Anne Chassagnol

Picturebooks ' Anne

Camille Joseph

Andrée-Anne Kekeh-Dika

Penser

'Anatomy of a Broken Line: Why Zigzags Matter in Picturebooks'

Anatomy of a Broken Line: Why Zigzags Matter in Picturebooks

Anne Chassagnol Shapes are at the core of pre-school picturebooks. Children at this stage are experimenting with letter formation and basic arithmetic. Identifying contours, colours, volumes and textures is a necessary step towards the acquisition of visual literacy. Numerous studies have been devoted to the role of geometrical motifs as a way of developing "children's cognitive engagement with mathematics" [START_REF] Elia | The Role of Pictures in Picture Books on Children's Cognitive Engagement with Mathematics[END_REF], some focusing more specifically on round shapes and the infinite possibilities of a circle to teach geometry while others rely on quadrilaterals to explore angles, symmetry and volume. English children's literature has always been influenced by authors who were themselves trained in mathematics, such as Lewis Carroll, whose publications in that field under the academic name of Charles Lutwidge Dodgson include

The Fifth Book of Euclid Treated Algebraically (1868), A Syllabus of Plane Algebraical Geometry (1860), Euclid andHis Modern Rivals (1879), andCuriosa Mathematica (1888). His literary work is infused with logic and geometrical concepts. Through the Looking-Glass (1871) is laid out as a chess conundrum, inspired by the knight's tour problem (Abeles, Deleuze). In this famous mathematical problem, the knight starts on a random square on the chessboard and the tour de force consists in finding the best sequence of moves to ensure that he will visit all the squares just once. Geometry also plays a key role in the publications of A. A. Milne who studied mathematics at Trinity College, Cambridge1 . The American writer, Norton Juster, author of The Phantom Tollbooth (1961), was not a mathematician per se but his training as an architect gave him an appreciation of geometry to which he paid homage in The Dot and the Line: A Romance in Lower Mathematics (1963).

If traditional geometric patterns are readily available in concept books for pre-schoolers, few academic studies of children's books have been devoted to "the strength of line" [START_REF] Hintz | Reading Children's Literature. A Critical Introduction[END_REF], the most basic and abstract geometrical figure. Being both ubiquitous and utterly versatile, the line, which can morph into an infinity of shapesa wiggle, a segment, a curve, a straight line or a zigzagis highly underrated. The father of sequential art, the Swiss Rodolphe Töpffer (1799Töpffer ( -1846)), invented a new type of visual narrative in his Histoires en étampes, or Stories in Etchings. He also published a lesser-known book two years before his death, Voyages en zigzag (1844) in which he promoted school trips to educate children outside the classroom. These trips did not involve a traditional tour, let alone a Grand Tour, but excursions through walks in the Alps. Zigzags, for Töpffer became a form of geographical pedagogythe art of meandering. They reveal the importance of rambling through physical space but also on the page, by connecting the text and its illustrations, by allowing fragmented, interrupted narratives through a series of digressions (Le Men,[39][40], just like the squiggle in

The Life and Opinions of Tristram Shandy (1759) offers a visual reflection on the art of storytelling. Tim Ingold's remark: "Lines are everywhere" and "subsume all the aspects of everyday human activity" (Ingold, 2007, 1) is true of children's literature, as the motif of the line seems to have become a sub-genre in itself [START_REF] Greene | When A Line Bends… A Shape Begins[END_REF][START_REF] Whitman | Lines that Wiggle, Mapplewood[END_REF][START_REF] Bossio | A Line[END_REF][START_REF] Lee | Lines[END_REF]. Lines appear in a variety of texts for young children, from alphabet books (Sharon Werner's Alphabeasties and Other Amazing Types, 2009), interactive books (Andrew Drew and Drew, 2012 by Barney Saltzberg), to concept books (Shapes, Shapes, Shapes, 1996, by Tana Hoban;Length, 1995, by Henry Arthur Pluckrose) and counting books (1,2,3 San Francisco: A Cool Counting Book, 2019, by Puck), challenging spatial skills and mathematical deductions. In Follow the Line (2006) or A Line Can Be (2015) by Laura Ljungkvist, readers are invited to run their finger along an uninterrupted trail, tracing words, solving riddles along the way, while eventually questioning the reliability of the line. Harold and the Purple Crayon (1955) by Crockett Johnson is emblematic of these picturebooks in which the whims of the pencil give shape to the story.

From a linguistic point of view, the polysemy of the term "line" points towards a rather fluid definition, which conveys a variety of meanings from a rope, to a track, a border or a direction.

A line, as in "a line of argument", can also be interpreted as a pattern of reasoning, which suggests that it is both visual and narrative. By connecting two points, a line establishes a link between two separate units, giving them a sense of direction, the beginning of a storyline.

If the definition of a line seems straightforward, how could we define a broken line? In his pioneering essay on the impact of geometrical forces in painting, Point and Line to Plane (1926), Wassily Kandinsky attempts to define a broken line, also called an "angular line", which is composed of at least two parts, as "the result of two forces which have discontinued their action after a single thrust." (Kandinsky,69). Do broken lines pertain to the category of "graphic codes" such as "blurs, motion lines, and the distortion of perspective" (Nikolajeva,139)? This article examines lines in picturebooks, more specifically broken lines, zigzags and discontinuous or disconnected lines that meet at oblique angles. The term "picturebook" is intentionally written here in one word and refers to Martin Salisbury and Morag Styles' definition where image and text cannot be dissociated and should be understood as: the particular use of sequential imagery, usually in tandem with a small number of words, to convey meaning. In contrast to the illustrated book, where pictures enhance, decorate and amplify, in the picturebook the visual text will often carry much of the narrative responsibility. In most cases, the meaning emerges through the interplay of word and image, neither of which would make sense when experienced independently of the other. (Salisbury and Styles,7) Broken lines, far from being a graphic mistake, represent a key stage in early literacy. Using Maria Montessori's concept of "psychogeometry", I will argue that one of the reasons why early picturebooks offer so many illustrations based on irregular lines, crosshatching or zigzags is because they introduce oblique shapes that will help pre-schoolers to identity the letters of the alphabet, preparing them, in turn, to form letters on a page. The second section will focus on the narrative value of broken lines and how they function as visual metaphors that are indicative of a disruption in the plot. Drawing on Scott McCloud's analysis in Understanding Comics. The Invisible Art (1993) devoted to the pictorial grammar of comics, I will investigate the "expressive potential of broken lines" (124). This section will contend that the sudden interruption of the line can be a comic mechanism to reactivate the story or can point to the discrepancy between the text and the image to signal dramatic tension. Finally, using the theory of visual perception and graphic design, I would like to suggest that broken lines, in their discontinuity, are essentially dynamic, thereby "implying more energy, more activity." (Nodelman,161). This third and final part will identify a series of picturebooks in which broken lines animate the page by resorting to different strategies that either produce an illusion of movement by bringing back to life retro instruments pre-dating cinema, or, by faking digital media, mimicking ipads and computer games.

Lines and Letters: The Geometry of Early Literacy (Kress, 17)

The omnipresence of lines in picturebooks for the very young and concept books is not incidental or purely aesthetic. Zigzags, snaky, wiggly and crenelated lines, are intentionally used as a way of stimulating children's early visual literacy.

Maria Montessori (1870Montessori ( -1952)), the influential Italian educator, pioneered an educational method based on training the senses. Initially conceived to suit the needs of mentally challenged children, she later adapted it to help pre-school toddlers. The very shape of the educational toys she used had a unique place in her pedagogical approach. She designed a series of self-instructive sensorial materialgeometric insets, stacking cubes, miniature wooden stairsencouraging children to experience geometry for themselves, before they were taught key concepts. Geometry plays a pivotal role in her method. It is mentioned in most of her publications, from Psychogeometry (1911), to Dr. Montessori's Own Handbook (1914), up to The Montessori Elementary Material:

The Advanced Montessori Method (1917) 2 . Geometry and the impact of lines in space also stand at the core of the Swiss constructivist Jean Piaget's theory of cognitive development. In The Child's Conception of Space (1956) or The Child's Conception of Geometry (1960), Piaget conducted a series of empirical experiments on children on the impact of linear order, angular measurements, straight and divided lines, with the intention of improving their mathematical reasoning by manipulating geometrical shapes [START_REF] Montessori | The Absorbent Mind[END_REF] upbringing in a mixed-raced family, exposing her budding passion for mathsa perfect cross between her father's artistic side and her mother's scientific mind, the second part offers an addendum, a self-reflective embedded book in the form of a fake maths diary. It is not exactly an exercise book but rather a copy of the notes that the narrator has assembled. The final pages encourage the reader to flick through the picturebook backwards to identify the fractals, polygons, and other solid forms on each page.

While Montessori crafted new "didactic materials" (Montessori, 1914, 18) to help primary school children acquire a sense of geometry through "conceptual manipulation" (Zuckerman, 2010, 130), she also focused on developing babies' "specially refined form of sensitiveness" (Montessori, 1967, 48). In the 1950s, students attending The Assistant School of Montessori Infancy in Rome combined the work of the avant-garde graphic designer Bruno Munari , particularly his "machine inutile", and the discoveries of Maria Montessori to create "The Bilancina Munari" (Munari's multi-sensorial mobile for newborn babies)4 . Bruno Munari was well aware himself of the recent advances in contemporary childhood education theories (Campagnaro,82). The mobile which is still very much in use today in many nurseries is composed of an articulated structure with three supporting horizontal stems, holding four geometrical shapes in black and white. The principle is simple: once suspended above the crib, the shapes interact, producing visual stimuli through animated broken lines, illustrating

Montessori's belief that the visual repetition of a motif appeals to "the absorbent mind" of a child.

The child has a different relation to his environment from ours. Adults admire their environment; they can remember it and think about it; but the child absorbs it. The things he sees are not just remembered, they form part of his soul. He incarnates in himself all in the world about him that his eyes see and his ears hear. In us the same things produce no change, but the child is transformed by them. (Montessori, 1997, 48) The lines and geometric shapes Maria Montessori designed for her didactic materials proved to be instrumental in her approach to child development. As the cortical maturation is in its early stages after birth, neonates find it easier to focus on geometrical, high contrast, shapes, especially when they appear in black and white [START_REF] Banks | Infant Pattern Vision: A New Approach Based on the Contrast Sensitivity Function[END_REF][START_REF] Bornstein | Development of Infancy. A Contemporary Introduction[END_REF]. Shapes are often the first geometrical patterns that pre-verbal infants encounter in picturebooks, particularly early concept books. According to Bettina Kummerling-Meibauer and Jörg Meibauer, these books target children between twelve to eighteen months and show single objects from the child's surroundings, such as a ball, a doll, an apple or a chair ( 14). They are designed to encourage visual literacy and develop the child's first lexicon (15). Discontinuous lines therefore function as iconic symbols. A jagged shape might represent a hungry predator. While a green triangular form might stand for a Christmas tree, regular broken segments might suggest a flight of steps, therefore indicating a movement upwards or downward. More generally, zigzags illustrate an iconic visual language. The Noisy Book (2010) by Soledad Bravi is aimed at children from 0 to 4 years old, each double page is composed of the visual description of a sound (a crying baby, a whistle, a kiss) and its textual explanation. Zigzags illustrate the irregular cracking tone of the drum ("the drum goes ratatat"), they depict the electrifying mane of a dangerous predator ("the lion goes roar"), or the explosive sound of the firecracker ("the firecracker goes boom") and in doing so they transcend the visual and the textual by introducing concepts such as boldness, fear, surprise or power. Alphabet books offer an interesting transition in early literacy as they typically match a letter to a sound. In ABC (2015) by Xavier Deneux, "A" is for Alligator and "B" is for bird. Nothing groundbreaking so far in terms of pairing, except that it might not be as simple as it seems. The image incorporates visual games, calling the attention of readers who are expected to detect the A-shaped mouth, teeth, and scales of the reptile but also to mentally fit the misplaced letter back where it belongs. This involves identifying its outline as well as its interior shape. Some alphabetical letters are hollow so that the readers can actually run their finger along the contours to experience the shape of the letters, preparing them for the next step, not only to spell but also to write. Other picturebooks such as Lines that Wiggle (2009), encourage pre-schoolers to become more aware of line patterns in their immediate surrounding, in order to build up motor The schematic representation of objects and concepts in early children's books is a necessary journey toward the acquisition of reading and writing skills. Tracing lines facilitates pencil control during letter formations; it helps keep the size of the letter within the boundary of the page, while the exercise can also be praised for instilling a sense of text directionality. In fact, from as early as the XVIth century up to the XIXth century, breaking the line as part of a writing exercise was one of the principles of samplers, which were used as an educational method, on textile rather than paper, to teach young girls the art of needlework, particularly as the practice of the marking of house linen remained a family tradition (Goggin,318). The etymology of the word "sampler" goes back to the Latin "exemplum" and the French "essamplaire", an example referring to the owner's mastery of domestic crafts, ranging from drawing, darning, to embroidery. It was used both as an exercise in stitching per se and as a catalogue of motifs to pass on to future generations5 . In certain embroidery patterns, the needlework is based on "a glottographic system, one that provides visible representations of spoken language utterances, for example, alphabets or verses." (Sampson,29). Subsequently, from the XVIIIth century onwards, samplers were indicative of a lesser-known social history of femininity. They document the intimate history of young women via significant datesbaptism, weddings, the birth or death of a sibling, initials, as well as the types of psalms learnt, while referring to local buildings and historical events. If samplers were initially a way of ensuring women a means of existence, and of achieving domestic literacy, they became a method, a grammar of ornaments, recycling repetitive motifs,alphabets, crosses and Christian symbols, naturalistic motifs, rosettes, interlacing symbols, multiplication tablesto teach girls to write using a needle. As they became "an educational tool, lettering displaced pattern, literacy rather than stitchery took precedence." (Goggin,324;Humphrey,6). Learning to write, learning to stitch, implied mastering the art of breaking the line by cross-stitching. Raiexplores the narrative potential of a simple straight line (Cavalier, 2011, 211). It stands, on a metaphorical level, as the tyrannical relationship between a creator and its creature. A male figure, M. Linea, drawn out of a white uninterrupted line on a piece of paper, rises to life and addresses his creatorwho is only visible via his hand holding a penciland implores him to carry on drawing so that he can continue to exist. The segmentation of the line becomes the site of an ontological reflection on being and becoming, but it also questions what happens between the segments, in the void, where the viewer is not granted access. The series relies on a comic mechanism alternating between the different moods of the cartoonistfeeling either lazy, uninspired, slow, slightly jaded or, on the contrary, highly creativeand the character begging In Tomi Ungerer's iconic picturebook, The Three Robbers (1962), broken lines offer a dramatic visual counterpoint to the traditional narrative of the orphan girl. When one of the brigands fractures the wheel of the vehicle carrying a screaming child, broken lines multiply within the illustration, conveying a sense of impending danger. The elegant position of the brigand's body, on tiptoes, delicately holding a massive weapon, and the zigzag motif on his head-dress, read as disruptive signs. A "Z" reappears on one of the men's hat when he kidnaps the little girl, who is safely tucked under his coat. The alphabetical letter ironically associates him with the famous masked hero, sly as a fox, stepping out of the night to rescue the innocent. In this example, the broken line heralds the upcoming sudden twist in the plot, taking the narrative off course, to deploy a new storyline. According to W.J.T. Mitchell, lines mark the vision of the narrative, and in that respect, the wheel appears as a revelatory "spatial form": "One of our most common spatial equivalent of literary action is the image of the wheel whose revolutions mark not just the time-line but the fortunes of the hero." (Mitchell,288). In Ungerer's tale, the attentive reader is already privy to the fact that "the orphan named Tiffany" was heading in the wrong direction and that her destiny is about to change ("She was on her way to live with a wicked aunt. Tiffany was delighted to meet the robbers."). Ironically, as the etymology suggests, the apparition of their little protégée announces the robbers' new career path as Tiffany comes from the Greek "theophania", meaning "manifestation of God", in other words, Tiffany is the brigands' epiphany. The discontinuous line corresponds here to the unexpected discontinuity in the narrative. The treacherous title was all but a diversionthe Three Robbers are no villains after all. Lines are equally deceptive in the wordless picturebook Caged (2018) by Duncan Annand. The complex architectural plot is mimetically erected like a building, through a series of beams that are being deployed rather laboriously by two amateurish builders. The book weaves simultaneously three different linear sequences of events: a blue bird on a branch building a nest; two eccentric gentlemen erecting a gigantic aviary; and a flock of parrots trapped in cages. The reader is visually challenged by the multiplication of acentric lines on the page cluttered with trees, poles, saws and wired cageswhich seem to branch out into contrapuntal storylines to create a dramatic effect. The fragile equilibrium between the horizontal and the vertical poles reinforces a sense of imminent instability (Kandinsky, 138). On the right page, two men are invested in an anti-ecological economy, destroying the surrounding nature to sustain their project, ultimately burning the forest down to trap animals in cages. Meanwhile, on the left page, the bird's nest develops organically, recycling the leftover twigs to lay her eggs. On a cognitive level, the reader has to connect the lines and match the story on each page to make sense of it all. In that respect, Caged is a picturebook version of a game of stacking blocks, as it stimulates spatial reasoning through geometry [START_REF] Casey | The Development of Spatial Skills Through Interventions Involving Block Building Activities[END_REF]. The construction of the birdcage is done by adding five different units, which eventually collapse when the blue bird sits at the apex. The complex juxtaposition of irregular lines, enmeshed and unstable, creates a narrative tension while revealing the potential playfulness of lines that are prone to gravity and can eventually yield. The sheer pleasure of the picturebook lies in its playful apparatus: the reader is invited to witness the structure being assembled, rather painstakingly, for the sheer pleasure of seeing it give way a few pages later. (Treiman,3). Broken lines are not a graphic whim: as an expression of visual communication, they signal an accident and mark the graphic deviation of the narrative.

The

Interactive Lines as Visual Forces: Energy, Rhythm and Speed

If the motif of the broken lines may be understood as a didactic tool, however, it cannot be reduced to an exercise in developing literary skills. As we have seen, zigzags and oblique angles are powerful unexpected signs that visually notify a counterpoint in the story. Geometrical shapes in a picture are also "dramatic visual forces" (Bang, xiii); lines, colours, sizes are displayed in order to affect the viewer in a particular way. In Picture This. How Pictures Work, Molly Brand uses the basic example of the oral tale, Little Red Riding Hood, to show how the choice of illustrations, their shape, their position, their size, might affect the readers and their understanding of the story. Her take is particularly original as she uses geometrical shapes to teach the art of visual storytelling. By contrasting the impact of line, colour, size, length and width, along with perspective, she determines the emotional effect each may have on the reader. She uses the components of a well-known fairy-tale (the girl, the wolf, the trees, etc) that are symbolized by a geometric shape. When she raises the question of drawing the forest, she suggests using tilted lines rather than vertical rectangles because "(d)iagonal lines give a feeling of movement or tension to the picture" (Bang,24). It would be incorrect to assume that broken lines can only suggest fracture or disruption; broken lines, zigzags, chevrons, have an energy of their own that electrify the image and create a visual palpitation on the page.

In Point and Line to Plane (1926), Wassily Kandinsky, who was interested in exploring new modes of expressions and particularly the combination of the five senses through synaesthesia, demonstrates how different lines project different emotions on canvas, depending on their positions. The opposite directions at work in an angular line can be understood as a graphic expression of a vibration. Kandinsky recalls that, in music "the line supplies the greatest means of expression", its variation in width suggesting the modulation of the pitch of the instruments (99). The regular and repetitive pattern of broken lines creates a sense of tempo: zigzags produce a music of their own. In Whistle forWillie (1964), Ezra Jack Keats, the author of A Snowy Day (1963) and winner of the Caldecott Medal, one of the most prestigious American children's book awards, relates a small-scale experience in a deceptively simple form as the story retraces the relentless efforts of a boy called Peter to whistle for his dog, Willie. (Schwarcz,183). Keats uses collage, which was not entirely new in children's books: Sparkle and Spin (1957), by the graphic designer Paul Rand, is another example of a ground-breaking picturebook using this particular technique inspired by geometric shapes (Alderson,201). In Keats's picturebook, the pink jagged ornamental lines on Peter's jumper operate as graphic metaphors, mimicking his failed attempts ("Peter tried and tried to whistle, but he couldn't.") to produce a sound ("He blew till his cheeks were tired. But nothing happened."). The crescendo structure of the picturebook, based on the delayed expectation of the whistling sound, alternates between the absence of "visible sounds" on the pageno musical notes, nor onomatopoeiaand the visual modulation of the broken line, vibrating like an acoustic wave, striving to find a balance between high and low pitch. Peter's jumper seems to morph into an oscilloscope, allowing the reader to visually experience the perception of a sound about to be produced (Schwarcz,77). The zigzag pattern of Peter's top is visually efficient on two levels, not only does it reveal the inchoate sound effect, his attempts to whistle, but it also renders visible the otherwise invisible expression of the boy's emotionsfluctuating between failure, persistence, confidence, to self-empowerment6 . To produce this sound, Peter subjects his body to a series of sensorial experienceshe sets his body into a spin, hides in a cardboard, experiences isolation, defies gravity by running away from his shadow; he disappears from the page, before duplicating his image in a mirror "to practice whistling". The title itself with the unusual absence of typographic space between the preposition and the proper noun ("forWillie") seems to reproduce the lack of necessary pause to form the sound by holding one's breath. While zigzag patterns are omnipresent in the image, the text invites the reader to identify the "W" letter words and attempt to pronounce each occurrence on the page ("wished", "whistle", "whirled", "wink", "sidewalk", "grownup", "walked", "shadow", "who", "way"). On the iconic book cover, the zigzag motif of Peter's jumper points toward the dog, as the triangular shapes of the lines suggest a direction. In Reading Images, The Grammar of Visual Design, Gunther Kress and Theo Van Leeuwen have shown the visual semiotic effect of this geometrical triangular form:

the triangle, especially when tilted, is a (fused structure of a) participant and a vector, because it can convey directionality, point at things. The meanings it attracts are therefore less like 'qualities of being' than like processes. (..) Triangles are a 'symbol of generative power', and represent 'action, conflict, tension' (Kress,(55)(56). that is achieved through the use of broken lines. Describing a similar shape, Molly Bang concludes that explosive circles with spiky edges trapped in a rectangle produce a flashing effect:

The rectangular frame of the picture forms a separate world inside itself. The edges enclose our attention and force it inward, and we are (usually subconsciously) aware of the rectangle's center. The frame creates a sort of visual conflict that can be thought of as a radial force, directing our attention inward from the frame midpoint. (…) the center of the picture also radiates energy out of itself. You can sense this even more strongly when the picture frame is square or round. (Bang,76) Although a zigzag line might be perceived as an erratic, unintentional geometrical figure, its multi-directionality, its versatility, allowing a mere segment to transform into a variety of shapes (a stratified soil, an arborescence, a volume, a grid, or the complex metallic structure of a building), is indicative of its dynamic quality. apparent that what is visible is only a fraction of the image: what they have seen is not only partial but static. The red round tree with its complex system of indented leaves, on closer inspection, is not an element of vegetation but rather a peacock spreading its tail. The enmeshed pile of twigs that, at first glance, seemed organic and unstructured turns out to be a very effective decoy, teeming with life, to lure predators away. The cross-section unveils a social architecture designed to optimize the ant's nest. The Great Journey is in fact a double expedition. It follows the migration of a bird from page 2 to page 24 through various landscapes but it also implies a nonlinear reading protocol, as the reader has to scour the page, both vertically and horizontally, to identify the hidden pictures in blue under the red drawings and then proceed to connect them to each other, as they operate in a sequential mode.

Interlaced stripes can also produce a much more dynamic image, as it is the case in kinetic art, which relies on an optical illusion and distorts the viewer's perception by juxtaposing, or superimposing, black and white geometrical lines7 . In his pop-up books, Gallop! (2007), Waddle! (2009) and Kick! (2008), Rufus Butler Seder relies on visual trickery to create an impression of movement by simply asking the reader to turn the page. The mechanism recalls the origins of animation, particularly Victorian devices, which allowed a sequence of images spinning on a cylinder to form a single image. Seder's books also rely on the practice of "Ombro-cinéma", which he refers to in each of the subtitles, "a book in scanimation". "Ombro-cinéma" is a slightly more modern invention dating from the late 1920s and based on a barrier-grid system of animation. To activate the image, the reader needs to slide a rectangular piece of transparent paper with vertical lines on hatched silhouettes. Once the reader turns the page, the horse in Gallop! is set into motion when the two drawings made of interrupted hatching lines overlap, thereby creating an illusion of motion. Wordless storytelling is not particularly unique as such. What is novel here is the combination of two storylines: a Carrollian Through-the-Looking-Glass narrative with a Conradian Heart of Darkness scenario in an environment, apparently deprived of oxygen, and abandoned by humans.

8 An anaglyph is a stereoscopic motion or still picture in which the right component of a composite image usually red in colour is superposed on the left component in a contrasting colour to produce a three-dimensional effect when viewed through correspondingly coloured filtered in the form of spectacles. (Merriam-Webster)

Jim's journey in his clunky spacesuit starts when he chases a dragon-fly through a mirror. The initial passage into another dimension is signalled by the reflection of a Z frame door, revealing a portrait of another man with a surprisingly similar suit and stereoscopic glasses. The title of the book ironically points to Jim's curiosity, while the double reflexive nature of this mysterious visitor in the mirror sparks the reader' inquisitivenesscould he be the protagonist? Did he journey there and back again? Could there be someone else out there? While the glasses are a useful aid for perceiving an additional dimensionthe picturebook remains undecipherable without their helpthe turning of the page fails to catch up with the central character who is always one step ahead of the reader, hiding within the decor. As soon as Jim goes through the looking-glass, there is a change of paradigm, from the graphic novel to the protocol of the video game. The character is set into motion through situationsan escape room, visually impaired sets, infested waterscommonly found in videogames, each corresponding to a different level of the game. Within each page, broken lines, whether they appear through diagonal lianas or oblique pillars, hinder the progression of the character who is on a course to complete his mission.

This hypnotizing journey, which started with a dizzying sense of seeing too much, of catching a glimpse of a dimension that cannot be achieved with paper, ends with an overwhelming sensation of heat. The mirror is cracked, lines lacerate the page, as the light is diffracted and radiates from the lighthouse. Matthias Picard's wordless masterpiece illustrates the multi-sensory quality of broken lines. Jim's curious journey is a voyage out, beyond paper.

  3 .

Figure 1 :

 1 Figure 1 : Miguel Tanco, Compte sur moi, 2019. Traduction de Cécile Hermellin © Éditions Gallimard Jeunesse.

Figure 2 :

 2 Figure 2 : Xavier Deneux, ABC, 2015 © Chronicle Books.

  skills and prepare them for letter tracing. Not aBox (2006) or Not a Stick (2008), both by Antoinette Portis, prompt children to question the versatility of a simple object that can morph into a variety of items, at a time when they start experimenting with pencil strokes and the idea of breaking lines to write in lowercase and uppercase letters. Zigzags and interrupted lines are a common feature of worksheets and teaching resources. They help children master the ligature of the "W" or the slanted line of the letter "N".

  Aesthetics of Accident: Zigzags, Narrative Tensions & Visual CommunicationIf the manipulation of motifs of broken lines is a key stage in the acquisition of writing skills and literacy, what might be the value of the discontinuous line from a narrative point of view? Or is it the other way round? Does the derailment of the narrative produce, on a graphic level, a series of broken lines? According to Maria Nikolajeva and Carole Scott, "the concept of the thread itself" is "metaphorically used so often to represent the thread of the story or narrative progression."(Nikolajeva, 10). This is typically what is at stake in Serge Bloch's I Can't Wait (2005), The Big Adventure of A Little Line (2015), or in Extra Yarn (2012) by Marc Barnett. The question is not so much what happens when we follow the line but rather what happens if the line is interrupted. Does the broken line function as a structuring narrative device and if so, how does it operate?

Figure 3 :

 3 Figure 3 : Osvaldo Cavandoli, La linea, episode 101, 1978 © La Rai.

  to be animated into existence. Each three-minute episode is structured on a what-happens-next scenario when the line is unexpectedly cut off. La linea exemplifies what Scott McCloud, in his picture plane pyramid of pictorial vocabulary, defines as "iconic characters": "Most comics art lies near the bottomthat is along the iconic abstraction side where every line has a meaning.Near the line, but not necessarily on it! For even the most straightforward little cartoon character has a 'meaningless' line or two!(McCloud, 51)." The cartoon is technically silent but, here, the sequence of events is punctuated by the mumbling protagonist who exhorts the cartoonist to keep on drawing. As Vivian Carol Sobchack suggests: "The line constructs not only a figural narrative but also a narrative of figuration and thus provides an exhilarating meta commentary"(Sobchack, 257). Each episode ends with the character going off piste, in the abyss, as he is kicked off the page by his creator until the next episode. The broken line, in this case, is a visual storytelling device, based on comic relief mechanisms, used at the climax of the narrative arc to reboot the narrative process.

Figure 4 :

 4 Figure 4 : Tomi Ungerer, The Three Robbers 1962/ Les Trois brigands, 1968 © L'École des loisirs.

Figure 5 :

 5 Figure 5 : Duncan Annand, Caged, 2018 © Tiny Owl.

Figure 6 :

 6 Figure 6 : Dr. Seuss, On Beyond Zebra!, 1955 © Random House.

Figure 7 :

 7 Figure 7 : Ezra Jack Keats, Whistle forWillie, 1964, / Un garçon sachant siffler, 2012 © Didier Jeunesse.

Figure 8 :

 8 Figure 8 : David A. Carter, White Noise, 2009 © Tate Publishing. Photo © Blog Lines Between.

Figure 9 :

 9 Figure 9 : Agathe Demois, The Great Journey, 2015. La Grande Traversée, 2014 © Seuil jeunesse.

Figure 10 :

 10 Figure 10 : Michaël Leblond and Frédérique Bertrand, Paris en Pyjamarama, 2014 © Éditions du Rouergue © Photo ohpopup.canalblog.com.

Figure 11 :

 11 Figure 11 : Matthias Picard, Jim Curious. Voyage à travers la jungle, 2019 © Éditions 2024.

Figure 12 :

 12 Figure 12 : Victor Hussenot, The Land of Lines, 2017/ Au pays des lignes, 2014 © La joie de lire.

  The story of Count on Me (2019) by Miguel Tanco revives the heritage of Maria Montessori. It reflects the importance of experiencing and visualizing maths in everyday life. It focuses on a

	little girl's obsession with lines. She detects mathematical concepts that materialize in her
	immediate surroundings. While the first part unfolds as a presentation of the narrator's
	2 See Montessori, Psico geometría : el estudo de la geometrica basado en la psicologia infantil, Barcelona, Casa
	Edit, Araluce, 1934. Montessori, Manuale di pedagogica scientifica, Napoli, A. Morano, 1921. Montessori,
	L'autoeducazione nelle scuole elementari, Roma, Loescher, 1916. Montessori, L'autoeducazione nelle scuole
	elementari. Roma, Loescher, 1916.
	3 Piaget, La Représentation de l'espace chez l'enfant, Paris, PUF, 1948. Piaget, La Géométrie spontanée de l'enfant,
	Paris, PUF, 1948.

See the poem "Lines and Squares" (1924) and Eeyore's interest in letters in The House at PoohCorner (1928).

I am particularly indebted to Marnie Campagnaro for her precious help on tracing the origin of what is wrongly referred to as the 'Montessori Mobile' and for pointing out to me Silvana Quattrocchi Montanaro's article, "Protogeometria: forma e dimensione nell'esperienza dei bambini nei primi anni", Vita Dell'Infanzia., n°11-12, 2005, p.47-54.

See the website of The Victoria & Albert Museum: "Embroidery -A History of Needlework Samplers" https://www.vam.ac.uk/articles/embroidery-a-history-of-needlework-samplers (Accessed on January, 11 2020)

See more specifically McCloud, Chapter 5, 'Living in Line', Understanding Comics. The Invisible Art, New York,HarperCollins, 1993, p.118-137.

see https://www.tate.org.uk/art/art-terms/k/kinetic-art(Accessed January, 13 2020)

Other interactive picturebooks mimic the protocols of digital media. In the wordless picturebook

The Land of Lines (2017) by Victor Hussenot, the sequential visual narrative is inspired by the culture of video games 9 . The story, told initially in bichromy, relates the adventures of a boy, in blue, and a girl, in red, as they progress in a landscape of broken lines that assemble themselves and recompose, according to a different pattern on each page. The irregularity of the line affects the context, by modifying volumes and perspectives, to produce a variety of intricate gamescapes. Broken lines can also morph into a series of obstacles that the characters need to overcome, if they wish to proceed forward. The repetitive use of zigzags partakes of the gamification of the story progressing in a non-linear, interactive dimension, as the game is played on a dual level (Egenfeldt-Nielsen, 7): the red and blue figures need to find a way out, while the reader is invited to locate them on the page and follow their tactical moves. Broken lines, as a graphic tool, create a visual dynamic that animates the image. The actions of the characters, requiring speed when they fall, slide, or jump, are often accompanied by a short line underlining part of their body, emphasizing a gesture, its intensity, or an emotion which seems to transform the figure into a GIF that can loop continuously. This is even more apparent in the second part of the book, when Hussenot introduces a third colour: each figure with its distinctive, flashing tonalities, red, blue or yellow, can be individually located on the page. In The Land of Lines (2017), Hussenot transposes the codes of video games and uses irregular lines to shape and pace the story, and by doing so he creates a fiction within the fictitious space, which consists in letting the readers believe that they have become players, rather than readers. From the author's perspective, broken lines offer an illusion of improvisation; while from the reader's point of view, they "map out the networks of possible player choices in a scene" and simulate his immersion in a parallel media.

Zigzags have become a fundamental component of a particular immersive category of picturebooks that attempt to reinvent the traditional text-image articulation by using digital media techniques largely based on broken lines. Jim Curious, The Land of Lines, Gallop!, or The Great Journey, all require "a nontrivial effort […] to allow the reader to traverse the text." (Aarseth,1).

They either require the use of an optical instrument (a pair of glasses or a sheet of rhodoid) to read the image or they imply a specific strategy to navigate through the book. Readers are no longer passive, they have no other choice but to gamble their way forward, and decide whether they wish to follow a line to proceed further. The broken line, as a geometrical motif, no longer stands for a mishap, on the contrary, it is a prop that empowers the reader. 9 See Hussenot, https://www.du9.org/entretien/victor-hussenot/ (Accessed January, 13 2020)


Ons Aouedi

Kandaraj Piamrat

Guillaume Muller

Kamal Singh

Federated Semi-Supervised Learning for Attack Detection in Industrial Internet of Things

Keywords: Data privacy, federated learning, machine learning, deep learning, semi-supervised learning, intrusion detection

Security has become a critical issue for Industry 4.0 due to different emerging cyber-security threats. Recently, many Deep Learning (DL) approaches have focused on intrusion detection. However, such approaches often require sending data to a central entity. This in turn raises concerns related to privacy, efficiency, and latency. Despite the huge amount of data generated by the Internet of Things (IoT) devices in Industry 4.0, it is difficult to get labeled data, because data labeling is costly and time-consuming. This poses many challenges for several DL approaches, which require labeled data. In order to deal with these issues, new approaches should be adopted. This paper proposes a novel federated semi-supervised learning scheme, that takes advantage of both unlabeled and labeled data in a federated way. First, an AutoEncoder (AE) is trained on each device (using unlabeled local/private data) to learn the representative and low-dimensional features. Then, a cloud server aggregates these models into a global AE using Federated Learning (FL). Finally, the cloud server composes a supervised neural network, by adding fully connected layers (FCN) to the global encoder (the first part of the global AE) and trains the resulting model using publicly available labeled data. Extensive case studies on two real-world industrial datasets demonstrate that our model: (a) ensures that no local private data is exchanged; (b) detects attacks with high classification performance, (c) works even when only a few amounts of labeled data are available; (d) has low communication overhead.

I. INTRODUCTION

Recent advances in communication and Internet of Things (IoT) technology enable the realization of the Industrial Internet of Things (IIoT). IIoT refers to the networking and connection of production equipment, instruments, products, and sensors, with the industrial and manufacturing applications [START_REF] Nguyen | Federated Learning for Industrial Internet of Things in future industries[END_REF]. In IIoT, a large number of smart devices and sensors act in the background to collect the environment and user data. However, these devices are vulnerable to several cyberattacks [START_REF] Mishra | Internet of Things applications, security challenges, attacks, Intrusion Detection, and future visions: A systematic review[END_REF] where the attackers can intercept and analyze some sensitive data [START_REF] Wang | Secure-enhanced Federated Learning for AI-empowered electric vehicle energy prediction[END_REF]. As a consequence, there is an urgent need to design efficient attack detection mechanisms. As a counter measure, more intelligent methods need to be designed and deployed in the network. Recently, Machine/Deep Learning (ML/DL) models have shown potential due to their ability to analyze the traffic and extract knowledge out of it. ML/DL models can automatically diagnose and detect attacks using Ons Aouedi and Kandaraj Piamrat are with LS2N, Nantes University, France, (e-mail: {firstname.lastname}@ls2n.fr).

Guillaume Muller and Kamal Singh are with Univ Jean Monnet, IOGS, CNRS, UMR 5516, LaHC, F -42023 Saint-Etienne, France, (e-mail: {firstname.lastname}@univ-st-etienne.fr flow and packet-based features. In fact, building DL models consists of three steps: i) data capture and labeling; ii) data pre-processing, and iii) model training. The first step requires that data are manually labeled after being captured, which is a highly expensive and time-consuming process. The second and third steps require the data owners to send their private data to a central entity for pre-processing and model training. This raises privacy issues as confidential data might need to be shared in the process.

Data labeling and unsupervised learning. Semisupervised learning can exploit abundant unlabeled data in combination with a small amount of labeled data, thus has received a lot of attention recently to solve the problem of labeling data. The unsupervised learning captures the most relevant features from the unlabeled data in order to provide a better representation of the data than the initial raw data (i.e., input data) itself. After capturing these features, the supervised training uses the labeled data to fine-tune model parameters and constructs the final model. The final model can then be used for example to classify the traffic as either benign or an attack.

Data privacy and Federated Learning (FL). In the Industry 4.0 environment, the datasets are distributed across the IIoT devices, where the data of each device is limited in diversity and quantity. Moreover, due to privacy concerns, the devices may not be willing to share their data with a central entity. Besides, privacy concerns, collecting the data to a central entity could lead to significant communication overhead as well as introduce network congestion [START_REF] Nguyen | Federated Learning for Industrial Internet of Things in future industries[END_REF]. All this makes it impractical to collect raw data from different IIoT devices. Solving the above issues requires a collaborative approach where the intrusion detection system is built while leaving the data at the location of its production. Federated Learning solves exactly this problem. FL does not need to move the data to a central entity. It is an iterative learning process where the model performance can be improved during each communication round [START_REF] Mcmahan | Communication-efficient learning of deep networks from decentralized data[END_REF]. More specifically, the IIoT devices (e.g., industrial machines or robots) using FL can train a model locally. Then, they can just send the model parameters instead of the raw data to a central entity, which can aggregate the different received models.

In our work, we address both the problems of data labeling and privacy at the same time, by combining semi-supervised learning with FL. This allows the IIoT devices to participate in the training process without labeling and sending their data to a central entity and in turn increase the scalability and robustness of IIoT applications as well as overcome data limitation problems and privacy concerns.

A. Related work

In this section, we first present some representative solutions of recently proposed approaches for intrusion detection based on conventional ML/DL models (Section I-A1). Then, we present the recent achievements of state-of-the-art approaches that proposed intrusion detection systems using FL-based architecture (Section I-A2).

1) Conventional Machine/Deep Learning models: ML/DL models offer a way to detect attacks efficiently, as they can find useful patterns in data in a reasonable time. This ability to generalize from examples also enables them to better adapt to new kinds of attacks than handcrafted Intrusion Detection Systems (IDS). In this context, Ge et al. [START_REF] Ge | Deep Learning-based Intrusion Detection for IoT networks[END_REF] proposed a novel intrusion detection approach for IoT systems using a DL model. Specifically, a feed-forward neural network (FNN) has been used for binary and multi-class classification for four categories of attacks, which are reconnaissance, DoS, DDoS, and information theft attacks. The experimental results demonstrated that the proposed method outperforms Support Vector Machine (SVM) in terms of accuracy and training time.

At the same time, several IDS based on the distributed architecture of fog computing nodes and ML/DL models were proposed, for example, Kumar et al. [START_REF] Kumar | A distributed ensemble design based Intrusion Detection System using fog computing to protect the Internet of Things networks[END_REF] proposed a novel ensemble learning for intrusion detection using fog nodes. It is composed of two levels. In the first level, three supervised models are trained on the initial data. Then, in the second level, Random Forests (RF) have been used to combine the output of the first level and provide the final classification ("attack" or "benign").

However, as new types of attacks emerge every day, this brings new challenges that need to be tackled. In fact, labeling data is often difficult and time-consuming. Therefore, the researchers started to use a semi-supervised learning model where they can exploit both labeled and unlabeled data. In this context, Aamir et al. [START_REF] Aamir | Clustering based Semi-Supervised Machine Learning for DDoS attack classification[END_REF] proposed a semi-supervised model for DDoS attack detection. In the first step, principal component analysis (PCA) was used for data reduction and feature extraction. In the second step, a clustering algorithm was used to label the data based on their cluster. Finally, after label assignment, several supervised models are applied for training and attack detection. The experimental results show that the RF model is more accurate than K-Nearest Neighbors (KNN) and SVM. In the same context, Hara et al. [START_REF] Hara | Intrusion Detection System using Semi-Supervised Learning with Adversarial Autoencoder[END_REF] proposed an IDS system using a semi-supervised learning model. They used an adversarial auto-encoder (AAE) to extract the relevant features. The experimental results demonstrate that the proposed model can achieve high accuracy with a minimal number of labeled data as compared to deep neural networks. To evaluate the performance of their solution, the authors used NSL-KDD dataset, which is quite an old dataset and can miss modern network behaviors.

All the above works are based on the centralized training system where data are collected and processed in a single central server. This type of solution can be computationally expensive, time-consuming, prone to threats, and problems of information security and can cause leakage of confidential data. In contrast, through the use of FL, our model tries to detect attacks without breaching data privacy as well as reducing the communication and storage overhead as much as possible by sending only model parameters to a central server instead of the raw private data.

2) Federated Learning: FL was proposed to ensure data privacy during ML/DL model training. For instance, Nguyen et al. [START_REF] Nguyen | D ÏoT: A federated selflearning anomaly detection system for IoT[END_REF] introduced an FL system for detecting compromised IoT devices, called D ÏoT. Gated Recurrent Units (GRUs) have been used as a DL model. This is the first system that deployed FL for IDS. It consists of two components, which are security gateways and IoT security services. Security Gateways use the local data to train the local models and IoT security service aggregates the local models into a global model.

Taheri et al. [START_REF] Taheri | FED-IIoT: A robust federated malware detection architecture in industrial IoT[END_REF] introduced an FL approach to detect Android malware in IIoT, called Fed-IIoT. Specifically, Fed-IIoT uses two generative adversarial networks (GAN) models to generate adversarial data and inject them into the dataset. The server uses a GAN in order to detect a malicious model and to delete the poisoned data. Fed-IIoT has been tested on three IoT datasets and the results confirm the higher attack detection performance of their approach.

Mothukuri et al. [START_REF] Mothukuri | Federated Learningbased anomaly detection for IoT security attacks[END_REF] proposed an ensemble FL-based attack detection and classification in IoT networks. The authors combine RF and GRUs models to construct their ensemble. In other words, they used RF in order to combine the predictions from the GRUs model to further improve the classification performance of the FL approach. The experimental results demonstrate a minimized error rate in predicting attacks and a reduced number of false alarms in comparison to the centralized ML approaches.

Similarly, Attota et al. [START_REF] Attota | An ensemble multi-view Federated Learning Intrusion Detection for IoT[END_REF] proposed an ensemble FLbased intrusion detection, called MV-FLID. Specifically, the authors have developed three FNN models for three views (i.e., Biflow View, Packet View, and Uniflow View), and the outcomes of these models are sent to an ensembler model (RF), which combines the predictions of these models and classifies the instances. The results show that the FL approach can outperform the Non-FL one. Li et al. [START_REF] Li | DeepFed: Federated Deep Learning for Intrusion Detection in Industrial Cyber-Physical Systems[END_REF] proposed a federated DL scheme in cyber-physical systems (CPS), called DeepFed. A combination of Convolutional Neural Networks (CNN) and GRUs has been used for intrusion detection. The experimental results on a real industrial dataset demonstrate the high accuracy of DeepFed as compared to some other FL approaches. Popoola et al. [START_REF] Popoola | Federated deep Learning for Zero-Day botnet attack detection in IoT edge devices[END_REF] proposed FL model for zeroday attack detection in IoT edge devices. The experimental results demonstrate that FL outperforms the state-of-the-art methods in terms of data privacy, communication overhead, and memory storage space as well as the attack detection accuracy. In the same direction, Fan et al. [START_REF] Fan | IoTDefender: A Federated transfer Learning intrusion detection framework for 5G IoT[END_REF] proposed another model for IDS in 5G IoT network using federated transfer learning, called IoTDefender. They use FL to aggregate information without data sharing and transfer learning to ensure a personalized model for each IoT network. In addition, Ferrag et al. [START_REF] Ferrag | Federated Deep Learning for cyber security in the Internet of Things: Concepts, applications[END_REF] presented a comprehensive survey including experimental analysis of FL approaches for cybersecurity, in the IoT domain, using several datasets. Finally, Sarhan et al. [START_REF] Sarhan | A cyber threat intelligence sharing scheme based on Federated Learning for Network Intrusion Detection[END_REF] proposed a collaborative FL approach for IDS using a DL-based model. Their results demonstrated that FL achieved similar performance with that of the centralized approach in the binary and multi-classification scenarios.

The summary of the above works demonstrates the effectiveness of FL for IDS however as shown in Table I none of the existing works exploits the unlabeled data during the FL training process. The integration of both FL and semisupervised learning for intrusion detection is what currently lacks in the state of the art and that our model addresses.

B. Motivations and Key Contributions

The motivations behind the use of the federated semisupervised learning model are as follows. Although semisupervised learning has been used in the literature for IDS, it is often limited by the scale and hence can not generalize very well. Secondly, user data protection is recently accentuated by several international regulatory policies, which restrict data access and protect medical data privacy. For example, the General Data Protection Regulation in European Union (GDPR) 1 completely redefines the data management policy. Thirdly, labeling data can be costly and time-consuming, whereas using unlabeled data can also provide valuable information for the model and in turn boost the model classification performance. Therefore, the integration of FL and semi-supervised learning for the attack detection task is a promising direction. However, the large body of literature presented above has not explored the combination of FL and semi-supervised learning for attack detection.

1 https://gdpr-info.eu/issues/data-protection-officer/ To address these limitations, we propose an FL-based semisupervised model for ensuring privacy as well as for taking advantage of the unlabeled and labeled data in the IIoT environment. More specifically, we train a model using only a small amount of labeled data combined with more abundant unlabeled data. Additionally, our proposal can enhance the model performance through the collaboration of data sources which are the distributed IIoT devices. The main contributions of our research are as follows:

• We propose a federated semi-supervised approach for intrusion detection, by integrating FL and semi-supervised learning in one model which is used by the distributed IIoT devices. • We use the AE as the unsupervised model at the IIoT devices in order to reduce the size of local models and to decrease the communication overhead. • We use the cloud server as follows. It first generates the global AE using FL. Next, the cloud server adds additional layers to the global encoder block and trains the resulting neural network further using supervised learning with publicly available labeled data. • We further deploy a joint-announcement protocol [START_REF] Liu | Privacy-preserving traffic flow prediction: A Federated Learning approach[END_REF], which uses clients' random selection in order to decrease the communication overhead. • We perform an extensive simulation using two industrial datasets, which are benchmark datasets, as well as we compare our approach against some state-of-the-art approaches.

C. Paper Structure

The rest of this paper is organized as follows. Section II-A, firstly presents essential background, then Section II-B introduces our FL-based semi-supervised algorithm for intrusion and attack classification. Experimental settings and results as well as the dataset are presented in Section III. Discussion and analysis of the results and the future directions are provided in Section IV. Finally, the conclusion is given in Section V.

II. PROPOSED SOLUTION

This section presents the main components of our system as well as the architecture of the FL-based semi-supervised learning.

A. Components 1) AutoEncoder: AE is an unsupervised neural network model. It consists of two blocks, (i) encoder and (ii) decoder. The encoder takes input data and maps it to a hidden representation (latent representation), called code. Then, the decoder uses the hidden representation to reconstruct the input. The main purpose of the AE is dimensionality reduction of the input data by extracting important features as the latent variable vector. For more details about AE, please refer to [START_REF] Guo | Deep Learning for visual understanding: A review[END_REF].

AE structure with single hidden layer is formulated as:

encoder : Z = f (W 1 X + b 1 ) (1)
decoder : where X = (x 1 , x 2 , . . . , x n ) is the input vector, and Z = (z 1 , z 2 , . . . , z m ) is the vector known as latent space which in turn is extracted from the input X, X ′ = (x ′ 1 , x ′ 2 , . . . , x ′ n ) is the output reconstruction of the input X, where n is the dimension of the input vector and m is the number of code units. W 1 and b 1 are the weight matrix and bias between the input layer and the second layer (i.e., code). W 2 and b 2 are the weight matrix and bias between the second and the output layer; f (.) is the activation function.

X ′ = f (W 2 Z + b 2 ) (2)
The difference between X and X ′ is usually called the reconstruction error RE, which is represented in the form of a cost function that the model tries to reduce during the training process. The cost function of the AE is computed using Equation 3, where the parameter set is denoted by

θ = {W 1 , b 1 , W 2 , b 2 }. J(θ) = n i=1 RE(x i , x ′ i ) (3)
2) Federated Learning: FL attempts to answer the following main question: Can we train the model without the need to transfer data over to a central location? the FL concept, the data is maintained where it is generated and no raw data gets exchanged. In other words, FL is a distributed machine learning framework where the data entities/clients collaborate to jointly learn a global model (e.g. intrusion detection) without sacrificing the privacy of the end-users as much as possible. Given K clients {C 1 , . . . , C K } and their respective local data {D 1 , . . . , D K }, the FL framework builds a model through the collaboration of several clients n ≤ K without exposing their data to others. For this task, the Federated Averaging (FedAvg) algorithm [START_REF] Mcmahan | Communication-efficient learning of deep networks from decentralized data[END_REF] is the most simple and frequently used one; hence, it will be used with our model. By making devices exchange model parameters rather than their raw local private data, the FL process limits the communication and storage overhead.

B. Methodology

Our objective is to train a semi-supervised model using only a small amount of labeled data while preserving data privacy. Figure 1 and Algorithm 1 present the architecture and the main idea of our model. Table II presents the notations used in Algorithm 1.

Our model consists of two parts: the clients' side and the server-side. The clients perform the model pre-training using their unlabeled data and the server fine-tunes the global parameters using its limited labeled data (Figure 1 (a)). In step 1 in Figure 1 (b), the server selects a random subset of clients, n ≤ K, that will participate in the learning process and send them the initial AE model. Then, on the clients' side (IIoT devices), the AE model is trained for a selected number of epochs using the clients' unlabeled data with the objective of reducing the reconstruction error (step 2). Here, the total unlabeled data are randomly distributed across the clients. In an IIoT context, clients are often far from human reach, thus accessing them to label their data is a difficult and impractical task. Thus, to be more realistic, in this work we consider that the client data are fully unlabeled. Then, the clients send their local models to the cloud server for global aggregation using the FedAvg algorithm (step 3). On the server-side, the AE is aggregated (step 4), then the decoder is removed and a Fully Connected Network (FCN) layer is attached to the encoder layers in order to fine-tune the model parameters for the supervised learning using a limited labeled data located on the server (step 5 and 6). Specifically, the server uses its limited labeled data for supervised learning, and thus unlike the classical FL, in our case, the server is not only used for the model aggregation but also for supervised learning. Finally, the server sends back the global AE to the clients for a further update as well as the trained supervised model for inference (classification of the attacks) on the network data of the devices (step 7). It is important to note that (i) no raw data is exchanged between the clients and the central server, and (ii) the supervised model is trained through domain-specific public datasets or laboratory data located on the server without privacy concerns. for e c = 1 to E c do 7:

Algorithm 1: Learning procedure. 1: Input: Public labeled Dataset D l = {x i , y i } (i = 1, 2, . . . , n; x i ∈ X; y i ∈ Y ); Private unlabeled dataset D u k = {x i } (k = 1, 2, . . . , K; x i ∈ X k ), R, E c , E s , K, r k / * ---
Update local AE parameters θ j using D u j 8:

end for

9:

Send updated θ j to the Server 10:

end for / * ---Server side ---* / for e s = 1 to E s do 15:

Train H using D l 16:

end for 17:

Send models θ and H to the clients 18: end for In our federated semi-supervised learning model, the computational complexity is related to the computations at each IIoT device or client (which performs the unsupervised learning) and at the FL server (supervised learning), plus their communications. The global aggregation results in insignificant computational costs. Thus, here we focus on analyzing the computational complexity related to learning at each device and the FL server. We assume that L AE and L H are the number of layers of the AE model and supervised model,

) × E c × L AE × N 2 ).
Similarly, the complexity of the supervised learning on the FL server is O(size

(D l ) × E s × L H × N 2 ).

III. EXPERIMENTS AND PERFORMANCE EVALUATIONS

In this section, we evaluate the performance of our model through extensive experiments and discuss the results.

A. Data Description

To validate the effectiveness of our proposition, we used a gas pipeline SCADA system dataset, which is a benchmark dataset for security research [START_REF] Morris | Industrial control system traffic data sets for intrusion detection research[END_REF]. It was released by the Mississippi State University in 2014. This dataset consists of 26 features and 1 label, the label contains eight possible values, benign and seven different types of attacks. The possible values for the label are presented in Table III. In addition, as this dataset consists of different features with values in different scales, we have standardized our data.

B. Experimental Setup

In this study, we split the dataset into two subsets: train (80%) and test (20%). We use Python as a programming language and Scikit-learn for the conventional models and PyTorch for DL models. All experiments are run on a fourcore Intel® Core™ i7-6700 CPU@3.40GHz processor, and 32GB of RAM. Table IV summarizes the model parameters and their selected settings in our simulations. The simulation of a partially-labeled dataset from this fully labeled data has been done by randomly selecting rows from the training set and removing their labels. It is important to note that during this experiment, we fixed the amount of labeled data and vary only the amount of unlabeled data. As the dataset is imbalanced (there are much more examples of some types of attacks than others), we use the F1-score (i.e., the harmonic mean of precision and recall) as an evaluation metric, together with the accurracy). In addition, in our experiments, we compute the average of the evaluation metric from five runs. 1) Impact of the number of selected clients: Here, we explore the impact of different numbers of clients on the classification performance of our model, while holding K fixed (K= 100). Before training, we distribute the unlabeled data into equally one hundred parts, meaning that each client gets nb rows(D u )/K observations randomly. Since in real-time situations, the probability of client failures is significant [START_REF] Agrawal | Federated Learning for Intrusion Detection System: Concepts, challenges and future directions[END_REF] we use the Joint-Announcement Protocol (JAP) to avoid the problem of client failure and communication overhead. Given the ratio of clients (r k ), JAP selects randomly the clients that participate in the i th training round [START_REF] Liu | Privacy-preserving traffic flow prediction: A Federated Learning approach[END_REF]. As shown in Table V we evaluate the performance of our classifier under different number of client n, where n ∈ {10, 30, 100}.

With n = 100 (i.e., the total number of clients) the classification performance of our model degrades little. Our model gets the best results when n = 10. This may be attributed to the fact that a large number of clients can increase the diversity of the AE model across the clients and degrade the classification performance. Further, we compare the communication overhead with and without JAP. Here we set the number of clients to n = 10 using JAP and n = 100 without JAP. As present in figure 2, we can notice that using JAP can reduce the communication overhead. Specifically, in every communication round, JAP with n = 10 reduces the overhead by 90%. This demonstrates the critical impact of the clients' number on the FL system. Moreover, as seen from Figure 3 the local epoch does not only impact the classification performance of the model but also the communication overhead. Specifically, here we show the communication overhead of our model under different frequency updates. It can be seen that increasing the number of local epochs in our model can decrease the communication overhead of the FL model. For example, when E c = 20, the clients send only 1 time their model to the sever. However, with E c = 5, and E c = 1, they send their model to the server 4 and 20 times, respectively. Therefore, since the sensors in IIoT systems generate large volumes of data, increasing the local epochs improves the attack classification task at the same time it decreases the communication overhead; hence, improving the network overall performance.

3) Impact of communication rounds: Here, we evaluate the classification performance (accuracy and F1-score) of our model under the different numbers of communication rounds R. As presented in Figure 4, the performance of the model increases with more interaction/round between the server and the clients. Also, it can be seen that the model converges quickly and its accuracy starts to be stable after 2 rounds. This may be attributed to the fact that the AE trained 1551-3203 (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. on the clients provide some pre-trained layers, which capture relevant features. However, we can see that increasing the number of rounds does not always lead to better performance. This is because the model can overfit with large R.

Fig. 4: Classification performance under different communication rounds 4) Impact of the unlabeled data available on the clients: To evaluate the impact of the quantity of unlabeled data on the model performance, we further fix the amount of labeled data and change only the quantity of unlabeled data distributed to the clients. To do so, we trained our model using different ratios of unlabeled samples R u :

R u = nb rows(D u ) nb rows(D l ) (4)
Where nb rows(D l ) (resp. nb rows(D u )) represents the amount labeled (resp. unlabeled) data.

It can be seen from Figure 5 that increasing the size of unlabeled data improves the performance of the whole model. These results are attributed to the fact that accessing more (diverse) unlabeled data provides informative characteristics to find a more discriminatory latent space and, in turn, our model benefits from these data and boosts its performance to classify unseen observations. Consequently, our model can fit very well the Industry 4.0 context where unlabeled data is often abundant and easily available.

D. Performance against other models

In order to further validate the effectiveness of our model, we compare its performance against the Non-FL semisupervised model, different supervised models as well as some state-of-the-art models.

T F L = R × (E c + E s ) (5)
T nF L = E c + E s (6)
In comparison with the non-FL model, our model has the best results in terms of accuracy and F1-score. This is because the communication rounds help our model to improve the AE model and in turn extract more relevant features that have been used during the supervised model training. Further, since the communication overhead is a critical factor in the IIoT environment, we compare our model against the Non-FL model in a such factor. Specifically, the communication overhead of both models are calculated as follows (C F L -resp. C nF L -is the communication overhead of our model -resp. the non-FL version of our model): where size(D u ) is the size of the private unlabeled data exchanged between the server and the clients in the Non-FL training, size(AE) is the size of the model located on the client and exchanged between the client and the server in each communication round, n is the number of the selected clients, and R is the total communication rounds, size(H) is the size of the supervised model distributed by the server to client for traffic classification. It can be seen from Figure 6 that centralized learning is always expensive and thanks to the FL, the communication overhead is reduced by 50%. This is because the clients need to upload their unlabeled data to constitute a central database (Equation 8) while with our model, the clients and cloud server exchange only the models' parameters. In addition, since the AE compresses the client local data this can reduce the size of the parameters communicated with the FL server. 2) Comparison with supervised models: In order to evaluate the effectiveness of the proposed model, we compare its performance to several supervised ML methods. It can be seen from Table VIII that our model has a good performance in detecting benign network traffic and the different attacks. This is thanks to the use of deep architecture, which covers benign and attacks patterns. Specifically, the F1-score of benign of our model is 6%, 20%, 21%, 21%, 27% better than AdaBoost, LightGBM, EXTree, RF, and DT, respectively. This demonstrates that our model is more practical in the Industry 4.0 context as it will trigger fewer false alarms. On the other hand, it achieves a greater F1-score for the classification of the attack, except the DoS and especially with NMRI attack and this is because the observations amount of this attack is relatively low in the labeled set. From these experiments, we can conclude that our model is better at covering all the attacks at once. Also, using unlabeled data, which is often abundant and easily available improves the classification performance.

C F L = n × R × (2 × size(AE) + size(H)) (7)
C nF L = size(D u ) (8)
3) Performance against state-of-the-art models: To validate the effectiveness of the proposed model, we also compare its performance against some state-of-the-art schemes using the gas pipeline dataset. The experimental results of these schemes are presented in Table IX and they include a simple model with fully labeled data [START_REF] Anton | Anomalybased Intrusion Detection in industrial data with SVM and Random Forests[END_REF], DL-based ensemble model using fully labeled data [START_REF] Huda | Securing the operations in SCADA-IoT platform based industrial control system using ensemble of Deep Belief Networks[END_REF], a semi-supervised model without FL [START_REF] Chang | Anomaly Detection for industrial control systems using k-means and Convolutional AutoEncoder[END_REF], and a supervised ensemble FL scheme [START_REF] Li | DeepFed: Federated Deep Learning for Intrusion Detection in Industrial Cyber-Physical Systems[END_REF]. Note that we have selected these works because of their variety. Anton et al. [START_REF] Anton | Anomalybased Intrusion Detection in industrial data with SVM and Random Forests[END_REF] used SVM for intrusion detection. Huda et al. [START_REF] Huda | Securing the operations in SCADA-IoT platform based industrial control system using ensemble of Deep Belief Networks[END_REF] proposed an ensemble Deep Belief Network (DBN) model for attack classification. In particular, different structures of DBN are combined to construct an ensemble of DBNs and the final classification is decided based on a majority voting scheme. Also, Chang et al. [START_REF] Chang | Anomaly Detection for industrial control systems using k-means and Convolutional AutoEncoder[END_REF] proposed an ensemble semisupervised model using the k-means and convolutional autoencoder (CAE) methods. Using this ensemble, the test data is predicted as normal only if the predicted outputs of k-means and CAE methods are normal. Recently, Li et al. [START_REF] Li | DeepFed: Federated Deep Learning for Intrusion Detection in Industrial Cyber-Physical Systems[END_REF] proposed a novel FL model, called DeepFed. DeepFed is an ensemble model that trains CNN and GRU in a federated way to detect the attacks. We can observe from Table IX that the intrusion detection model based on FL including our model and DeepFed scheme [START_REF] Li | DeepFed: Federated Deep Learning for Intrusion Detection in Industrial Cyber-Physical Systems[END_REF] incur the best results. This is due to the communication round that improves the performance of traffic classification to some extent. More specifically, although our model uses a few amounts (only 25%) of labeled data during the training task, it still achieves a competitive accuracy as compared with DeepFed. Moreover, with our approach, the clients only train the AE model, which in turn is a less complex model as compared to the CNN-GRU models used with the DeepFed scheme. Note that a fair comparison for our model will be to compare it with only semi-supervised schemes. Nonetheless, we include some fully supervised schemes in our comparison for reference. 4) Experiments on the second dataset: We have also tested our model against its non-FL version on the second dataset, which is also an open dataset and has been released by Mississippi State University's lab in 2014 [START_REF] Morris | Industrial control system traffic data sets for intrusion detection research[END_REF]. The traffic in this dataset corresponds to the water storage tank control system and consists of 23 features. The label contains eight possible values, benign and seven different types of attacks, same as the gas pipeline dataset.

From the simulation results presented in Table X and Figure 7, we can see that our model performs better than the non-FL model in terms of accuracy, F1-score as well as communication overhead. Specifically, the improvement in terms of communication overhead becomes more significant with this dataset (reduction by almost 75%), as it contains more traffic than the gas pipeline system and hence sending row data to the central entity becomes more expansive. This shows that our model is suitable for a real IIoT scenario because industrial machines and robots can generate tremendous amounts of traffic. IV. DISCUSSION In this study, a semi-supervised FL model is proposed for attack and intrusion detection for the Industry 4.0 environment that makes use of both unlabeled and labeled data. We have analyzed the impact of the different parameters on the performance of the proposed model. First, the evaluation demonstrates that this model performs well even with a limited amount of labels. It automatically provides feature extraction without human intervention and avoids time-wasting for labeling data as maximum as possible. Second, communication overhead, storage requirements have been reduced thanks to the use of FL. Also, we have demonstrated that the local epochs and the clients' selection process play a critical role in communication overhead improvement. Third, using joint announcement protocol addresses the problem of communication overhead and the failure of some clients as well as alleviates the out-of-sync issue. Therefore, the communication overhead can be reduced by reducing the number of the selected clients n, reducing the frequency model update using a large number of local client epochs E c . In addition, taking the advantage of FL our model solves the dilemma of data sharing.

Last but not least, to show the features of our model, we have compared our model against its Non-FL setting, some state-of-the-art models including, simple, ensemble, semisupervised, and FL models. Also, we applied our model on a second dataset and evaluated its performance in terms of accuracy, F1-score, and communication overhead. The presented results show that our model achieves higher classification with less communication overhead as well as without privacy concerns.

Limitations of proposed scheme Although our model uses the joint-announcement protocol, the random client selection can increase the training time and communication cost due to the clients who become stragglers. To handle these issues, more intelligent client selection algorithms are needed. One idea could be to use reinforcement learning, which can be a promising solution, to learn client selection based on the learning performance. Also, as our model still need labels in future research, we can investigate the fully unsupervised FL model by looking at the AE reconstruction error. Finally, to ensure a secure model transmission, we can use some mechanisms such as differential privacy [START_REF] Lian | COFEL: Communicationefficient and optimized Federated Learning with Local Differential Privacy[END_REF] or blockchain [START_REF] Wang | Blockchain-based reliable and efficient certificateless signature for IIoT devices[END_REF].

V. CONCLUSION

In this paper, a federated semi-supervised learning model has been proposed. This model uses a limited amount of labeled data and a huge amount of unlabeled data without privacy concerns. Also, unlike the classical FL model, with our model, the server is not only used for the model aggregation task, but also for supervised learning. The proposed model has been evaluated in terms of its ability to identify the network traffic and different attacks. Moreover, we have analyzed the performance of our model while varying different factors. Using two real datasets, the experimental results demonstrate that the support of unlabeled data for the training process can enhance the performance of the learned model as well as decrease the communication overhead. Also, our numerical simulations showed that the proposed federated semisupervised model with only 25% labeled data can achieve competitive results, compared to the state-of-the-art schemes.

1 2 4

 14 Fig. 1: The network architecture and communication process of our Federated semi-supervised proposal for IIoT.

  First, the central server specifies the hyperparameters of the global model and the training process. Then, it disseminates the initial global model to several clients. Based on this model, each client updates the global model locally for a selected number of epochs using its own data. Once the local training is finished, these clients send their own model updates to the central server that aggregates these updates and computes the global model without exchanging data. These steps are repeated until the global model achieves satisfactory accuracy. The aim of the global aggregation is to aggregate the local models of different clients.

2 : 5 :

 25 Server side ---* / Send initial global model θ = θ 0 to clients 3: i = 1 to R do 4: n= r k * K for j = 1 to n do in parallel / * ---Client side ---* / 6:

C.

  Performance under different factors In this section, we study the performance (in terms of accuracy, F1-score, and communication overhead) of the proposed model under different factors, which are: fraction of the selected clients r k , local client epochs E c , communication rounds R, and amount of unlabeled data.

Fig. 2 :

 2 Fig. 2: Overhead of the proposed model without/with JAP. 2) Impact of the local epochs: In this subsection, we study the performance of our model in terms of F1-score and communication overhead using different values for the clients' local epochs E c , used during the AE training (client level). As shown in Table VI, increasing the number of E c = 20 improves the performance of our model. This may be attributed to the fact that increasing the local epoch of the AE helps to find more relevant features which, in turn, boosts the performance of the supervised model. However, increasing this parameter also increases the training time.

7 Fig. 3 :

 73 Fig. 3: Comparison in terms of communication overhead.

Fig. 5 :

 5 Fig. 5: F1-score with various R u .

1 )

 1 Comparison with Non-FL model: Evaluation results of our proposed model in comparison to the non-FL model are presented in Table VII. With the non-FL model, the clients/devices need to send their data to a central server in order to train the AE and the supervised models. The below formulas give the training process of our model and the non-FL model where R = 18 (total communication rounds), E c = 20 (client AE epochs) and E s = 100 (server supervised learning epochs). T F L (resp. T nF L ) is the training process of our FL model (resp. the equivalent non-FL version of our model).

Fig. 6 :

 6 Fig. 6: Comparison in terms of communication overhead.

Fig. 7 :

 7 Fig. 7: Comparison in terms of communication overhead.

TABLE I :

 I Summary of existing works.

			Characteristics
	Schemes	Federated	Semi-	Communication
		learning	supervised	overhead
			learning	minimisation
	[9] Ensemble FL-based attacks	✓
	detection (GRUs, RF)
	[10] MV-FLID: ensemble	✓
	FL-based intrusion detection
	(FNN, RF)
	[8] Semi-supervised model-		✓
	based intrusion detection
	(AAE)
	[11] Ensemble FL based at-	✓
	tacks detection (CNN, GRUs)
	[12] DL-based intrusion detec-	✓
	tion (GRUs)
	[13] Federated transfer learn-	✓
	ing based intrusion detection
	(CNN)
	[7] Semi-supervised DDoS at-		✓
	tack detection (PCA, K-means,
	RF)
	[14] FL-based attacks detec-	✓		✓
	tion (DNN)
	[15] FL-based attacks detec-	✓
	tion (DNN, CNN, RNN)
	[16] FL-based attacks detec-	✓
	tion (DNN)
	[17] FL-based Android mal-	✓
	ware detection (GAN)
	Our model (Federated semi-	✓	✓	✓
	supervised attacks detection
	(AE, FCN))

11 :

 11 Aggregate {θ} j=1...n with FedAvg into θ

	12:	Extract encoder
	13:

Concatenate encoder and FCNLayer into H 14:

TABLE II :

 II List of notations used in our model.

	Notation	Meaning
	K	Total number of clients
	r k	fraction of the clients randomly selected from K
	n	The number of clients selected at each round
	Ec	Local clients epochs
	Es	Local server epochs
	R	Total number of rounds
	H	Supervised model
	θ 0	Initial AE parameters
	respectively. For simplicity, we assume that each layer has
	N neurons. The computational complexity comes from back-
	propagation algorithm. To calculate new weights for each
	layer, we require matrix multiplications for applying gradients,
	which is the most complex part of the process. Assuming
	simplest matrix multiplication algorithm, the complexity of
	multiplying two matrices (with N rows and size(D u k ) columns into size(D u k ) columns and N rows) is O(N ×size(D u k )×N ).
	Considering the training epochs and the number of layers, the
	computational complexity of the AE at each client becomes
	O(size(D u k

TABLE III :

 III Dataset description.

	Label	Description	# Total observations
	Benign Normal traffic	61,156
	NMRI Naive malicious response injection	2,763
	CMRI Complex malicious response injection	15,466
	MSCI	Malicious state command injection	782
	MPCI	Malicious parameter command injection	7,637
	MFCI	Malicious function command injection	573
	DoS	Denial of service	1,837
	Rec	Reconnaissance	6,805

TABLE IV :

 IV Implementation parameters.

	Deep Learning
	Deep learning tool	PyTorch
	DL algorithms	Autoencoder (AE)
		Fully Connected Layer (FCN)
	AE hidden layers	8
	encoder neurons	[Input, 20, 15, 10]
	FCN layers	1
	Activation functions	ReLu (AE), Softmax (FCN)
	Optimizer	Adam
	Learning rate	0.001
	Batch size	64
	Dropout	0.01
	Federated Learning
	FL server	1
	Nb clients	100
	Clients used in federated updates	10%
	AE epochs (client)	20
	FCN epochs (server)	100
	Communication round	18

TABLE V :

 V F1-score for various numbers of clients.

	C k	10	30	100
	F1-score	88.14% ±0.16	88.07% ± 0.07	87.91% ± 0.24

TABLE VI :

 VI F1-score for various numbers of epochs on the client-side.

	Ec	20	5	1
	F1-score	88.14% ± 0.16	88.01% ± 0.18	87% ± 0.17

TABLE VII :

 VII Performance of the proposed model against the Non-FL model.

	Metric	Non-FL model	Proposed model
	Accuracy	95.4% ±0.34	95.84% ±0.02
	F1-score	86.7% ± 0.44	88.14% ±0.16

TABLE VIII :

 VIII F1-score comparison of our model vs. supervised models for the identification of normal vs. attack traffic.

	Model	Benign	NMRI	CMRI	MSCI	MPCI	MFCI	DoS	Rec
	DT	0.70	0.32	0.94	0.94	0.97	0.98	0.98	1
	RF	0.76	0.38	0.96	0.95	0.97	0.98	0.98	1
	EXTree	0.76	0.49	0.92	0.95	0.95	0.97	0.87	1
	AdaBoost	0.91	0.89	0.92	0.93	0.95	0.97	0.96	1
	LightGBM	0.77	0.50	0.96	0.96	0.97	0.99	0.95	1
	Proposed model	0.97	0.20	0.96	0.97	0.98	0.99	0.97	1

TABLE IX :

 IX Overall performance analysis of the proposed model with existing schemes.

	Type	Ref.	FL	Accuracy	Precision	Recall
		[22]		92.5	78.2	93.6
	Supervised	[23]		95.6	85.36	85.53
		[11]	✓	99.20	98.85	97.45
		[24]		95.53	95.43	83.52
	Semi-supervised	Our model	✓	95.84	97.89	87.15

  1551-3203 (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TII.2022.3156642, IEEE Transactions on Industrial Informatics

TABLE X :

 X Performance of the proposed model against the Non-FL model.

	Metric	Non-FL model	Proposed model
	Accuracy	90.41 ±0.24	90.73 ±0.08
	F1-score	85.99 ±0.34	86.41 ±0.09

Authorized licensed use limited to: University of Nantes. Downloaded on March 21,2022 at 08:28:59 UTC from IEEE Xplore. Restrictions apply.

1551-3203 (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TII.2022.3156642, IEEE Transactions on Industrial Informatics

Authorized licensed use limited to: University of Nantes. Downloaded on March 21,2022 at 08:28:59 UTC from IEEE Xplore. Restrictions apply.

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