J Roussel

P Ravier

C Capdessus
email: cecile.capdessus@univ-orleans.fr

M Jabloun
email: meryem.jabloun@univ-orleans.fr

Vibration Analysis for Damage Detection in Ultra filtration Fibers

Keywords: Water filtration, bubbles, transient signals, detection, vibration analysis

This paper concerns the monitoring of water ultra-filtration devices by vibration analysis. Damaged devices are characterized by the presence of air bubbles that can be heard by a trained service engineer. The present study investigates the feasibility of an automated detector that could complete the engineer's expertness. A description of the filtration devices is given and the main features of the transients to be detected and the ambient noise are brought to light. Based on these features, six detectors are proposed. They are tested and compared through Monte Carlo simulations over data built from real life signals.

Introduction

Ultrafiltration using hollow fibers is a commonly used method for drinking water production because it allows removing micro particles up to ten nanometers and pathogens without using chemicals. Fibers are organized like spaghettis inside ultrafiltration modules (tubes). High pressure water is injected through the fibers. Part of it runs through the fibers membranes and reaches the inside of the module as pure as source water. However, due to the high water pressure, fibers may be damaged, causing the dirty water to reach the module and be mixed with the pure water. Some integrity tests have been investigated that induce the dismantling of modules [START_REF] Guo | Low-pressure membrane integrity tests for drinking water treatment: A review[END_REF], which requires time and money. Moreover water analysis methods do not allow determine which module is damaged within the set of modules [START_REF] Guo | Low-pressure membrane integrity tests for drinking water treatment: A review[END_REF].

The diagnosis technique used in practice consists in injecting air into the fibers within a water filled module. The air is normally stopped by the fiber membrane but in the presence of a damaged fiber, the air passes through this damaged fiber creating some bubbles inside the module. This bubble emission exhibits a recognizable acoustic signature, a transient signal, which can be used for detection purpose. Currently skilled technicians listen to these bubble sounds in order to report on the module state but the subjectivity of this test may induce both detection errors and improper estimation of damages.

Time and money could be saved with an on-line based fault diagnosis system that would be able to automatically detect bubbles and therefore to identify damaged modules. Such a system would not need to stop water production but would only require integration of equipments into or on the modules. However, for this system, a correct identification of damaged modules is essential and depends on the quality of the transient detectors.

The purpose of this study is hence to investigate for an efficient automated bubble detection technique.

To fulfill this goal, in-situ measurements are carried out for precise transients and noise characterization. Transient types can therefore be identified. Appropriate detectors are then proposed and their performance is evaluated.

The experimental setup is described in section 2. In section 3, both noise and transients are characterized. In section 4, performance detection curves are presented for each transient type with the proposed detection methods and detection curves are shown on a real life signal.

Experimental Setup

Sensor selection

Microphone seems to be a natural way to replace human ear. However, the noisy environment of the modules makes accurate measurements difficult. In addition, the pressure wave crosses multiple medium interfaces: water to module wall, module wall to air and air to microphone membrane. Accelerometers actually appear to be better fitted to the case. Firstly, they reduce the number of interfaces by two: water to wall and wall to accelerometer. Secondly, they reproduce what happens when the technician sticks his ear to the module, i.e. the vibration is directly transmitted to the inner ear by the cranium bones. For those reasons and based on preliminary experiments, the system has been chosen to be based on vibration analysis rather than sound analysis.

The selected sensor is a high sensitivity integrated circuit piezoelectric accelerometer (ICP) Kistler 8786A5 with a range of ±5 g, a sensitivity of 1000 mV/g and a bandwidth of 6 kHz. The sensor is fixed on the module with a special wax.

Acquisition Chain

The data acquisition setup is composed of the following parts (Figure 1):

-A high sensitivity accelerometer.

-A signal conditioner MEIRI ME2680 with an ICP sensor amplifier card ME26C-ICP. The amplifier has an adjustable gain between 1 and 20, an adjustable output range up to 10V and a bandwidth of 30 kHz.

-A National Instrument data acquisition card NI PCI-4472 composed of eight 24-bit analog to digital delta-sigma converters at a maximum of 102400 samples per second, with an alias-free bandwidth of 45 kHz.

-A computer with Labview environment.

Measurements

Preliminary measurements on damaged modules show a maximum useful frequency at 5 kHz. So, in order to maintain a good signal quality, a sampling frequency of 50 kHz has been chosen. The voltage gain of instrumentation amplifier has been adjusted to maximize the output voltage at 10 V, with a value of 13.33. So we have a full measurement range of 750 mg. The 24 bits analog to digital converter provides a quantum of 0.089 µg.

A database has been established with damaged and undamaged modules measurements during the integrity test process. Each measurement is associated to an information file with the module serial number and its location inside the plant. On each damaged module, two measurements have been made, one before repairing and one afterwards.

The identification of damaged modules is primarily based on the presence of bubbles that have to be detected as transient signals buried in ambient noise. In order to develop detection procedures in the classical binary hypothesis testing way, the noise and added transients have to be characterized. First, in order to characterize the noise inside a module, some records have been made from undamaged modules. A statistical study performed over temporal signals shows a Gaussian distribution of this background noise as displayed in Figure 2. The standard deviation may vary according to the production or module position but is generally around 10 to 50 mg.

Noise and Transient Signals Characterization

Background noise

The power spectral density of the records, using Welch's averaged periodogram with a 10 ms rectangular window, exhibits two pink noises at -60 dB/dec up to 800 Hz and -10 dB/dec at higher frequencies (Figure 3). Figure 4 displays the time-frequency representation of this noise calculated with a short-time Fourier transform using a Hanning window of 1 s.

Time-Frequency (dB) -Window length = 1 s

Engine noise

We can also note the presence of constant frequencies along some signals. Figure 5 and Figure 6 show the spectrograms of an undamaged module with some components at 300, 800, 900, 1400, 1700, 2800 and 3400 Hz. This unfavorable case is not permanent since all these frequencies are induced by the pumps used during the filtration process.

Transients

Secondly, we have identified phenomena inside damaged modules: different kinds of transients were observed. We have classified them into five categories, depending on their time-frequency features and the corresponding sound that can be heard when sending the vibration signal to a loudspeaker. In what follows, these five categories will be denoted types {A, B, C, D, E} and labeled {"glou-glou", "slup", "tic-tic", "tac-tac", "taptap"} sounds, respectively. Figure 7 In Figures 9-a

Proposed detection techniques and test method

Based on noise and signal features, several detectors can be proposed. First, as can be seen on the transients spectrograms as well as on their temporal representation, the bubbles produce local bursts of energy. The simplest detector is thus based on the energy calculation. Second, it was shown both by time and frequency analysis and by ear study coupled with different band pass-filters, that bubbles are located above 800 Hz, while 0.05 0.1 0 50 100 the noise exhibits very strong energy at low frequencies. Therefore, a high-pass filter can be applied to the signal prior to the energy detector. Third, the magnitude distribution of many transients deviates from Gaussianity as it is shown on the histogram plots of section 3.2, whereas the noise exhibits the Gaussian property. Thus higher order statistics, for example the fourth-order cumulant, can be used to measure deviations from Gaussianity. Fourth, the signals of interest are naturally non-stationary, so that time-frequency or time/scale representations are dedicated tools to properly analyze these data. We tested adaptive segmentation in the time domain using Malvar wavelet decomposition [Rav98] and adaptive segmentation in the frequency domain using wavelet packet decomposition [Rav01].

The selected detectors as well as the test method will be more precisely described in the following parts, while the experimental results will be given in the last part of this section.

Energy detector

Let us denote by the n th sample of the observation, and N the total number of samples. The energy is adaptively estimated using the following expression:

1 • 1 , 2; 1
where the initial energy is set as 1 1 . The adaptive step can be any µ 0; 1 , and it is to be chosen by taking into account the duration of the transients. In what follows, this step is fixed to µ 0.1 for all experiments.

Energy detector with prior filtering

The preliminary study showed that transients are located above 800 Hz, while the noise energy dramatically decreases with frequency. Therefore it is interesting to apply a high-pass filter to the data prior to the detection. The second detector is thus the combination of a butterworth high-pass filter at frequency 500 Hz followed by the energy detector.

Fourth-order moment with prior filtering

This detector aims at detecting deviations from Gaussian property. A second-order estimator (energy estimator) is not sufficient to detect nonGaussian events. For that purpose, higher-order statistics are often used. In particular, the normalized kurtosis value which is a normalized value of the fourth-order cumulant, is known to be zero when the process is Gaussian. Let us recall the definition of the normalized kurtosis value . For any zero-mean random variable desc t ribing he process, writes:

3 2 with representing the mathematical expectation. The normalization factor is the energy whose online estimation may thoroughly impact the kurtosis estimated value. A simple way to estimate is to normalize the variable with respect to the total energy of the observation before estimation. So, for this detector, the estimation is reduced th to e simple online fourth-order moment:

1 1 , 2; 3 •
where the initial moment is set as 1 1 . This detector may be seen as a complementary tool to the online energy detector: the periods where the energy is constant may be associated to high values of indicating nonGaussian periods, i.e. the presence of transients. This third detector is then a combination of a butterworth high-pass filter at frequency 500 Hz followed by the online fourth-order detector, the step being fixed to µ 0.1.

Fourth-order cumulant with prior filterin g

Considering a zero-mean random variable describing the process, its fourth-order cumulant is expressed as:

3 4
The online estimation of the basically requires the estimation of the fourth-order moment as well as the second-order moment (energy). So, estimation errors are not only due to the fourth-order moment estimator but also to the second-order moment estimator. To get a direct online estimation of the fourth-order cumulant and to make the estimation robust with respect to the second-order moment estimation, the adaptive version of this estimator described in [START_REF] Amblard | Adaptive estimation of the fourth-order cumulant of a white stochastic process[END_REF] is processed:

• 1 5 1 • 3 1 1
In the below presented experiments, the steps are fixed to 0.05 and 0.01.

For this fourth detector, the 500 Hz high-pass filter is also applied prior to the calculation of the fourth-order cumulant.

Wavelet transform and kurtosis

In the frequency domain, transients produce local high coefficients because of their oscillatory components. The non-Gaussian property of the transients is therefore enhanced in the transformed domain whereas the noise which is Gaussian remains Gaussian. This property is used to get an adaptive segmentation of the data in the time domain using Malvar wavelet decomposition. This segmentation acts like a pre-detection of transient signals. Indeed, the idea is to get large temporal segments for stationary periods corresponding to the noise only condition (with Gaussian coefficients) whereas short temporal segments are obtained when nonstationary periods appear due to the presence of any transient (with non-Gaussian coefficients). The segmentation is a data driven procedure that is achieved according to the Gaussianity nature of wavelet coefficients, using the split and merge algorithm [START_REF] Wickerhauser | Adapted wavelet analysis from theory to software[END_REF]. The total record duration must have a number of samples that is a power of two in order to consider a dyadic partition of the sequence. At the beginning, the finest equal size temporal segmentation is considered and wavelet coefficients are computed. At the next level, wavelet coefficients are computed for twice longer segments until reaching the total sequence duration. To get a non redundant time frequency representation, a selection of wavelet coefficients (i.e. nodes in the tree structure) is achieved. The algorithm is the following: at a given level, if the wavelet coefficients of two consecutive segments are Gaussian, then merge them. Otherwise let them dissociated. In this procedure, the fourth-order cumulant is measured to zero for making the Gaussian / non-Gaussian decision. Practically, the comparison to zero is achieved using the Byenaymé-Tchebychev inequality, considering the number of coefficients at the current level and a confidence percentage value in the decision making (90% taken in the simulations). The same procedure continues with the following two by two segments, at the considered level. When splitting, the highest cumulant node value is marked (except at the leaf level where the two nodes are marked) and replaces the father node value. This is for keeping a local high non-Gaussianity trace. This procedure is repeated from the leafs (32 samples segments) to the root (1 segment with all the samples) of the tree structure. The marked nodes permit to select the best adapted wavelet decomposition basis. Then the kurtosis values of the retained coefficients are computed to form the detection curve [Rav98].

Wavelet packet decomposition

According to [START_REF] Ravier | Wavelet packets and de-noising based on higher-order-statistics for transient detection[END_REF], the wavelet packet decomposition acts like a filtering that is adapted to the statistical data content. The algorithm steps are described as follows. First, a denoising procedure based on wavelet packet decomposition is achieved. This procedure only keeps frequency bands that are non-Gaussian, which allows dealing with the important low frequency components due to the pink noise. A denoised version of the data is computed by inverse transforming selected coefficients of the wavelet packet decomposition. Both wavelet packet decomposition and coefficient selection are based on a kurtosis computation criterion. Secondly, adaptive energy values are evaluated along the time on the denoised signal yielding a temporal detection curve. The Daubechies wavelet (order 16), a decomposition depth of 8 and a confidence percentage of 99% have been fixed for the simulations.

Test method

The performance of the proposed detectors is evaluated with the Receiver Operating Characteristic (ROC) curves, through a Monte Carlo set of experiments. The ROC curve is the representation of the detection rate as a function of the false alarm rate . Any detection threshold applied at the output of the receiver is associated to a value of the couple ( , ). The ROC curve is the location of the ( , ) points obtained when changing the detection threshold [START_REF] Whalen | Detection of signals in noise[END_REF]. The experiments are carried out for all types of transients presented in section 3.2. For the experiments, each type of transient is defined as a unique pattern which is a "clean" transient extracted f the tions of a damaged module. rom vibra

In order to estimate the and probabilities, the noise only condition and the transient plus noise condition have to be created. The detector under test is then applied to a set of 250 synthetic observations or with a desired signal-to-noise ratio (SNR). For each observation, the noise sequence is a temporal slice extracted from the vibrations of undamaged modules. A detection is decided when the receiver output is greater than a given threshold. Counting the number of detections gives an estimation of the and values under the noise only condition and the transient plus noise condition respectively.

A typical detector under test is characterized by a ROC curve which is located between the blue curve (ideal detector) and the red curve (poorest detector which is a chance line) as illustrated in Figure 10. A global efficiency measure of the detector can be evaluated as the area located between the curve of the detector under test and the poorest detector, divided by the area of the upper triangle of the graph. For an ideal detector, this parameter would be equal to one, whereas for the worst possible detector it would be zero. This parameter allows an easy comparison between different detectors and will be denoted by "mean performance" of the detector.

Experimental results

The ROC curves of all the proposed detectors are presented on Figure 11 for SNR=0 dB. The mean performance of the detectors are given on Figure 12. Without any surprise, rather poor results are obtained with the energy detector which uses few of the available information about transient features. It can be noted from these figures that high-pass filtering the data efficiently improves the results obtained by this very simple method. After filtering, the basic use of fourth-order moment does not improve the detection performance with respect to the use of second-order moment. Taking into account both statistics in the robust fourth-order cumulant estimation greatly helps improving the detection performance, especially for A and B transient types. For all the transient types except to the type E, the Malvar wavelet decomposition permits a better detection than the temporal detectors. This can be explained by the fact that the type E transient is more widely spread in the time-frequency plane than the others preventing a correct adapted wavelet decomposition. Note also that none of the methods really makes better than of the high-pass filtering and energy computation method. Due to a difficult time-frequency signature, the other proposed methods have no positive effect on type E transient detection. The results concerning the wavelet packet are disappointing (except for the type A transient). Indeed, the method sometimes suppresses frequency bands that correspond to transient content and is also not able to systematically suppress the low energy frequency bands.

All these results are summed up on Figure 12. The wavelet packet results and have not been reported on this figure. The best method among those that have been tested remains the Malvar wavelet transform detector, whose detection rate is above 96% in all cases and reaches 99% for the type A transient. The fourth-order cumulant is also very interesting since the results look very similar. This method has the advantage to be computationally low-cost but suffers from the parameters adjustment. For example, the type B "slup" transient and the E "tap-tap" transient show comparable results, i.e. 96% and 95% respectively. With the parameters choice 0.01 and 0.005 (instead of 0.05 and 0.01 previously), the results are very different, i.e. 89% and 98% respectively (not shown here).

Note that the 0dB SNR simulates very poor operating conditions, since the real SNR is most of the time far above that value. Three detectors have been applied on a real data sequence. On the example shown on Figure 13, the Malvar wavelet decomposition method clearly shows the presence of weak transients in comparison with the simple energy detector, even after band-pass filtering in the 200-2000 Hz frequency band. The presence of transients is accredited by human expertise and corroborated by time-frequency representations.

Conclusion

In this paper, we presented a feasibility study concerning the development of an automated diagnosis device for water ultrafiltration modules. Several detectors have been proposed for the detection of transient signals that are distinctive of damaged filtration fibers. Some Monte Carlo simulations have been performed over simulated signals constructed from real life recordings. These simulations showed that one of the detectors based on Malvar wavelet decomposition gives better results than any other for most of the transients to be detected. Providing the parameters are properly tuned, the fourth-order cumulant based detector also allows detection of very weak transients in ambient noise. However, this kind of test is not sufficient and needs testing the algorithms on long time continuous real data sequences. This approach needs the creation of annotated database which is not an easy task because of uncertainties in human diagnosis. The work is under progress. Moreover, at the time being, the tested method cannot identify the class of the detected transients. Other methods are under investigation for this classification problem.

The presented results are preliminary since occurrence behaviors of the transients have not been taken into account for a precise diagnosis of the modules. The occurrence of transients may be continuous along the time with a specific mean delay between two occurrences. On the contrary, when a bubble appears, a volley of transients may be sporadically detected. Lastly, other indicators like the air pressure decrease velocity during the integrity test procedure, can be merged to the transients detection procedure in order to improve the decision making.

Figure 2 :

 2 Figure 2: Histogram of the temporal vibration recorded on an undamaged device (20 s duration). The blue plot represents the data histogram while the red one represents Gaussian fit of the histogram, .

Figure 3 :

 3 Figure 3: Noise Power Spectral Density, estimated on 20 s duration records. The sampling frequency is 50 kHz.

Figure 4 :

 4 Figure 4: Noise Spectrogram.

Figure 5 :

 5 Figure 5: Noise Spectrogram (550 to 5000 Hz). The observed pure tones are caused by pumps.

Figure 6 :Frequency

 6 Figure 6: Noise Spectrogram (550 to 2000 Hz).Pumps are the major contributions to the observed pure tones.

  is a spectrogram example of a damaged module. Repetitive events are mostly observed. Depending on the types, the events can either be narrow band mono-component transients (Figure 8-a) or multi-component transients (Figure 8-b). Mono-component transients are composed of one central frequency which may vary (Figure 8-a, center frequency of 1 kHz ± 200 Hz). Multi-component transients are constituted of several local central frequencies that are synchronized or not (Figure 8-b, center frequency between 200 Hz and 2 kHz). Note that the sounds produced by these multi component transients are characteristic of bubble pops.

Figure 7 :Figure 8 :

 78 Figure 7: Spectrogram of one damaged module.

  to 9-e the five different types of transients are exposed through their temporal representation, their spectrogram and their temporal histogram. In many cases, the distribution is far from the Gaussian one (cases when the normalized kurtosis value is far from the 0 value[START_REF] Nikias | Higher-Order spectra analysis[END_REF]).

Figure 9 Figure 9 Figure 9 Figure 9 Figure 9

 99999 Figure 9 -a: Spectrogram and temporal histogram of the type A transient (glou-glou), 4.1

Figure 10 :

 10 Figure10: Example of a typical ROC curve (green line). This curve is located between the ideal curve (blue line) and the poorest one (red line). This latter is denoted chance line in the text.

Figure 11 :

 11 Figure 11: ROC curves of the proposed detectors for the five types of transients, SNR = 0 dB.

Figure 12 :

 12 Figure 12: Mean performance of the tested detectors for each type of transient, SNR=0dB.

Figure 13 :

 13 Figure 13: (a) Time representation of a damaged module (b) Energy of the Malvar wavelet decomposition basis coefficients (c) Energy of the time representation (d) Energy of the time representation after 200-2000 Hz bandpass filtering.


Sébastien Destercke

Handling bipolar knowledge with imprecise probabilities

Information is said to be bipolar when it has a positive and a negative part. The problem of representing and processing such bipolar information has recently received a lot of attention in uncertainty theories. In this paper, we are concerned with the representation of asymmetric bipolarity, i.e., with situations where positive and negative information are unrelated and processed in parallel. In this latter case, positive information consists in observations of experiment results showing what values are possible, while negative information consists in constraints (e.g., provided by an expert) restricting the range of possible variable values. Up to now, there are no proposition as to how such bipolar information can be treated in the framework of imprecise probability theory, i.e., when information is represented by convex sets of probabilities. In this paper, we propose the basis of such a framework, and provide some illustrative examples.

. Our proposition as to how bipolar information should be processed with such representations is then detailed in other subsections.

, or convex sets of probabilities, are very general models of uncertainty that encompass most known uncertainty models. They therefore

Introduction

Information about a given variable usually comes in different forms and from various sources. Recently, there has been a growing interest in the handling of bipolar information. Information is bipolar when one can differentiate between a positive and a negative part in the information. Such information usually concerns either evidences about the true value assumed by an ill-known variable, or preferences expressed by one or more agent. In this paper, we are concerned with the first type of information. One can consider at least three different types of bipolarity (See Dubois and Prade [START_REF] Dubois | An overview of the asymmetric bipolar representation of positive and negative information in possibility theory[END_REF] for more details). The first one, called symmetric univariate, models bipolarity by the use of an univariate scale and can be represented by the means of classical probability measures. The second one, called a symmetric bivariate scale, handle two separate unipolar scales (positive and negative) that refer to the same information and are usually linked by some duality relation. Lower and upper previsions [START_REF] Walley | Statistical reasoning with imprecise Probabilities[END_REF][START_REF] Miranda | A survey of the theory of coherent lower previsions[END_REF] are examples of this kind of bipolarity, as well as other uncertainty models encompassed by this representation (Belief and plausibility functions [START_REF] Shafer | A mathematical Theory of Evidence[END_REF], possibility and necessity measures [START_REF] Dubois | Possibility Theory: An Approach to Computerized Processing of Uncertainty[END_REF]).

The last type of bipolarity, coined as asymmetric or heterogeneous, is the one addressed in this paper. Such bipolarity arise when negative and positive information parts are two unrelated kinds of information that have to be processed in parallel: one asserting what is impossible (negative information), the other what can exist (positive information). Negative information can correspond to constraints over possible values expressed by physical laws, expert opinions, etc. Examples, observation and measurements are instances of positive information. Note that the two kinds of information are effectively unrelated, as an expert opinion (or a model prediction) declaring some values as impossible does not imply that all the others can or will be observed, hence the need for asymmetry. This need is further confirmed by some psychological studies [START_REF] Cacioppo | The affect system: architecture and operating characteristics[END_REF] supporting the fact that the brain processes differently positive and negative information.

The notion of bipolarity have been declined in a number of application areas and theoretical frameworks: multicriteria decision making [START_REF] Grabisch | Bi-capacities I: definition, Möbius transform and interaction. II: The choquet integral[END_REF], conflict resolution in argumentative frameworks [START_REF] Amgoud | On bipolarity in argumentation frameworks[END_REF], uncertainty and preferences representation [START_REF] Dubois | An overview of the asymmetric bipolar representation of positive and negative information in possibility theory[END_REF], spatial reasoning [START_REF] Bloch | Fuzzy and bipolar mathematical morphology, applications in spatial reasoning[END_REF], database querying [START_REF] Dubois | Bipolarity in flexible querying[END_REF], . . . . The processing of bipolar information within uncertainty theories has already been discussed in the frameworks of possibility theory [START_REF] Dubois | An overview of the asymmetric bipolar representation of positive and negative information in possibility theory[END_REF] and of the transferable belief model [START_REF] Smets | The canonical decomposition of a weighted belief[END_REF]. However, to the best of our knowledge, the processing of bipolar information when information is modelled by convex sets of probabilities, or so-called credal sets [START_REF] Levi | The Enterprise of Knowledge[END_REF], has not been considered so far.

The purpose of this paper is to lay out a basic framework to represent and process bipolar information when information is modelled by such credal sets. The idea behind this framework is simple and can be summarised in two main steps:

1. the first step consists in collecting negative and positive information separately, and to represent each corpus of information by separate credal sets;

2. the second step consists in merging positive and negative information in a single representation, possibly coping with conflicting or new information in the process.

After recalling the basics of credal sets [START_REF] Levi | The Enterprise of Knowledge[END_REF] and their relation with Walley's [START_REF] Walley | Statistical reasoning with imprecise Probabilities[END_REF] lower previsions, Section 2 presents our proposition, i.e. how bipolar information can be modeled and processed with credal sets. Section 3 then provides some illustrative examples using some practical imprecise probabilistic representations (i.e., p-boxes, probability intervals and possibility distributions).

Handling bipolar information with credal setss

provide an attractive and unifying framework to model and reason under uncertainty. As uncertainty representations, they are equivalent to Walley's [START_REF] Walley | Statistical reasoning with imprecise Probabilities[END_REF] coherent lower previsions, that extends de Finetti's previsions [START_REF] Finetti | Theory of probability[END_REF] by integrating imprecision to them.

Credal sets and lower previsions

In this paper, we consider a variable X assuming its values on a space X made of mutually exclusive elements and whose exact value is ill-known or uncertain. We also assume that this uncertainty is modelled by the means of a credal set P, or convex set of probability distributions over X . We denote by L(X ) the set of real-valued bounded functions on X . Given a function f ∈ L(X ), one can compute the lower and upper expectations E P (f ), E P (f ) induced by P such that

E P (f ) = inf p∈P E p (f ) E P (f ) = sup p∈P E p (f ),
where p is a probability distribution over X and E p (f ) the expected value of f w.r.t. p. These two values are dual, in the sense that E P (f ) = -E P (-f ).

Thanks to this duality, one can only work with one of the two mappings (usually E).

Alternatively, one can start from a lower mapping P : K → R from a subset K ⊆ L(X ), and consider the induced credal set P(P ) such that

P P = {p ∈ P X |(∀f ∈ K)(E p (f ) ≥ P (f ))}.
with P X the set of all probability mass functions over P X . In his theory of lower previsions [START_REF] Walley | Statistical reasoning with imprecise Probabilities[END_REF], Walley starts from the mapping P that he calls lower prevision. He interprets P (f ) as the supremum buying price for the uncertain reward f . A lower prevision P is then said to avoid sure loss iff P P = ∅, and to be coherent if the lower expectation E P P (f ) = P (f ) coincides with P for every f ∈ K (i.e., P is the lower envelope of P P ). He also shows that coherent lower previsions and credal sets have the same expressive power (in the sense that any credal can be identified by a unique lower prevision, and vice versa). Given a credal set P, its lower (resp. upper) probability of an event A, denoted by P P (A) (resp. P P (A)), corresponds to the lower (resp. upper) expectation of the indicator function 1 (A) of the event , that takes value one on A and zero elsewhere. By duality, we have P P (A) = 1 -P P (A c ).

Note that credal sets and coherent lower previsions are very general models, in the sense that they encompass most of the other uncertainty models proposed in the literature [START_REF] Walley | Towards a unified theory of imprecise probability[END_REF]. In particular both necessity measures of possibility theory [START_REF] Dubois | Possibility Theory: An Approach to Computerized Processing of Uncertainty[END_REF] and belief measures of evidence theory [START_REF] Shafer | A mathematical Theory of Evidence[END_REF] correspond to particular classes of lower probabilities inducing specific credal sets. We now detail the two main steps of bipolar information handling: information representation and merging.

Collecting and representing bipolar information

Similarly to what is done in possibility theory [START_REF] Dubois | An overview of the asymmetric bipolar representation of positive and negative information in possibility theory[END_REF] and other frameworks [START_REF] Grabisch | Bi-capacities I: definition, Möbius transform and interaction. II: The choquet integral[END_REF][START_REF] Smets | The canonical decomposition of a weighted belief[END_REF], we propose to model positive and negative information by using two separate models of our chosen framework. That is, positive information is modelled by a credal set P + , while negative information is modelled by another credal set P -.

Negative information (P -): Negative information expresses constraints about the value X can assume. It rules out some values, considering them as impossible, or less likely than others. It can come from expert opinion about some particular events, or from the answer of some model for which X is the output value. The negative credal set P -corresponding to such information will typically be induced by a collection of expectation bounds over a set of chosen functions1 f 1 , . . . , f k ∈ L(X ), in the form

P (f i ) ≤ x∈X f (x)p(x) ≤ P (f i ). (1)
In many situations, functions f 1 , . . . , f k will be some indicator functions of events A 1 , . . . , A k , and the negative information will consists in lower and upper bounds over such events. Note that pieces of negative information are treated conjunctively, in the sense that we consider the credal set induced by all constraints (1) at once. This means that the more we accumulate negative information, the more precise is P -. We assume here that P -= ∅ (i.e., the lower prevision P given by Eq. (1) avoids sure loss).

Positive information (P + ): Positive information consists most of the time in a set of observations or of experiment results. Here, we consider that they consists in a set of m data or observations {x 1 , . . . , x m }. However, these data or observations alone are not sufficient by themselves to obtain a credal set P + . A classical means to build a credal set P + from these data is to associate the m observations with a statistical model and a learning process. For example, multinomial data can be associated to the well-known Imprecise Dirichlet model [START_REF] Bernard | An introduction to the imprecise dirichlet model[END_REF], or to some confidence intervals derived from empirical frequencies [START_REF] Denoeux | Constructing belief functions from sample data using multinomial confidence regions[END_REF]. Again, such models usually become more and more precise as more data are accumulated and we can consider that positive information is accumulated conjunctively as well. Such a behaviour can be explained by the fact that the space X is composed of mutually exclusive elements, meaning that observing one value more often makes the others less likely.

Remark 1 There are cases where either positive or negative information should be combined disjunctively instead of conjunctively. Smets [START_REF] Smets | The canonical decomposition of a weighted belief[END_REF], when combining reasons to belief and reasons not to belief, propose a rule that combines disjunctively negative information and conjunctively positive information. However, he works at a different level from ours, since we work directly with knowledge (i.e., information) about variables, and not with knowledge about evidences (i.e., meta-information).

In their possibilistic approach, Dubois and Prade [START_REF] Dubois | An overview of the asymmetric bipolar representation of positive and negative information in possibility theory[END_REF] also work directly with knowledge about variables, but propose to combine positive information disjunctively and negative information conjunctively. However, their proposition concerns variables taking their values on a conjunctive space X , i.e., the true value of X can be several values of X (in their example, the opening hours of a museum). In that case, it appears natural to combine disjunctively positive information, as observing a particular value does not make the others less likely.

Merging bipolar information

Once negative and positive information have been collected and represented, it is desirable to combine them into a unique representation, making the most of each information part. Such a unique representation then allows one to infer more precise conclusions than the conclusions one could have infer from each information part alone. This unique representation should also be coherent with both positive and negative information.

As both negative and positive information are modelled by the means of credal sets, it seems natural to merge them through a conjunctive combination operator. The final representation is then the credal set P +∩-:= P + ∩ P - resulting from the conjunction of P + and P -, provided the set P +∩-= ∅ is not empty. Note that this conjunction can again be associated to operations done on lower previsions (see [START_REF] Walley | The elicitation and aggregation of beliefs[END_REF][START_REF] De Cooman | Coherent lower previsions in system modelling : products and aggregation rules[END_REF] for details). Note that a simple necessary condition for P +∩-= ∅ is the following:

Proposition 1 P +∩-= ∅ implies that, for any f ∈ L(X ), max(E P -(f ), E P + (f )) ≤ min(E P -(f ), E P + (f ))
Proof Immediate, since for any probability distribution p ∈ P +∩-, we have

max(E P -(f ), E P + (f )) ≤ E p (f ) ≤ min(E P -(f ), E P + (f ))
Now, it may be possible that positive and negative information conflict with each other, i.e. that P +∩-= ∅. In such a case, it is desirable to restore consistency through some revision process. As in [START_REF] Dubois | An overview of the asymmetric bipolar representation of positive and negative information in possibility theory[END_REF], we propose to weaken one type of information to restore consistency. Given a parameter ∈ [0, 1] and a credal set P, let us first define the -discounted credal set P as

P = { p P + (1 -)p|p P ∈ P, p ∈ P X }. (2)
When positive and negative information conflict, then it seems reasonable to weaken one of the two state of information by a value such that they are no longer conflicting. Note that the value is often interpreted as the reliability of the given information. When dealing with knowledge about the value of a variable, direct observations are usually judged more reliable than negative information (as this latter one often comes from models or experts), and it seems more reasonable to revise P -rather than P + . A solution to restore consistency is to consider the minimal value * such that P -, * is consistent with P + , i.e., * = min

{ ∈ [0, 1]|P -, * ∩ P + = ∅} (3)
and then take P -, * ∩ P -as our final state of knowledge. However, note that the above (minimal) revision can lead to an overly precise final information state (as will be shown in the examples of Section 3), and one may well consider some value ≥ * (for instance, coming from some previous reliability assessment about the information sources).

In principle, the same revision process can be applied to P + , instead of P -. As pointed out in [START_REF] Dubois | An overview of the asymmetric bipolar representation of positive and negative information in possibility theory[END_REF], this strategy is often more suited to cases where information represent preferences rather than knowledge about a variable. However, it could be used in the case of knowledge representation when the reliability of observations is questionable.

Revising knowledge with new piece of information

Another case where keeping track of positive and negative information separately can be of importance is when we learn a new piece of information from some source. This new piece of information can be positive information, i.e., new observations, or negative information, given as a credal set P - new described by new constraints.

In both cases, we can argue that it is desirable to add this new piece of information to its proper corpus of knowledge before merging the information, rather than adding the new piece of information to the merged information. Namely, it is desirable to revise the model P + by considering old and new observations when the new information is positive, and to revise P -into P ,- such that P ,-= P - new ∩ P -when the new information is negative. Indeed, the following situation may happen: the new piece of information P - new is not conflicting with P -, i.e. P - new ∩ P -= ∅, while it is conflicting with the merged knowledge, in the sense that P - new ∩ P +∩-= ∅. Thus, there would be no problem in adding this new piece of information to P -, since it does not conflict with it, while it would be conflicting with the merged information P +∩-.

In this latter case, a safe behaviour would be to merge disjunctively the new piece of information with the merged one (i.e., compute P - new ∪ P +∩-), as one would no longer be able to tell the difference between negative and positive information in P +∩-. This could lead to a possible bigger loss of information than if we had added P - new directly to its proper corpus of information (again, this will be illustrated in the examples of Section 3). Moreover, the two final representations would not be the same.

Illustrative examples

The above framework to deal with bipolar information is quite general, in the sense that it can be applied to any credal set. However, computations with such generic models can be fastidious, and it is often desirable to work with simple models for which computations are more tractable. In this section, we provide some illustrative examples using some popular imprecise probabilistic models, for which the above scheme can be easily applied. The chosen examples represent situations that are likely to be encountered in practical situations (e.g., risk analysis, classification problems, . . . ). We provide for each example a means to build a representation from positive and negative information. We then detail how the information merging can be done with the representations, including how revision can be performed in case of conflicting information, and finally we provide a numerical example.

p-boxes

A p-box [START_REF] Ferson | Constructing probability boxes and dempster-shafer structures[END_REF], denoted by [F , F ] and defined on the real line R, is a pair of lower and upper cumulative distributions describing our uncertainty about the value of a given random variable. They consist in lower and upper probabilities given over events of the type (-∞, x]. P-boxes are very often used in risk analysis where variable values are ill-known. A p-box induce the credal set

P [F ,F ] such that P [F ,F ] = {p ∈ P R |∀x ∈ R, F (x) ≤ P ([-∞, x]) ≤ F (x)}
Positive information: Following [START_REF] Ferson | Constructing probability boxes and dempster-shafer structures[END_REF], it is possible to derive a p-box from a set of (i.i.d.) observations (x 1 , . . . , x m ) by using Kolmogorov-Smirnov confidence limits to define bounds around the empirical distribution F m , making no assumption about the distribution form. The distribution F m is defined as

F m (x) =                0 for x ≤ x (1) . . . i/n for x (i) ≤ x ≤ x (i+1)
. . .

1 for x (m) ≤ x
where x (i) are the ordered sample values (i.e., x (i) ≤ x (j) if (i) ≤ (j)). Given the unknown distribution F , we denote by D KS the maximal deviation such that

D KS = max {|F (x (i) ) -i /m|, |F (x (i) ) -i-1 /m||i = 1, . . . , m}.
D KS is a random variable whose exact distribution is unknown, but Kolmogorov has shown that √ mD KS has a limiting distribution that allows to define, for each confidence level α ∈

[0, 1], a value D m (α) such that P (D KS ≤ D m (α)) ≤ 1 -α. Given a level α, we then define a p-box [F m , F m ] such that F m = max(0, F m -D m (α)) and F m = min(1, F m + D m (α))

We denote by [F , F ]

+ the p-box generated by the accumulation of positive information, and P + [F ,F ] the credal sets induced by this p-box.

Negative information: Asking percentile estimations to an expert is a classical elicitation method [START_REF] Cooke | Experts in uncertainty[END_REF]. However, the reliability of precisely estimated percentiles may be questionable, and an expert may be more comfortable in providing interval estimations rather than point estimations. Here, we consider that an expert can give imprecise evaluations of percentiles, that is a set of increasing values (x 1 , . . . , x m ) with x i ∈ R is fixed and the expert provides estimations of the probabilities of events2 [-∞, x i ] for i = 1, . . . , m. Other ways by which a p-box could be built using negative information include using bounds resulting from some uncertainty propagation through a physical model [START_REF]Uncertainty analysis based on probability bounds (p-box) approach in probabilistic safety assessment[END_REF][START_REF] Baudrit | Joint propagation and exploitation of probabilistic and possibilistic information in risk assessment[END_REF]. We denote by [F , F ] -the p-box shaped after negative information, and P -

[F ,F ]
the induced credal set.

Merging:

When both P + [F ,F ] and P - [F ,F ] are induced by p-boxes, it is known [18] that the credal set P + [F ,F ] ∩ P - [F ,F ] is still induced by a p-box [F , F ] -∩+ such that [F , F ] -∩+ = [max{F -, F + }, min{F -, F + }].
This means that in the particular case of p-boxes, calculations of the final representation is straightforward, at least in the case where positive and negative information are not conflicting. Treating exactly conflicting information is more difficult, as applying Eq. (2) to a credal set induced by a p-box does not usually result in another credal set induced by a p-box. However, given a value , the p-box [F , F ] = [F , F ] such that F = F and F = F + 1 -gives an outer approximation of P . Such a procedure simply comes down to only retain from P the information concerning the lower and upper probabilities on events [-∞, x].

Numerical example: Assume that variable X evolves between [0, 16] and that 10 samples (1; 1.5; 3; 3.5; 4; 6; 10; 11; 14; 15) have been collected concerning this variable. Given a classical confidence level α of 0.95, the value D 10 (0.95) = 0.40925.

An expert also provides his opinion about the probabilities that the variable is lower than the values 4, 8, 12, 16, in the form of the following lower and upper bounds: [0, 0.2], [0.1, 0.3], [0.5, 0.7].

Figure 1 displays the p-boxes resulting from negative and positive information, as well as the resulting merged p-box. In this case, they are non-conflicting.

probability intervals

Probability intervals [START_REF] De Campos | Probability intervals: a tool for uncertain reasoning[END_REF] are lower/upper probability bounds given on singletons x ∈ X . Probability intervals can thus be described by a set L = {[l(x), u(x)]|x ∈ X } of intervals. They induce a credal set P L such that

P L = {p ∈ P X |∀x ∈ X , l(x) ≤ p(x) ≤ u(x)}.
Necessary and sufficient conditions for probability intervals to induce a nonempty credal set and to be exact lower/upper probabilistic bounds are provided by [START_REF] De Campos | Probability intervals: a tool for uncertain reasoning[END_REF]. They can be summarized by the conditions that, ∀x ∈ X , u(x) + y∈X \x l(y) ≤ 1 and l(x) + y∈X \x u(y) ≥ 1

Positive information: There exist mutliple models to compute confidence bounds on multinomial data with only a limited number of samples. This can be done, for instance, by considering statistical confidence intervals over multinomial data [START_REF] Goodman | On simultaneous confidence intervals for multinomial proportions[END_REF] or by using the so-called Imprecise Dirichlet Model (IDM) [START_REF] Bernard | An introduction to the imprecise dirichlet model[END_REF]. In this paper, we have retained the latter option, which is the commonest when working with imprecise probabilities. Let m be the total number of observations, m(x) the number of times an element x ∈ X has been observed, and s ∈ [0, ∞] a positive hyperparameter. Probability intervals derived from the IDM are such that, for anyx ∈ X ,

l(x) = m(x) m + s and u(x) = m(x) + s m + s . (4)
We will denote by L + the set of probability intervals obtained in this way, and by P + L the induced credal set.

Negative information: As for p-boxes, negative information can be provided by some experts or by a propagation through a model such as a decision tree [START_REF] Abellan | Using the total uncertainty criterion for building classification trees[END_REF] or a credal network [START_REF] Cozman | Credal networks[END_REF]. We denote by L -the obtained probability intervals and by P - L the induced credal set.

x 1 x 2 x 3 u 0.3 0.9 0.2 l 0.1 0.7 0 Merging: As in the case of p-boxes, the credal set P + L ∩P - L resulting from the merging of two probability intervals is again induced by a probability interval. Probability intervals L +∩-inducing P + L ∩ P - L are such that, ∀x ∈ X ,

x
l +∩-(x) = max{l + (x), l -(x), 1 - y∈X \x u + (y), 1 - y∈X \x u -(y)} u +∩-(x) = min{u + (x), u -(x), 1 - y∈X \x l + (y), 1 - y∈X \x l -(y)}.
Also note that the result of Eq (2), when applied to probability intervals L, result in a credal set still induced by probability intervals L such that, ∀x ∈ X , l (x) = l(x) and u (x) = u(x) + 1 -, therefore, in the specific case of probability intervals, the proposed framework can be exactly applied without much computational costs.

Numerical example: We consider a 3-element space X = {x 1 , x 2 , x 3 } on which are defined our probability intervals, as such a space gives us the opportunity to represent credal sets in barycentric coordinates (the space of all probability mass functions is represented by a simplex where each vertex correspond to an element of the space. Each point in the simplex then represents a probability mass p, where the mass p(x) allocated to an element x is proportional to the distance from the point p to the edge opposed to the vertex corresponding to x). Assume that the observed samples are such that m = 8 with m(x 1 ) = 1, m(x 2 ) = 7, m(x 3 ) = 0. To model positive information, we use the IDM with a parameter s = 2 and apply Eq. ( 4) to obtain the probability intervals L + . Negative information is assumed to come from an expert opinion given as a set of lower and upper bounds L -. The two probability intervals and the reulsting merged intervals L +∩-are summarised in Table 1 and pictured in Figure 2.

To illustrate the problem of coping with new information, consider now that a new expert (i.e. negative information) provides some complementary opinion in the shape of probability intervals L - new that are summarized in new probability intervals are in conflict with L +∩-, but not with L -alone, and we can thus revise our negative information with this new piece of information by computing L - new ∩ L -. Positive and negative information now conflicts. Using Eq (3), we obtain the value * = 0.6. The discounted probability intervals (L - new ∩ L -) * and the merged one are summarized in Table 2. Figure 3 illustrates the complete process that makes them consistent again. As pointed out in Section 2, the result is very precise (the probability of x 2 is precisely known) and it would perhaps be safer to adopt a strategy providing more cautious inferences (such as taking ≥ * or simply ignoring the new piece of information if its reliability is questionable).

comonotonic possibility distributions and clouds

A possibility distribution is a mapping π : X → [0, 1] such that ∃x, π(x) = 1.
From a possibility distribution [START_REF] De Cooman | Supremum-preserving upper probabilities[END_REF][START_REF] Dubois | When upper probabilities are possibility measures[END_REF] can be defined a possibility measure such that Π(A) = sup x∈A π(x). In this paper, possibility measures are interpreted as upper probabilities, and a possibility distribution induces the credal set The lower probability induced by a possibility measure is called a necessity measure. Possibility measures are popular imprecise probabilistic models, due to their simplicity and the fact that they can be interpreted (and elicited) as sets of lower confidence bounds given over collection of nested sets, as recalls the next proposition [START_REF] Couso | The necessity of the strong alpha-cuts of a fuzzy set[END_REF]:

P π = {P ∈ P X |P (A) ≤ Π(A)}. x1 x2 x3 P - L ∩ P - Lnew P + L (P - L ∩ P - Lnew ) * P + L ∩ (P - L ∩ P - Lnew ) *
Proposition 2 Given a distribution π, a probability distribution P is in P π fi and only if ∀α ∈ [0, 1], 1 -α ≤ P ({x ∈ X |π(x) > α}). (5)
However, possibility distributions alone can hardly be used in our bipolar context. Indeed the intersection of two credal sets induced by possibility distributions does not usually result in a credal set induced by another possibility distribution, nor in a credal set that can be approximated by a possibility distribution without much loss of information. Also possibility distributions, although attractive, remain of limited expressiveness as they do not allow for instance to treat the case where an expert provides both lower and upper confidence bounds over nested intervals. This is why we will need a richer representation, namely clouds [START_REF] Neumaier | Clouds, fuzzy sets and probability intervals[END_REF], to deal with this example and situation. Clouds are recent imprecise probabilistic representation, defined as follows:

Definition 1 A cloud on a space X is a pair of mappings [π, δ] such that δ ≤ π, and there is at least one element x ∈ X s.t. π(x) = 1 and one element y ∈ X s.t. δ(y) = 0.

After Neumaier [START_REF] Neumaier | Clouds, fuzzy sets and probability intervals[END_REF], a cloud [π, δ] induces a probability family P [π,δ] s.t.

P [π,δ] = {P ∈ P X |P (δ(x) ≥ α) ≤ 1 -α ≤ P (π(x) > α)}.
Note that we retrieve Eq. ( 5) when δ = 0, in which case the cloud [π, δ] is equivalent to the possibility distribution π alone. It is also known [START_REF] Destercke | Unifying practical uncertainty representations: II clouds[END_REF] that P [π,δ] = P π ∩ P 1-δ , where 1 -δ is a possibility distribution and P 1-δ its induced credal set. Clouds therefore allow us to represent the merging of two distinct possibility distributions, provided they satisfy Definition 1. However, this does not solve the problem of information merging when either negative or positive information is represented originally by clouds. In this case, the subfamily of comonotonic clouds presents important practical advantages.

Definition 2 A cloud is said to be comonotonic if ∀x, y ∈ X , π(x) ≤ π(y) ⇒ δ(x) ≤ δ(y).

From a practical standpoint, comonotonic clouds are particularly attractive models. When they take a finite number of values, the credal set P [π,δ] is induced by a set of lower and upper probabilistic bounds given over a collection of nested subsets ∅ = A 0 ⊂ A 1 ⊂ . . . ⊂ A N = X that have the following forms

α i ≤ P (A i ) ≤ β i
and the corresponding cloud is such that, for all x ∈ X , δ(x) = 1 -β i and π

(x) = 1 -α i-1 with x ∈ A i \ A i-1 .
Positive information: As possibility distributions can represent nested confidence intervals, they are natural candidate to represent information derived from statistical inequalities such as Chebyshev inequality [START_REF] Baudrit | Practical representations of incomplete probabilistic knowledge[END_REF]. They are also adequate representations when only few measurements are available [START_REF] Mauris | Inferring a possibility distribution from very few measurements[END_REF]. In this paper, we consider that enough data are available so as to provide good estimations of the mean value µ and the variance σ of variable X, and that Chebyshev inequality can be used. The possibility distribution π + induced by this inequality is such that

π + (µ -k • σ) = π + (µ + k • σ) = 1 if k < 1 1 k 2 if k ≥ 1. (6)
We denote by P + π the credal set induced by the distribution π + .

Negative information: Clouds and their interpretation in terms of bounds over nested sets gives a convenient way to elicit information from experts [START_REF] Fuchs | Potential based clouds in robust design optimization[END_REF].

We consider here that a collection of intervals centred around x * are provided to the expert, and that he is asked for confidence bounds around these intervals. Such information generates a cloud [π -, δ -], and we denote by P - [π,δ] the induced credal set.

Merging: When both π + , δ -and π -are all comonotonic, the credal set

P + π ∩ P - [π,δ] is again a cloud [π -∩+ , δ -∩+ ] [18]
such that δ -∩+ = δ -and π -∩+ = min(π + , π -). Conflict happens when there is an element x ∈ X such that δ -∩+ (x) > π -∩+ (x).

In case of conflict, applying Eq.( 2) to a cloud [π, δ] does not result in a credal set induced by a cloud, while it does in the case of possibility distribution (hence, if the cloud is reduced to a single distribution, this problem does not happen). However, as for p-boxes, given a value , the cloud [π, δ] such that π = π+1-and δ = δ gives an outer approximation of P [π,δ] . Other revision processes have been proposed in literature [START_REF] Benferhat | Bipolar possibility theory in preference modelling: representation, fusion and optimal solutions[END_REF], but they seem difficult to interpret in terms of imprecise probabilities and credal sets.

Note that assuming comonotonicity of all functions π + , δ -and π -may appear unreasonable in some situations: for instance, an expert most plausible value for X (i.e., the value x * around which nested intervals are built) may differ from a measurement obtained by a sensor or an estimated mean value. In such a case, the merging result is no longer representable by a cloud but still remains a lower probability (computing the merged representation then requires heavier computational efforts). However, it is always possible to consider a weakening of the information so that clouds are made comonotonic, but the incurred information loss can be important.

Numerical example:

We assume that estimations from data of the mean and variance are respectively µ = 30 and σ = 5. Distribution π + can then be obtained by using Equation [START_REF] Benferhat | Bipolar possibility theory in preference modelling: representation, fusion and optimal solutions[END_REF].

Let us now assume that an expert has provided the following constraints about the value of X (knowing the estimated mean, one can propose this mean as a starting point to the expert):

0.3 ≤ P (X ∈ [28, 32]) ≤ 0.6, 0.7 ≤ P (X ∈ [20, 40]) ≤ 0.9, 1 ≤ P (X ∈ [15, 45]) ≤ 1.
These opinions and constraints can be represented as a cloud [π -, δ -], and both information can then be merged. The whole process is illustrated in Fig. 4. In this case, there is no conflict between positive and negative information.

Linking clouds and bipolar possibility theory: let us explore a bit more the links between clouds and bipolar possibility theory [START_REF] Dubois | An overview of the asymmetric bipolar representation of positive and negative information in possibility theory[END_REF]. In bipolar possibility theory, two distributions are defined, π and δ, and to them are associated four measures: the classical possibility and necessity measures, respectively defined on any event A ⊆ X as

Π(A) = sup x∈A π(x) and N (A) = 1 -Π(A c ) = inf x∈A c (1 -π(x)),
and two other measures [START_REF] Dubois | Possibility theory:qualitative and quantitative aspects[END_REF], called guaranteed possibility (∆) and potential certainty (∇), and respectively defined on any event A ⊆ X as

∆(A) = inf x∈A δ(x) and ∇(A) = 1 -∆(A c ) = 1 -inf x∈A c δ(x)
. Now, let us consider that a cloud [π, δ] models the intersection of two credal sets modelled by possibility distributions π (negative information) and 1 -δ (positive information). We can then compute lower and upper probabilities of P π for any event A ⊆ X

P Pη (A) = Π η (A) = sup x∈A η(x) = Π(A),
P Pη (A) = N η (A) = inf x∈A c 1 -η(x) = N (A)
, where Π and N are the measures induced by the initial distribution π. On the other hand we have that the lower and upper probabilities of P 1-δ are such that

P P 1-δ (A) = Π 1-δ (A) = sup x∈A (1 -δ(x)) = 1 -inf x∈A δ(x) = 1 -∆(A) = ∇(A c ). P P 1-δ (A) = N 1-δ (A) = inf x∈A c (δ(x)) = 1 -∇(A) = ∆(A c
). Clearly, ∇ and ∆ respectively play the role of upper and lower probability measures of positive information, and provided P [π,δ] = ∅, we have the inequality (using Prop 1)

max(N (A), ∆(A c )) ≤ min(∇(A c ), Π(A)).
Note that this inequality is different from the following inequality max(N (A), ∆(A)) ≤ min(Π(A), ∇(A)), proved in [START_REF] Dubois | Possibility theory:qualitative and quantitative aspects[END_REF] in the case where δ = π in the setting of possibility theory. However, both inequalities confirms the role of ∆ as a lower uncertainty measure and of ∇ as an upper uncertainty measure.

Although clouds are formally equivalent to interval-valued fuzzy sets intuitionistic fuzzy sets [START_REF] Atanassov | Intuitionistic fuzzy sets[END_REF], their link with the pair π, 1 -δ is only apparent (as noted in [START_REF] Dubois | An overview of the asymmetric bipolar representation of positive and negative information in possibility theory[END_REF]). This comes from the fact that the two approaches models different things (e.g., the so-called mirror cloud [START_REF] Neumaier | Clouds, fuzzy sets and probability intervals[END_REF] [1 -δ, 1 -π] corresponds to negation for intuitionistic fuzzy sets, while in the current approach it models exactly the same information as the cloud [π, δ]).

Conclusion

We have proposed a framework to handle bipolar asymmetric information in the framework of imprecise probabilities, when this information concerns knowledge about the value assumed by a variable. This framework mainly comes down to consider two separate credal sets, one for positive information and one for negative information, and to merge them conjunctively to obtain a final representation. This is in the spirit of other propositions made in other uncertainty theories to handle bipolar information, where two separate (positive and negative) representations are also merged to give a final representation. We then have illustrated our proposition with some specific uncertainty representations that are easier to handle than generic ones, and for which the proposed framework applies easily.

This work is a first step towards the modelling and handling of bipolar information within the theory of imprecise probabilities. It still has to be compared with other approaches proposed within the frameworks of possibility theory and evidence theory (although some elements have been given here, a deeper investigation would be needed). One of the main difference with these two latter approaches is that both positive and negative information are here combined conjunctively, and give more precise models as more information becomes available. However, since we're working with variable taking their values on spaces made of mutually exclusive elements, such a behaviour is not counter-intuitive.

There are other situations where the use of bipolar information in imprecise probabilities could be of interest, among which:

• the case where information concerns probabilities themselves, for example in the case of linguistic assessments [START_REF] De Cooman | A behavioural model for vague probability assessments[END_REF]. This would require to connect the present proposition with axioms governing lower previsions, possibly progressing towards an operational definition of bipolarity in terms of betting strategy;

• the case where information concerns not knowledge but preferences, that is when credal sets or lower previsions are used not to express uncertainty but rather to express some preferences or utilities between different criteria of an agent. With this objective in mind, the model of desirable gambles, which have been recently considered as a solution to multicriteria decision problems [START_REF] Utkin | Multi-criteria decision making with a special type of information about importance of groups of criteria[END_REF] and extends both credal sets and lower previsions, seems particularly interesting to study.

Figure 1 :

 1 Figure 1: Illustrative example: p-boxes

Figure 3 :

 3 Figure 3: New information and merging of conflicting information

Fig. 4 .Fig. 4 .

 44 Fig. 4.A Distribution π + using Chebyshev inequality (µ = 30 and σ = 5)

Figure 4 :

 4 Figure 4: Cloud and possibility distributions and their merging result.

Table 1 :

 1 Probability intervals L + (above left), L -(above right) and L +∩- (bottom middle)

			1 x 2 x 3
		u	0.4 0.8 0.3
		l	0.2 0.4 0
		x 1 x 2 x 3
	u	0.3 0.8 0.1
	l	0.2 0.7 0

Table 2

 2

	. This

Table 2 :

 2 New probability intervals L -

new (above left), new information state L - new ∩ L -(above right), discounted intervals (L - new ∩ L -) * (below left) and merged intervals P + L ∩ (P - L ∩ P - Lnew ) * (below right)

For example, functions corresponding to some chosen events, or to moments such as the mean or the variance.

Note that the reverse can also be done, i.e., the expert is given a set of probability values from 0 to 1 and is asked to provide intervals in which these percentiles fall.