Impersonal constructions in Slavic

Irina Kor Chahine & Marguerite Guiraud-Weber University Cote d'Azur -CNRS, BCL / Aix-Marseille University, France

The issue of the impersonal-one of the main issues in Slavic languages-can be considered in a variety of ways, i.e. it may refer to grammatical person, verbal syntax, or voice usage. An extensive bibliography has been developed on this question (cf. first typologies of impersonal constructions in [START_REF] Creissels | Impersonal and related constructions: a typological approach[END_REF][START_REF] Malchukov | Towards a typology of impersonal constructions. A semantic map approach[END_REF][START_REF] Malchukov | Towards a typology of impersonal constructions. A semantic map approach[END_REF]. Impersonal constructions constitute only a part of this vast domain. While in some languages, such as Arabic, Persian, or Japanese, impersonal constructions are relatively rare, in Slavic languages, they turn out to be especially abundant. However, even in the Slavic languages this domain is varied. Among five Slavic languages belonging to three different groups, namely Russian, Polish, Czech, Serbo-Croatian1 , and Bulgarian, it is in Russian that the system of impersonal constructions is most developed. These constructions exist to a lesser extent in other Slavic languages, but their development does not always appear to be a result of geographical language contact.

Impersonal constructions have been well known since antiquity; in particular they are well attested in Greek and Latin. In modern European languages, among non-referential markers of impersonal constructions we can point out it in English (it's raining), es in German (es regnet), or even the nonreferential il in French (il pleut) and ça in colloquial French (ça tonne) [START_REF] Maillard | L'impersonnel français de "il" à "ça[END_REF]. Often used to express meteorological phenomena or sensations, these constructions are based on verbal syntax that allows either a personal reading, or an impersonal one, depending on context (je me souviens…/il me souvient…). In Slavic languages, the domain of impersonal constructions is probably the richest in comparison to other European languages. This domain is very common and comprises various syntactic models with a large range of semantics, including expressing existence, sensations, physical or mental states, object of communication, and modality. Moreover, these constructions cannot be considered exclusively in the context of verbal syntax, because the core predicate can be expressed not only by a verb, but also by a predicative form (noun or adverb serving as a predicate), or even by the entire construction.

Below general issues associated with impersonal constructions in Slavic languages will be briefly discussed and presented in an overview of this vast domain.

Impersonal constructions: A problem of definition

Impersonal constructions are attested in all Slavic languages, although their variants do not appear in every language, and their frequency is not the same in different languages. Therefore, it is a heterogeneous domain, despite the fact that they originated from the same archaic Balto-Slavic model derived from the Indo-European one. Their heterogeneity and diversity have been examined in multiple studies devoted to this issue; however, Slavic impersonal constructions have not yet received a completely exhaustive description.

Diachronic studies have brought to light a variety of ways the impersonal constructions have evolved in each language. Historical data show that very often an identical or quasi-identical structure in two different languages is not necessarily the same linguistic phenomenon, and that visible resemblances may be misleading. It is therefore necessary to acknowledge that the role an impersonal construction plays in the system of one language may not be the same that it plays in another. This represents a serious difficulty for potential comparison between languages.

Another difficulty concerns terminology. The term impersonal refers to various syntactic constructions, and, according to the linguistic schools and traditions in each country, is used differently. As early as in 1883 Franz Miklosich introduced the term of "subjectlose Sätze" to underline the absence of subject as a common property of these constructions, and since then, this term has known some development, in particular in the Czech and Polish schools. However, when referring to the notion of "sentences without a subject," we have to know what subject is, because this notion is not always clearly defined and remains implicit in many studies. An attempt to provide further clarification consists in talking of sentences without the nominative case, since nominative is typically the case of the subject [START_REF] Wolińska | Konstrukcje bezmianownikowe we wspȯłczesnej polszczyźnie[END_REF][START_REF] Guiraud-Weber | Les propositions sans nominatif en russe moderne[END_REF]. Despite the inconvenience of the term impersonal, it has finally become accepted and universally recognized, in particular within studies written in Russian and English. It should be noted, however, that this is a purely conventional term, and many scholars have shown its inadequacy [START_REF] Zolotova | Očerk funkcional'nogo sintaksisa russkogo jazyka[END_REF][START_REF] Creissels | Approche des constructions asubjectales, improprement désignées comme impersonnelles[END_REF]Malchukov & Siewierska 2011, and others).

The issue of subject

The conventional term of impersonal construction is now used, on the one hand, merely as a label applying to all sentences that do not conform to the basic pattern of « Subject in Nominative + agreed Predicate ». On the other hand, it is used in a narrow sense for designating constructions that one would call purely impersonal, and which one cannot classify according to morphological properties of the predicate alone (cf. Galkina-Fedoruk 1958 and other scholars pursuing this tradition), but according to syntactic and semantic properties of impersonal constructions as a particular syntactic structure. In this structure, in addition to the predicate, there is very often a recurring nominal element in an oblique case. The latter refers to the object or person concerned by the events designated by the predicate. And in this case, the non-agreed predicate takes the form of 3rd person of singular in present or the form of neuter then it is in past tense. The lack of nominative, which is the only case appropriate for designating voluntary action by an animated agent, is semantically justified. If the events designated by the predicate refer to a person, the person will be expressed by an oblique case and play the role of a simple observer, a passive participant, or, at best, an experiencer, but never of an active agent.

It has long been observed that some oblique cases in impersonal constructions behave similarly to a nominative subject. This is especially the case for dative (type (1a)), to a lesser extent for accusative (type (1b)), and even for instrumental (type (1c)). These similarities are at the origin of a special term in Russian linguistics, namely sub"ekt ('semantic or logical subject'), which is opposed to podležaščee ('grammatical subject'). The frequency of these terms in Russian linguistics used in all Russian textbooks and papers shows the importance of this phenomenon in the Russian language, which explains why some linguists attempt to describe syntax in semantic rather than in formal terms [START_REF] Zolotova | Očerk funkcional'nogo sintaksisa russkogo jazyka[END_REF].

Studies devoted to the notion of subject in different world languages, in particular those by Edward [START_REF] Keenan | Towards a universal definition of subject[END_REF] and Bernard [START_REF] Comrie | Language Universals and Linguistic Typology[END_REF], have shown that there is a certain number of morphological, syntactic, and semantic properties that allow for identifying the subject, and that a number of languages contain subject-like obliques with behavioural subject properties [START_REF] Seržant | The Diachronic Typology of Non-canonical Subjects[END_REF]). These features are not the same in all languages, but could be similar. If, in the case of Russian, we temporarily renounce to define the subject in discrete terms (as an argument that agrees with the verb in number, person and gender), it becomes apparent that certain subject properties (such as indispensability, control over the reflexive pronouns, position in front of the predicate, topic of the sentence) not only characterize the subject in nominative, but can occur in other oblique cases, in which case the latter plays the role of the argument enabling the predicate quality (see Table 1). This approach requires a scalar definition of the subject, such as some arguments will be considered more "subject" than others and will be placed closer to the nominative in a subjectival scale. From this point of view, it becomes possible to justify the linguistic insight that underlies the sub"ekt / podležaščee dichotomy. This approach has been adopted for Russian in [START_REF] Testelec | Vvedenie v obščij sintaksis[END_REF]Guiraud-Weber 2003a. It supports preferential treatment of certain nominal arguments in oblique cases within impersonal constructions because these arguments express the quality of the predicate.

However, only Nominative (or its syntactic equivalents) can be considered as grammatical subject in Slavic. The choice of other cases is determined by general semantics of the construction. Thus, on examples from Slavic languages we can see that Genitive indicates the absence or non-existence; Dative, and sometimes Accusative case, designates physical and mental state; Instrumental characterizes the inanimate destructive force, and Locative -a space. In addition, the two latter cases can be used with prepositions to designate other specific situations [START_REF] Guiraud-Weber | Les propositions sans nominatif en russe moderne[END_REF]. This fact shows that not only the lexis, but also the syntactic structure takes part in forming the meaning of the construction.

Evolution of impersonal constructions in Russian and other Slavic languages

The domain of impersonal constructions in Russian is very broad, especially in comparison to other Slavic languages. Over many centuries it has continued to become richer. Diachronic studies have demonstrated the following trends:

1. Expansion of the negative existential constructions (see (2a)). Lexical means of predicates in these constructions have been continually renewed ever since the XVIII th century (Švedova 1964). This expansion, involving a great number of verbs in modern Russian, is described in a number of publications [START_REF] Ickovič | Očerki sintaksičeskoj normy[END_REF][START_REF] Babby | Existential Sentences and Negation in Russian[END_REF]Padučeva 1992;Guiraud-Weber 2003b ;Partee & Borščev 2007); 2. Emergence, at the end of XIX th century, of constructions with the nominal argument "s + instrumental" (see (2b)), obviously related to the deletion of the desemantized subject, like delo 'fact', položenie 'situation', vopros 'question' [START_REF] Zolotova | K razvitiju složno-padežnyx konstrukcij[END_REF];

3. Decline of passive impersonal constructions and their substitution with impersonal constructions using the nominal constituent "o + locative" (see ( 2c)) (Švedova 1964) In West Slavic languages, in particular in Polish and Czech, the domain of strictly impersonal constructions appears to be less extensive; while constructions with indefinite human subjects-"manconstructions"-are abundant and diverse. In South Slavic languages, like in Serbo-Croatian, the domain of impersonal constructions is more limited, and it is especially restricted in Bulgarian because of the absence of noun declension but dative impersonal constructions have wider derivational schemas compared to Russian. What follows is a brief survey of the main types of impersonal constructions in Slavic languages.

Main types of impersonal constructions in Slavic

Constructions describing environment

Impersonal constructions describing environment exist in all Slavic languages. These impersonal constructions include predicates (4) and some impersonal verbs (5) describing environment. They regularly occur without any nominal support ( 5), but necessarily refer to a space, often where the speaker is located (according to deictic rules). In such impersonal constructions, the situations are always localizable, unlike non-localizable situations depicting feelings, emotions, beliefs, or psychical states of individuals (cf. [START_REF] Boguslavskij | Lingvističeskij processor i lokativnye obstojatel'stva[END_REF]. Thus, in this type of impersonal construction, the nominal element referring to location (4a) is not just an adverbial phrase, which, by definition, could be deleted. It semantically represents necessary support to the predicate, without which there would be no message. This first type of constructions is usually accounted for as "meteorological phenomena" (cf. [START_REF] Galkina-Fedoruk | Bezličnye predloženija v sovremennom russkom jazyke[END_REF][START_REF] Birjulin | Semantika i sintaksis russkogo impersonala: Verba meteorogica i ix diatezy[END_REF]) even though it also comprises constructions like (6) having nothing to do with weather events. They simply describe the space concerned by the conversation, like construction (7) where the space is characterized from the olfactory perspective. Thus, the latter construction should also be included in this category [START_REF] Guiraud-Weber | O sintaksičeskoj prirode konstrukcij tipa: 'V komnate paxnet jablokami[END_REF]

Genitive constructions

Constructions of negative existence with genitive are extremely frequent in Russian (see ( 9) below) and involve a large number of verbs, including verbs that are not traditionally perceived as verbs of existence (vyjti 'to go out', projti 'to pass though', vypast' 'to fall', rasti 'to grow', etc.). In Polish, genitive may occur only in the existential constructions with the verbs być and zostać (both of which also function as auxiliary verbs). The verb 'to be' contains two forms: być and mieć, functioning in complementary distribution (10). No other verb can serve as a predicate of existence (Guiraud-Weber 2003b; Guiraud-Weber & Zaremba 2007). In Serbo-Croatian, existential constructions use genitive when the sentence is negative but have a tendency to use this case, in its partitive meaning, in affirmative sentences with uncountable nouns (11). However this construction is only limited to one single verb of existence represented by forms of the verbs imati -biti, used in complementary distribution like in Polish: the verb imati 'to have' is used in present tense, and biti 'to be' in other verbal forms. In contrast, in Czech, constructions of negative existence with genitive have disappeared. Their decline began in the 17th century, and is now complete [START_REF] Hausenblas | Vývoj předmĕtového genitivu v češtině[END_REF]. Even negative constructions with the verb 'to be' use nominative in Czech ( 12) (Žaža 1978), and the variant with genitive sounds archaic (12': Otce nebylo doma). The nominative is also used in Bulgarian where the genitive case stopped its independing existance in the old Bulgarian even before its complete disappearance in modern Bulgarian [START_REF] Feuillet | Réflexions sur la perte des cas en bulgare[END_REF]

Dative constructions

Dative constructions in Slavic languages are the most diverse and heterogeneous type of impersonal constructions. They differ in their derivational lexical schemas, semantics, and syntactic structures. In Serbo-Croatian we can find here both adverbal and noun predicates (13c': = Зло му је. Zlo mu je.) (Vojvodić, pc). Nowadays, dative impersonal constructions in -o are much more common in Bulgarian than in Russian because they can be derived not only from an adjective (cf. гадно ми е, отвратително ми е, мъчно ми е, болно ми е, тъпо ми е (coll.)), but also from a participle (active or passive): spokojno i usmixnato mi e 'I feel calm and joyfull' [START_REF] Gradinarova | Dativnaja model' s predikativom na -o v russkom i bolgarskom jazykax na fone eë paradigmatičeskix svjazej[END_REF]. As for the forms indicating an emotional or psychological state of sadness, fear, or disgust, like sramno, strašno, they are seldom used as adverbal predicates with a dative but they are common as noun predicates with accusative case (see below). (ibid.).

Constructions of emotional or psychological state

Constructions of predisposition to action

In this section have to be considered impersonal constructions with a nominal constituent "do + genitive", which occur with negative predicate in most Slavic languages (14). Moreover, in Czech, in Serbo-Croatian, and to a lesser extent in Polish (15), this construction can also appear in an affirmative form [START_REF] Mrázek | O semantičeskom podxode k sintaksisu i genitivnom sub"ekte glagola byt[END_REF][START_REF] Žaža | Ešče raz k xarakteristike otricatel'nyx konstrukcij v russkom jazyke[END_REF][START_REF] Tanasić | Jedan tip besubjekatske rečenice u srpskom jeziku[END_REF] Impersonal constructions describing a predisposition to action with a dative experiencer are also common in Slavic languages with an impersonal intransitive verb. They are restricted to some verbal predicates designating humain or mental activity, or volition, used in pronominal form, although their nonpronominal forms appear in personal constructions with a nominative subject: cf. Ja (zdes') splju xorošo. 'I sleep well (here).' ≈ Mne (zdes') xorošo spitsja. 'I feel like sleeping well (here)'. While some transitive verbs can appear in these constructions, the latter don't comprise a direct object argument, except in Polish and in Bulgarian (16b, e). ( 16 'I feel like learning (foreign language).' If these constructions are not accompanied by an adverb, they describe the attitude of the experiencer toward a state or action denoted by the verb, while constructions including an adverb denote an assessment of the experiencer's state in the process of activity (Fici & Zhukova 2012). In their negative conterparts, a negative particle cannot be used with adverb. ( 17 In Serbian, the affirmative construction of this type can be transformed to "do+genitive" construction (see above) without any semantic changes (16d': Djeci je nešto bilo do spavanja.) (Tanasić 1998: 92) due to the frequent usage of deverbal nouns in it. However, in Russian, this kind of transformation may be possible only with negative constructions but it is not systematic (cf. 17a': Rossii ne do sna, *ne do žizni).

Constructions with modal predicate

Impersonal constructions with the modal predicate in Russian commonly use a few predicates. They are mainly nado, nužno 'necessary', možno 'possible/allowed' and nel'zja 'may/can not' which are used more often (18a-b). ( 18 While this kind of impersonal constructions exists in other Slavic languages (18c-d), they tend to be replaced by personal constructions (19,20). Thus, most impersonal constructions with modal predicate in West Slavic languages have been replaced by personal constructions with loan-translations from German müssen, dürfen, and haben. This is the case of Polish and Czech where the verbs pl. musieć and mieć, from one side, and cz. muset, smĕt, and mít from another, require a nominative subject (19) [START_REF] Wolińska | Konstrukcje bezmianownikowe we wspȯłczesnej polszczyźnie[END_REF][START_REF] Weiss | Polsko-niemieckie paralele w zakresie czasownik.w modalnych[END_REF][START_REF] Weiss | Konstrukcje typu: mieć (coś) do załatwienia, być do załatwienia oraz dać (coś) do załatwienia: Stopień ich sfrazeologizowania dawniej i dziś[END_REF][START_REF] Žaža | K jednomu typu českych a ruskych vět s funkčnim predikativem[END_REF]. In Bulgarian, the nominal argument isn't expressed; the subject is marked by a verbal flexion, therefore in a personal construction (20).

Infinitive constructions

Originally occurring in Old Slavic, Russian infinitive constructions like ( 21) are common nowadays. Such constructions combine modal and temporal (future) structural meanings. Their usage, however, remains syntactically and stylistically limited [START_REF] Veyrenc | Les propositions infinitives en russe[END_REF][START_REF] Maurice | Der modale Infinitiv in der modernen russichen Standardsprache[END_REF][START_REF] Fortuin | Polysemy or Monosemy: Interpretation of the Imperative and the Dative-infinitive Construction in Russian[END_REF][START_REF] Israeli | Dative-infinitive constructions in Russian : taxonomy and semantics[END_REF]). ( 21 Dative infinitive constructions are neither common nor productive in other Slavic languages. They have completely disappeared in Bulgarian and Czech (Žaža 1998). Only some of their remnants remain in Serbo-Croatian and Polish (22) [START_REF] Wolińska | Konstrukcje bezmianownikowe we wspȯłczesnej polszczyźnie[END_REF][START_REF] Weiss | Infinitif et datif en polonais contemporain: Un couple malheureux?[END_REF]. In Serbo-Croatian these constructions are mostly found in poetic and archaic style: their function is now transmitted by 'da + present' construct, which does not combine with dative (22b': = Šta treba da činimo) (Vojvodić 2007: 575). Instead of the missing infinitive constructions, Slavic languages use other morphosyntactic forms, like future and past tenses, subjunctive mood, future or present of modal predicate (srb. morati, moći, trebati, etc.; pl. mieć, móc, mosieć, etc.) combined with a dependent infinitive, and so on. (ibid.). ( 22 Thus, there is an obvious disparity between impersonal constructions with a dative experiencer that are widely used in Slavic languages on the one hand, and, on the other hand, infinitive impersonal constructions with dative attested only in some Slavic languages. The both constructions also present functional differences. This leads some scholars to suggest that these constructions contain two different types of dative "resulting from distinct case-marking strategies" [START_REF] Franks | Parameters of Slavic Morphosyntax[END_REF].

Causative constructions

Accusative constructions

The accusative argument is also common in impersonal constructions in Slavic languages. However, these constructions with accusative are limited only to the designation of a physical illness or reaction and to metaphoric uses. Unlike other Slavic languages, Russian constructions with accusative are less common with only a limited number of verbs denoting mostly physical diseases; there are impersonal verbs like lixoradit', znobit' 'to tremble', tošnit' 'to vomit', and personal transitive verbs like lomit' 'to feel pain' or trjasti 'to tremble' (23a). In other Slavic languages, accusative constructions can, however, include not only a verbal predicate, but a predicative as well. So, in Bulgarian, as well as in Serbian ( 24), accusative constructions appear frequently with nominal predicates indicating an emotional state of sadness, fear, and disgust (gnus go e, strax go e) [START_REF] Gradinarova | Dativnaja model' s predikativom na -o v russkom i bolgarskom jazykax na fone eë paradigmatičeskix svjazej[END_REF], while Russian requires for this semantics a dative constituent (emu strašno 'he is scared', see above). On the other hand, this type of impersonal constructions is not attested in Czech (Holk 1983: 151), where there is a personal construction, which includes an experiencer-possessor in dative (hrdlo se mi svíralo 'my throat was clutching'). ( 23 The accusative argument should not be confused with a direct object in accusative because the impersonal construction uses an intransitive verb. This type of accusative is thus an argument of an intransitive verb. In contrast, in the construction (25a) below, the accusative kryšu 'roof' is a regular direct object of the transitive verb sorvat' 'to tore off'.

External force constructions

Constructions depicting damage caused by an external force,-a natural phenomenon or an unknown force,-occur in Russian with the instrumental case (25), while they use no nominal argument in a corresponding construction in West Slavic languages (26). In Serbian, where the instrumental case used in a subject fonction disapeared very early [START_REF] Ivić | Značenja srpskohrvatskog instrumentala i njihov razvoj[END_REF] According to most scholars of comparative Slavic studies, the "impersonal" instrumental, which is used, as we know, for a meteorological phenomenon or a force endowed with its own energy, takes a central place in East Slavic languages, in particular in Russian, but it has a rather peripheral status in other Slavic languages [START_REF] Mrázek | Státní pedagogické nakladatelství[END_REF][START_REF] Ivić | On the origin of the Russian sentence type: (Ego) zavalilo snegom[END_REF]. From the structural point of view, we can refer to R. Mrázek, who was aware of the specificity of the "impersonal" instrumental and its structurally indispensability, and compared it to a subject [START_REF] Mrázek | Státní pedagogické nakladatelství[END_REF], although this form should be placed at the far end of the subjectival scale (see above, § 2). On the other hand, from the typological point of view, it is possible to compare these models to ergative constructions that assign an oblique case to the subject of transitive verbs. But the neutralization of the role of agent is proper not only to ergative and impersonal but also to passive constructions (see § 6) (for impersonals and related constructions in Slavic languages, see Kor Chahine (ed.) 2013).

Constructions with indefinite subjects

It is worth mentioning another problem related to identifying impersonal constructions. In order to identify strictly impersonal constructions, we must eliminate all instances when the subject is missing due to the influence of context or because of pragmatic, lexical, and stylistic constraints, i.e. all elliptical or deleted subjects. The cases of zero subject must be eliminated as well. The ellipsis and zero subject always denote a significant absence. The elliptical subject may, however, be reinstated, which is not possible for zero subject: no lexeme can substitute a zero subject without changing the meaning of the construction. Among constructions without a nominative subject, i.e. impersonal constructions "in a large sense," are those types which refer to a generic or indeterminate subject. These constructions have different characteristics in each Slavic language, and their typology is yet to be examined.

There are two well-known variants of such constructions: constructions with zero subject, designating a person or a group of indeterminate persons, and, in principle excluding the speaker; and constructions including a generic human subject, among them the speaker. These constructions are common not only to Slavic domain but also to other languages of Europe [START_REF] Siewierska | Overlap and complementarity in reference impersonals : man-constructions vs. third person plural-impersonals in the languages of Europe[END_REF]). Both constructions, unlike impersonal constructions in a narrow sense, generally include an action verb. They also have a very real, although indeterminate, agent.

Constructions with zero subject

Constructions with zero subject occur in most Slavic languages. While the impersonal construction strictly speaking has a predicate in the 3rd person singular, which can be considered a mark of non-agreement, in constructions with indefinite subject, the verb is in the 3rd person plural and refers to a human subject, which can be determined by a local argument or by the situation.

(28) a. Ru. На заводе бастовали. Na zavode bastovali. at factory:loc striked:3pl 'There was a strike at the factory.' b. Pl. W fabryce strajkowali. at factory:loc striked:3pl 'There was a strike at the factory.' c. Cz. V továrně stávkovali. at factory:loc striked:3pl 'There was a strike at the factory.' d. S-Cr. У фабрици cу штраjковали. U fabrici su štrajkovali. at factory:loc are:3pl striked:3pl 'There was a strike at the factory.' e. Bg. В завода стачкуваха.

V zavoda stačkuvaxa. at factory striked:3pl 'There was a strike at the factory.'

Constructions with generic human subject

Constructions referring to a generic human subject, or "universal person", exist only in Russian. They contain a verbal predicate in the 2nd person singular which includes any referent comprising the speaker : (29) Ru. Никогда не знаешь, что может понадобиться в дороге.

(NKRJa 2015) Nikogda ne znaeš', čto možet ponadobit'sja v doroge. 'One never knows what can be useful on a trip.' Most Slavic languages express the generic human subject by constructions comprising a word etymologically related to the lexeme 'man' (such as Man in German or on in French) (pl. człowiek, bg. čovek, cz. človĕk, s.-cr. čov(j)ek). For this reason, these constructions are used to call « man-constructions » (cf. Siewierska 2011). However, these Slavic constructions are personal by their syntactic structure. (30) a. Pl. Człowiek nie wie, co ma robić.

'One does not know what to do.' b. Cz. Človĕk nikdy neví.

'One never knows.' c. S-Cr. Чов(ј)ек никад не зна шта га чека сутра. Čov(j)ek nikad ne zna šta ga čeka sutra.

Impersonal and related constructions

The absence of agent or its occasionally minor status makes impersonal constructions closer to other models that share some of their features. This often makes identifying these constructions difficult. While similarities between impersonal and passive constructions can be observed on the synchronic level, they can be seen on the diachronic level as well. In the history of Slavic languages, the development of impersonal constructions is often related to passive constructions. Thus, in Polish, the passive suffix -no/-to became an impersonal marker [START_REF] Pisarkowa | Historia składni języka polskiego[END_REF], which can be found in constructions like (35), with a direct object making the passive reading impossible. Today, they function as active constructions and are very common. (35) Podano herbatę. served:part.neut tea:acc 'Tea was served. ' Similarly, in Russian, the relatively late emergence of impersonal constructions with instrumental, such as (25a) may have originated from the expansion of the instrumental agent in passive constructions. In Old Russian, the agent was expressed by the form "ot + genitive," which subsequently became outdated and fell out of use [START_REF] Ivić | On the origin of the Russian sentence type: (Ego) zavalilo snegom[END_REF]. It is interesting to note that in similar modern Serbo-Croatian structures the "real" passive construction is used: an inanimate agent is encoded by the instrumental (36a), while an animate agent is encoded by the phrase "od + genitive" (36b). ( 36 Furthermore, there are similarities between passive constructions and constructions with "o + locative", such as (37), which replaced the former impersonal passive (see § 3 above) (Švedova 1964) The classification of these constructions among impersonal constructions is not always unanimous. In this construction, the postfix -sja attached to a verb can be seen as a morphological marker of impersonal subject originating from a verbal derivation, which is often confused with the process of passivization (see various types of passive constructions in [START_REF] Plungjan | Obščaja morfologija. Vvedenie v problematiku[END_REF].

In addition to these examples of direct links between the impersonals and passive, it should be noted that the domain of impersonal constructions has been gradually expanding at the expense of the passive. Thus, in Russian, in constructions with zero subject including a predicate in the 3rd person plural, the role of the agent is minimal, which makes them closer to passive constructions where the agent plays a peripheral role. Some scholars think that this construction compensates for an absent position in the passive paradigm, which does not possess all aspect and tense forms [START_REF] Wiemer | Analityczne passivum w języku rosyjskim i polskim[END_REF]. Besides, it is also possible to view impersonal constructions as a type of construction where the voice opposition is neutralized.

On the other hand, the relationship between impersonal and reflexive constructions is based on operations that affect the predicate-argument structure. From a certain point of view, this operation, which detransitivizes a transitive construction, can be linked to the phenomenon observed in ergative languages where this type of construction contains a dedicative anti-passive marker.

(

Table 1 .

 1 Ranking of Russian nominal constituants in a subjectal scale (inspired byGuiraud-Weber 2003a).

	Subjectal features	Prototypical subject	Nominative Dative Genitive Accusative Instrumental
	Predicative link	+	+	+	+	+	+
	Agreement	+	+	-	-	-	-
	Initial position	+	+	+	+	+	±
	Animated	+	±	+	±	+	-
	Agent	+	±	-	-	-	-
	Control of reflexive	+	+	+	-	-	-
	Control of gerund	+	+	+	-	-(+)	-

  th century has had the semantic function of experiencer replacing the construction "dlja + genitive" (see (3a-b)). During the XIX th century, serious confusion prevailed in this domain(Švedova 1964,), and variants with dlja are very rare in today's Russian (about Russian dative impersonal constructions in XVIII th -XXI nd s. also see[START_REF] Say | Leksičeskie mexanizmy grammatičeskogo drejfa: konstrukcii s dativnym sub"ektom v russkom jazyke XVIII-XXI vekov[END_REF].

	(2) a. Его	не видно.
	Ego	ne vidno.
	he:acc neg visible:pred
	'He is not visible.'
	b. С	деньгами было		плохо.
	S	den'gami bylo		ploxo.
	prep money:inst was:neut tight:pred
	'Money was tight.'
	c. Об	этом говорилось	в газетах.
	Ob	ètom	govorilos'	v gazetax.
	about this:loc discussed:neut.ref in newspapers:loc
	'This was discussed in the newspapers.'
	(3) a. Для	него	казалось	странным…
	Dlja nego	kazalos'		strannym…
	for	he:gen seemed:neut strange:inst
	'It seemed strange to him.'
	b. Для	него	было	приятно видеть…
	Dlja nego	bylo	prijatno	videt'…
	for	he:gen was:neut nice:pred see:inf
	'It was nice for him to see.'
	c. Ему	казалось	странным…
	Emu	kazalos'	strannym…
	he:dat seemed:neut strange:inst
	'It seemed strange to him.'
	d. Ему	было	приятно видеть…
	Emu	bylo	prijatno	videt'…

; 4. Lexical and syntactic reorganization of constructions with dative argument (see (3c-d)), which since the XIX he:dat was:neut nice:pred see:inf 'It was nice for him to see.'

  . The noun in genitive denoting an action is used to be a deverbative.

	'Peter does not feel like laughing.'
	d. S-Cr. Ко пева зло не мисли, а ко мисли, ниjе му	до песме.	(Tanasić 1998: 89)
	Ko peva zlo ne misli, a ko misli,	nije mu	do pesme.
				is.neg:3sg he:dat prep song:gen
	'He didn't think singing bad, and if he did, he did not feel like singing.'
	e. Bg. Не ми е		до шега.
	Ne mi e		do šega.
	neg I:dat is:3sg prep joke:gen
	'I do not feel like joking.'
	(15) a. Pl. Jest mi	do płaczu.			(Zaremba, pc)
	is:3sg I:dat prep cry:gen
	'I feel like crying.'
	b. Cz. Petrovi bylo	do	smíchu.
	Peter:dat was:neut prep laughter:gen
	'Peter felt like laughing.'
	c. S-Cr. Њему jе		било	до круне	Немањића, а	не до	оних костиjу.(Tanasić 1998: 89)
	Njemu je		bilo	do krune	Nemanjića, a	ne do	onih kostiju.
	he:dat is:3sg was:neut prep crown:gen N.:gen,	conj neg prep their borns:gen
	'He cared about Nemanić's crown, not about their borns.'
	(14) a. Ru. Когда в стране хаос, обычным людям	не до	политики.	(NKRJa 2011)
	kogda v strane xaos, obyčnym	ljudjam	ne do	politiki.
			ordinary:dat people:dat neg prep politics:gen
	'Then a country faces a chaotic situation, ordinary people don't care about politics.'
	b. Pl. Nie jest mi	do	śmiechu.			(Zaremba, pc)
	neg is:3sg I:dat prep laughter:gen
	'I don't feel like laughing.'
	c. Cz. Petrovi nebylo		do smíchu.
	Peter:dat was:neut.neg prep laughter:gen

  , the external force constructions are usually nominative active constructions, and are rarely passive (27).

	(25) Russian
	a. Бурей	залило	двор	и	здания.	(NKRJa 2009)
	Burej	zalilo	dvor	i	zdanija.
	storm:inst flooded:neut courtyard:acc conj buidings:acc
	'The storm flooded the courtyard and buildings.'
	b. На ноге, повыше колена, осколком вырвало		кусок	мяса.	(NKRJa 1994)
	Na noge, povyše kolena, oskolkom	vyrvalo		kusok	mjasa.
			shrapnel:inst tore.off:neut piece:acc flesh:gen
	'The shrapnel tore a piece of flesh off his leg, above the knee.'
	(26) a. Pl. Zasypało	drogę.
	covered:neut road:acc
	'The road was covered.'
	b. Cz. Souseda	ranilo.
	neighbor:acc wounded:neut
	'The neighbor was wounded.'
	(27) Serbian
	a. Цело небо	су	прекрили	облаци.		(Babić 2015: 106)
	Celo nebo	su	prekrili	oblaci.
	all sky:acc are:3pl covered:3pl clouds:nom
	'The entire sky was covered with clouds.'
	b. Цело небо	је	прекривено		облацима.	(Vojvodić, pc)
	Celo nebo	je	prekriveno		oblacima.
	all	sky:nom is:3sg covered:part.pst.neut	clouds:inst
	'The entire sky was covered with clouds.'

  :nom is:3sg awarded:part.m by side:gen president:gen 'He was awarded by the president.'

	) Serbo-Croatian				(Vojvodić, pc)
	a. Пут		jе	затрпан	снегом.
	Put		je	zatrpan	snegom.
	road:nom.m is:3sg covered:part.m snow:inst
	'The road is covered with snow.'
	b. Он	jе	одликован	од стране председника.
	On	je	odlikovan	od strane	predsednika.

he

  in books:loc and in films:loc about love:loc discussed:neut.ref very much 'There was a lot of talks about love in books and movies.'

								.
	(37) В книгах и	в кино	о	любви	говорилось	очень много.	(NKRJa 1970)
	V knigax	i	v kino	o	ljubvi	govorilos'	očen' mnogo.

Here we do not make a distinction between Serbian and Croatian.

Nacional'nyj Korpus Russkogo Jazyka (National Russian Corpus) -ruscorpora.ru

'One never knows what to expect tomorrow.' d. Bg. Čovek ne znae nikoga kakvo može da se sluči.

'One never knows what might happen'.

Constructions with indefinite subject

A third variant of these impersonal constructions, constructions with indefinite subject, realizes in Russian in constructions like (31) which not to be confused with strictly impersonal constructions. The zero subject here can either refer to a indefinite person or to an animal, as well as to an indefinite object [START_REF] Georgieva | K voprosu o granicax bezličnyx predloženij russkogo jazyka. Učenye zapiski Moskovskogo Gosudarstvennogo Pedagogičeskogo Instituta[END_REF](Georgieva , 1978)). In other words, it does not necessarily refer to a person (like in the two types presented above), and its predicate is in the 3rd person singular. One can observe a certain correspondence of such constructions with the constructions existing in other Slavic languages that also use a verb in the 3rd person singular without an explicit subject, but in this case the zero subject refers to a person (not to an inanimate object), and very often to the speaker. Such constructions always use a reflexive verb. It's worth to note that in colloquial Czech, a reflexive marker can also be applied to an entier chain of coordinated verbs (34b). ( 32 It is precisely these forms of indefinite subject that have not been lexicalized in Slavic languages.


Ghazaleh Khodabandelou
email: ghazaleh@lptmc.jussieu.fr

Mohammad Mehdi Ebadzadeh

Fuzzy Neural Network with Support Vector-based Learning for Classification and Regression

Keywords: Fuzzy neural network (FNN), Support vector machine (SVM), Adaptive fuzzy kernel, Classification, Regression

Fuzzy Neural Network (FNN) and Support Vector Machine (SVM) are two prominent and powerful learning models broadly used for classification and regression. FNN has a significant local representation and human reasoning advantage. However, the drawback of such a network is that the focal point of the learning algorithms (e.g., backpropagation) is minimizing empirical risk. In contrary to FNN, SVM emphasizes simultaneously on minimizing empirical and expected risks, which theoretically leads to an excellent generalization performance power. Nevertheless, this depends on determining an adequate kernel function which is consistent with the data properties. Some approaches have been proposed to take advantage of one to deal with disadvantage of the other by combining SVM kernel and FFN. In this paper, we show that a Takagi-Sugeno-Kang (TSK)-type-based fuzzy neural network is, in fact, equivalent to an SVM with an adaptive kernel based on fuzzy rules generated in this FNN. Consequently, it is possible to learn the last layer of the FNN using the concepts of SVM and thus taking the advantage of SVM in generalization. In fact, on the one side, the proposed method is an SVM with an adaptive kernel based on fuzzy rules and on the other side, it is a TSK-FNN with SVM-based learning. Therefore, the FNN with SVM-based learning has the benefit of SVM which is minimizing both learning and testing errors. Besides, such FNN does not have the disadvantages of the SVM which is finding an appropriate kernel and training the kernel parameters. As a matter of fact, the defined kernel in FNN is an adaptive kernel based on data characteristics, which is derived from the fuzzy rules generated by the FNN itself. As a result, it is expected that the results of the proposed method for classification and regression outperform the results of SVM with the conventional kernels and fuzzy neural network trained in the conventional way. This claim is confirmed with the results obtained in this paper.

Introduction

The two most salient techniques for pattern classification and regression are the Fuzzy neural network (FNN) and support vector machine (SVM). SVM is a kernel-based machine learning method which is based on the statistical learning theory [START_REF] Vapnik | The nature of statistical learning theory[END_REF]. The focus of SVM, in contrary to FNN, is simultaneously on minimizing empirical and expected risks. This theoretically leads to an excellent generalization performance strength. Nevertheless, this highly depends on determining an adequate kernel function that is consistent with the data properties [START_REF] Cristianini | An introduction to support vector machines and other kernel-based learning methods[END_REF]. Determining manually an appropriate kernel function or adjusting its parameters is an exhaustive task since it requires specific knowledge of the field. This issue has been addressed in several studies by automatically identifying the proper kernel using prior knowledge [START_REF] Chapelle | Choosing multiple parameters for support vector machines[END_REF][START_REF] Cristianini | On kernel-target alignment[END_REF][START_REF] Lanckriet | Learning the kernel matrix with semidefinite programming[END_REF][START_REF] Ong | Learning the kernel with hyperkernels[END_REF][START_REF] Rakotomamonjy | Simplemkl[END_REF].

A fuzzy system can estimate any continuous real function specified on a bounded domain as shown in [START_REF] Kosko | Fuzzy systems as universal approximators[END_REF]. Using a collection of fuzzy rules the fuzzy system covers the function graph in input-output data. Integration of fuzzy system to neural networks has given birth to hybrid fuzzy neural networks (FNNs) [START_REF] Nauck | Function approximation by nefprox[END_REF]. These networks are considered as a particular type of neural networks due to their similarity to neural networks. The learning process of FNN is usually performed using different algorithms in which neurons weights represent rules in the fuzzy system. In this regards, the adaptive-network-based fuzzy inference systems (ANFIS) has been proposed in [START_REF] Jang | Anfis: adaptive-network-based fuzzy inference system[END_REF] where the system is formulated based on both specified input-output space and humans' cognition. This system has yielded significant performance and since has been extensively used in modeling and handling nonlinear systems [START_REF] Schilling | Approximation of nonlinear systems with radial basis function neural networks[END_REF]. ANFIS uses Takagi-Sugeno-Kang fuzzy system [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF] with combining gradient decent and least square method in the learning procedure [START_REF] Nauck | Neuro-fuzzy systems for function approximation[END_REF]. Nevertheless, ANFIS is affected by the curse of dimensionality when faced with the large number of input dimension (i.e., the number of fuzzy rules). Therefore, it is necessary to integrate into the learning algorithm a reliable method to reduce the number of fuzzy rules. FNNs have some advantages such as human-like reasoning and remarkable local representation [START_REF] Ebadzadeh | Cfnn: correlated fuzzy neural network[END_REF][START_REF] Lin | Neural fuzzy systems: a neuro-fuzzy synergism to intelligent systems[END_REF]. However, the drawback of such networks is that their learning algorithms (i.e., back-propagation) only focus on minimizing empirical risk [START_REF] Chai | Mamdani model based adaptive neural fuzzy inference system and its application[END_REF][START_REF] Gabrys | General fuzzy min-max neural network for clustering and classification[END_REF]. Furthermore, a major challenge of existing FNN methods is dealing with a large number of fuzzy rules. Therefore, several approaches have been proposed in the literature for optimizing the number of rules in hybrid FNN such as dividing the input space by some unsupervised algorithms [START_REF] Dickerson | Fuzzy function approximation with ellipsoidal rules[END_REF][START_REF] Kroll | Identification of functional fuzzy models using multidimensional reference fuzzy sets[END_REF] or by clustering algorithms [START_REF] Klawonn | Constructing a fuzzy controller from data[END_REF][START_REF] Oh | Polynomial-based radial basis function neural networks (p-rbf nns) realized with the aid of particle swarm optimization[END_REF]. Some methods proposed to consider both input and output data to determine fuzzy rules [START_REF] González | A new clustering technique for function approximation[END_REF][START_REF] Wang | Efficient neuro-fuzzy control systems for autonomous underwater vehicle control[END_REF]. In [START_REF] Chuang | Robust tsk fuzzy modeling for function approximation with outliers[END_REF][START_REF] Leng | An on-line algorithm for creating self-organizing fuzzy neural networks[END_REF] the cluster number has been manually specified, which is not guaranteed to return an optimal solution. All these methods are human-in-the-loop systems which require prior establishment of fuzzy rules numbers by an expert. Consequently, to overcome this issue dynamic adaptive methods have been proposed in several studies [START_REF] Chuang | Robust tsk fuzzy modeling for function approximation with outliers[END_REF][START_REF] Chai | Mamdani model based adaptive neural fuzzy inference system and its application[END_REF][START_REF] Ebadzadeh | Cfnn: correlated fuzzy neural network[END_REF][START_REF] Hsu | A comparison of methods for multiclass support vector machines[END_REF][START_REF] Ester | A density-based algorithm for discovering clusters in large spatial databases with noise[END_REF]. Some efforts have been made to surmount the challenges in SVM and FNN by leveraging the properties of both techniques (i.e., by developing a theoretical foundation of FNN using the SVM learning theory) [START_REF] Chen | Support vector learning for fuzzy rule-based classification systems[END_REF][START_REF] Chiang | Support vector learning mechanism for fuzzy rule-based modeling: a new approach[END_REF][START_REF] Juang | A self-organizing ts-type fuzzy network with support vector learning and its application to classification problems[END_REF][START_REF] Lin | Fuzzy neural network classification design using support vector machine[END_REF][START_REF] Lin | Support-vector-based fuzzy neural network for pattern classification[END_REF]. Despite the fact that all these methods use an elimination procedure to reduce the number of fuzzy rules, they still require a high number of fuzzy rules to achieve high learning performance. Besides, it is crucial that these fuzzy rules be provided automatically by an adaptive fuzzy system.

In order to address the aforementioned challenges, we leverage the underlying relation between SVM kernel and FNN. We propose a new combinatorial method, so-called Fuzzy Neural Network with Support Vector-based learning (FNN-SVBL), to take advantage of this underlying relation. In this paper, we show that a TSK-type fuzzy neural network is, in fact, equivalent to an SVM with an adaptive kernel based on fuzzy rules generated by TSK-FNN. Therefore, it is possible to learn the last layer of the TSK-FNN using the concepts of SVM and thus obtain the benefits of SVM in generalization. In fact, on the one side, the proposed method is an SVM with an adaptive kernel based on fuzzy rules and on the other side, it is a TSK-FNN with SVM-based learning in the last layer. Therefore, the FNN with SVM-based learning has the advantages of SVM, which minimizes both learning and testing errors. Besides, such FNN does not have the disadvantages of the SVM which is finding an appropriate kernel and training the kernel parameters. As a result, the defined kernel in FNN is an adaptive kernel based on data characteristics, which is derived from the fuzzy rules generated by the FNN itself. We use a Gaussian Mixture Model (GMM) to generate the fuzzy rules of FNN-SVBL. The number of fuzzy rules for each dataset is determined as in [START_REF] Ganji | Support vector-based fuzzy classifier with adaptive kernel[END_REF]. For evaluating the classification performance of FNN-SVBL we use some well-known benchmark datasets. These datasets are accessible at the UCI repository of machine learning databases [START_REF] Blake | UCI Machine Learning Repository[END_REF] and the Statlog collection [START_REF] Dj | Machine learning, neural and statistical classification[END_REF].

Also, we use two simulated datasets for the regression performance assessment. Finally, we assess the sensitivity of the proposed method to the number of fuzzy rules. As the experimental results delineate the classification and regression performances of FNN-SVBL are competitive or even superior to stateof-the-art methods while the fuzzy rules number is substantially lower than those methods. The remaining sections of the paper are organized as follows. In Section 2, we discuss briefly some related previous works. Section 3 and 4 briefly reviews the SVM learning theory and the GMM algorithm, respectively. In Section 5, we describe the fuzzy neural network. In Section 6 we explain the learning method based on support vector for the last layer of the FNN-SVBL. In Section 7, we present and discuss in detail, some experimental results and comparisons. Finally, in Section 8, we conclude the paper.

Related works

In this section, we review the state-of-the-art of existing work focusing on combination of fuzzy theory techniques and kernel learning.

Positive-definite kernel functions on fuzzy sets

Some approaches have been dealing with popular machine learning problems such as classification [START_REF] Chen | Support vector learning for fuzzy rule-based classification systems[END_REF][START_REF] Lin | Support-vector-based fuzzy neural network for pattern classification[END_REF], clustering [START_REF] Huang | Multiple kernel fuzzy clustering[END_REF][START_REF] Kanzawa | On kernel fuzzy c-means for data with tolerance using explicit mapping for kernel data analysis[END_REF], feature extraction [START_REF] Wang | Fuzzy maximum scatter discriminant analysis with kernel methods[END_REF], discriminant analysis [START_REF] Heo | Robust kernel discriminant analysis using fuzzy memberships[END_REF] by combining fuzzy techniques with positive definite kernels. Incorporating non-vectorial fuzzy data into a normed vectorial space such as kernel Hilbert space (RKHS) has given birth to some innovating methods [START_REF] Guevara | Supervised machine learning with kernel embeddings of fuzzy sets and probability measures[END_REF][START_REF] Guevara | Positive definite kernel functions on fuzzy sets[END_REF], which use a positive definite kernel function specified on fuzzy sets. The kernel proposed in [START_REF] Guevara | Cross product kernels for fuzzy set similarity[END_REF] measures the resemblance between fuzzy sets in RKHS. It has been demonstrated in this approach that such a kernel is a special case of the convolution kernel. An improvement of [START_REF] Zhang | Fuzzy clustering neural network (fcnn): competitive learning and parallel architecture[END_REF] has been proposed in [START_REF] Wang | Robust fuzzy clustering neural network based on -insensitive loss function[END_REF] to deal with the sensitivity problem of clustering in the presence of outliers.

SVM kernel learning

Several methods of SVM kernel learning have been proposed in the literature. These methods aim at learning the kernel automatically. Also, they can be used to determine the kernel function of SVM.

The approach proposed in [START_REF] Ong | Learning the kernel with hyperkernels[END_REF] has chosen a classic kernel in order to specify the kernel parameters by optimizing its quality function. Some methods have been defined the quality function as class separability [START_REF] Xiong | Optimizing the kernel in the empirical feature space[END_REF][START_REF] Zhong | Optimizing the gaussian kernel function with the formulated kernel target alignment criterion for two-class pattern classification[END_REF], Bayesian probabilities [START_REF] Sollich | Bayesian methods for support vector machines: Evidence and predictive class probabilities[END_REF][START_REF] Khan | Sn-svm: a sparse nonparametric support vector machine classifier[END_REF], generalization error [START_REF] Chapelle | Choosing multiple parameters for support vector machines[END_REF], alignment-based techniques [START_REF] Cortes | Two-stage learning kernel algorithms[END_REF][START_REF] Cristianini | On kernel-target alignment[END_REF] or cross-validation error [START_REF] Lee | An efficient method for computing leave-one-out error in support vector machines with gaussian kernels[END_REF]. Certain approaches have been combined the classic kernels [START_REF] Aiolli | Easymkl: a scalable multiple kernel learning algorithm[END_REF][START_REF] Rakotomamonjy | Simplemkl[END_REF][START_REF] Kloft | Lp-norm multiple kernel learning[END_REF]. The combination coefficients can be estimated through convex optimization. However, this approach is time-consuming and resource intensive. An inductive approach, known as multiple kernel learning, consists in learning a combined kernel from a restricted classic kernels set [START_REF] Argyriou | Learning convex combinations of continuously parameterized basic kernels[END_REF][START_REF] Jawanpuria | Generalized hierarchical kernel learning[END_REF][START_REF] Lanckriet | Learning the kernel matrix with semidefinite programming[END_REF]. Nevertheless, there are a limited learnable kernels class. In [START_REF] Lin | Support-vector-based fuzzy neural network for pattern classification[END_REF] a customized similarity measurement function has been used to estimate the kernel matrix inputs. Some methods have been proposed to set up the SVM kernel function [START_REF] Chapelle | Choosing multiple parameters for support vector machines[END_REF][START_REF] Lee | An efficient method for computing leave-one-out error in support vector machines with gaussian kernels[END_REF][START_REF] Sollich | Bayesian methods for support vector machines: Evidence and predictive class probabilities[END_REF][START_REF] Lin | Support-vector-based fuzzy neural network for pattern classification[END_REF].

Combination of fuzzy systems and SVM

A widely known method, called FSVMs, takes advantage of both SVM and fuzzy systems by redefining the SVM optimization problem using fuzzy theory. In this method, a fuzzy membership value is applied to each input point such that these latter participate differently to the learning of decision surface [START_REF] Lin | Fuzzy support vector machines[END_REF]. Similarly, some methods have been proposed to include the membership value of each input point in the penalty term [START_REF] Batuwita | Fsvm-cil: fuzzy support vector machines for class imbalance learning[END_REF][START_REF] Wang | A new fuzzy support vector machine to evaluate credit risk[END_REF]. The proposed method in [START_REF] Batuwita | Fsvm-cil: fuzzy support vector machines for class imbalance learning[END_REF] attempts to improve FSVMs for handling the class imbalance in the presence of outliers and noise. Although using these methods contributes to reduce the class imbalance effect, the principal issue is finding a way to assess an adequate fuzzy membership [START_REF] Fan | Entropy-based fuzzy support vector machine for imbalanced datasets[END_REF]. An interval type-2 fuzzy support vector machine has been proposed in [START_REF] Ekong | Classification of epilepsy seizure phase using interval type-2 fuzzy support vector machines[END_REF] to classify patients to three classes. In [START_REF] Esme | Fuzzy c-means based support vector machines classifier for perfume recognition[END_REF] a hybrid classifier has been proposed which is composed of unsupervised fuzzy clustering c-mean and supervised support vector machine. In a multiclass context, an approach has been proposed in [START_REF] Tsujinishi | Fuzzy least squares support vector machines for multiclass problems[END_REF] to determine classes in "unclassifiable regions". This approach consists in specifying a fuzzy membership function for each class found on the SVM separating hyperplane. In [START_REF] Wu | A fuzzy support vector machine algorithm for classification based on a novel pim fuzzy clustering method[END_REF] the authors have studied a new clustering algorithm to achieve a more solid fuzzy membership for fuzzy SVM. They have proposed to ameliorate the partition index maximization by obtaining appropriate data centroid. The fuzzy SVM has been used to influence the classification decision function.

SVM-based fuzzy neural networks

Several approaches, including the proposed method in this paper, focus on expanding a theoretical foundation for FNN based on the SVM learning theory. In [START_REF] Lin | Support-vector-based fuzzy neural network for pattern classification[END_REF] a support vector-based fuzzy neural network has been proposed for patterns classification. First, the fuzzy rules and membership functions were defined using clustering rules. Second, the parameters of FNN were estimated by the SVM. Finally, the fuzzy rules were chosen by the proposed fuzzy rule method by omitting irrelevant fuzzy rules. An SVM has been used with a Gaussian kernel on a task training set in [START_REF] Chen | Support vector learning for fuzzy rule-based classification systems[END_REF][START_REF] Chiang | Support vector learning mechanism for fuzzy rule-based modeling: a new approach[END_REF]. In these approaches, the extracted support vectors were used to generate fuzzy rules of FNN. The proposed approach in [START_REF] Lin | Fuzzy neural network classification design using support vector machine[END_REF] has tackled the problem of minimizing the training and testing errors simultaneously in a pattern classification context. To this end, a support-vector-based fuzzy neural network classification was proposed using jointly the classification strength of SVM and the efficient human-like reasoning of FNN. However, the main drawback of these approaches [START_REF] Lin | Fuzzy neural network classification design using support vector machine[END_REF][START_REF] Chiang | Support vector learning mechanism for fuzzy rule-based modeling: a new approach[END_REF][START_REF] Chen | Support vector learning for fuzzy rule-based classification systems[END_REF] is when the fuzzy rules number becomes equivalent to the support vectors number, which is generally very large. In all aforementioned methods, the consequent of fuzzy rules is of the fuzzy singleton type. A Takagi-Sugeno fuzzy system for a classification objective has been proposed in [START_REF] Juang | A ts fuzzy system learned through a support vector machine in principal component space for real-time object detection[END_REF] which uses a linear SVM to tune the PCA parameters. A fuzzy system was proposed in [START_REF] Juang | A self-organizing ts-type fuzzy network with support vector learning and its application to classification problems[END_REF] which is generated by the crossbreeding of fuzzy clustering and SVM. Here, the assumption is that a fuzzy system with Takagi-Sugeno outperforms a fuzzy system with the fuzzy singletons, as shown in [START_REF] Juang | A tsk-type recurrent fuzzy network for dynamic systems processing by neural network and genetic algorithms[END_REF][START_REF] Juang | An online self-constructing neural fuzzy inference network and its applications[END_REF]. In [START_REF] Guevara | Kernel functions in takagi-sugeno-kang fuzzy system with nonsingleton fuzzy input[END_REF] the authors have studied the relationship among non singleton fuzzy input, TSK fuzzy systems, and positive definite kernels.

Support vector machine

The learning theory of SVM has been developed in [START_REF] Vapnik | The nature of statistical learning theory[END_REF]. In this section, we describe the learning process of SVM in a binary classification issue. Let S = {(X 1 , y 1 ), (X 2 , y 2 ), • • • , (X n , y n )} be a set of training data, where X i ∈ R d . Each input X i is a d-dimensional real vector and y i ∈ {-1, +1} stands for the input labels. SVM is a classifier specified by a separating hyperplane which tends to maximize the margin width between two classes. For linearly separable data the hyperplane is defined as:

y i (W T X i + b) ≥ 1 -ε i , i = {1, • • • , n}, ε i ≥ 0 (1)
where

W ∈ R d , b ∈ R and ε i ≥ 0.
The slack variable is presented by ε i which is the smallest nonnegative number satisfying Eq.( 1). In cases where values of ε i > 1 the data are misclassified. Therefore, to discover the separating hyperplane having the maximum margin, the constrained optimization problem with a differentiable objective function can be solved as follows:

arg min 1 2 W 2 + C n i=1 ε i subject to y i (W T X i + b) ≥ 1 -ε i , i = {1, • • • , n} (2)
where ε i stands for a slack variable for the ith sample, C is a cost parameter and n i=1 ε i stands for an upper bound of the expected training error (the primal problem). By constructing a Lagrangian representation for the above quadratic optimization problem, one obtains the following dual problem:

arg max( n i=1 α i - 1 2 n i=1 n j=1 α i α j y i y j X T i X j ) subject to n i=1 α i y i = 0, α i ≥ 0, i = {1, • • • , n} (3)
where α i are Lagrange multipliers and the dual maximization problem is a quadratic function of the α i subject to linear constraints. One obtains α i by solving Eq.( 3) so that the final hyperplane decision function is achieved as follow:

f (X) = sign{W T X + b} = sign    X i ∈SV α i y i X T i X + b    ( 4
)
where b is the offset and SV are the support vectors, with the nonzero α i values in the training samples set. The derivation process is explained in detail in [START_REF] Ong | Learning the kernel with hyperkernels[END_REF]. In order to maximize the separability among classes mainly for non-linearly separable data, one can use a mapping function φ(X) to transform the data into a higher dimensional feature space. To do so, X i in Eq.( 1), ( 2), ( 3) and ( 4) can be transformed into φ(X i ) and X in Eq.( 4) into φ(X). The SVM allows for defining implicitly the mapping function by applying a kernel function such that K(X i , X j ) = φ(X i ) T φ(X j ). Therefore, the dual optimization problem in Eq.( 3) can be transformed into its kernel form. The final decision function is defined as follows:

f (X) = sign    X i ∈SV α i y i k(X i , X) + b    (5)
where b can be established by the Karush-Kuhn-Tucker (KKT) complementary slackness conditions [START_REF] Vapnik | The nature of statistical learning theory[END_REF].

Gaussian mixture model clustering algorithm

Clustering algorithms can be widely categorized into two classes: model-based and non-modelbased algorithms. Model-based clustering makes an assumption that an underlying model has generated the data and attempts to discover the original model from the data. This kind of clustering is based on a limited mixture of distributions in which each mixture component is related to a different cluster [START_REF] Fraley | Model-based clustering, discriminant analysis, and density estimation[END_REF]. A commonly used algorithm in model-based clustering context is the Expectation-Maximization (EM) algorithm which allows estimating the finite mixture models [START_REF] Fraley | How many clusters? which clustering method? answers via model-based cluster analysis[END_REF]. The Bayesian information criterion is widely used to determine the number of components in a Gaussian Mixture.

Other approaches are cross-validation or heuristic-based greedy [START_REF] Vlassis | A greedy em algorithm for gaussian mixture learning[END_REF]. In non-model-based clustering algorithms, which are based on the distance and similarity between clusters, the assumption is the data do not follow a particular model [START_REF] Karypis | Chameleon: Hierarchical clustering using dynamic modeling[END_REF][START_REF] Ester | A density-based algorithm for discovering clusters in large spatial databases with noise[END_REF]. In this paper, we use the Gaussian mixture model algorithm (GMM) that is a probabilistic model assuming all the data points are generated from a mixture of a limited number of Gaussian distributions with unknown parameters. A GMM is usually used to estimate the distribution of random variables as a sum of several Gaussians. An iterative procedure optimizes the Gaussians parameters (e.g., variance, average, and amplitude) via a maximum likelihood criterion. This procedure is done using the EM algorithm. It should be pointed out that for the sake of comparison, we use also Fuzzy c-means (FCM) and Hierarchical Cluster Analysis (HCA) for both classification and regression. The results are reported in section 7.5.

Fuzzy neural network

A fuzzy system consists of a collection of fuzzy rules utilizing human knowledge processing ability. Fuzzy neural networks (FNNs) exploit both the fuzzy system and neural networks abilities. On the one hand, FNNs learning algorithm is inspired by the ability of neural network theory in learning from processes, on the other hand, FNNs use fuzzy systems reasoning in handling unknown information.

Fuzzy rules are composed of different preconditions assessing the association of the input variables to fuzzy sets and one consequence (i.e., fuzzy singleton, fuzzy sets or TSK-type consequence). As shown in [START_REF] Juang | An online self-constructing neural fuzzy inference network and its applications[END_REF][START_REF] Juang | A tsk-type recurrent fuzzy network for dynamic systems processing by neural network and genetic algorithms[END_REF] facing sophisticated problems a FNN acquiescent with TSK-type consequence usually outperforms fuzzy singleton or fuzzy set consequence. Each rule in a TSK-type fuzzy system is defined as follows:

Rule c : IF x 1 is A c1 AND • • • AND x d is A cd THEN y = β c0 + d i β ci x i , c = {1, • • • , n c } (6)
where A ci represents the fuzzy set of the input variables x i , β ci is a real number, n c is the number of fuzzy rules and d is the number of dimensions. As depicted in Fig. 1, the network structure of the proposed model (FNN-SVBL) is a FNN with fuzzy rules consequences of TSK-type. FNN-SVBL consists of five layers including the data input, membership function, rule, consequent and output layers, respectively. The first step is to use GMM clustering algorithm to construct the FNN fuzzy rules by dividing input space into overlapping clusters. In fact, the generated clusters correspond to rules in the rule-base. This step is necessary to tune the parameters of FNN-SVBL by constructing the SVM kernel. The clustering of input space has to be applied on the basis that the points in a cluster have the same values or belong to the same class of the output variable. To this end, first, the training set is partitioned into two subsets using class information such that each subset comprises the same class samples. Second, GMM is applied to each subset to obtain two categories of clusters. In the training phase, the class information is explicitly considered to construct fuzzy rules. In the testing phase, the input data influence directly fuzzy rules with no need for such information. The parameters of the Gaussian function for a cluster c in the input space are represented by µ c and Σ c , which stand for mean and diagonal covariance matrix of samples, respectively. Thereby, the Gaussian function is represented as follows:

g c (X) = exp -γ(X -µ c ) T Σ -1 c (X -µ c ) (7)
where the parameter γ stands for the inverse of the radius of influence of samples selected by the model as support vectors. The preconditions of the fuzzy rules are determined using the clusters generated in the previous step. Since GMM generates orthogonal clusters, the membership functions can be achieved by projecting them onto the input variables in the cluster space.

Let I (L) and O (L) be the input and output variables of a node in layer L of the FNN-SVBL network, respectively. In what follows, we describe the basic functions and the signal propagation in each layer of the network: Layer 1 -Input layer. This layer is the input layer and each node represents an input variable. In this layer, the input values are directly forwarded towards the next layer. The output of this layer is defined as:

O (1) i = I (1) i = x i , (8)
where x i is the ith input variables of the network.

Layer 2 -Membership function layer. In this layer, nodes represent membership functions corresponding to a fuzzy set of the input variables. Indeed, the membership functions evaluate the membership degree of input values to that fuzzy set. The membership value of each node is calculated as follows:

O (2) c i = exp -(x i -µ c i ) 2 σ 2 c i = M c i (x i ), (9)
where O

c i stands for the output of the ith membership function of the cth cluster (rule).

210 Layer 3 -Rule layer. In this layer, nodes represent fuzzy logic rules and carry out fuzzy preconditioning of the rules. In fact, the output of this layer represents the firing force of a fuzzy rule. The AND operation is applied to all nodes of this layer as follows:

O (3) c = d i=1 I (3) c i = d i=1 O (2) c i = d i=1 M c i (x i ), ( 10
)
When X is the network input vector, g c (X) in Eq.( 7) is equivalent to O

c . Thus, the Eq.( 10) can be defined as follows:

O (3) c = d i=1 exp - (x i -µ c i ) 2 σ 2 c i = exp - d i=1 (x i -µ c i ) 2 σ 2 c i = exp -(X -µ c ) T Σ -1 c (X -µ c ) = g c (X), (11)
where Σ c is a diagonal matrix such that each diagonal element of Σ c ii = σ

(2)

c i .
Layer 4 -Consequent layer. In this layer, the TSK-type consequence of the fuzzy rules are generated by the nodes. A linear combination of the input vector β T c X is applied to cth rule (node), where β c represents the network parameter to be adjusted by SVM. The output value of the cth node is as follows:

O (4) c = β c0 + β T c X × exp -(X -µ c ) T Σ -1 c (X -µ c ) . ( 12
)
Layer 5 -Output layer. In this layer, the global output is calculated as the weighted sum of all the inputs with a bias term. The output of this layer is as follows:

O (5) = nc c=1 β c0 + β T c X × exp -(X -µ c ) T Σ -1 c (X -µ c ) + b = nc c=1 β c0 + β T c X × g c (X) + b (13)
Consequently, the decision function of FNN-FVBL network can be written as follows:

f (X) = sign nc c=1 β c0 + β T c X × g c (X) + b (14)
In the next section, it will be shown that the Eq.( 14) can be equivalent to the Eq.( 5), or, in other words, the parameters of β in the FNN can be written in terms of the parameters of α in the SVM.

Therefore, instead of learning the values of β in the FNN, one can learn the values of α in the SVM, and then, based on their underlying relationship, the values of β can be obtained from the values of α.

Proposed method

In this section, we show that the Eq.( 14) can be equivalent to the Eq.( 5), or in other terms, a FNN equivalent to an SVM with an adaptive fuzzy rule-based kernel. If we consider the Eq.( 14) obtained in the previous section, which is the output equation of the FNN, then we have:

f (X) = nc c=1 (β c0 + β T c X)g c (X) + b (15)
with considering :

β c0 = SV i=1 α i y i g c (X i ), β c = SV i=1 α i y i g c (X i )X i (16)
by replacing Eq.( 16) in Eq.( 15), we obtain:

f (X) = nc c=1 SV i=1 α i y i g c (X i )g c (X)(1 + X T i X) + b = SV i=1 α i y i (1 + X T i X) nc c=1 g c (X i )g c (X) (17)
with comparing Eq.( 5) and Eq.( 17) then:

K(X i , X) = (1 + X T i X) nc c=1 g c (X i )g c (X) (18)
where K : IR d × IR d → IR + 0 , functions g c (X i ) and g c (X) determine the membership degree of a sample to cluster c and n c represents the number of clusters.

The kernel K(X i , X) is adaptive since the number of Gaussian functions in g c (X i ) and g c (X) along with their parameters are automatically determined according to data distribution of the problem. Indeed, in the clustering phase, the data distribution is approximated by a set of Gaussian functions.

Experimental results

In this section, we first explain the benchmark datasets used to evaluate the method robustness. In order to evaluate the classification and regression performance, we then use conventional metrics. Last, we verify parameter sensitivity of FNN-SVBL on some datasets.

Classification datasets

In order to perform experiments for classification problem, we use six widely-used benchmark datasets accessible at the Statlog collection [START_REF] Dj | Machine learning, neural and statistical classification[END_REF] and the UCI repository of machine learning databases [START_REF] Blake | UCI Machine Learning Repository[END_REF]. Hereafter, we describe briefly the chosen datasets. We The Iris, Glass, and Wine datasets are selected from the UCI Repository. The Iris is the well-known dataset for pattern recognition problems. It contains 150 equally distributed samples each of them is characterized by four attributes information: septal length, septal width, petal length and petal width in centimeter. This dataset includes 3 classes of Iris flowering plant referring to a type of the plant, named Setosa, Virginica, and Versicolor. The Glass dataset consists of 214 instances defined by 10 continuous attributes (including an Id). The dataset is categorized into six classes and the data are the results of a chemical analysis of wines grown in the same region. The 178 samples are classified into three classes as the data are obtained from three different cultivars. The data are represented by 13 attributes since the analysis confirmed the presence of 13 constituents found in each of the three types of wines. The classes are separable, though RDA [START_REF] Aeberhard | The classification performance of rda[END_REF] has reached 100% correct classification.

Regression datasets

In this section, the performance of the proposed method is assessed using two simulated datasets: nonlinear two-inputs function and a nonlinear dynamic system identification.

Nonlinear sinc function

The non-linear sinc function is defined as follows:

f (x, y) = Z = sin(x)sin(y) xy , x ∈ [-10, • • • , 10], y ∈ [-10, • • • , 10] (19)
Using the non-linear sinc function a set of 250 samples is uniformly generated. This data are divided into 125 samples for each of training and test sets. Figure 2a depicts the surface of the training samples.

Nonlinear dynamic system identification

The Nonlinear dynamic system is defined as follows:

y(t + 1) = y(t)y(t -1)[y(t) + 2.5] 1 + y 2 (t) + y 2 (t -1) + u(t) t ∈ [1, 200], y(0) = 0, y(1) = 0 ( 20
)
where u(t) = sin( 2πt 25 ). The Eq.( 20) can be described in a form of three-input one output function:

ŷ(t + 1) = f (y(t), y(t -1), u(t)) (21)
A set of 200 data are selected as a training set and 200 data as a testing set. Figure 3a shows the surface from training points.

Dataset C γ C γ C γ C C Iris 2 -1 2 -1 2 12 2 -7 2 12 2 -9 2 12 2 -1 Vehicle 2 -3 2 -1 2 6 2 -4 2 9 2 -3 2 11 2 -1 DNA 2 -11 2 -1 2 -3 2 -8 2 3 2 -6 2 4 nr * Satimage 2 -9 2 -1 2 -1 2 -7 2 4 2 0 2 6 nr Glass 2 -1 2 -1 2 11 2 -4 2 12 2 -4 nr nr Wine 2 -6 2 -1 2 3 2 -4 2 7
2 -10 nr nr * nr: is not reported.

Performance comparison

In this section, we compare the performance of FNN-SVBL with some of the state-of-the-art methods in terms of number of fuzzy rules or support vectors as well as the accuracy and Root Means Square Error (RMSE), which is defined as

N i=1 (x i -xi ) 2 N
. For both datasets used in classification and regression problems, we carry out the following steps: first, we normalized the datasets using z-score normalization formula: x i -µ σ , where x i is the datasets samples, µ and σ stand for mean and standard deviation of samples, respectively. Note that datasets with binary attribute (e.g., DNA) are not normalized. This scaling adjusts the values measured on different scales to a common one. Second, to evaluate the quality of our trained model, we use 10fold cross-validation. To this end, we estimate generalized accuracy and RMSE for classification and regression problems, respectively, by varying the cost parameters

C = [2 -21 , • • • , 2 21 ].
The kernel parameter γ is set to 2 -1 since there is no changes in its other values. This procedure is applied over the training sets with tenfold cross-validation for each value of C. Then, to assess the quality of the model over unseen data, we apply the trained model with C that reaches the optimal average cross-validation rate over the test sets. These parameters are reported in Table 1 for all datasets.

For datasets that do not explicitly contain test sets (e.g., Iris, Vehicle, Glass, and Wine), we split the whole dataset into halves as training and test sets. The proposed method (FNN-SVBL) is then applied over the training sets of these datasets using the number of fuzzy rules inspired from [START_REF] Ganji | Support vector-based fuzzy classifier with adaptive kernel[END_REF].

We report the optimal kernel and cost parameters for FNN-SVBL and the other methods in Table 1. The number of fuzzy rules/support vectors are reported in Table 2. The smallest and the largest numbers of fuzzy rules belong to Iris and Satimage, respectively. In general, FNN-SVBL and SVFC-AK sketch the lowest fuzzy rules compared with the other methods.

The proposed method is then compared to some prominent related work in classification problem such as SVFC-AK [START_REF] Ganji | Support vector-based fuzzy classifier with adaptive kernel[END_REF], RBF kernel-based SVM [START_REF] Hsu | A comparison of methods for multiclass support vector machines[END_REF], support vector-based fuzzy neural network (SVFNN), self-organizing TS-type fuzzy network with support vector learning (SOTFN-SV) [START_REF] Lin | Support-vector-based fuzzy neural network for pattern classification[END_REF], support vector learning (SOTFN-SV) [START_REF] Juang | A tsk-type recurrent fuzzy network for dynamic systems processing by neural network and genetic algorithms[END_REF] and FNN classifier [START_REF] Berthold | Constructive training of probabilistic neural networks[END_REF][START_REF] Lee | An efficient method for computing leave-one-out error in support vector machines with gaussian kernels[END_REF]. The results of this comparison are displayed in Fig. 4. Overall, the method FNN-SVBL achieves the highest classification performance among all methods while having the smallest number of fuzzy rules along with SVFC-AK. For Iris dataset, FNN-SVBL achieves the same error rate (1.3%) as SVFC-AK and they both have the lowest error rate compared to other methods. For DNA dataset, FNN-SVBL obtains an error rate of 4.1% which is slightly lower than the methods SVFC-AK (4.3%) and RBF-SVM (4.6%). For Glass dataset, FNN-SVBL achieves an error rate of 3.74%. This is significantly lower than the error rate of SVFC-AK (23.9%) and RBF-SVM (28.5%), while using only 7 fuzzy rules comparing to 113 support vectors for RBF-SVM. For wine dataset, SVFC-AK has no classification error using only 3 rules. With the same number of rules FNN-SVBL achieves an error rate of 1.12%. Nevertheless, RBF-SVM obtains an error rate of 0.6% using 56 support vectors. For the Vehicle dataset, whereas the error rate of SVFC-AK is the smallest one (10.3%) by using only 9 rules, SOTFN-SV reports an error rate of 11.1% using 221 rules. For the Satimage dataset, while RBF-SVM obtains the uppermost classification performance (8.7%), it has the largest number of support vectors (1611) among all methods. In contrast, FNN-SVBL reports a slightly higher error rate (8.9%) using only 78 fuzzy rules, which is much lower than both the number of rules of RBF-SVM and the number of support vectors of SVFNN. Consequently, the results in Table 3 and Fig. 4 demonstrate that the accuracy of our method (FNN-SVBL) is competitive and even superior to state-of-the-art methods. The strength of our method is also in the number of its fuzzy rules, which is significantly lower than the related work. The results reveal the effectiveness of the proposed adaptive kernel since by using fuzzy rules, FNN-SVBL can efficiently augment class separability in a kernel space.

F N N -S V B L S V F C -A K R B F -S V M S V F N N S O T F N -S V F N N
For evaluation of FNN-SVBL performance in regression problem, we apply the method FNN-SVBL on the training set of nonlinear sinc function and Nonlinear dynamic system identification datasets according to the procedure explained earlier in this section. The resulting output is shown in Fig. 2b and 3b, respectively. The number of fuzzy rules is inspired from methods in [START_REF] Malek | Three new fuzzy neural networks learning algorithms based on clustering, training error and genetic algorithm[END_REF] which represent the lowest RMSE for test sets in the related work. Table 4 and 5 report for both datasets, the numbers of neurons/fuzzy rules as well as the results of RMSE for training and test sets for FNN-SVBL and the related work. For the sake of clarity, the RMSE results are also displayed in Fig. 5a and5a. For sinc function dataset, the number of rules used in FNN-SVBL is determined to 14 which is the lowest among all methods. For the dynamic system, the number of rules used in FNN-SVBL is determined to 9 which is comparable with methods such as [START_REF] Khayat | A novel hybrid algorithm for creating self-organizing fuzzy neural networks[END_REF] or space partitioning [START_REF] Malek | Three new fuzzy neural networks learning algorithms based on clustering, training error and genetic algorithm[END_REF]. FNN-SVBL represents the highest regression performance for sinc function dataset with a training score of 0.0121 and a testing score equals to 0.0201. The same trend is observable for dynamic system dataset. The results of our method over training and test sets outperform the other methods. We implement our method using the Python library. The computational experiments are carried out with a GTX 1070 Ti GPU based on Linux Fedora operating system.

F N N -S V B L A N F IS S O F N N S O F N N G A M e a n -s h if t K N N S p
F N N -S V B L O L S R B F A F S D F N N G D F N N F a r a g S O F N N S O F N N G A K h a y a t M e a n -s h i f t K N N S p

Fuzzy rules Sensitivity

The parameters of FNN-SVBL is the number of fuzzy rules (clusters). Thus, the number of fuzzy rules is the only parameter to be investigated for sensitivity analysis. For this experimental study, we use Iris, Vehicle, DNA, Glass, and wine datasets for both classification and regression problems. To evaluate the sensitivity of FNN-SVBL to the number of fuzzy rules in a classification problem, we estimate the generalized accuracy using different number of rules in the range of [START_REF] Cristianini | An introduction to support vector machines and other kernel-based learning methods[END_REF] the best performance belongs to the value 9 where the sensitivity tends to variate moderately. For the remaining values, the sensitivity stays steady.

In all datasets, the accuracy is ascending before and descending after the number of fuzzy rules chosen for our study. This complies with the choice of the number of fuzzy rules. Correspondingly, the effect of variation of fuzzy rules number on classification and regression performance of FNN-SVBL is predictable. Hence, using a cross-validation process the optimal values of fuzzy rules can be chosen by investigating a limited set of candidate values.

Clustering Methods

In this section, we address the problem of choosing the clustering method. To do so, we apply three most commonly used clustering GMM, FCM and HCA to obtain the FNN fuzzy rules (clusters). The number of clusters, as mentioned before, is inspired from [START_REF] Ganji | Support vector-based fuzzy classifier with adaptive kernel[END_REF]. Then, the performance of FNN-SVBL is measured over the training sets of 10 benchmark datasets (for both classification and regression purposes) selected from UCI repository [START_REF] Blake | UCI Machine Learning Repository[END_REF].

For classification purpose, the five benchmark datasets that are used for the study include student academics performance dataset, bank marketing, poker hand, human activity and breast cancer Wisconsin (diagnostic). The student academics performance dataset is composed of 22 attributes spanned across 300 instances. These multivariate attributes are chosen from different social, economic and academic characteristics such as students' gender, parents' degree, profession and wage, marital situation, social security type, students' major, to name a few. The classification goal is to predict the end semester students' performance (i.e. good, average or poor). The bank marketing dataset has 45,211 instances and 17 attributes. The input variables are related to social and economic situation of bank client data such as profession, marital and educational situation, housing, age and loan. Here, the aim of classification is to predict whether the client subscribes a bank product or not. The human activity recognition dataset contains information related to certain human activities (i.e. walking, walking-upstairs, walking-downstairs, sitting, standing, laying) that are collected using Smartphones. The dataset has 10,299 instances and 561 attributes providing the input data from sensor signals. The poker hand dataset consists of records for poker hands where each card is defined using two attributes of suit and rank for a total of 1,025,010 instances and 11 attributes. Here, the classification goal is to predict 10 poker hands class (ordinal) such as royal flush, full house, two pairs, straight flush. The breast cancer dataset comprises 32 attributes describing characteristics of 569 digitized images (instances) obtained from several patients' breast mass. Some of these attributes are radius, texture, compactness, fractal dimension or smoothness. The aim of classification is to diagnose whether the examined breast mass is malignant or benign.

For regression purpose, the five datasets that are used for the study include air quality [START_REF] Vito | On field calibration of an electronic nose for benzene estimation in an urban pollution monitoring scenario[END_REF], daily demand forecasting [START_REF] Ferreira | Study on daily demand forecasting orders using artificial neural network[END_REF], energy efficiency [START_REF] Tsanas | Accurate quantitative estimation of energy performance of residential buildings using statistical machine learning tools[END_REF], yacht hydrodynamics and individual household electric power consumption (IHEPC). The air quality dataset contains time-series information about gas concentrations in a city collected from sensor devices. It has 9,358 instances and 15 attributes such as time, date, temperature, humidity, sensor response. The daily demand forecasting dataset comprises information regarding different orders collected from a logistics company during two months. The data are recorded on a daily basis (60 instances) for 13 attributes such as order type, banking orders, urgent or non-urgent order, day of the week. The energy efficiency includes information on energy analysis of 12 different building that comprises 768 instances and eight features such as heating load, wall and surface area. The response values are heating and cooling load. The yacht hydrodynamics dataset is composed of information about dimensions and velocity of 308 yachts in order to estimate their hydrodynamic performance from seven attributes. The IHEPC dataset contains 2,075,259 samples obtained from electric power consumption of a household. It has nine attributes such as date, time, the active and reactive power, intensity and voltage. As depicted in 7a, although GMM and FCM show similar scores in terms of accuracy for all datasets, the overall accuracy obtained by GMM is greater than FCM and HCA. This observation remains valid for regression scores. Generally, the method GMM reports the scores slightly lower than FCM and strictly lower than HCA (see 7b).

Conclusion

In this paper, we show that a TSK-type fuzzy neural network (TSK-FNN) can be equivalent to a support vector machine with an adaptive fuzzy kernel. It is also proved that this adaptive fuzzy kernel is generated by fuzzy rules used in the fuzzy neural network. In other words, we show that the weights in the latest layer of the TSK-FNN can be written based on the support vector machine parameters. Therefore, in order to obtain the weights in the latest layer of FNN, instead of using the gradient or pseudo-inverse method, one obtains the support vector machine parameters on the basis of which the weights of the latest layer of the TSK-FNN can be achieved. On the one hand, since FNN-SVBL is an SVM-based learning method, it takes advantage of SVM, which is the generalization ability (i.e., minimization of the test set error). This advantage is experimentally confirmed as reported in the paper. The proposed method achieves a higher performance compared to the state-of-the-art methods in terms of classification accuracy and regression error of the test sets. On the other hand, the proposed method outperforms the conventional SVM with standard kernels. In such kernels, choosing an appropriate kernel and learning kernel parameters are usually done empirically which are both cost-intensive and not optimal. While the kernel of the proposed method is based on data space since it is generated using fuzzy rules. As a result, it is expected that the results of the proposed method have a higher performance compared to an SVM with the standard kernel. The experimental outcomes of applying the proposed method for classification and regression and comparing their results with the conventional SVM with the standard kernels confirm this expectation. As reported in the paper, the proposed method demonstrates better results compared with the standard kernels in terms of the classification accuracy and regression error. Since the proposed method uses fuzzy rules to generate a kernel, and the kernel is, in fact, a similarity measure, the proposed method is not sensitive to the number of rules, and it is only sufficient that the number of rules is such that it can measure the similarity of each data with the support vectors. The insensitivity of the proposed method to the number of rules is also apparent from the experiments conducted for classification and regression. As shown in these results, there is no significant change for a different number of rules. In the future work, a conceivable way is to investigate the other fuzzy neural networks to know whether it is possible to find an adaptive kernel for a fuzzy neural network that can convert it to an SVM with a particular kernel.

Figure 1 :

 1 Figure 1: The network structure of the proposed model (FNN-SVBL).

  select DNA, Vehicle and Satimage datasets from Statlog collection. The DNA dataset is a collection of 3186 feature vectors with 180 sequential DNA nucleotide positions as attributes. The dataset is split into 2000 samples for training and 1186 samples for testing. The data are classified into three classes. The DNA examples are taken from Genbank 64.1. The Vehicle dataset consists of 846 samples distributed among four classes. Each sample is represented by 18 input attributes. The Satimage dataset consists of the Landsat multispectral values of pixels in 3 × 3 neighborhoods in a satellite image. It contains 4435 instances for the training set and 2000 instances for the test set. The dataset is classified into six classes. Each sample is composed of 36 input numerical attributes in the range between 0 to 255.

Figure 2 :

 2 Figure 2: Nonlinear sinc surface data.

Figure 3 :

 3 Figure 3: Nonlinear dynamic system data.

Figure 4 :

 4 Figure 4: Comparison of classification error rate between FNN-SVBL and the related work methods.

Figure 5 :

 5 Figure 5: Comparison of FNN-SVBL with the related work results.

Figure 6a displays the

  Figure 6a displays the classification accuracy of FNN-SVBL for different number of fuzzy rules.As it is shown, for Iris dataset the changes of fuzzy rules number dimly affect the accuracy. For Vehicle dataset, whereas the accuracy reaches its maximum value for 9, it is gradually declined for the values 2, 6, and [12, • • • , 15]. The sensitivity varies slightly for the values smaller than 5 as well as in the range between 10 and 13. For DNA dataset, the sensitivity is almost constant in the range of [4, • • • , 9]. However, it achieves the best performance for the values 3 and 14. For Glass problem, there are more fluctuations than the other datasets though the overall sensitivity variations remain low. For Wine problem, the sensitivity to the fuzzy rules number is quasi-constant, except for the value 3, which is representing the highest accuracy score among all values. Overall, FNN-SVBL shows a slight sensitivity to the variations of the number of fuzzy rules for the classification problem.

Figure

  Figure6adepicts the RMSE for the regression problem for different number of fuzzy rules. For sinc function dataset, while the RMSE score decreases for the value 14, it shows the highest score for the value 10. The global sensitivity is low excluding for the value 14. For dynamic system dataset,

Figure 6 :

 6 Figure 6: The accuracy and RMSE of FNN-SVBL for different number of fuzzy rules.

Figure 7 :

 7 Figure 7: The comparison of FNN-SVBL performance in classification and regression for 3 clustering methods (GMM, FCM and HCA).

Table 1 :

 1 The optimum cost parameter C and kernel parameter γ for the datasets used for classification.

	FNN-SVBL	SVFC-AK[33]	RBF-SVM[26]	SVFNN[32]	SOTFN-SV[64]

Table 2 :

 2 Number of support vectors (SV) or/and fuzzy rules (FR) for FNN-SVBL and the related work methods.

	Dataset FNN-SVBL SVFC-AK RBF-SVM SVFNN SOTFN-SV FNN
		(FR)	(FR)	(SV)	(FR)	(FR)	(FR)
	Iris	3	3	16	9	4	nr *
	Vehicle	9	9	302	125	221	nr
	DNA	3	3	967	334	nr	nr
	Satimage	78	78	1611	889	nr	nr
	Glass	7	7	113	nr	nr	nr
	Wine	3	3	56	nr	nr	nr

* nr: is not reported.

Table 3 :

 3 Classification error rate obtained for different datasets by FNN-SVBL and the related work methods.

	Dataset FNN-SVBL SVFC-AK RBF-SVM SVFNN SOTFN-SV FNN
	Iris	1.3	1.3	2.7	4	2.8	4.3
	Vehicle	10.3	11.9	13.4	14.5	11.1	29.9
	DNA	4.1	4.3	8.7	9.2	nr	8.9
	Glass	3.74	23.9	28.5	nr	nr	nr
	Wine	1.12	0.1	0.6	nr	nr	nr

* nr: is not reported.

  , • • • , 16]. Similarly,

Table 4 :

 4 Comparison of RMSE results for the training and testing sets for sinc function.

	Models	Neurons Training RMSE Testing RMSE
	FNN-SVBL	14	0.0121	0.0201
	ANFIS [10]	16	nr	nr *
	SOFNN [25]	14	0.0217	0.0860
	SOFNNGA [75]	14	0.0173	0.0567
	Mean shift [73]	18	0.0163	0.0571
	KNN [73]	14	0.0168	0.0231
	Space partitioning [73]	24	0.0161	0.2603

* nr: the result is not reported.

Table 5 :

 5 Comparison of RMSE results for the training and testing sets for nonlinear dynamic system identification. nr: the result is not reported.to specify sensitivity of FNN-SVBL to the number of fuzzy rules in regression problem, we evaluate RMSE for the testing set using different number of rules in the range of[6, • • • , 18]. The training sets of all datasets are scaled in the range of [-1, 1]. To measure classification accuracy and RMSE for the datasets, which do not contain testing sets explicitly, we conduct a tenfold cross-validation on the whole training sets.

	Models	Neurons Training RMSE Testing RMSE
	FNN-SVBL	9	0.0051	0.0042
	OLS	65	0.0288	nr *
	RBFAFS	35	0.1384	nr *
	DFNN [76]	6	0.0283	nr *
	GDFNN [77]	6	0.0241	nr *
	Farag	75	0.193	0.201
	SOFNN [25]	5	0.0157	0.0151
	SOFNNGA [75]	4	0.0159	0.0146
	Khayat [74]	4	0.0147	0.0141
	Mean shift [73]	5	0.0137	0.0127
	KNN [73]	4	0.0150	0.0131
	Space partitioning [73]	9	0.0065	0.0055

*