Elsa Tolone
email: elsa.tolone@univ-paris-est.fr

Stavroula Voyatzi
email: voyatzi@univ-mlv.fr

Claude Martineau
email: martinea@univ-mlv.fr

Matthieu Constant
email: mconstan@univ-mlv.fr

Extending the adverbial coverage of a French morphological lexicon

Keywords: morhological lexicon, adverb, paraphrase

 and compound adverbs (Gross, 1986b; Gross, 1986a). This work exploits fine-grained linguistic information provided in existing resources. The resulting resource is reviewed in order to delete duplicates and is freely available under the LGPL-LR license.

Introduction

Recognising adverbs such as extrêmement 'extremely' and à long terme 'in the long run' in texts is likely to be useful for information retrieval and extraction because of the information that some of these adverbs convey. Adverbs, or more generally circumstantial complements, have often been overlooked in the compilation of lexical resources (Nølke, 1990) (p. 3). Several reasons explain this lack of interest. Firstly, adverbs are usually felt as less useful than nouns for information retrieval and extraction. Secondly, compound adverbs in particular are difficult to distinguish from prepositional phrases assuming other syntactic functions, such as arguments or noun modifiers: the distinction is hardly correlated to any material markers in texts and lies in complex linguistic notions [START_REF] Villavicencio | Learning to distinguish pp arguments from adjuncts[END_REF][START_REF] Merlo | Generalised pp-attachment disambiguation using corpus-based linguistic diagnostics[END_REF]. The availability of large-coverage lexicons providing lexical, syntactic and semantic information is essential in order to gain insight on the recognition and analysis of adverbs, including the dual problems of variability and ambiguity (Laporte and Voyatzi, 2008). In addition, it is likely to help solving prepositional phrase attachment during shallow or deep parsing [START_REF] Agirre | Improving parsing and pp attachment performance with sense information[END_REF]. In this paper, we propose a method for extending the coverage of the French morphological lexicon DELA (Dictionnaires Electroniques du LADL / LADL electronic dictionaries) with respect to adverbial entries. These adverbs were extracted from LGLex, a NLP-oriented syntactic resource for French, which in its turn contains all adverbs extracted from the Lexicon-Grammar tables (hereafter LG tables) of both simple adverbs ending in -ment '-ly' [START_REF] Molinier | Grammaire des adverbes : description des formes en -ment[END_REF] and compound adverbs (Gross, 1986b;Gross, 1986a). The paper is organized as follows. Section 2. provides an overview of the resources used in our work. Section 3. describes the method used to enhance DELA thanks to the integration of the new entries of LGLex, which are extracted from LG tables. Section 4. is focused on the generation of complex adverbial entries with variables. In section 5., we report and discuss the obtained results. Finally, in sec-tion 6. we point out several possible extensions and avenues for future research.

Resources

The LG tables of adverbs

LG tables are currently one of the major sources of lexical and syntactic information for the French language 1 . Their development was initiated as early as the 1970s by Maurice Gross, at the LADL [START_REF] Gross | Méthodes en syntaxe : Régimes des constructions complétives[END_REF], and then at the LIGM, University Paris-Est [START_REF] Boons | La structure des phrases simples en franc ¸ais : Constructions intransitives[END_REF][START_REF] Guillet | La structure des phrases simples en franc ¸ais : Les constructions transitives locatives[END_REF]. Lexical information is represented as tables. Each table puts together elements of a given lexical-grammatical category (for a given language) that share a certain number of defining features, which usually concern subcategorization information. These elements form a class. Tables are represented as matrices: each row corresponds to a lexical item of the corresponding class; each column lists a feature that may be valid or not for the different members of the class; at the intersection of a row and a column, the symbol + (resp. -) indicates that the feature corresponding to the column is valid (resp. not valid) for the lexical entry corresponding to the row. The resources described in this paper correspond to the LG tables of both simple and compound adverbs, in which previously implicit features have been made explicit 2 for more convenient use in NLP. All tables are fully available 3 under 1 LG tables are available in several languages. With regard to adverbs, LG tables exist in English (Gross, 1986b), German (Seelbach, 1990), Spanish [START_REF] Blanco | Quelques remarques sur un dictionnaire électronique d'adverbes composés en espagnol[END_REF]Català, 1998), Italian (Gioia, 2001), Portuguese [START_REF] Baptista | Some families of compound temporal adverbs in portuguese[END_REF], Korean (Jung, 2005) and Modern Greek [START_REF] Voyatzi | Description morpho-syntaxique et sémantique des adverbes figés en vue d'un système d'analyse automatique des textes grecs[END_REF].

2 In order to make previous implicit features explicit, a table of classes has been created [START_REF] Tolone | Les tables du Lexique-Grammaire au format TAL[END_REF][START_REF] Tolone | Analyse syntaxique à l'aide des tables du Lexique-Grammaire du franc ¸ais[END_REF]. Its role is to assign features when their value is constant over a class, e.g. class definition features. Each row stands for a class and each column stands for a feature. Each cell corresponds to the validity of a feature in a class. In particular, the table of French adverb classes is composed of 32 different classes and 164 features.

3 http://infolingu.univ-mlv.fr/english > Language Resources > Lexicon-Grammar > Download.

a free license (LGPL-LR). In French, there are two resources of adverbs that follow different principles both in classification and in representation within the Lexicon-Grammar framework (Tolone et al., 2010). That is, first, tables of simple adverbs ending in -ment '-ly' [START_REF] Moliner | Étude syntaxique et sémantique des adverbes de manière en -ment[END_REF][START_REF] Molinier | Grammaire des adverbes : description des formes en -ment[END_REF], which are mainly derived from adjectives and, secondly, tables of compound adverbs4 (Gross, 1986b;Gross, 1986a). In both tables, there are encoded 3,203 simple and 7,284 compound entries assuming an adverbial function in discourse.

Figure 1

The syntactic lexicon LGLex

The current version of French LG tables enables the use of their lexical data in NLP tools [START_REF] Tolone | Les tables du Lexique-Grammaire au format TAL[END_REF]. To this end, the tables have been converted into an interchange format, based on the same linguistic concepts as those handled in the tables. This conversion is based on LGExtract: a generic tool for generating a syntactic lexicon for NLP from the LG tables [START_REF] Constant | A generic tool to generate a for NLP from Lexicon-Grammar tables[END_REF]. It relies, first off, on a global table of classes in which we added the missing features and, second, on a single extraction script including all operations related to each feature to be performed for all tables. Thanks to LGExtract, a French lexicon for NLP has been generated from all LG tables and for most lexicalgrammatical categories: verbs, predicative nouns, idioms and adverbs. This syntactic lexicon is named LGLex (Constant and Tolone, 2010;[START_REF] Tolone | Analyse syntaxique à l'aide des tables du Lexique-Grammaire du franc ¸ais[END_REF]. It is manually evaluated and freely available5 under the LGPL-LR license in both plain text format and XML. Each entry of the lexicon includes three sections:

1. section Lexical-information identifies the lexical entry and gives the category of each lexical component.

For instance, the flexible entry à échéance :Adj (including the adverb à échéance courte 'short-term'), which is encoded in table PCA (see 2.1.), contains the categories Pr ép, C and Adj (Det and Modif pr é-adj are empty). We added the information of paraphrases, other structures and other entries with intensification;

2. section Arguments gives information about the arguments of the predicate: for instance, the subject argument N0, assigned to the predicate that may be modified by the entry à échéance :Adj is a human or a non human noun phrase, represented by N0 =: Nhum and N0 =: N-hum;

3. section Constructions enumerates the identifiers of all constructions of the lexical entry (e.g. N0 V Adv W or Adv parlant, P) 6 and of all internal morphosyntactic structures, that is Adv for all simple adverbs or Pr ép Det C Modif pr é-adj Adj for compound adverbs like à échéance :Adj, but also Pr ép Det Modif pr é-adj Adj C for its variant with the permutation of the noun C, e.g. à :Adj échéance (including the adverb à courte échéance 'short-term'), and Pr ép Det C for its variant without prepositional noun phrase modifier, e.g. à échéance 'at expiry date'.

LGLex is currently composed of 13,872 verbal entries (from 67 tables), 14,271 nominal entries (from 81 tables), 39,628 idioms (from 69 tables) and 10,492 adverbial entries (from 32 tables) of which 3,207 are simple adverbs (from 16 tables) and 7,285 are compound adverbs (from 16 tables).

In order to enrich the French DELA, we first extended

LGLex with respect to adverbial entries by using various types of features that are encoded in the tables of both simple and compound adverbs [START_REF] Tolone | Extending the adverbial coverage of a NLP oriented resource for French[END_REF]. We added 11,351 entries (+108%), so the lexicon is now composed of 21,843 adverbial entries in total.

The morphological lexicon DELA

The French morphological lexicon DELA (Dictionnaires Électroniques du LADL / LADL electronic dictionaries) describes the simple and compound lexical units of French and provides the corresponding grammatical, semantic and inflectional information. It is freely available 7 , and is currently composed of 683,824 simple entries and 108,436 entries. Each entry is represented in its canonical form and contains an inflectional code which allows to automatically generate all inflected forms of the entry using a graph. When we process a corpus with the Unitex software 8 we can apply directly the DELA lexicon. Unitex generates the lexicon of all inflected forms (called DELAF) present in texts; then, it tags them.

An entry of a DELAF is a line of text that ends in a newline and conforms to the following syntax: paresseuse,paresseux.A+d+z1:fs 'lazy'

The different elements of this line are:

• paresseuse is the inflected form of the entry; it is mandatory; paresseux is the canonical form (lemma) of the entry. For nouns and adjectives (in French), it is usually the masculine singular form; for verbs, it is the infinitive. This information may be left out as in the following example:

paresseux,.A+d+z1:ms

This means that the canonical form is the same as the inflected form. The canonical form is separated from the inflected form by a comma.

• A+d+z1 is the sequence of grammatical and semantic information. In our example, A designates an adjective and d indicates that the adjective occurs after the noun.

7 http://infolingu.univ-mlv.fr/english > Language Resources > Dictionaries > Download.

8 http://igm.univ-mlv.fr/ ˜unitex/

The codes +z1, +z2 and +z3 indicate the language register (this information is optional): +z1 is used for general language (for example, blague 'joke'), +z2 for specialized language (for example, disquette 'floppy disk') and +z3 for very specialized (or technical) language (for example, sérialisation 'serialization').

Each entry must have at least one grammatical or semantic code, separated from the canonical form by a period. If there are more codes, these are separated by the + character.

• :fs is an inflectional code which indicates that the noun is feminine singular. Inflectional codes are used to describe gender, number, declination, and conjugation. This information is optional. An inflectional code is made up of one or more characters that represent one information each. Inflectional codes have to be separated by the : character, for instance in an entry like the following:

adverses,adverse.A+d+z1:mp:fp 'opposite'

The : character is interpreted as a logical OR. Thus, :mp:fp means "masculine plural" or "feminine plural".

Extending DELA

We completed DELA with the new adverbial entries added in LGLex in order to evaluate their accuracy by means of both a corpus-annotation practice and a detailed comparison with related work by (Laporte et al., 2008). Using this method, we managed to increase the number of the adverbial entries in the morphological lexicon.

Adverbial variants in LGLex

The first step of our method consisted of representing various types of features, already encoded in LG tables, as variants of the adverbial entries present in LGLex. These features describe paraphrases ( à Adv parler, P or N0 V W de (fac ¸on+mani ère) Adj), substructures (Pr ép1 Det1 C1 derived from the basic structure Pr ép1 Det1 C1 Pr ép2 C2), or intensified structures (plus Adv) 9 . 9 Entries with intensification are included in the lexicon when their combination does not seem to obey regular rules of syntactic and semantic compositionality. For instance, particulièrement 'particularly' is a quantifier in the following example and, thus, refuses intensification: Elle est (*très+*plus) particulièrement grande pour son âge 'She is (*very+*more) particularly tall for her age'. On the contrary, it can be intensified when it is used as a focus adverb: Les adolescents, (tout+plus) particulièrement les filles, sont grands pour leur âge 'Teens, (more) particularly girls, are tall for their age'. However, this is not the case of the focus adverbs essentiellement 'essentially', principalement 'basically', notamment 'notably'. Thus, plus particulièrement 'more particularly' is considered as an idiomatic compound, as opposed to an open series of identical forms with a different syntax. It is represented in LGLex as a variant of the focus adverbial entry particulièrement 'particularly'.

Hence we added the following fields to lexical-info :

• paraphrases, for instance, à franchement parler 'frankly speaking' and de (manière+fac ¸on) franche 'in a frank way' for the adverb franchement 'frankly' ;

• other-structures, for instance, jusqu'à la fin 'until the end' for the adverb jusqu'à la fin des (=de les) temps 'until the end of time' ;

• other-ID, refering to other entries with intensification, for instance, plus particulièrement 'more particularly' for the adverb particulièrement 'particularly'.

The following example is taken from the lexicon LGLex10 and concerns the adverbial entry paresseusement 'lazily', which is encoded in table ADVMS of subject oriented manner adverbs. Four new adverbs were added in section lexical-info and are represented as paraphrases, followed by their internal morphosyntactic structure in struc-tureAdv of section all-constructions :

ID=P_advms_643;status=completed lexical-info= [cat="adv", exprF=[adv=[notperm=[complete= "paresseusement"]]], paraphrases=(adv="de facon <paresseux>", adv="d'une facon <paresseux>", adv="de maniere <paresseux>", adv="d'une maniere <paresseux>"), autres-ID=(), autres-structures=()] args=() all-constructions=[ structureAdv=(construction="base::Adv", construction="o::de facon Adj", construction="o::d'une facon Adj", construction="o::de maniere Adj", construction="o::d'une maniere Adj"), absolute=(construction="true::N0 V Adv W", construction="true::Adv, N0 V W", construction="o::N0hum V W de (E+une) (facon+maniere) Adj"), relative=()]

In LGLex, adverbial variants do not form new entries, but are only considered and represented as variants. If one wants to find in texts the various forms of an adverb using DELA, they cannot do so unless these forms are added in the lexicon and are associated to a canonical (or standard) form.

Conversion into DELA format

In order to produce adverbial entries in DELA format from the adverbial variants that have been added in LGLex11 , we followed the following steps:

• First, a few specific treatments had to be performed only once. For example, the paraphrasal feature du point de vue de Ddef Ndomaine 'from the point of view of Ndomain', present in table ADVPM of viewpoint adverbs, requires to encode explicitely the definite determiner (Ddef) associated to the domain noun (Ndomaine), which is encoded in the table. To do so, we added and encoded a new column for Ddef;

• Second, we retrieve the forms of the lexicalgrammatical categories that are described inside angle brackets in various constructions from DELAF. For instance, all verbal manner adverbs ending in -ment '-ly' (e.g. paresseusement 'lazily') have the paraphrase de (E+une) (fac ¸on+mani ère) Adj 'in a Adjective way'. In order to reconstruct the adverbial entry, it is necessary to associate to each simple verbal manner adverb the correspondent adjective (Adj) in the feminine singular form (here, paresseuse, cf. 2.3.);

• Third, we extract the list of adverbial variants from the three following fields present in LGLex: paraphrases, other-structures and other-ID;

• Then, we must do a few language-dependent substitutions which are of common use in French. For instance, in jusqu'à la fin de les temps 'until the end of time', the preposition de 'of' and the definite determiner les have to be merged into a single form, and so de les becomes des;

• Finally, we convert this list into DELA format, linking all variants of a given adverb to its canonical (or standard) form (complete), and specifying the grammatical category ADV, followed by the name of the morphosyntactic or syntactico-semantic class (found in ID) in which it is encoded. All this information is provided in LGLex (cf. 3.1.).

For instance, the four variants associated in LGLex to the adverbial entry paresseusement 'lazily' (see 3.1.), produce in DELA format:

de fac ¸on paresseuse, paresseusement.ADV+advms d'une fac ¸on paresseuse, paresseusement.ADV+advms de manière paresseuse, paresseusement.ADV+advms d'une manière paresseuse, paresseusement.ADV+advms 'in a lazy way' Moreover, we added the 24 missing adjectives in DELA12 followed by their inflectional code. This allowed us to construct the corresponding variants with the feminine form of the adjectives, such as:

de fac ¸on abrégée,abrégément.ADV+advmv 'concisely'

Then, we filtered the entries in order to remove duplicates. In fact, some entries are exactly the same. We have 20,757 entries without variables and 830 entries with variables. The first step consists in integrating all entries without variables in the existing DELA. See the section 4. for the treatment of entries with variables.

3.3. Integration in the existing DELA All these variants need to be added to the existing DELA, which currently contains 9,036 entries, after merging the two lists and verifying that no variant is already present in DELA. In order to produce the new DELA of adverbs, we followed the following steps:

• First, we removed from the existing DELA the adverbial entries of the morpho-syntactic class PECO as they describe idiomatic adjectival phrases rather than adverbs:

lent comme une tortue,.ADV+PECO+z1 'slow as a turtle'

• Then, we lowercased the LG We sorted and compared all pairs of entries in order to detect any differencies between them with respect to the language register;

• Finally, we merged the entries encoded in the 16 LG tables of simple adverbs with the relevant entries present in the existing DELA in order to add the information about the language register. For instance, abominablement 'abominably' already exists in current DELA but is followed only by the language register code +z1; after merging the two entries, we obtained a complete new entry:

Graph dictionary

There are 830 entries with variables13 (cf. 3.2.) which are represented in the 16 LG tables of compound adverbs.

In order to produce dictionary entries from these data we used what we call a graph dictionary. This is a sort of transducer which calls upon subgraphs and so it is capable to dynamically produce new dictionary entries in a format similar to that of DELA.

Variables present in adverbial entries can be expressed, according to their type, in two different ways:

• Variables such as Adj (adjective), N (noun) and N-hum (non human noun) are represented in the form of lexical masks, respectively <A>, <N> and <N∼hum>. They use lexical items previously recognized by other resources (such as dictionaries or graph dictionaries);

• All other variables are transformed into a call to a subgraph of the same name. So, we created 171 subgraphs (they are empty by default but some subgraphs already exist). For example, variable :DNUM is transformed into a call to the subgraph DNUM that recognizes numerical determiners whether they are expressed in numbers or letters.

For instance, the entries described in the end of section 3.2. are represented as follows:

par temps <A>,.ADV+pca à <A> échéance,à échéance <A>.ADV+pca à :DNUM franc près,.ADV+pca

The graph dictionary corresponding to these entries is given in the Figure 2. The compact notation ",." (cf. 2.3.) indicates that the inflected form and the canonical form are identical. The notation <franc> refers to all inflected forms of the noun. The boxes "A(" and ")A" allow to capture the value of the adjective in order to reproduce it inside the canonical form given in output by "$A$". The grey box represents a call to a subgraph.

During a corpus processing, the analysis of the sequences par temps sec 'in dry weather', par temps pluvieux 'in rainy weather', à brève échéance 'in the short term' and à cinq francs près 'to the very last penny' produce the four following entries in DELA format: par temps sec,.ADV+pca par temps pluvieux,.ADV+pca à brève échéance,à échéance brève.ADV+pca à cinq francs près,.ADV+pca

In order to generate dictionary entries in DELA format from the LGLex adverbial entries with variables we need to create manually all 171 subgraphs. Some subgraphs already exist: DNUM, Ntps, POSS.

Results

Table 1 We enhanced the DELA with 13,528 entries, and we have 830 entries with variables which enable the generation of more entries by using the graph dictionary method. The results are quite satisfactory as we obtain 150% new entries in the lexicon only by exploiting precise linguistic information of high coverage, which is freely available in existing resources.

Conclusion and future work

At a time when the lack of large-scale lexical syntactic resources for French impedes on NLP research, we have shown the interest of using fine-grained linguistic information, which is provided in existing resources, in order to enrich or diversify their content. This work led to an increase of 150% of the adverbial entries in DELA, and this method is being tested in other languages such as Modern Greek [START_REF] Voyatzi | Description morpho-syntaxique et sémantique des adverbes figés en vue d'un système d'analyse automatique des textes grecs[END_REF]. These encouraging results confirm it is worthwhile exploiting features such as paraphrases. Therefore, we plan to complete the LG tables in that direction, starting, for example, with the table of verbal manner adverbs: Adj-ment = en tout Nabstrait =: amicalement = en toute amitié 'friendly' 'in all friendship' Adj-ment = par Nmoyen communication =: téléphoniquement = par téléphone 'by telephone' 'by telephone' Furthermore, we plan to convert the new adverbial entries into the Lefff format [START_REF] Sagot | The Lefff, a freely available and large-coverage morphological and syntactic lexicon for French[END_REF], in order to integrate them into a parser, following similar work by [START_REF] Tolone | Using Lexicon-Grammar tables for French verbs in a large-coverage parser[END_REF] and [START_REF] Tolone | Analyse syntaxique à l'aide des tables du Lexique-Grammaire du franc ¸ais[END_REF]. Finally, we can also consider enhancing the French Wordnet with respect to adverbial entries [START_REF] Sagot | Extending the adverbial coverage of a french wordnet[END_REF].

Figure 1 :

 1 Figure 1: Compound adverbs of table PCA

Figure 2 :

 2 Figure 2: Example of a graph dictionary

  displays a sample of the table PCA which is defined by the morphosyntactic structure Preposition, Determiner, Constrained noun, Pre-adjectival Modifier, Adjective. In this table, each row corresponds to a lexical item with adverbial function, and each column corresponds to:• one of the components in the morphosyntactic structure of the items, i.e. features with identifiers Pr ép, Det, C, Modif pr é-adj, and Adj;

	• a syntactic feature holding binary values, for ex-
	ample: Pr ép Det Modif pr é-adj Adj C describes the
	possible permutation (without loss of information)
	of the adjectival phrase represented in this table as
	Modif pr é-adj Adj; moreover, Neg obl encodes the
	constraint that the adverbial occurs obligatorily in a
	negative clause;
	• a semantic feature holding binary values, for instance,
	Conjonction points out whether the compound adverb
	has a connector function in discourse, i.e. it links the
	clause in which it occurs with the previous clause as,
	for example, dans le cas contraire 'otherwise';
	• an item of information provided as an aid to help hu-
	man readers find examples of sentences containing the
	compound adverb: features with identifiers Ppv and
	Pr édicat type give an example of a verbal predicate
	that combines commonly with the adverb.
	Some entries in the table are flexible, like à échéance :Adj,
	which contains the variable :Adj (adjective) in order to
	recognize adverbs like à échéance courte 'short-term',	à
	échéance longue 'long-term', etc.

  For example, ces temps derniers 'recently', defined by the morphosyntactic structure Pr ép Det C Modif pr é-adj Adj and encoded in table PCA, can also take the form ces derniers temps due to the permutation of the adjective derniers 'recent'. This latter is already encoded in table PAC:We can also evoke the case of the deleting relation that associates different adverbs of table PCDC. For example, the adverbs dans l'état actuel des choses 'in the current state of things' and dans l'état actuel des connaissances 'in the current state of knowledge' are both encoded in table PCDC. They both accept the substructure dans l'état actuel 'in the current state', which is obtained after deletion of the prepositional noun phrase modifiers des choses 'of things' and des connaissances 'of knowledge', and without loss of information:Last, some errors in the new entries are due to the way initial adverbial entries are encoded in tables. Considering the adverb à cette heure-ci 'at the present time' represented in table PCA: the noun component heure 'time' is encoded together with the hyphen, and thus form an amalgam that is automatically reproduced in the substructure à cette heure-'at this time'. So, we deleted hyphens:

	Some entries contain variables like :Adj (adjective) or
	:DNUM (numerical determiner) (see 2.1.):
	par temps :Adj,.ADV+pca
	à :Adj échéance,à échéance :Adj.ADV+pca
	à :DNUM franc près,.ADV+pca
	ces temps derniers,.ADV+pca
	ces derniers temps,ces temps derniers.ADV+pca
	ces derniers temps,.ADV+pac
	dans l'état actuel des choses,.ADV+pcdc
	dans l'état actuel,dans l'état actuel des choses.ADV+pcdc
	dans l'état actuel des connaissances,.ADV+pcdc
	dans l'état actuel,dans l'état actuel des connais-
	sances.ADV+pcdc
	In fact, each substructure provides information about
	the corresponding entry in LGLex, and so the generated
	substructures in DELA format are filtered automatically in
	order to delete duplicates. We obtained 21,587 new entries
	(22,850 new entries before removing duplicates) in DELA
	format.
	à cette heure-ci,.ADV+pca
	à cette heure-,à cette heure-ci.ADV+pca
	In addition, some spaces had to be deleted. We mention
	the previous example, written à cette heure-ci, or the entry
	y compris 'including' ending by a space:
	y compris ,.ADV+pcpn
	These corrections are necessary for two reasons: to im-
	prove the quality of entries and to compare with the current
	version of DELA.

Table 1 :

 1 shows the number of the initial adverbial entries in DELA and the new entries extracted from LGLex: Number of entries in DELA

	without	with
	variables variables

According to(Laporte and Voyatzi, 2008) (p. 31) "a phrase composed of several words is considered to be a multiword expression if some or all of its components are tied together, that is, if their combination does not obey productive rules of syntactic and semantic compositionality". This criterion ensures a complementarity between lexicon and grammar. In other words, it tends to ensure that any combination of linguistic elements which is correct in the language, but is not represented in common syntacticsemantic grammars, should be stored in lexicons.

http://infolingu.univ-mlv.fr/english > Language Resources > Lexicon-Grammar > Download.

Symbols with obvious interpretation are used such as: Pr ép (preposition), Det (determiner), Adj (adjective), Modif pr é-adj (pre-adjectival modifier), N (noun), V (verb), Conj (conjunction), W (a range of verbal complements), and C (noun tied with the rest of the adverbial structure).

This is an extract of the version 3.4 that takes into account the paraphrases which are encoded in LG tables, notably, the associated lexical features.

 11 Another possibility is to construct graphs used in Unitex to generate these variants in DELA format. But graphs must be constructed manually. So, it is better to generate new entries with these paraphrases and other structures that have been already added in LGLex rather than create graphs to generate them.

These adjectives are: abrégé, affaireux, amiteux, amitieux, aucun, bijournalier, componctueux, considéré, drôlatique, exaspéré, flâneux, gent, goujat, indévot, intra-utérin, irrévérent, marmiteux, maupiteux, médiatif, peineux, plaignard, recta, révérent, salaud.

We corrected the names of variables directly in the tables of LG in order to delete duplicates. So, we have 174 differents names of variables in total.


Elsa Tolone
email: elsa.tolone@univ-paris-est.fr

Benoît Sagot
email: benoit.sagot@inria.fr

Éric Villemonte

La Clergerie
email: eric.de_la_clergerie@inria.fr

Evaluating and improving syntactic lexica by plugging them within a parser

Keywords: syntactic lexica, parsing, error mining

We present some evaluation results for four French syntactic lexica, obtained through their conversion to the Alexina format used by the Lefff lexicon (Sagot, 2010), and their integration within the large-coverage TAG-based FRMG parser (de La Clergerie, 2005). The evaluations are run on two test corpora, annotated with two distinct annotation formats, namely EASy/Passage chunks and relations and CoNLL dependencies. The information provided by the evaluation results provide valuable feedback about the four lexica. Moreover, when coupled with error mining techniques, they allow us to identify how these lexica might be improved.

Introduction

The development of a large-scale symbolic parsing system is generally achieved by co-developing its various components in a consistent way, in order to ensure good integration and consistency. This is the case, for example, concerning the development of the French syntactic lexicon Lefff [START_REF] Sagot | The Lefff, a freely available and large-coverage morphological and syntactic lexicon for French[END_REF] together with that of the FRMG grammar and parser for French [START_REF] De | From metagrammars to factorized TAG/TIG parsers[END_REF]. However, several syntactic lexica have been developed for French, sometimes for decades. An example thereof are Lexicon-Grammar tables [START_REF] Gross | Méthodes en syntaxe : Régimes des constructions complétives[END_REF]Boons et al., 1976b), although they were built with no or poor integration in large-scale NLP systems. Such a resource contains rich and valuable information, but its usefulness for real NLP applications has never been thouroughly assessed. This is also the case for the valency lexicon DICOVALENCE (van den [START_REF] Van Den Eynde | Le dictionnaire de valence DICOVALENCE : manuel d'utilisation[END_REF], although to a lesser extent. Therefore, after converting these resources in the same lexical model as the Lefff, we performed a preliminary integration of these resources within FRMG. We also evaluated NewLefff, a new experimental version of the Lefff that benefits, among other things, from its merging with DICO-VALENCE [START_REF] Sagot | Merging syntactic lexica: the case for French verbs[END_REF]. Following previous preliminary results (Tolone et al., 2011), we were able to evaluate these lexica on two reference corpora based on two different annotation schemes, namely the EasyDev corpus [START_REF] Paroubek | The PASSAGE syntactic representation[END_REF] and on the CONLL dependency version of the French TreeBank [START_REF] Candito | Statistical french dependency parsing: treebank conversion and first results[END_REF]. As a side effect, these experiments also show that it is possible to switch lexica in a lexicalized parser like FRMG, at the cost of a relatively small decrease in performances.

Lexica

The Lefff

Our reference and baseline lexicon is Lefff [START_REF] Sagot | The Lefff, a freely available and large-coverage morphological and syntactic lexicon for French[END_REF], a large coverage morphosyntactic and syntactic lexicon for French. 1 As mentioned before, Lefff was specifically de-1 Freely available at http://gforge.inria.fr/projects/alexina/ veloped for NLP tasks, and in particular to be used in conjunction with the FRMG parser. The Lefff relies on the Alexina framework for the acquisition and modeling of morphological and syntactic lexica. To represent lexical information, an Alexina lexicon relies on a two-level architecture:

• the intensional lexicon associates (among others) an inflection table and a canonical sub-categorization frame with each entry and lists all possible redistributions from this frame;

• the compilation of the intensional lexicon into an extensional lexicon builds different entries for each inflected form of the lemma and every possible redistribution.

The current version of the Lefff (version 3.1)2 contains only one entry for the lemma vérifier 'verify', 'validate'. Here is a simplified version of this entry:

vérifier1 Lemma;v;<Suj:cln|sn, Obj:(cla|qcompl|scompl|sinf|sn)>; @CtrlSujObj, @CompSubj cat=v; %ppp employé comme adj,%actif,%passif, %se moyen impersonnel,%passif impersonnel

It describes a transitive verb entry whose arguments have the syntactic functions Suj and Obj listed between angle brackets. 3 The subject might be realized as a nominative clitic (cln) or a noun phrase (sn), whereas the direct object can be realized as an accusative clitic (cla), a noun phrase (sn), an infinitive (sinf, with a control phenomenon expressed by @CtrlSujObj), a finite clause (scompl, in the subjunctive mood because of @CompSubj) or an indirect interrogative clause (qcompl). Finally, this verb entry allows for the functional redistributions past participle used as an adjective, active (the default distribution), impersonal middle-voice "se" construction, impersonal passive, and passive.

The Lefff 3.1 contains 7,108 verbal entries corresponding to 6,827 distinct lemmas, and 112,118 entries covering all other categories. Detailed figures are given in Table 1. 4

Other syntactic lexica

Besides Lefff, we have considered three other lexica, whose verbal entries are combined with the non-verbal Lefff entries:

• LGLex: 5 this lexicon results from a two-step conversion of the Lexicon-Grammar tables [START_REF] Gross | Méthodes en syntaxe : Régimes des constructions complétives[END_REF]Boons et al., 1976b), a rich syntactic lexical resource developed over several decades although not with an NLP orientation. A first conversion was made to get a fully electronic version of this lexicon into the LGLex format [START_REF] Constant | A generic tool to generate a lexicon for NLP from Lexicon-Grammar tables[END_REF], opening the way to a second conversion to the Alexina format (Tolone and Sagot, 2011). The result is a wide-coverage lexicon with, often, many entries for each verbal lemma, associated to several meanings and valency frames.

LGLex contains 13,867 verbal entries corresponding to 5,738 distinct lemmas, as well 12,696 entries for predicative nouns corresponding to 8,531 distinct nominal lemmas;

• DICOVALENCE: 6 this lexicon (van den [START_REF] Van Den Eynde | Le dictionnaire de valence DICOVALENCE : manuel d'utilisation[END_REF] follows the Pronominal Approach [START_REF] Blanche-Benveniste | Pronom et syntaxe. L'approche pronominale et son application au franc ¸ais[END_REF] for characterizing verb valency frames and for defining fine-grained entries (several entries per lemma). This mediumcoverage resource contains 8,313 verbal entries corresponding to 3,738 distinct lemmas;

• NewLefff: 7 this experimental new version of Lefff targets more semantically-oriented finer-grained entries, while still preserving Lefff's wide coverage. This lexicon is the result of two extension steps: (1) the automatic extraction, interpretation, conversion and integration or merging of denominal and deadjectival verbal entries in -iser and -ifier from the LVF lexicon [START_REF] Sagot | Description et analyse des verbes désadjectivaux et dénominaux en -ifier et -iser. Arena Romanistica[END_REF]; (2) the automatic merging of Lefff entries and DICOVALENCE entries through the comparison of their valency frames [START_REF] Sagot | Merging syntactic lexica: the case for French verbs[END_REF], then completed by a phase of manual validation on the 100 most frequent lemmas and on all dubious lemma 8 (those lemma who got more entries 4 The category "other" includes all kinds of conjunctions, determiners, interjections, punctuation marks, pronouns, prefixes and suffixes, as well as special entries for named entities and unknown words.

5 Freely available at http://infolingu.univ-mlv.fr/english/ > Language Resources > Lexicon-Grammar > Download 6 Freely available at http://bach.arts.kuleuven.be/dicovalence/ 7 Freely available at http://gforge.inria.fr/projects/alexina/. For obtaining NewLefff, v new files in the Lefff 3.1 package must be compiled and used instead of v and v-phd files.

8 505 verbal lemmas, corresponding to 986 entries.

than originally in both input lexica). NewLefff contains 12,613 verbal entries corresponding to 7,933 distinct lemmas.

Table 2 provides some figures about the coverage and granularity of each lexicon for verbal entries, showing that our four lexica actually cover an diverse spectrum of configurations (medium to large coverage, small to large granularity). For the experiments, all lexica use the non-verbal entries of Lefff in addition of their own verbal entries. In consequence, the differences between the lexica arise from the verbal entries.

Lexica

FRMG

FRMG 9 (de La Clergerie, 2005) is a large-coverage symbolic grammar and parser for French. In fact, the acronym FRMG denotes resources that cover several representation levels. The most abstract level corresponds to a linguistically motivated modular and hierarchical meta-grammar. This meta-grammar is used to generate a compact (featurebased) Tree Adjoining Grammar (TAG) [START_REF] Aravind | Tree Adjunct Grammars[END_REF] containing around 300 factorized elementary trees, including only 35 trees anchored by verbs. Despite its compactness, the grammar exhibits a wide coverage, thanks to factorization operators, such as disjunction and guards, used in the trees to allow many possible tree traversals.

The grammar is compiled into an efficient chart-based parser, also named FRMG, which is able to return both full parses (whenever possible) or sequences of partial parses (otherwise) as shared dependency forests. The forests may then be disambiguated using heuristic-based rules to get the best dependency trees. Finally, these trees may be converted to various output formats, including the EASy format and the CONLL format. FRMG benefits from the extended domain of locality provided by TAG trees, with, for instance, the possibility to capture all the components of a verb valency frame through the nodes of a single elementary tree. However, it also implies that a TAG grammar like FRMG works best when coupled with a lexicon that provides such rich lexical information. There is also the need to propagate this information from the words to the trees. Concretely, each tree of FRMG is associated with an hypertag [START_REF] Kinyon | Hypertags[END_REF], a feature-structure resuming (in the case of a verbal tree) the various frames and argument realizations covered by the tree. Similarly, each verbal entry (but this is also true for other categories) has an hypertag derived from its lexical information. Anchoring a verbal tree by a verbal entry involves the unification of both hypertags. Figure 1

                 arg0 fun suj kind subj | - pcas - arg1 fun obj kind obj | scomp | - pcas - arg2 fun objà kind prepobj | - pcas à | - refl - ctrsubj suj imp -                 

EASy Evaluation

Our first evaluation was conducted on the EasyDev corpus, a small corpus of around 4000 sentences used during the first EASy French parsing evaluation campaign and covering various document styles (journalistic, literacy, medical, mail, speech, etc.). The corpus is annotated with the EASy format [START_REF] Paroubek | Data, Annotations and Measures in EASy, the Evaluation Campaign for Parsers of French[END_REF][START_REF] Paroubek | The PASSAGE syntactic representation[END_REF], a mix of 6 kinds of chunks and 14 kinds of dependencies between forms or chunks, as illustrated by Figure 2 (with ovals for chunks and diamonds for dependencies). Table 3 shows the performances of the various lexica, on this Easy-Dev corpus. The coverage column indicates the rate of full parses (keeping in mind that the almost all remaining sentences get partial parses), and shows a clear decrease for DICOVALENCE and smaller ones for LGLex and NewLefff. We retrieve similar results in terms of F-measure on the chunks and dependencies. Finally, the fact that LGLex is both a wide-coverage and very fine-grained lexicon has a clear impact on parsing time. Figure 3 shows the F-measure for some of the EASy verbal dependencies, namely SUJ-V for the subject-verb relation, AUX-V for the auxiliary-verb relation, COD-V for the object-verb relation, CPL-V for the complement-verb relation (with no distinction between argument and adjuncts), and ATB-SO for the subject or object attributes. We have also added MOD-N for noun-modifiers, this relation being the most numerous one and being partly related to verbs through past and present participles on nouns. Again, we observe a slight decrease for all new lexica versus Lefff, more marked on some relations like COD-V for

LGLex and ATB-SO for LGLex and DICOVALENCE.

CoNLL Evaluation

For our second evaluation, we used the version of the French TreeBank (journalistic style) [START_REF] Abeillé | Building a treebank for French[END_REF] converted by [START_REF] Candito | Statistical french dependency parsing: treebank conversion and first results[END_REF] into the CoNLL dependency format, a format now largely used in international parsing evaluation campaigns [START_REF] Nivre | The CoNLL 2007 shared task on dependency parsing[END_REF]. This version of the French TreeBank has already been used to train and compare several statistical parsers [START_REF] Candito | Statistical french dependency parsing: treebank conversion and first results[END_REF], thus providing us baselines to evaluate FRMG and the various lexica. Note however that our results are still preliminary.

The CoNLL format relies on a fine-grained set of verbal dependencies, with in particular for verbal dependencies:

• the distinction between several kinds of auxiliaries: aux-tps (temporal auxiliaries), aux-pass (passive constructions), aux-caus (causative constructions);

SUJ-V AUX-V COD-V CPL-V ATB-SO MOD-N 0

Lefff NewLefff

LGLex DICOVALENCE Old DICOVALENCE Figure 3: F-mesures for some verbal EASy relations

• suj obj for the subjects and objects (but noting that the obj relation is also used in non-verbal cases);

• relations for the prepositional objects, with a obj for those introduced by à (to), de obj for those introduced by de (of ), and p obj for the remaining ones;

• relations for the attributes, with ato for the attributes of objects and ats for those of subjects;

• aff for affixes, actually verb clitics not covered by the above-mentioned relations;

• mod for verb modifiers such as adverbs (noting that again this relation is also used for non verbal cases)

Table 4 shows that all lexica got very good full parse coverage (on journalistic style) and emphasizes again the speed problems for LGLex (on relatively long and complex sentences, with a mean length average of 27 words vs 19.3 for EasyDev). Again, we note a slight decrease for the alternative lexica in terms of Labeled Attachment Score (LAS). We also note that all FRMG versions are still a few points below state-of-the-art statistical parsers, for instance MST [START_REF] Candito | Statistical french dependency parsing: treebank conversion and first results[END_REF]. A finer analysis in terms of recall and precision at the dependency level shows a contrasted landscape (Figure 4), with LGLex or more specifically NewLefff being sometimes better either in recall (aff, a obj) or precision (ato, aux caus). More generally, recall is relatively good but we observe precision problems. We conjecture that the finer granularity of LGLex and also of NewLefff tends to wrongly select rare valency frames for some medium to high frequency verbs, frames that are strongly favored by the heuristic-based FRMG disambiguation algorithm, leading to confusion between verb arguments (obj, a obj, de obj, p obj) and modifiers (mod, dep).

Cover. LAS Time

Lefff

LGLex NewLefff Dicovalence

Figure 4: CONLL verbal relations

Error Mining

The evaluation already provides interesting feedback to identify the strong and weak points of a lexicon (as illustrated by DICOVALENCE with the ATB-SO relation). However, to get information at the level of a verb or of an entry, we rely on error mining techniques (Sagot and Villemonte de La Clergerie, 2006). More precisely, the basic idea is to identify suspect lexical entries by mining the full parse failures on a large corpus, based on the following intuition:

A form is suspect if it occurs more often than expected in non-full-parsable sentences, in cooccurrence with non-suspect forms.

The mathematical formulation of this intuition leads to a fix-point iterative algorithm, close to EM (Expectation-Maximization), which may be used to also return, for each suspect w, a set of sentences in which w is suspected to be the cause of failure for a full parse. Suspects, their lexical entries, as well as associated sentences may then be browsed in a web interface to quickly identify the errors or lacks in the lexical entries. This idea and its implementation may be directly used to track the errors in any new lexicon L. However, it is also interesting to contrast L with Lefff, considered as reference, which may be achieved through a slight rephrasing, as follows:

A verb is suspect for lexicon L if it occurs more often than expected in sentences not full-parsable when using L but that received a full parse when using Lefff, in cooccurrence with non-suspect verbs.

The modified algorithm was then used on a larger corpus of 100K sentences (1.6M words), named CPJ (Corpus Passage Jouet), and comprising various style of documents (encyclopedic with Wikipedia, literacy with Wikisource, news with AFP, and discourse with Europarl). We have started exploiting the results for (former versions of)

LGLex and, to a lesser extent, NewLefff, and have identified several kinds of errors for some entries. The interest of this approach is that it can be applied to very large corpora, and we plan to do it, to overcome lexical data sparseness. Here, we provide some analysis of the data provided by the algorithm, with an emphasis on LGLex.

LGLex

We analyzed the first 15 suspicious verbs in LGLex in order to determine where the errors come from. We indicate the number of failed sentences for each verb between parentheses and we give one example. In total, there are 212 failed sentences for this selection of verbs:

• Some entries do not appear in the tables:

mixer 'mix' (7) in Mixé par Jimi Hazel, assisté de Bruce Calder, enregistré chez Jimi à l' "Electric Lady Studios" à New York: this entry is encoded in table 36S but with the meaning 'blend' (Max mixe les carottes (et+avec) les navets dans un mixeur). We added this entry to table 32PL (Max a mixé les sons), which has the defining feature <N0 V N1>, with an encoding similar to the entry mélanger 'blend' (Max a mélangé les (étiquettes+cartes+couleurs));

zapper 'omit' (4) in Elle a également "déploré" la mémoire de "plus en plus sélective" de la jeune femme, "qui zappe les détails qui font désordre": this entry appear in table 35L but with the meaning 'channel hop' (Max zappe de la 1ère et la 2ème chaîne). We added this entry to table 32R2 (Max a zappé un (repas+paragraphe)), which has the defining feature <N0 V N1>, with an encoding similar to the entry sauter 'skip', 'miss' (Max a sauté (un repas+une étape+une ligne));

réaffirmer 'reaffirm' (28) in Nous réaffirmons la nécessité de consulter les sans-abri et leurs or-ganisations sur les programmes européens pertinents. We can add the feature <re-V> for the entry affirmer 'affirm' of table 9 (Max a affirmé à Luc qu'il viendrait) 10 , with the defining feature <N0 V N1 à N1>, which accepts the construction <N0 V N1>, in order to recognize the entry réaffirmer; réélire 'reelect' (10) in Helmut Kohl est réélu au poste de Chancelier fédéral. We can add the feature <re-V> for the entry élire 'elect' of table 39 (On a élu Mac (E+comme) député), with the defining feature <N0 V N1 N2>, in order to recognize the entry réélire; the pronominal form se réimplanter 're-establish itself' (5) in Celles-ci cherchent toujours à se réimplanter dans la zone", a relevé M. Besson. The pronominal form s'implanter 'establish itself' does not appear either in the tables. We can add the feature <re-V> for the entry implanter 'establish' in table 38LD (On a implanté une usine dans cette région), with the defining feature <N0 V N1 Loc N2 destination>, in order to recognize the entry réimplanter 'reestablish'. Then, we can add the feature <se V> (or more precisely, <N1 se V W>) to accept the pronominal forms s'implanter and se réimplanter (Une usine s'est implantée dans la zone) 11 .

• Some entries appear in the tables but are not encoded (∼) or have been corrected:

susciter 'spark off' (41) in A d'autres niveaux, les propositions sur la table suscitent de sérieuses objections and recruter 'recruit' (14) in 80 intérimaires ont déjà été recrutés, pour assurer les commandes: these two entries appear in table 38R (Ceci a suscité une (vive réplique+réaction) chez Luc and Max a recruté Luc (comme+sur un poste de) lecteur), which has the defining feature <N0 V N1 Loc N2>, but are not encoded. This implies that no other construction are accepted, whereas the construction <N0 V N1> appear in the table and allows the erasure of the second complement. We encoded this construction as +; réprouver 'reprove', 'reprobate' (11) in Dieu ne réprouve donc personne: this entry is encoded 10 The entry affirmer appears also in table 32R3 Max a affirmé sa (position+résolution) but the difference is that it doesn't accept the intrinsic completive as here (Paul a affirmé (la nécessité de+qu'il était nécessaire de) venir).

11 We could add the entry s'implanter to table 35L but we don't because it shares exactly the same meaning with the entry implanter in table 38LD. This case is different between the entry fixer 'screw' in table 38LD (Max a fixé le tableau au mur (E+avec des vis)) and the entry se fixer 'settle' in table 35ST (Paul s'est fixé dans le midi), which have different meanings. We don't add the entry fixer to table 38LD with the meaning 'settle' because the transitive construction is not accepted (?(On+ceci) a fixé Paul dans le midi).

in table 12 (Max réprouve qu'Ida boive), which has the defining feature <N0 V N1 de N2>. When we modified the defining features [START_REF] Tolone | Analyse syntaxique à l'aide des tables du Lexique-Grammaire du franc ¸ais[END_REF], we replaced it by <N0 V N1>.

• Other entries are encoded in the tables but with obligatory complements which do not appear in the sentences of corpus:

- 38LHR (Le roi campe ses troupes dans la plaine), which has the defining feature <N0 V N1 Loc N2>, and it accepts the construction <N1 V W>. This means that an object N1 can become the subject of a sentence with conservation of its other objects. Therefore, this corresponds to the construction <N1 V Loc N2> (Ses troupes campent dans la plaine), without allowing the erasure of the locative complement.

• Last, some specific cases:

rediriger 'redirect' (50) in wrong sentences, such as deux cent cinquante-troisredirige ici;

consoler 'comfort' (6) in sentences with clitic pronominalization of the object, such as Elle essayait de le consoler: this entry is encoded in table 32R1 (Max console le chagrin de Luc), which has the defining feature <N0 V N1>, without allowing the clitic pronominalization of the first complement (we can add the feature like <N1 = Ppv =: le>).

The previous examples show (a) that some entries appear in the tables but are not encoded and therefore we have to encode them, and (b) that some other entries are missing, with several cases to be distinguished:

1. the entry should be added as a new entry. It can be a new verb or a different meaning of an existing verb (cf. mixer and zapper);

2. the entry re-V (or ré-V) has a meaning which can be derived by a "simple" use of the verb V (cf. réaffirmer and réélire): we should add a column <re-V> to all tables and encode it for all entries. Indeed, those without a derivable meaning have been added like another entry (entries which they do not mean faire une deuxième fois 'do twice'), such as revendre 'retail' that does not mean vendre une deuxième fois 'sell twice' or retomber 'come down' (La balle retombe) which is not tomber une deuxième fois 'fall twice';

3. the entry se V has a meaning which can be derived by a transitive use of the verb V (cf. s'implanter): we have at least 5 different cases (Boons et al., 1976a) (p.120-163), so we should add 5 columns <se V> to all tables and encode it for all entries: for instance, se regarder 'look at oneself' (Paul se regarde dans la glace), se mentir 'lie to one another' (Paul et Marie se mentent), s'étonner 'be surprised' (Paul s'étonne de mon silence), se laver 'wash' (Paul se lave les pieds), se manger 'eat' 'be served' (Le roti se mange froid). Indeed, only the intrinsic pronominals, which are not linked by a transitive use, have been added as a new entry: for instance, s'évanouir 'faint' (Paul s'est évanoui);

4. the entry se V and re-V is a combination of the two previous cases (cf. se réimplanter);

5. the entry dé-V has a meaning which can be derived by a verb V (no example in this selection of verbs): we should add a column <dé-V> to all tables and encode it for all entries. Indeed, only the uses different from faire l'action inverse 'do the opposite action' have been added like another entry. For instance, dévisser has an entry for the meaning 'fall' (L'alpiniste a dévissé), but not for the meaning 'unscrew' (dévisser une vis), which is the opposite action of 'screw on' (visser une vis).

As we show, some features are also missing, including some that encode the erasure or the clitic pronominalization of certain complements. Indeed, we can allow as erasures the complements that are in the defining features, and we should add these features to the appropriate tables and encode them for all entries in all tables. For instance, the entry donner 'give' in table 36DT can accept the erasure of all complements if the context allows it:

• <N0 V N1 Prep N2>: Paul donne du fric aux associations • <N0 V N1>: Paul donne du fric • <N0 V Prep N2>: Paul donne aux associations • <N0 V>: Paul donne souvent In conclusion, error mining enables us to detect and correct many errors in LGLex but we should manually analyze all 613 suspicious verbs in all 2,623 concerned sentences and all corrections have to be done manually, which represents an important (but rewarding) effort.

NewLefff

Error mining results on parses produced by FRMG with NewLefff on the large CPJ corpus were also investigated, although less thoroughly than for LGLex. It turned out that one verbal lemma was ranked by far the highest among all dubious lemmas, namely estimer 'consider'. Out of 569 sentences containing a form of this verb, as many as 200 could not receive a full parse. By looking at some of these failed sentences, we were able to quickly identify the following problem: the entry for estimer 'consider'12 lacked clausal realizations for the direct object (finite clause and infinitive clause). We also spotted a few other errors concerning s'attendre à 'expect' (missing control information), the attributive entry for savoir 'know' (missing clausal realizations for the object), inciter 'encourage, prompt' (missing infinitive realization for the subject), réitérer when meaning 'repeat' (missing clausal realization for the object), se résoudre à 'resolve to' (missing clitic and finite clause realizations for the indirect object), and others.

Conclusion

We have presented some preliminary but promising evaluation results for several lexica, obtained through their integration within a lexicalized deep TAG parser. Clearly, even if good, the results show that some efforts of adaptation remain to be done to improve the integration and to better exploit the richness of these lexica. Error mining techniques should help us to achieve this objective, and should also help us to identify the strong and weak points of each lexicon, which should lead to a new generation of better quality lexica, freely available and ready to use in large scale NLP systems. We would like also to mention very recent results showing that partially supervised learning techniques may be used to boost the performance of FRMG disambiguation to reach a LAS of 85.1% when using Lefff, to be compared with the 82.2% presented in this paper -and much closer to MST, a stochastic parser specifically trained on the French Treebank. It remains to be tested whether this improved disambiguator leads as such to similar gains when using the other lexica, or whether the learning phase has to be done for each of them.

Figure 1 :

 1 Figure 1: Hypertag for promettre 'promise'

Figure 2 :

 2 Figure 2: Sentence sample in Passage format 'Since a few weeks, relations between the two sides are deteriorating.'

Table 1 :

 1 Quantitative data about the Lefff hypertag associated with one of the entries for promettre 'promise' corresponding to the frame "(arg0) promises

	shows the

(arg1) to-(arg2)" with a control of (optional) arg1 by the subject arg0, and nominal or clausal realization for the (optional) object argument arg1.

Table 3 and

 3 Figure 3 also provide data for the first tried Alexina version of DICOVALENCE (dubbed "Old DICOVA-

	Lexicon	Cover. Chunks (%) (%)	Rels (%)	Time (s)
	Lefff	83.45	89.03 66.76	0.35
	NewLefff	82.19	88.74 66.09	0.55
	LGLex	80.61	87.89 63.19	1.10
	DICOVALENCE	71.44	88.08 64.49	0.38
	Old DICOVALENCE	65.69	87.06 62.72	0.42

LENCE"). We can observe a much weaker coverage and very poor performances for the ATB-SO relation. Because of these figures, we were led to investigate and correct the conversion script that generates the Alexina version of DI-COVALENCE, resulting in much better results parsing results.

Table 3 :

 3 Overall performances on EasyDev

Table 4 :

 4 Overall evaluation on French Tree Bank (test part)

	Lexicon	(%)	(%)	(s)
	Lefff NewLefff LGLex DICOVALENCE	89.53 82.21 88.76 81.36 86.73 78.75 75.28 79.38	0.61 0.94 1.95 0.69
	MST		-88.20	-

The Lefff

3.1 package includes v new files that are not considered as being part of the Lefff yet. In fact, replacing v and v-phd files by v new files leads to what is called NewLefff in this paper. See below for more details.3 The different syntactic functions used in the Lefff are: Suj (subject), Obj (direct object), Objà (indirect object canonically introduced by preposition "à"), Objde (indirect object canonically introduced by preposition "de"), Loc (locative), Dloc (delocative), Att (attribute), Obl or Obl2 (other oblique arguments).

Freely available at http://mgkit.gforge.inria.fr/

In NewLefff, estimer has three entries, that corresponds to the meanings 'consider', 'estimate' and 'esteem'.


E Sarrouy

O Dessombz
email: olivier.dessombz@ec-lyon.fr

J-J Sinou

Stochastic study of a non-linear self-excited system with friction

Keywords: friction, stochastic, polynomial chaos, eigenvalue problem, limit cycles

This paper proposes two methods based on the Polynomial Chaos to carry out the stochastic study of a self-excited non-linear system with friction which is commonly used to represent brake-squeal phenomenon. These methods are illustrated using three uncertain configurations and validated using comparison with Monte Carlo simulation results. First, the stability of the static equilibrium point is examined by computing stochastic eigenvalues. Then, for unstable ranges of the equilibrium point, a constrained harmonic balance method is developed to determine subsequent limit cycles in the deterministic case; it is then adapted to the stochastic case. This demonstrates the effectiveness of the methods to fit complex eigenmodes as well as limit cycles dispersion with a good accuracy.

Introduction

This work follows the study conducted in (Sinou and Jézéquel, 2007b) which investigated the influence of damping and friction coefficient on the stability of the equilibrium point using an analytical deterministic criterion.

The main results of this study are recalled in Section 2.1. Our objective is to propose a stochastic approach for such a system and to demonstrate the ability of the proposed method to handle systems with friction in a more general view. Brake systems are indeed hard to characterize accurately because of the high variability of operating conditions [START_REF] Ibrahim | Friction-induced vibration, chatter, squeal, and chaospart i: Mechanics of contact and friction[END_REF]. A stochastic approach seems then highly adequate for design of robust products. If Monte Carlo (MC) simulations can be carried out for small models, it is way too expensive for larger systems such as finite element models used in industrial context. Moreover, brake squeal arises as the static equilibrium becomes unstable [START_REF] Kinkaid | Automotive disc brake squeal[END_REF]. In this case, the system -being non-linearconverges toward limit cycles whose amplitude and frequency are unknown: these systems are part of the self-excited systems family. Characterization of these limit cycles is then an important point to be taken care of.

The method proposed herein, based on the expansion of stochastic elements on the Polynomial Chaos, is much less expensive than MC simulations when it comes to larger systems and does not share convergence radius problems as methods relying on Taylor series do [START_REF] Ghanem | Stochastic finite elements: a spectral approach -Revised Edition[END_REF].

Some recent work uses Polynomial Chaos expansion and derivatives such as

Multi-Element generalized Polynomial Chaos (MEgPC) [START_REF] Wan | An adaptive multi-element generalized polynomial chaos method for stochastic differential equations[END_REF] to demonstrate stability of equilibria of stochastic systems using the Lyapunov function [START_REF] Fisher | Stability analysis of stochastic systems using polynomial chaos[END_REF][START_REF] Nechak | A polynomial chaos approach to the robust analysis of the dynamic behaviour of friction systems[END_REF] or to compute limit cycles in the time domain [START_REF] Nechak | Prediction of random self frictioninduced vibrations in uncertain dry friction systems using a multi-element generalized polynomial chaos approach[END_REF] when the equilibrium loses stability. Recent papers propose evaluation of Hopf bifurcation point (when stability changes) using MEgPC for a system with dry friction when one parameter (the friction coefficient) varies but does not investigate the non-linear effects in the unstable range (see [START_REF] Nechak | Non-intrusive generalized polynomial chaos for the robust stability analysis of uncertain nonlinear dynamic friction systems[END_REF] for a 2-dofs system and [START_REF] Sarrouy | Piecewise polynomial chaos expansion with an application to brake squeal of a linear brake system[END_REF] for a finite element model of a brake).

Our first objective is to show that using a proper decomposition of eigenmodes, Polynomial Chaos provides accurate results on eigenvalues dispersion resulting from one or more uncertain parameters and hence on the stability of equilibria for a system undergoing multiple uncertainty sources. The second point of this paper addresses the dispersion of limit cycles that occur in the unstable ranges which are responsible for brake squeal. To this end, a constrained harmonic balance method is proposed. This modified harmonic balance method overcomes the problem of dealing with an unknown cycle period and long term integration problems that arise when using a time integration scheme as in [START_REF] Nechak | Prediction of random self frictioninduced vibrations in uncertain dry friction systems using a multi-element generalized polynomial chaos approach[END_REF]. The proposed method suits both deterministic and stochastic cases.

The paper is composed as follows: Section 2 first exposes the model and its equations along with main results from previous deterministic study. The three uncertain configurations that will be numerically processed are then detailed. Section 3 is devoted to theoretical discussion. First, computation of eigenmodes using a Polynomial Chaos expansion is explained. Then, the method for computing the limit cycles is exposed in the deterministic context and adapted to the stochastic approach. Finally, numerical results for the three uncertain cases illustrate efficiency of the methods and physical phenomena are analysed.

System deterministic study

A simple system similar to the one studied in (Sinou and Jézéquel, 2007b) is considered. This kind of phenomenological model was introduced by [START_REF] Hultén | Drum brake squeal -a self-exciting mechanism with constant friction[END_REF] in order to study squeal vibration in drum brakes and is sufficient to investigate friction-induced vibrations. After a brief description, main deterministic results are recalled. Then, three uncertain configurations used as application material are described.

System description

The system considered is a simple mass linked to a moving band by two non-linear springs attached to plates (k 1 + k nl 1 and k 2 + k nl 2 ) in orthogonal directions . The friction coefficient at contact between the plates and the band is denoted µ. Damping is also introduced through c 1 and c 2 dashpots.

A schematic is presented in Fig. 1. Such a low order model may not contain all the necessary information to provide fully reliable predictions as stated in recent studies (Butlin andWoodhouse, 2009, 2010). This non-linear 2 degrees of freedom model will nonetheless provide an acceptable method application support whose results may be compared to the conclusions of Sinou and Jézéquel (2007b). The proposed method could be applied without any modification to larger models.

m k nl 1 k nl 2 k 1 k 2 c 1 c 2 q 1 q 2 µ µ
Figure 1: Mass with double rubbers: plan Equations describing the system dynamic behaviour are:

  1 0 0 1   M q +   η 1 ω 01 0 0 η 2 ω 02   D q +   ω 2 01 -µω 2 02 µω 2 01 ω 2 02   K q+    ϕ 1 q 3 1 -µϕ 2 q 3 2 µϕ 1 q 3 1 + ϕ 2 q 3 2    f nl (t,q, q) =    0 0    (1)
where

ω 0i = k i /m, η i = c i / √ mk i and ϕ i = k nl i /m. Base value set is ω 01 = 2π × 1000 rad.s -1 , ω 02 = 2π × 600 rad.s -1 , η 1 = 0.06, η 2 = 0.02, ϕ 1 = 1 • 10 11 m -2 .s -2 , ϕ 2 = 1 • 10 9 m -2
.s -2 and µ = 0.398. q size will be denoted n = 2.

As demonstrated in (Sinou and Jézéquel, 2007b,a), this system has a null equilibrium (q * = 0 and q * = 0) whose stability depends on the friction parameter µ. Stability is determined by real parts of eigenvalues of the tangent system at equilibrium point:

Mq + (D + D q f nl (t, q * , q * )) q + (K + D q f nl (t, q * , q * )) q = 0 (2)
where D q f nl (t, q * , q * ) and D q f nl (t, q * , q * ) stand for derivatives of f nl with respect to q and q at equilibrium point (q * , q * ). It turns out from f nl expression that D q f nl (t, q * , q * ) = 0 as the non-linear terms do not depend on q. Moreover, the equilibrium having null coordinates, D q f nl (t, q * , q * ) = 0 too. Then, the tangent system can be expressed in the following augmented form:

  K 0 0 M   B ẋ -   0 K -K -D   A x = 0, x =    q q    (3)
The corresponding eigenvalue problem with a normalization condition which defines eigenvectors u k uniquely is

Au k = λ k Bu k and u T k Bu k = 1, 1 ≤ k ≤ 2n = 4 (4)
Whenever one of the eigenvalue λ k has a positive real part, the system is unstable.

For µ ranges for which instability occurs, the mode being unstable depends on the (η 1 ω 1 )/(η 2 ω 2 ) ratio. Non-linear springs lead to existence of limit cycles for these ranges, whose frequency is a priori unknown. Numerical determination of these limit cycles and of their frequencies is explained in Section 3.2.

Uncertain configurations

Three different uncertain configurations will be dealt with in Section 4, each of them having a different purpose. They are described below, including numerical differences to the base value set previously stated.

Dispersion of parameters are expressed as functions of independent random variables ξ i which follow a standard normal distribution. This is a classical distribution and matches the basic theoretical frame of Polynomial Chaos.

The first uncertain configuration considers dispersion on the friction coefficient µ. This will illustrate consequences of µ dispersion on the system stability and will be used to validate the method by comparing PC results to MC simulations. For this case, one will use

µ(ξ) = µ + ∆µξ (5)
with µ = 0.398 and ∆µ = 5 100 µ = 0.0199. This dispersion impacts K matrix as well as the non-linear contribution f nl .

The chosen set of values puts the system close to the stability limit.

The second configuration focuses on (η 2 ω 02 )/(η 1 ω 01 ) ratio action with regard to the mode becoming unstable. To this end, a µ value is chosen so as to ensure having an unstable static equilibrium. The ratio is varied by taking into account dispersion on η 2 .

η 2 (ξ) = η 2 + ∆η 2 ξ (6)
with η 2 = 0.034 and ∆η 2 = 14.7 100 η 2 = 0.005, that is a standard deviation equal to 14.7% of the mean value η 2 for η 2 . Moreover, η 1 = 0.02 and µ = 0.55. The (η 2 ω 02 )/(η 1 ω 01 ) nominal ratio is then equal to 1.02.

Variation of η 2 implies variation of damping matrix D only.

Finally the two previous parameters undergo dispersion at the same time depending on two different random variables ξ 1 and ξ 2 . This will prove that both methods can handle configurations with multiple uncertain parameters.

µ(ξ) =µ + ∆µξ 1 (7a) η 2 (ξ) =η 2 + ∆η 2 ξ 2 (7b)
with µ = 0.56, ∆µ = 1 100 µ = 0.0056, η 2 = 0.0333 and ∆η 2 = 1 100 η 2 = 0.000333. Moreover, η 1 = 0.02 as in the previous case that is (η 2 ω 02 )/(η 1 ω 01 ) = 1. Theses variations affect K and D matrices as well as the non-linear forces f nl .

Theoretical discussion

Introducing uncertainty on some parameters modifies the dynamic equation (1) and uncertain quantities are to be added:

M + M(ξ) q + D + D(ξ) q + K + K(ξ) q + fnl (t, q, q; ξ) = 0 (8)
where tilde quantities depend on a set of random variables ξ used to describe uncertainty. We assume that these variables follow a standard normal distribution.

The randomness of the inputs has consequences on eigenvalues. The method used to evaluate their dispersion is exposed in the next subsection.

Furthermore, as stated in Section 2.1, limit cycles appear when equilibria become unstable. These cycles are also affected by dispersion of the inputs. Section 3.2 proposes a method to compute these limit cycles with unknown period in the deterministic case and its adaptation to the stochastic case.

Eigenvalue problem for equilibria

Using the same transformation that the one applied to (1) to get (4), the stochastic eigenvalue problem with a normalization condition is deduced from (8):

A + Ã(ξ) ũk = λk B + B(ξ) ũk , ũT k B + B ũk = 1, 1 ≤ k ≤ 2n (9)
where tilde matrices are added to introduce uncertainty and ũk and λk respectively denote stochastic eigenvectors and eigenvalues.

As stated in the introduction, the method at stake relies on the expansion of stochastic eigenmodes onto the Polynomial Chaos (PC). This expansion consists in decomposing uncertain quantities on a basis of Hermite multivariate polynomials denoted ψ ψ n as introduced by [START_REF] Ghanem | Stochastic finite elements: a spectral approach -Revised Edition[END_REF].

These polynomials are most appropriate for Gaussian stochastic processes.

The computation of these polynomials and the appropriate scalar product are recalled in Appendix A.

Stochastic eigenvalues and eigenvectors of augmented system (9) are here decomposed on both the deterministic augmented eigenmodes and the polynomial chaos using complex weights:

λk (ξ) = λ k N s=1 (k) a s + j (k) b s ψ ψ s (ξ) (10) ũk (ξ) = P p=1 (k) γp (ξ) + j (k) μp (ξ) u p (11) with (k) γp (ξ) = N n=1 (k) γ n p ψ ψ n (ξ) and (k) μp (ξ) = N n=1 (k) µ n p ψ ψ n (ξ) (12) giving ũk (ξ) = P p=1 N n=1 (k) γ n p + j (k) µ n p ψ ψ n (ξ) u p ( 13
)
where the coefficients (k) γ n p , (k) µ n p , (k) a s and (k) b s are real and j denotes the imaginary unit (j 2 = -1). ψ ψ 1 will conventionally denote the constant polynomial equal to 1; means of polynomials with index greater than or equal to 2 are then null. This decomposition using P deterministic eigenvalues and eigenvectors is a generalization of what is proposed in [START_REF] Dessombz | Analyse dynamique de structures comportant des paramètres incertains (Dynamic analysis of structures with uncertain parameters[END_REF][START_REF] Dessombz | Analysis of stochastic structures: perturbation method and projection on homogeneous chaos[END_REF]. Using complex weights, real and imaginary parts of stochastic eigenvalues and eigenvectors can evolve independently. In this paper, all the deterministic modes are used for projection of stochastic eigenvectors, P = 2n.

Using this decomposition for eigenmodes, one also applies it to the stochastic matrices of Eq. ( 9). This new writing of matrices can be achieved by using a Karhunen-Loève decomposition or assuming them so regarding experimental results:

à = N A n=2 A n ψ ψ n (ξ) and B = N B n=2 B n ψ ψ n (ξ) (14)
Having ψ ψ 1 (ξ) = 1, it is associated with the deterministic components A and B and only polynomials with zero mean are considered for the decomposition of these matrices.

To get the final system of equations that leads to the unknowns (k) γ n p ,

µ n p , (k) a s and (k) b s once solved, Eqs. ( 9) are projected onto the polynomial chaos basis ψ ψ k (ξ), 1 ≤ k ≤ N using the dedicated scalar product defined in Appendix (A.5). During this step, one has to compute scalar products of the form < ψ ψ i 1 . . . ψ ψ i j , ψ ψ n >. Let us point out that these quantities can be evaluated and stored once for all, being reused for further studies and that many of them are null, implying simplification of the resulting set of equations. This is an advantage provided by the chosen intrusive approach compared to non intrusive approaches which require the use of quadrature formula to evaluate the unknown coefficients. To overcome the problem of dealing with complex valued functions, real and imaginary parts of the subsequent equations are separated. This generates 2P (P + 1)N non-linear (quadratic) equations, that is as many as unknowns. This set of equations can be solved through a general non-linear solver relying for example on the Levenberg-Marquardt scheme.

Once the PC expansion is obtained, stochastic indicators such as mean or standard deviation are computed by evaluating eigenmodes using their PC decomposition on a large MC sample.

Constrained HBM for limit cycles determination

Deterministic case

As recalled previously, the self-excited non-linear system tends to draw limit cycles when the static equilibrium is unstable. Coming from a mode coupling, these limit cycles have an unknown period. To determine their properties while avoiding a direct time-integration process, a well-known method is the harmonic balance method (HBM) which consists in a Galerkin approach using a basis of trigonometric functions. However this method is designed to approximate periodic solutions with known period [START_REF] Cameron | An alternating frequency/time domain method for calculating the steady-state response of nonlinear dynamic systems[END_REF][START_REF] Von Groll | The harmonic balance method with arclength continuation in rotor/stator contact problems[END_REF]. Briefly, the time vector q(t) is developed in a truncated Fourier series:

q(t) = a 0 √ 2 + K k=1 (a k cos(kωt) + b k sin(kωt)) (15)
where coefficients a k and b k are real vectors with same size as q, and ω = 2π/T is the limit cycle angular frequency. This development imposes q and q as well.

Once reinjected in the dynamical equation, the time variable is eliminated by projecting the system of equations onto the functions 1/ √ 2, cos(kωt) and sin(kωt) using the scalar product

< f, g >= 2 T T t=0 f (t)g(t)dt (16)
If q size is n, this generates a square system of equations with size n(2K +1).

Yet, when angular frequency ω is unknown, the system is not square anymore and one usually wishes to add an equation or remove an unknown.

Work presented in [START_REF] Coudeyras | A new treatment for predicting the self-excited vibrations of nonlinear systems with frictional interfaces: The constrained harmonic balance method, with application to disc brake squeal[END_REF] explores the first solution by adding a constraint on real parts of eigenvalues of the tangent system. This procedure requires a lot of computation and does not fit further developments when uncertainties is introduced. Therefore, this study proposes to explore the second solution which consists in removing an unknown by setting it to a given value as suggested in [START_REF] Seydel | From Equilibrium to Chaos, Practical Bifurcation and Stability Analysis[END_REF]. It will be referred to as CHBM for Constrained Harmonic Balance Method. It is an interesting improvement of usual HBM procedure that provides a way to catch limit cycles of self-excited non-linear systems which is hardly treated in the literature.

For the current system, q(t) is developed with only one harmonic and no constant term (for centred solutions approximation only); an unknown is removed by assigning a given value a * 1 to the first component of a vector:

q(t) = a cos(ωt) + b sin(ωt), a =    a * 1 a 2    , b =    b 1 b 2    (17)
Injecting this decomposition in the dynamic equation ( 1) and projecting the resulting set of equations onto cos(ωt) and sin(ωt) functions using scalar product ( 16), one finally gets the algebraic set

K -ω 2 M a + ωDb + 3 4    (a * 1 3 + a * 1 b 2 1 )ϕ 1 -(a 3 2 + a 2 b 2 2 )µϕ 2 (a * 1 3 + a * 1 b 2 1 )µϕ 1 + (a 3 2 + a 2 b 2 2 )ϕ 2    = 0 (18a) K -ω 2 M b -ωDa + 3 4    (a * 1 2 b 1 + b 3 1 )ϕ 1 -(a 2 2 b 2 + b 3 2 )µϕ 2 (a * 1 2 b 1 + b 3 1 )µϕ 1 + (a 2 2 b 2 + b 3 2 )ϕ 2    = 0 (18b)
where the four unknowns are a 2 , b 1 , b 2 and ω.

Stochastic case

When some parameters are uncertain, the limit cycles are affected too.

Dispersion is then approximated using an expansion of CHBM unknowns on the polynomial chaos:

ã2 (ξ) = N ′ n=1 α n 2 ψ ψ n (ξ) , b1 (ξ) = N ′ n=1 β n 1 ψ ψ n (ξ), ω(ξ) = N ′ n=1 ̟ n ψ ψ n (ξ), b2 (ξ) = N ′ n=1 β n 2 ψ ψ n (ξ), (19)
where decomposition coefficients α n 2 , β n 1 , β n 2 and ̟ n are real. These approximations are injected into the CHBM deterministic equations (18) where uncertainty on matrices and non-linear function have been introduced. For example, for the third uncertain configuration processed in Section 4.2 (i.e. uncertainties for the friction coefficient µ and the damping ratio η 2 ), one gets

(K + K(ξ)) -ω2 (ξ)M    a * 1 ã2 (ξ)    + ω(ξ)(D + D(ξ))    b1 (ξ) b2 (ξ)    + 3 4    (a * 1 3 + a * 1 b2 1 (ξ))ϕ 1 -(ã 3 2 (ξ) + ã2 (ξ) b2 2 (ξ))(µ + ξ 1 ∆µ)ϕ 2 (a * 1 3 + a * 1 b2 1 (ξ))(µ + ξ 1 ∆µ)ϕ 1 + (ã 3 2 (ξ) + ã2 (ξ) b2 2 (ξ))ϕ 2    = 0 (20a) (K + K(ξ)) -ω2 (ξ)M    b1 (ξ) b2 (ξ)    -ω(ξ)(D + D(ξ))    a * 1 ã2 (ξ)    + 3 4    (a * 1 2 b1 (ξ) + b3 1 (ξ))ϕ 1 -(ã 2 2 (ξ) b2 (ξ) + b3 2 (ξ))(µ + ξ 1 ∆µ)ϕ 2 (a * 1 2 b1 (ξ) + b3 1 (ξ))(µ + ξ 1 ∆µ)ϕ 1 + (ã 2 2 (ξ) b2 (ξ) + b3 2 (ξ))ϕ 2    = 0 (20b) with K(ξ) = ξ 1 ∆µ   0 -ω 2 02 ω 2 01 0   (21a) D(ξ) = ξ 2 ∆η 2   0 0 0 ω 02   (21b)
The final set of 4N ′ equations is obtained by projecting this system onto the Hermite polynomials ψ ψ n , 1 ≤ n ≤ N ′ using the associate scalar product (A.5). The final system of equations providing the decomposition of CHBM coefficients on PC is non-linear and solved using an appropriate algorithm.

Results and Discussion

This part is dedicated to application of the methods exposed in the previous section. Each of the three proposed cases permits the illustration of different physical phenomena and is a support for showing suitability of the approaches for accurate stochastic studies. and in an equivalent way for real parts. MC and PC superscripts denote values obtained using direct Monte Carlo and PC expansion evaluation respectively. This error is computed for each mode (1 and 2) and each point ξ (j) of the sample, leading to the proposed histograms.

One can see that error on imaginary part is almost negligible whatever the PC order is. On the contrary, the real part of eigenvalue for mode 1 really takes advantage on the higher orders. Finally, order D = 3 gives accurate enough results with an error mostly lower than 5 percent.

Moreover, these eigenvalues being mainly used to predict the stability of the equilibrium, the number of stable and unstable occurrences is summed up in Table 1. Once again, order 2 and order 3 PC expansions reproduce these counts with very little difference to the Monte Carlo reference.

This comparison shows that even for great dispersion of the input value (standard deviation being 5 % of the nominal value with a Gaussian repartition, it means that 99% of the values are within a range with width equal to 30% of the nominal value), the PC expansion method suits representation of eigenvalues dispersion and so the system stability and instability rate with a satisfying accuracy.

Case 2: dispersion on damping ratio η 2 -Limit cycles study

The aim of this configuration being to monitor η 2 influence on the mode becoming unstable and the subsequent limit cycles, the friction coefficient µ is chosen great enough to ensure unstable equilibria whatever η 2 value may be.

First, eigenmodes PC expansion coefficients were evaluated using an order 3 decomposition. Then a sample of 10000 points was used to evaluate eigenvalues. Results are presented in Fig. 3. They confirm that the instability may come from either mode 1 or mode 2 depending on the value of η 2 . In a second time, stress is put on the limit cycles. Their dispersion is evaluated using the CHBM method exposed in Section 3.2 using a * 1 = 2 • 10 -3 . Different orders for PC expansion are tested and compared to deterministic results for 11 points ξ (j) equally spaced between -3 and +3. For each variable a 2 , b 1 , b 2 and ω, the absolute relative error between the value obtained using PC expansion and the one obtained using direct calculation is evalu-ated. Results are summed up in Fig. 4. This shows that order 3 and order 4 expansions give very accurate results. The deterministic results were previously checked using them as initial conditions for a time integration with duration T = 2π/ω for each of the 11 test points and verifying that initial and final states were almost identical. Order 4 PC expansion is retained for expansion and MC simulation. This confirms that error is mostly lower than 1 % for each variable and hence that PC expansion is an accurate method to describe stochastic CHBM coefficients. Moreover, direct computation for a sample with 10000 points required 190 s while PC method only used 0.60 s for evaluation of PC expansion of coefficients and 0.24 s for evaluation of the 10000 realizations: direct MC simulation is about 226 times longer than the proposed method.

Finally, PC expansion with order 4 is used to rebuild cycles over time to visually check occupation rate in different spaces as depicted on Fig. 6. The procedure used to build these portraits is the following: for each point ξ (j) , q and q vectors are evaluated at 1000 time points τ k = 2π/1000 × (k -1) using formula:

q (j) k =    a * 1 ã2 (ξ (j) )    cos(τ k ) +    b1 (ξ (j) ) b2 (ξ (j) )    sin(τ k ) (23a) q(j) k = ω(ξ (j) )   -    a * 1 ã2 (ξ (j) )    sin(τ k ) +    b1 (ξ (j) ) b2 (ξ (j) )    cos(τ k )   (23b)
Then, to build (q 1 , q 2 ) portrait for example, minimum and maximum values for each variable are evaluated and a grid is used with 251 bins in each direction to divide (q 1 , q 2 ) space. For each cycle q

(j) k , 1 ≤ k ≤ 1000, q (j) k
at time τ k is located in the grid. If it is located in the same bin as the previous time point q (j)

k-1 , it is not counted; otherwise, +1 is added to the bin weight. This avoids affecting too much weight to a bin because of high time discretization. When all the cycles are processed, the weights are divided by the number of cycles (10000) and multiplied by 100: this gives for each grid point a percentage of occupation.

As can be seen on Fig. 6 (a), two modes can contribute when the equilibrium is unstable resulting in two different main orientations for the cycles. (m) (a) (q 1 , q 2 ) portrait For this last configuration, two independent random variables are used.

First, eigenvalues are decomposed on PC using an order 2 expansion. This involves the following polynomials: ψ ψ 1 (ξ) = 1, ψ ψ 2 (ξ) = ξ 1 , ψ ψ 3 (ξ) = ξ 2 , ψ ψ 4 (ξ) = ξ 2 1 -1, ψ ψ 5 (ξ) = ξ 1 ξ 2 and ψ ψ 6 (ξ) = ξ 2 2 -1. Using the random sample of 10000 points represented in Fig. 7 (a), one can rebuild the eigenvalues and plot the location probability (in percent) of the unstable eigenvalue in the complex plane. It is also possible to distinguish between the occurrences for which mode number 1 becomes unstable and the ones for which instability comes from mode number 2 (Fig. 7 (b) and (c)). It turns out that for this configuration, all the occurrences are relative to an unstable equilibrium.

Among the 10000 points, 5375 become unstable because of mode 1 and 4625 because of mode 2.

After the stability study, the limit cycles were fitted using an order 2 PC expansion. Using the same procedure as the one described previously, one can draw portraits in different spaces as depicted in Fig. 8. This time, dispersion being smaller than in the second case, one cannot clearly identify very different behaviours because mode 1 and mode 2 keep being very close from one another.

Conclusions

Two methods designed for stochastic study of systems with friction were exposed. The first one provides a way to evaluate complex eigenmodes dispersion by using an expansion of stochastic eigenmodes on both the deterministic eigenmodes and the Polynomial Chaos. The second method evaluates (m) (a) (q 1 , q 2 ) portrait

4. 1 .

 1 Case 1: dispersion on friction coefficient µ -Stability analysis Variation of the friction coefficient µ has effect on the static equilibrium stability. Therefore, focus is put on accuracy of eigenvalues and especially their real part. First, PC coefficients are evaluated for different orders D of expansion, namely 1 ≤ D ≤ 3. Having only one random variable ξ, the PC basis used consists in N= D + 1 polynomials ψ ψ n , 1 ≤ n ≤ N where ψ ψ 1 (ξ) = 1, ψ ψ 2 (ξ) = ξ, ψ ψ 3 (ξ) = ξ 2 -1 and ψ ψ 4 (ξ) = ξ 3 -3ξ.Then a set of 10000 sample points ξ (j) is generated (using a standard normal distribution) and used to evaluate λk (ξ (j) ), 1 ≤ k ≤ 4, 1 ≤ j ≤ 10000 using the different PC approximations as well as direct Monte Carlo computation. This last set of values is considered as the reference set. Note that is has been verified that µ values resulting from this sample made physical sense.Results are showed onFig. 2. On this figure, only eigenvalues with positive imaginary part are displayed, others being directly complex conjugate. The left panes are devoted to the imaginary parts and the right ones to the real parts. Fig. 2 (a) displays Monte Carlo simulations results while the three lower blocks (b) to (d) display histograms of absolute relative error computed as follow for relative error on imaginary part e rel = Im λ PC -Im λ MC |Im (λ MC )| × 100 (22)

Figure 2 :

 2 Figure 2: Uncertain µ: comparison of PC expansion results with MC simulations. Left half: quantities relative to imaginary part of eigenvalues. Right half: quantities relative to real part of eigenvalues. (a) MC simulations; (b) to (d) histograms of absolute relative error (in percent) between PC and MC results for each mode and different PC orders D. Light blue: mode 1; dark blue: mode 2.

Figure 3 :

 3 Figure 3: Uncertain η 2 : unstable mode histograms. (a) Unstable mode number; (b) Imaginary part and (c) Real part of unstable eigenvalue. Light blue: mode 1, dark blue: mode 2

Figure 4 :

 4 Figure 4: Uncertain η 2 : CHBM PCE results comparison with deterministic results for different PC expansion orders using absolute relative error in percent.-•-: order D = 2; -+-: order D = 3; -×-: order D = 4.

Figure 5 :

 5 Figure 5: Uncertain η 2 : CHBM order 4 PC expansion results comparison with MC deterministic results. MC results and absolute relative error (in percent) for (a) a 2 , (b) b 1 , (c) b 2 and (d) ω coefficients.

Figure 6 :

 6 Figure 6: Uncertain η 2 : dispersion in different spaces rebuilt from PC expansion with order 4.

Figure 7 :

 7 Figure 7: Uncertain η 2 and µ: unstable mode dispersion; (a) Random sample; (b) Unstable mode 1 and (c) unstable mode 2 eigenvalue in complex plane

Figure 8 :

 8 Figure 8: Uncertain η 2 and µ: dispersion in different spaces rebuilt from PC expansion with order 4.

Table 1 :

 1 Uncertain µ: comparison of PC expansion (PCE) results with MC simulation (stable occurrences counts)

		MC PCE order 1 PCE order 2 PCE order 3
	Stable	5187	4705	5185	5213
	Unstable 4813	5295	4815	4787

Acknowledgements

The authors gratefully acknowledge the financial support of the French National Research Agency through the Young Researcher program ANR-07-JCJC-0059-01-CSD2.

dispersion on limit cycles amplitudes and frequencies. It can been seen as an improvement of the usual HBM combined with a constraint that provides a way to deal with limit cycles having an unknown period and a Polynomial Chaos expansion of its coefficients.

Through three uncertain configurations, they were demonstrated to be accurate and efficient, using comparison with direct calculation, even in the case of wide uncertainty ranges.

A. Hermite multivariate polynomials

Hermite polynomials (one dimension) can be defined using a derivative

or recursively (formula from (Sudret and Der Kiureghian, 2000, Section 3.2)):

The general formula for multidimensional Hermite polynomials is

but the formula used in practice to compute the multivariate Hermite polynomials is the one found in (Sudret and Der Kiureghian, 2000, Chap. 3, Section 3.1): for Q random variables, the multivariate Hermite polynomials of order d can be expressed as

polynomials with degree d. That is, for Q random variables, there are N polynomials with degree less than or equal to D:

These polynomials are orthogonal with respect to the following scalar product

× g(ξ 1 , . . . , ξ Q )e -ξ T ξ/2 dξ 1 . . . dξ Q (A.5)