L F Lopes Oliveira

R Rossi

A R Bosco De Magalhães

J G Peixoto De Faria

M C Nemes

Continuous Monitoring of Dynamical Systems and Master Equations

We illustrate the equivalence between the non-unitary evolution of an open quantum system governed by a Markovian master equation and a process of continuous measurements involving this system. We investigate a system of two coupled modes, only one of them interacting with external degrees of freedom, represented, in the first case, by a finite number of harmonic oscillators, and, in the second, by a sequence of atoms where each one interacts with a single mode during a limited time. Two distinct regimes appear, one of them corresponding to a Zeno-like behavior in the limit of large dissipation.

The decoherence program is one of the most successfull theroretical proposals devised to give suitable responses to the problem of quantum-to-classical transition (see [1] and references therein) within the scope of the quantum theory itself. According to this program, isolated systems must be considered as mere idealization and the emergence of classicality from quantum mechanics is a consequence of the unavoidable interaction between a physical system and its environment.

The net effect of this interaction is the disappearance of the local quantum correlations between preferred states1 . The early formulations of this program assign the mechanism of decoherence to the continuous monitoring of the system of interest by the surrounding particles [2][3][4][5]. These formulations were devised to explain the quasilocalization of macroscopic bodies or symmetry breaking in microscopic systems, like the observation of a well defined chirality in optical isomers.

One of the widespread ways to obtain the effective dynamics of an open quantum system considers the microscopic model describing the coupling between the system of interest and the degrees of freedom of its environment. In general, the number of these environmental degrees of freedom considered is very large, such that the irreversibility of the effective dynamics could be guaranteed. In the limit of infinite external degrees of freedom, the environment is suitably modeled by a thermal reservoir and the effective dynamics of the system of interest is governed by a master equation obtained by tracing these external degrees of freedom out [START_REF] Kubo | Statistical Physics II: Nonequilibrium Statistical Physics[END_REF].

Another well known effect related to continuous measurements is the Quantum Zeno Effect (QZE). In 1977 B. Misra and E. C. G. Sudarshan reported an intriguing result related to measurements in Quantum Mechanics [7]. They showed that a sequence of projective measurements on a system inhibits its time evolution. In the limit of continuous measurements the evolution is completely frozen. Similarities with one of the paradoxes proposed by Zeno of Elea, who intend to show that movement is theoretically impossible, motivated the authors to name the quantum effect after the Greek philosopher. Originally the Quantum Zeno Effect was called the Quantum Zeno Paradox, because it was supposed to show the theoretical impossibility of a "movement" (the quantum evolution) of a decaying particle in a bubble chamber.

The Quantum Zeno Effect (QZE) plays an important role in Quantum Mechanics and enormous literature about this topic has been produced over the last thirty years [8]. After the realization of the pioneer experiment [9] on the effect, which showed the inhibition of transitions between quantum states by means of frequent observations, the QZE became the center of fervorous debates [10,11] about the necessity of the projection postulate on the measurement description. New approaches have been proposed [10,12] and the initial association between the QZE and the projection postulate was no longer a necessary ingredient. Nowadays the literature on this subject is extensive and includes relation between QZE and quantum jumps [13,14], quantum Zeno dynamics [15,16], its implementation in the system of microwave cavities [17], and semiclassical evolution for coupled systems obtained by frequent Zeno-like measurements [18]. With the increasing interest in quantum computation, QZE has become also a tool for the development of protocols on quantum state protection [19][20][21], that are important for the implementation of quantum computation.

Although originally the QZE was considered as a result of a sequence of measurements on the system of interest, the effect has also been predicted in different contexts, as an example we quote: In Ref. [22] the authors suggested that a manifestation of the QZE appears in the relaxation of optical activity in a medium. In Ref. [23] a relation between the QZE and the coherence properties of a two-state system subjected to random influences in a medium is presented. In Ref. [24,25] the authors also show that the QZE can be induced by other physical interactions.

In the present work we study two completely distinct approaches to the dynamics of quantum open systems. We consider the system composed by two coupled modes interacting unitarily, where only one of them is coupled to external degrees of freedom. We show that the Zeno like effect similar to the one shown in [25] is present in both dynamics. Firstly we couple to the system of interest a dynamical environment over which we have control (interactions, number of components, etc.). Secondly we simulate an experimental situation where the system is continuously monitored by a probe system, again controlling interaction time, and other parameters.

We show that not only the conjecture is correct in the situations presented, but also a Zeno like effect can be obtained for both cases, when the influence of the external system is sufficiently strong. Precisely the same effect has been observed by modeling the tunneling of a photon between two cavities, one of them whose dissipation is governed by a master equation [25].

I. COUPLING WITH A FINITE NUMBER OF HARMONIC OSCILLATORS

In this section we consider the system of two linearly coupled harmonic oscillators, one of them coupled to an environment composed by a finite number of harmonic oscillators according to the Hamiltonian

Ĥ = ĤS + ĤR + Ĥint , ĤS = ω 1 â † 1 â1 + ω † 2 â † 2 â2 + G â † 1 â2 + â † 2 â1 , (1)
ĤR = N k=3 ω † k â † k âk , Ĥint = N k=3 γ k â † 1 âk + â † k â1 ,
where â † 1 and â † 2 (â 1 and â2 ) are creation (annihilation) bosonic operators for the modes of interest M 1 and M 2 , and â † k and âk refer to the environment for k ranging from k = 3 to N. Defining

A =           â1 â2 â3 . . . âN           , A D =           â † 1 â † 2 â † 3 . . . â † N           , H =           ω 1 G γ 3 • • • γ N G ω 2 0 • • • 0 γ 3 0 ω 3 • • • 0 . . . . . . . . . . . . . . . γ N 0 0 • • • ω N           , ( 2
)
we can write the Hamiltonian Ĥ in a matrix form:

Ĥ = A T D HA. (3)
In order to preserve the hermiticity of the operator Ĥ, the matrix H must be hermitian. Without loss of generality, we can consider it real and symmetric. Thus, there is an orthogonal matrix P such that

P T HP =         λ 1 0 • • • 0 0 λ 2 • • • 0 . . . . . . . . . . . . 0 0 • • • λ N         = D, (4)
where the λ k are real numbers, which may be used to write the Hamiltonian in a diagonal form:

Ĥ = B T D DB = M k=1 λ k b † k bk , (5)
where

B = P T A =         b1 b2 . . . bN         , B D = P T A D =         b † 1 b † 2 . . . b † N         . ( 6
)
Using the orthogonality of P, it is easy to show that canonical commutation relations hold for bk

and b † k : bk , b † j = δ kj , bk , bj = b † k , b † j = 0. (7)
The modes related to the operators âk shall be called the original modes; the ones concerning to bk will be referred to as the normal modes.

In what follows, we assume that the main modes M 1 and M 2 are resonant and we consider that the matrix H is a function of five positive parameters N, Ω, G, ∆ and Γ:

H =                       Ω G Γ √ N -2 Γ √ N -2 Γ √ N -2 Γ √ N -2 • • • Γ √ N -2 Γ √ N -2 G Ω 0 0 0 0 • • • 0 0 Γ √ N -2 0 Ω + ∆ N -2 0 0 0 • • • 0 0 Γ √ N -2 0 0 Ω -∆ N -2 0 0 • • • 0 0 Γ √ N -2 0 0 0 Ω + 2 ∆ N -2 0 • • • 0 0 Γ √ N -2 0 0 0 0 Ω -2 ∆ N -2 • • • 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . Γ √ N -2 0 0 0 0 0 • • • Ω + ∆ 2 0 Γ √ N -2 0 0 0 0 0 • • • 0 Ω -∆ 2                       . ( 8
)
The parameter Ω gives the frequency of the oscillators of interest, which are coupled according to the constant G. The total number of environmental oscillators is (N -2). Their frequencies are distributed in a finite interval around Ω defined with the help of the parameter ∆:

[Ω -∆/2, Ω + ∆/2]. If ∆ is large enough, the environmental modes out of this interval have negligible effects over the oscillators of interest [26]. The environmental frequencies are equally spaced distributed above and below the central frequency Ω. This choice is not essential for the dynamics induced and for derivation of master equations: in fact, randomly distributed frequencies lead, when N is large, to equivalent results. In derivations of master equations, it is usually assumed reservoirs with infinite frequency range and dense spectrum. Infinite frequency range is approximated by increasing ∆, and dense spectrum by decreasing ∆/ (N -2); in order to achieve both conditions simultaneously, we must have large values of ∆ and N. In this limit, a master equation may derived from the Hamiltonian specified by Eq. ( 8). The parameter Γ is related to the strength of the coupling with the environment. The choice of γ k being proportional to 1/ √ N -2 is consistent with assumptions usually performed in derivation of master equations that guarantee finite decay rates in the thermodynamic limit N -→ ∞. Of course, all γ k do not have to be equal for such derivations.

Since the bk are linear combinations of the âk , they share the same vacuum state |0 . In order to investigate the dynamics of the system plus environment in the space of one excitation, let us define the states

|θ k = â † k |0 , |φ k = b † k |0 , k = 1 to N. (9)
Using Eqs. ( 6), we see that they are connected by

|θ k = N j=1 P k,j |φ j , |φ k = N j=1 P j,k |θ j , (10)
where P i,j is the element of the matrix P in the i-th line and j-th column. Using these relations and observing that Ĥ |φ k = λ k |φ k , we can calculate the evolution of the states with one excitation in the original modes as

e -i Ĥt |θ k = N j=1 P k,j e -i Ĥt |φ j = N l=1 N j=1 P k,j P l,j e -iλ j t |θ l . ( 11
)
The probability of finding one excitation in mode M 2 if it is initially in this mode is

p (t) = N j=1 P 2 2,j e -iλ j t 2 . (12)
In Ref. [25], two regimes were found for the dynamics of such a probability calculated by using the master equation corresponding to Hamiltonian (1) with resonant modes M 1 and M 2 :

d dt ρ = κ 2â 1 ρâ † 1 -ρâ † 1 â1 -â † 1 â1 ρ (13)
-i Ω â † 1 â1 + â † 2 â2 , ρ + G â † 2 â1 + â † 1 â2 , ρ .
Here, the density operator ρ stands for the state of the composed system M 1 + M 2 . In one of the regimes, the increasing of the dissipation constant κ of mode M 1 leads to the decreasing of the permanence of the excitation in mode M 2 . This is expected, since mode M 2 is connected to the environment only through mode M 1 . The other regime was called Zeno regime: there, the increasing of the interaction of M 1 with the environment inhibits the transition of the excitation from M 2 to M 1 , leading to the enhancement of the probability of finding the excitation in M 2 .

The turning point between these regimes occurs for κ = 2G, where G is the unitary coupling constant between the modes of interest. In the Appendix, we show that such a turning point corresponds to Γ = 2∆G/π, what is corroborated by Figs. (1a) and (2a). The occurrence of two regimes may be understood with the help of the following analytically calculable cases: for Γ = 0, p (t) = (1 + cos (2Gt)) /2; for G = 0, p (t) = 1. Depending on the relation between Γ and G, the dynamics can be approached to one of these limiting cases. By comparing Figs.

(1a) and (2a) with Figs. (1b) and (2b), respectively, we see that the Hamiltonian and the master equation approaches exhibit good agreement. In order to achieve such an agreement, we have to pay attention to two aspects concerning the parameter ∆: it must be large enough so that environmental modes with frequencies out of the interval [Ω -∆/2, Ω + ∆/2] have negligible action on the system (as pointed out in Ref. [26], the relevant environmental modes are the ones with frequencies around the frequencies of the normal modes of the system Ω ± G); the ratio ∆/ (N -2) must be small, allowing the use of the limit of dense spectrum.

II. SEQUENCE OF MEASUREMENTS

In this section we study the dynamics of two resonant coupled modes (M 1 , M 2 ) and N atoms interacting, one at the time, with mode M 1 . The sequence of interacting atoms represents, in the limit of instantaneous interactions, a continuous measurement of the M 1 excitation number. The investigation shows that the two regimes reported in Ref. [25] and in the previous section are also present if a sequence of atomic interactions is performed on mode M 1 . These results illustrate the relation between continuous measurement of a quantum system and the dynamics governed by a master equation.

The results are obtained by numerical simulations, where we consider, as in Ref. [25], the system of modes M 1 and M 2 in the initial state |0 1 , 1 2 . A sequence of two level atoms interacts with mode M 1 . The atoms are prepared in the ground state |g and interact with mode M 1 one at the time. The Hamiltonian of the global system for the interaction of the k-th atom is given by

Ĥ(k) = Ω(â † 1 â1 + â † 2 â2 ) + G(â † 1 â2 + â † 2 â1 ) + Ω 2 σ(k) z + g(â † 1 σ(k) -+ â1 σ(k) + ), (14)
where â † 1 (â 1 ) and â † 2 (â 2 ) are creation (annihilation) operators for modes M 1 and M 2 , Ω their frequency, G the modes coupling constant, σ(k)

z = |e (k) e (k) | -|g (k) g (k) |, σ(k) -= |g (k) e (k) |, σ(k) + = |e (k) g (k)
| and g the coupling constant for the interaction between k-th atom and mode M 1 . Here, g (k) and e (k) stand for the ground and the excited states of the k-th atom, respectively.

After each interaction we perform the trace over the corresponding atomic system, i.e., we do not consider the final state of the atoms. We also assume that the coupling constant g scales as 1/ √ t int , where t int is the interaction time of each atom [START_REF] Jacobs | Topics in quantum measurement and quantum noise[END_REF]. The overall effect of these atomic interactions with a cavity mode is a dissipative evolution of the mode. The effective dissipative constant related to this process is κ = g 2 t int .

In Figs. (1c) and (2c) we shown two regimes for the probability of finding the excitation in mode M 2 (P(t)). In the dissipative regime (κ < 2G), the increasing of the dissipative constant κ leads to the decreasing of P(t). In the second regime (κ > 2G), the increasing of the dissipative constant κ leads to the increasing of P(t), preserving then the excitation in mode M 2 . It is worth to note that the agreement with the master equation results is reached in the limit of vanishing interaction time, t int . The increasing of interaction time leads the curves away from the ones obtained by the master equation.

III. CONCLUSION

In the present work we investigate the two regimes of the system of two coupled modes, induced by two completely different dynamics. In the first one we consider the mode M 1 linearly coupled to a finite number of harmonic oscillators, and in the second one we consider such mode interacting with N atoms, one at the time. Both dynamics, in appropriate limits, can describe the regimes obtained in [25] using a master equation. In the first model, when the number of harmonic oscillators goes to infinity the coupling between them and the system of interest can simulate the interaction with the environment. In the second model, the interaction with N atoms, when the interaction time goes to zero, simulates continuous measurements. Therefore, the present results illustrate the idea that the interaction between system of interest and environment can be interpreted as continuous measurement on the system of interest. The results for both dynamics were obtained using numerical simulations. As no approximations were used in the calculations, the analysis of intermediate scenarios, out of the master equations limits, is possible.

IV. ACKNOWLEDGEMENTS

This work was partially supported by the brazilian agencies CNPq and FAPEMIG. t int with η = 2 (solid line), η = 1/2 (dashed line) and η = 1/8 (dotted line). For sake of comparison, in the plots (a) and (c), a curve concerning a given value of η corresponds to the curve related to the same value of κ/G in plot (b).

[1] M. Schlosshauer, Rev. Mod. Phys. 76, 1267 (2005).

  FIG. 1: (a) Evolution of the occupation of mode M 2 according to the Hamiltonian approach for N = 500, Ω = 100G, ∆ = 20G and Γ = η∆G π with η = 2 (solid line), η = 8 (dashed line) and η = 32 (dotted line). (b) Evolution of the occupation of mode M 2 according to the master equation approach for κ/G = 2 (solid line), κ/G = 8 (dashed line) and κ/G = 32 (dotted line).

Figure

  Figure fromRef.[25]. (c) Evolution of the occupation of mode M 2 during a sequence of measurements probing the presence of the excitation in mode M 1 for g = 2ηG t int with η = 2 (solid line), η = 8 (dashed line) and η = 32 (dotted line). For sake of comparison, in the plots (a) and (c), a curve concerning a given value of η corresponds to the curve related to the same value of κ/G in plot (b).

FIG. 2 :

 2 FIG. 2: (a) Evolution of the occupation of mode M 2 according to the Hamiltonian approach for N = 500, Ω = 100G, ∆ = 20G and Γ = η∆G π with η = 2 (solid line), η = 1/2 (dashed line) and η = 1/8 (dotted line). (b) Evolution of the occupation of mode M 2 according to the master equation approach for κ/G = 2 (solid line), κ/G = 1/2 (dashed line) and κ/G = 1/8 (dotted line). Figure from Ref. [25]. (c) Evolution of the occupation of mode M 2 during a sequence of measurements probing the presence of the excitation in mode M 1 for g = 2ηGt int with η = 2 (solid line), η = 1/2 (dashed line) and η = 1/8 (dotted line). For sake of comparison, in the plots

To be exact, the quantum correlations does not disappear. In fact, they are carried from the system to the environment and turned out inaccessible.

Appendix A: Establishing the relation between the effective dissipation constant and the parameters used in the Hamiltonian approach

The master equation employed in Ref. [25] is given by Eq. (13). It describes the dynamics of two linearly coupled resonant modes, one of them interacting with an environment at zero absolute temperature. Such a master equation may be derived as an approximation of the dynamics emerging from the Hamiltonian Ĥ, leading, under the specifications in Eq. ( 8), to

where ν j = j∆/ (N -2) and τ c is a value of |τ | beyond which the summations above are negligible [26]. For N sufficiently large, we can take the limit of dense spectrum, resulting in

Since the integrand is assumed to be negligible for |τ | > τ c , we change ±τ c for ±∞, which leads to

Observing that the term related to δ (ω + G) vanishes, we find, for ∆/ (N -2) < G < ∆/2, κ = πΓ 2 ∆ .

The transition to the Zeno regime were found in Ref. [25] for κ = 2G. This corresponds to )


Frédéric Blanchard
email: frederic.blanchard@univ-reims.fr

Philippe Vautrot

Herman Akdag

Michel Herbin

Data Representativeness Based on Fuzzy Set Theory

Keywords: Data Representativeness, Fuzzy Set Theory, Robustness, Data Analysis, Clustering, Multidimensional data, Multivariate Analysis, Ordinal Analysis, Rank Transformation

This paper presents an original definition of data representativeness. The representativeness of each datum in a dataset is a meaningful notion quantified by a degree computed by aggregating fuzzy subsets. These fuzzy subsets are obtained by fuzzifying data in a robust way. We illustrate the usefulness of the representativeness by presenting applications for statistical location estimation,and for cluster analysis.

Introduction

In the past few years the growing computing capabilities and the improvement of data acquisition have led to the need for managing new kinds of data. The extraction of information from datasets is a crucial task. Data-mining and data analysis theories offer methods and techniques that allow exploration, description and explanation of these datasets.

In the context of statistical theory data sets are generally considered as representative cross-sections of theoretical populations. More precisely, most of the involved tools consist in estimating the parameters of a hypothetical underlying distribution of the sample. Classical instances of location estimators are the median and the mean. The standard deviation is a common dispersion estimator. These tools predominantly study the probability distributions and are most often based on assumptions about them. We assume that classical statistics are often not representative of datasets.

The lack of data is another critical point. Data usually contain intrinsic uncertainty or imprecision induced by the acquisition, the nature of the data etc. In addition, real datasets often contain outliers and noisy data. Finally, it 1 is usually impossible to make the assumptions on the data distributions needed by classical statistical and probability tools [START_REF] Friedman | The use of ranks to avoid the assumption of normality implicit in the analysis of variance[END_REF], [START_REF] Bonett | Approximate confidence interval for standard deviation of nonnormal distributions[END_REF]. To deal with these problems, some efficient concepts can be used. The theory of fuzzy sets and the possibilistic approach, are common ways to deal with imprecision and uncertainty. As said before, the sensivity to outliers is an element that often leads the most common statistical methods to fail. The concept of robustness, inherent to the non-parametric methods, offers "resistance" to outliers. More generally, non-parametric concepts like ranks and order statistics [START_REF] Galambos | Order statistics of samples from multivariate distributions[END_REF], [START_REF] Savage | Contributions to the theory of rank order statistics: Computation rules for probabilities of rank orders[END_REF], [START_REF] Slaby | Expectation of rank statistics under setup of stochastic inequalities[END_REF] permit both to beat the curse of outliers and noisy data, and need no assumption on the data distributions.

Our goal is to propose a method that extracts information from data relatively to the given dataset. The proposed tool has to be as insensitive as possible to outliers and to deal with lack of data. As an important constraint, we wanted to develop a non-parametric method needing no assumption on the distribution of data [START_REF] Conover | Rank transformations as a bridge between parametric and nonparametric statistics[END_REF], [START_REF] Friedman | The use of ranks to avoid the assumption of normality implicit in the analysis of variance[END_REF].

To address this question we have defined a new theoretical notion expressing the representativeness of data, relative to a given dataset. This new notion affects to each datum a quantitative information that expresses how this datum is representative in the dataset. In other words, it can be viewed as a way to answer the question: "How reprentative is one datum in his sample ?". By adapting the semantics, we introduce different applications of this notion. By searching the "best representant" in a sample we define a new location estimator of the underlying distribution. By finding some best representant of subgroup, and linking data with these "centers", we propose an algorithm for cluster analysis. How the representativeness is constructed ? Our concept of representativeness is defined by using different efficient and eprouved concepts. The goal of our methodology is to compute a degree of representativeness for each datum of a dataset. This degree is computed as the aggregation of fuzzy subsets [START_REF] Yager | On ordered weighted averaging aggregation operators in multicriteria decisionmaking[END_REF] associated to data. The first step of our method consists in fuzzifying data [START_REF] Zadeh | Fuzzy sets and systems[END_REF], [START_REF] Bouchon-Meunier | La logique floue[END_REF]. This fuzzification is not classical. We choose a robust technique that uses first a rank transformation (see [START_REF] Scheffe | Non-parametric estimation. i. validation of order statistics[END_REF], [START_REF] Mansouri | Multifactor analysis of variance based on the aligned rank transform technique[END_REF], [START_REF] Headrick | An investigation of the rank transformation in multiple regression[END_REF]) of the dissimilarity between data. This technique makes our method free of assumption about the distribution [START_REF] Friedman | The use of ranks to avoid the assumption of normality implicit in the analysis of variance[END_REF], [START_REF] Conover | Rank transformations as a bridge between parametric and nonparametric statistics[END_REF]. This way is really different from a classical fuzzy ranking or fuzzy ordering approach [START_REF] Flaig | Fuzzy ranking: theory and applications[END_REF], [START_REF] Barner | Fuzzy ordering theory and its use in filter generalization[END_REF]. The aggregation is then realized using an OWA operator defined by Yager in [START_REF] Yager | On ordered weighted averaging aggregation operators in multicriteria decisionmaking[END_REF]. This aggregation operator brings flexibility and permits to attenuate the effect of outliers. Finally, we obtain for each element of the initial dataset a value that quantifies how this element is representative in his set. In this approach, the computation of representativeness is made exclusively on the finite input dataset and not in an hypothetic underlying space. Although we are not concerned by the problems of sampling, estimation or interpolation.

To illustrate the interest of our contribution we present its use in different theoretical contexts. First we can use the representativeness as a location estimator. The simulations and the experimentations proove that our induced statistic is efficient and gives better results than the median and other classical statistics. Its efficiency with different kind of distributions and in particular with assymetric distributions is presented. Secondly we illustrate the use of the representativeness for data clustering [START_REF] Hartigan | Clustering Algorithms[END_REF], [START_REF] Jain | Data clustering: a review[END_REF], [START_REF] Murtagh | A survey of recent advances in hierarchical clustering algorithms which use cluster centres[END_REF]. Our concept permits to link data and to define clusters. One advantage of our approach is that the separability between clusters is not assumed to be linear.

The next section of this paper presents the theoretical basis and definitions of our contribution. In section 3, we illustrate the use of the notion in two major applications: statistical description of a sample, and data clustering. Finally we discuss the different choices we made and propose a conclusion for this paper.

Theory

We decribe in this section the theoretic basis of our concept which is based on the fuzzy set theory.

The concept of fuzziness was introduced in 1965 [START_REF] Zadeh | Fuzzy sets and systems[END_REF] to deal with imprecision and vagueness. It allows for instance to represent and process imprecise statements like "Bob is young", "Alice is tall", or rules like "if an obstacle is close, then brake is immediately".

In a data analysis problem, the nature of data depends on the domain of application. We can have to analyze financial data, spatial data, or image data for instance. In most of cases there is an intrinsic imprecision induced by the acquisition technique. For example, in the case of image data, there is an imprecision caused by the limitations of the captors. According to the resolution of a captor, each pixel of a photography matches to different area's size of the captured scene. Therefore, the information contained in each pixel could be disturbed and photos are sometimes blured or noisy. For each kind of data, it's necessary to consider and to manage imprecision and uncertainty. Fuzzy sets and possibiliy theory are a common and efficient way to deal with this problem.

We introduce in this paper a notion of fuzziness of data relatively to its dataset. This notion allows us to define a kind of fuzziness for data, depending on the attributes of the dataset. This fuzzification step leads then to the definition of data representativeness in a dataset.

Data as fuzzy sets : a robust representation

As said before, real data are usually imprecise and the observations in datasets can be subsequently represented as fuzzy sets.

Let Ω be a dataset formed by n observations. Ω = {x 1 , x 2 , . . . , x n }. We decide to represent each x i (for each i ∈ I = {1..n}) as a Fuzzy subset. We note xi the corresponding fuzzy set. The membership function associated to each fuzzy set xi is not defined as usual [START_REF] Bouchon-Meunier | La logique floue[END_REF], [START_REF] Bezdek | Pattern Recognition with Fuzzy Objective Function Algorithms[END_REF]. We chose to define these membership function with the intention of dealing with outliers and with datasets where data are distributed according to various and heterogenous underlying probability distributions. In clustering problems [START_REF] Cutrona | A new fuzzy clustering technique based on pdf estimation[END_REF], [START_REF] Guha | Rock: a robust clustering algorithm for categorical attributes[END_REF], [START_REF] Jain | Data clustering: a review[END_REF], [START_REF] Krishnapuram | A possibilistic approach to clustering[END_REF], real datasets contain often outliers and noise. The data clusters are generally not well separated [START_REF] Dunn | A fuzzy relative of the isodata process and its use in detecting compact well-separated clusters[END_REF] and the corresponding probability density functions could be very dissimilar (effectives could be very different and density very dissemblable). Non paramatric tools are classical to construct more robust methods. Rank transformation of data is one these ways [START_REF] Conover | Rank transformations as a bridge between parametric and nonparametric statistics[END_REF], [START_REF] David | Order Statistics[END_REF], [START_REF] Friedman | The use of ranks to avoid the assumption of normality implicit in the analysis of variance[END_REF]. We decided to use this transformation when fuzzifying data. You will note too that our fuzzification is deeply different from the concept of fuzzy rank statistic [START_REF] Denoeux | Nonparametric rank-based statistics and significance tests for fuzzy data[END_REF] despite the fact that the theoretical tools involved are quite the same. The rank transformation is introduced at the beginning of our method and is used in the computation of the membership function of each fuzzy subset xi .

We suppose that we dispose of a dissimilarty measure on the set on Ω. Let δ denote this dissimilarity measure. Therefore, we can consider the induced weak orders family ( i ) i=1..n such that ∀i ∈ [1..n], i is defined by :

∀j, k ∈ [1..n], x j i x k ⇔ δ(x i , x j ) ≤ δ(x i , x k )
Note : a relation on a set is called weak order if it is reflexive and transitive.

We can then obtain from this weak orders family the rank table (R(i, j)) i,j=1..n defined by : R(i, j) = σ -1 j (i) where (σ i ) i=1..n is the permutation family on Ω such as :

∀i ∈ [1, n], x σi(1) i x σi(2) i . . . i x σi(n)
In other words, the i th column of R contains the ranks of the elements of Ω when sorting them by ascending value of dissimilarity measure with x i . This operation could be seen as a rank transformation that is a common way to make data analysis procedures more robust. The rank transformation consists in replacing each quantitative value of a variable in a multivariate dataset by the rank of this value. This transformation is frequently used and classical techniques like PCA have corresponding induced "nonparametric" methods. In our case, we consider that the dataset is the given dissimilarity table. Each element is a datum and each datum is considerd as a descriptor too. Let we describe this point. Let D = (d i,j ) i=1..n,j=1..n denote this dissimilarity table of the dataset (x i ) i=1..n . The element of the i th row and the j th column, d i,j , is the dissimilarity measure between x i and x j . Each datum x j induces a set of n real values corresponding to the j th column of D and is viewed as a variable. The rank transformation replaces each of these real values by its rank when the set (d k,j ), k = 1..n is ordered.

The first advantage of rank transformation is that the resulting methods are less sensitive to outliers. In fact one abnormal value generates high distances that could disturb a procedure base on the least square for instance. The corresponding ranks contain however no extreme values.

The second interest is that we make no assumption (of normality for instance). Ranks are uniformaly distributed. This point is attrative by notably allowing to deal with initially non symetric distributions.

According to this rank transformation we choose to consider that each datum of the dataset Ω is a fuzzy set where fuzzy membership function is a discrete function of the ranks of the data. If we consider the fuzzy set associated to x i ∈ Ω, the degrees of membership to xi are defined by :

µ xi (x j ) = g(σ i (j)) = g(R j,i ) (1)
where g is a discrete monotonically decreasing function defined on [1, n] and such that 0 ≤ g ≤ 1.

The subsequently particularity of the obtained function is that its evaluation depends on the ranks of data. Each fuzzy datum xi is defined by using the same function on the ranks. We call this function the shadow of the generated fuzzy subset. If we consider the classical representation of membership function, we can observe that the shapes of the fuzzy data are however different. The choice of the shadow and his characteristics are presented in the following.

The shadow of the membership functions has to be choiced at the beginning of the processus. The role of this function could be compared to the kernel function in functional estimation problems.

In our context, it can be viewed as a scoring function. For instance, the value of g(R ji ) can be considered as a score assigned by xi to x j . This score, assigned by x i to x k , is also a transformation of the rank assigned by x i to x j . The value of this score is the degree of membership of x j to xi . The constraints on the shadow function g are easy to explain and interpret. For a given fuzzy data xi we want that the higher the rank of an element x j the smaller its score. In other words, if the rank of x j is small its degree of membership to xi has to be small. It justifies why g is forced to be decreasing. Therefore, since the result of g applied to a rank values is a degree of membership, g has to take values in [0, 1] Let g be the Gaussian function: g(r) = e -(r-1) 2 s 2 (where s ∈ R is the standard deviation of the Gaussian function g). Experimentally, we can affirm that this Gaussian shadow is a satisfying choice by default. Other choices are proposed in the discussion part.

In this section we have presented our particular data fuzzification procedure where each element of the dataset is represented as one fuzzy set. We will now introduce a new notion by representing the whole data set as a fuzzy set itself.

The dataset as a particular fuzzy set

In this section we define the whole data set as a fuzzy set computed using the precedent step. This data set can be viewed as the aggregation of the n fuzzy sets associated to data. After presenting the theoretical definition we propose an application, and the interpretability of the notions will be exposed.

We define now a membership function of the fuzzy dataset Ω (the so called fuzzified Ω). This function is evaluated on each datum x i . Let us describe this function evaluated on the datum x i ∈ Ω. Let (µ x1 (x i ), µ x2 (x i ), . . . , µ xn (x i )) and w = (w 1 , . . . , w n ) with each w i ∈ [0, 1] and w i = 1. We consider an ordered weighted averaging operator (OWA) [START_REF] Yager | On ordered weighted averaging aggregation operators in multicriteria decisionmaking[END_REF] of dimension n, F w :

F w : R n → R : (y 1 , y 2 , . . . , y n ) → n j=1
w j .y j and we define : ∀i ∈ [1..n],

µ Ω(x i ) = F w ((µ x1 (x i ), µ x2 (x i ), . . . , µ xn (x i ))) (2)
µ Ω(x i ) is the degree of membership of x i in the fuzzy set Ω.

Degree of representativeness

We have defined a particular way of representing data. It consists in representing each datum as a fuzzy set, and the whole dataset as another fuzzy set. It allows to represents data individualy and globally using the relevantness of fuzzy set theory.

We can see the degree of membership to Ω as a notion of representativeness resulting from the given dissimilarity measure. We can say that the high the degree of membership of one datum to the fuzzy dataset, the more representative the considered datum. This notion of representativeness is relative to the dataset.

It leads us to define the degree of representativeness.

Definition 1

The degree of representativeness DR of one datum in Ω is defined by : DR(x i ) = µ Ω(x i )

We have exposed bellow a new framework for fuzzifying data. This approach can be relevant in the context of data analysis and we present in this section an example of application for the problem of unsupervised data clustering. The use and the interest of the degree of representativeness will be exposed in section 3.

Illustrations

After exposing the theoretical concept of our contribution, we present different kind of use in the field of data analysis. The first illustration introduce a purpose of representativeness in the problem of location estimation of a distribution. The second one illustrate the interest of representativeness in clustering problem.

The Best Representative : A robust location estimator

Let us consider the problem of location estimation [START_REF] Huber | Robust estimation of a location parameter[END_REF], [START_REF] Beran | Robust location estimates[END_REF]. The goal of location estimation is to extract one datum or one value from the data sample or data space, that reflects the "location" of the data sample. In other terms, location estimation consists in finding one observation that is as representative of the sample as possible. A classical method consists in computing one statistic from the sample [START_REF] Huber | Robust estimation of a location parameter[END_REF], [START_REF] Beran | Robust location estimates[END_REF], [START_REF] Thomas | Use of the bootstrap in robust estimation of location[END_REF]. This statistic can be viewed as the estimation of one parameter of the underlying probability distribution [START_REF] Huber | Robust estimation of a location parameter[END_REF]. The well known mean and median are the probably the most used statistics to achieve it. The median statistic is a robust location estimator contrary to the mean. It means that in presence of outliers, the mean becomes often non significative contrary to median that remains efficient. The robustness [START_REF] Thomas | Use of the bootstrap in robust estimation of location[END_REF] is an important characteristics that makes the mean statistic obsolete when working on real data.

The best representative of a sample

We propose to use our representativeness notion as an objective function to be optimized in order to determine the best representantive of a sample. Trivially, the best representative of a data sample is the element of the set that is owning the maximum degree of representativeness.

Thus we define the statistic we called best representative as follow :

Definition 2
The best representative of a data sample Ω = {x 1 , . . . , x n } is the element of Ω whose degree of representativeness RD(x i ) is the highest :

x BR = arg max xi∈Ω DR(x i )
Let us illustrate the interest and the characteristics of our statistic in some simulations.

Examples

The first point we want to highlight is the robustness [START_REF] Rousseeuw | Robust Regression and Outlier Detection[END_REF] our the best representative of a sample. To exhibit it, we simulate a one dimensional data sample composed by one population distributed according to a non-centred χ 2 law (with 4 freedom degrees) , and an uniform noise. The noise represent 75% of the total sample. The figure 1 shows the distribution of the sample and the values of the best representative statistic versus the mean and the median statistics. We observe that the best representative lies in the "real" population contrary to the mean statistic which value shifts to the noise. As it's well known, the median is sufficiently robust to "resists" to this kind of noise. The second characteristic of our statistics is the meaning it keeps when the sample is formed by several mixed subsamples. In fact, contrary to the mean and the median which only shows the central location of a sample, our statistic brings additional information. The best representative of a sample enhances the location information with information of representativeness. The figures 2 and 3 illustrate this point.

The figure 2 represents the distribution of a simulated random sample composed of two chi-square distributed subsamples (χ 2 (4) and χ 2 (24)). It shows that the median and the mean of the sample give the central location of the whole data sample. They do not consider the two different subsamples contrary to the best representative that is located in the "middle" of one of the two subdistributions. The best representative corresponds to one element of the initial data sample that is as more representative as possible. This result is illustrated in a second example (figure 3). In this third example, the sample is constitued of three subsamples that represent respectively 400, 200, and 100 elements of the sample. The median and the mean represent two elements of the minority subpopulation contrary to the best representative that is chosed in the majoritary sub-population. This specificity is a real advantage in many cases. In fact, it permits to take account of the structure of the data sample. Classical statistics -that only traduce a central tendency of the sample-extract elements or compute values that are not representative in this given sample. If we consider for example a problem of consumer behavior, our statistic offers to determine the person which is the most representative in the sample of studied consumers. The median-people or the average-people are not necesseraly representative (median) or can be abstract (mean) (note that in the case of the mean, the results could be a "virtual average behavior" that corresponds to nobody in the sample).

A data clustering procedure based on representativeness

We will now describe another application of the representativeness degree in a sample. This application offers a clustering procedure based on the degree of representativeness seen as an objective function to be optimized.

Principle

The principle lies on a classical approach in data clustering [START_REF] Hartigan | Clustering Algorithms[END_REF], [START_REF] Jain | Data clustering: a review[END_REF]. In the cluster analysis field, we can find many algorithms based on the probability density function. The idea is to condider the underlying probability density function of the sample. Most of methods consists in estimating this density function and to associate each datum to one mode of the estimated function. This association induces a partition of the sample in clusters. More details about density based clustering algorithms can be find in [START_REF] Sander | Density-based clustering in spatial databases: The algorithm gdbscan and its applications[END_REF], [START_REF] Comaniciu | Distribution free decomposition of multivariate data[END_REF], [START_REF] Herbin | Estimation of the number of clusters and influence zones[END_REF], [START_REF] Cutrona | A new fuzzy clustering technique based on pdf estimation[END_REF].

The idea of our application is to use the degree of representativeness of the data instead of the values of the estimated probability density function at these data.

This choice permits to keep the major advantage of these techniques which make no assumption on the shape of the clusters. That means for instance that it does not suppose that the clusters are spherical (it 's the case with the k-means algorithm for example).

The second advantage of our technique is its effectiveness with clusters which are distributed according to various densities and with clusters with different effectives, contrary to the classical techniques cited above.

The procedure we use is described as follow :

Algorithm 1 Representativeness Based Clustering Procedure Require: The data sample {x 1 , x 2 , . . . , x n }, ε the radius of a sphere S ε (c) centered in c. Ensure: The associated mode (AM) of each point of the data sample for all point x i of the data sample do

AM (x i ) ← x i repeat AM (x i ) ← arg max x k ∈Sε(AM (xi)) DR(x k )
until stabilization return AM (x i ) the mode associated to the point x i end for

We will present our proposition in action in the following example.

Exemple

Let Ω be the two dimensional data sample whose graphical representation is presented in the figure 4 Despite the simplicity of the procedure, we constate immediately that our method permits to deal with the non linearity of the separation between clusters.

Discussion

We have presented below the theoretical aspects of our contribution, we discuss in this section some critical points and some choices we made.

Input data

The first point we have to discuss is the nature of the input data. Our method to represent data starts with using dissimilarity between data. The only essential precondition on the dataset is to have a way to compare data. This not-restrictive background allows our technique to be used with many kind of data and especially with multivariate data or non numerical data. Therefore, as said in introduction of this paper, our method works on -and only on-the given data. The working set is the input dataspace. The computation of the representativeness is not depending on any visionary or unsupported assumptions about an underlying space of the data. Consequently, the use of this technique presents some practical interest. In our clustering application for instance, the representatives (the centers) of the clusters are extracted from the dataset. So they could be considered as "real" epitomes or "real" prototypes of the clusters, contrary to classical method that provides an imaginary mean representative (e.g. k-means).

The second point concerns the choice of the function that computes the degrees of membership to data as fuzzy sets from the ranks. We have called this function the shadow of the membership functions.

Shadow of membership function

The choice of the function is large but can be inspired or leaded from classical kernel functions or classical fuzzy membership functions (see for instance [START_REF] Cheng | Mean shift, mode seeking, and clustering[END_REF]) :

Let r be a rank value R i,j ,

• Gaussian shadow :

g(r) = e -(r-1) 2 s 2 s ∈ R • flate shadow : f (r) = 1 if r ≤ k 0 if r > k k ∈ N * • Epanechnikov shadow : k(x) = k -r 2 if r 2 ≤ k 0 if r 2 > k k ∈ N *
Graphic illustrations of each of these shapes are presented in the table 2. The choice of the shadows is not so critical. It permits to give more flexibility to procedure for a better adaptability to the nature of data.

Gaussian shape flate shape Epanechnikov shape triangular shape Table 2: Shapes examples

In our applications, we obtains good results with different shapes. The classical Gaussian function is usually a good choice and could thus be considered as an interessant choice by default. A different choice could be made according to the nature of the considered data of for particular applications.

Weights vector in OWA

The choice of the weights is very subsequent and could be guided by the input sample. We propose to design the weighting vector according to the trapezoid profil represented on the figure 5. Let us explain the way we choose. We consider that the first weights have to be null (I 1 on the figure 5). This constraint allows to ignore isolated data or very small group of data, when calculating the degree of membership of this data in the sample. In other words, it offers to treat the case of outliers. The second parameter permits to define the threshold of maximal contribution of one datum. So in this interval, the weights are maximum (I 2 on the Figure 5). Finally we consider that after a choosed threshold, the weights are null, i.e. the contribution of the datum is insignificant. Between these ranges, the weights are lineary calculated (I 3 on the Figure 5).

After discussing the different possible choices of the methodology, we present a different way of seeing our theoretical contribution. The idea is to expose another point of view on the defined notions by presenting these notions as objects in a different framework or context. In fact, our "fuzzy approach" can be expressed in a social choice theoretic context or in the framework of "preferences".

A different point of view

The degrees of membership to the sample fuzzy set can be interpreted as global scoring values affected to each datum by the set of all the others. This global score of one element is the result of the aggregation of the scores affected by each others. This point of view permits to make a link with the field of the social choice theory. In fact we can see the degrees of membership to each fuzzy data as the expression of individual preferences. On the other hand, the degrees of membership to the fuzzy data sample is close to the notions of collective

Conclusion

We have presented in this paper the notion of representativeness in a data sample. The framework and the theoretical basis of this concept are the fuzzy set theory and data analysis. Our mathematical definition of the representativeness of data permits to address to the question "How one datum is representative in its data sample ?". The most immediate application of this approach is to construct a satistical location estimator which is robust and which keeps meaning and significance according to the underlying distribution. The use of representativeness in clustering problem is natural. By associating each datum to one "good representant", we create naturally a partition of the data set. Thus the representativeness notion provides a usefull tool for cluster analysis.

Figure 1 :

 1 Figure 1: Best representative Vs Median Vs Mean -Data with 75% noise

Figure 2 :

 2 Figure 2: Best representative Vs Median Vs Mean -2 subsamples

Figure 3 :

 3 Figure 3: Best representative Vs Median Vs Mean -3 subsamples

  (Top left). After computing the representativeness of each datum (represented in the figure 4(Top right)), we use the process described in the algorithm 1 to affect each datum to one mode of the representativeness function. At each iteration of the process, we shifts from the current point to the best representative of the neighborhood. So we obtain for each datum a path to the mode in the data sample. The figure 4(Bottom Left) represents the graph obtained by drawing these pathes on the representation of the representativeness. The so called graph is composed by two trees which define the data clusters. In fact, by grouping data which have the same associated mode, we obtain the data clusters. Each final representative is the best representative of his induced cluster. The figure 4(Bottom right) represents the labelized data according to the obtaineed clusters.

Figure 4 :

 4 Figure 4: Illustration of using representtivity in a clustering process. Top Left : the two dimensional data sample. Top Right : 3D representation of the representativeness of each datum of the data sample (representativeness as elevation). Bottom Left : pathes obtained by iterative association to local best representative. Bottom Right : clusters obtained after labelization.

Figure 5 :

 5 Figure 5: Construction of weight vectors for OWA procedure

Table 1 :

 1 Numerical values of the statistics in the three simulated examples

			1 Example of fig. 2 Example of fig. 3
	Mean	-16, 76	10, 19	16, 85
	Median	-11, 66	4, 99	7, 10
	Best representative	0, 44	3, 89	4, 67