±d-sin ( eo,±1 -e-1,±1) +sin ( eo,±2-eo,±d Sin ( e±2,0-e±1,0) +Sin ( e±1,1 -e±1,0)-Sin ( e±1,0-e±1,-1) 0, 0. ;;; Solutions for ( 5)-( 7) are computed by the Newton method, starting from the initial guess en,m = lXn,m• The initial guess C/Jn,m = Rn,me;sen,m, can be used to compute a vortex solution of ( 2) by the Newton method, with Eqs. ( 3) and ( 4) acting as boundary conditions. The continuation technique used is a regular parametric continuation, i.e., for each value of C (starting at the anti-continuum limit), we converge to the exact numerical solution, and then we increment the value of C by a small 8C, using the converged solution of the previous step as a good initial guess for the solution of the next one (i.e., of the case with C + 8C).

NUMERICAL COMPUTATION OF SINGLE VORTICES

In this section we analyze the existence and stability of discrete vortices centered on a single site, as a function of the coupling strength C for fixed-end boundary conditions. The stability of the discrete vortex solutions is studied assuming small perturbations in the form of blflm,n = exp( -it) [Pn,m exp( -iAt) +qn,m exp(iA *t )], the onset of instability indicated by the emergence of Im(A) # 0; A in this setting denotes the perturbation eigenfrequency. Note that it is sufficient to consider the case m = 1 for stability computations, because this case can always be recovered by rescaling time.

The stability analysis is done fully for the case of configurations of size 41 x 41. Subsequently, the procedure is repeated with matrices of size 81 x 81, 121 x 121, 161 x 161 and 201 x 201. For the latter sizes, it is no longer possible to compute all the eigenvalues at once by a full matrix diagonalization. So, we exploit the sparsity of the system with an Arnoldi iterative diagonalization routine to calculate eigenvalues and corresponding (approximate) eigenvectors.

Figure 1 shows, for s = 1 and s = 2 vortices, the profile llfln,m 1 2 = I C/Jn,m 1 2 = R~,m, the angles en,m, and the spectral plane of the stability eigenfrequencies. In all cases, C = 0.2 is shown, which corresponds to unstable solutions. Vortices with S = 1 and S = 2 are, respectively, stable for C < Ccr ~ 0.0395 and C < Ccr ~ 0.0425. This instability, highlighted in Fig. 2 can be rationalized by analogy with the corresponding stability calculations in the case of dark solitons [11]. At C = Ccr the system experiences a Hamiltonian Hopf bifurcation. When C increases, a cascade of Hopf bifurcations takes place due to the interaction of a localized mode with extended modes, as it was observed in one-dimensional dark solitons [11,16]. This cascade implies the existence of stability windows between inverse Hopfbifurcations and direct Hopf bifurcations.

For S = 1 vortices, each one of the bifurcations takes place for decreasing IRe( A) I when C grows, and, in consequence, the bifurcations cease at a given value of C, as IRe( A) I of the localized mode is smaller than that of the lowest extended mode frequency [however, in the infinite domain limit, this eventual restabilization would not take place but for the limit of C ---7 oo ]. This fact is illustrated in Fig. 2. When the lattice size tends to infinity (N ---7 oo ), the linear mode band extends from zero to infinity and becomes dense; thus, we conjecture that these restabilization windows should disappear at this limit. In an effort to numerically suggest this point, we have considered lattices of up to 201 x 201 sites for the S = 1 and S = 2 vortices and have shown the growth rate of the corresponding instabilities in the right panels of Fig. 2. The maximum growth rate (i.e. the largest imaginary part of the stability eigenfrequencies) takes place at C ~ 0.115 for S = 1 and S = 2 and being Im(A,) ~ 0.0845 (0.0782) for S = 1 (S = 2). Further insight could be gained by means of a Lyapunov-Schmidt stability analysis [17]. We will consider in the near future this stability study. Another perspective is to consider saturable nonlinearity instead of the cubic one.

  --;Jtlfln,m lfln,m + el..l.lfln,m = 0,[START_REF] Pitaevskii | Bose-Einstein Condensation[END_REF]

  x N lattice with (n,m) E r = { -M, ... ,M} 2 (N =2M+ 1), equipped with fixed-end boundary conditions given below. We set C/Jn,m = Rn,m eien,m, with: and Rn,m = 1 en,m = SlXn,m for Max(lnl, lml) = M, for Max(lnl, lml) = M. { arctan(m/n) + 3 ; lXn,m = arctan(m/n) + ~ ~(1-sign(m))

-FIGURE 1 .

 1 FIGURE 1. Vortex solitons with C = 0.2 and S = 1 (top row) and S = 2 (bottom row). (Left panels) density profile; (center panels) angular dependence; (right panels) spectral plane of stability eigenfrequencies [recall that the presence of eigenfrequencies with non-vanishing imaginary part denotes instability].

FIGURE 2 .

 2 FIGURE 2. Stability for S = 1 (top row) and S = 2 (bottom row) vortices. (Left and center panels) Real part of the stability eigenfrequencies. These panels show zooms of two different regions. (Right panels) Largest imaginary part of the stability eigenfrequencies as a function of the coupling strength C. This corresponds to the instability growth rate.


T Lecerf

J L Kop

B Dauvier

Exploratory network analysis of the French Wechsler Intelligence Scale for children-Fourth Edition (WISC-IV)

❑ Material:

▪ The 10 core subtests of the WISC-IV and the optional subtest Picture Completion were administered to children (PC was not administered to 19 children).

❑ Analyses:

❑ Within the network approach, we used the R-package exploratory graph analysis (EGA) to estimate the number of dimensions. ❑ A Gaussian Graphical Model (GGM) was used to model the undirected network between subtests. Centrality indices were computed from the network.

▪ In GGM, an edge is the partial correlations between 2 variables, controlling for all other associations in the network. ▪ We used the R-package qgraph that includes the LASSO regularization to return a sparse network model. ▪ We used the R-package bootnet to assess the stability of the network structure and centrality indices.

❑ Exploratory and Confirmatory Factor Analysis are frequently used to uncover underlying latent variables that explain the structural covariation in the data. ❑ The psychometric network modeling suggests to move away from the latent variable perspective, and considers that dimensions arise from the interactions between entities in a network. ▪ Estimation of the number of dimensions underlying the WISC-IV using network psychometrics. ▪ Estimation of the network; edges represent the relations between the subtests of the WISC-IV. ▪ Estimation of the centrality of each variable, to determine the overall connectivity of a subtest, and the importance of each score.

❑ Network approach conceptualizes psychological dimensions as networks of psychological variables that directly interact with one another (without assuming a latent variable). ❑ Exploratory Graph Analysis (EGA) revealed 4 clusters. ❑ The clusters of nodes VC, WM, and PS in the network are equal with the 3 latent variables with factor analysis. Network analysis showed that the fourth dimension is more "VisuoSpatial" than "Perceptual Reasoning". ❑ The three most central node in the network are Similarities, Vocabulary and Block Design. This finding is partially consistent with factor analysis that showed that g loadings of Similarities and Vocabulary tasks were the highest. ❑ In sum, broad abilities and general intelligence can be conceptualized as networks of basic components.

  SGP SSP conference, 2017, Lausanne; thierry.lecerf@unige.ch ❑ Sample: ▪ 483 French-speaking Swiss children aged from 7 to 12 years. The sample included 230 males (mean age = 9.50, SD=1.30) and 253 females (mean age = 9.54, SD=1.29).

▪❑

  Network includes nodes (psychological variables) connected by edges (statistical relationships). ▪ Models focused on the estimation of direct relationships between observed variables. Analysis of the structure of the French WISC-IV by applying a network approach.

VC: Verbal Comprehension PR: Perceptual Reasoning WM: Working Memory PS: Processing Speed

				❑ EGA suggested to retain 4
				dimensions.
				❑ Green lines = positive
				correlations
				❑ Thicker edge = larger
				correlation
				▪ Betweenness: how many
				shortest paths between 2
				nodes go through the node
				in question. Includes
				direct and indirect
				connections of a subtests.
				A subtest with high
				betweenness is more
				important in connecting
				nodes.
				▪ Closeness: how strongly a
				node is indirectly
				connected to other nodes
				in the network.
				▪ Node strength: how
				strongly a node is directly
				connected to other nodes
				in the network.
	SI: Similarities	BD: Block Design	DS: Digit Span	CD: Coding
	VO: Vocabulary	MR: Matrix Reasoning LN: Letter-number	SS: Symbol Search
	CO: Comprehension	PCo: Picture
		Concepts
		(PC: Picture
		Completion)

❑ Four clusters emerged: VC (SI, VO, CO); WM (LN, DS); PS (SS, CD); and VisuoSpatial (VS with BD, PC, MR) ▪ Cluster : group of connected nodes ❑ PCo is related with VC, WM, and VS.


domain: info spi

text:
Sonda Ammar Bouhamed

Imene Khanfir Kallel

Ronald R Yager
email: yager@panix.com

Éloi Bossé

Basel Solaiman
email: basel.solaiman@imt-atlantique.fr

An intelligent quality-based approach to fusing multi-source possibilistic information

Keywords: Information fusion, quality of information, possibility theory, Gini entropy, source credibility, uncertainty

Recently, Yager and Petry were proposing a quality-based methodology to combine data provided by multiple probabilistic sources to improve the quality of information for decision-makers. This paper offers a sort of companion paper that adapts this methodology to possibilistic sources. Possibility theory is particularly well suited to cope with incomplete information from poor-data sources. The methodology and algorithms used for the probabilistic approach are adapted for the possibilistic case. Both approaches are then compared by the means of a numerical example and four experimental benchmark datasets: one, the IRIS data set, being data-poorer than the three other ones (Diabetes dataset, Glass dataset and Liver-disorder dataset). A vector representation is introduced for a possibility distribution as in the probabilistic case and, the Gini's formulation of entropy is being used. However, the Gini's entropy has to be used differently than with the probabilistic case. This has an impact on the selection of subsets. A fusion scheme is designed to select the best-quality subsets according to two information quality factors: quantity of information and source credibility. Results obtained from comparison of both approaches on the four experimental benchmarks confirm the superiority of the possibilistic approach in the presence of information scarcity or incompleteness.

Introduction

Yager and Petry [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF] propose a quality-based methodology to combine data provided by multiple sources in order to improve the quality of information essential to decision-makers in the execution of their tasks. A task can be the estimation of a parameter or, for instance, to perform an inference process about the occurrence of events. Yager and Petry's approach is schematized in Figure 1 using four interrelated functional blocks (a-d):

(a)

Modelling of the data and information sources;

(b) Quality-based information criteria;

(c) Ranking of subsets ranking;

(d) Fusion and Subset Selection Process.

The methodology proposed by Yager & Petry is making use of quality-based criteria (block b) in a fusion process (block c) to identify the most valuable set of information (block d) to be used in the execution of a task. For instance, the task is to infer a parametric description for an object, a physical process or an event,

given measurements tainted with uncertainty. A measure is being defined to quantify the notion of source credibility that is exploited in the fusion process to provide high-quality fused values for decision-making with reduced uncertainty based on the selection the best credible subset of the sources. Full details concerning Figure 1 blocks (a-d) can be found in [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF].

Probability theory is a powerful modelling tool to represent empirical knowledge about random phenomena.

The empirical knowledge is generally obtained by sensor observations (hard sensors) and probability theory is an ideal tool to formalize that kind of uncertainty where evidence is based on outcomes with enough independent random experiments. However, in the problem of multi-source information fusion where the information can come not only from hard sensors but also from soft sources such as expert knowledge, contextual knowledge, computing with words [START_REF] Herrera | Computing with words in decision making: foundations, trends and prospects[END_REF] and human-source in general [START_REF] Blasch | High Level Information Fusion (HLIF): Survey of models, issues, and grand challenges[END_REF][START_REF] Hall | Information Fusion[END_REF] , then the information is better modelled either by imprecise probabilities [START_REF] Walley | Statistical Reasoning With Imprecise Probabilities[END_REF], by evidence theory [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF], by fuzzy sets [START_REF] Zadeh | Fuzzy Sets as the Basis for a Theory of Possibility[END_REF] and namely, in this paper, by a possibilistic approach [START_REF] Yager | Hard and soft information fusion using measures[END_REF]. This paper considers the case of possibilistic sources. It follows the same methodology than in [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF] depicted in Figure 1 but the blocks (a-d) have now been translated with the mathematical tools of the possibility theory as shown in Figure 2. In addition, a comparison of both approaches, probabilistic (Figure 1) and possibilistic (Figure 2), is being done through the processing of four experimental benchmark datasets: IRIS, Diabetes, Glass and Liver-disorder.

The paper is organized as follows. Section 2 discusses the related work and scope. Sections 3 to 7 have corresponding sections in [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF]. Section 3 is about vector representation of possibility distributions. Section 4 details the use of entropy with possibility distributions. Section 5 is about information in maximally certain and uncertain distribution. Section 6 discusses the uniform fusion of possibility distributions while Section 7 follows on weighted average fusion. Finally, Section 8 presents the results of a comparison of both approaches: probabilitic [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF] and possibilitic. For convenience, some important notations with their descriptions are presented in Table 1.

Table 1. List of important notations with their related descriptions.

Notations Descriptions 𝜋

The Possibility distribution 𝜋(𝑗)

Possibility measure, where 𝑗 = 1 to 𝑛 𝑛

The size of the possibility distribution 𝜃 [START_REF] Das | High-Level Data Fusion[END_REF] The angle between two possibility distributions 𝜋 1 and 𝜋 2 𝑐𝑜𝑚𝑝(𝜋 1 , 𝜋 2 ) The compatibility degree between 𝜋 1 and 𝜋 2 𝑐𝑜𝑛𝑓(𝜋 1 , 𝜋 2 ) The conflict degree between two possibility distributions 𝜋 1 and 𝜋 2 𝐶𝑟𝑒𝑑(𝜋)

The credibility measure of a possibility distribution π 𝑆𝐸

The separability measure 𝑃 Probability distribution 𝑝(𝑗)

Probability measure, where 𝑗 = 1 𝑡𝑜 𝑛 𝐺(𝜋)

The Gini's entropy of a possibility distribution π ‖𝜋‖

The norm of a possibility distribution π 1 ‖𝜋‖ 2 The 𝑁𝑒𝑔𝐸𝑛𝑡 (Negative Entropy) of a possibility distribution π ‖𝑃‖ 2 The 𝑁𝑒𝑔𝐸𝑛𝑡 (Negative Entropy) of a probability distribution P 𝑤 Weight value 𝑅 A linear aggregation 𝑡

The number of the information sources 𝑍 A set of possibility distributions 𝐵 A subset of possibility distributions from 𝑍 𝜋 𝐵

The possibility distribution of the subset 𝐵 𝐷𝑜𝑚(𝐵 𝑖 , 𝐵 𝑗 ) A Boolean variable corresponding to the dominance concept 𝑁𝐷 Non-dominated subsets 𝑚

The number of features Figure 2. An intelligent quality-based approach for possibilistic sources

Scope and Related Work

Several research studies have been proposed in the literature in the field of multi-source data and information fusion systems for low-level fusion (sensors-level) [START_REF] Liggins | Handbook of Multisensor Data Fusion: Theory and Practice[END_REF][START_REF] Hall | Mathematical Techniques in Multisensor Data Fusion[END_REF][START_REF] Khaleghi | Multisensor data fusion: A review of the state-of-the-art[END_REF] and for high-level fusion [START_REF] Hall | Information Fusion[END_REF][START_REF] Das | High-Level Data Fusion[END_REF][START_REF] Blasch | High-Level Information Fusion Management and Systems Design[END_REF]. This list is far from being exhaustive. Most of these studies are focused on 'fusion systems' and consider much more functions than the 'merging' or the 'combining' rules of a fusion system. Regarding the merging rules, an excellent synthesis of the basic principles for 'combining' pieces of imperfect information, regardless of the representation formalism (sets, logic, partial orders, possibility theory, belief functions or imprecise probabilities), is presented in [START_REF] Dubois | The basic principles of uncertain information fusion. An organised review of merging rules in different representation frameworks[END_REF]. They propose to rank the pieces of information to be combined in terms of their relative plausibility as well as to identify impossible values. The relative information content and the mutual consistency of information pieces that affect the performance of the fusion process are also being considered.

When we refer to an "intelligent quality-based" approach, what is meant by "intelligent" is the representation of contextual information items that are being taken into account in the knowledge chain processing from data to information to knowledge to decisions and actions [START_REF] Snidaro | Context-Enhanced Information Fusion[END_REF]. One of the most complete treatments (gathering-processing-combining-decision-making) along this chain has been presented in [START_REF] Tacnet | New integrated and multiscale decision-aiding framework in a context of imperfect information: application to the assessment of torrent checkdams' effectiveness[END_REF][START_REF] Tacnet | A two-step fusion process for multi-criteria decision applied to natural hazards in mountains[END_REF] for the risk management of natural hazards using beliefs functions and multicriteria decision-making.

The book [START_REF] Bossé | Information Quality in Information Fusion and Decision Making (Information Fusion and Data Science[END_REF] provides a formal foundation and implementation strategies allowing to incorporate information quality into the information fusion processes to various decision support applications for reallife scenarios such as remote sensing, medicine, automated driving, environmental protection, crime analysis, intelligence, defense and security. In [START_REF] Calderwood | Context-dependent combination of sensor information in Dempster-Shafer theory for BDI[END_REF], the authors describe the whole process of modelling uncertain sensor information. The Dempster-Shafer (DS) theory was chosen to model uncertain sensor information. The authors identify suitable measures from DS theory for determining quality criteria to select subsets on which Dempster's rule of combination is being applied.

Finally, the proposed paper can be considered as a companion paper to [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF] in the framework of possibilistic sources. It extends the capacity to treat more types of uncertainty (e.g., fuzziness, vague) and then provides, by the means of a fusion process, a better quality information to decision-makers.

Vector representation of possibility distributions

Possibility distributions can also be represented as a n-dimensional vector 𝜋 = [𝜋(1), 𝜋(2), … , 𝜋(𝑛)]. Here, 𝜋 is a normal possibility distribution (max(𝜋) = 1) on the space Ω = {𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 } where 𝜋(𝑗) is the possibility of occurrence of 𝑥 𝑗 . This vector has all its components within the unit interval, 𝜋(𝑗) ∈ [0,1]. A possibility distribution and its sum can be higher than 1, so that, ∑ 𝜋(𝑗) ∈ [0, 𝑀] 𝑛 𝑗=1

with 𝑀 ≥ 1. In addition, if 𝜋 1 and 𝜋 2 are two possibility vectors on the space Ω then their dot product, < 𝜋 1 , 𝜋 2 > , a scalar value, is:

< 𝜋 1 , 𝜋 2 > = ∑ 𝜋 1 (𝑗)𝜋 2 (𝑗) 𝑛 𝑗=1 (1)
In the case where 𝜋 1 and 𝜋 2 are normal possibility distributions then 0 ≤ (< 𝜋 1 , 𝜋 2 >) ≤ 1. When 𝜋 1 and 𝜋 2 are identical, their dot product is:

< 𝜋 1 , 𝜋 2 > = ∑(𝜋 1 (𝑗)) 2 𝑛 𝑗=1 (2)
In the following, the dot product will simply be notated as 𝜋 𝑖 𝜋 𝑘 . The norm of a vector is its self dot product, also known as the Euclidean length:

‖𝜋‖ = √𝜋𝜋 = (∑(𝜋(𝑗)) 2 𝑛 𝑗=1 ) 1/2 = (< 𝜋, π >) 1 2 ⁄ (3)
Using special properties of the possibility distribution vector, 𝜋(𝑗) ∈ [0,1] and 1 ≤ ∑ 𝜋(𝑗) ≤ 𝑛 𝑛 𝑗=1

, it can be shown that:

- Figure 3 illustrates a possibility vector in the two-dimensional case (n=2). In this figure, each possibility distribution is represented as a point. Non-normal distributions have a vector whose its extremity is "strictly" inside the square. Normal distributions have obligatorily one extremity on a side of the square.

In
If 𝜋 1 and 𝜋 2 are two possibility vectors, it is known that the cosine of the angle 𝜃 12 between them is expressed as in Table 2. This cosine is also the dot product of 𝜋 1 and 𝜋 2 divided by the vector respective norms. The interpretation of the dot product is similar to what is presented in [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF]. As in [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF], cos 𝜃 12 can be used as measure of the degree of compatibility, comp, between the two possibility distributions; see Table 2. The closer the value of 𝑐𝑜𝑚𝑝(𝜋 1 , 𝜋 2 ) is to one, the more compatible possibility distributions are.

Furthermore, 1 -𝑐𝑜𝑚𝑝(𝜋 1 , 𝜋 2 ), denoted 𝑐𝑜𝑛𝑓(𝜋 1 , 𝜋 2 ), is the degree of conflict between two possibility distributions.

-If 𝜋 1 and 𝜋 2 are orthogonal then 𝑐𝑜𝑚𝑝(𝜋 1 , 𝜋 2 ) = 0 and 𝑐𝑜𝑛𝑓(𝜋 1 , 𝜋 2 ) = 1.

-If 𝜋 1 and 𝜋 2 are coincident then 𝑐𝑜𝑚𝑝(𝜋 1 , 𝜋 2 ) = 1 and 𝑐𝑜𝑛𝑓(𝜋 1 , 𝜋 2 ) = 0.

𝑐𝑜𝑠( 𝜃 12 ) = 𝑝 1 𝑝 2 ‖𝑝 1 ‖‖𝑝 2 ‖ 𝑐𝑜𝑠( 𝜃 12 ) = 𝜋 1 𝜋 2 ‖𝜋 1 ‖‖𝜋 2 ‖ 𝑐𝑜𝑚𝑝(𝑝 1 , 𝑝 2 ) = 𝑝 1 𝑝 2 ‖𝑝 1 ‖‖𝑝 2 ‖
, with

𝑐𝑜𝑚𝑝(𝑝 1 , 𝑝 2 ) ∈ [0,1] 𝑐𝑜𝑚𝑝(𝜋 1 , 𝜋 2 ) = 𝜋 1 𝜋 2 ‖𝜋 1 ‖‖𝜋 2 ‖
, with

𝑐𝑜𝑚𝑝(𝜋 1 , 𝜋 2 ) ∈ [0,1], < 𝑃 1 , 𝑃 2 >= ∑ 𝑃 1 (𝑗)𝑃 2 (𝑗) 𝑛 𝑗=1 = 1 𝑛 ∑ 𝑃 2 (𝑗) = 𝑛 𝑗=1 1 𝑛 𝑐𝑜𝑚𝑝(𝑃 1 , 𝑃 2 ) = 1 𝑛 ‖𝑃 2 ‖ (𝑛 1 2 ) = ( 1 𝑛 ) 1 2 ⁄ ‖𝑃 2 ‖ = 1 √𝑛 1 ‖𝑃 2 ‖ < 𝜋 1 , 𝜋 2 > = ∑ 𝜋 1 (𝑗)𝜋 2 (𝑗) 𝑛 𝑗=1 = 1 ∑ 𝜋 2 (𝑗) 𝑛 𝑗=1
, where ∑ 𝜋 2 (𝑗)

𝑛 𝑗=1 ∈ [0, 𝑛] 𝑐𝑜𝑚𝑝(𝜋 1 , 𝜋 2 ) = ∑ 𝜋 2 (𝑗) 𝑛 𝑗=1 ‖𝜋 2 ‖ (𝑛 1 2 ) = 1 ‖𝜋 2 ‖ 1 (𝑛 1/2 ) ∑ 𝜋 2 (𝑗) 𝑛 𝑗=1 = 1 ‖𝜋 2 ‖ 1 √𝑛 ∑ 𝜋 2 (𝑗) 𝑛 𝑗=1
We see that the conflict takes its maximal value when 𝜋 1 and 𝜋 2 are orthogonal and takes its minimum value when 𝜋 1 and 𝜋 2 are coincident.

-Complete knowledge (C.K): consider 𝜋 1 and 𝜋 2 being two certain possibility distributions on the same

element i like ∃𝜋 1 (𝑖) = 1, 𝜋 2 (𝑖) = 1, ∀𝜋 1 (𝑗) = 0, 𝜋 2 (𝑗) = 0 𝑗 ≠ 𝑖, 𝑗 = 1 𝑡𝑜 𝑛 so 𝑐𝑜𝑚𝑝(𝜋 1 , 𝜋 2 ) = 1
and 𝑐𝑜𝑛𝑓(𝜋 1 , 𝜋 2 ) = 0.

-Total ignorance (T.I): if one of the distributions, 𝜋 1 , has 𝜋 1 (𝑗) = 1, ∀𝑗 = 1, … , 𝑛.. In this case, . For this case, the comparative equations are given in the last row of Table 2.

There are two special cases of 𝜋 2 that worth to be commented.

-The case of total ignorance (T.I): if 𝜋 2 is a uniform possibility distribution, all 𝜋 2 (𝑗) = 1, 𝑗 = 1 𝑡𝑜 𝑛.

In this case,

The use of entropy with possibility distributions

Entropy is an important concept for measuring uncertainty associated with a probability distribution. With 𝑃, a probability distribution defined on the space Ω = {𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 } with 𝑝 𝑗 , the probability associated with 𝑥 𝑗 , then the common measure of entropy is the Shannon entropy defined as: According to Klir [START_REF] Klir | Information-Preserving Probability-Possibility Transformations[END_REF], maintaining a link between a possibility and a probability distribution is important.

𝐻(𝑃) = -∑ 𝑝(𝑗) 𝑛 𝑗=1 ln( 𝑝(𝑗)) (4)
The uncertainty of a possibility distribution 𝑈(𝜋) and the entropy of a probability distribution 𝐻(𝑃) must be equal, so 𝑈(𝜋) = 𝐻(𝑃).

Finally, 𝑈(𝜋) ∈ [0, ln(𝑛)], and the maximal uncertainty happens when all 𝜋(𝑗) = 1 so 𝑈(𝜋) = ln(𝑛). The minimal uncertainty occurs when one 𝜋(𝑗) = 1 and all other 𝜋(𝑗) = 0, hence 𝑈(𝜋) = 0. Larger is 𝑈(𝜋), larger is the uncertainty. The other side of uncertainty is certainty or information. Smaller 𝑈(𝜋) is then more information is convoyed by a possibility distribution. This is desirable for decision-making. The maximal entropy occurs when all 𝑝(𝑗) = 1 𝑛 ⁄ then 𝐻(𝑃) = ln( 𝑛). The minimal entropy occurs when one 𝑝(𝑗) = 1 and all other 𝑝(𝑗) = 0 then 𝐻(𝑃) = 0. For Shannon entropy, we also have 𝐻(𝑃) ∈ [0, ln(𝑛)]. For the same reasons evoked in [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF], we use an alternative formulation for entropy called the Gini's entropy defined in [START_REF] Dubois | A notion of comparative probabilistic entropy based on the possibilistic specificity ordering[END_REF][START_REF] Elmore | Information Theory Applications in Soft Computing[END_REF] for the probabilistic framework and given in Table 3. By analogy, we define the Gini's entropy for a possibility distribution π on  by : 𝐺(𝜋) = 1 -𝑵𝒆𝒈𝑬𝒏𝒕, where 𝑵𝒆𝒈𝑬𝒏𝒕 = The bigger 𝐺(𝜋) is then the larger uncertainty is conveyed by the possibility distribution. The closer 𝐺(𝜋) is to zero, the more certainty resides in the knowledge so more information is provided. Thus, to increase certainty or information, we must decrease 𝐺(𝜋) and, according to equation (8), increase 𝑁𝑒𝑔𝐸𝑛𝑡.

𝑁𝑒𝑔𝐸𝑛𝑡 is defined differently for probabilistic and possibilistic frameworks, as it is shown in Table 3.

However, it has the same variation domain, being Let us now examine the variation of 𝜋 2 with respect to the quantity of information contained in the possibility distribution. Assume 𝜋 1 and 𝜋 2 are two possibility vectors on the space  and that the relations between them are given as in

𝐺(𝑝) = 1 -∑(𝑝(𝑗)) 2 𝑛 𝑗=1 𝐺(𝜋) = 1 - 1 ∑ (𝜋(𝑗)) 2 𝑛 𝑗=1 𝑝 2 (1) = 𝑝 1 (1) -𝛼 𝑝 2 (2) = 𝑝 1 (2) + 𝛼 𝑝 2 (𝑗) = 𝑝 1 (𝑗) 𝑓𝑜𝑟 𝑗 = 3 𝑡𝑜 𝑛 For 𝛼 ≥ 0. We have ∑ (𝑝 2 (𝑗)) 2 𝑛 𝑗=1 = (𝑝 1 (1) -𝛼) 2 + (𝑝 1 (2) + 𝛼) 2 + ∑(𝑝 1 (𝑗)) 2 𝑛 𝑗=3 Since (𝑝 1 (1) -𝛼) 2 = (𝑝 1 (1)) 2 -2𝛼𝑝 1 (1) + 𝛼 2 and (𝑝 1 (2) -𝛼) 2 = (𝑝 1 (2)) 2 -2𝛼𝑝 1 (2) + 𝛼 2 then we get ∑ (𝑝 2 (𝑗)) 2 𝑛 𝑗=1 -∑ (𝑝 1 (𝑗)) 2 = 𝑛 𝑗=1 2𝛼 (𝑝 1 (2) -𝑝 1 (1)) 2 + 2𝛼 2 = 2𝛼 ((𝑝 1 (2) -𝑝 1 (1)) + 𝛼) 𝜋 2 (1) = 𝜋 1 (1) -𝛼 𝜋 2 (2) = 𝜋 1 (2) + 𝛼 𝜋 2 (𝑗) = 𝜋 1 (𝑗) 𝑓𝑜𝑟 𝑗 = 3 𝑡𝑜 𝑛
For 𝛼 ≥ 0. We have

1 ∑ (𝜋(𝑗)) 2 𝑛 𝑗=1 = (𝜋 1 (1) -𝛼) 2 + (𝜋 1 (2) + 𝛼) 2 + 1 ∑ (𝜋(𝑗)) 2 𝑛 𝑗=3 Since (𝜋 1 (1) -𝛼) 2 = (𝜋 1 (1)) 2 -2𝛼𝜋 1 (1) + 𝛼 2
and

(𝜋 1 (2) -𝛼) 2 = (𝜋 1 (2)) 2 -2𝛼𝜋 1 (2) + 𝛼 2 then we get 1 ∑ (𝜋 2 (𝑗)) 2 𝑛 𝑗=1 - 1 ∑ (𝜋 1 (𝑗)) 2 𝑛 𝑗=1 = 2𝛼(𝜋 1 (2) -𝜋 1 (1)) 2 + 2𝛼 2 = 2𝛼 ((𝜋 1 (2) -𝜋 1 (1)) + 𝛼)

Information in maximally certain and uncertain distribution

Two probability distributions on 𝑋: 𝑃 = [𝑝 1 , 𝑝 2 , ⋯ , 𝑝 𝑛 ] and 𝑄 = [𝑞 1 , 𝑞 2 , ⋯ , 𝑞 𝑛 ] , are considered in Section 4 of Yager and Petry [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF]. The objective is to calculate the information associated with the linear aggregation R, that is: 𝑅 = 𝑤 1 𝑃 + 𝑤 2 𝑄, R is a probability distribution when 𝑤 1 + 𝑤 2 = 1. For each component 𝑟 𝑗 of 𝑅, we have, 𝑟 𝑗 = (𝑤 1 𝑝 𝑗 + 𝑤 2 𝑞 𝑗 ). The details are in [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF]. For the possibilistic case, the information, associated with 𝜋, according to the value of the weights 𝑤 1 and 𝑤 2 , is calculated in order to study the impact of the weights on the distribution 𝜋. Three different cases are considered:

1. 𝜋 1 and 𝜋 2 are two certain possibility distributions, 2. 𝜋 1 and 𝜋 2 are both maximally uncertain distributions, and 3. 𝜋 1 is a distribution completely certain and 𝜋 2 is a distribution completely uncertain.

To perform the information associated with 𝜋, we calculate:

‖𝜋‖ 2 = ∑(𝑤 1 𝜋 1 (𝑗) + 𝑤 2 𝜋 2 (𝑗)) 2 𝑛 𝑗=1 = ∑ (𝑤 1 2 (𝜋 1 (𝑗)) 2 + 𝑤 2 2 (𝜋 2 (𝑗)) 2 + 2𝑤 1 𝑤 2 𝜋 1 (𝑗)𝜋 2 (𝑗)) 𝑛 𝑗=1 (9)
‖𝜋‖ 2 = 𝑤 1 2 ∑ (𝜋 1 (𝑗)) 2 + 𝑤 2 2 ∑ (𝜋 2 (𝑗)) 2 𝑛 𝑗=1 + 2𝑤 1 𝑤 2 ∑ 𝜋 1 (𝑗)𝜋 2 (𝑗) 𝑛 𝑗=1 𝑛 𝑗=1 (10) ‖𝜋‖ 2 = 𝑤 1 2 ‖𝜋 1 ‖ 2 + 𝑤 2 2 ‖𝜋 2 ‖ 2 + 2𝑤 1 𝑤 2 < 𝜋 1 , 𝜋 2 > (11)
We do not repeat here all derivations for the possibilistic representation when they are similar to those of the probabilistic framework. We only list in Table 5 where there are differences. We invite the reader to consult Section 4 of Yager and Petry [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF] for details. If 𝑤 𝑖 ∈ [0,1] and ∑ 𝑤 𝑖 𝑡 𝑖=1

= 1, then p is also a probability distribution vector.

(Case when

𝑤 1 = 𝑤 1 = 1 2 ⁄ ) 𝜋 = ∑ 𝑤 𝑖 𝜋 𝑖 𝑡 𝑖=1 with 𝜋(𝑗) = ∑ 𝑤 𝑖 𝜋 𝑖 (𝑗) 𝑡 𝑖=1
If 𝑤 𝑖 ∈ [0,1] and ∑ 𝑤 𝑖 𝑡 𝑖=1

= 1, then π is also a possibility distribution vector.

(Case when

𝑤 1 = 𝑤 1 = 1 2 ⁄ )
If 𝑃 1 and 𝑃 2 are two certain probability distributions, then ‖𝑃 1 ‖ = 1 and ‖𝑃 2 ‖ = 1 and there are two cases of interest.

1) If 𝑃 1 and 𝑃 2 are completely compatible, 𝑃 1 (𝑗) = 𝑃 2 (𝑗) = 1 for some j. Here 𝑐𝑜𝑠( 𝑃 1 , 𝑃 2 ) = 1 and

‖𝑃‖ 2 = 1 4 (1) + 1 4 (1) + 1 2 (1) = 1. 2) If 𝑃 1 and 𝑃 2 are completely conflicting, 𝑃 1 (𝑗) = 1, 𝑃 2 (𝑘) = 1 for 𝑗 ≠ k, then 𝑐𝑜𝑠( 𝑃 1 , 𝑃 2 ) = 0 and ‖𝑃‖ 2 = 1 4 + 1 4 + 0 = 1 2 .
If 𝜋 1 and 𝜋 2 are two certain possibility distributions, then ‖𝜋 1 ‖ = 1 and ‖𝜋 2 ‖ = 1 and there are two cases of interest.

1) If 𝜋 1 and 𝜋 2 are completely compatible, 𝜋 1 (𝑗) = 𝜋 2 (𝑗) = 1 for some j. Here 𝑐𝑜𝑠( 𝜋 1 , 𝜋 2 ) = 1 and

‖𝜋‖ 2 = 1 4 (1) + 1 4 (1) + 1 2 (1) = 1 2) If 𝜋 1 and 𝜋 2 are completely conflicting, 𝜋 1 (𝑗) = 1, 𝜋 2 (𝑘) = 1 for 𝑗 ≠ k, then 𝑐𝑜𝑠( 𝜋 1 , 𝜋 2 ) = 0 and ‖𝜋‖ 2 = 1 4 + 1 4 + 0 = 1 2 . If 𝑃 1 and 𝑃 2 are both maximally uncertain distributions, they have 𝑃 1 (𝑗) = 𝑃 2 (𝑗) = 1 𝑛 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗. Here ‖𝑃 1 ‖ = ‖𝑃 2 ‖ = (1 𝑛 ⁄ ) 1 2
⁄ and since we have shown in this case that

𝑐𝑜𝑠( 𝑃 1 , 𝑃 2 ) = 1 then ‖𝑃‖ 2 = 1 4 1 𝑛 + 1 4 1 𝑛 + 1 2 ( 1 𝑛 ) 1/2 ( 1 𝑛 ) 1/2 1 = 1 𝑛
If 𝜋 1 and 𝜋 2 are both maximally uncertain distributions, they have 𝜋 1 (𝑗) = 𝜋 2 (𝑗) = 1 for all j.

Here‖𝜋 1 ‖ = ‖𝜋 2 ‖ = (𝑛) 1 2
⁄ and since we have shown in this case that 𝑐𝑜𝑠( 𝜋 1 , 𝜋 2 ) = 1 then

‖𝜋‖ 2 = 1 4 (𝑛) + 1 4 (𝑛) + 1 2 (𝑛 1 2 ) (𝑛 1
2 ) 1 = 𝑛 .

If 𝑃 1 is completely certain distribution,

‖𝑃 1 ‖ = 1, and 𝑃 2 is completely uncertain distribution, ‖𝑃 2 ‖ = ( 1 𝑛 ) 1/2 , then 𝑃 1 𝑃 2 = 1 𝑛 and ‖𝑃‖ 2 = 1 4 (1) + 1 4 1 𝑛 + 1 2 1 𝑛 = 3+𝑛 4𝑛 If 𝜋 1 is completely certain distribution, ‖𝜋 1 ‖ = 1,
and 𝜋 2 is completely uncertain distribution,

‖𝜋 2 ‖ = (𝑛) 1 2 ⁄ , then < 𝜋 1 , 𝜋 2 > = 1 and ‖𝜋‖ 2 = 1 4 (1) + 1 4 (𝑛) + 1 2 (1) = 1+𝑛+2 4 = 3+𝑛 4 .
Let us now consider the general case where we have t possibility distributions 𝜋 1 , 𝜋 2 , … , 𝜋 𝑡 where

𝜋 𝑖 = [𝜋 𝑖 (1), 𝜋 𝑖 (2), … , 𝜋 𝑖 (𝑛) ] with 𝜋 = 1 𝑡 ∑ 𝜋 𝑖 𝑡 𝑖=1
and where each component of 𝜋 is being defined as:

𝜋(𝑗) = 1 𝑡 ∑ 𝜋 𝑖 (𝑗) 𝑡 𝑖=1
. Table 6. Comparative equations associated with the calculation of ‖𝜋‖ 2 for t distributions Probabilistic Framework Possibilistic Framework

‖𝑃‖ 2 = ∑ (𝑝(𝑗)) 2 𝑛 𝑗=1 = ∑ ( 1 𝑡 2 ) (𝑝 1 (𝑗) + 𝑛 𝑗=1 𝑝 2 (𝑗)+. . . +𝑝 𝑖 (𝑗)) 2 ‖𝑃‖ 2 = ∑ 1 𝑡 2 (‖𝑃 𝑖 ‖) 2 + ∑ ∑ 1 𝑡 2 < 𝑃 𝑖 , 𝑃 𝑘 > 𝑡 𝑘=1 𝑘≠𝑖 𝑡 𝑖=1 𝑛 𝑖=1 ‖𝑃‖ 2 = ∑ 1 𝑡 2 (‖𝑃 𝑖 ‖) 2 + 2 𝑡 2 ∑ ∑ < 𝑃 𝑖 , 𝑃 𝑘 > 𝑡 𝑘=𝑖+1 𝑡-1 𝑖=1 𝑛 𝑖=1 ‖𝜋‖ 2 = ∑ (𝜋(𝑗)) 2 𝑛 𝑗=1 = ∑ ( 1 𝑡 2 ) (𝜋 1 (𝑗) + 𝑛 𝑗=1 𝜋 2 (𝑗)+. . . +𝜋 𝑖 (𝑗)) 2 ‖𝜋‖ 2 = ∑ 1 𝑡 2 (‖𝜋 𝑖 ‖) 2 + ∑ ∑ 1 𝑡 2 < 𝜋 𝑖 , 𝜋 𝑘 > 𝑡 𝑘=1 𝑘≠𝑖 𝑡 𝑖=1 𝑛 𝑖=1 ‖𝜋‖ 2 = ∑ 1 𝑡 2 (‖𝜋 𝑖 ‖) 2 + 2 𝑡 2 ∑ ∑ < 𝜋 𝑖 , 𝜋 𝑘 > 𝑡 𝑘=𝑖+1 𝑡-1 𝑖=1 𝑛 𝑖=1
Let us take a look at the formula

‖𝜋‖ 2 = ∑ 1 𝑡 2 (‖𝜋 𝑖 ‖) 2 + 2 𝑡 2 ∑ ∑ < 𝜋 𝑖 , 𝜋 𝑘 > 𝑡 𝑘=𝑖+1 𝑡-1 𝑖=1 𝑛 𝑖=1
and consider the situation where we have two categories of possibility distributions: complete knowledge and total ignorance distributions:

-One being a certainty distribution; only one of its components is one (complete knowledge).

-The other is a pure uncertainty distribution; here, all elements are 1 (total ignorance).

First, for 𝜋 𝑖 that has certainty, then 𝑁𝑒𝑔𝐸𝑛𝑡 = 1. In contrast, for any 𝜋 𝑖 that is pure uncertainty, we have shown that 𝑁𝑒𝑔𝐸𝑛𝑡 = 1 𝑛 . Table 7 below gives results for the calculations of the dot product. ]

‖𝑃‖ 2 = 1 𝑡 2 [𝑡 1 + ∑ 𝑔 𝑗 (𝑔 𝑗 -1) 𝑛 𝑗=1 + 1 𝑛 ((𝑡 -𝑡 1 )(𝑡 + 𝑡 1 ))] ‖𝑃‖ 2 = 1 𝑡 2 [𝑡 1 + ∑ 𝑔 𝑗 (𝑔 𝑗 -1) 𝑛 𝑗=1 + 1 𝑛 (𝑡 2 -𝑡 1 2 )]
In the special case where all the pure certain distributions agree, 𝑔 1 = 𝑡 1 and all other 𝑔 𝑗 = 0 , we get

‖𝑃‖ 2 = 1 𝑡 2 [𝑡 1 + 𝑡 1 (𝑡 1 -1) + 1 𝑛 (𝑡 2 -𝑡 1 2 )] ‖𝑃‖ 2 = 1 𝑡 2 [𝑡 1 2 + 1 𝑛 (𝑡 2 -𝑡 1 2 )] 𝑁𝑒𝑔𝐸𝑛𝑡 = 1 𝑡 2 [𝑡 1 2 + 1 𝑛 (𝑡 2 -𝑡 1 2 )] ‖𝜋‖ 2 = 1 𝑡 2 [𝑡 1 + 𝑛(𝑡 -𝑡 1 ) + 𝑛((𝑡 -𝑡 1 )(𝑡 + 𝑡 1 - 1)) + ∑ 𝑔 𝑗 (𝑔 𝑗 -1) 𝑛 𝑗=1 ] ‖𝜋‖ 2 = 1 𝑡 2 [𝑡 1 + ∑ 𝑔 𝑗 (𝑔 𝑗 -1) 𝑛 𝑗=1 + 𝑛((𝑡 -𝑡 1 )(𝑡 + 𝑡 1 ))] ‖𝜋‖ 2 = 1 𝑡 2 [𝑡 1 + ∑ 𝑔 𝑗 (𝑔 𝑗 -1) 𝑛 𝑗=1 + 𝑛(𝑡 2 -𝑡 1 2 )]
In the special case where all the pure certain distributions agree, 𝑔 1 = 𝑡 1 and all other 𝑔 𝑗 = 0 we get

‖𝜋‖ 2 = 1 𝑡 2 [𝑡 1 + 𝑡 1 (𝑡 1 -1) + 𝑛(𝑡 2 -𝑡 1 2 )] ‖𝜋‖ 2 = 1 𝑡 2 [𝑡 1 2 + 𝑛(𝑡 2 -𝑡 1 2 )] 𝑁𝑒𝑔𝐸𝑛𝑡 = 𝑡 2 [𝑡 1 2 + 𝑛(𝑡 2 -𝑡 1 2 )]

Uniform fusion of possibility distribution

The fusion of multi-source probabilistic information has been detailed in Section 5 of Yager and Petry [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF].

This section discusses the main differences between both approaches. Assume 𝑋 is a variable that takes its value in the space Ω = {𝑥 1 , 𝑥 2 , … , 𝑥 𝑡 }. Let 𝜋 𝑖 be a possibility distribution on  representing the information provided by source i regarding the value of 𝑋 so that, each 𝜋 𝑖 = [𝜋 𝑖 [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF], 𝜋 𝑖 (2), … , 𝜋 𝑖 (𝑛) ] with t possibility distributions, 𝜋 𝑖 , for 𝑖 = 1 𝑡𝑜 𝑡, then the basic uniform fusion of these distributions is a possibility distribution π as given in Table 9.

Table 9. Equations for the fusion of distributions for both frameworks Probabilistic Framework Possibilistic Framework

𝑃 = 1 𝑡 ∑ 𝑃 𝑖 𝑡 𝑖=1 with 𝑃(𝑗) = 1 𝑡 ∑ 𝑃 𝑖 (𝑗) 𝑡 𝑖=1 𝜋 = 1 𝑡 ∑ 𝜋 𝑖 𝑡 𝑖=1 with 𝜋(𝑗) = 1 𝑡 ∑ 𝜋 𝑖 (𝑗) 𝑡 𝑖=1 ‖𝑃‖ 2 = 1 𝑡 2 [∑‖𝑃 𝑖 ‖ 2 + 2 ∑ ∑ < 𝑃 𝑖 . 𝑃 𝑘 > 𝑡 𝑘=𝑖+1 𝑡-1 𝑖=1 𝑡 𝑖=1 ] ‖𝜋‖ 2 = 1 𝑡 2 [∑‖𝜋 𝑖 ‖ 2 + 2 ∑ ∑ < 𝜋 𝑖 . 𝜋 𝑘 > 𝑡 𝑘=𝑖+1 𝑡-1 𝑖=1 𝑡 𝑖=1 ]
The total number of terms being combined:

𝑡 + 2(𝑡)(𝑡-1) 2 = 𝑡 2 and 𝑁𝑒𝑔𝐸𝑛𝑡 = ‖𝑃‖ 2 ∈ [0,1].
The total number of terms being combined:

𝑡 + 2(𝑡)(𝑡-1) 2 = 𝑡 2 and 𝑁𝑒𝑔𝐸𝑛𝑡 ∈ [0, 1 𝑛 ] 𝐴𝑣𝑒𝐶𝑜𝑛𝑓(𝑃 𝑖 , 𝑃 𝑘 ) = 2 (𝑡)(𝑡 -1) ∑ ∑ (1 - 𝑃 𝑖 𝑃 𝑘 ‖𝑃 𝑖 ‖‖𝑃 𝑘 ‖ ) 𝑡 𝑘=𝑖+1 𝑡-1 𝑖=1 = 1 - 2 (𝑡)(𝑡 -1) ∑ ∑ ( 𝑃 𝑖 𝑃 𝑘 ‖𝑃 𝑖 ‖‖𝑃 𝑘 ‖ ) 𝑡 𝑘=𝑖+1 𝑡-1 𝑖=1 𝐴𝑣𝑒𝐶𝑜𝑛𝑓(𝜋 𝑖 , 𝜋 𝑘 ) = 2 𝑡(𝑡 -1) [∑ ∑ (1 - 𝜋 𝑖 𝜋 𝑘 ‖𝜋 𝑖 ‖‖𝜋 𝑘 ‖ ) 𝑡 𝑘=𝑖+1 𝑡-1 𝑖=1 ] = 1 - 2 𝑡(𝑡 -1) [∑ ∑ ( 𝜋 𝑖 𝜋 𝑘 ‖𝜋 𝑖 ‖‖𝜋 𝑘 ‖ ) 𝑡 𝑘=𝑖+1 𝑡-1 𝑖=1 ]
The objective of fusing t possibility distributions, 𝜋 𝑖 , 𝑓𝑜𝑟 𝑖 = 1 𝑡𝑜 𝑡, provided by multiple sources, is to obtain a fused estimate that contains the largest amount of information about the value of 𝑋. That means to obtain a large value of 𝑁𝑒𝑔𝐸𝑛𝑡. As in the case of the probabilistic framework, pairing possibility distributions that are non-conflicting, i.e. a 𝜋 𝑖 𝜋 𝑘 large, tend to increase the information. Alternatively, those pairs with small compatibility, i.e. a 𝜋 𝑖 𝜋 𝑘 small, tend to increase 𝑁𝑒𝑔𝐸𝑛𝑡 as well. The explication is given in Yager and Petry [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF] as that the former pairs affect the value ( 1 𝑡 2 ⁄ ). We note that 𝑡 2 = 𝑡 + 𝑡(𝑡 -1) is the number of possibility distributions plus the number of pairs. Thus while a conflicting pair does not affect too much the sum ∑ ∑ (𝜋 𝑖 𝜋 𝑘 )

𝑡 𝑘=𝑖+1 𝑡-1 𝑖=1
, it can however augment the value of 𝑁𝑒𝑔𝐸𝑛𝑡 since it is counted in the term 𝑡 2 .

Yager and Petry [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF] point out that to only fuse the possibility distributions that have a high compatibility provokes a loss of credibility of the fusion. They suggest the use of a set measure that indicates the credibility of a fusion based on a simple weighted average of a subset of the distributions. This can be applied to possibility distributions as well.

The fusion process considers the possibility distribution subset 𝐵 of the most credible distributions. Thus, the set measure 𝐶𝑟𝑒𝑑: 2 𝑍 → [0,1] is defined such that for any subset 𝐵 of 𝑍, 𝐶𝑟𝑒𝑑(𝐵) indicates the credibility of a fusion based on only the distributions in 𝐵. The credibility of a possibility distribution subset requires the following natural properties:

1. 𝐶𝑟𝑒𝑑(∅) = 0 ;
2. 𝐶𝑟𝑒𝑑(𝑍) = 1 ; and 3. 𝐼𝑓 𝐴 ⊆ 𝐵 𝑡ℎ𝑒𝑛 𝐶𝑟𝑒𝑑(𝐴) ≤ 𝐶𝑟𝑒𝑑(𝐵). The search and rescue example of Yager and Petry [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF] that consists of three distributions, and which represent information on four spatial locations, is being reused here for the possibilistic case. The decision is about finding the location where to rescue? The sources of probabilistic information are, hypothetically, from UAVs, surveillance aircraft, and human sources. For the purpose of comparison let us state that the sources of possibilistic information on the same locations are from humans. Table 10 shows the comparison.

As expected, it does not gives the same results since we use normalized distributions in the possibilistic framework. The choice of using a framework or another is based upon the nature and characteristics of the information sources. Consequently, the analysis presented in Section 5 of Yager & Petry on the use of 𝐶𝑜𝑛𝑓 and 𝑁𝑒𝑔𝐸𝑛𝑡 for the probabilistic framework also applies here to the possibilistic framework.

=     ( ) ( ) 8050 . , , 3 2 1 =     NegEnt

On weighted average fusion

The previous section shows how to use the distributions and their information content as an initial step for choices of possibility distributions. In this section, we discuss how to use the credibility, 𝐶𝑟𝑒𝑑, along with 𝑁𝑒𝑔𝐸𝑛𝑡 for the final fusion taking into account the credibility of fused values using different subsets of 𝑍 = {𝜋 1 , 𝜋 2 , . . . , 𝜋 𝑡 }. Given a subset 𝐵 of possibility distributions from 𝑍 we can calculate the associated fused value, 𝜋 𝐵 . In particular if |𝐵| is the number of distributions in 𝐵 then 𝜋 𝐵 = 1 |𝐵| ∑ 𝜋 𝑖 𝜋 𝑖 ∈𝐵 , and ‖𝜋 𝐵 ‖ 2 ∈ [0,1] is given by:

‖𝜋 𝐵 ‖ 2 = 1 |𝐵| 2 [ ∑ ‖𝜋 𝑖 ‖ 2 + 2 ∑ ∑ 𝜋 𝑖 𝜋 𝑘 𝑡 𝑘=𝑖+1 𝜋 𝑘 ∈𝐵 𝑡-1 𝑖=1 𝜋 𝑖 ∈𝐵 𝜋 𝑖 ∈𝐵 ] (12)
The measure, 𝐶𝑟𝑒𝑑, provides the credibility associated with 𝜋 𝐵 and its range is given by 𝐶𝑟𝑒𝑑(𝐵) ∈ [0,1].

The decision problem is now to find a fused value that has both high values for 𝐶𝑟𝑒𝑑(𝐵) and for 𝑁𝑒𝑔𝐸𝑛𝑡(𝜋 𝐵 ). To select which subset to use, authors in [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF] use the dominance concept related to the credibility and quantity of information. It is expressed as the predicate 𝐷𝑜𝑚(𝐵 𝑖 , 𝐵 𝑗 ):

𝐷𝑜𝑚(𝐵 𝑖 , 𝐵 𝑗 ) = [(𝐶𝑟𝑒𝑑(𝐵 𝑖 ) ≥ (𝐶𝑟𝑒𝑑(𝐵 𝑗 )) ∧ (𝑁𝑒𝑔𝐸𝑛𝑡 ≥ 𝑁𝑒𝑔𝐸𝑛𝑡) ∧ 𝐶𝑜𝑛𝑑(>, <)] (13)
Where 𝐶𝑜𝑛𝑑(>) is true only if at least one of the "≥"is ">. If 𝐷𝑜𝑚(𝐵 𝑖 , 𝐵 𝑗 ) is true then 𝐵 𝑖 dominates 𝐵 𝑗 and so 𝐵 𝑗 can be removed from consideration. Based on dominance rules, we have a collection of non-dominated fusions where each fusion is determined from a subset of space 𝑍 denoted as 𝐵 1 , . . . , 𝐵 𝑟 . For each subset 𝐵 𝑗 , we can calculate its associated fusion 𝜋 𝐵 𝑗 , also its 𝑁𝑒𝑔𝐸𝑛𝑡(𝐵 𝑗 ) value, its 𝑁𝑒𝑔𝐸𝑛𝑡(𝜋 𝐵 𝑗 ) and its credibility, 𝐶𝑟𝑒𝑑(𝐵 𝑗 ). The information about 𝑁𝑒𝑔𝐸𝑛𝑡(𝐵) and credibility values are being used to select among these possible fusions, the 𝜋 𝐵 𝑗 . In the following, we consider how to choose the final possibility distribution based both on the information quantity and on credibility using the example of the previous section.

The collection of relevant possibility distributions is 𝑍 = {𝜋 1 , 𝜋 2 , 𝜋 3 }. So there are seven subsets of 𝑍 to consider:

𝐵 1 = {𝜋 1 }, 𝐵 2 = {𝜋 2 }, 𝐵 3 = {𝜋 3 }, 𝐵 4 = {𝜋 1 , 𝜋 2 }, 𝐵 5 = {𝜋 1 , 𝜋 3 }, 𝐵 6 = {𝜋 2 , 𝜋 3 }, 𝐵 7 =
{𝜋 1 , 𝜋 2 , 𝜋 3 }. In Table 11, we compare credibility functions proposed in [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF] for the probabilistic case with the credibility function obtained for the possibilistic approach. The dominance principle (Eq.13) is applied to get the results of Table 11. The results obtained with the possibilistic approach use the follow statements:

-𝑁𝑒𝑔𝐸𝑛𝑡(𝐵 4 ) > 𝑁𝑒𝑔𝐸𝑛𝑡(𝐵 2 ), 𝑁𝑒𝑔𝐸𝑛𝑡(𝐵 7 ) > 𝑁𝑒𝑔𝐸𝑛𝑡(𝐵 5 ), 𝑁𝑒𝑔𝐸𝑛𝑡(𝐵 7 ) > 𝑁𝑒𝑔𝐸𝑛𝑡(𝐵 6 ) -(𝐶𝑟𝑒𝑑(𝐵 4 ) > 𝐶𝑟𝑒𝑑(𝐵 2 ), 𝐶𝑟𝑒𝑑(𝐵 7 ) > 𝐶𝑟𝑒𝑑(𝐵 5 ), 𝐶𝑟𝑒𝑑(𝐵 7 ) > 𝐶𝑟𝑒𝑑(𝐵 6 ))

-𝐵 1 , 𝐵 3 , 𝐵 4 , 𝐵 7 dominate all other subsets 𝐵 2 , 𝐵 5 , 𝐵 6 ;

A quick look at the results obtained using Yager & Petry [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF] credibility functions as compared to the possibilistic approach shows that the most best subsets are selected using the possibilistic framework. Yager and Petry [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF] propose another approach for obtaining the quality of a fusion by the use of prioritized aggregation introduced by Yager [START_REF] Yager | On prioritized multiple-criteria aggregation[END_REF] where a score associated with a fusion is a weighted sum of its credibility and 𝑁𝑒𝑔𝐸𝑛𝑡. In our case, we start with the set 𝑍 = {𝜋 1 , 𝜋 2 , . . . , 𝜋 𝑡 } and choose a subset 𝐵 𝑖 of 𝑍.

The fusion is performed by taking an equally weighted aggregation of the possibility distributions in 𝐵 𝑖 .

In [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF], the authors suggest a more general approach based on a non-uniform weighted average of the elements in 𝑍. Here, we propose a new weighting approach. The weighted factor of a possibility distribution is denoted by 𝑤, which is determined by both the credibility 𝐶𝑟𝑒𝑑 and the separability SE. The weighting factor, w, is defined as:

𝑤 = 1 2 (𝐶𝑟𝑒𝑑 + 𝐶𝑟𝑒𝑑 ⋅ 𝑆𝐸 -𝐶𝑟𝑒𝑑 ) (14)
The factor 1 2 ⁄ is used to normalized 𝑤 and to guarantee that 0 ≤ 𝑤 ≤ 1. The separability degree of two possibility distributions is obtained based on Manhattan distance as:

𝑆𝐸 = 1 - ( 1 𝑛 -1 ∑(𝜋 𝑖 -𝜋 𝑗 ) 𝑛 𝑗=1 𝑗≠𝑖 ) (15)
Finally, the functional processes for an intelligent quality-based approach to fuse multi-source possibilistic information can be summarized as shown in Figure 5 as an '8-step' methodology. This methodology is being used, in the next section, to conduct an empirical comparison between both frameworks.

Empirical results

This section details the calculations of the possibilistic approach to get the results presented in the right column of Table 11 as well as to illustrate the functional processing of Figure 5. A numerical example is shown first. The numerical example is the same than the one presented in the previous section. The second part shows an experimentation with four benchmark experimental datasets: IRIS Fisher data set [START_REF] Fisher | IRIS data set[END_REF], Pima

Indians Diabetes [START_REF] Lekkas | Evolving fuzzy medical diagnosis of Pima Indians diabetes and of dermatological diseases[END_REF][START_REF] Dataset | Pima Indians Diabetes dataset[END_REF], Glass data set [START_REF] Bache | Glass Data Set Available[END_REF] and Liver-disorders [START_REF] Bache | Liver-disorders data set[END_REF] that have different degrees of information quality namely about information incompleteness.

Numerical example

The set 𝑍, of possibility distributions are considered as the sources of information, 𝑍 = {𝜋 1 , 𝜋 possibility distribution has a small conflict with respect to other distributions, then, that distribution should contribute more on the final fusion results than the other distributions that show high conflicts. A weighting function must then be defined to reflect that. To determine the value of each weight, a correlation coefficient between two possibility distributions is used that leads to the definition of a credibility measure for each distribution.

Using the same numerical example of the previous section, the 8-step methodology, illustrated in Figure 5, is unfolded the following way:

Step 1: Computation of the credibility measure for each possibility distribution based on one single source of knowledge

The credibility (𝐶𝑟𝑒𝑑) of possibility distribution represents the similarity among distributions. The credibility values for the numerical example are computed accordingly to Table 11 and are respectively:

𝐶𝑟𝑒𝑑(𝜋 1 ) = 0.7103; 𝐶𝑟𝑒𝑑(𝜋 2 ) = 0.7389; 𝐶𝑟𝑒𝑑(𝜋 3 ) = 0.4785

The credibility measures show that the distributions, 𝜋 1 and 𝜋 2 , are the most credible.

Step 2: Computation of the separability measure for each possibility distribution

This step is to define weights for each possibility distribution based on the distribution credibility and its discriminating power that is represented by the separability measure.

For the search and rescue example, the values of each degree of separability are:

𝑆𝐸(𝜋 1 ) = 0.8333; 𝑆𝐸(𝜋 2 ) = 0.8667; 𝑆𝐸(𝜋 3 ) = 0.7667

Step It is defined by: 𝑤 = 1 2 (𝐶𝑟𝑒𝑑 + 𝐶𝑟𝑒𝑑 ⋅ 𝑆𝐸 -𝐶𝑟𝑒𝑑 ). The factor 1 2 ⁄ is used to normalize 𝑤 and to guarantee that 0 ≤ 𝑤 ≤ 1. In our example, the weights are as follow:

𝑤 1 (𝜋 1 ) = 0.7594; 𝑤 2 (𝜋 2 ) = 0.7801; 𝑤 3 (𝜋 3 ) = 0.5109

Step 4: Definition of all subsets of the distribution set Z

The example consists of 3 information sources, consequently 7 subsets are being constructed:

𝑍 𝐵 1 = {𝑓 1 } 𝐵 2 = {𝑓 2 } 𝐵 3 = {𝑓 3 } | 𝐵 4 = {𝑓 1 , 𝑓 2 } 𝐵 5 = {𝑓 1 , 𝑓 3 } 𝐵 6 = {𝑓 2 , 𝑓 3 } | 𝐵 7 = {𝑓 1 , 𝑓 2 , 𝑓 3 } | (16)
Step 5: Fusion of the weighted possibility distributions of each subset

The corresponding distributions of the different subsets (𝐵 𝑘=1 𝑡𝑜 7 ) are given as follow. This step performs the evaluation of the quality of each subset and selects only subsets that have a large degree of 𝑁𝑒𝑔𝐸𝑛𝑡. The information quantity (𝑁𝑒𝑔𝐸𝑛𝑡) of each weighted subset is presented in Figure 6. We remark the following:

1. The subset 𝐵 𝑖 with the largest 𝑁𝑒𝑔𝐸𝑛𝑡 provides the most information (in Figure 6, 𝐵 1 𝑎𝑛𝑑 𝐵 3 ).

For example, the subset 𝐵 1 gives a large 𝑁𝑒𝑔𝐸𝑛𝑡 value since 𝐵 4 fuses 𝜋 1 and 𝜋 2 that the first distribution (𝜋 1 ) and the second distribution (𝜋 2 ) have a small value of conflict.

2. Fusing a subset 𝐵 𝑖 and 𝐵 𝑗 with a large value of conflict provides a subset with a small 𝑁𝑒𝑔𝐸𝑛𝑡.

For example, the fusion of 𝜋 1 (represented by 𝐵 1 ) or 𝜋 2 (represented by 𝐵 2 ) with 𝜋 3

(represented by 𝐵 3 ) gives a small value of 𝑁𝑒𝑔𝐸𝑛𝑡 as shown by 𝐵 5 , 𝐵 6 .

Step 7: Computation of the credibility measure for each subset (𝑩 𝒌=𝟏 𝒕𝒐 𝟕 )

Figure 7 provides the information about the credibility (𝐶𝑟𝑒𝑑) of fused subsets, (𝑩 𝒌=𝟏 𝒕𝒐 𝟕 ). We can conclude based on this figure that the subset as 𝐵 5 , 𝐵 6 and 𝐵 7 containing at least one credible distribution may have a high credibility value. To verify the quality of a subset, it is possible to calculate a score associated with a fusion that is a weighted sum of its credibility and its information quantity (𝑁𝑒𝑔𝐸𝑛𝑡). More formally the score of the fusion based on the subset 𝐵 is defined as:

𝑆𝑐𝑜𝑟𝑒(𝐵 𝑘 ) = 𝑤 1 𝐶𝑟𝑒𝑑(𝐵 𝑘 ) + 𝑤 2 𝑁𝑒𝑔𝐸𝑛𝑡 (18) 𝑤 1 𝑤 2 = 𝐶𝑟𝑒𝑑(𝐵 𝑘 ) 1 (19)
Normalizing the weights so that 𝑤 1 + 𝑤 2 = 1 , we get

𝑆𝑐𝑜𝑟𝑒(𝐵 𝑘 ) = 𝐶𝑟𝑒𝑑(𝐵 𝑘 ) 1 + 𝐶𝑟𝑒𝑑(𝐵 𝑘 ) (1 + 𝑁𝑒𝑔𝐸𝑛𝑡) (20)
The score, associated with each subset, is illustrated in Figure 9. Subsets selected according to the dominance rule are represented by the blue bars. Subsets with the highest scores are being selected according to the dominance principle. The complexity of the proposed method depends on the complexity of the credibility measure, the complexity of the separability and the complexity of the information quantity measure. The complexity of these concepts is of :

-O(log2(t)), while processed for each source of the t sources of knowledge.

-O(2 t ), while processed for all source of knowledge subsets.

The resulting complexity is equal to the maximum of [O(log2(t)), O(2 t ) ] that corresponds to O(2 t ).

Experimental datasets as benchmark

This section presents a comparison between both approaches (probabilistic, possibilistic) following the 8steps methodology of Figure 5 but this time on real dataset benchmarks. Four dataset benchmarks are being used: IRIS Fisher data set [START_REF] Fisher | IRIS data set[END_REF], Pima Indians Diabetes [START_REF] Lekkas | Evolving fuzzy medical diagnosis of Pima Indians diabetes and of dermatological diseases[END_REF][START_REF] Dataset | Pima Indians Diabetes dataset[END_REF], Glass data set [START_REF] Bache | Glass Data Set Available[END_REF] and Liver-disorders

Probabilistic and possibilistic modeling of the information sources

Prior to apply the 8-step methodology of Figure 5, the information sources have to be modelled either by possibility or probability distributions. The construction of distributions is illustrated in Figure 10 for the IRIS data set for both frameworks: four features and three classes (color curves). The estimation process of probability distribution 𝑝 𝑗,𝑖 from feature 𝑓 𝑗 uses a histogram method. Then, the possibility distribution 𝜋 𝑗,𝑖 is inferred from the probability distribution based on the approach of Dubois and Prade [START_REF] Dubois | On possibility/probability transformations[END_REF].

For both probability and possibility modeling, the resolution of the histogram data that corresponds to the number of points 𝑛 (bins) selected to represent the distributions is crucial since it directly impacts on the distribution's shape. For example, in Figure 10 the number of bins used to represent the distributions is 10.

The total number of available samples is determinant in the choice of the number of bins in the histogram to represent classes according to features. Indeed, with a small amount of data, a large number of bins is not possible since the modeling of information associated with the feature becomes incorrect. The question now is what would be the adequate number of points to represent a feature?

To answer this question, a method based on the computation of the Shapley Index [START_REF] Pinar | Measures of the Shapley index for learning lower complexity fuzzy integrals[END_REF] is proposed in this study. The Shapley index is generally used to assess the quality of the feature discrimination for the resulting models. The information associated to a class according to feature values is modeled according to five different numbers of points 𝑛 selected to quantify feature values, namely 𝑛 = 5, 8, 10, 12 and 15 points.

The Shapley Index is then calculated corresponding to each number of 𝑛. Figure 11 shows the results of the Shapley class discrimination values on the Iris dataset for the possibility modeling. In [START_REF] Bouhamed | Feature selection in possibilistic modeling[END_REF], the authors have demonstrated that the feature with the highest Shapley value is the one that has the largest discriminating factor between classes. Figure 11 shows that the highest Shapley value is obtained for feature 4 as 𝑛 = 10.

So, the number of points chosen to represent the probability and the possibility distributions has been fixed

to 𝑛 = 10.

With this value of 𝑛 = 10, a comparison of a possibility class and a probability class is illustrated in Figure 10. It shows that the possibilistic framework gives distributions that have a greater potential to discriminate between classes (e.g. feature 2 and feature 4) than for a probabilistic framework.

Comparison of the 8-step Probabilistic-Possibilistic methodology on real cases

The 8-step methodology of Figure 5 is now being compared but on real cases for both: the possibilistic and the probabilistic frameworks. The 8-steps methodology is listed as:

Step 1: Computation of the credibility measure for each distribution

Step 2: Computation of the separability measure for each distribution

Step 3: Computation of the weights for distributions

Step 4: Definition of all subsets of each set 𝒁 𝒊, 𝐢=𝟏 𝐭𝐨 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐜𝐥𝐚𝐬𝐬𝐞𝐬

Step 5: Initial fusion of the weighted distributions of each subset

Step 6: Computation of the information quantity (𝑵𝒆𝒈𝑬𝒏𝒕) in each subset (𝑩 𝒊,𝒌, 𝒌=𝟏 𝒕𝒐 𝟐 𝒎 , 𝒎=𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒇𝒆𝒂𝒕𝒖𝒓𝒆𝒔 )

Step 7: Computation of a credibility measure for each subset (𝑩 𝒊,𝒌 )

Step 8: Selection of the best subsets for the final fusion for each set 𝒁 𝒊 Before applying the methodology, let us give a primary idea of the dataset information quality. The information quantity (𝑁𝑒𝑔𝐸𝑛𝑡) has been calculated for all distributions. Results are tabulated in Figure 12.

The information quantity (𝑁𝑒𝑔𝐸𝑛𝑡) obtained with the possibilistic framework gives larger values than with the probabilistic framework. For the four benchmarks, and based only on 𝑁𝑒𝑔𝐸𝑛𝑡, the possibilistic framework seems to offer a better modelling of the information sources and its imperfections than the probabilistic one. Let us now apply the methodology according to both frameworks.

Step 1: Computation of the credibility measure for each distribution

The credibility degrees obtained with the possibilistic and the probabilistic frameworks are presented in Figure 13. We note different behaviors for credibility values in probabilistic and possibilistic frameworks.

The credibility is defined as an average correlation value. The mean value is a linear operation. However, the correlation is calculated based on the cosine of the angle between the distributions. It is known that the cosine is a non-linear operation. So that the correlation is nonlinear. That nonlinearity is being observed, Step 2: Computation of the separability measure for each distribution

The separability degrees obtained in the possibilistic framework and the probabilistic framework are presented in Figure 14. This operation is linear, so it presents similar behavior for both frameworks. Figure 14 shows that the separability degree between classes for all features is very close to each other in probability framework. However, we observe a much better degree of separability with the possibilistic framework, which allows us to better distinguish between classes. Step 5: Initial fusion of the weighted distributions of each subset

For each class 𝐶 𝑖 , once the weight of each feature and the different subsets 𝐵 𝑖,𝑘 are defined, we proceed to the fusion of the weighted distributions of each subset. Then, the obtained distributions are used to select the best subset (s), for each class 𝐶 𝑖 .

In next steps (Step6, step 7 and step8), we present by figures (Figure 16, Figure 17 and Figure 18) only results obtained for Iris dataset because it contains a low number of subsets which facilitate results visualisation and interpretation.

Step 6: Computation of the information quantity (𝑵𝒆𝒈𝑬𝒏𝒕) in each subset (𝑩 𝒊,𝒌, 𝒌=𝟏 𝒕𝒐 𝟐 𝒎 , 𝒎=𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒇𝒆𝒂𝒕𝒖𝒓𝒆𝒔 )

The information quantity (𝑁𝑒𝑔𝐸𝑛𝑡) obtained for each subset in the possibilistic and probabilistic frameworks are presented in Figure 16. In addition, the difference between the largest value of 𝑁𝑒𝑔𝐸𝑛𝑡 and the smallest one is very small in the probabilistic framework unlike the possibilistic framework where the difference is more important. The value of 𝑁𝑒𝑔𝐸𝑛𝑡 for class 3 (yellow curve) is always the weakest one. 12, Table 13, Table 14 and Table 15 present the selected subsets respectively for the four benchmarks (Iris, Diabetes, Glass, Liver-disorders) to be considered in the final fusion step. The selection process is based on the dominance rule that uses 𝐶𝑟𝑒𝑑 and 𝑁𝑒𝑔𝐸𝑛𝑡 values. Table 12, Table 13, Table 14 and Table 15 show the results. The selected subsets are not the same in all classes for both frameworks. To verify the quality of a subset, a score associated with a result of fusion is calculated as in Section 7.1.

The score associated with each subset is illustrated in Figure 19. Selected subsets according to the dominance rule are circled in red. Subsets with the highest scores are selected by the dominance rule irrespective of the framework. 16. This advantage is quite important for the IRIS database, this is mainly due to the reduced size of this database. In fact, as we have previously noted, the IRIS database consists of only 150 samples that are equally divided between the different classes. Then, the samples from each class are further divided into two parts, the first one for training and the other for testing. We can then conclude that the number of samples is insufficient to build and learn a probabilistic model. On the other hand, possibilistic theory and SVM classifier have been able to cope with that poor-data small sample size situation to build a model capable of generalizing the classification problem of IRIS.

The Diabetes database, Glass database and Liver-disorders database contain more samples than the IRIS database. The number of samples seems enough to construct a probabilistic model that gives similar recognition rates that with possibilistic modeling and with SVM classifier (79.3% versus 80.6% and 78.1%

for Diabetes database, 94.5% versus 97.3% and 95.3% for Glass database and 66.6% versus 69.9% and 67.3% for Liver-disorders database). The possibility theory is an uncertainty theory that deal better with incomplete information [START_REF] Agarwal | Possibility theory versus probability theory in fuzzy measure theory[END_REF] than the probability theory and the classical SVM classifier. This is even more confirmed in a scenario where we deliberately impoverish the Diabetes dataset, the Glass dataset and Liverdisorders dataset. The poorer Diabetes dataset version contains only 50 samples for each class. The poorer Glass dataset and Liver-disorders dataset versions contain only 20 samples for each class. The recognition rates, illustrated in Table 16 for the impoverished datasets, show the advantage of the possibility theory in data modeling compared to the probability theory and the SVM classifier in poor data environments (information incompleteness). Same kind of advantages have been confirmed from results that have been obtained for different kind of applications namely in pattern recognition and image segmentation in the processing of poor-quality images [START_REF] Bouhamed | Feature selection in possibilistic modeling[END_REF][START_REF] Kallel | An iterative possibilistic knowledge diffusion approach for blind medical image segmentation[END_REF][START_REF] Alsahwa | Iterative refinement of possibility distributions by learning for pixel-based classification[END_REF][START_REF] Alsahwa | A method of spatial unmixing based on possibilistic similarity in soft pattern classification[END_REF][START_REF] Kallel | The use of contextual spatial knowledge for low-quality image segmentation[END_REF].

It is worth mentioning that the size of databases in Table 16 does not exceed 800 samples. We notice from Table 16 that for some classes, there is no big difference between the number of samples used before and after the database impoverishment scheme. As an example, the number of samples from Class 2 of the Diabetes database, initially 50, is reduced to 30 after the impoverishment process. Similarly, in the case of Class 2 of the Glass database, the number is 15 samples before the reduction process and 12 samples after. We also observe an imbalance between the number of samples for the classes within the same database. To overcome these inherent difficulties to classification of real databases, we propose to make a performance comparison with a very large database where the classes are balanced with respect to number of samples. The classification rate is calculated using the principle of cross validation (10 times). An average classification rate for 1000 draws of each size has been calculated. Figure 20 illustrates the performance achieved by the three approaches for each size. The figure is presented in two parts (Portion A, Portion B) where "Portion A" is for large size samples (50000, 20000, 10000, 5000, 2500, 1000) and "Portion B" is for small size samples (500, 250, 100, 50, 20, 15, 10 and 5).

In "Portion A" of figure 20, the probabilistic approach gives excellent results in terms of classification rates as well as for the possibilistic approach. The performance obtained with the classical SVM is less good as compared to the two other ones in "Portion A" but it is known that SVM is better designed for small size cases as exhibited in "Portion B". In "Portion B", the small size cases, the possibilistic approach outperforms gradually the other two methods as the data impoverishment augments. This is under the condition where the possibilistic method can be exploited at best. Finally, it is worth mentioning that the possibilistic method demonstrates the best potential of the three in a context where we would need to reduce the size of the data while maintaining excellent performances.

Conclusion

This paper offers a companion paper to Yager and Petry who proposed a quality-based methodology to combine data provided by multiple probabilistic sources to improve quality of information for decisionmakers. In this paper, we adapted the Yager-Petry 8-steps methodology, not identified as such in Yager and Petry, to be used with multiple possibilistic sources. The paper offers a side by side development of the possibilistic approach with respect to the probabilistic one where both approaches are compared by the means of four experimental benchmark datasets: one being data-poorer (IRIS data set) than the others one (Diabetes dataset, Glass dataset and Liver-disorders dataset). Results obtained from comparison of both approaches confirm the superiority of the possibilistic approach in the presence of less complete information. Considering the results of this paper, incompleteness of information is the only aspect that we can objectively bring concluding remarks upon. Other aspects will obviously have to be taken into account in the choice of a modelling approach in the presence of a diversity of information sources (hard and soft) such as the nature and characteristics of the sources as well as the specifics of the domain of applications.

Figure 1 .

 1 Figure 1. Intelligent quality-based approach of Yager & Petry for probabilistic sources

-

  the case of total ignorance (T.I): The maximal value of ‖𝜋‖ occurs when all 𝜋(𝑗) = 1, 𝑗 = 1 𝑡𝑜 𝑛 so ‖𝜋‖ = √𝑛 then ‖π‖ 2 = n. In the case of complete knowledge (C.K): ∃𝜋(𝑘) = 1, 𝑘 ∈ [1, ⋯ , 𝑛] , ∀𝜋(𝑗) = 0, 𝑗 ≠ 𝑘, 𝑗 = 1 𝑡𝑜 𝑛 then ‖𝜋‖ = 1 and ‖π‖ 2 = 1.

Figure 3 .

 3 Figure 3. Angle between possibilistic vectors

-= 1 .

 1 ‖𝜋 2 ‖ = √𝑛 and ∑ 𝜋 2 (𝑗) The case of complete knowledge (C.K): if 𝜋 2 is a certain possibility distribution, ∃𝜋 2 (𝑗) = 1, for one element, then ‖𝜋 2 ‖ = 1 and ∑ 𝜋 2 (𝑗) 𝑛 𝑗=1 In this case, 𝑐𝑜𝑚𝑝(𝜋 1 , 𝜋 2 ) = 1 √𝑛 . We illustrate three cases of possibility distributions (when n=2) in Figure 4: 1) orthogonality between two distributions, 2) complete knowledge and, 3) the total ignorance distributions. Certain possibility distributions, 𝜋 1 and 𝜋 2 , are completely conflicting and 𝜋 3 is an uncertain possibility distribution. Then, in this case, the distributions are: 𝜋 1 (1) = 1, 𝜋 1 (2) = 0, 𝜋 2 (1) = 0, 𝜋 2 (2) = 1, 𝜋 3 (1) = 1, 𝜋 3 (2) = 1.

Figure 4 .

 4 Figure 4. Illustration of orthogonal, complete knowledge and the total ignorance distributions

1 ‖𝜋‖ 2 -

 12 and ‖𝜋‖ 2 = ∑ (𝜋(𝑗)) It can be shown that: -In the case of total ignorance (T.I): The Gini's entropy assumes its minimal value 𝐺(𝜋) = 1 -1 𝑛 when all 𝜋(𝑗) = 1, 𝑗 = 1, … , 𝑛., In the case of complete knowledge (C.K): The Gini's entropy assumes its maximal value 𝐺(𝜋) = 0 when only one 𝜋(𝑗) = 1 and all other 𝜋(𝑗) = 0.

  [

1 𝑛, 1 ]

 11 and even the same behaviour according to information quantity variations. In both probabilistic and possibilistic frameworks, 𝑁𝑒𝑔𝐸𝑛𝑡 is linked to information quantity. Table 3. Domains of variation of 𝑵𝒆𝒈𝑬𝒏𝒕 for both

𝐶1(𝐵 1 ) 1 -

 11 = 𝐶1(𝐵 2 ) = 𝐶1(𝐵 3 ) = 0 𝐶1(𝐵 4 ) = 𝐶1(𝐵 5 ) = 𝐶1(𝐵 6 ) = 𝐶1(𝐵 7 ) = Based on correlation matrix 𝑐𝑜𝑟𝑟(𝜋 1 , 𝜋 2 ). 1 ) = 𝐶2(𝐵 2 ) = 𝐶2(𝐵 3 ) = 0 ; N/A 𝐷𝑜𝑚 -

Figure 5 .

 5 Figure 5. The functional processing (8-step methodology) of the possibilistic approach

3 :

 3 Computation of the weights for possibility distributionsThe determination of weights for the distributions is related to the credibility 𝐶𝑟𝑒𝑑 and the separability measure SE which are two information quality-based factors. If both factors can be taken into consideration, then a better selection of distribution subsets will result. The credibility and separability weights can be derived based on possibility distributions alone, requiring no extra a priori knowledge. The weight factor of a possibility distribution is denoted by 𝑤, which is determined by both credibility 𝐶𝑟𝑒𝑑 and separability SE.

Figure 6 .

 6 Figure 6. Information quantity (𝑵𝒆𝒈𝑬𝒏𝒕) in each weighted subset

Figure 7 . 8 :Figure 8

 788 Figure 7. Credibility values of each subset Step 8: Selection of the best subsets for the final fusion Figure 8 presents the selected subsets to be considered in the final fusion. The selection process is based on the dominance concept that uses credibility, (Cred), and negative Entropy (𝑁𝑒𝑔𝐸𝑛𝑡) values. The obtained results confirm that all subsets have a good information quantity except the subsets 𝐵 2 , 𝐵 5 , 𝐵 6 .

Figure 9 .

 9 Figure 9. Score values of all subsets

[

  28]. The Iris dataset consists of 3 classes, 4 attributes and 150 samples, with 50 samples for each class. In this paper, the first 35 samples of each class have been used for training. Each class 𝐶 𝑖 is being represented by the set 𝑍 𝑖 = {𝜋 1,𝑖 , 𝜋 2,𝑖 , 𝜋 3,𝑖 , 𝜋 4,𝑖 } where 𝜋 𝑗,𝑖 is the distribution corresponding to a feature 𝑓 𝑗 . The Pima Indians Diabetes is a dataset consisting of 2 classes, 8 attributes and 768 samples, with 500 samples for the first class and 268 samples for the second one. In this paper, the last 100 samples of the first class and the last 50 samples of the second class have been used for training. For the Pima Indians Diabetes database, each class 𝐶 𝑖 is represented by the set 𝑍 𝑖 = {𝜋 1,𝑖 , 𝜋 2,𝑖 , 𝜋 3,𝑖 , 𝜋 4,𝑖 , 𝜋 5,𝑖 , 𝜋 6,𝑖 , 𝜋 7,𝑖 , 𝜋 8,𝑖 } where 𝜋 𝑗,𝑖 is the distribution corresponding to the feature 𝑓 𝑗 . The Glass dataset consists of 2 classes, 9 attributes and 214 samples, where 163 samples are for the first class and 51 samples for the second one. In this paper, 58 samples of the first class and 15 samples of the second class are used for training. Using the Glass database, each class C i is represented by the set Z i = {𝜋 1,𝑖 , 𝜋

Figure 10 .Figure 11 .

 1011 Figure 10. Class representations in probabilistic and possibilistic frameworks using the Iris dataset

Figure 12 .

 12 Figure 12. Information quantities (𝑵𝒆𝒈𝑬𝒏𝒕) in (a) probabilistic framework and (b) possibilistic framework

particularly for feature 4 ofFigure 13 .

 413 Figure 13. Credibility degrees in (a) probabilistic framework and (b) possibilistic framework

Figure 14 . 3 :Figure 15 . 4 :

 143154 Figure 14. Separability degrees in (a) probabilistic framework and (b) possibilistic framework Step 3: Computation of the weights for distributionsWeights have been estimated based on credibility and separability degrees. That explains the non-linearity in Figure15. By looking at Figure12, on the information quantity (𝑁𝑒𝑔𝐸𝑛𝑡), and Figure15, on the weight values, we can see that the weight computation is more sensitive to the variation of the information quantity in the possibilistic framework than the probabilistic framework. For instance, in the probabilistic modeling of the Iris dataset, 𝑁𝑒𝑔𝐸𝑛𝑡 generated by the feature 2 for class 3 is larger than the one obtained by the same feature in class 2. The weight given for this feature in class 3 is smaller than the one given for the same feature in class 2. In the possibilistic modeling, the 𝑁𝑒𝑔𝐸𝑛𝑡 generated by the same feature 2 for class 3 is larger than the one obtained by the same feature in class 2. The weight given for this feature in class 3 is larger than the one given for the same feature in class 2.In the probabilistic modeling of the Diabetes dataset, 𝑁𝑒𝑔𝐸𝑛𝑡 generated by feature 8 for the first class is larger than 𝑁𝑒𝑔𝐸𝑛𝑡 for the second class. Contrariwise, the weight given for feature 8 in the first class is greatly smaller than the one given for the second class. In the possibilistic modeling, for the same feature 8 of the same Diabetes dataset, the value 𝑁𝑒𝑔𝐸𝑛𝑡 generated by feature 8 for the first class is greatly larger than the 𝑁𝑒𝑔𝐸𝑛𝑡 for the second class. Also, the weight given for feature 8 in the first class is greatly larger than the one given for the second class.

Furthermore, the behaviorFigure 16 . 7 :

 167 Figure 16. Information quantity (𝑵𝒆𝒈𝑬𝒏𝒕) of subsets in (a) probabilistic framework and (b) possibilistic framework for Iris dataset Step 7: Computation of the credibility measure for each subset (𝑩 𝒌=𝟏 𝒕𝒐 𝑵 )The credibility measure obtained for each subset in the possibilistic and probabilistic frameworks are presented in Figure17. It is clear that 𝐶𝑟𝑒𝑑 behavior is nonlinear when comparing both frameworks. For example, the first class 𝐶𝑟𝑒𝑑 value is always below the other classes in the probabilistic framework, but in the possibilistic framework and especially for the fourth and the ninth subsets, the first class 𝐶𝑟𝑒𝑑 value goes up to situate between the second and the third class.

Figure 17 .Figure18.

 17 Figure 17. Credibility measures of subsets in (a) probabilistic framework and (b) possibilistic framework for Iris dataset

Figure 19 .

 19 Figure 19. Scores of subsets in (a) probabilistic framework and (b) possibilistic framework for Iris dataset Finally, the recognition rate is calculated based on SVM (Support Vector Method) classifier using 10-fold cross-validation based on all features, when the SVM classifier is applied without subset selection, or the union of the selected subsets for each class whether in the probabilistic or possibilistic framework. The use of the possibility theory has shown its advantages in data modeling compared to the probability theory and to the SVM classifier as illustrated in Table16. This advantage is quite important for the IRIS database, this

Table 2 .

 2 Comparative equations associated with a vector representation under both frameworks

	Probabilistic Framework	Possibilistic Framework

  ‖𝜋 1 ‖ = √𝑛. Consider now 𝑐𝑜𝑚𝑝(𝜋 1 , 𝜋 2 ) where 𝜋 2 is that uniform possibility distribution with 𝑐𝑜𝑚𝑝(𝜋 1 , 𝜋 2 ) =

	𝜋 1 𝜋 2
	‖𝜋 1 ‖‖𝜋 2 ‖

Table 4 .

 4 Table 4 below. In fact, if you replace respectively 𝑝 1 by 𝜋 1 and 𝑝 2 by 𝜋 2 as shown in Table 4, the analysis provided in Section 3 of Yager & Petry [1] stands as well for a possibilistic framework. Comparative equations associated with the use of entropy under both frameworks

	Probabilistic Framework	Possibilistic Framework

Table 5 .

 5 Comparative equations associated with the calculation of ‖𝜋‖ 2

	Probabilistic Framework	Possibilistic Framework
	𝑃 = ∑ 𝑤 𝑖 𝑝 𝑖 𝑡 𝑖=1	with 𝑝(𝑗) = ∑ 𝑤 𝑖 𝑝 𝑖 (𝑗) 𝑡 𝑖=1

Table 8

 8 below presents the calculations of 𝑁𝑒𝑔𝐸𝑛𝑡 in a mixed case i.e. that 𝑡 1 of the distributions are pure certainty and 𝑡 2 are pure uncertainty.

Table 7 .

 7 Comparative equations associated with the dot product under both frameworks Probabilistic Framework Possibilistic Framework If both 𝑝 𝑖 and 𝑝 𝑘 represent certainty then if they agree on the certainty element < 𝑝 𝑖 , 𝑝 𝑘 >= 1 and if they disagree, then < 𝑝 𝑖 , 𝑝 𝑘 > = 0. If both 𝜋 𝑖 and 𝜋 𝑘 are certainty then if they agree on the certainty element < 𝜋 𝑖 , 𝜋 𝑘 > = 1 and if they disagree < 𝜋 𝑖 , 𝜋 𝑘 > = 0. If one of < 𝜋 𝑖 , 𝜋 𝑘 > is pure uncertainty, for example 𝜋 𝑖 , then: -if the other distribution 𝜋 𝑘 is certainty, we have < 𝜋 𝑖 , 𝜋 𝑘 > = ∑ 𝜋 𝑖 (𝑗)𝜋 𝑘 (𝑗) -if the other distribution is all pure uncertainty, we get < 𝜋 𝑖 , 𝜋 𝑘 > = ∑ 𝜋 𝑖 (𝑗)𝜋 𝑘 (𝑗) -if we have only one possibility distribution that is certain then < 𝜋 𝑖 , 𝜋 𝑘 > = 1 or -if all possibility distributions are certain and they agree on the certainty element then < 𝜋 𝑖 , 𝜋 𝑘 > = 1.

	If one of < 𝑝 𝑖 , 𝑝 𝑘 > is pure uncertainty,
	for example 𝑝 𝑖 , then:
	-if the other distribution 𝑝 𝑘 is certainty, we have
	< 𝑝 𝑖 , 𝑝 𝑘 >= ∑ 𝑝 𝑖 (𝑗)𝑝 𝑘 (𝑗) 𝑛 𝑗=1	= 1	1 𝑛 =	1 𝑛 .	𝑛 𝑗=1	= 1.
	-if the other distribution is all pure uncertainty, we
	get < 𝑝 𝑖 , 𝑝 𝑘 >= ∑ 𝑝 𝑖 (𝑗)𝑝 𝑘 (𝑗) 𝑛 𝑗=1	=	𝑛 𝑛 2 =	1 𝑛 .	𝑛 𝑗=1	= 𝑛..
	-if one of 𝑝 𝑖 or 𝑝 𝑘 is pure uncertainty then
	< 𝑝 𝑖 , 𝑝 𝑘 >=	1 𝑛	.

Table 8 .

 8 Comparative equations associated with ‖𝜋‖ 2 in mixed case under both frameworks

		Probabilistic Framework	Possibilistic Framework
	‖𝑃‖ 2 =	1 𝑡 2 [𝑡 1 +	1 𝑛	(𝑡 -𝑡 1 ) +	1 𝑛	((𝑡 -𝑡 1 )(𝑡 + 𝑡 1 -
	1)) + ∑ 𝑔 𝑗 (𝑔 𝑗 -1) 𝑛 𝑗=1

Table 10 .

 10 Calculations of conflicts and information for the example given in Yager and Petry[START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF]

	Probabilistic Framework		Possibilistic Framework
	𝑃 1 : (. 5, .2, .2, .1); 𝑃 2 : (. 4, .3, .2, .1); 𝑃 3 : (. 1, .2, .1, .6 )		𝜋 1 : (. 5, .2, .2, .1); 𝜋 2 : (. 4, .3, .2, .1); 𝜋 3 : (. 1, .2, .1, .6 )	Normalisation	𝜋 1 : (1, .2, .2, .1); 𝜋 3 : (. 1, .2, .1,1 ) 𝜋 2 : (1, .3, .2, .1);
	‖𝑃 1 ‖ = (√. 34) = .583; 𝑁𝑒𝑔𝐸𝑛𝑡 = .34	‖𝜋 1 ‖ = (√1.09) = 1.0440; 𝑁𝑒𝑔𝐸𝑛𝑡 =0.9174
	‖𝑃 2 ‖ = (√. 3) = .547; 𝑁𝑒𝑔𝐸𝑛𝑡 = .3		‖𝜋 2 ‖ = (√1.14) = 1.0677; 𝑁𝑒𝑔𝐸𝑛𝑡 = 0.8772
	‖𝑃 3 ‖ = (√. 42) = .648; 𝑁𝑒𝑔𝐸𝑛𝑡 = .42	‖𝜋 3 ‖ = (√1.06) = 1.0296; 𝑁𝑒𝑔𝐸𝑛𝑡 = 0.9434
	𝑃 1 ⋅ 𝑃 2 = .31; 𝑃 1 ⋅ 𝑃 3 = .17; 𝑃 2 ⋅ 𝑃 3 = .18	𝜋 1 ⋅ 𝜋 2 = 1.11; 𝜋 1 ⋅ 𝜋 3 = .26; 𝜋 2 ⋅ 𝜋 3 = .28
	Conf(𝑃 1 , 𝑃 2 ) = 1 -= 1 -.975 = .025 . 31 . 583 * .547 = 1 -Conf(𝑃 1 , 𝑃 3 ) = 1 -. 17 . 583 * .648 = 1 -= 1 -.450 = .550	. 31 . 318 . 17 . 378	1.11 1.0440 * 1.0677 = 1 -Conf(𝜋 1 , 𝜋 2 ) = 1 -1.11 = 1 -0.9958 1.11478 = 0.0042

Table 11 .

 11 Credibility functions used for both frameworks

	Probabilistic Framework	Possibilistic Framework
	First credibility function: at least two
	distributions included in a subset is used.

  𝜋 𝐵 𝑘 (𝜋 𝑖 , 𝜋 𝑗 ) = (𝑤 𝑖 . 𝜋 𝑖 + 𝑤 𝑗 . 𝜋 𝑗 )/2 𝑤 1 . 𝜋 1 + 𝑤 2 . 𝜋 2 + 𝑤 3 . 𝜋 3 ) 3 ⁄ = {0.2476,0.1627,0.1197,0.1535}

	(17)
	𝐵 1 = 𝑤 1 . 𝜋 1 = {0.3797,0.1519,0.1519,0.0759}
	𝐵 2 = 𝑤 2 . 𝜋 2 = {0.3120,0.2340,0.1560,0.0780}
	𝐵 3 = 𝑤 3 . 𝜋 3 = {0.0511,0.1022,0.0511,0.3065}
	𝐵 4 = (𝑤 1 . 𝜋 1 + 𝑤 2 . 𝜋 2 ) 2 ⁄ = {0.3459,0.1929,0.1539,0.0770}
	𝐵 5 = (𝑤 1 . 𝜋 1 + 𝑤 3 . 𝜋 3 ) 2 ⁄ = {0.2154,0.1270,0.1015,0.1912}
	𝐵 6 = (𝑤 2 . 𝜋 2 + 𝑤 3 . 𝜋 3 ) 2 ⁄ = {0.1816,0.1681,0.1036,0.1923}
	𝐵 7 = (

Step 6: Computation of the information quantity (𝑵𝒆𝒈𝑬𝒏𝒕) in each subset (𝑩 𝒌=𝟏 𝒕𝒐 𝟕 )

  5 , 𝐵 6 .

	𝑆𝑢𝑏𝑠𝑒𝑡 𝐶𝑟𝑒𝑑	|	𝐵 1 0.9174	|	𝐵 2 0.8772	|	𝐵 3 0.9434 |	𝐵 4 0.8989	|	𝐵 5 0.4848	|	𝐵 6 0.4796	|	𝐵 7 0.8050 |
	𝑁𝑒𝑔𝐸𝑛𝑡		0.7937		0.7946		0.4668	0.7952		0.8392		0.8420		0.8697
	⇓
	𝑛𝑜𝑛 -𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑒𝑑 𝑠𝑢𝑏𝑠𝑒𝑡𝑠: 𝑁𝐷 = {𝐵 1 , 𝐵 3 , 𝐵 4 , 𝐵 7 }
					Figure 8. Subsets selection

  2,𝑖 , 𝜋 3,𝑖 , 𝜋 4,𝑖 , 𝜋 5,𝑖 , 𝜋 6,𝑖 , 𝜋 7,𝑖 , 𝜋 8,𝑖 , 𝜋 9,𝑖 } where 𝜋 𝑗,𝑖 is the distribution corresponding to feature f j .The Liver-disorders is a dataset consisting of 2 classes, 6 attributes and 326 samples, whose 138 samples for the first class and 188 samples for the second one. Using the Liver-disorders database, each class C i is represented by the set 𝑍 𝑖 = {𝜋 1,𝑖 , 𝜋 2,𝑖 , 𝜋 3,𝑖 , 𝜋 4,𝑖 , 𝜋 5,𝑖 , 𝜋 6,𝑖 , 𝜋 7,𝑖 , 𝜋 8,𝑖 } where 𝜋 𝑗,𝑖 is the distribution corresponding to feature 𝑓 𝑗 .

Table 14 .

 14 Selected subsets of Glass dataset 𝐵 11 , 𝐵 67 , 𝐵 150 , 𝐵 272 , 𝐵 282 , 𝐵 292 , 𝐵 297 , 𝐵 336 , 𝐵 398 , 𝐵 400 , 𝐵 405 , 𝐵 419 , 𝐵 511 } 𝐵 6 , 𝐵 8 , 𝐵 14 , 𝐵 21 , 𝐵 23 , 𝐵 24 , 𝐵 43 , 𝐵 60 , 𝐵 63 , 𝐵 65 , 𝐵 106 , 𝐵 113 , 𝐵 115 , 𝐵 116 , 𝐵 143 , 𝐵 214 , 𝐵 217 , 𝐵 224 , 𝐵 484 , 𝐵 486 , 𝐵 500 , 𝐵 508 , 𝐵 510 , 𝐵 511 } 𝐵 9 , 𝐵 31 , 𝐵 36 , 𝐵 37 , 𝐵 43 , 𝐵 45 , 𝐵 93 , 𝐵 122 , 𝐵 123 , 𝐵 125 , 𝐵 129 , 𝐵 211 , 𝐵 230 , 𝐵 236 , 𝐵 245 , 𝐵 247 , 𝐵 250 , 𝐵 255 , 𝐵 302 , 𝐵 338 , 𝐵 367 , 𝐵 368 , 𝐵 377 , 𝐵 426 , 𝐵 446 , 𝐵 451 , 𝐵 456 , 𝐵 457 , 𝐵 492 , 𝐵 496 , 𝐵 507 , 𝐵 511 }

		Possibilistic framework
		𝐂 𝟏 : 𝐖𝐢𝐧𝐝𝐨𝐰 -𝐆𝐥𝐚𝐬𝐬
	𝑁𝐷 = {𝐵 3 , 𝐂 𝟐 : 𝐍𝐨𝐧 -𝐖𝐢𝐧𝐝𝐨𝐰 -𝐆𝐥𝐚𝐬𝐬
		𝑁𝐷 = {𝐵 219 , 𝐵 443 , 𝐵 484 , 𝐵 496 , 𝐵 500 , 𝐵 511 }
		Probabilistic framework
		𝐂 𝟏 : 𝐖𝐢𝐧𝐝𝐨𝐰 -𝐆𝐥𝐚𝐬𝐬
	𝑁𝐷 = {	𝐵 2 , 𝐂 𝟐 : 𝐍𝐨𝐧 -𝐖𝐢𝐧𝐝𝐨𝐰 -𝐆𝐥𝐚𝐬𝐬
	𝐵 8 ,
	𝑁𝐷 = {

Table 15 .

 15 Selected subsets of Liver-disorders dataset 𝐵 6 , 𝐵 15 , 𝐵 16 , 𝐵 19 , 𝐵 20 , 𝐵 31 , 𝐵 34 , 𝐵 37 , 𝐵 40 , 𝐵 41 , 𝐵 56 , 𝐵 62 , 𝐵 63 } 𝐂 𝟐 : 𝐃𝐞𝐮𝐱 𝑁𝐷 = {𝐵 3 , 𝐵 5 , 𝐵 15 , 𝐵 28 , 𝐵 31 , 𝐵 37 , 𝐵 40 , 𝐵 46 , 𝐵 62 , 𝐵 63 }

	Possibilistic framework
	𝐂 𝟏 : 𝐔𝐧
	𝑁𝐷 = {𝐵 23 , 𝐵 42 , 𝐵 44 , 𝐵 57 , 𝐵 58 , 𝐵 63 }
	𝐂 𝟐 : 𝐃𝐞𝐮𝐱
	𝑁𝐷 = {𝐵 1 , 𝐵 10 , 𝐵 22 , 𝐵 26 , 𝐵 29 , 𝐵 35 , 𝐵 57 , 𝐵 63 }
	Probabilistic framework
	𝐂 𝟏 : 𝐔𝐧
	𝑁𝐷 = {𝐵 5 ,

Table 16 .

 16 Performance comparison of the possibilistic modelling with the probabilistic one and SVM classifier

		Iris	Diabetes	Impoverished	Glass	Impoverished	Liver-	Impoverished
		dataset	dataset	Diabetes	dataset	Glass dataset	disorders	Liver-disorders
				dataset			dataset	dataset
	Feature	4	8	8	9	9	6	6
	number
	Class	3	2	2	2	2	2	2
	number
	Total	50	500 /C1	50	163 /C1	20	138 /C1	20
	sample size		268 /C2		51 /C2		188 /C2
	/ Class
	Training	35	100 /C1	30	58 /C1	12		12
	sample size		50 /C2		15 /C2
	/ Class
	Possibilistic 96%	80.6%	73%	97.3%	100%	69.9%	71.5%
	Probabilistic 80%	79.3%	63%	94.5%	92.8%	66.6%	64%
	SVM	96%	78.1%	69%	95.3%	92.5%	67.3%	69%
	Classifier

Comparison of Probabilistic-Possibilistic methodology on large sample size

In this section, a large database, namely Skin dataset, is considered. The Skin database has two classes: the Skin class and the non-skin class, both described by three attributes [38] [39]. The total sample size of this base is 245057, of which 50859 are skin samples and 194198 are non-skin samples. The imbalance between the two classes, Skin and non-Skin, is also present in this database. The database is balanced by taking 50 000 samples for each class. The impoverishment process is implemented as follows: starting with 40 000 samples per class down to 5 samples per class. The sizes considered are then: 50 000, 20 000, 10 000, 5000, 2500, 20, 15, 10 and 5. For each size, a Monte Carlo process (1000 draws) has been applied to obtain results for the three approaches: the Yager & Petry [START_REF] Yager | An intelligent quality-based approach to fusing multi-source probabilistic information[END_REF] probabilistic approach, its corresponding possibilistic approach (as proposed in this paper) and a classical SVM classifier.

The following steps have been performed to compare the three approaches:

• Select data subsets, by the possibilistic approach taking into account the quality of the attributes, and then applying the SVM approach in the classification stage;

• Select data subsets, using the probabilistic approach taking into account the quality of the attributes, and then applying the SVM approach in the classification stage.

• Use of the three attributes of the database, without subset selection and not taking into account of attribute quality, and then applying the SVM approach in the classification stage.