François Suffix

François Recanati
email: recanati@ens.fr

Keywords:

In earlier work on so-called moderate relativism, I distinguished three semantic levels: (i) the meaning of the sentence, (ii) the lekton (a typically 'relativized' proposition, true at some situations and false at others), and (iii) the Austinian proposition (the lekton together with a topic situation serving as circumstance of evaluation). The lekton can be construed as a property of situations or a type of situation. The Austinian proposition is true iff the topic situation is of the type corresponding to the lekton. In his contribution to this volume, Max Kölbel expresses a few worries about my framework. First, he finds the psychological considerations I offer in support of the intermediate notion (the lekton) insufficient: a properly semantic justification is needed, he argues (and he provides one). Second, he worries about my thesis that the lekton is 'fully articulated', because it conflicts with the contextualist claim (defended by myself in many writings) that 'what is said' is porous and hospitable to unarticulated constituents. Third, he discusses potentially unwelcome implications of my view in connection with faultless disagreement. In this response I consider the three issues raised by Kölbel, along with his suggestions for tackling them. I endorse Kölbel's semantic argument for the lekton while dismissing his objection to the claim of full articulatedness. Regarding faultless disagreement, I attempt to make sense of it from a classical expressivist standpoint.

U N C O R R E C T E D P R O O F
In his contribution to this volume, Max Kölbel expresses a few worries about my framework. First, he finds the psychological considerations I offer in support of the intermediate notion (the lekton) insufficient: a properly semantic justification is needed, he argues (and he provides one). Second, he worries about my thesis that the lekton is 'fully articulated', because it conflicts with the contextualist claim (defended by myself in many writings) that 'what is said' is porous and hospitable to unarticulated constituents. Third, he discusses potentially unwelcome implications of my view in connection with faultless disagreement.

In this response I consider the three issues raised by Kölbel, along with his suggestions for tackling them. I endorse Kölbel's semantic argument for the lekton while dismissing his objection to the claim of full articulatedness. Regarding faultless disagreement, I attempt to make sense of it from a classical expressivist standpoint.

Keywords Unarticulated constituents • lekton • Austinian proposition • Relativized propositions • Faultless disagreement

In Perspectival Thought, I distinguish three semantic levels: (i) the meaning of the sentence, (ii) the lekton (a typically 'relativized' proposition, true at some situations and false at others), and (iii) the Austinian proposition (the lekton together with a topic situation serving as circumstance of evaluation). The lekton can be construed

U N C O R R E C T E D P R O O F F. Recanati
as a property of situations or a type of situation. The Austinian proposition is true iff the topic situation is of the type corresponding to the lekton. This analysis is more complex than the standard analysis which only distinguishes the meaning of the sentence and the (classical, unrelativized) proposition that it expresses. My analysis posits an intermediate level, the lekton. The lekton is referentially loaded, in contrast to sentence meaning, but it falls short of determining absolute truth-conditions. The lekton falls short of determining absolute truthconditions because some determinants of truth-conditions are left unarticulated and are provided by the situation. The lekton itself is 'fully articulated': nothing is in the lekton that does not correspond to some element in the sentence. What is not articulated can still affect truth-conditions, but only via the topic situation (which serves as circumstance of evaluation for the lekton). So, in 'it is raining', the place and time are unarticulated (that's the difference with 'it's raining here' and 'it's raining now'), but they still affect truth-conditions via the topic situation: if the topic situation is the situation here and now, the utterance is true iff it is raining in that situation (here and now). So the truth-conditions of 'it is raining', 'it is raining here' and 'it is raining now' are the same, but the lekta differ: an element that is unarticulated in 'it is raining' is articulated and incorporated into the lekton in 'it is raining here' and 'it is raining now'. The difference has psychological significance, I argue, and it is missed by theories which only appeal to two levels (sentence meaning and absolute truth-conditions).

Although sympathetic, Max Kölbel expresses a few worries. First, he finds the psychological considerations I offer in support of the intermediate notion (the lekton) insufficient: a properly semantic justification is needed, he argues (and he provides one). Second, he worries about the idea that the lekton is 'fully articulated', because it conflicts with the contextualist claim (defended by myself in Literal Meaning) that 'what is said' is porous and hospitable to unarticulated constituents. Third, he discusses potentially unwelcome implications of my view in connection with faultless disagreement. I am grateful to Kölbel for raising these issues and offering friendly amendments or extensions of my view for dealing with them. I will consider the three issues in turn, along with Kölbel's suggestions.

1.

AQ1

Kölbel is not convinced by the psychological reasons I offer for distinguishing the lekta in 'it is raining' and 'it is raining now'. I say that the thought 'it is raining now' requires on the part of the subject an explicit representation of the time of thought, while the thinker of 'it is raining' may be 'temporally innocent' (i.e. not even able to entertain such representations). Kölbel objects that the subject who thinks 'it is raining' and the subject who thinks 'it is raining now' can both be credited with time awareness. I agree, but I think one should distinguish between different sorts of time awareness, and also distinguish between actuality and requirement. What the thought 'it is raining now' requires of its subject is an explicit representation of the time of thought, contrasted with other times. Such an explicit and contrastive representation is objective in the sense that the present time is viewed as one particular time among others. The subject who thinks 'It is raining now' is implicitly contrasting the present time, a rainy one, with some other time(s): the property of being a time at which rain occurs is ascribed to the former in contrast to the latter. The idea that the thinker of 'It is raining' might be temporally innocent is a colourful way of saying that 'It is raining' does not require such an explicit, objective representation of time; that thought is available also to those subjects who lack the advanced conceptual ability to think about times qua elements of an objective chronology (e.g. animals), but who are nevertheless able to register the weather in their environment. Such subjects can still think 'It is raining'. Their thought 'concerns' the time at which they occur but are not 'about' that time, in contrast to 'it is raining now', which is about the time of thought.

In his work on the self notion, Perry has insisted on the difference between primitive self-knowledge based on special, egocentric ways of gaining information and acting, and objective self-knowledge based on the ability to think of oneself also from a third-person point of view, as one person among others.1 I agree with Perry:

The distinction between the two kinds of representation is essential to the theory of indexical thought. The self is unarticulated in primitive self-knowledge, while objective self-knowledge rests on deploying a self concept (or a self file) which explicitly represents the subject of thought [START_REF] Perry | Thought without representation[END_REF]). I agree with Perry that such a mental file requires 'the capacity to pool the information I get in normally otherdirected ways that happens to be about myself, with the primitive self-knowledge I have concerning myself' (Perry 2014: 29). Because of that extra requirement, objective self-knowledge is more demanding than primitive self-knowledge, which it presupposes. Certain de se thoughts, which I call 'implicit', express primitive selfknowledge and are available to any subject endowed with primitive self-knowledge; other de se thoughts, which I call 'explicit', require possessing a self file based on the capacity Perry talks about: the capacity to pool first person and third person information about oneself. These two levels are similar to those we find in the temporal case. The time is explicitly represented only in 'it's raining now' (and the place only in 'It is raining here'). In 'it's raining' it is left unarticulated, as Prior suggested. 2 Temporal or locational concepts are deployed in one case, not in the other.

With respect to implicit de se thoughts, Lewis's theory applies: the thinking subject serves as 'circumstance of evaluation' for the de se content, construed as a property which the subject self-ascribes [START_REF] Lewis | Attitudes De Dicto and De se[END_REF]). But explicit de se thoughts are a different matter: when a self file is deployed, the thought is explicitly about the sub-

U N C O R R E C T E D P R O O F F. Recanati
ject, who is incorporated into the lekton. The self occurs 'as object' in the thought, while in implicit de se thought it merely occurs 'as subject' [START_REF] Wittgenstein | The blue and brown books[END_REF].

Be that as it may, Kölbel offers a semantic justification for holding that 'it is raining' and 'it is raining here' (or 'it is raining now') carry distinct lekta. The indexicals 'now' and 'here' are rigid and their reference cannot be shifted by circumstantial operators. In the case of 'it's raining', the place of evaluation can be shifted: 'it is raining' is typically evaluated at the place of utterance, but in 'somewhere, it is raining' the place of utterance drops out of the picture: it does not survive the shift induced by the operator 'somewhere'. Following Dummett, Evans, Lewis and others, we can use that criterion to distinguish the lekton (Dummett's 'ingredient sense'; Lewis's 'semantic value') and the full assertoric content (the Austinian proposition, in my framework). 3 The assertoric content of 'it's raining' and 'it's raining here' are the same, these authors argue, but their embedding profiles are different. If we prefix 'it is raining' with 'somewhere', the place of utterance no longer affects the truth-conditions. It does not survive operator-shifting, so it must be left out of the lekton. Not so with 'it is raining here': the place of utterance is fixed by the indexical as an element of content that survives operator shifting ('somewhere it is raining here'). As Kölbel points out, the unshiftability of indexicals provides ample justification for including the reference of indexicals in the lekton. In 'It's raining', the place affects the truth-conditions via the topic situation (the situation here and now). In 'it is raining here' it is a constituent of the lekton.

I fully endorse Kölbel's semantic criterion. My three-level framework is inherited from Kaplan, whose 'contents' are relativized to time and world. Kaplanian contents contrast both with sentence meaning (which is referentially unloaded) and full assertoric content (what we get when we specify the circumstance with respect to which the content is to be evaluated). Now the main justification adduced by Kaplan for his intermediate notion of content is the rigidity/unshiftability of indexicals, whose referential content survives operator-shifting. I agree with Kölbel that that property is fundamental, and I think it is instantiated also in the realm of thought.

Indexical concepts are unshiftable, in the sense that if you deploy such a concept in thinking a counterfactual thought or holding a counterfactual attitude, the reference of the indexical concept survives the shift away from the actual world induced by the counterfactual attitude. In the counterfactual attitudes, the reference of indexical concepts is fixed by actual world relations, yet the referents may be rationally ascribed properties incompatible with their standing in such relations [START_REF] Ninan | Imagination, content, and the self[END_REF]. That means that the individuals found in the actual environment are transported in imagination into possible worlds where these very individuals may or may not instantiate the environmental properties which make it possible for us to think about them in the first place.

3 See Dummett (1973: 446-47;1981: 572-74;1993: 47-50), [START_REF] Evans | Reference and contingency[END_REF] and [START_REF] Lewis | Index, context and content[END_REF].

U N C O R R E C T E D P R O O F
13 About the lekton: Response to Max Kölbel

2.

I say that the lekton is fully articulated, and Kölbel concludes that what I called 'what is said' in previous writings should not be equated to the lekton, despite the fact that 'lekton' means 'what is said' in Greek. The reason why the lekton cannot be equated to what is said is that, according to my earlier account, what is said is 'nonminimal'-it is affected by modulation processes such as free enrichment-while, according to Perspectival Thought, the lekton cannot contain any 'unarticulated constituent'. Whatever unarticulated constituents there are are constituents of the complete content, or Austinian proposition: they are aspects of the circumstance of evaluation. So it seems that my earlier 'what is said' should be equated to the Austinian proposition rather than to the lekton. Or, Kölbel suggests, perhaps it should be equated to the utterance's truth-conditions, which correspond to an equivalence class of Austinian propositions.

Kölbel objects to all this, on the grounds that, for various reasons, the lekton should play the roles which I ascribe to 'what is said' (input to the inferential process of implicature generation, content of the attitudes and of speech acts etc.); so it is unfortunate that they wind up not being the same thing. If we fix the theory so that they come out identical, as Kölbel suggests we should, then, given my insistence that the lekton is fully articulated, we arrive at a position strikingly different from that which I defend in Literal Meaning. According to the position we arrive at, Kölbel says, the Austinian proposition is non-minimal (it contains unarticulated constituents) but the lekton just is the minimal proposition which I deride in Literal Meaning as having no psychological reality and no interesting role to play in the theory. To sum up, there is, to say the least, a significant tension between Literal Meaning and Perspectival Thought.

I deny that there is any such tension. I agree with Kölbel that the lekton should be equated to what is said for all the reasons he gives. Indeed their being identical is one of the reasons why I use that term 'lekton' which means 'what is said' (and additionally conveys the suggestion of semantic incompleteness, due to the use of the term by Stoic logicians in connection with tensed propositions). But how can they be identical if one of them is minimal and the other one not?

Well, I never said that the lekton was minimal, and I don't think it is. The lekton is the rich 'what is said' of Literal Meaning. How, then, can I maintain that the lekton is fully articulated, that whatever is in the lekton must correspond to some constituent in the sentence that expresses that lekton? Is not free enrichment, the paradigmatic modulation process, the provision of unarticulated constituents?

No it is not. In free enrichment some aspect of meaning is contributed in a top down manner by the context. This is often interpreted as the provision of 'unarticulated constituents', but that is not my interpretation. Free enrichment typically corresponds to a process of specifization, through which we make the contextual meaning of a lexical item more specific than its literal (conventional) meaning. Is this a matter of providing unarticulated constituents? No. The contextualmeaning,

U N C O R R E C T E D P R O O F F. Recanati
resulting from free enrichment, is not unarticulated, because it corresponds to something in the sentence, namely the lexical item whose meaning has been made contextually more specific.

Consider a sentence like 'he opened the door'. This is understood in context as a particular kind of opening-the standard kind rather than an opening of the door with a scalpel (as when we open a wound). In context the opening may be taken to involve a particular instrument (as in 'he took out his key and opened the door'). The kind of opening at stake, the instrument etc.-all these contextual aspects of the interpretation result from modulating (more specifically, 'enriching') the lexical meaning of 'open' by providing further specifications. That process of free enrichment is optional and context-driven, in contrast to saturation, which is linguistically mandated; yet the contextually specific content contributed by the word is articulated, by definition, since it is the contextually specific content contributed by the word. My claim that the lekton is fully articulated is therefore compatible with its being non-minimal, i.e. affected by free processes of modulation.4

3.

I discuss faultless disagreement in Perspectival Thought, but the conclusions I reach are very tentative. I start from the idea that agreement and disagreement are over the complete content (involving the lekton and the situation of evaluation). If two people entertain the same lekton but evaluate it with respect to distinct situations, they don't disagree even if they ascribe opposite truth-values to the utterance: I call you on the phone, and commenting upon my situation I say 'It is raining'. If you say 'No, it isn't', meaning that there is no rain in your situation, there is misunderstanding rather than genuine disagreement. Or, adapting Barwise's example, suppose that Holmes says 'The salt is left of the pepper', and Watson, speaking from his own perspective, replies 'No it is not'. Clearly, there is no substantive disagreement here. If each of them is talking about his own perspective, there is misunderstanding rather than genuine disagreement.

The same considerations apply to the temporal case. At time t, you say 'It is raining'. Later, when the sun is shining again, you say 'It is not raining'. You cannot conclude 'so I was wrong'. Here, as Richard points out, genuine disagreement can only be about temporally specific contents. (Recanati 2007: 90-91) Faultless disagreement seems to be a counterexample: people who disagree about e.g. matters of taste seem to evaluate the same lekton (e.g. the relativized proposition that vegemite is tasty) with respect to their respective standards of taste, and end up with distinct truth-values when the standards are sufficiently different. Thus vegemite is tasty to you, but not to me. You say 'Vegemite is tasty'; I respond: 'No,

U N C O R R E C T E D P R O O F
13 About the lekton: Response to Max Kölbel it is disgusting'. The disagreement here seems genuine, but there is no complete content which the discussants share and over which they disagree. They only share the lekton (that vegemite is tasty) but evaluate it with respect to their respective standards. The complete contents of their respective thoughts are therefore different: subject A evaluates the lekton with respect to A's standards of taste (and ends up with the value 'true'), while B evaluates the same lekton with respect to B's different standards (and ends up with the value 'false'). So what is going on? What explains the difference between the 'it's raining' case, in which there is no genuine disagreement, and the vegemite case, in which it seems that there is?

In my framework 'vegemite is tasty' expresses a relativized proposition, whose truth or falsity is relative to a judge (a standard of taste). The judge (the person for whom vegemite is tasty) is incorporated into the lekton in 'vegemite is tasty to me', but in 'vegemite is tasty' it is left unarticulated and is contributed at the Austinian proposition level. There is another difference between 'vegemite is tasty and 'vegemite is tasty to me', however. 'Vegemite is tasty' has an objective flavour (in many contexts at least)5 while 'vegemite is tasty to me' or 'I find it tasty' are more subjective. According to Perspectival Thought, what accounts for the objective flavour of 'vegemite is tasty' is the fact that the relevant judge is the community to which both the speaker and the hearer belong. What exactly counts as 'the community' is a highly context-sensitive matter, however, and this gives us a good deal of flexibility in the analysis of particular examples. This leaves many possible options for analysing alleged cases of faultless disagreement. First, when A says 'vegemite is tasty' and B responds 'it is not', A may be wrong in presupposing that her standard of state regarding vegemite is shared by the community. B's dissenting voice suggests that the standard may actually not be shared. If that is so, then A is at fault (and must retreat to the subjective statement 'I find it tasty'). Second, A is free to maintain his statement regardless of B's dissension, by suitably adjusting 'the community' and excluding B from it. For example, A may judge that B departs, by his bad taste, from the standards of the community. If this is true, then, from A's point of view, B is at fault. However we interpret the case, the disagreement is not faultless.

Kölbel points out an implication of my view: whenever A wrongly presupposes that her standard of taste is shared by the community, she fails to express a complete Austinian proposition, for the consensual standard of taste she invokes, and against which she evaluates the lekton as true, does not actually exist. As he writes,

The context of utterance fails to determine a situation of evaluation. It's like uttering 'it is raining', intending to say something concerning Cockaigne (a mythical place where, when it trains, it rains cheese). (Kölbel,p. 19)

U N C O R R E C T E D P R O O F F. Recanati
On that basis Kölbel raises the following objection:

According to Recanati's account, we cannot, when talking to one another, correctly assert that the picture is beautiful. For if we do so we commit a failure of presupposition similar to that of someone who asserts that it's raining intending to say something concerning Cockaigne. But no such failure seems to be involved. We take it to be quite normal to have diverging views on the beauty of pictures, but do not think that this prevents us from successfully asserting to one another that pictures are or are not beautiful. On Recanati's view, our assertions will lack a semantically complete content. So at the very least Recanati's account would have to allow that when we are talking about matters of taste, failing to assert something with a semantically complete content is not in any way a linguistic failing. For this type of situation seems to be absolutely standard. (Kölbel p. 20) I find the objection interesting and worthy of a detailed response, based on a distinction between three types of case.

As Kölbel must admit, there are cases for which the notion of failure straightforwardly applies. The speaker often retreats to a weaker position (the subjective statement 'I find it tasty') when she discovers that the hearer dissents. That retreat reveals that the speaker considers herself to have been at fault when she made the stronger, objective claim.

On my account, however, presupposition failure (hence lack of determinate content) is only one option, corresponding to the case where A is at fault. 'Adjusting' the target community so as to make B's dissension irrelevant and rejecting his standard of taste as faulty (i.e. contrary to the community's standard) is another option, which need not involve any failure on the speaker's part. Of course, if the speaker is right, it is the hearer who is now at fault. I agree with Kölbel that there are also cases of a third type: cases of 'faultless disagreement' such that no failure whatsoever is involved on the part of either speaker or hearer. The protagonists may enter an episode of sustained disagreement in which both parties (re)affirm diverging views without ever retreating to a weaker, subjective statement of their taste.

Such cases raise a problem for my framework, Kölbel says. Each protagonist vacuously refers to a communal standard that is being negociated and is not yet established. It follows that no determinate Austinian proposition is expressed (this is like referring to Cockaigne). For that reason, the utterance is, or should be, neither true nor false. But this conclusion-that judgements of taste in situations of sustained disagreement do not have a complete content and fail to determine a truthvalue-seems incompatible with the observation that such cases are 'absolutely standard' and involve no linguistic failing. But I don't think the conclusion and the observation are actually incompatible. Following Kölbel's own suggestion, I want to 'allow that when we are talking about matters of taste, failing to assert something with a semantically complete content is not in any way a linguistic failing'. I agree that sustained disagreement is faultless-it involves no failing of any sort. It is faultless because no standard of taste is antecedently given as the standard of the local community formed by the protagonists; it is up for grabs (up for negociation) what the communal standards are. Each protagonist tries to impose his or her standards, to make them the standards

U N C O R R E C T E D P R O O F
13 About the lekton: Response to Max Kölbel of the local community. Each protagonist is entitled to do so, so there is no 'failure' (no 'fault') when one does. Still, I suggest, the utterances of the protagonists lack absolute truth-conditions. I propose that, in sustained disagreement about matters of taste, the interlocutors's moves and countermoves ('vegemite is tasty', 'no, it is disgusting') are (i) neither true nor false, yet (ii) perfectly felicitous (involving no 'failure'). That conjunctive property-truth-valueless yet felicitous-is not unheard of: imperative utterances also have it. The lack of truth and falsity entails infelicity for assertions, which aim at truth, but it does not entail infelicity for other speech acts. So it is worth asking which speech act is performed by the utterances at stake.

In sustained disagreement the interlocutors negociate the standards for the local community they form. I propose that their utterances (e.g. 'vegemite is tasty') are not assertions but prescriptions, inviting the hearer to adopt standards with respect to which the lekton is true. The speaker offers her own standards as the coordinative standard for the community. A counter-proposal by the hearer is a fully legitimate move: the hearer may resist the speaker's invitation and offer his own standard as the one they should adopt. So the disagreement persists and it is faultless.6 I conclude that 'vegemite is tasty' can be either an assertion (in which case disagreement cannot be faultless) or a prescription (in which case it can be). If it is an assertion, the context must determine a topic situation with respect to which it can be evaluated. The typical lack of such a determinate situation in episodes of disagreement about taste, where the communal standard is unsettled, is a problem if the utterance is an assertion; but it raises no problem if the utterance is a prescription. (We can also, and presumably should, accept that there are mixed cases, indeed a whole continuum of cases with an assertive and an prescriptive component; but I will ignore this complication here.)

When the utterance is a prescription the content of the speech act is a relativized proposition (lekton): that vegemite is tasty. The speaker tries to get the hearer (and the local community more generally) to adopt standards which make that lekton true. The lekton has truth-at conditions (it is true at certain standards and false at others) but it carries no absolute truth-conditions in these circumstances, for lack of a communal standard accepted by all parties. The conversation aims at establishing such a standard. Until it is settled, no determinate, stable situation of evaluation is provided to turn the lekton into a full Austinian proposition. The speech act is nondefective, however. It has a determinate force and a determinate content (the lekton) but, like orders, it does not allow for truth-evaluation.

  lekton: Response to Max Kölbel

I

  think this proposal, of the classical expressivist variety, fares better than Kölbel's. Kölbel thinks cases of faultless disagreement are cases in which A and B cannot accept the lekton endorsed by the other (from their own perspective) without changing their mind. But that characterization fails to exclude cases of misunderstanding like the example I gave earlier: I call you on the phone, provide section heading here and subsequence occurrences. AQ2. Please update the reference Cohen and Kehler (forthcoming) if possible. AQ3. Please update page range for[START_REF] Kölbel | About concerns[END_REF].

For a recent statement of Perry's view, see his Romanell lecture(Perry

2014).2 "Tensed propositions are understood as directly or indirectly characterising the unmentioned time of utterance"(Prior 1977: 30).

The claim of full articulatedness amounts to a ban on so-called 'topmost modulation' (seeRecanati 2010: 22-23). For a recent argument that there is topmost modulation (hence that the lekton is not fully articulated), see Cohen and Kehler forthcoming.

There are exceptions. In episodic uses referring to a particular experience ('it was tasty') the judge is the experiencer rather than any community.

This is similar to MacFarlane's suggestions regarding 'context coordination' (see PerspectivalThought,.

This paper benefitted from support from the French Agence Nationale de la Recherche under grant agreement n ı ANR-10-LABX-0087 IEC and grant agreement n ı ANR-10-IDEX-0001-02 PSL.

and commenting upon my situation I say 'It is raining'; you respond 'No, it isn't', meaning that there is no rain in your situation. In that example, there is misunderstanding rather than genuine disagreement; but that is also a case in which the interlocutors cannot accept the lekton endorsed by the other (from their own perspective) without changing their mind. 7


Pablo Juesas

Emmanuel Ramasso
email: emmanuel.ramasso@femto-st.fr

Ascertainment-adjusted parameter estimation approach to improve robustness against misspecification of health monitoring methods

Keywords: Time-series, noisy and uncertain labels, weighted distribution theory, health monitoring, CMAPSS datasets

Condition monitoring aims at ensuring system safety which is a fundamental requirement for industrial applications and that has become an inescapable social demand. This objective is attained by instrumenting the system and developing data analytics methods such as statistical models able to turn data into relevant knowledge. One difficulty is to be able to correctly estimate the parameters of those methods based on time-series data. This paper suggests the use of the Weighted Distribution Theory together with the Expectation-Maximization algorithm to improve parameter estimation in statistical models with latent variables with an application to health monotonic under uncertainty. The improvement of estimates is made possible by incorporating uncertain and possibly noisy prior knowledge on latent variables in a sound manner. The latent variables are exploited to build a degradation model of dynamical system represented as a sequence of discrete states. Examples on Gaussian Mixture Models, Hidden Markov Models (HMM) with discrete and continuous outputs are presented on both simulated data and benchmarks using the turbofan engine datasets. A focus on the application of a discrete HMM to health monitoring under uncertainty allows to emphasize the interest of the proposed approach in presence of different operating conditions and fault modes. It is shown that the proposed model depicts high robustness in presence of noisy and uncertain prior.

Introduction

The statistical representation of multi-dimensional time-series originating from a dynamical system consists in finding a concise and meaningful mathematical model that can be easily interpreted and used to undertand the behavior of the system. It is an important problem encountered in a wide range of real applications such as localization and mapping for mobile robot exploring an unknown environment [START_REF] Reineking | Belief functions: Theory and algorithms[END_REF][START_REF] Reineking | Particle filtering in the Dempster-Shafer theory[END_REF], structural health monitoring under different loading conditions [START_REF] Abbiati | Time-dependent estimators for on-line monitoring of full-scale structures under ambient excitation[END_REF][START_REF] Spiridonakos | Non-stationary random vibration modelling and analysis via functional series time-dependent ARMA (FS-TARMA) models -a critical survey[END_REF][START_REF] Ramasso | Unsupervised consensus clustering of acoustic emission time-series for robust damage sequence estimation in composites[END_REF], forecasting and prognostics of various systems [START_REF] Van Den Berg | Financial markets analysis by using a probabilistic fuzzy modelling approach[END_REF][START_REF] Serir | E2GKpro: An evidential evolving multi-modeling approach for system behavior prediction with applications[END_REF][START_REF] Froelich | Evolutionary learning of fuzzy grey cognitive maps for the forecasting of multivariate, interval-valued time series[END_REF][START_REF] Liu | A novel method using adaptive hidden semi-markov model for multi-sensor monitoring equipment health prognosis[END_REF] or human motion analysis [START_REF] Ramasso | Belief scheduler based on model failure detection in the TBM framework. application to human activity recognition[END_REF][START_REF] Gong | Structured time series analysis for human action segmentation and recognition[END_REF]. The context of real-world applications generally involves temporal processes subject to uncertainty which is generally managed either by the theory of belief functions, set-membership approaches or probability theory [START_REF] Klir | Uncertainty-Based Information: Elements of Generalized Information Theory[END_REF][START_REF] Smets | The transferable belief model[END_REF][START_REF] Beer | Special issue of mechanical systems and signal processing imprecise probabilitieswhat can they add to engineering analyses?[END_REF][START_REF] Beer | Editorial: Engineering analysis with vague and imprecise information[END_REF].

This paper is focused on the management of uncertainty in statistical models of time-series originating from in-situ monitoring of the health state of industrial equipements. The monitoring is ensured by sensors that continuously record data (observations) based on which the current and future degradation levels of the system has to be infered for detection (or diagnostics) and prognostics purposes [START_REF] Vachtsevanos | Intelligent Fault Diagnosis and Prognosis for Engineering Systems[END_REF].

The literature on data-driven Prognostics and Health Monitoring (PHM) is mainly focused on supervised or unsupervised models [START_REF] Tibaduiza | A study of two unsupervised data driven statistical methodologies for detecting and classifying damages in structural health monitoring[END_REF][START_REF] Mechbal | A probabilistic multi-class classifier for structural health monitoring[END_REF][START_REF] Liu | A novel method using adaptive hidden semi-markov model for multi-sensor monitoring equipment health prognosis[END_REF][START_REF] Ramasso | Performance benchmarking and analysis of prognostic methods for CMAPSS datasets[END_REF]: For training, the degradation level (or state) is either known precisely or hidden and, during testing, the current and future levels are infered from sensor data. In situations where the degradation level can be represented as a sequence of discrete states, statistical models with discrete latent variables have been widely used. Hidden Markov Model (HMM) [START_REF] Rabiner | A tutorial on hidden Markov models and selected applications in speech recognition[END_REF][START_REF] Renooij | Efficient sensitivity analysis in hidden Markov models[END_REF] represents one of those models. HMMs have indeed been widely exploited for PHM [START_REF] Ma | A new method for discrimination between fault and magnetizing inrush current using HMM[END_REF][START_REF] Chinnam | Autonomous diagnostics and prognostics through competitive learning driven HMM-based clustering[END_REF][START_REF] Dong | A segmental hidden semi-Markov model (HSMM)based diagnostics and prognostics framework and methodology[END_REF][START_REF] Moghaddass | Equipment Degradation Diagnostics and Prognostics Under a Multistate Deterioration Process[END_REF][START_REF] Liu | A novel method using adaptive hidden semi-markov model for multi-sensor monitoring equipment health prognosis[END_REF][START_REF] Ramasso | Contribution of belief functions to hidden markov models with an application to fault diagnosis[END_REF] where it is assumed that the degradation of the equipment follows a doubly stochastic process: One for the dynamics of the hidden states and one to account for the distribution of the observations. Generally, the generative form of HMM is used: One model is built for each possible degradation level, and, during monitoring, a similarity-based approach is applied to find the likeliest model, that is then used to infer the current state (detection and diagnostics) and the future trends (forecasting and prognostics). We propose an alternative in using HMM for PHM by considering discriminative learning where the parameters are estimated with the aim to improve the classification into degradation levels.

Compared to classical generative approaches for HMM-based PHM, and more generally to latent model-based PHM, the proposed approach is based on the idea that sensor data related to different degradation levels have to be put together in a training dataset. Each level is then assigned a soft label : It allows one to use uncertain or noisy labels [START_REF] Côme | Learning from partially supervised data using mixture models and belief functions[END_REF] according to the quantity and the quality of the prior knowledge about the degradation level. By this way, some of the latent variables may now represent one degradation level and multiple levels can share common feature subspaces through the use of uncertain labels.

The solution for incorporating soft labels in models with latent variables is based on a modification of the conditional expectation of the log-likelihood in the Expectation-Maximization algorithm by using the Weighted Distribution Theory [START_REF] Patil | Weighted distributions[END_REF]. An application of this criterion is shown for Gaussian Mixture Models, continuous and discrete HMM. A particular attention is paid to Maximum Likelihood Estimation (MLE) of parameters in HMM with discrete-valued observations (DHMM) in presence of soft labels.

Standard DHMMs have been used in the past in many applications [START_REF] Murphy | Dynamic Bayesian Networks: Representation, inference and learning[END_REF][START_REF] Yu | Hidden semi-Markov models[END_REF][START_REF] Ranjan | HMM modeling for OFDM-BER performance[END_REF][START_REF] Chen | A framework for video event classification by modeling temporal context of multimodal features using HMM[END_REF][START_REF] Baldi | Hidden Markov models of biological primary sequence information[END_REF][START_REF] Ma | A new method for discrimination between fault and magnetizing inrush current using HMM[END_REF] and in particular for noisy speech and character recognition [START_REF] Bourlard | A continuous speech recognition system embedding MLP into HMM[END_REF][START_REF] Kosaka | Robust Speech recognition and understanding[END_REF][START_REF] Elms | The advantage of using an HMM-based approach for faxed word recognition[END_REF][START_REF] Dehghan | Handwritten farsi (arabic) word recognition: a holistic approach using discrete HMM[END_REF]. In the context of PHM, DHMM has also been widely used, for instance in [START_REF] Liu | Predictive modeling for intelligent maintenance in complex semiconductor manufacturing processes[END_REF] for predictive modeling dedicated to intelligent maintenance in complex semiconductor manufacturing processes, in [START_REF] Menon | Incipient fault detection and diagnosis in turbine engines using hidden Markov models[END_REF] for incipient fault detection and diagnosis in turbine engines, in [START_REF] Altan | Hierarchical HMM-based failure isolation for cognitive robots[END_REF] for failure isolation for cognitive robots and in [START_REF] Dorj | A Bayesian Hidden Markov Model-based approach for anomaly detection in electronic systems[END_REF] for anomaly detection in electronic systems.

While the inference phase is very similar between HMM with continuous (CHMM) and discrete observations (DHMM), the learning phase presents a fundamental difference since the observations in the latter are discrete. The learning phase for huge datasets is thus generally faster for the DHMM by using matrix encoding. Moreover, it has been shown that the DHMM may be better in presence of noise with unknown (non Gaussian) characteristics [START_REF] Kosaka | Robust Speech recognition and understanding[END_REF].

The contribution of this paper is two-fold:

• We propose a framework for PHM based on latent variables. It is based on the Expectation-Maximization algorithm and the Weighted Distribution Theory. This work is inspired from a previous work proposed by T. Denoeux [START_REF] Denoeux | Maximum likelihood estimation from uncertain data in the belief function framework[END_REF] based on Dempster-Shafer's theory of belief functions and plausibility weights. The main difference with the present work is the consideration of almost unrestricted weights (satisfying only positiveness) based on the work of Patil [START_REF] Patil | Weighted distributions[END_REF].

• This framework is used to improve the performance of DHMM for PHM. An application to turbofan health monitoring is presented where we evaluate the sensitivity of DHMM to vector quantization with respect to the quality and quantity of prior.

The problem of incorporating partial knowledge about latent variables is discussed in Section 2. Section 3 presents the application of this method to DHMM. Section 4 is dedicated to the analysis of the proposed model on several datasets.

Incorporating prior knowledge on latent variables in EM

The Expectation-Maximization algorithm (EM) [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF] allows to estimate the parameters of a statistical model with latent variables. It has been adapted in [START_REF] Denoeux | Maximum likelihood estimation from uncertain data in the belief function framework[END_REF] in order to take uncertain prior information into account. The prior is supposed to be represented by a collection of T Basic Belief Assignments (BBA) denoted as m = {m

Q(θ, θ (q) ) = E θ (q) [log L(θ; z)|x, m] = y pl(y)p(y|x, θ (q) ) log L(θ; z) L(θ; pl) (1)
where pl represents the prior knowledge on latent variables y encoded by contour functions in place of BBA m and computed as: pl : Ω y → [0, 1] such that pl(ω) = P l({ω}) with P l(C) = D∩C =∅ m(D), C ⊆ Ω y . The quantity L(θ; pl) represents the amount of conflict between both the model (parameterized by θ and evaluated on the observed data x) and the prior (represented by pl).

As it can be observed, the uncertain prior play a role of weight on the posterior distribution in the auxiliary function. This can be interpreted as an adjustment of the distribution over hidden states due to, for example, model misspecification or to improve data interpretation.

Proposition 1. Eq. 1 is valid for any positive weights w in place of plausibilities:

Q(θ, θ (q) ) = E θ (q) [log L(θ; z)|x, w] = y w(y)p(y|x, θ (q) ) log L(θ; z) E θ [w(z)] (2)
The proof is based on the Weighted Distribution Theory (WDT) which is a general framework for adjusting probabilities. Patil [START_REF] Patil | Weighted distributions[END_REF] illustrated the WDT with practical examples and the reader may also be interested by the paper of Iyengar and Zhao [START_REF] Iyengar | Maximum likelihood estimation for weighted distributions[END_REF] for a use in maximum likelihood, and by the paper of Nielsen and Signorovitch [START_REF] Nielsen | Correcting for ascertainment biases when analyzing SNP data: applications to the estimation of linkage disequilibrium[END_REF] with an application to single nucleotide polymorphisms discovery.

More specifically, the random variable Z is supposed to be governed by a probability density function f (z; θ) (for instance the posterior in Eq. 2). The weight function w(z) allows one to weigh the pdf f when Z = z [START_REF] Patil | Weighted distributions[END_REF]:

f w (z; θ) = f (z; θ)w(z) E θ [w(Z)] s.c. w(z) ≥ 0, 0 < E θ [w(Z)] < ∞ (3)
The weights are not considered as probabilities and only have to be positive [START_REF] Patil | Weighted distributions[END_REF]. It is interesting to make a parallel between the denominators of Eq. 1 and 2. Indeed, in the former case, it has been shown that [START_REF] Denoeux | Maximum likelihood estimation from uncertain data in the belief function framework[END_REF]:

L(θ; pl) = E θ [pl(Z)] (4)
which is similar to the denominator of Eq. 3. Taking the conditional expectation of the complete-data log-likelihood given partial knowledge on states encoded as weights w = {w 1 . . . w T } thus leads to a similar expression as Eq. 2.

Learning with soft labels in DHMM

This section shows how to use the proposed model to learn parameters in a Discrete HMM (DHMM). For health monitoring, it follows the plot chart depicted in Figure 1.

[Figure 1 about here.]

Model and notations

In a DHMM, the observed data take the form of a sequence of discrete symbols x = {x 1 . . . x T }, x t ∈ {1 . . . V } with V the number of symbols, and where t = 1 . . . T is a discrete time index. The model assumes that the sequence has been generated by a stochastic (first-order) Markov process with hidden states y = {y 1 , . . . y T }, with y t ∈ {1 . . . K}, K is the number of hidden states, and Ω y = {ω 1 . . . ω K } represents the set of states. The distribution of the output observation x t at time t depends only on y t . The complete data is denoted as z = (x, y).

A DHMM is described by the following parameters [START_REF] Rabiner | A tutorial on hidden Markov models and selected applications in speech recognition[END_REF]:

• Prior probabilities Π Π Π = {π 1 , ..., π k , ..., π K }, where π k = P (y 1 = k) is the probability of being in state k at t = 1 ;

• Transition probabilities A = [a kl ],
where

a k,l = P (y t = l|y t-1 = k), (k, l) ∈ {1, ..., K} 2 (5)
is the probability for being in state l at time t given that it was in state k at t -1 with l a kl = 1;

• Observation symbol probabilities B = [b kv ] (emission matrix) where

b k,v = P (x t = v|y t = k), k ∈ {1, ..., K}, v ∈ {1, ..., V } (6)
is the probability for being in state k at time t and observing symbol v

with v b k,v = 1
The set of parameters in a DHMM is denoted as:

θ = (A, B, Π Π Π) (7)

Learning procedure based on soft labels

The weighted conditional expectation of the log-likelihood given partial knowledge on states for DHMM (Eq. 2) requires to express the likelihood over hidden and observed variables which, in the DHMM, is given by [START_REF] Rabiner | A tutorial on hidden Markov models and selected applications in speech recognition[END_REF]:

L(θ; z) = p(y 1 ; Π) T t=2 p(y t |y t-1 ; A) T t=1 p(x t |y t ; B) (8a) = K k=1 π y 1,k k T t=2 k,l a y t-1,k y t,l k,l T t=1 K k=1 V v=1 b y t,k k,v (8b)
This form makes use of the fact that the latent variables are multinomial discrete variables [START_REF] Bishop | Pattern recognition and machine learning[END_REF], therefore y tk = 1 if state k is true at time t. The conditional expectation of the log-likelihood given partial knowledge on states obtained at iteration q becomes:

Q(θ, θ (q) ) = K k=1 γ (q) 1,k log π k + T t=2 k,l ξ (q) t-1,k;t,l log a k,l + T t=1 K k=1 V v=1 γ (q) t,k log b k,v (9)
where

γ t,k = E θ (q) [y t,k |x, w] and ξ t-1,k;t,l = E θ (q) [y t-1,k y t,l |x, w]
represent the mathematical expectations with respect to the posterior distribution over hidden states given by:

ξ (q) t-1,t;k,l = α (q) t-1,k b l,v w t,l a (q) k,l β t,l L(θ (q) ; w) (10)
γ (q) t,k = α (q) t,k β (q) t,k L(θ (q) ; w) (11)
and

L(θ (q) ; w) = K k=1 α T,k (12)
The variables α

(q) t,k and β (q)
t,k can be recursively computed as

α (q) 1k = π (q) k w 1,k b k,v (13a)
α (q) tk = b k,v w t,k j α (q) t-1,j a (q) j,k (13b) β (q)
T k = 1 (13c)

β (q) tk = l β (q) t+1,l b l,v w t+1,l a (q) k,l (13d)
for t = 2 . . . T . We can observe that the soft labels w do not change the conditional expectation compared to [START_REF] Ramasso | Making use of partial knowledge about hidden states in HMMs: an approach based on belief functions[END_REF] except for the emission probabilities (third term of Eq. 9). Therefore, the reestimation of parameters A and Π Π Π are given by:

π (q+1) k =γ (q) 1,k (14)
a (q+1) kl = T t=2 ξ (q) t-1,k;t,l T t=2 K l =1 ξ (q) t-1,k;t,l (15)
and the MLE of the emission probability model B is given by applying a similar reasoning as in the unsupervised learning of DHMM [START_REF] Rabiner | A tutorial on hidden Markov models and selected applications in speech recognition[END_REF] adapted as proposed in [START_REF] Ramasso | Making use of partial knowledge about hidden states in HMMs: an approach based on belief functions[END_REF] to take into account the uncertain prior on hidden states leading to

b (q+1) k,v = T t=1 γ (q) t,k 1{x t = v} T t=1 γ (q) t,k (16)
The expression of γ is given by Eq. 11 that makes use of the partial knowledge about hidden states in discrete HMM.

Applications

A first simulated dataset is used to illustrate the behavior of the proposed model (Eq. 2) for Gaussian Mixture Models, CPHMM and DPHMM with respect to different weights. Those three models are available as Matlab codes which allow to reproducing all results of this paper: http://www.mathworks.com/matlabcentral/profile/authors/7468430

The second simulated dataset allows us to study the effect of the soft labels on the both the performance and on the quantization of continuous variables. Finally, complex benchmarking datasets provided by the NASA PCOE [START_REF] Saxena | Damage propagation modeling for aircraft engine run-to-failure simulation[END_REF] are used to illustrate the performance of the proposed model for diagnostics.

In the three applications, the sequences of states estimated (for instance, in order to represent a degradation) with the proposed approach (WDT applied on various models such as DHMM) are compared to a ground truth using the Adjusted Rand Index (ARI) [START_REF] Vinh | Information theoretic measures for clusterings comparison: Is a correction for chance necessary?[END_REF]. This performance indicator, widely used for unsupervised classification, allows to evaluating the impact of the quality and quantity of labels. It tends to 1 (resp. 0) if the recognition rate is perfect (resp. bad).

Toy example dedicated to the application of the methodology on various

models from the literature with comparison The interest of using arbitrary weights in Eq. 2 is illustrated on three different models: partially-Hidden Markov Models with continuous observations (CPHMM) [START_REF] Ramasso | Making use of partial knowledge about hidden states in HMMs: an approach based on belief functions[END_REF], the DHMM presented above and the Gaussian Mixture Model (PGMM) proposed in [START_REF] Côme | Learning from partially supervised data using mixture models and belief functions[END_REF].

A 2-dimensional dataset was generated using the following model made of K = 3 clusters:

Π Π Π = (1/3, 1/3, 1/3) , A =   0.6 0.1 0.3 0.1 0.6 0.3 0.15 0.15 0.7   ( 17
)
µ 1 = 4 0 , µ 2 = 0 4 , µ 3 = 2 2 (18) Σ 1 = diag 7 7 , Σ 2 = diag 7 7 , Σ 3 = diag 2 2 (19)
A set of N=300 samples is drawn from this CPHMM and illustrated in Figure 2.

[Figure 2 about here.]

The particularity of this dataset is to be composed of two distinct groups (clusters) plus a third one located on purpose at the middle of both previous ones in order to simulate a fuzzy frontier. The centers of the three clusters are estimated by using the ECM algorithm [START_REF] Masson | ECM: An evidential version of the fuzzy C-means algorithm[END_REF] with K=3 (and with the default parameters proposed in [START_REF] Masson | ECM: An evidential version of the fuzzy C-means algorithm[END_REF]). The interest of using this algorithm here is to get the credal partition, denoted as m Ω i , Ω = {c 1 , c 2 , c 3 }, i = 1 . . . N , that gives the membership degree (a BBA) of each data point to each cluster and to all subsets of clusters. The credal partition is assumed to represent a prior knowledge that can be used in the three models.

To be used with Eq. 2, we need to transform the credal partition into a set of weights assigned to singleton clusters. For that, we proceed as proposed in [START_REF] Masson | ECM: An evidential version of the fuzzy C-means algorithm[END_REF] by transforming the partition into three types of weights:

BBA are belief masses on singleton clusters where weights are equal to

w i (c k ) ≡ m Ω i (c k );
PL are plausibilities on singleton clusters where weights are equal to

w i (c k ) ≡ pl Ω i (c k );
BETP are pignistic probabilities on singleton clusters [START_REF] Smets | The transferable belief model[END_REF] where weights are equal to

w i (c k ) ≡ BetP Ω i (c k ).
ECM was run 5 times with random initializations of centers, and the model (position of centers) leading to the highest ARI.

The ARI is used to compare the partition obtained by maximizing the BBA on singletons with the real partition obtained by sampling the previous model (the correct cluster is known). By doing so, we explicitly make favourable the choice of BBA as weights. The performance is thus expected to be higher all the more than the plausibility and the pignistic transformations are computed by transfering the belief masses on some subsets towards singletons: Conversely to BBA, the frontier artificially added through the third cluster is expected to collapse with those two transformations.

Learning in CPHMM was performed as in [START_REF] Ramasso | Making use of partial knowledge about hidden states in HMMs: an approach based on belief functions[END_REF] except that the weights took the three forms presented above (BBA, pl and BetP). The states were estimated by maximizing the posterior distribution on states (γ-variable) for both the DPHMM (DHMM with soft labels) and the CPHMM. For the DPHMM, quantization was performed by a Kmeans by minimizing the sum of squareddistance between points and clusters [START_REF] Rabiner | A tutorial on hidden Markov models and selected applications in speech recognition[END_REF].

[Figure 3 about here.] Figure 3 shows the evolution of the boxplot of ARI over 50 sets of 1000 samples for the CPHMM and for the PGMM learned with plausibility as initially required in E2M (Evidential EM) algorithm [START_REF] Denoeux | Maximum likelihood estimation from uncertain data in the belief function framework[END_REF] using Eq. 1 (E2M+pl), with pignistic probabilities (E2M+betp) and with the belief masses (E2M+BBA). The performance of ECM is also shown in order to evaluate the performance obtained with the introduction of transitions (which are explicitly used for sampling). The results with noisy labels is also depicted: Noisy labels are obtained by assigning a binary value (0 or 1) for the maximum of BBA, pl or BetP over each cluster and for each data point (for instance

m Ω i (c k ) = 1 if k = argmax j m Ω
i (c j ), 0 otherwise). Results are globally quite low for all models. This is due to the fact that one of the clusters is localized at the very middle of the other clusters with high overlapping. The comparison between the three models has thus to be performed relatively. The absolute performance is indeed not the main purpose of this experiment which is to show that using arbitrary weights can be useful. A deep analysis of the performances is proposed in next sections.

Figure 3 shows that the performance is better when using the belief masses by a factor of almost 2 on average, compared to pignistic probabilities and plausibilities. This is an important result which goes in favor of considering the Weighted Distribution Theory to incorporate the most relevant information in the learning phase when using E2M.

Furthermore, the DPHMM behaves similarly with noisy or uncertain labels on this particular dataset whereas the CPHMM and PGMM with noisy labels provides lower performance compared to uncertain labels. This result was also underlined in [START_REF] Ramasso | Making use of partial knowledge about hidden states in HMMs: an approach based on belief functions[END_REF] for CPHMM. The CPHMM provides the best overall result (37% ± 2%) when used with the BBA.

Impact of soft labels on quantization in DPHMM

The goal of the following experiments is to study the effect of soft labels on both the performance and the quantization in the newly proposed DPHMM. For that, we consider the same continuous-valued data as in [START_REF] Ramasso | Making use of partial knowledge about hidden states in HMMs: an approach based on belief functions[END_REF] generated with a continuous HMM with 3 states in 3 dimensions.

The Kmeans algorithm was applied for vector quantization [START_REF] Rabiner | A tutorial on hidden Markov models and selected applications in speech recognition[END_REF] in order to obtain discrete symbols. To cope with the sensitivity of Kmeans with respect to the initialization, 10 runs were performed and the best partition was selected according to the Davies-Bouldin index [START_REF] Davies | A cluster separation measure. Pattern Analysis and Machine Intelligence[END_REF].

Two different experiments were carried out to study the influence of "label imprecision" and "labeling error" on the performance of the DPHMM. In each case, different numbers of clusters (N ) were tested to estimate the impact of the quantization on the performance.

Influence of label imprecision on the performance and on the quantization

We proceed as proposed in [START_REF] Ramasso | Making use of partial knowledge about hidden states in HMMs: an approach based on belief functions[END_REF] to study how the imprecision of knowledge on hidden states affects the performance of the DPHMM. A learning sequence of length T was generated using the model presented in [START_REF] Ramasso | Making use of partial knowledge about hidden states in HMMs: an approach based on belief functions[END_REF]. Imprecision on labels was generated as pl tk = 1 if y t = k, or ν otherwise, where ν represents the nonspecificity coefficient (ν = 1 corresponds to a classical DPHMM model with no information available on hidden states, ν = 0 represents a fully supervised learning situation).

To assess the quality of learning, a testing dataset of 1000 observations was generated following the same distribution. The most probable state sequence was computed using the maximum of the γ variable (Eq. 11). It was observed that the Viterbi algorithm may lead to bad results when the transition matrix (Eq. 15) or the emission matrix (Eq. 16) are ill-conditioned (presence of 0). This instability was not observed important in the continuous version of this algorithm [START_REF] Ramasso | Making use of partial knowledge about hidden states in HMMs: an approach based on belief functions[END_REF].

The precision of the predicted state sequences was assessed using the ARI. The whole experiment (data generation, clustering and learning) was repeated 30 times, for different number of clusters and for different length of the learning sequence T = 100 and T = 300.

Results are shown in Figure 4. We can observe that results degrade from the fully supervised (ν = 0) to the fully unsupervised (ν = 1) case for most of tests. As T increases, the estimation is improved in terms of better ARI and lower variance on results. It allows to decrease the impact of both the quantization phase (ie of N ) and of the corruption process (ie of ν).

In addition to the computational problem observed in the Viterbi algorithm mentioned above, we observed that the number of iterations in the learning phase should be low (5 -10). Otherwise, the convergence leads to very specific emission matrices (Eq. 16) which ultimately lead to overfitting. This problem is mainly due to the size of the training sequence which is probably too small. It can be remarked that this problem was not encountered in the continuous version [START_REF] Ramasso | Making use of partial knowledge about hidden states in HMMs: an approach based on belief functions[END_REF].

[Figure 4 about here.]

Influence of labeling error on the performance and on the quantization

In the previous experiment information on hidden states was assumed to be always exact, i.e. the true state had always the largest plausibility value. In this part, we proceed as proposed in [START_REF] Côme | Learning from partially supervised data using mixture models and belief functions[END_REF][START_REF] Ramasso | Making use of partial knowledge about hidden states in HMMs: an approach based on belief functions[END_REF] where at each time step t of a sequence, an error probability q t was drawn randomly from a beta distribution with mean ρ and standard deviation 0.2. With probability q t , the state y t was then replaced by a completely random value ỹt (with a uniform distribution over possible states). The plausibilities pl tk were determined as pl tk = P (y t = k|ỹ t ) = q t /K + 1 -q t if ỹt = k, or q t /K otherwise.

Uncertain labels generated in this way are more imprecise when the error probability is high. It simulates a more realistic and complex situation compared to the previous section.

Training and test data sets were generated as in the previous section, and results were evaluated in the same way. For each randomly generated data set, the DPHMM was applied with uncertain labels pl tk and noisy labels ỹtk .

Figures 5 and6 show the ARI as a function of mean error ρ for uncertain and noisy labels respectively, for different number of clusters and sequence length. As previously, a degradation of the segmentation quality is observed when the mean error probability ρ increases, for both uncertain and noisy labels. Moreover, increasing T from 100 to 300 does improve the quality. It can also be observed that results are better with noisy labels than with uncertain ones. The explanation is that the observation matrix is better filled with noisy but certain information than with uncertain (fuzzy), the latter tends to smooth the values of the matrix. This is in contrast with results obtained with continuous HMM reported in [START_REF] Ramasso | Making use of partial knowledge about hidden states in HMMs: an approach based on belief functions[END_REF] in which the distributions are better fitted with noisyfuzzy information than with noisy-certain ones. This is also in agreement with previous experiments.

[Figure 5

. Description of the data

The turbofan datasets were generated using the CMAPSS simulation environment that represents an engine model of the 90,000 lb thrust class [START_REF] Frederick | User's guide for the commercial modular aero-propulsion system simulation[END_REF][START_REF] Saxena | Damage propagation modeling for aircraft engine run-to-failure simulation[END_REF]. A number of editable input parameters was used to specify operational profile, closed-loop controllers, environmental conditions (various altitudes and temperatures). Some efficiency parameters were modified to simulate various degradations in different sections of the engine system. Selected fault injection parameters were varied to simulate continuous degradation trends. Data from various parts of the system were collected to record effects of degradations on 21 sensor measurements and provide time-series exhibiting degradation behaviors in multiple units.

These datasets possess unique characteristics that make them very useful and suitable for developing prognostic and health monitoring algorithms [START_REF] Ramasso | Performance benchmarking and analysis of prognostic methods for CMAPSS datasets[END_REF]: Multidimensional response from a complex non-linear system, high levels of noise, effects of faults and operational conditions, and plenty of units were simulated with high variability.

In the present paper, the learning datasets of the four turbofan datasets were used to illustrate the DPHMM on a complex system. The characteristics of the datasets are described in Table 1. It can be observed that dataset #1 is the simplest one with one operating condition (OC) and one fault mode. Datasets #2 and #4 are the most complex datasets with six OC and one or two fault modes. Dataset #3 presents two fault modes and one OC.

The state-of-the-art results on those datasets are presented in [START_REF] Ramasso | Performance benchmarking and analysis of prognostic methods for CMAPSS datasets[END_REF]. According to this review, only the publication [START_REF] Ramasso | Making use of partial knowledge about hidden states in HMMs: an approach based on belief functions[END_REF] provided results using soft labels for health monitoring purposes. However, it makes use of particular features for dataset #1 only. In the present work, we apply a more general methodology where a health indicator is extracted as presented in [START_REF] Ramasso | Investigating computational geometry for failure prognostics[END_REF] for each training data in each dataset. Then, soft labels are introduced (described below) and the influence of noise and uncertainty on those labels are analyzed.

[Table 1

about here.]

Each dataset is made of a certain number of trajectories with different length and 21 sensor measurements, with a total amount of about 700 training degradation trajectories (sum of values in Table 1). From sensor measurements, a health indicator is built for each trajectory as proposed in [START_REF] Ramasso | Investigating computational geometry for failure prognostics[END_REF]. The health indicators (HI) for all trajectories and all datasets are depicted in Figure 7. We can observe that datasets #2 and #4 are made of about 500 trajectories with high variability in terms of noise and length as compared to dataset #1. These four sets of trajectories are used for both training and testing data to illustrate the influence of labeling errors in the DPHMM.

[Figure 7 about here.]

Generation of the ground truth

Those datasets have been used for clustering or classification purposes in many papers as reported in [START_REF] Ramasso | Performance benchmarking and analysis of prognostic methods for CMAPSS datasets[END_REF]. In order to get a ground truth to both feed the DPHMM and make comparison, a way to automatically generate three states from each trajectory is described in [START_REF] Ramasso | Segmentation of CMAPSS health indicators into discrete states for sequence-based classification and prediction purposes[END_REF] with Matlab codes.

The generation of the state sequences corresponds to a labeling which may be corrupted by errors due, for example, to the noise on the HI (impacted by OC and fault modes), and to the threshold (10% on the absolute error). To evaluate the influence of labeling errors, we proceed as in Section 4.3 (by drawing randomly from a beta distribution and replacing the label by a random value) and considering noisy labels. It was indeed observed that the DPHMM performed badly in presence of uncertain labels for this particular application but better with noisy labels.

The influence of labeling errors was studied for the four datasets using a leave-one-out cross-validation over all training instances for each CMAPSS dataset. Since the HI are continuous, a quantization was performed to generate symbols used in the DPHMM by simply converting the floats into integers as follows:

HI discrete (t) ← HI continuous (t) × N (20)
where N is a parameter playing a similar role as the number of clusters in usual quantization methods [START_REF] Rabiner | A tutorial on hidden Markov models and selected applications in speech recognition[END_REF]. The influence of N is studied in the next tests. Figure 8, 9, 10 and 11 represent boxplots of the ARI with respect to both the labeling error rate and to the value of N for the four datasets. For dataset #1 (Fig. 8), the best performance is obtained for the supervised case with N = 5 yielding 88%, then it decreases monotonically with respect to the labeling error but always remains above the unsupervised case (ρ = 1) with 66%. This performance is similar to the state-of-the-art results using soft labels presented in [START_REF] Ramasso | Making use of partial knowledge about hidden states in HMMs: an approach based on belief functions[END_REF]. For dataset #2 (Fig. 9), the performance equals 83% for the supervised case and 63% for the unsupervised case. The performances decreases monotonically with an increasing noise level except for N = 15. For dataset #3 (Fig. 10), the best performance is obtained for the supervised case with N = 5 yielding 82%. For dataset #4 (Fig. 11), a similar behavior to dataset #2 can be observed. The performance for the supervised case is similar to dataset #2 (84.5%) despite one additional fault mode. The results for this last dataset shows the impact of the combination of fault modes and operating conditions on the performance.

[Figure 8 [Figure 11 about here.] N = 5 yields the best performance for all datasets (over about 700 degradation trajectories). The DPHMM with partial noisy knowledge depicts robustness to the labeling error on the four datasets. Moreover, the performance in the supervised case is generally expectedly better than the unsupervised case. The performance then generally decreases as the level of labeling error increases.

Large values of N leads to a particular results where the performance is almost independent of the noise level. It can be pointed out that the ARI was created for corrected-for-chance evaluation which means that random labelling of states should tend to 0. Therefore, the performance obtained with large N is not due to randomness. Moreover, the performance is better than with N = 3. An explanation is given hereafter, remembering that large N implies a high number of symbols. In addition to the noise level on the HI together with the labeling error rate, the estimation of the emission probability model (Eq. 16) is impacted but the performance still remains satisfying (above N = 3) since there are always some symbols which allows to represent the sequence. It can also be pointed out that, in presence of more symbols, the model requires more data to estimate the emission matrix. This behavior is confirmed by the interquartile ranges which are important for N = 15 for datasets #1 and #3 which are made of 100 trajectories (Tab. 1) whereas the interquartile ranges are lower for datasets #2 and #4 made of about 260 trajectories.

Conclusion

The Weighted Distribution Theory (WDT) has been initially proposed to cope with the fact that the probability of a particular observation entering in a data sample gets multiplied by some non-negative weight function w. The data are thus naturally distributed according to f w (Eq. 3) instead of f . Since model specification is of paramount interest in data analysis, such bias should be taken into account.

This paper suggests a means to incorporate noisy and uncertain prior knowledge on latent variables in a sound manner using the Expectation-Maximization algorithm together with the WDT. An application to discrete Hidden Markov Models (DHMM) is developed and illustrated for system's health monitoring. Illustrations are also provided for Gaussian Mixture Models and HMM with continuous outputs.

The interest of such an approach is to cope with model selection for classification with maximum likelihood learning. Instead of considering one model per class, all classes are considered simultaneously and the definition of the classes may be imperfect (noise on labels, uncertainty). The learning phase based on WDT and EM then finds a compromise between the data and the prior.

The first experiment allows to demonstrate the relevance of considering possibly arbitrary weights compared to the algorithm proposed in [START_REF] Denoeux | Maximum likelihood estimation from uncertain data in the belief function framework[END_REF]. The second and third experiments show the DPHMM model behaved better when considering noisy labels than uncertain labels compared to continuous HMM, which seems in agreement with the model. It is also demonstrated that the quantization (transformation of continuous to discrete values) in the DPHMM applied to continuous data does not have so much impact on classification results when selected appropriately, using for instance validity index as proposed in this paper. It is practically interesting since DPHMM are less complex.

The third experiment was performed on a complex benchmark representing health indicators originating from a turbofan engine simulator developed at NASA. These tests shown that the proposed model depicted robustness to labelling errors when the level of noise on the initial data was reasonable. A labeling error rate up to 50-60% yields good results. Moreover, the same parameterization of the quantization phase applied to all datasets provided the best results, demonstrating that the model is reasonably robust to quantization on different datasets.

The integration of prior in a sound manner for the modelling of real systems is of key interest, in both data-driven and physics-based approaches. Indeed, the amount of tests required to correctly pave a complex feature space and deduce particular behavior should be sufficiently large to expect a correct identification of parameters. This is for instance the case for identification of failure modes in complex mechanical systems. The integration of prior is all the more important in presence of non stationary systems submitted to external factors (generally hidden and unknown) which creates intermittent faults characterized by sudden changes in system parameters. Making use of prior appears as a means to reduce the number of experiments and include information about external factors with respect to time.

The Bayesian approach is an inescapable way to integrate uncertain prior knowledge on parameters [START_REF] Bishop | Pattern recognition and machine learning[END_REF]. The main advantage is the possibility to penalize models with too many parameters with the aim to reduce overfitting which can appear in maximum likelihood approaches. As a consequence, this allows to improving model selection which is particularly important for health monitoring [START_REF] Vanik | Bayesian probabilistic approach to structural health monitoring[END_REF][START_REF] Yan | A novel bayesian approach for structural model updating utilizing statistical modal information from multiple setups[END_REF]. The influence of the prior on results is however not often deeply studied as proposed in this paper.

While the Bayesian approach makes generally use of prior on parameters, the proposed WDT-based approach is focused on the integration of prior on discrete latent variables which appears as a different means to adjust indirectly the distributions on parameters. The proposed approach could be used in Bayesian formulations to give a hand in improving model specifications by driving the estimation using both prior on parameters and on latent variables. We aim at exploring this path in addition to the application of the WDT approaches to other Markov switching and statistical models with latent variables especially dedicated to nonlinear and non-stationary systems monitoring and prognostics [START_REF] Kan | A review on prognostic techniques for non-stationary and non-linear rotating systems[END_REF].

Ωy 1 ,

 1 . . . m Ωy T } defined on the set of discrete states Ω y with A⊆Ωy m Ωy t (A) = 1, and such that m Ωy t (∅) = 0 (normalized BBA). The adaptation of EM concerns the auxiliary function (conditional expectation of the log-likelihood) to be maximized at each iteration q:

  about here.] [Figure 6 about here.] 4.4. Application of DPHMM to turbofan engine diagnostics 4.4.1

  about here.] [Figure 9 about here.] [Figure 10 about here.]

2 State 1 (

 21 healthy mode) State 2 (degraded mode) State 3 (fuzzy frontier between modes)

Figure 2 :

 2 Figure 2: Distribution of signal points for the first dataset

Figure 3 :

 3 Figure 3: Boxplots of ARI for GMM and HMM (continuous and discrete) learned with E2M based on different weights.

Figure 4 :

 4 Figure 4: Medians of the adjusted Rand index as function of the nonspecificity coefficient over 30 repetitions for different number of clusters (3, 5, 15) and with time-series of different length (100, 300).

Figure 5 :

 5 Figure 5: Medians of the adjusted Rand index as a function of the labelling error with uncertain labels over 30 repetitions for different number of clusters (3, 5, 15) and with time-series of different length (100, 300).

Figure 6 :

 6 Figure 6: Medians of the adjusted Rand index as a function of the labelling error with noisy labels over 30 repetitions for different number of clusters (3, 5, 15) and with time-series of different length (100, 300).

Figure 8 :

 8 Figure 8: Dataset #1: Boxplot of ARI for different numbers of symbols (N, used in quantization).

Figure 9 :

 9 Figure 9: Dataset #2, with operating conditions modeling: Boxplot of ARI for different numbers of symbols (N, used in quantization).

Figure 10 :

 10 Figure 10: Dataset #3: Boxplot of ARI for different numbers of symbols (N, used in quantization).

Figure 11 :

 11 Figure 11: Dataset #4, with operating conditions modeling: Boxplot of ARI for different numbers of symbols (N, used in quantization).

Acknowledgement

This work has been carried out in the framework of the Laboratory of Excellence ACTION through the program "Investments for the future" managed by the National Agency for Research (references ANR-11-LABX-01-01). The authors are grateful to the Région Franche-Comté and "Bpifrance financement" supporting the SMART COMPOSITES Project in the framework of FRI2. The authors would like to thank the reviewers who provided valuable comments.

List of Figures