Stéphanie Bernard

Thomas Hardy

CHAPTER THREE AUTHORITY AND DISPLACEMENT: THOMAS HARDY AND THE AESTHETICS OF MODERNITY

his last novel Jude the Obscure 1 in 1895. Young Jude Fawley dreams of going away from the village of Marygreen, "a small sleepy place" (JO, 6) where he is brought up by his old aunt. As he grows up, the protagonist moves from town to town without seeming to find a place to settle down and feel at home. He is a wandering soul that does not belong anywhere, so his story could be called a tale of displacement.

Movement and displacement inspire and shape the narrative. The chapter headlines point out the spatial dimension of Jude's itinerary through life: starting "at Marygreen", then taking place "at Christminster", it ends "at Christminster again".

Such relentless displacement suggests that something dysfunctions in the symbolic order that should rule over and secure social and individual lives. As a consequence it raises questions of identity, belonging, and authority. For the continual displacement of the protagonist in the novelbe it in terms of location or ideology-makes it increasingly tricky to grasp who he is. The delineation of his character gets blurred and his identity seems in danger of dislocation; as the story unfolds and comes to its end, he slowly and tragically disappears from the text.

Such danger also seems to threaten the figure of the author behind the writing of Jude the Obscure: the narrative voice has become discreet,

Victorian values are being questioned and a sense of pervading doubt seems to testify to the first instants of modernism.

Reading Jude the Obscure can thus be a trying experience, as it means going through moments of extreme sadness, loss and horror. The reader has to decide on the meaning or absence of meaning in the text; he or she has to contemplate and even acknowledge indeterminacy. For one is made to wonder, with the unfolding of the narrative, if man can decide of the moments and movements of life. Is not one the captive of one's own destiny? Can one escape the inexorable displacement of one's being through time, that space that makes up one's existence?

Displacement in space

In Jude the Obscure, Thomas Hardy invites his reader to follow the main character as he wanders through the South West of England, from Dorchester to Oxford. The different villages or towns the protagonist comes to live in mark out his life story and each new location represents a new phase in his experience. In other words, the timeline of Jude's existence crystallizes into a succession of places.

These several towns, the villages like Marygreen in Jude the Obscure-or Marlott in Tess of the D'Urbervilles2 -, the forests and the plains, gradually come to make up what Hardy called "Wessex". Every place in that invented region is given a fictional name: Dorchester is Casterbridge, Oxford is Christminster-while London or Bath retain their original names as they are situated outside Wessex.

The word Wessex was first used by Hardy in Far From the Madding Crowd3 in 1874, more precisely in the expression "Wessex nooks" (FMC, 114). The limits of the imaginary region widened with each novel and were eventually set with the last three works: The Woodlanders (1887), Tess of the D'Urbervilles (1891) and Jude the Obscure (1895). In the years 1895-1896, the publication of Hardy's prose in sixteen volumes was entitled Wessex Novels. The map of Wessex, presented in all the editions of Hardy's works from 1895 onwards, has since then allowed the reader to visualise the scope of that fictitious universe inspired by Dorset and Hardy's life there.

Wessex is a way for the author to introduce distance between the world of the novel and the real world. It is based on real places but the identification is hindered by the invention of new names and a new map. The theme of displacement is perceptible in that authorial move from the reality of the South West of England to imagined Wessex-from a concrete and locatable region to a fictitious world.

Spatial displacement is combined with, and reinforced by, temporal displacement in so far as Wessex is strikingly out of tune with the notion of progress.

The word Wessex was, until the later quarter of the nineteenth century, a purely historical term defining the south-western region of the island of Britain that had been ruled by the West Saxons in the early Middle-Ages. [...] Hardy unearthed the word and used it in his novels and poems. 4 Wessex is a re-creation by Hardy, who uses this old and forgotten term to build a frame for narrative action in his novels and short stories, but also for poetic writing, and to produce an impression of timelessness.

Nevertheless, in Hardy's late novels especially, the rural setting in which the protagonists are placed is no idyllic refuge against society or industrialization. In Tess of the D'Urbervilles, as the heroine walks towards her new working place, the landscape seems to be well preserved "for the railways which engirdled this interior tract of country had never yet struck across it" (TD, 107). However such a description announces the landscapes in Jude the Obscure which do look "engirdled" by the railways, as the characters keep travelling to and fro by train rather than on foot.

Besides, even in the pastoral surroundings of Tess of the D'Urbervilles, the dangers of mechanization are clearly underlined through the episode of the threshing machine. On one "inexpressive" March morning, Tess is working in a field at Flintcomb-Ash Farm in a bleak and harsh environment. It is the threshing period and the new red machine used to thresh the thatch only quickens the pace of work and increases the physical strain put upon the workers. The narrator describes it as the red tyrant that the women had come to serve-a timber-framed construction, with straps and wheels appertaining-the threshing machine, which, whilst it was going, kept up a despotic demand upon the endurance of their muscles and nerves. (TD, 315) The threat of dehumanization is intense as the threshing machine tends to transform the men and women that use it into machines themselves, having no time to stop, or rest, or even talk: [...] the perspiring ones at the machine, including Tess, could not lighten their duties by the exchange of many words. It was the ceaselessness of the work which tried her so severely, and began to make her wish she had never come to Flintcomb-Ash. [...] for Tess there was no respite; for, as the drum never stopped, the man who fed it could not stop, and she, who had to supply the man with untied sheaves, could not stop either [...]. (TD, 316-317) Flintcomb-Ash becomes the symbol of Tess's slow descent towards desolation and ruin, contrasting with the farm at Talbothays which had been the place of her hopes and of regained happiness. Each different place in the novel, therefore, bears significance in the heroine's life; the final scene at Stonehenge endows her story with sacredness and gives it the grandeur of tragedy. The next and final step will be reached with Jude the Obscure whose protagonist, being born and living in North Wessex, is getting ever closer to the confines of the fictional universe shaped by Hardy work after work.

With the successive novels indeed, the limits of Wessex become gradually fixed. While the map of Wessex is getting more precise and definite, the narratives come to be enclosed in a sort of iron curtain that seems to imprison the characters. Hardy's last novel ends in Christminster, the emblematic city on the extreme Northern limit of Wessex. Such movement towards closure strengthens the impression of a jail-like environment for Jude and could be seen as the expression of Hardy's own experience: the setting of the frontiers of that geographical and fictitious background in the novels (alongside, by the way, the intrusion of the railway into Wessex) coincides with the end of Hardy's career as a novelist.

Displacement as alienation

His main character in his last novel, Jude, appears to be alienated within a world that is nonetheless well-known to him. In spite of his being born and bred at the heart of Wessex, he appears never to belong there. Right from the beginning, Jude is out of place: his aunt, who is in charge of the orphan child, wishes he had died with his parents: "It would ha' been a blessing if Goddy-mighty had took thee too, wi' thay mother and father, poor useless boy!" (JO, 9) Jude is a poor useless boy indeed, who fails to scare the birds that are eating the crop in the field he is supposed to watch and becomes the victim of the farmer's fury. A poor useless boy, whose dreams of education and erudition remain illusory and unreachable.

This fundamental inadequacy of the character prompts the tragic end of the novel and leads to the blurring of the protagonist's identity. Indeed, Jude will die alone in Christminster, just a few streets away from the university he dreamt of entering, and surrounded by the dusty books he had read and studied to no avail. In spite of strenuous work to acquire scholarship and be equal to the goal he has set himself to reach, he is not allowed to become a student because he does not belong to the right class. A professor in the prestigious university writes to him: "[…] judging from your description of yourself as a working-man, I venture to think that you will have a much better chance of success in life by remaining in your own sphere and sticking to your trade than by adopting any other course." (JO, 140) This discrepancy between Jude's aspirations and his social status, between his dreams and the reality he lives in bring to mind the notion of displacement. Jude is dis-placed, not in the right place. He and Sue, the woman he loves, have lived too early: "Our ideas were fifty years too soon to be any good to us", he says. (JO, 389) The failure of his schemes and hopes and the loss of his loved one signify his inability to become the learned and successful man he could have been if he had lived in a different society or at a different period. The passage in which Jude can say the Creed in Latin better than the undergraduates who are present in the public house illustrates that point: " 'Good! Excellent Latin!' cried one of the undergraduates, who, however, had not the slightest conception of a single word." (JO, 145) The author's ironical tone underlines the fact that the protagonist's knowledge, though impressive, is totally useless, as Jude himself insists: "You pack of fools!" he cried. "Which one of you knows whether I have said it or no? It might have been the Ratcatcher's Daughter in double Dutch for all that your besotted heads can tell! See what I have brought myself to-the crew I have come among!" (JO, 146) That's why, when being back in Christminster on the day students receive their diplomas and having deciphered a Latin inscription on a wall, Jude exclaims that he is "a frightful example of what not to do." (JO, 389) Spatial displacement and social inadequacy both illustrate and trigger the dislocation of the character's identity: who is Jude after all? Where does he really come from? Isn't he first and foremost a bereaved child? And where do his wanderings lead him but to his death? As Jude's dreams are incessantly thwarted-never accessible, ever receding further-and his course is repeatedly diverted, tragedy becomes his destiny.

The tragic in Jude the Obscure, therefore, highlights and underscores the theme of displacement which generates indeterminacy: the difficulty one has to grasp and define the identity of the main character shatters the vision one gets of the world Jude inhabits. While in Tess of the D'Urbervilles the narrator maintained the illusion that her fate was in the hands of the "President of the Immortals" (TD, 384), in Hardy's last novel meaning gets blurred and the world appears to be the creation of "the First Cause" that "worked automatically5 like a somnambulist, and not reflectively like a sage". (JO, 409) Meaning is equally blurred under the influence of other texts that interfere with the narrative voice. Among these the Bible holds pride of place. Numerous biblical quotations punctuate the narrative. Jude's final words are citations from the Book of Job: "Let the day perish wherein I was born, and the night in which it was said, There is a man child conceived." [...] "Let that day be darkness [...]." "Why died I not from the womb?" [...] "Wherefore is light given to him in misery, and life unto the bitter in soul?" 6 (JO, 484).

These ultimate utterances insist on the impossibility for Jude to know who he is and why he was born. He appears like a nineteenth-century Job whose obscure destiny raises the questions of the existence of God and the meaning and origin of life.

Other texts by Hardy also contribute to warping our judgement of the protagonist and rendering his identity unclear: the heroine of Tess of the D'Urbervilles is another double of Jude, a feminine counterpart to the masculine tragic hero. The similarity brings the reader to wonder about the stirs and urges everything" in a poem entitled "The Convergence of the Twain" and alluding to the sinking of the Titanic (Thomas Hardy. The Complete Poems. London: Macmillan, 2001, 306). It also finds an echo in the opening of the Dynasts (Volume 1 of 2. ReadHowYouWant, 2007, 10), Hardy's epic drama about the Napoleonic Wars (1904-1908):

"[…] like a knitter drowsed, Whose fingers play in skilled unmindfulness, The Will has woven with an absent heed Since life first was; and ever will so weave." 6 Jude quotes the Book of Job 3: 3-4, 11, 20. limits between genres. There is something undeniably feminine in Jude's character, accounting in part for his vulnerability, just as there is something undeniably masculine in Tess, as she decides of her own fate in becoming a murderess.

Polyphony

The uncertain, wavering identity of the main character in Jude the Obscure foreshadows the questioning of the figure of the author: Hardy was a nineteenth century novelist with such works as Under the Greenwood Tree, A Pair of Blue Eyes, Far from the Madding Crowd, but a twentieth century poet as the Collected Poems well show. Thomas Hardy gradually entered the modern era, perforce because of his longevity, which made him a witness of the industrial revolution, as well as the tragedy of the Titanic and the First World War, but also by choice as a writer who tried different genres. The reading of the last two novels, just like that of some of the short stories, gives intimations of that evolution in his writing.

In her study of Hardy's short stories, Nathalie Bantz insists that if Hardy started writing these texts "under the reign of Queen Victoria", the last ones were produced "at the dawn of the Edwardian period", 7 in the early days of modernism. These later works do convey "a sense of disillusionment and fragmentation" 8 that is typical of modernism. Moreover, the form of the short stories-especially the limited use of an extra-diegetic narrator and the extended recourse to indirect speech by the characters-suggests that Hardy drew away from the traditional form of fiction more present in the novels, giving birth to polyphonic texts. The fading voice of the narrator, who is both present in and absent from the texts, signals a more and more modern style in Hardy's writing.

The polyphonic quality that can be found in Hardy's short stories tends to be a distinctive feature of the novel according to Mikhaël Bakhtin in his enlightening study on Dostoevsky: the dramatic quality of a work makes it dialogic-another word for polyphonic-by allowing the characters' voices to resound and stand by themselves instead of being muffled by or 7 "La nouvelle hardyenne voit le jour sous le règne de la reine Victoria et s'achève à l'aube de la période édouardienne, au moment où se développe le courant littéraire moderniste […] subjected to the authorial voice. Dialogism is not only an aspect of the writing of fiction but it is also an aspect of language itself: Words always contain a dialogic quality, embodying a dialogue between different meanings and applications. Bakhtin's dialogism undermines any argument for final and unquestionable positions, since every position within language is a space of dialogic forces rather than monologic truth.9

The presence of an omniscient narrator in Hardy's novels makes it difficult to assert that the works are not monologic at first. Far From the Madding Crowd, for instance, can disturb the reader because of the recurrent generalizing remarks about women, money or life as a whole. Yet, the narrator becomes less and less intrusive, novel after novel. The narrative voice is more subdued and the figure of the intrusive author is much less noticeable in Tess of the D'Urbervilles or Jude the Obscure: the tragic dimension of the characters' lives in these two stories speaks for itself; at other times the narrator chooses to refer to classical texts-the Bible, Latin, Greek or even English poets-to give a more impersonal turn to the sayings.

Jude the Obscure is precisely characterized by a large number of intertextual occurrences. The last one of these is the quotation from the Book of Job which we have already mentioned: Jude's voice is doubled by that of the biblical character. The end of the novel is undeniably polyphonic because of this intertextuality and because "[a]n occasional word, as from someone making a speech, floated from the open windows of the Theatre across to this quiet corner […]." (JO, 489) The narrator's own voice fades away while other voices fill the textual space; his words seem to be "populated-overpopulated-with the intentions of others",10 to quote Bakhtin.

The last lines of the novel are a piece of dialogue between Jude's villainous wife named Arabella and a secondary character, Mrs Edlin. The very last words are pronounced by Arabella: while the whole story attracts the reader's attention to Jude's exceptional and terrible destiny, the ending enables an unsympathetic gaze to be cast on the narrative and deprives both the narrator and the protagonist of a voice. The text enacts the displacement of Jude, who is definitely alienated from this world: it is only through death and disappearance-total displacement-that he can be freed from his sad fate and that there suddenly seems "to be a smile of some sort upon the marble features of Jude." (JO, 489) The polyphonic ending of this narrative that opens up to alien voices signals the attraction of modernism for Hardy and announces the closure of his career as a novelist. The dark and at times unbearable writing of Jude the Obscure testifies to the impossibility for the author to continue writing fiction and the necessity to turn to poetry, a genre that allows for harsher realities to be put into words while avoiding scandal, as Hardy himself wrote in his autobiography: "If Galileo had said in verse that the world moved, the Inquisition might have let him alone". 11 An aesthetic displacement is therefore perceptible, from one periodthat of the Victorian novel-to another-that of modern fiction; from the convention of the happy ending to the more subversive rewriting of tragedy; from one genre-fiction-to another-poetry. How then can one grasp the identity of an author who can let out so many different art forms (Hardy was also an architect and he could draw very well) and so many voices? How can authority be retained amidst such authorial instability?

Displacement and authority: the question of authorship

The death of Jude, his displacement outside the fictional world of the novel, and the silencing of the narrative voice raise the question of the authority of the author-of authority and authorship.

The generic and stylistic uncertainty which is a characteristic of Hardy's work has often made it intricate to decide about the quality of his writing. In an essay on Thomas Hardy in The Common Reader, Virginia Woolf, who had met the man and appeared to be very much impressed by him, nevertheless described his novels as being "full of inequalities". 12 Although she praised Hardy's prose, she recognized that his style was particularly "difficult to analyse" because it was so imbued with what she called "that wild spirit of poetry". 13 The following quotation that underlines the unevenness and the brilliance of his craft at the same time, can help us grasp how difficult it can be to attempt to get a clear and definite view of the author and his work:

11 Florence Emily Hardy. The Life of Thomas Hardy, 1840-1928. London: Macmillan, 1962, 285. 12 Virginia Woolf. The Common Reader, Second Series. The Hogarth Press, (1932) 1965, 248. 13 Ibid., 256-257.

There is a feeling that Hardy's genius was obstinate and perverse; first one gift would have its way with him and then another. They would not consent to run together easily in harness. Such indeed was likely to be the fate of a writer who was at once poet and realist, a faithful son of field and down, yet tormented by the doubts and despondencies bred of booklearning; a lover of old ways and plain countrymen, yet doomed to see the faith and flesh of his forefathers turn to thin and spectral transparencies before his eyes. 14 The author's polymorphous personality invites us to ask: "who is Hardy?" This mystery inspired Siegfried Sassoon to write a poem entitled "At Max Gate"15 which was first published in 1950, that is to say more than thirty years after the young poet met the old man and more than twenty years after Hardy's death. The poem suggests how the old man managed to remain an enigmatic character despite his success and fame.

"At Max Gate" Old Mr Hardy, upright in his chair, Courteous to visiting acquaintance chatted With unaloof alertness while he patted The sheep dog whose society he preferred. He wore an air of never having heard That there was much that needed putting right. Hardy, the Wessex wizard, wasn't there. Good care was taken to keep him out of sight.

Head propped on hand, he sat with me alone, Silent, the log fire flickering on his face. Here was the seer whose words the world had known. Someone had taken Mr Hardy's place.

The scathing author who called his heroine Tess "a pure woman" in spite of her being an unmarried mother, the writer who sympathised with the disturbing and even-according to D.H. Lawrence-"frightful" 16 character of Sue Bridehead as a representative of the "New Woman", led a quiet and secluded life in his Dorset house. Although he believed that "a marriage should be dissolvable as soon as it becomes a cruelty to either of the parties-being then essentially and morally no marriage […]"17 (JO, viii)-, he continued living in the same house as his first wife until her death in 1912, although they had been estranged from each other for years.

The old man's appearance displaces and conceals-rather than replaces-the figure of the author. Sassoon's poem calls to mind a similar movement towards concealment that can be traced in Hardy's autobiography: the third person came to displace the "I" that should have been used, as his second wife, Florence Dugdale Hardy, who was his former secretary, wrote The Life of Thomas Hardy (1928-1930) under his dictation. Thomas Hardy appears in it as "he". His authorship is disguised under the pretence of being written about; his actual authority over the facts retailed is unconfessed and allegedly displaced unto his wife.

In other words, Thomas Hardy fuels the feeling of ambiguity that some readers expressed particularly strongly after the release of Jude the Obscure. Such is the case with Margaret Oliphant, the nineteenth century Scottish writer and critic, who reacted quite vehemently on the publication of the novel: "I do not know, however, for what audience Mr. Hardy intends his last work […] whose tendency throughout is so shameful […]",18 she wrote in 1896. To her, the disgraceful content reflects the shameful mind of the writer and spoils the potential artistic qualities of the work that is unsuitable for "decent houses in England and America". In other words, she identifies the author and his text. D.F. Hannigan, another contemporary critic of Hardy, adopted a very different position regarding Hardy and his work:

The plot […] of Jude the Obscure has been sketched, and, indeed, misrepresented, by so many of the smug journalistic critics of this book, that it is better to let all intelligent and honest readers find out the true story of Jude Fawley for themselves by reading the novel. It is certainly "strong meat", but there is nothing prurient, nothing artificial in this work; it is human in the widest sense of that comprehensive word. 19 The critic here stresses the necessity to accept the notion of displacement that makes it impossible to read the novel as a monologic text. The work encourages misrepresentations and misunderstandings precisely because the author has resigned his authority over the meaning of the tale and leaves it to the reader to decide. The "author-God", to quote Roland Barthes, has withdrawn and the text appears to speak for itself.

Displacement and authority: the question of God

In his ground-breaking essay, Roland Barthes contends that The Death of the Author signifies that, in a modern text, there is no "single 'theological' meaning (the 'message' of the Author-God)". 20 The displacement or "alienation" of the author turns the act of writing into "a pure gesture of inscription (and not of expression)" that "has no other origin than language itself". 21 It is in the act of reading that meaning reappears: "the birth of the reader must be at the cost of the death of the Author". More recently, Frances Fortier and Andrée Mercier have argued that narrative authority depends on "the couple formed by the speaker and the addressee". 22 It is "the result of interaction or dialogue with the reader" 23 that generates authority rather than the unique voice of the author.

The quotations apply to twentieth and twenty-first century writings that challenge the concept of a single voice and a unified vision purveyed by an omniscient narrator as in nineteenth century fiction. However it is also relevant when applied to less recent but clearly polyphonic novels where the voice becomes plural as in Jude the Obscure. Hardy the author seems to withdraw his authority by silencing his main representative-the narrator-as well as his creature-the protagonist. Those doubles of the author, so similar at times that it is difficult to discriminate between 20 Roland Barthes. "The Death of the Author." in Stephen Heath (ed.). Image-Music-Text. London. Fontana, 1977. Reprinted in Séan Burke (ed.) them, 24 give way to the intrusive gaze and striking voice of Arabella. The reader is invited to interact indeed, in order to look for Jude behind the screen of his sarcastic wife's judgement in a novel that bears his name, and to look for the narrator behind that veil of unpoetic words in a fictional work of high poetic quality.

Authority, therefore, is undeniably displaced from the traditional figure of the author to a new reality that implies the writing and the reading of the text. Although anti-theological discourse was still highly controversial when Jude the Obscure was published, the Author-God does eventually disappear from the novel, announcing the death of God that Hardy represented some years later in a poem entitled "God's Funeral" in Satires of Circumstances. 25 The story of that god is in reality the history of humanity. The denunciation is aimed at man who appears to be, in a striking and ironical role reversal, the creator of the divinity, urged by a need to revere and believe, while in the process the god of the poem is deprived of any sacredness or grandeur.

The symbolical value of the poem is great: one cannot ignore the echo of Nietzsche's declaration that "God is dead. God remains dead. And we have killed him." 26 The displacement of the focus onto man and his idolatry may attenuate the boldness of the image in Hardy's rhyme, but by relocating the core of the subject onto earth, the author highlights the absence of transcendence and the extremism of man's tragedy-in other words he voices the anxieties of modern man. "God's Funeral" breeds a feeling of doubt and sadness-rather than shock or opposition-when the speaker of the poem eventually joins the procession of mourners following the coffin, "'twixt the gleam and gloom" (CP, 329).

The mixture of attraction and repulsion towards religion-the impression of a perverted interpretation of the biblical text, of what Harold Bloom would call "misprision"-27 is typical of Hardy's writing. He lost faith as a young man living in London but he kept regretting the loss. The

  .": Nathalie Bantz. Les Nouvelles de Thomas Hardy, Stratégies narratives d'une écriture sous contrainte. Paris: Honoré Champion, 2011, 251 (my translation). 8 "Modernism." Encyclopaedia Britannica. Encyclopaedia Britannica Online Academic Edition. Encyclopaedia Britannica Inc., 2013. Web. 21 Oct. 2014.

<http://www.britannica.com/EBchecked/topic/387266/Modernism>.

  "[…] le couple que forment le locuteur et l'allocutaire": Frances Fortier et Andrée Mercier. "L'Autorité narrative en question dans le roman contemporain", in Emmanuel Bouju (dir.). L'Autorité en littérature. Rennes: Presses Universitaires de Rennes, 2010, 113 (my translation). 23 "[…] le produit d'une interaction ou d'un dialogue avec le lecteur": ibid., 113 (my translation).

	. Authorship:
	From Plato to the Postmodern: A Reader. Edinburgh: Edinburgh University Press,
	1995, 128.
	21 Ibid.

22

Thomas Hardy. Jude The Obscure. Harmondsworth: Penguin, (1895[START_REF] Grellet | A Handbook of Literary Terms[END_REF]. All subsequent references to this book in this article will be to that edition, which will also be abbreviated as JO.

Thomas Hardy. Tess of the D'Urbervilles. Oxford: Oxford University Press, (1891) 1998. All subsequent references to this book in this article will be to that edition, which will also be abbreviated as TD.

Thomas Hardy. Far From the Madding Crowd. New York and London: Norton, (1895) 1986. All subsequent references to this book in this article will be to that edition, which will also be abbreviated as FMC.

Simon Gatrell, "Wessex." in Dale Kramer (ed.). The Cambridge Companion to Thomas Hardy. Cambridge: Cambridge University Press, 1999, 19.

The "First Cause" can be related to what Hardy will call the "Immanent Will that

Graham Allen. Intertextuality. London: Routledge, 2000, 211.

Mikhail Bakhtin. The Dialogic Imagination: Four Essays. Austin: University of Texas Press, 1992, 294.

Ibid., 247.

Max Gate is the name of Hardy's house, located in Dorchester, which he himself designed in 1885 and where he lived until his death in 1928.

D. H. Lawrence. Study of Thomas Hardy and Other Essays. Cambridge: Cambridge University Press, 1988, 109.

The character of Sue voices the same opinion in Jude the Obscure: "If a marriage ceremony is a religious thing, it is possibly wrong; but if it is a sordid contract, based on material convenience in householding, rating, and taxing, and the inheritance of land and money by children, making it necessary that the male parent should be known-which it seems to be-why surely a person may say, or even proclaim upon the housetops, that it hurts and grieves him or her?"(JO, 250)

Margaret Oliphant. Blackwood's Magazine, January 1896, quoted in Thomas Hardy. Jude the Obscure. New York & London: Norton,

1999, 379. 19 D.F. Hannigan. The Westminster Review, January 1896, quoted in Thomas Hardy. Jude the Obscure. New York & London: Norton, 1999, 389.

We will not go into the useful but tricky notions of author, implied author or frame narrator in this article. As Françoise Grellet suggests in A Handbook of Literary Terms (Paris: Hachette Supérieur, 1996, 102), they are among the various " 'voices' in the narration". The interferences in the difficulty to distinguish between them contribute to making a work dialogic.

Thomas Hardy. The Complete Poems. James Gibson (ed.). London:Macmillan, 2001, 326-329.

Friedrich Nietzsche. The Gay Science. Book III, section 125, 1882, in Walter Kaufmann (ed.). New York:Vintage, 1974, 181.

Harold Bloom. The Anxiety of Influence: A Theory of Poetry. London, Oxford, New York: Oxford University Pr ess, 1973.

centrality of the Scriptures in his writing illustrates this nostalgic and ambivalent feeling towards religion. It also reflects the displacement of authority and the ambiguity of authorship in Hardy. He seems both to assert his position as the creator of his works, recurrently introducing a first-person speaker in the poems, and to efface himself by displacing himself outside the world of his fiction. The years he spent as a recluse at Max Gate recall the elegiac tone of the writings, both celebrating the absence of God as the author of life, and mourning the loss.

" 'What is meant by the word Author?' one might ask with Michel Foucault by way of conclusion, 'What is an Author?' " [START_REF] Bennett | Andrew Bennet quotes Michel Foucault[END_REF] Andrew Bennett answers that he (or she) is "that intriguing figure, the author, that 'nobody' who holds for us such fascination". 29 The remark is true of Hardy, whose personality was as fascinating as his novels if we rely on the testimonies of those who met him and admired him: Woolf, Sassoon, Galsworthy, Ford … His last novel, Jude the Obscure, keeps the mystery alive: the obscurity mentioned in the title is met by the darkness of the tale and by the sudden disappearance of the author of fiction, who then devoted himself to poetry.

Hardy's multiple personality and craft calls to mind Bakhtin's remark that: language, for the individual consciousness, lies on the borderline between oneself and the other... The word in language is half someone else's. It becomes one's "own" only when the speaker populates it with his own intentions, his own accent, when he appropriates the word, adapting it to his own semantic and expressive intention. Prior to this moment of appropriation, the word does not exist in a neutral and impersonal language... but rather it exists in other people's mouths, in other people's contexts, serving other people's intentions; it is from there that one must take the word, and make it one's own. [START_REF] Bakhtin | The Dialogic Imagination: Four Essays[END_REF] Hardy's style lies on the borderline between the artist and the text, the creator and the created world, the author and the reader, between tradition and modernity. It is the vivid style of an author that is not authoritarian but fluctuates through time in a slow whirl of incessant displacement.


Adrien Hadj-Salah
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email: mikael.capelle@irt-saintexupery.com

Schedule Earth Observation satellites with Deep Reinforcement Learning

Optical Earth observation satellites acquire images world-wide, covering up to several million square kilometers every day. The complexity of scheduling acquisitions for such systems increases exponentially when considering the interoperability of several satellite constellations together with the uncertainties from weather forecasts. In order to deliver valid images to customers as fast as possible, it is crucial to acquire cloud-free images. Depending on weather forecasts, up to 50% of images acquired by operational satellites can be trashed due to excessive cloud covers, showing there is room for improvement. We propose an acquisition scheduling approach based on Deep Reinforcement Learning and experiment on a simplified environment. We find that it challenges classical methods relying on human-expert heuristic.

Introduction

Earth Observation (EO) systems acquire cloud-free images and deliver them to customers worldwide on a daily basis. Requests come in a variety of size and constraints, from the urgent monitoring of small areas to large area coverage. In this work we are particularly interested in the latter case, with requests covering whole countries or even continents. Depending on weather conditions, such requests may take several months to complete, even with multiple satellites.

In order to shorten the time required to fulfill requests, the mission orchestrator shall schedule acquisitions with both a short and a long-term strategy. Determining a strategy robust to an uncertain environment is a complex task, this is why current solutions mainly consist of heuristics configured by humanexperts. This paper demonstrates that Reinforcement Learning (RL) might be well-suited for such a challenge. RL has proven to be of great value since these algorithms have mastered several games such as Pong on Atari 2600 [START_REF] Mnih | Playing atari with deep reinforcement learning[END_REF], Go with AlphaGo [START_REF] Silver | Mastering the game of go without human knowledge[END_REF]) and more recently Starcraft [START_REF] Arulkumaran | Alphastar: An evolutionary computation perspective[END_REF].

c 2019 All rights reserved.

2 Scheduling acquisitions for Earth observation systems

2.1 Single satellite acquisition scheduling EO satellites carry optical instruments which are able to take acquisitions with a specific width, called swath, and a maximum length depending on the satellite models. The capacity of the satellites to take multiple images along their orbit track is related to their agility [START_REF] Lemaître | Selecting and scheduling observations of agile satellites[END_REF].

Due to limited swath and acquisition length, a large area must be split into tiles called meshes. For instance, considering the Pleiades satellites, covering France requires thousands of meshes. A satellite overflying an area is able to acquire a sub-part of those meshes due to its limited agility. With sun-synchronous orbit, revisit of a ground point takes days which explains the importance of mesh selection [START_REF] Gleyzes | Pleiades system architecture and main performances[END_REF].

The satellite schedule is computed on ground by the Mission Planning Facility (MPF), where an optimization algorithm selects the top-ranked acquisitions and ensures the kinematic feasibility of the attitude maneuvers.

Interoperable EO systems scheduling for large-area coverage

The trend of EO systems is toward large constellations of heterogeneous satellites. For instance, Airbus Intelligence, operating the well-known SPOT and Pleaides satellites, will soon manage a new system of 4 satellites (Pleaides NEO). Dealing with multiple EO systems needs both human expertise and algorithms to dispatch requests over the satellites and to deliver end customers on time.

We approach the constellation scheduling by having an orchestrator responsible for request ranking towards each MPF. The orchestrator analyzes a large-scope of data (e.g., forecasts, access opportunities) to optimize the global schedule, while each MPF has a narrowed and short term vision of their single (or dual) satellite scheduling. Additionally, we focus in this paper on requests consisting in a large area (countries, continents). Such requests usually contain several hundreds of meshes to acquire over long periods (up to several months).

The two main contributors to the overall uncertainty on the time to completion are: firstly the weather conditions at the time of acquisition, which can only be forecasted, and secondly the presence of other requests within the systems, arriving at an unknown rate.

This explains our focus on RL algorithms which have the capacity to learn new strategies, robust to uncertainties, while challenging traditional approaches.

Reinforcement Learning approach

In Reinforcement Learning, an agent learns how to behave through trial-and-error interactions with a dynamic environment. The actions the agent takes are decided by a policy, which can be seen as a function mapping information from the environment to actions. The goal of reinforcement learning algorithms is to find an optimal policy, i.e., a policy that maximizes the reward of the agent over the long-term.

Recently, deep neural networks have proven to be efficient for finding policies. Several deep-RL algorithms are actively studied to solve complex sequential decision-making problems. Among the best-known methods, one can cite value-based algorithms such as DQN, Rainbow [START_REF] Hessel | Rainbow: Combining improvements in deep reinforcement learning[END_REF], policy-based algorithms such as REINFORCE [START_REF] Sutton | Policy gradient methods for reinforcement learning with function approximation[END_REF] or actor-critic methods such as A2C [START_REF] Mnih | Asynchronous methods for deep reinforcement learning[END_REF] or PPO [START_REF] Schulman | Proximal policy optimization algorithms[END_REF].

Problem simplification

In order to evaluate the benefits of Reinforcement Learning, we propose a simplified environment.

We consider that all satellites have the same swath, thus the tessellation (i.e., the meshes) of the area is the same for all satellites. We also assume that each satellite can acquire at most one mesh per pass over the considered area. A satellite pass occurs when it overflies the large-area request on a given orbit. The planned mesh is validated or rejected depending on actual cloud cover observations at the time of acquisition. We do not consider uncertainties related to the load of our system, i.e., satellites are always fully available.

The area of interest (AOI) is enclosed in a rectangular box -considering a Mercator projection -containing N lat ×N lon meshes. Since some meshes of this grid mesh may not belong to the AOI, we define M = {m k : 1 ≤ k ≤ K}, the set of meshes to acquire.

For each pass t ∈ N, we denote by M t ⊆ M the subset of meshes in the AOI that can be acquired by the corresponding satellite knowing its orbit and agility.

We denote by c a t (m) and c f t (m) the actual and forecast cloud cover above mesh m during pass t.

Problem formulation

The given problem can be formalized as a Markov Decision Process (MDP) which is an intuitive and fundamental formulation for RL [START_REF] Bensana | Dealing with uncertainty when managing an earth observation satellite[END_REF]).

An agent interacts with the environment by taking actions from a legal set of actions. The agent purpose is to acquire M as quickly as possible. For each step t, only one mesh can be selected. The chosen mesh is then validated or rejected depending on weather conditions.

The state space S, the discrete action space A ⊂ N, the stochastic discrete-time transition function P and the reward function R define the underlying MDP: M =< S, A, P , R >.

The horizon is considered finite. Therefore, there is a finite number of discrete time steps t during an episode. Each episode comprises a maximum of T ∈ N * steps. The state space S is defined as:

S = S status × S time × S passes
where S status = {0, 1} N lat ×N lon characterizes the status of each mesh (i.e., already validated or to acquire), S time ⊂ R encodes the date of the current satellite pass t and S passes ⊂ R d describes all pass dates, accessible meshes M t and related weather forecasts.

The goal is to find a policy π : S → A that maximizes the expected discounted reward over the finite horizon:

T t=0 γ t R(s t , π(s t ), s t+1 ) (1)
where 0 ≤ γ < 1 is the discount factor and s ∈ S.

Action space

At each discrete step t, the learning agent takes an action. A step corresponds to a satellite pass over the AOI and the action is to pick up a single mesh to acquire during this pass.

A = {0, 1, . . . , K}
We denote a k the action selecting the mesh m k . Note that |A| = K+1 because there is one more "do nothing" action available for the agent.

Observation space

At a given step, the agent perceives only useful and available information about the environment. The problem is generalized to a Partially Observable Markov Decision Problem (POMDP).

The observation space O provides information about the mesh status and their validation probability for the following N pass passes, including the current pass for which the agent shall select a mesh. The validation probability of a mesh depends on weather forecast accuracy, as detailed in Section 3.6. Thus, an observation is a tensor with a shape (N lat , N lon , N pass + 1).

The observation can be seen as a stack of N lat × N lon matrices. Each frame (i.e., 2D matrix) encodes information for all tiles of the grid mesh. This representation preserves spatial information and enables the use of Convolutional Neural Networks.

The validation frame encodes the status of each mesh: validated (0) or to be validated (1). We denote the validation frame space O status = S status .

The validation probability frames belong to the space O p = [0, 1] N lat ×N lon ×Npass . They encode the probability p t to acquire and validate each mesh for each pass in chronological order from time step t. For a given mesh m and a given pass n ∈ {1, . . . , N pass } at the step t:

p t (m, n) = 0 if m / ∈ M t P(c a tn (m) ≤ c max | c f tn (m)
) otherwise with c max the total cloud cover validation threshold. t n = t + n -1 is the time related to the pass n knowing that the current time step is t.

We can now define O = O p × O status

Reward

A reward is given to the agent at each step. The value of the reward depends on the status of the chosen mesh before and after this step. R : S × A × S → R gives rewards for particular transitions between states.

R(s t , a k , s t+1 ) = 1 if m k is newly validated 0 otherwise
This dense reward encourages the agent to reduce the completion time with a discount factor γ < 1 (1).

Transition function

P : S × A × S → [0, 1] denotes the transition function.

P (s t , a, s t+1 ) = P(s t+1 |s t , a)

For each transition, the current state is updated. s time ∈ S time takes the value of the next pass date in the chronological order. s passes ∈ S passes remains the same during the whole episode. s status ∈ S status is updated if the selected mesh is validated:

P(s status t+1 (m k ) = 0 | s status t (m k ) = 1, a k ) = p t (m k , n)
where s status t (m k ) is the status of m k at t. This probability is computed considering the following weather model:

c a t (m) = c f t (m) + χ(m) with χ(m) ∼ N (c f t (m), σ(c f t (m)) 2 ) and σ(x) = u × x + v
σ is a linear function computing a representative deviation between forecast and observed data.

Experiments

Based on the hypotheses from Section 3.1, we implement a simulator using the OpenAI Gym framework.

Scenario

To evaluate the agents, we choose mainland France as our area of interest. It is an interesting case to study because one mesh selection can have an important impact on the mission length due to the territory climate variability.

We consider 4 satellites with a common 60 km swath, implying K = 122 meshes for our tesselation. Each scenario begins at a random date.

We use the ERA-Interim dataset [START_REF] Dee | The ERA-Interim reanalysis: Configuration and performance of the data assimilation system[END_REF]) which provides total cloud cover observations on a 0.5 • × 0.5 • grid to compute the observation space. In the weather model, u is fixed to 0.1 and v to 0.2. c max is set to 20% for all scenarios.

Reference algorithms

In order to benchmark the performances of our agent, we define a random agent and a heuristic that selects one mesh among M t for each time step t:

• Random that selects the mesh randomly among accessible meshes at each pass. • Heuristic that selects the mesh with the highest trade-off score between short-term and long-term probabilities p t :

p t (m, 1) + α   1 - 1 N pass -1 Npass n=2 β n p t (m, n)  
where α is the weight on future passes and β the discount factor that favors near future passes. The best performances are reached with (α = 1, β = 0.99) for N pass = 20.

Train and test methodology

To avoid overfitting, we use a train and test split methodology on the weather data. Training is done using data from the years 2013 and 2014, while testing is done with data from 2015.

We concentrate our experiments on the A2C algorithm which gives the best results. We train A2C agents using two observation spaces: one with a short-term vision (N pass = 1) and one with a long-term vision (N pass = 20). Those A2C agents are respectively named A2C-1 and A2C-20. We use the A2C implementation from the OpenAI baselines framework [START_REF] Dhariwal | Openai baselines[END_REF]) and train agents during 3 × 10 7 steps using 16 parallel environments (∼ 30 hours using a K80 GPU and 8 vCPUs). Other hyper-parameters are set to default values.

We use a neural network architecture made of a convolution block followed by a dense block with two heads: one to estimate the state value and one to estimate the policy distribution. The convolution block contains three convolutional layers with decreasing kernel sizes (7 × 7, 3 × 3, 1 × 1), 128 filters per layer and ReLU activation functions. The value and policy heads are only made of a dense layer with respectively one unit and K + 1 units.

Figure 1 shows the mean length of the last 100 episodes as a function of the number of network weight updates for A2C-1 and A2C-20. In our environment, the length of an episode directly relates to the completion time of the area. We set a maximum number of 10 × K time steps before resetting the environment 0 1 2 3 to avoid too long episodes when the policy does not perform well. The performances of the trained agents converge close to the heuristic one. Best results are achieved with A2C-20.

During testing phase, we select days from 2015 as starting dates. For each date we assess the performances of the models and the reference algorithms. We repeat the operation using 3 different weather seeds (3 × 365 runs in total). Table 1 presents statistics on the episode length for the different agents. We find that for both agents the transfer on the new weather data went well.

A2C-20 still provides the best results winning the heuristic in almost 80% of the cases. It confirms the intuition that a long term strategy is necessary to optimize time-to-completion.

Conclusion

This paper demonstrates how Reinforcement Learning can be used in Earth Observation satellites scheduling in order to reduce the time-to-completion of large-area requests. The computed network has been trained to rank the requests and dispatch them to the satellites. In a series of simulation-based experiments, the proposed method challenges the state-of-the-art heuristics.

In future research, we aim to improve the simulation representativeness in order to pave the way for a potential industrial transfer.

Table 1 :

 1 Mean, median, standard deviation of the episode lengths for the different agents.

	•10 5


Carl Frélicot
email: cfrelico@univ-lr.fr

A robust semi-supervised EM-based clustering algorithm with a reject option

In this paper, we address the problem of semisupervision in the framework of parametric clustering by using labeled and unlabeled data together. Clustering algorithms can take advantage from few labeled instances in order to tune parameters, improve convergence and overcome local extrema due to bad initialization. We extend a robust parametric clustering algorithm able to manage outlier rejection to the semi-supervision approach. This is achieved by modifying the Expectation-Maximization algorithm. The proposed method shows good performance with respect to data structure discovering, even facing to outliers.

Introducing partial supervision in clustering algorithms

Clustering is an important task for exploratory data analysis. It aims at searching for structure in data on the basis on some similarity measure. This process can be done through an extremum search for an appropriate objective function, e.g Fuzzy C-Means (FCM) algorithm in [START_REF] Bezdek | Pattern Recognition with Fuzzy Objective Function Algorithms[END_REF] or Expectation-Maximization (EM) approach in [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm (with discussion)[END_REF]. However, the complexity of space to be observed generally imposes the use of sub-optimal methods leading to local extrema. Thus, a good choice of parameter values and especially of initial conditions are essential for the method success. Partial supervision occurs when both unsupervised and supervised examples are available. Unsupervised data are generally easy to obtain whereas supervised one can need a costly expertise. The main idea of partial supervision is to increase the estimation accuracy of classes parameters and avoid local extrema of the objective function. Let us illustrate how semi-supervision can help to overcome these problems on two examples. Left part of Fig. 1 shows the clustering result of two relatively well-separated groups of points and some outlying points using the FCM algorithm under poor initialization. It fails because of an attempt to obtain separated clusters instead of compact ones. Clusters provided by the CEM algorithm [START_REF] Celeux | A classification EM algorithm for clustering and two stochastic versions[END_REF] on a XOR-type configuration are shown in Fig. 1 (right part). Here again, data structure is not recovered despite 50 random initializations (the best result is shown). Specifying appropriate points as supervised ones allows to correctly cluster these two data sets. Many popular clustering algorithms have been extended to partial supervision including Fuzzy C-Means in [START_REF] Pedrycz | Fuzzy clustering with partial supervision[END_REF], [START_REF] Bensaid | Partially supervised clustering for image segmentation[END_REF], Hierarchical Clustering Algorithms in [START_REF] Amar | Semi-supervised hierarchical clustering algorithms[END_REF], Point-Prototype Clustering in [START_REF] Bensaid | Semi-supervised point-prototype clustering[END_REF], or Support Vector Machines in [START_REF] Bennett | Semi-supervised support vector machines[END_REF]. In this paper, we focus on the EM approach which is parametric.

A semi-supervision version of parametric clustering with EM

Finite mixtures are commonly used in pattern recognition for unsupervised learning (clustering). From the statistical point of view of clustering, data are assumed to arise from an unknown probability density function f (pdf). Finite mixtures models are a powerful and convenient approach to make an approximate decomposition of f :

f (x) = c k=1 π k f (x|θ k )
where π k and f (x|θ k ) are respectively the mixing proportions (prior probabilities summing up to one) and the classconditional densities parameterized by theta k . Note that we do not address here the problem of the number c of components involved. Let x = {x 1 , x 2 , . . . , x n } be a set of n independent and identically distributed samples from a ccomponent mixture. The log-likelihood function is

logL(Θ, x) = n i=1 log c k=1 π k f (x i |θ k ) (1)
where Θ = [π 1 , . . . , π c-1 , θ 1 , . . . , θ c ] stands for the model parameters to be estimated (notice that π c is redundant as

π c = 1 - c-1 k=1 π k ).
The estimator Θ that maximizes log L(Θ, x) cannot be found analytically. A possible approach is the EM algorithm which is a general framework to incomplete data problems. From this point of view, an observed data x i can be regarded as being incomplete where the missing part is the true class labelling. Therefore, one can form complete data y i = (x i , z i ) by introducing for all unlabeled examples x i the realization z i = (z i1 , . . . , z ic ) of a c-dimensional random variable Z representing the labels of x i , i.e. z ik is equal to 1 when x i arises from the k th component and 0 otherwise. Z is generally supposed to be a multinomial random variable parameterized by π = (π 1 , . . . , π c ) T . Let z = {z 1 , x 2 , . . . , z n } denotes a set of realizations of Z. Under these assumptions, the complete log-likelihood is defined as

logL c (Θ, x, z) = n i=1 c k=1 z ik log (π k f (x i |Θ k )) (2)
In the context of partial supervision, logL c can be split up into unlabeled and labeled part (x u x d = x):

logL c (Θ, x, z) = xi∈x u c k=1 z ik log (π k f (x i |Θ k )) x u unlabeled examples + xi∈x d c k=1 z ik log (π k f (x i |Θ k )) x d labeled examples (3)
The EM algorithm is a hill-climbing procedure that finds a local maximum of logL c (Θ, x, z). It proceeds in two steps:

• E-Step : Computation of the conditional expectation of logL c , given x and current parameter estimates Θ(t) :

Q(Θ| Θ(t) ) ≡ E[logL c (Θ, x, z)|x, Θ(t) ] (4)
As Q(Θ|Θ (t) ) is a linear combination of ẑik , we just have to compute

ẑik = E[z ik |x i , Θ(t) ] = πk f (x i | Θk ) c l=1 πl f (x i | Θl ) (5)
for all unlabeled examples. For supervised ones, z i are not estimated but fixed by the expert (possibly not binary).

• M-Step : Update parameters estimates to maximize

Q(Θ|Θ (t) ) Θ(t+1) = arg max Θ (Q(Θ| Θ(t) )) (6)
This computation depends on the theoretical model for classes to be tracked. We propose to use a model that is robust enough to cope with outliers.

A robust approach with a reject option

Theoretical model for classes

When a normal mixture model is assumed, the parameters Θ k of each component are the mean µ k and the covariance matrix Σ k of cluster C k . In [START_REF] Saint-Jean | Clustering with EM: complex models vs. robust estimation[END_REF], we have proposed to model each cluster C k as a mixture of two normal subcomponents:

f (x|C k ) = (1 -γ k )N (x; µ k , Σ k ) (A) + γ k N (x; µ k , α k Σ k ) (B) (7)
First term (A) intends to track cluster kernel points while second term (B) allows to take into account surrounding outliers via a multiplicative coefficient. Parameters γ k and α k control respectively the combination of the two subcomponents and the spread of the second one by modifying its covariance. Parameters γ k has to be taken in [0, 0.5] to penalize (B) and α k must be greater than 1. This model corresponds to an ǫ-contamination assumption which is the basis of the Student t-distribution construction (see [START_REF] Mclachlan | Finite Mixture Models[END_REF],p. 223).

Estimation of classes parameters

The key point of the approach we propose is the use of different estimates for the same random variables µ k , Σ k for (A) and for (B). In the following, μk , Σk stand for robust estimates (A) whereas μk , Σk denotes standard estimates (B). The principle of this approach is to allow standard estimates to be disturbed by outliers while robust estimates concentrate on class-kernel parameters. Robust estimates are provided through an iterative procedure, quite similar to the reweighted least-squares, that uses robust M-estimates [START_REF] Huber | Robust Statistics[END_REF]. Such an estimator is an influence function where the weight w i associated to the i th sample x i decreases as its distance to the cluster prototype increases (e.g Mahalanobis distance

d 2 M = (x -µ k ) T Σ -1 k (x -µ k )).
The area of influence is controlled by a threshold, say h: the larger h, the more samples are involved in the estimation process. Fig. 2 shows how weight w i is related to distance d M and threshold h of the so-called Huber M-estimator. Algorithm 1 describes the whole estimation process replacing parameters updating equations involved during the M-Step of classical EM.

π(t+1) k = 2 x i ∈x u ẑ(t) ik + 2 x i ∈x d ẑik card(x u )+card(x d ) μk,0 = μ(t+1) k = 2 x i ∈x u ẑ(t) ik xi+ 2 x i ∈x d z ik xi 2 x i ∈x u ẑ(t) ik + 2 x i ∈x d z ik ; Σk,0 = Σ(t+1) k = 2 x i ∈x u ẑ(t)
ik Sik,0 + 2

x i ∈x d z ik Sik,0

2 x i ∈χ u ẑ(t) ik + 2 x i ∈x d z ik ; with Sik,0 = (x i -μk,0 )(x i -μk,0 ) T ; repeat τ ← τ + 1; for x i ∈ x u x d do e i,τ = d M (x i ; μk,τ-1 , Σk,τ-1 ); w i,τ = ψ(ei,τ ,h) ei,τ ; μk,τ = 2 x i ∈x u wi,τ ẑ(t) ik xi+ 2 x i ∈x d wi,τ z ik xi 2 x i ∈x u wi,τ ẑ(t) ik + 2 x i ∈x d wi,τ z ik ; Σk,τ = 2 x i ∈x u wi,τ ẑ(t) ik Sik,τ + 2 x i ∈x d wi,τ z ik Sik,τ 2
x i ∈x u wi,τ ẑ(t) ik + 2

x i ∈x d wi,τ z ik ;

with Sik,τ = (x iμk,τ )(x iμk,τ ) T ;

until Stop Criterion; μ(t+1) k = μk,τ ; Σ(t+1) k = Σk,τ ;
Index t refers to the current M-Step while τ refers to the robust one. The more τ , the less points are used in the robust estimation process. One can need to stop this loop by using a criterion involving an upper bound on either τ or κ the rate of samples having a quite zero weight. It can be shown that the property of monotonous increase of log-likelihood (see Eq. 1) no more holds because of the iterative estimation process which results in approximating instead of strictly maximizing it. We propose to update threshold h of the Huber M-estimator during the iterative estimation with the help of labeled samples, e.g. by taking the maximum distance between supervised points and the mean estimates of their true class :

h k = max xi∈x d (d M (x i , C k ), h min ) (8)
where h min denotes a minimum value insuring sufficient statistics.

Decision rule and outliers rejection

Outlier rejection is performed during the affectation step by the way of an additional (c+1) th cluster, called the reject cluster, which is created by summing all (B)-terms. Then, the mixture decomposition is as follows:

f (x) = c k=1 πk (1 -γ k ) N (x; μk , Σk ) c clusters + c k=1 πk γ k N (x; μk , α k Σk ) rejection class (9)
Affectation is carried out according the Maximum A Posteriori criterion (MAP) among the (c+1) clusters:

P (C k |x) = πk (1 -γ k ) N (x; μk , Σk ) f (x) (10)
P (Reject|x) = c k=1 πk γ k N (x; μk , α k Σk ) f (x) (11)

Experiments

The first result we present deals with the two artificial data sets shown in Fig. 1 which structure is not correctly detected when totally non robust or non supervised algorithms are used. Fig. 3 shows the clusters obtained with the method we propose. Parameters γ k and α k are randomly chosen in the intervals mentioned in the previous section. For the left case 4.82% of points are supervised (•-marked) while 3.1% for the right one. In both cases, the proposed method provide clusters that look good and correctly reject outliers ( -marked). We have experimented the proposed method on well-known real data sets from the UCI repository 1 with various random settings of parameters γ k in [0.05, 0.5], α k in [START_REF] Amar | Semi-supervised hierarchical clustering algorithms[END_REF]20], up to 50 different initializations and supervision rate s varying from 0% (no supervision) to 100% (total supervision). Some particular results are reported in Table 1. In order to compare with other published results, e.g. by [START_REF] Labzour | Improved semisupervised point-prototype clustering algorithms[END_REF] (partial supervision without rejection), we make a resubstitution according to the MAP criterion. As expected, a low s is generally sufficient to significantly reduce error rates P e and to significantly increase success rates P c without making the reject rates P r too high. More generally, the obtained results are comparable with those of supervised methods 2 . From Table 1, it is worthy of note that P c can be used as a validation index when setting s = 100%, e.g. Wine data are known to be easily separable while Pima Indian Diabetes data are not. For the Iris data, it is fruitless to try to improve P c over a particular low value of s. In [START_REF] Labzour | Improved semisupervised point-prototype clustering algorithms[END_REF], an error rate of 3.33% has been obtained for a supervision rate of 10% and taking original classes proportions into account while selecting supervised points. Without this assumption, we divide this error by more than two (1.33%) for about twice less supervision (5.33%).

Conclusion

In this article, the advantage of the semi-supervision in context of clustering multivariate data is recalled. We propose a new algorithm based on EM which differs from the classical approach by introducing of a reject option that makes it possible to deal with outliers, as well as the use of a robust M-estimator. We focussed here on the Huber M-estimator, but others can be used. Experiments results obtained on various artificial and real datasets show that the

Figure 1 .

 1 Figure 1. Two examples where a clustering method always fail -FCM (left), CEM (right)

H 1 :

 1 Parameters update (M-Step) Input: x = {x 1 , . . . , x n }, ẑik current estimates of P (C k |x i ), h the M-estimator threshold having ψ as influence function τ ← 0;

Figure 2 .

 2 Figure 2. Huber M-estimator: weight as a function of distance and threshold h

Figure 3 .

 3 Figure 3. Two examples where the proposed method success

Table 1 .

 1 1 http://www.ics.uci.edu/mlearn/MLRepository.html 2 http://www.phys.uni.torun.pl/kmk/projects/datasets.html Real datasets -Selected results method achieve performance equivalent or better than other semi-supervised clustering approaches.

	Dataset	s	Pc	Pe	Pr
		0.%	66.93%	29.04%	4.04%
	Pima Indians Diabetes N=768,C=2,P=8	6.77%	70.18%	29.82%	0.%
		100.%	76.56%	23.44%	0.%
		0.%	96.67%	2.66%	0.67%
	Iris N=150,C=3,P=4	5.33%	98.66%	1.33%	0.%
		100.%	98.66%	1.33%	0.%
		0.%	89.32%	5.62%	5.06%
	Wine N=178,C=3,P=13	1.78%	96.07%	1.12%	2.81%
		100.%	100.%	0.%	0.%