Margot Leclair
email: margot.leclair@mines-paristech.fr

Cédric Dalmasso

Trade-offs with Economy: How Creative Workers engage with Market-based Roles

Les industries créatives ont la particularité de conjuguer, au sein d'un même secteur d'activité, les enjeux du monde de la création et ceux du monde des affaires. La littérature en sciences de gestion a depuis longtemps mis en lumière les tensions pouvant exister entre rationalité créative et rationalité économique et la manière dont les acteurs créatifs s'efforcent

2 de gérer les pressions économiques pour protéger leurs espaces de création. Dans cette communication nous abordons cette thématique au travers d'une méthodologie originale, une ethnographie de l'activité quotidienne, en prenant comme objet d'étude un acteur singulier ayant la particularité de conjuguer la direction artistique et la gérance de son entreprise. Nous mettons en évidence qu'il existe une asymétrie dans la manière dont cet acteur s'engage dans les rôles marchands d'acheteur et de vendeur. S'il s'investit dans le rôle d'acheteur et que ce rôle est perçu comme une ressource au processus créatif, il se distancie en revanche du rôle de vendeur qu'il appréhende comme une contrainte. L'analyse de cette attitude différenciée nous permet de qualifier l'acteur créatif d'entrepreneur non conventionnel et de mettre en lumière une posture particulière quant à la place donnée à la recherche de la performance économique.

Il ne s'agit pas ici de chercher à maximiser le profit et à accumuler les ressources dans le but de faire croître l'activité, mais au contraire d'assurer la simple reproduction des ressources afin de rendre possible la continuité de l'activité tout en se protégeant d'un accroissement des contraintes économiques. Il apparaît alors qu'il n'existe pas une opposition indépassable entre rationalité économique et rationalité créative mais que l'enjeu consiste, pour les acteurs et plus globalement pour les entreprises créatives, dans l'articulation astucieuse des deux rationalités afin d'assurer tout à la fois la soutenabilité économique de l'activité et la pérennité de l'activité créative.

Mots-clés : Industries créatives, pratiques créatives, pratiques économiques, rôles marchands

Trade-offs with Economy: How Creative Workers engage with Market-based Roles

INTRODUCTION

This paper departs from the research literature that underlines the on-going debate arising within creative companies, between creative rationales on the one hand and economic rationales on the other hand [START_REF] De Fillippi | Introduction to paradoxes of creativity: managerial and organizational challenges in the cultural economy[END_REF]Hesmondhalgh, 2013;[START_REF] Lampel | Balancing Act: Learning form Organizational Practices in Cultural Industries[END_REF][START_REF] Linstead | Turning Rebellion into Money: The Clash, Creativity and Resistance to Commodification[END_REF]. Creative industries represent an iconic field for investigating such paradoxes and tensions creative actors have to deal with [START_REF] De Fillippi | Introduction to paradoxes of creativity: managerial and organizational challenges in the cultural economy[END_REF]. Known as "particular for the need to appease art and business" (Jones et al., 2005), those industries are organized around the production and circulation of "non-material goods directed at a public of consumers for whom they generally serve an aesthetic or expressive, rather than clearly utilitarian function" (Hirsch, 1972: 641). The conflicts and tensions between the imperative of a relentless creation of new genres, formats and products on the one hand, and economic viability on the other hand occur within the creative economy in a most striking fashion [START_REF] De Fillippi | Introduction to paradoxes of creativity: managerial and organizational challenges in the cultural economy[END_REF]. Most creative actors have to operate both within and through economic rules and boundaries to effect creative propositions.

In this context, scholars have noted that so-called creative individuals of those industries tend to resist or disregard economic preoccupation [START_REF] Caves | Creative Industries: Contracts between Art and Commerce[END_REF]Jones et al., 2016;[START_REF] Linstead | Turning Rebellion into Money: The Clash, Creativity and Resistance to Commodification[END_REF]). Yet research that explains interactions and ways of working within creative contexts as consequences of these conflicting tensions is still scarce (Austin, Hjorth & Hessel, 2017). In the paper we describe our effort to address some of the shortcomings of existing theory by taking up the following research question: How do actors involved into the creative process deal with market-based activities? By following a creative entrepreneur in her daily routine, we explore the unlikely conversation between profit maximization and creative forces 4 (Austin, Hjorth & Hessel, 2017;[START_REF] Eikhof | For art's sake ! Artistic and economic logics in creative production[END_REF][START_REF] Lampel | Balancing Act: Learning form Organizational Practices in Cultural Industries[END_REF], how those supposedly opposing forces play out on a day-to-day basis.

We structure our paper as follows. First, we describe how current research discusses the organization of conflict between economic and creative influences for creative workers.

We then present the research setting and describe the everyday doings of creative workers at Maria Maliusi, the fashion house that provided the occasion for our study. We follow that with a description of our ethnographic research approach, and then present our key findings.

We arrive at findings that describe an asymmetry in market-based roles, with creative workers engaging with the buyer role while dis-engaging from the seller role. We highlight how market-based activities are lived as resourceful in the buyer role but constraining in the seller role. We continue by underlining how creative workers deal with economy through trade-offs and a peculiar way of performing the market, behaving as some sort of unconventional entrepreneur, not trying to reach absolute performance. Rather than seeking accumulation (of the assets), with the growth of the company, creative workers we followed mostly seek reproduction (of the assets), main objective being to simply allow the continuity of creation without necessarily growing up. Nuanced implications about creative work within economic interests then arise; we return to the literature to describe how our findings and theorizing open the discussion on the economic actions and attitudes of creative agents at work.

CONCEPTUAL FRAMEWORK

The creative industries comprise "industries supplying goods and services that we mostly associate with cultural, artistic, or simply entertainment value" (Caves, 2000: 1). Such creative products and services are simultaneously artistic creations and economic outcomes (Caves, 2002), and this twofold nature (i.e. business versus art) challenges organizational practices and work dynamics [START_REF] Eikhof | For art's sake ! Artistic and economic logics in creative production[END_REF].

CREATIVE WORK AND ECONOMIC IMPERATIVES

When it comes to the practical business of creating and selling creative goods, firms must proceed with both polarities in mind. Undeniably, a balance has to be found as too much economic concern can kill creativity. If firms only pursue the goal of mass entertainment they could lose sight of artistic values. And vice versa. If artistic values dominate, economic survival dictates that market realities cannot be ignored indefinitely. 'Uncreative' activities surrounding creativity are also vital to the creative process [START_REF] Bilton | Book Review-Managing creativity: exploring the paradox[END_REF][START_REF] Bilton | Book Review-Managing creativity: exploring the paradox[END_REF].

Still, several scholars mention the gap between managerial tools and creative values (see for example Guillet de [START_REF] Guillet De Monthoux | The art firm[END_REF]Howkins, 2001;[START_REF] Menger | Artistic Labor Markets and Careers[END_REF]. The creative and the business sub-systems have different interests and priorities. If we are to believe the traditional textbooks of management, business world would be dominated by rationality, planning and control [START_REF] Koivunen | Creativity and the contemporary economy[END_REF]. On a daily basis, creative workers experience and manage identity tensions between "artistry" and a "more business-like identity that supports firm performance" (Gotsi et. al, 2010). Following this line of thought, the creative actor becomes the opposite of his/her business counterpart, whether it pertains to the identity, discourses or logics of action [START_REF] Flew | The Creative Industries: Culture & Policy[END_REF]. [START_REF] Koivunen | On Creativity, Art & Economy[END_REF] reveals the contradiction between the two figures that are 'the genius' and 'the manager', the 'bohemian-artist' versus the 'conservative organisation-man'. The underlying assumptions of the two areas that are creativity and economy would be contradictory and clash regularly [START_REF] Koivunen | On Creativity, Art & Economy[END_REF]. [START_REF] Flew | The Creative Industries: Culture & Policy[END_REF] talks about the unpredictable nature of creativity in terms of financial gain, and how organizations position themselves with the sole aim to manage, without actually resolving, the uncertainty and risks issues. He highlights the dispute between creative ideals and risks that markets and their flexibility involve, especially with the 'winner takes all' system of creative industries. This is especially so in the contemporary 'dispositif of creativity' [START_REF] Reckwitz | Creativity as Dispositif[END_REF]:

creative industries have to face tensions resulting from this simultaneous articulation of trade constraints with creativity, especially in the contemporary Reckwitz argues that nowadays creativity is rebranded as the engine of post-industrial economies. Creativity and so-called creatives have become desirable, socially and economically. Yet as Ross indicates, this "new creative class" [START_REF] Florida | The rise of the creative class: and how it's transforming work, leisure, community and everyday life[END_REF], despite its "cool hip-hop heaven", has to come to terms with the price to pay for their no-collar jobs (Ross, 2004). The hype of the creative industries is indeed embedded in a certain neoliberal political and economic paradigm, which entails more and more economisation of creative practices. Such economisation of practices reflects on the notion of rationalization, defined by Tschang (2007: 989) as "the predominant focus on business interests or productivity-oriented production processes, usually at the expense of creativity". The evolution of creative industries, Tschang argues, tends to be driven by a deeper, continuing tension between forces for creativity and forces for rational (e.g.: business) interests. And this rationalization context has notably reduced the individual's creative scope.

As a consequence of those far-off professional ethos, creative people are said to have a negative vision of management [START_REF] Paalumäki | Leadership, Creative Work and Experience: The Case of Design[END_REF], or even reject the association of their art to the market, at the risk of tarnishing its purity [START_REF] Heikkilä | Se on vanhanaikainen se asetelma, ettei talous kuuluisi taiteeseen[END_REF].

Becker (1982) depicts creative actors as "mavericks", unique extreme cases who violate established conventions in a given creative industry. Creative agents often rebel against efforts to direct them towards managerial objectives [START_REF] Florida | The rise of the creative class: and how it's transforming work, leisure, community and everyday life[END_REF][START_REF] Sutton | The weird rules of creativity[END_REF].

On that topic also, [START_REF] Linstead | Turning Rebellion into Money: The Clash, Creativity and Resistance to Commodification[END_REF] worked on the understanding of the historical tension between creative employees and commercial duties, and the continuity of contradiction. He mentions the ambivalence of the relation between commercialization and creativity, with organizations that value creativity but at the same time devise ways to kill it.

Consequently, this intrinsic contradiction of creative industries is reflected into the creative process, and translated into a dilemma for creative actors who have to deal with both values in their day-to-day practices. Linstead talks about a "dilemma of commodification" for creative artists who work in a commercial system. [START_REF] Eikhof | For art's sake ! Artistic and economic logics in creative production[END_REF] underline how the economic practices inevitably dominate the creative practices: the quantifiable business rationale is stronger and more robust than the artistic rationale, which is more vague and nonmeasurable. In a similar way, other creative industries studies on non-profit professional theatres [START_REF] Voss | Linking Organizational Values to Relationships with External Constituents: A Study of Nonprofit Professional Theatres[END_REF] or also on Hollywood's film studios [START_REF] Epstein | The Big Picture: Thew Logic of Money and Power in Hollywood[END_REF][START_REF] Mezias | Resource partitioning and the founding of specialist firms: The American feature film industry, 1912-1929[END_REF] have demonstrated that the tensions between creative and financial plans are most of the time settled by the domination of one specific force -the pursuit of financial security and stability.

In light of the reflections presented above, we now understand that fusion between economy and creativity becomes the daily experience of creative sectors. We also understand how creative workers are part of a creative neo-liberal 'dispositif' that entails economic and commercial pressures. Research on creative organizations often highlights a concern that economic influences on creative work might crowd out creative influences. How this concern can be managed, however, is not well understood (Austin et al, 2017).

THE 'MISSING LINK'

Our ambition with this overview was neither to condemn nor celebrate the creative industries, but to remind how the literature holds on to the ambivalence of creative work.

Within existing research on creative work, the divergent perspectives of economic and aesthetic agents are frequently mentioned. We read about economic and creative imperatives, paving their way all along. Both imperatives are needed and reunited in a conversation that appears very unlikely (Austin, Hjorth & Hessel, 2017). However this ongoing form of conversation is vital, providing the ongoing context for various activities aimed at producing periodic creative outcomes. Cohabitation seems indispensable. The purpose of this paper is then to acknowledge for such activities more precisely, unpack that missing link and unravel the threads that link economic and creative rationales all the way.

The question of how contracts work between creativity and economy is nested within the larger question of the relationships between creative actors and commercial inputs (Austin, Hjorth & Hessel, 2017;[START_REF] Caves | Creative Industries: Contracts between Art and Commerce[END_REF]. Following up from previous studies, [START_REF] Thompson | From conception to consumption: creativity and the missing managerial link[END_REF] point to a gap in understanding of the inner workings of creative firms, a 'missing link between conception and consumption', which does "leave a gap where concrete analysis of work should be" (p. 625). What seems to be missing here are the concrete work practices that unify the lucrative venture with its creative vision. How do economy and aesthetics become conversant ? (Austin et. al, 2017). It is not clear how the existence of multiple forces can translate into a certain stability for creative actors, more precisely in the work they accomplish through their day-to-day activities. To date, research on the relationship between creative and business motives has lacked a sound theoretical model of individual action in creative production [START_REF] Eikhof | For art's sake ! Artistic and economic logics in creative production[END_REF].

By specifically addressing creative workers' practices, we thus recognize the centrality of human actions to organizational outcomes, which also reflects an increasing recognition of the importance of practices in the ongoing operations of organizations. Building on the insights outlined above, our aim in this research is to consider how particular practices can allow the daily work of creative actors within lucrative organizations. Over the past two decades, organization scholars have argued the need for a practice turn (Barley andKunda, 2001, Corradi et al., 2010), thereby implying that theories need to be constructed based on what happens in practice rather than researchers' conceptualizations.

RESEARCH CONTEXT AND METHOD

ETHNOGRAPHIC WORK IN A FASHION-DESIGN STUDIO

Fashion industry is sitting at the boundary between a commercial and a creative enterprise (Caves, 2002). Fashion products have cultural, symbolic and economic value and navigate a complex system of creators, producers, arbiters and diffusers before being purchased at retail by consumers (Caves, 2002). At the organizational level, the product development process is governed by a series of creative and commercial judgments. Designers speculate about broad, socio-cultural aspects of clothing such as fashion trends, customer lifestyle and brand aesthetic; while simultaneously relying on personal attributes such as taste, creative abilities and commercial judgement as they move through the creative process. In this economy based on fabrics, textures and bodily experiences, the concern for efficiency, control and commerciality is necessary. Although biannual fashion shows of New York, London, Milan or Paris are the opportunity for fashion designers to demonstrate their artistic talents and dazzle the public, fashion houses are also battling daily against very concrete decisions, like the fixing of their selling prices, the geographic localization of their factories, the definition of their distribution channels or also the implementation of their advertising campaign [START_REF] Godart | Managing creatives: Paradoxical approaches to identity regulation[END_REF].

Ethnography -or, to emphasize its processual nature: ethnographying [START_REF] Tota | Ethnographying public memory: The commemorative genre for the victims of terrorism in Italy[END_REF] typically means having a prolonged and intensive engagement with the research setting, following actors, issues and materials as they move through time and space. In this paper we focus our study on one specific fashion house. First author spent three months in the studio of the Parisian designer Maria Maliusi 1 as an intern, helping out with various daily activities.

The ethnographical method was first prescribed for the study of organisations in a special issue of the journal Administrative Science Quarterly on qualitative methods [START_REF] Van Maanen | Reclaiming Qualitative Methods for Organizational Research: A Preface[END_REF]. For this paper, doing ethnography is approached as a way to illustrate the complexity surrounding the many and scattered practical activities attached to creativity within creative industries. Going on the field through ethnography was thus an opportunity to go beyond the knowledge of ordinary practitioners that could have been reached through interviews. With this in mind, an ethnographic work helped us seize the more 'miniature', moving and diffuse aspects of creativity, the more unpredictable and vague aspects of the work, and the changing relationships between individuals. It was in that sense an opportunity to reintegrate the informal relations, perspectives contradictions, and symbolic aspects of interaction in the analysis. And through all that, go beyond the economy versus creativity dichotomy that is difficult to avoid when interviewing people.

FUNCTIONING OF THE HOUSE

The company Maria Maliusi is an eponymous brand, in the name of its founder Maria. In the findings, transcript of the field tries to be authentic and honest as possible, as to provide the most accurate image of what is going on in a design studio on a daily basis.

Vignettes will allow the reader to spot the semantic shifts we operated between the underlined extracts and the interpretations given. In that so, it is an explicit way to make the interpretive journey seen. The analysis was based on an iterative process of reading and scrutinising the material -diary was encoded in a loose manner-and a constant reflection on what emerged as intelligible in the field by participating in it and for us as researchers. In the end, the process was more messy than linear, representing a critical and reflexive style of working. Empirical observations were related to emerging and already present ideas and information, about creativity, organization and economy, by reflexively moving back and forth between theories and data in a continuous and moving manner.

FINDINGS

Designers at work constantly navigate the intersections of creative and economic rationales.

They are at once making creative decisions, interpreting and constructing taste; and simultaneously making economic decisions around appropriate market transactions. There lies the key part of their work, the paradox of selling creativity: they construct economic value where, initially, none exists.

Alongside the creative process, business side is constantly calling. As the founder, CEO, artistic director or designer-entrepreneur behind the fashion house Maria Maliusi, Maria must continuously perform, represent and personalize her label. And it takes a lot of discipline: send the e-mails, remember the deadlines, negotiate with factories, answer sales inquiries, send collection photos to the fairs, update facebook, meet buyers, and so on. The aim for Maria is to get through the contemporary economic system. The label Maria Maliusi would not survive very long if her propositions ever failed to sell through.

Overall, the job is a lot about knowing how to manage a portfolio of projects evolving at different rates and levels. Organizing manufacturing doings, sending e-mails, working from home, calculating production costs, coordinating private sells, working late, carrying out reminder calls, performing commercial at fairs, adapting to clients, setting up photo shoots, taking care of the website, packing final orders, planning business trips...and the list goes on and on. Two main market-based activities stand out in this long list: the buyer role and the seller role. We might here refer to Harrison White's conception of the industry (1981,2002), represented as a flow of goods produced by fashion houses, which serve as interface between "upstream" suppliers, and "downstream" consumers. As a designer, Maria has to select and purchase stock or materials (mainly fabrics) for her business. As a designer as well, she also has to sell her designs to actual or potential buyers. Those two roles delineate her perimeter as a fashion designer placing her designs onto the market. In what follows, we unfold the various actions that she takes in studio, revealing how she engages in the buyer role but dis-engages from the seller role.

ENGAGE IN THE BUYER ROLE

At the start of each new collection, Maria orders small quantities of fabrics that she likes, from which she will develop prototypes. Every new collection is made of a given number of pieces that take the form of prototypes. Maria started all by herself 20 years ago. Her work has been out for some time now, with two collections per year (Fall/Winter & Spring/Summer). To find new fabric, every six months Maria goes to fabric fairs. Going to the fabric fair is a big event in the timeline of a collection and a big amount of excitement and inspiration usually goes with it.

Diary extract I'm talking about fabrics with Maria. She goes to fabric fairs every six months, and it is a big event. At those fairs prices are not displayed: "above all you want to find a fabric, not a price!" (Maria). She likes going there, getting to touch, handle, be surprised by new as well as traditional fabrics. So she selects tons of samples and a few days after make up her mind on the one she prefers, and asks for the price to the suppliers by email. Sometimes it is too expensive so she goes back to the samples she gathered to select new ones. At fairs she takes many, many, many samples. Then she'll do some sorting, progressively. She says "in the fair after a while, you saturate. It's too much fabric. Like in a perfume shop, you can't take it anymore. It's good that you can take samples and choose in the following weeks". Then once Maria enquired about prices and minimum amounts of quantity to be ordered (fabric suppliers always set a minimum amount of quantity to order), she asks for a "coupe type".

The coupe type is the first piece of fabric submitted (by suppliers) to the designer and which serves as a reference for future production. This piece of fabric will enable the two designers in the studio to achieve the prototype piece. And all those prototype-pieces will constitute a whole collection, in four months.

Maria and Helen, the two designers, often talk about those fabric fairs, and the fabric they saw there, and how it made them think of this other thing, and that, and so on. Going to fairs is the first step, fabric the main source of inspiration. Interestingly, fabrics seem to bring the renewal, encountered materials at fairs have the power to create images that involuntarily or voluntarily act upon their thinking.

This morning I asked Helen how they find the will to start a new collection, knowing that at one point in time they have to start the next collection (eg summer 18), they also have to Maria and Helen listen, sometimes they don't. In the following episode for example, another manufacturer suggests to order from a cheaper supplier.

I am going with Maria at Wonni [the manufacturer] to discover the fabric that Wonni just received from the supplier. "Discover" is the proper term. Four months ago, Maria did choose a specific fabric (with parallel lines) to prototype a T-shirt, and there were many orders for this T-shirt at fairs. After that, Wanda told her he knew a supplier that had the same fabric for half the cost. So she took 300 metres of this less expensive fabric for the production. She is so anxious about this choice she keeps saying it was a mistake: "maybe it was a stupid move. I'm afraid the fabric is not going to be of good quality". She did not feel very comfortable having to change supplier but had to cover her costs. That's what made her make up her mind. At Wonni we discover the fabric and luckily it is of good quality. Maria handles it, cuts it, checks its robustness for a long time before then slowly smiling and starting to imagine even more pieces with this fabric, drawing lines in the air to imagine the future piece in volume.

And this other moment:

I ask Maria if she already thought of using cashmere. "Impossible, it's too expensive. But aren't people ready to pay good money for high quality cashmere (me)? -No, because now mass market brands sell some for less than 100 euros (Maria). -But how do they manage with their costs? (me). -They manufacture in China (Maria).

Maria and Helen always have in mind that they need to cover their costs, but they don't always act accordingly.

Maria starts cutting a skirt in a grey jersey. Then she realizes that she loses a lot of fabric while cutting the pattern. The different pieces of the pattern are adjusted to the fabric while cutting it, but sometimes it's hard to fill the space between the pieces, which generates a lot In those extracts from the field, we notice how the creative process is not excluded from the buying role, it is rather part of it. Creative process starts at the fabric fairs. Fabric is considered with respect to its price obviously but also with respect to its evocative power.

From the beginning of the designing process (fabric enter the studio), to the end of it (clothes leave the studio), we observe an increase in power. At the beginning, Maria takes time, tries and experiments. Economic choices also pave the way but designing is sensed viscerally.

Intensity is registered as Maria moves and dives into new forms and possibilities for the fabric, maintaining a sort of suspense along the way. But the closer the deadline is, the faster it goes, and less and less time is devoted to experimenting. After a few months there is no time left, she needs to sell her designs.

DIS-ENGAGE IN THE SELLER ROLE

The result of 4 months of designing is around 60 prototypes that set the tone for every season.

Once the prototypes are realized (or rather when there is no time left, explaining the 'around' before 60 pieces), they make use of the services of an external photographer to establish the "carnet des collections" (the collection book). The modelist Leo also comes on stage, handling patterns and other gradations (adjustments of sizes for manufacturers).

In fairs both designers show the prototypes to buyers from all over the world. Regular Once the orders are confirmed, Maria recontacts the textile producers to order the appropriate quantity following the number of pieces to produce (as previously seen, she cannot afford a surplus or a lack of textile). Textiles are directly sent to manufacturers with the various patterns. While the outfits are being produced, Maria develops the next collection.

From now and then she would also organize private sales in the studio.

The more the fair is approaching, the more the garment in-the-making, filled in aesthetic knowing, becomes a garment ready-to-sell, associated to more commercial rationales. Toward the end, when the collection is almost complete, Maria starts mentioning the buyers -she never did beforehand-and their potential reaction to the new collection. She herself recognises that the designing process has now come to an end. It is done, there is nothing they can change. What's interesting is the apparition of those ghosts-actors -such as buyers-that progressively materialize.

From diary

It is now time to fix prices. Helen says "fixing prices, it stresses me out! I mean, it concretizes the work, you know. So it becomes real, and necessarily it's stressful". Fairs are coming, and I can feel that the atmosphere in the studio is changing. Clothes now shift to enter the business side. I say it to Helen, who approves. She tells me she really does not like going to fairs: "it's just a bad time to pass". Labels are put on garments, garments are ironed and prepared for upcoming fairs. While the deadline for fairs approaches, business interests make their presence felt in the studio.

Work-intense practices now focus on the presentation of the collection. In that moment, creative studio-objects become high-end fashion commodities. Maria gets ready to move on to the outside world and take on the professional cape. Fairs cannot be re-scheduled or even delayed. The new collection of prototypes, which, up to that point, continuously moved in between various geographical locations and spaces of production, is now coordinated and presented to a prominent and leading crowd of buyers in specific exhibition spaces. They will after that get produced in numbers according to orders at those fairs, to end up being shipped all over the world.

Time has come to put much care into presenting the garments. And naturally this goes through its 'outside', by adding worth and significance through the many other affective interacting elements, such as hair and make-up -for pictures-or accessories -to present next to garments on the clothing rack at the fair. Prototypes from the brand new collection are blended with suitable accessories, edgy jewellery and hair and make up styles that altogether definitely enhance the collection and move audiences (Huopalainen, 2016).

The collection now looks and appears professional. More and more space is given to the shining and affective aura around the clothes (ready-for-buyers), reducing or at least rendering less visible the improvisational reality of spontaneous actions that are for the most part associated with creative actions. So it is a real solidification of design that develops towards the end, according to the events of daily life in the studio. The process of crystallization starts with an initial state of designing that is very messy to end up in a much more rationalized way, marked by measuring techniques (to calculate how much fabric will be necessary for production), by the fixing of prices, by the grading (adapting for different sizes) and by the constitution of silhouettes (presentation of clothes to the buyers).

Big buyers might even have an influence on the line, sometimes asking for specific designs for example.

End of the day: I am working on the threads and ironing of the special models for one of the big buyers that ordered a dress without sleeves. Helen explained to me that during fairs, big buyers sometimes ask for specific designs: this blue dress in beige, this short skirt in long, this shirt without sleeves, etc. Of course Maria and Helen often find senseless what demanding buyers ask for, but follow their will as they cannot possibly refuse their financial support. In a similar way, some pieces might be dropped after the fairs if they do no catch enough attention -meaning orders-from buyers... Playing the part of the seller becomes also very concrete when buyers come in the studio. Maria could be joyful to present her offer but she would rather avoid it if she could, and stay out of the seller role. Such observation might be illustrated from that moment when an appointment was scheduled with a buyer from Canada (in the studio). Maria was so late she barely saw the buyer and Helen had to take care of it all by her own.

At
I come in the studio with Maria, around 11. The buyer is here and has been here for one hour.

She [the buyer] is already done with her selection. Maria does not ask about the buyer's order or opinion on the new line. She will have a look at her order, but only after on the order form, once the buyer has left. She is not curious about her opinion on the new collection. She does not ask any question to the buyer who came all the way from Canada! She is totally avoiding any small talk. I am very intrigued, she does not make any effort. It is obvious that she looks forward to the end of the visit. And when, at last, the buyer leaves, she looks very relieved. (...) It really feels like buyers are not welcome around the clothes. How many times in the studio did I hear things like "buyers really think they are all allowed" or also "buyers are just carpet merchants, they do not care at all about design"... The designers are tough when it come to buyers. By extension, there is no curiosity whatsoever for the final customers: the clients, women and men, at the end of the chain, who buy the clothes. Neither Maria nor Helen are very interested in their clients. They design clothes according to their creative desires with the fabric, but not according to any predefined or typical ideal of a client in mind.

At the end of the day, Maria dissociates herself and her designs from the company. This is very clear, as her brand name (the name of the fashion house) is Maria Maliusi, whereas the company's name is Mal. She willingly gave two different names. So when she speaks of the business, she talks about the company Mal., and sometimes it almost feels as if it were not hers. Some business-related activities actually happen in parallel, as may the emails between the financial person and the accounting person show: Maria is in copy but does not intervene in the exchange and communication between them. Mal. happens in autonomy.

Far from mass manufacture, less than 2000 pieces are then produced by season, all of which are made in France. With those medium-size series, Maria Maliusi, the brand, is currently available at more than 50 points of sale across the world. With no need for runway presentations of directly-operated boutiques, and no desire to refine her brand image or design a marketing strategy, Maria maintains, as much as possible, distance from a commercial role.

Her brand name, in fact, is her own moniker written in lowercase letters.

Similarly, she could try to communicate more on her private sales, attract some more people. But she is not interested in doing it. It annoys her. As a consequence, it's essentially her friends, or friends of friends that come to her private sales. The database of clients to whom she sends her invitations (by mail) to private sales are only those friends -and friends of friends. She never took the time to add her real clients (from different stores where she is sold), that would be much interested by those private sales. She often opposes this statement (not trying to reach absolute performance) with the constructed accounts of other fashion labels, created "only to sell" -in her own words. In the following extract from diary, Maria strongly disapproves:

Maria talks to me: "For designers nowadays it's very difficult. Designing cannot happen without marketing and communication". Maria seems very pessimistic, saying that designing does not really happen anymore. "Look at xxx [a famous brand], we hear about them a lot, but don't you think that they also have difficulties, seeing that their manufacturer is the same as ours? Trust me, that explains why their designs are far from being amazing, they're obsessed with profit".

Work intensification is another typical consequence of rationalization processes that is resisted at Maria Maliusi. Produce more would mean grow bigger and Maria does not want that. Invest in marketing or communication would mean less time for designing and she does not want that either. This is also what allows her to maintain as much control as possible on her designs. Helen the assistant designer once suggested making some pictures of the clothes for a very little amount of money, with a friend of hers who is in a photography school, Maria said no. Those anti-performance behaviours and associations, added to Maria's unwillingness to stand out raise questions.

Building on the key scenes previously presented, we can draw a number of analytical conclusions. What we notice is how creative and economic rationales co-exist during the whole process, and especially during the two market-based roles we identified. However, we witnessed how designers engage in the buyer role but dis-engage in the seller role. To lend form to the theoretical refinements that are the objective of this study, we here develop two main ideas to illuminate and give added meaning to our findings.

DEVELOPING THEORETICAL CONCEPTS FROM FINDINGS

First idea is the asymmetry in market-based roles, how designers engage in the buyer role but not in the seller role. Second idea is the unconventional entrepreneur behaviour, how designers seek the reproduction of their assets rather than their accumulation, performing the market in their very own way.

THE ASYMMETRY IN MARKET-BASED ROLES: ENGAGING IN THE BUYER ROLE WHILE DIS-ENGAGING IN THE SELLER ROLE

Maria and Helen, the designers, enter the creative process at the stage of the fabric. The transformative act that the designing activity implies starts once they handle the fabric for the next collection. They can start anew once the fabric has arrived. The moment when they buy the fabric is a moment of maximum potentiality saturated with openness. They experiment, they interact. The overall process is associated to creative rationales, possibilities are there.

Things are never fully finished, but remain in a process of construction and transformation.

At some point however such process starts to close because there is a deadline. They have to go to fairs and sell the line. Garments become wearable and need to seduce. We go then from the 'garment' stage to the 'wear' stage. Interestingly, Maria seems to confine herself as much as possible to the fabric-to-garment (first step of the process) transformation. She is not interested in presenting her offer, nor in interacting with the clients (buyers) or the final clients (users). In a way, we could say that the creation process is not extended to the sales process (the relationship to the buyer). Maria could be joyful to present her offer but in reality she would rather avoid it if she could.

Doing fashion includes pretending to make huge efforts appear 'effortless', while convincing critical gatekeepers of the design team's intelligence and taste (Huopalainen, 2016). This is definitely not the case here. There is no interest in the final use(r) of the garment. Functionality is part of the thing and Maria knows it. She knows that in the end, the garment is worn. Still, she does not make any effort to convince critical gatekeepers (such as buyers) of her taste. So while she is pleased to talk to the upstream suppliers, those who sell the fabric, she avoids buyers. She is certainly happier to talk to those who sell fabrics (mostly Italian sellers) than those who buy garments. She performs as a buyer, dealing with fabric, prices, negotiations; but not as a seller, avoiding as much as possible to play the sales agent.

There is an asymmetry in market-based roles, a lack of equivalence between how designers engage with the buyer role and the seller role. Playing her part as an interface (White, 1981(White, , 2002)), Maria nevertheless gets the least involved as possible with the downstream clients.

From the findings, it appears that this is because everything remains open in the buying role.

This occasion, when they dispose of the fabric, is an episode of maximum openness.

Designers at that moment are like bees around honey. The emulation taking place around the new fabric is like a fuel; they get the kick out of it, out of the rawness of the material, the possibilities to experiment, and so on. On the contrary, the other side is a closing of things:

they have to harness, make use of this, give it a certain direction, give it clarity. They have to weave with constraints, more and more.

In other words, the designers we observed don't want to be at the end, they want to be at the opening. That would be why Maria confines herself to the initial stages, as much as possible. Buy the fabric means opening, sell the garment means closing. Accordingly, a passion for opening is observed as well as a fear of closing. The seller role is not attached anymore to potentiality. This is the ultimate scene, and exam. It is scary for her, her design is being examined. There is no joy in presenting her offer; but rather a sort of fear to conclude, to terminate. She is very nervous and wishes she could avoid that moment of finalizing. As previously seen, some powerful buyers might ask her to have the piece she initially designed in blue in beige: such requests become a threat to her integrity and designer priorities.

So we inquired into the reactions of creative actors with regard to the potential tensions, pressures and constraints encountered during the creative process within rationalization forces [START_REF] Tschang | Balancing the Tensions Between Rationalization and Creativity[END_REF]. What we witnessed on the field is a designer that performs the buyer role but not the seller role. Yet in the end, Maria wants to play and keeps playing, she does not refuse the idea of selling own work. Obviously she is not resisting the stream of income coming out of the selling. More precisely, it is the idea of continuously seeking more profit with it that she rejects. In those moments, what is resisted is not the commercial logic per se but rather the capitalistic one.

Economic practices are lived as resourceful in the buyer role but constraining in the seller role. In the buyer role, to buy (even with a limited budget) is seen as resourceful. Designers are able to make some compromises, dealing with fabric costs and all the while launch the new collection. They do not resist commercial orders that intrude without fundamentally polluting the creative experience. In the seller role, to sell and to make profit with more sales is experienced as constraining, actions such as marketing or commercial zeal put an end to creative motives. Designers perform in the buyer role (carries a connotation of openness) but not in the seller role (carries a connotation of closing). They confine themselves to the first transformation steps, where the two logics (commercial and creative) are coupled, whereas in the final steps those same logics become distinct.

THE UNCONVENTIONAL ENTREPRENEUR: SEEKING THE REPRODUCTION OF ASSETS RATHER THAN THEIR ACCUMULATION, FOR THE SAKE OF CREATION

What we witnessed on the field are anti-performance behaviours and associations from designers, an unwillingness to stand out. We described how Maria cultivates a balanced engagement with various perspectives by reaching a 'happy medium' in terms of production.

A 'happy medium', with 2000 pieces by season, no more. Such a threshold she does not want to cross. She does not want to reach the potential clients she could get, let alone commit to marketing actions and plans. In the eyes of Maria, being successful means to move from one day to the other, from one collection to the other, rather than growth, fame or fortune.

In so doing, it seems that the two designers consider designing throughout a conscious posture against established consumerist agendas, and with an active disinterest in profit, growth or additional earnings -as long as they get the necessary earnings to keep on designing. For Maria, branding, communicating or even think positioning is unneeded. Keep a reasonable size of a business is what matters. In everyday business life, production and sales are the key (sell own products), and always gain the upper hand over more elaborated practices (think about an advertising campaign to access more customers). Her primary objective is not to increase the market share of the house.

Interestingly, the concept of efficiency is not so settled in the studio. In traditional companies, efficiency has become an objective to reach in itself, justifying all the efforts put into managerial rationalization. On the contrary, in the studio processes are far from being optimised. Inventory is hardly managed and it's common that they rush outside to get new sewing threads in one colour or another once they run out of it. In that sense, it appears as if Maria would be an 'anti-hero', or at least very unconventional, not trying to reach absolute performance. She makes a clear distinction between selling her designs on the short run so as to be able to live out of it; and seeking assets accumulation with her brand. She owns her company and play the game of the market through market competition. Yet, profit motives for the sake of profit are rejected. The ultimate goal for the fashion house is not to make money but rather to keep going. The reason for her business's existence is rather continuity of creation, when all the costs come in balance, that turning a profit.

Resources are not always being allocated efficiently. For some designs, there is no profit involved. Within capitalistic thinking, it is a sign that the labour and assets devoted to its production are misdirected. The value of the resources used up in making the garment looks greater than the value of the article itself. But in this case there is more to it: the satisfaction of a good -as in aesthetically satisfying-design. Following capitalistic reasoning, profit let companies know whether an item is worth producing. Theoretically, in free and competitive markets, maximising profits ensures that resources are not wasted. But once again there is more to it, and wasted resources from a capitalist point of view do no often correspond to the fashion house ways of doing internally. The aesthetic added-value do not correspond to traditional control functions, and classical lines of argument on production costs might be fatal for the creative impulse surrounding the designing of a new collection.

To conclude, we enquired about how market-based roles are embraced by creative workers. What we witnessed on the field is not only an asymmetry in market-based roles but also a differentiation between the commercial logic and the capitalistic one. Creative workers try to reach what we called a happy medium, the necessary amount of sales to keep up with their creative activity, willingly not aiming at optimal performance or more broadly any growth perspective. Those behaviours characterize what we call the unconventional entrepreneur.

DISCUSSION

Based on our study, we suggested that entrepreneurs of a certain kind of organization -one that creates economic value by creating creative value-aspire to reproduce rather than accumulate assets. The reason for that kind of organization's existence is rather continuity of creation, with all the costs in balance, that turning a profit.

To think other than the (already much covered) dichotomous approach of creative and economic rationales, our concept of unconventional entrepreneur helps address what we see as a theoretical deficit. We lack concepts or frameworks to help think about creative and economic aspects' interaction beyond their divergence. Even in the creative industries field, there are surprisingly few empirical studies that examine how creativity is managed and organized at work [START_REF] Warhurst | The missing middle: management in the creative industries[END_REF], and even fewer that unpack the economic incursion more specifically (Austin et al., 2017). We suggest extensions to theory by refusing the either-or thinking (De Fillipi et al., 2007), and voluntarily develop theory that we believe might be especially applicable to creative work, reflective of its distinctive characteristics [START_REF] Caves | Creative Industries: Contracts between Art and Commerce[END_REF]. Austin and his colleagues highlight the advantages of a theory specific to creative work and insist: 'a theory of creative work should invite us to take seriously concerns about the coherence of outcomes that combine conflicting influences and the fragile states that produce these ' (2017: 15). We believe our theoretical conceptualizations contribute new insights about work in creative firms in such contexts. More specifically, we nuance and precise what is understood by 'economic aspects' and reveal strategies that work through trade-offs, creative actors playing the game but not extensively, taking the tangent from management books' guidance. Observed behaviours are at odds with the traditional dominant models that are described, taught and as such performed by the managerial doxa [START_REF] Parker | Utopia and the organizational imagination: outopia[END_REF][START_REF] Hjorth | Organizational Entrepreneurship : With de Certeau on Creating Heteropias or Spaces for Play[END_REF]. What we demonstrate here is how desire and passion for creative action is precisely what organizes creative professionals in opposition to dominant managerial strategies.

Our position is not, in our view, incompatible with previously outlined theorizing.

Rather, we bring some nuance to the debate by specifying what is behind the economic understanding most literature depicts. We believe economic and creative perspectives can 'live with' (Austin et al., 2017) one another, and highlighted one way this can be done, through unconventional behaviours that allow a differentiation between commercial and capitalistic rationales. We showed how designers avoid compromising on their artistic integrity by thoroughly withdrawing from capitalistic imperatives, while still embracing commercial roles. It seems to us that several dimensions constitute the economic, commercial practices and capitalist practices are two of them. Obviously the intertwinement between dimensions is fluid and this is what allows the introduction and setting-up of trade-offs (agree on the commercial part but not the capitalistic), as well as the varying engagement with market-based roles (refusing to engage with the selling). In revealing so, we go beyond the idea that the economic inevitably colonizes and endangers the creative.

Hopefully our work offers novel ways of moving beyond the inherent pessimism of poststructuralist theory and its perceived emphasis on the domination of the subject by power (Fotaki et al., 2017). As previously seen, in modern organizations, and as a result of the industrial revolution's processes, the 'creative man' seems to be crowded out of the workplace by the 'economic man' (Hjorth, 2003). The economic man would be conditioned by the organization to carry out pre-determined activities to maintain control and predictability, hence constructing playfulness and passion as non-organizational (Hjorth, 2003). Yet the antiperformance behaviours that we identified in the findings derive from the passion for the creative practice, nurtured among the community of competent practitioners (other independent designers in our case) for whom and by whom the practice is enjoyed. It is the passion for the practice, the investment and trust in the creative production that enables the stake for such practices we identified to stand out meaningfully and powerfully for all those involved [START_REF] Contu | On boundaries and difference: Communities of practice and power relations in creative work[END_REF]. Hopefully this research will lead to new investigations interested in uncovering other mechanisms and/or features of engaging and dis-engaging that characterize creative action nested within economic canvas.

Austin, R., [START_REF] Hjorth | Organizational Entrepreneurship : With de Certeau on Creating Heteropias or Spaces for Play[END_REF]Hessel, S. (2017). How Aesthetics and Economy

Become Conversant in Creative Firms, Organization Studies, online 10.17

  Chief designer (Maria) is intensely active in all work phases and everyday doings of work: those related to design processes, fabric manipulations, manufacturing, sales, fairs, communication and so on. The company has been active for 20 years. Established in Paris, it is specialized into the sector of high-end design -Prêt-à-porter haut de gamme, with a turnover of 197 900 € in 2016. In the studio, it is only Maria, the founder and CEO of the brand, and Helen, an assistant designer. Helen carries a big part of the responsibility of the design processes and the everyday doings in the studio. They also work with an accountant, Kate (employee of EuroGestion, an accounting firm) and a model-maker (freelance), Leo.Maria also works with Samuel at FinanceMode -an accompanying structure on the financial side-, getting public grants and tax credit for instance, essential for the survival of the company. Daily interlocutors are also the subcontractors: the manufacturers. Maria deals with three manufacturers, one in Paris, one in the suburb of Paris, and one in Niort (west of France).Overall they achieve two collections per year, each one of them composed of about 60 pieces that are then produced according to purchase (in fairs). The success of a collection is what allows the next to be done. The collection is sold by going to fairs. They have to get it right. Each year, Maria and Helen go to fairs that take place twice a year: Tranoï (Paris), and exported for the most part abroad, in stores in Europe, in Asia, in the United-States and Middle East.

  send the previous collection to buyers(summer 17), and launch the just-finished collection (winter 17/18) with the manufacturers...which means 3 collections at a time. She replied instantly: "We go get the new fabrics...(...) and it is through those that we want to start again as soon as possible". Helen stops what she is doing to show me the various samples of fabric, explaining how they build the collection out of those initial purchases, describing the suppliers behind the samples -mostly Italian companies-, the characteristics of each fabric, the work they could imagine departing from those tiny tiny pieces of material... Buying the fabric is also often central in the discussions with manufacturers. Manufacturers often complained about the fabrics -commenting on their price. Several times Sofia (one of the manufacturer) told the designers that the fabrics they buy are too expensive. Sometimes

  customers (sales representatives for clothing stores worldwide) are always very curious to discover the new collection and order almost every season, provided they like the creative proposition. Fairs are also an opportunity to show the collections to sales representatives who do not know the brand. If it was not for them, Maria could definitely organize showrooms in her studio and invite her regular customers. Obviously the risk exists that those regulars grow tired of her work, or do not get hooked by the brand new collection, but it remains the most reliable commercial target.

  Barley, S.R. and Kunda, G. (2001) Bringing work back in. Organization Science 12 (1): 76-95. Becker, H. S. (1982). Art worlds. Berkeley, CA: University of California Press. Bilton, C. (2009). Relocating creativity in advertising : from aesthetic specialisation to strategic integration ? In Creativity, Innovation and the cultural economy. A.C.Pratt & P.Jeffcutt. London, Routledge : 1-20

  of scrap fabric. Maria hesitates." Do we stick to this fabric?". She goes next room check the price."15 euros for 1 linear metre". It's expensive. She hesitates, she could switch the grey jersey with a black cotton, far less expensive..."Ok. Let's keep on with the jersey, now that we started". The choice of fabrics does not always depend on prices. I can see that she definitely wants to work with jersey, manipulate jersey, find creative solutions from this jersey. What she has in mind and in the hands does not 'fit' with black cotton.

  fairs Maria is waiting for people's reactions, looking out for any appreciation on the collection. She cannot escape confronting what buyers think and feel about her creative proposition. She tries to catch any assessment or judgment and keeps wondering. Although now professional designer, it is not so easy for her to really form a clear opinion about how

her designs are received -commercial success not being the only indicator. Exposing a brand new collection is both exciting and anxiety-causing. If her affective makings do not resonate, she might be in trouble for the future.

Designers & Agents (New-York). As Maria does not have a store in her name, the collections are sold in concept-stores or in small clothing stores, where retailers are looking for innovative designers (in those fairs). On average 1500 pieces are produced each season, and


Emmanuel Vazquez
email: emmanuel.vazquez@supelec.fr

Eric Walter
email: eric.walter@lss.supelec.fr

Estimating derivatives and integrals with Kriging

This paper formalizes a methodology based on

Kriging, a technique developped by geostatisticians, for estimating derivatives and integrals of signals that are only known via possibly irregularly spaced and noisy observations. This finds direct applications, e.g., in system identification when differential algebra is used to express parameters as nonlinear functions of the inputs and outputs and their derivatives. The procedure is quite simple to implement, and allows confidence intervals on the predicted values to be derived.

I. INTRODUCTION

Let f * (x) be a real function defined on some compact set X ⊂ R d . The problem to be considered here is the estimation of the derivative (or the integral) of f * at any given x ∈ X from a finite set of observations S = {(x i , f obs xi ), i = 1, . . . , n}. These observations may be corrupted by (not necessarily white) noise, so f obs xi is not equal to f (x i ) in general. Moreover, the observations need not be regularly sampled. Such a problem is frequently encountered in system identification and control, for instance when algebraic differential methods are used to express parameters as nonlinear functions of the input and outputs and their derivatives, see, e.g., [1], [START_REF] Diop | A resettable extended Kalman filter based on numerical differentiation[END_REF].

The methodology to be presented is based upon techniques developed by Geostatisticians and known as Kriging and Intrinsic Kriging. The possibility of estimating derivatives and integrals via Kriging has already been suggested in the context of Geostatistics [START_REF] Chilès | Geostatistics: Modeling Spatial Uncertainty[END_REF] and we would like to call the attention of the control community on its simplicity, pertinence and performances. This paper will consist of two parts. The first one will briefly recall the theory of Kriging and Intrinsic Kriging, before considering the estimation of derivatives and integrals with these methods. To the best of our knowledge, the mathematical formalization of the prediction of derivatives and integrals with intrinsic Kriging had never been published. The second part of the paper provides illustrative examples.

II. FROM LINEAR PREDICTION TO THE ESTIMATION OF DERIVATIVES AND INTEGRALS

A. Random Processes and Kriging

Kriging can be used to approximate or interpolate data, just as splines do. See, for instance, [START_REF] Chilès | Geostatistics: Modeling Spatial Uncertainty[END_REF], [START_REF] Cressie | Statistics for Spatial Data[END_REF], [START_REF] Matheron | Principles of geostatistics[END_REF], [START_REF] Sacks | Design and analysis of computer experiments[END_REF]. As splines, it is a kernel regression method [START_REF] Schölkopf | Learning with Kernels[END_REF], [START_REF] Wahba | Spline Models for Observational Data[END_REF]. Its specificity is that the kernel is chosen after a statistical analysis of the data. Let (Ω, A, P) be a probability space and L 2 (Ω, A, P) be the set of second-order real-valued random variables defined on (Ω, A, P).

Definition 1: The set of all random variables F (ω, x) ∈ L 2 (Ω, A, P) obtained when x runs through X is called a second-order random process with parameter x ∈ X.

Let m(x) = E[F (x)] be the mean of F (x), and k(x,

y) = cov(F (x), F (y)) = E[(F (x) -m(x))(F (y) -m(y))] be its covariance function. The covariance function is positive since var n i=1 λ i F (x i ) = n i,j=1 λ i λ j k(x i , x j ) ≥ 0 , for all λ i ∈ R, x i ∈ X and n > 0.
Let F 1 (x), . . . , F q (x) be random processes defined on the same probability space and parameter space X. Let m α (x) be the mean of F α (x) and k α,β (x, y) be the co-

variance E[(F α (x) -m α (x))(F β (y) -m β (y))], α, β ∈ {1, . . . , q}. Note that k α,β (x, y) = k β,α (y, x) but in general k α,β (x, y) = k α,β (y, x).
Assume, for the time being, that f * (x) is a trajectory of F (x) (i.e., there exists ω ∈ Ω such that F (ω, x) = f * (x) for all x ∈ X) and that the observations are noisefree. Each f obs xi = f * (x i ) is then a realization of the random variable F (x i ). The data S can then be interpolated by predicting F (x) given the random variables F (x 1 ), . . . , F (x n ), which amounts to finding a function F (x) of F (x 1 ), . . . , F (x n ) that minimizes F (x) -F (x) in some sense. A predicted value is obtained by replacing the random variables F (x 1 ), . . . F (x n ) by their realizations f obs 1 , . . . , f obs n in the expression of F (x) (ω remaining uncertain). Consider the class of linear predictors, which can be written as

F (x) = n i=1 λ i,x F (x i ) .
Assume, moreover, that F (x) is a zero-mean process (m(x) = 0, ∀x ∈ X), an hypothesis that will be relaxed in Section II-B. The best predictor or Kriging predictor F (x) of F (x) is the orthogonal projection of F (x) onto the subspace

H S = span{F (x i ), i = 1, . . . , n} . Since (F (x) -F (x), F (x i )) L 2 (Ω,A,P) = 0 ⇒ k(x, x i ) - n j=1 λ j,x k(x j , x i ) = 0 , (1)
for i = 1, . . . , n, the scalars λ i,x , i = 1, . . . , n, are obtained by solving a linear system of equations.

Proposition 1 (Kriging): Let F 1 (x), . . . , F q (x) be second-order random processes, with zero mean, and covariance functions k α,β (x, y). The best predictor of F α (x) from F αi (x i ), i = 1, . . . , n, is the orthogonal projection of F α (x) onto H S = span{F αi (x i ), i = 1, . . . , n}, written as

Fα (x) = n i=1 λ i,x F αi (x i ) , (2)
where the λ i,x s are the solution of the linear system1 of equations

K λ x = k x . (3)
In (3), K is the n×n covariance matrix of the random vector F S = (F α1 (x 1 ), . . . , F αn (x n )) T , λ x = ( λ 1,x , . . . , λ n,x ) T is the vector of the Kriging coefficients, and k x = (k α,α1 (x, x 1 ), . . . , k α,αn (x, x n )) T is the covariance vector of F S and F α (x). Confidence intervals are obtained by evaluating the variance of F α (x) -Fα (x).

B. Intrinsic Kriging and Intrinsic Random Functions

Intrinsic Kriging (IK) [START_REF] Matheron | The intrinsic random functions, and their applications[END_REF] extends linear prediction to the case where the mean of F (x) is unknown. In this framework, the function f * generating the data is assumed to fluctuate around m(x), which can be written as a linear parametric function b T r(x), where b and r(x) are l-dimensional vectors. Let N be the vector space {b T r(x), b ∈ R l } and F (x) be a random process with mean m(x) ∈ N . The main idea of IK is to find some linear transformations of F (x) filtering out the mean so as to consider a zero-mean process again.

We first recall the notion of generalized random processes. Let Λ be the vector space of finite-support measures, i.e. the space of linear combinations n i=1 λ i δ xi , where δ x stands for the Dirac measure, such that for any B ⊂ X, δ x (B) equals one if x ∈ B and zero otherwise. Let ΛN ⊥ be the subset of the elements of Λ that vanish on N . Thus, λ ∈ Λ implies

λ, f := n i=1 λ i f (x i ) = 0 , ∀ f ∈ N . Definition 2: A symmetric function k : X × X → R is conditionally positive with respect to N if, for all λ ∈ ΛN ⊥ , k(λ, λ) ≥ 0, where k(λ, µ), λ, µ ∈ ΛN ⊥ , is defined by k(λ, µ) := n,m i,j=1 λ i µ j k(x i , y j ) .
If, moreover, k(λ, λ) = 0 implies λ = 0, for all λ ∈ ΛN ⊥ , then k(x, y) is conditionally positive definite.

Let F G (λ) be a linear application defined on ΛN ⊥ , with values in L 2 (Ω, A, P). Assume that F G (λ) is zero-mean for all λ and that cov[F G (λ), F G (µ)] = k(λ, µ), where k(x, y) is a conditionally positive definite function. Then, F G (λ) is called a generalized random process. Such a random process is no longer defined on X but on a space of measures, and k(x, y) is called a generalized covariance (see Section II-E). Let HN ⊥ be the subspace of L 2 (Ω, A, P) generated by F G (λ), λ ∈ ΛN ⊥ . Since random variables in HN ⊥ are zeromean, the inner product of L 2 (Ω, A, P) can be expressed in HN ⊥ as

(F G (λ), F G (µ)) L 2 (Ω,A,P) = k(λ, µ) = i,j λ i µ j k(x i , y j ) ,
where λ = i λ i δ xi and µ = j µ j δ yj are in ΛN ⊥ . Thus, the bilinear form k(λ, µ) endows ΛN ⊥ and HN ⊥ with a structure of pre-Hilbert space. The completions H N ⊥ and Λ N ⊥ of HN ⊥ and ΛN ⊥ under this inner product define isomorphic Hilbert spaces. F G (λ) can be extended on Λ N ⊥ by continuity. The generalized random process F G (λ) is used as a random model. Simplifying hypotheses are introduced in the next paragraph.

Intrinsic Random Functions (IRF) are obtained when generalized random processes are endowed with a stationarity property. IRF are flexible models to use since unknown means can be conveniently dealt with and stationarity makes the inference of the parameters of their (generalized) covariance function feasible. Let τ h : ΛN ⊥ → Λ be the translation operator such that for λ =

i λ i δ xi ∈ ΛN ⊥ , τ h λ = i λ i δ xi+h .
Assume that ΛN ⊥ is stable under translation. N must therefore be itself a stable space of functions under τ h . Assume further that the generalized covariance k(x, y) is invariant by translation. In the following, we shall write k(h) with h = xy instead of k(x, y), when the covariance is assumed to be stationary. Then τ h is continuous and can be uniquely extended on Λ N ⊥ .

Definition 3: Let F G (λ) be a zero-mean generalized random process defined on Λ N ⊥ , with stationary generalized covariance k(h). The random process h

→ F (τ h λ), λ ∈ Λ N ⊥ , is therefore weakly stationary. F G (λ), λ ∈ Λ N ⊥ , is then an Intrinsic Random Function.
The stability of N under the group of translations implies that N is necessarily a vector space of exponentialpolynomial functions [START_REF] Matheron | La théorie des fonctions aléatoires intrinsèques généralisées[END_REF]. Such a space is generated by functions that can be written as x l e a T x , where a is a real or complex vector, l is the vector-valued index (l 1 , . . . , l d ) and

x l = x l1 [1] • • • x l d [d]
. (For a vector-valued index l, we shall write |l| = l 1 + • • • + l q .) For N to be stable by linear combinations and translations, the monomials x l must form a complete basis. We restrict ourselves to the case where N is a vector space of polynomials of degree at most equal to l. Let N l = span{x l , ∀ l such that |l| ≤ l} and Λl = ΛN ⊥ l . Let Λ l be a completion of Λl under the inner product k(λ, µ).

If the IRF F G (λ) is defined on Λ l , F G (λ) is an IRF of order l,
or IRF(l) in short.

Proposition 2: Any IRF(l) is an IRF(l + 1).

Proof: This follows from the fact that the spaces Λ l are nested:

Λ l+1 ⊂ Λ l ,
and that any IRF F G (λ) defined on Λ l will be stationary on Λ l+1 .

Remark: Λ l can be viewed as a set of finite-difference (increment) type operators. The condition for λ = n i=1 λ i δ xi to be in Λ 0 can be expressed as

n i=1 λ i = 0. Thus, λ = n i=1 λ i (δ xi -δ x1
), so λ is a linear combination of increment measures δ xiδ x1 . For l > 0, generalized increments are obtained. Note that if X is a space of dimension d, a minimum of d+l l points have to be taken to fully specify an element of Λ l .

A generalized random process can be viewed as a class of equivalence of random processes defined on X with mean in N . If F (x), x ∈ X, is a second-order random process, with mean in N and covariance k(x, y), the linear application

F : ΛN ⊥ → H λ = n i=1 λ i δ xi → F (λ) = n i=1 λ i F (x i ) , extends F (x) on ΛN ⊥
, where H stands for the Hilbert space generated by F (x), x ∈ X. Since the mean of

F (x) is in N , F (λ), λ ∈ Λ N ⊥ ,
is a zero-mean random variable, as λ filters out any function of N . Assume that k(x, y) is positive definite. Then (λ, µ) ΛN ⊥ := (F (λ), F (µ)) H defines an inner product on ΛN ⊥ . Let Λ N ⊥ be the completion of ΛN ⊥ under this inner product and extend F (λ) on Λ N ⊥ by continuity. A generalized random process is thus obtained. The next paragraph indicates how the procedure may be reversed.

Definition 4: Let F G (λ) be a generalized random process defined on Λ N ⊥ . A second-order random process

F (x), x ∈ X, is a representation of F G (λ) if F G (λ) = F (λ), ∀ λ ∈ Λ N ⊥ . If F G (λ) is an IRF(l)
, its representations can be written explicitly by taking appropriate measures λ ∈ Λ l [START_REF] Matheron | The intrinsic random functions, and their applications[END_REF]. If F 0 (x) is any representation of F G (λ), other representations of F G (λ) can be written as

F (x) = F 0 (x) + q i=1 B i p i (x) , (4)
where the p i s form a basis of N l and the B i s are any secondorder random variables [START_REF] Matheron | The intrinsic random functions, and their applications[END_REF]. Thus, the representations of an IRF(l) constitutes a class of random processes with mean in N l .

C. Derivation

In this section the notion of derivative of an IRF is developed. The aim is to estimate the derivative of a function f * modeled by an IRF(l), which means that f * comprises an unknown polynomial drift of degree at most equal to l. To simplify presentation, assume that x ∈ R . Extension to the multi-dimensional case is straightforward.

Recall that a zero-mean stationary second-order random process F (x) with covariance function k(h) is mean-square differentiable at x if

F h (x) = 1 h (F (x + h) -F (x)) (5)
converges in mean square when h → 0. The limit exists if and only if k (2) (0) exists, and F (x) is then meansquare differentiable at all x. The limit process is called the derivative process and denoted by F (1) (x). Higher-order derivatives are obtained by iteration, and it is straightforward to check that

cov[F (q) (x), F (r) (y)] = (-1) (r) k (q+r) (x -y) . ( 6
)
Let F G (λ) be an IRF(l), with generalized covariance k(h). The difficulty for defining the derivative of F G (λ) lies in the fact that neither F G (λ) nor its derivatives can be defined point-wise. Thus, the notion of differentiability cannot be defined using the variance of an expression such as [START_REF] Mardia | Maximum likelihood estimation of models for residual covariance in spatial regression[END_REF].

To define a derivative, we must use elements of

Λ l . Since for λ = i λ i δ xi ∈ Λ l , τ h λ ∈ Λ l , ∀ h ∈ R, define λ h = 1 h (τ h λ -λ) ∈ Λ l . Definition 5: An IRF(l) F G (λ) is mean-square differen- tiable at λ ∈ Λ l if F G (λ h
) converges in mean square as h → 0. When the limit exists, it is denoted by F

(1) G (λ). Proposition 3: Let F G (λ) be an IRF(l), with generalized covariance k(h). If k (2) (h) exists for all h, then F G (λ) is mean-square differentiable for all λ in Λ l , in which case F

(1) G (λ) is an IRF(l) with generalized covariance -k (2) (h).

Proof:

Let λ = n i=1 λ i δ xi be in Λ l . Then F G (λ h ) 2 = 1 h 2 F G n i=1 λ i (δ xi+h -δ xi ) 2 = 1 h 2 n i,j=1 λ i λ j (2k(x i -x j ) -k(x i -x j + h) -k(x i -x j -h)) .
Assume further that k(h) is twice differentiable for all h ∈ R.

Then F G (λ h ) converges when h → 0 and

lim h→0 F G (λ h ) 2 = - n i,j=1 λ i λ j k (2) (x i -x j ) .

It follows that F

(1) G (λ) is a generalized random process on Λ l with zero-mean and generalized covariance -k (2) (h).

Remark that the convergence of F G (λ h ) in L 2 (Ω, A, P) when h → 0 is equivalent to the convergence of λ h in Λ l . Let λ (1) = lim h→0 λ h , if allowable. We shall then identify F

(1) G (λ) and F G (λ (1) ). Proposition 4: Let F G (λ) be an IRF(l) and F (x) be a representation. Then,

F (1) (x) is a representation of F (1) G (λ). Proof: For all λ ∈ Λ l , F (1) G (λ) = lim h→0 F G (λ h ) = lim h→0 F (λ h ) = F (λ (1) ) = F (1) (λ) .
Order r derivatives are denoted by F (r) G (λ) and λ (r) . Given

λ = i λ i δ (qi) xi and µ = j µ j δ (rj ) yj in Λ l , it is easy to check that cov[F G (λ), F G (µ)] = i,j (-1) rj λ i µ j k (qi+rj ) (x i -y j ) .
It becomes now possible to predict the derivatives of a representation of an IRF(l). The case where observations are corrupted by additive noise is directly studied. Observed values then correspond to realizations of the random variables F obs (x i ) = F (x i ) + N i , i = 1, . . . , n, where F (x) is an unknown representation of F G (λ), and the N i s are zero-mean random variables independent of F (x), with covariance matrix K N . When the noise is white, K N = σ 2 N I n . A linear predictor F (r) (x) of F (r) (x) can be written as

F (r) (x) = i λ i,x F obs (x i ) .
In IK the prediction error F (x) -F (x) is minimized under the constraint δ xλ i,x δ xi ∈ Λ l . To deal with derivatives, we similarly minimize var[F (r) (x) -F (r) (x)] under the constraint

δ (r) x - i λ i,x δ xi ∈ Λ l . (7)
The solution can be obtained using var[F (δ

(r) x - i λ i,x δ xi )] = var[F G (δ (r)
xi λ i,x δ xi )]. One can then check that the coefficients λ i,x , i = 1, . . . , n, are solutions of a system of linear equations, which can be written in matrix form as

K + K N P T P 0 λ x µ = k (r) x p (r) x , ( 8
)
where K is the n × n matrix of generalized covariances k(x ix j ), P = (x j i ) l,n i=0, j=1 is a (l + 1) × n matrix, µ is a vector of Lagrange coefficients, k (r)

x is a vector of size n with entries k (r) (xx i ) and p (r)

x is a vector of size l + 1 with entries (x i ) (r) , i = 0, . . . , l. Note that

(x i ) (r) = 0 if i < r i! (i-r)! x i-r if i ≥ r
The variance of the prediction error is given by

var[F (r) (x) -F (r) (x)] = -k (2r) (0) - λ µ T k (r) x p (r) x .
It can be used to assess confidence intervals, as illustrated in Section III.

D. Integration

This problem can be viewed as the prediction of a function f (x) from observations of its derivative. Formally, this is equivalent to the previous problem, with straightforward adaptation.

E. Choice and estimation of the covariance

Once the covariance function is chosen, the procedure of estimating a derivative is quite simple to implement, since it boils down to solving a linear system. The question of the choice of the covariance is now considered. Asymptotic results [START_REF] Stein | Interpolation of Spatial Data: Some Theory for Kriging[END_REF], [START_REF] Yakowitz | A comparison of Kriging with nonparametric regression methods[END_REF] suggest that satisfactory performance may be obtained even if the covariance is chosen incorrectly. It could therefore be argued that covariance choice is not an important issue. However, for satisfactory performance with a finite and often relatively small number of samples, a proper choice of the covariance turns out to be very important.

Any classical parametrized covariance can be used as a generalized covariance. For instance, [START_REF] Stein | Interpolation of Spatial Data: Some Theory for Kriging[END_REF] advocates the Matérn covariance, which can be written as

k(h) = σ 2 2 ν-1 Γ(ν) 2ν 1/2 h ρ ν K ν 2ν 1/2 h ρ ,
where K ν is the modified Bessel function of the second kind, ν > 0 controls the regularity (the differentiability) of the covariance at the origin, σ 2 > 0 is the variance (k(0) = σ 2 ), and ρ > 0 is a scale parameter.

Polynomial covariances are also a useful class of generalized covariances [START_REF] Matheron | The intrinsic random functions, and their applications[END_REF]. Given an order l, they can be written as

k(h) = l p=0 (-1) p+1 a p h 2p+1 with a p > 0, ∀p .
Note that this expression is linear in its parameters a p . For example, intrinsic Kriging based on a covariance written as -a 0 h gives a piecewise-linear interpolation.

We use Maximum Likelihood to estimate the vector θ of the parameters of a covariance k θ (x, y) when the mean of the covariance is known [START_REF] Mardia | Maximum likelihood estimation of models for residual covariance in spatial regression[END_REF]. Let F (x) be a zero-mean Gaussian random process with covariance k θ (x, y). Assume also that the observation noise is Gaussian. Let K(θ) be the covariance matrix of F S = [F (x 1 ), . . . , F (x n )] T and K N (θ ) be the covariance matrix of the random vector N of the measurement noise, assumed Gaussian, with zero mean and a covariance depending on some parameter vector θ .

To simplify presentation, take θ = [θ T , θ T ] T and K( θ) = K(θ) + K N (θ ). The log-likelihood of the data can then be written as

L(f obs | θ) = - n 2 log 2π - 1 2 log det K( θ) - 1 2 f obs T K( θ) -1 f obs . (9)
In the following paragraph we recall the Restricted Maximum Likelihood (REML) approach to estimating the covariance parameters of a random process with unknown mean. Instead of the likelihood function of the data, REML maximizes that of the increments (or generalized increments) of these data [START_REF] Stein | Interpolation of Spatial Data: Some Theory for Kriging[END_REF].

Let F G (λ) be a Gaussian IRF(l). Let F obs be the random observation vector, the sum of F S = [F (x 1 ), . . . , F (x n )] T , with F (x) a representation of F G (λ), and some zero-mean measurement noise vector. Let P = (x li j ) q,n i,j=1 be the q × n matrix of basis functions of N l evaluated on {x 1 , . . . , x n }. Since the dimension of N l is q, the dimension of the space of the measures with support {x 1 , . . . , x n } that cancel out the functions of N l is nq. Assume an n × (nq) matrix W with rank nq has been found, such that

P W = 0 .
(The columns of W are in the kernel of P .) The columns of W are therefore the coefficients of measures with support {x 1 , . . . , x n }, n j=1 w i,j δ xj ∈ Λ l . Then Z = W T F obs is a Gaussian random vector taking its values in R n-q , with zero mean and covariance matrix W T (K(θ) + K N (θ ))W = W T K( θ)W , where K(θ) is the generalized covariance matrix with entries k θ (x ix j ) and where K N (θ ) is the covariance matrix of the observation noise. The random vector Z is a contrast vector. The log-likelihood of the contrasts is given by

L(z | θ) = - n -q 2 log 2π - 1 2 log det(W T K( θ)W ) - 1 2 z T (W T K( θ)W ) -1 z . ( 10
)
Various methods may be employed to compute the matrix W . We favor the QR decomposition of

P T P T = (Q 1 | Q 2 ) R 0 , where (Q 1 | Q 2 )
is an n × n orthogonal matrix and R is a q × q upper triangular matrix. It is trivial to check that the columns of Q 2 form a basis of the kernel of P , so we may chose W = Q 2 . Note that W T W = I n-q .

III. EXAMPLES

A. Estimation of derivatives

Figure 1 represents a system output and its derivative. We see that the prediction of the derivative from a number of irregularly spaced noise-free observations of this output is satisfactory. Confidence intervals for this prediction can be provided, which is one of the advantages of the methodology advocated here. As could be expected, the uncertainty intervals are narrower close to the locations of the observations but, may be more surprisingly, when two observations are close enough, prediction is best between these observations. The experiment is repeated in Figure 2, now with the addition of noise on the observations. Again the prediction is quite satisfactory, and the potential applications of such predictors are numerous.

B. Black-box model with prior information on derivatives

The previous examples illustrate only partially the possibilities offered by Kriging for the prediction of the derivatives of a signal. In Figure 3, observations of both the function and its derivative are used. This makes it possible to improve prediction, for instance by taking into account the knowledge that the value of the derivative at time zero is zero. This is an opportunity for introducing some prior information in a black-box model [START_REF] Vazquez | Intrinsic Kriging and prior information[END_REF].

C. Approximation of a function from observations of its derivative -integration.

To conclude this section, let us apply Kriging to integration (Figure 4). Note that at least one observation of f is required in order to specify an initial condition. This approach may be extended to the numerical integration of partial differential equations. It may also be used to model nonlinear dynamical systems described by the state equation ẋ = f (x, u), which constitutes a promising perspective.

IV. CONCLUSIONS AND PERSPECTIVES

The methodology presented and formalized in this paper provides tools for solving two basic problems in control and signal processing, namely the differentiation and the integration of possibly multivariable signals that are only known via possibly noisy and irregularly sampled observations. It is based on intrinsic Kriging, a general-purpose technique for black-box modeling developed by geostatisticians during more than 50 years but still relatively ignored by the control community. As demonstrated by the examples treated, the resulting methodology is quite versatile and allows prior information, e.g., on boundary values of the derivative to be taken into account. Another distinctive feature of this statistically-based approach is that it allows confidence intervals on the predicted values of the derivative or integral to be provided. If the covariance function is chosen a priori, the technique requires only the solution of linear systems of equations. Better results, however, are to be expected if this covariance function is estimated from the data, for instance by maximum likelihood or Bayesian estimation. Numerical differentiation finds direct applications, e.g., in the framework of algebraic differential methods for parameter and state estimation, while the possibilities offered by numerical integration in the context of dynamical system modeling look very promising.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Top: f (x), x ∈ [0, 1], and 10 noise-free irregularly sampled observations (squares). Bottom: predicted value of the derivative of f (x) (bold solid line) from previous observations. True derivative (plain solid line) and 95% confidence intervals (dashed lines) are shown. Vertical bars indicate the positions of the observations.

Fig. 3 .

 3 Fig. 3. Approximation (bold solid line) of the sinc function (plain solid line) from noisy observations (squares) taking into account the derivative of the sinc function at positions indicated by the vertical bars.

  [1] M. Braci and S.Diop. On numerical algorithms for nonlinear estimation. In Proc. IEEE CDC 2003, pages 2896-2901, Hawaii, USA, 2003.

Fig. 4 .

 4 Fig. 4. Integration. Top: f (x) and 20 noisy irregularly sampled observations (squares). Bottom: approximation of the integral of f (x) (bold solid line) from previous observations knowing the value of the primitive at x 0 = 0.5 (indicated by a square). The true integral (plain solid line) and confidence intervals (dashed lines) are shown. Vertical bars indicate the positions of the observations.

K in (3) is generally a full rank matrix since covariances are most often positive definite functions. However, adapted solving techniques such as rank reduction must be used when the condition number of K is large, which may happen for instance when two observations are close in the space of factors.