Raphaël Robert-Bouchard
email: raphael_rb@hotmail.fr

Jérôme Dupire

Pierre Cubaud
email: cubaud@cnam.fr

Designing Indoor Tangible Games based on Fuzzy Localisation

Keywords: geolocation, zigbee, beacon, location-based, game design I

We present in this paper the development of an easy-to-adapt architecture for radio based, indoor geolocation; and the new opportunities it brings for game design. The weak precision of such systems, usually regarded as a problem, is considered here as a source of innovation for the design of pervasive games.

INTRODUCTION

The IEEE 802.15.4 standard allows for wireless connections of sensor networks at a range of several dozen meters. These networks were also used to support indoor geolocation attempts. But like other conventional radio protocols (WiFi, Bluetooth), it does not currently achieve high accuracy (i.e., close to the GPS outdoor's one). Usually considered as a weakness, this inaccuracy has been integrated into this project as the very component that could be used to bring innovation. Our aim was to design an indoor locationbased game, whose core gameplay would exploit this uncertainty inherent to the system. The proposed design space is between the board game and outdoor location based game. The advantages are the appropriation or re-appropriation of a known environment by turning it into a playground. Ultimately, we'll reach ease of deployment and the almost immediate launch of the game, since our environment will naturally be populated with communicating devices (smart objects, connected TVs, etc.) that may provide usable infrastructure to our system. After a review of previous works, we present the hardware architecture that we designed. Then we will describe the Scuba World game, developed with a specific configuration of the previous architecture. We then present the results of an evaluation experiment conducted for this game, before concluding.

II. RELATED WORK AND BACKGROUND

A. Location-based Game Design

Many outdoor location-based games have been studied in the scientific literature since the early 2000s. Can You See Me Know? [START_REF] Benford | Coping with uncertainty in a location-based game[END_REF] is a hunting game. Up to 15 players are pursued by three professional riders throughout the city. Hunted people "exist" only in a virtual world, whereas the runners, located by GPS, must physically go to the right location to catch them. Mogi [START_REF] Licope | Les usages émergents d'un jeu multijoueurs sur terminaux mobiles géolocalisés : les conséquences interactionnelles des "rencontres d'écran[END_REF] is a treasure search game. Placed by the game designers, treasures appear regularly in a virtual space, and the only way for the player to recover the treasure is to physically move through the city, located by a GPS system. Once in place, the player can collect the object. On his way, he will meet other treasure hunters with whom he can communicate, socialize and exchange treasures. Savannah [START_REF] Reid | Design for coincidence : Incorporating real world artifacts in location based games[END_REF] is the most cited educational location-based game on Google Scholar. Using a PDA, the game turns a school playground into a virtual African Savannah. Using sound and images from the real Savannah, the game brings players to act like a real lion pack to mark their territory and hunt prey. Ingress [START_REF] Hodson | Google's Ingress game is a gold mine for augmented reality[END_REF] is an augmented reality game created by Google. In this massively multiplayer game, the players are split into two camps and must conquer territories. These territories are pieces of the real environment and players must travel to strategic keypoints in the real world in order to fight for their control. Here geolocation is also done via GPS. Besides the fact that they use geolocation in their gameplay, these games share a common feature: they are all outside. The vast majority go through the GPS to allow accurate geolocation. However, there are lessons to be gained from these games, despite the difference in scale, as shown by a section on lessons learned from the analysis of geocaching done by Neustaedter et al. [START_REF] Neustaedter | Creating scalable locationbased games: lessons from Geocaching[END_REF]. Geocaching is an activity becoming increasingly popular over the years. Halfway between sport and gambling, geocaching is a treasure hunt in the world, consisting of 2 phases, theoretically independent. The cache phase consists of creating a "geocache" a box which can vary in form and content. Often (not always), it contains a notebook and pencil. This box is then hidden at a precise location, and whoever conceals it notes the precise GPS coordinates of the location. The searcher then goes on the geocaches website and publicly reveals the coordinates. The other phase of the game is the search. A person will try to find a geocache in coordinated it has recovered from the referral sites. The Geocache is usually well hidden. To go to the coordinates is often insufficient to find the object. Once the person has found it, she will write her name and the date in the associated diary and will put the object back in place to allow other players to do their own research. Neustaedter et al. [START_REF] Neustaedter | Creating scalable locationbased games: lessons from Geocaching[END_REF] chose to study this game because, after 14 years of existence, it is the only location-based game that managed to be sustainable and continue to see its number of users growing. Now, apart the Ingress beta, no other location-aware game has survived for more than 2 years. Having studied and even participated in the operation of geocaching for a year, they reached several interesting conclusions. First the authors have learned lessons in relation to games of the same type and same scale. Then, they extracted the general characteristics to differentiate types of location-based games: the type of usable game location, the complexity for the content of the game, the timing of the interaction between players and finally the type of entity that created the content of the game. These characteristics are quite close to those set out in [START_REF] Reid | Design for coincidence : Incorporating real world artifacts in location based games[END_REF] about location-based games. Although rare, some studies exist about the technical limitations of indoor location-based games, especially in terms of time and accuracy. In 2012, the most frenetic action game using geolocation, Sword fight, was created to repel these technical limitations [START_REF] Zhang | SwordFight: Enabling a New Class of Phone-to-Phone Action Games on Commodity Phones[END_REF]. There was a delay of a hundred millisecond minimum, giving the illusion that the system response was instantaneous. This system was accurate to the centimeter, even at a speed of 2 meters per second.

B. Uncertainty and Game Design

The object of research here is the spatial uncertainty as a gameplay catalyst. This uncertainty is the fact that the player has some doubts on certain borders in the game space that she's practicing. As far as we know, very little research has been published on this subject. Two papers have yet addressed the spatial uncertainty in the game, each approaching the concept in a radically different angle. In both cases, the games are some kind of game of cat and mouse using the GPS system and using the same standard: IEEE 802.11. In [START_REF] Benford | Coping with uncertainty in a location-based game[END_REF] Benford et al. consider the spatial uncertainty as a defect that must be managed. Uncertainty can be reduced by technology or by the game design, because some of its effects can be hidden to the player. On the contrary, her attention can be drawn on it, for her to take note of this uncertainty as something external to the game that she will have to deal with. In all cases, the aim is to manage the problem early, during the creation of the game and setting it up so that it interferes as little as possible in the users gaming experience. Chalmers et al. [START_REF] Chalmers | Seamful Games[END_REF] take a very different approach. GPS can sometimes have gray areas or "cold spots". In these places the player cannot be positioned using GPS. Addressing this phenomenon positively, the authors have experienced a game with two teams of players are made. Each team must pick up in the same neighborhood some coins scattered on a virtual map corresponding to the physical topology. Each player tries to steal the other players or to protect herself. Using a PDA, each player knows the position of the coins, of the other players, and the signal strength levels in the area. A possible strategy is then to lie in ambush, voluntarily in an area where GPS does not work, attempting to surprise a player and steal it. Although very short, this article has the merit of showing that a game can be developed based on the idea of using a technological "defect". We think that it can also be interesting for indoor games.

III. ARCHITECTURE

We describe here our generic architecture that allows for a fuzzy geolocation feature. The benefits of our proposal are a low cost and ease of implementation. This architecture consists of two objects: the beacons, on the one hand and the one or more mobile devices, on the other.

A. Geolocation methods

Range-based techniques typically exploit the Received Signal Strength Indication (RSSI) and the flight time while the range-free solutions estimate the location of sensor nodes by, either, exploiting the radio connectivity information among neighboring nodes, or exploiting the sensing capabilities that each sensor node possesses. Due to the distinct characteristics of these two approaches, we categorize the range-free localization schemes into: anchor-based schemes (which assume the presence of sensor nodes in the network that have knowledge about their location) and anchor-free schemes, which require no special sensor nodes for localization. Our proposal belongs to the category of range-free solution (anchor-based). In this context, several technologies are naturally candidates: WiFi, Bluetooth (BT), ZigBee (ZB). WiFi present best performances in terms of throughput and range. This is however not the criteria that interest us. Indeed, a low cost and a reduced range were privileged. The WiFi was thus not retained. The short range of other solutions mentioned above (BT, ZB) allows us to consider reduced radiation areas for each anchor, and be consistent with the scale of the areas identified through our system. To choose between these two technologies, we considered the energy consumption criterion, since all of the objects should be battery powered. We therefore choose the ZB solution. Our approach differs insofar as our anchor network is not spatially fixed by the constraints of associated sensor architecture. It will not support any useful communication. This allows us to position each anchor freely. Their positioning will only be driven by the needs identified in the game design (i.e., the deployment of gaming areas in the real environment).

B. Beacon

In our architecture, the anchor is equipped with a ZB module (XBee Series 1 from Digi) and is powered either by a battery or by a wall plug (Fig. 1). This alternative allows us to be completely independent of the presence of electrical outlets in the choice of the anchors location. The electronics consists of a simple voltage regulator with smoothing capacitors. We use only one functionality offered by the radio communication protocol: a particular node can send (broadcast) a request in order to make all the nodes within the emitting range respond (AT ND command: Node Discover). All nodes receiving this packet will always respond with a message containing some information about them, such as their MAC addresses or label. By periodically sending the AT ND command, one is thus able to know what nodes are within reach of the mobile node and to deduce relative positioning, as described in Fig. 2. Finally, two different types of modules, associated with two types of antennas have been used: the modules with a wire antenna (Fig. 2 top left) and the modules with a chip antenna (Fig. 2 top right). This difference allows us to adapt the different scopes of the two types of antenna to the context of the deployment, including the surface of the global game area and the game sub-areas to be delimited.

C. Mobile Device

The mobile device is the object that will be the interface between the game and the player. It is based on an Arduino UNO board, a ZB shield and a Xbee series 1 chip (from Digi). This device hosts a program on the one hand, triggering periodic emissions of the AT ND command and the treatment of the responses received (i.e., the comparison of the beacons that responded with a topographic model of the organization of these beacons) and, on the other hand, manages the user interface (UI). This UI can consist of LED's, or 7 segments LCD display, or any other device or combination of device manageable by the embedded microcontroller.

IV. THE SCUBA WORLD GAME DESIGN

A. The Game

The game can be played in a single room or inside an apartment / house. It is a two-player game. One player hides a few objects, "pearls" to find for the second player, and then places the beacons to define some "air bubbles". Pearls are small physical objects, unrelated to the electronic system and there are 6 of them to find during a game session. The air bubbles are invisible areas allowing the second player (called the fisherman) to breathe. On a second time, the purpose of the fisherman is to find the pearls scattered in the play area, while taking care not to choke. He has indeed a limited air supply when searching the room. To restore his air supply, he must return regularly in bubbles located in the room. The bubbles are areas of superposition of the radiation range of the 3 beacons. When the fisherman is in a bubble, he can stay as long he wants, and his air gauge fills up in 3 seconds. When the fisherman is out of this air bubble, he is considered snorkeling and air supply decreases. A full air supply makes the player autonomous for 1 minute and 35 seconds. After this duration, without any resupply, the player dies (game over). The metaphor of fishing and snorkeling was chosen for several reasons. First of all, it is a simple metaphor that speaks to everybody in terms of opportunities and constraints. It has also been long used in video games. In addition it helps the player to conceptualize this 3D space, the more or less shaped bubbles, with which he/she will interact regularly. And finally because the radio waves tend to be fluctuating [START_REF] Ganesan | Complex behavior at scale: An experimental study of lowpower wireless sensor networks[END_REF], as could be an air bubble lightly tossed by seabed.

B. Hardware

The object that the fisherman player holds during the game is a plastic box of 8 cm wide,15 cm long, 6 cm high (Fig. 3) containing all the electronics. The upper side is open, allowing the user to see the color diodes (LED). There are 4 LED: a yellow, two red and one green. The yellow LED serves as an alarm: when there are only 15 seconds left for the player to breath, it turns on. When the player has totally exhausted her air supply, it flashes, signifying the end of the game. The three other diodes are used for the identification of air bubbles. Each beacon is associated with a diode. When the mobile device is within the range of a beacon, the corresponding LED turns on. When the three LEDs are switched on, the player is in a bubble and its reserve fills up. There is no display for the remaining air. The player must estimate by herself the time she can remains away from the air bubble. The solid yellow LED is the only (and late) warning for her to know.

V. EXPERIMENT

The game Scuba World was tested in order to verify if there was some fun playing with it, and more generaly with rules based on spatial uncertainty.

A. Subjects, settings and protocol

Twenty unpaid adults participated to the experiment (mean age: 36.7 years, standard deviation σ=11.8), including 11 men and 9 women. All were students or staff from our university. Most of them don't work in a field related to game and none of them played a location-based game before the experiment. The tests were conducted in a (messy) laboratory office that measures 3m wide and 5m long. The briefing prior to the game session took place in a nearby office. The three beacons and the pearls were always placed in the same locations, in order to present the same initial situation to all participants. The beacons' position lead to the creation of an overlapping area in the center of the room. This configuration should cause the player to explore the periphery of the room, forcing him to get out of this bubble regularly. The pearls were hidden differently (from easy to difficult) allowing the player to discover them during the 10 minutes experience. Each participant played the game alone. The experimenter simulated the second player, by placing in advance the tags and pearls. The experimenter explained the game context to the participants and the rules: duration of apnea, air bubble system and the LED meanings. It was specified, since the prototype is radio-based, that the body can act as a screen and therefore sometimes a slight displacement of the player's body could have consequences on radio communication between beacons and the mobile device. The experimenter placed the player with the mobile device in the main air bubble. Then the game began. After 10 minutes, or if the player "died" before this time, the game ends. The player was then asked to answer a short questionnaire and then performed a self-confrontation, during the presentation of the session's recorded video.

B. Results

The purpose of the questionnaire was to have a feedback regarding the gaming experience. Desire to replay the game was graded on average 7.0 (σ=1.7) and game fun was graded 6.8 (σ=1.5), on a scale ranging from 1 (weakest) to 10 (strongest). Only 3 participants gave a rating of 4 to one of these two questions, and no lower rating was given. Fourteen participants wanted to place the pearls for the next player. This would tend to indicate that the game was generally appreciated by the participants. Eight participants reported they would recommend the game to everyone, and 8 specifically to children. Eight participants slightly underestimated and 5 participants slightly overestimated their play time. During the play sessions, participants were silent most of the time. However, during self-confrontation, most participants spoke spontaneously. The first topic discussed was often the air bubble strategy. For the 20 participants, many tactics have emerged. Some players have found those bubbles too small, others were more confident in their exploration once they knew for sure how to find a bubble. Four players preferred to neglect other potential bubbles to focus on one. One of them argued that due to the relatively small size of the room, he did not have to bother with another bubble than the one he knew. Conversely, 3 players have specifically sought several points of folds, often taking one that seemed closest to them. The others were more opportunistic, seeking a bubble if they thought they could find one near, if not back to the initial location. Overall, it seemed easy for the player to conceptualize this virtual space of air bubbles, or at least the rules to play with. The link between this virtual space and the player was the mobile device. This object has a very special meaning for the participants and it appeared difficult to most of them to drop it when looking for the pearls. Only 2 participants left the device during their session. Participants also noted the system stability problems (i.e., the fuzzy part). Because of the anisotropic behavior of waves, the areas covered by the beacons could fluctuate, which prompted most participants to be careful. The limits of the air supply have been reached only by one player.

VI. CONCLUSION AND FUTURE WORK

Uncertainty is at the core of any game. This project shows that a new form of uncertainty -spatial uncertainty -can be exploited when designing games. The indoor location system presented here can be developed at a low cost and in a very short time, opening up new opportunities for game designers. The design, development and testing of the game Scuba World demonstrate the technical feasibility and the innovative nature of this type of experience for the players. Today, radio networks constantly run around us, in our homes or in public buildings. By considering networks irregularities as a design option and combining it to connected objects, one can open a whole new area for game creation and play. For enhenced game experience, we now need to imagine objects, like pawns or even bigger toys, and design them in an adaptable way.

Fig. 1 .

 1 Fig. 1. ZigBee modules (XBee series 1 from Digi) wire antenna (top left) and chip antenna (top right) and PCB with power supply: wall plug revamped from a mobile phone (bottom left) and battery powered (bottom right)

Fig. 2 .

 2 Fig. 2. Range-free localisation (with beacons)

Fig. 3 .

 3 Fig. 3. The mobile device used in the Scuba World game


Elodie Choque

Vanessa Durrieu
email: vanessa.durrieu@ensiacet.fr

Isabelle Alric

José Raynal

Florence Mathieu

Impact of Spray-Drying on Biological Properties of Chitosan Matrices Supplemented with Antioxidant Fungal Extracts for Wine Applications

Keywords: Fungal metabolites, Chitosan, Microencapsulation, Spray-drying, Antioxidant Abbreviations ABTS

Black aspergilli produce many bioactive compounds: enzymes, organic acids, and secondary metabolites. One such fungus, Aspergillus tubingensis G131, isolated from French Mediterranean vineyards, produces secondary metabolites with antioxidant properties that can be extracted with ethanol. In this study, crude antioxidant extracts obtained from A. tubingensis G131 cultures were encapsulated with two types of chitosan matrix. Spray-drying was used to obtain dried particles from a dispersion of fungal crude extracts in a solution of the coating agent chitosan. This process appeared to be an efficient method for obtaining a dry extract with antioxidant activity. Three types of fungal extracts, with different antioxidant capacities, were produced: two different concentrations of crude extract and a semi-purified extract. In this study, the chitosan matrices for encapsulation were chosen on the basis of their antimicrobial activities for wine applications. Classical low molecular weight chitosan was compared with NoBrett Inside ® which is already used to prevent the development of Brettanomyces spp. in wine. The objective of this study was to confirm that both antioxidant (fungal extract) and antimicrobial (chitosan) properties were preserved after spray-drying. The combination of these two properties and the powder formulation of this entirely natural product would make it a good alternative to chemicals, such as sulfites, in the food and wine industries.

Introduction

One of the key challenges in the wine industry is controlling wine stability during production, aging, and after bottling. During wine aging, the principal objectives of the winemaker are preventing wine spoilage and stabilizing wine color by limiting oxidation. Sulfites are added at various steps in the winemaking process to achieve these objectives. However, it has been suggested that sulfites may have adverse effects on human health, such as pseudoallergies, and winemakers are therefore now trying to limit the use of sulfites in the winemaking process [START_REF] Mf | Review of sulfite sensitivity[END_REF][START_REF] Armentia | Adverse reactions to wine: think outside the bottle[END_REF][START_REF] Chagas | Assessment of potential side effects of common fining agents used for white wine protein stabilization[END_REF]. Indeed, wine consumers prefer high-quality less-processed wines [START_REF] Lockshin | Consumer behaviour for wine 2.0: a review since 2003 and future directions[END_REF][START_REF] Costanigro | The wine headache: consumer perceptions of sulfites and willingness to pay for nonsulfited wines[END_REF]. For winemakers to meet this demand, the development of new preservative agents or stabilization techniques is required. Natural products are currently defined as metabolites produced by living organisms and/ or naturally occurring in nature. Such metabolites from various organisms are capable of preventing microbial spoilage, and thus are a suitable alternative to the use of synthetic products [START_REF] Pozo-Bayón | Wine features related to safety and consumer health: an integrated perspective[END_REF]. Among them, chitosan meets these requirements, and the European Union (EU) and International Organization of Vine and Wine (OIV) have already approved its use for wine applications. The OIV/ OENO 338A/2009 (International Organization of Vine and Wine (OIV) 2009) resolution added the use of chitosan in winemaking to the International Code of Oenological Practice, specifying that the maximum dose of chitosan used to reduce ochratoxin A (OTA) levels must not exceed 500 g/hL. Chitosan is a linear polysaccharide composed of two repeating units (d-glucosamine (GlcN) and N-acetyld-glucosamine (GLcNAc)) randomly distributed along the polymer chain and linked by β(1-4)-bonds. Various activities have been demonstrated for different types of chitosan: as a fining and protein stabilizing agent, a preservative and an antimicrobial agent [START_REF] Chagas | Assessment of potential side effects of common fining agents used for white wine protein stabilization[END_REF][START_REF] Ferreira | The antimicrobial action of chitosan against the wine spoilage yeast Brettanomyces/Dekkera[END_REF][START_REF] Nunes | Chitosan-genipin film, a sustainable methodology for wine preservation[END_REF]. Chitosan has antimicrobial effects against lactic acid bacteria, acetic acid bacteria, fungi, and undesirable yeasts, such as Brettanomyces sp., during aging, but it is permissive for the growth of Saccharomyces species [START_REF] Ferreira | The antimicrobial action of chitosan against the wine spoilage yeast Brettanomyces/Dekkera[END_REF][START_REF] Nardi | Evaluating the impact of a fungal-origin chitosan preparation on Brettanomyces bruxellensis in the context of wine aging[END_REF][START_REF] Petrova | Effectiveness of chitosan preparations against Brettanomyces bruxellensis grown in culture media and red wines[END_REF][START_REF] Taillandier | Effect of a fungal chitosan preparation on Brettanomyces bruxellensis, a wine contaminant[END_REF][START_REF] Valera | Effect of chitosan and SO2 on viability of Acetobacter strains in wine[END_REF].

For wine aging applications, it would be useful to couple the antimicrobial properties of chitosan with antioxidant activity to stabilize wine color. In this study, an ethanol extract of Aspergillus tubingensis G131 was chosen. This black Aspergilli strain was isolated during a survey on the occurrence of OTA producers on grapes from various French vineyards [START_REF] Bouras | Occurrence of naphtho-gamma-pyrones-and ochratoxin A-producing fungi in French grapes and characterization of new naphtho-gamma-pyrone polyketide (aurasperone G) isolated from Aspergillus niger C-433[END_REF]. A. tubingensis G131, isolated from a French Mediterranean Vineyard, does not produce OTA [START_REF] Choque | Wholegenome sequencing of Aspergillus tubingensis G131 and overview of its secondary metabolism potential[END_REF]. An ethanol extract of a sporulated mycelial cake from this strain was found to be dark gold in color, due to the presence of melanin, and has interesting antioxidant properties. Equivalent extracts from other black Aspergilli strains have already been reported to have antioxidant activity. For example, a methanol extract from an A. niger strain was found to have beneficial effects on rat growth and hepatoprotective activity [START_REF] Rabache | Physiologic effects of the Aspergillus niger pigments. 2-Antioxygen property of the naphthogamma-pyrones estimated in the rat[END_REF]. Another organic extract of A. niger was shown to prevent lard oxidation [16]. These findings suggest that organic extracts of non-mycotoxigenic black Aspergilli are of potential interest for their use as preservatives in the food industry.

A suitable strategy for efficiently combining the antimicrobial properties of chitosan with the antioxidant activity of the chosen fungal extract was therefore required. Microencapsulation appeared to be an effective approach.

Interest in antioxidant encapsulation has recently increased, particularly in the food industry. This process makes it possible to increase the stability of bioactive compounds during processing and storage and prevents undesirable interactions with the other components of a formulation [START_REF] Mozafari | Recent trends in the lipid-based nanoencapsulation of antioxidants and their role in foods[END_REF][START_REF] Abbas | Comparison of major biocontrol strains of non-aflatoxigenic Aspergillus flavus for the reduction of aflatoxins and cyclopiazonic acid in maize[END_REF][START_REF] Munin | Encapsulation of natural polyphenolic compounds: a review[END_REF]. The protective mechanism of microencapsulation involves the formation of a membrane wall to enclose droplets or particles of the encapsulated ingredient, thereby protecting it, improving its stability, and making it possible to convert liquid preparations into powders, or to improve their dispersibility in water [START_REF] Gharsallaoui | Applications of spray-drying in microencapsulation of food ingredients: an overview[END_REF].

Various microencapsulation technologies, such as spraydrying, prilling, coacervation, extrusion, and in situ polymerization, are available. Spray-drying is one of the simplest and most widely used processes due to its relatively low cost and efficiency and the availability of appropriate industrial equipment [START_REF] Estevinho | Microencapsulation with chitosan by spray drying for industry applications-a review[END_REF]. Antioxidant encapsulation by the spraydrying method has already been described for various wall materials, such as plant proteins [START_REF] Pereira | Legumes seeds protein isolates in the production of ascorbic acid microparticles[END_REF][START_REF] Robert | Encapsulation of polyphenols and anthocyanins from pomegranate (Punica granatum) by spray drying[END_REF][START_REF] Nesterenko | The effect of vegetable protein modifications on the microencapsulation process[END_REF], milk proteins [START_REF] Rocha-Guzmán | Effect of high-pressure homogenization on the physical and antioxidant properties of Quercus resinosa infusions encapsulated by spray-drying[END_REF], various carbohydrates, including maltodextrins [START_REF] Ersus | Microencapsulation of anthocyanin pigments of black carrot (Daucus carota L.) by spray drier[END_REF][START_REF] Laine | Storage stability of microencapsulated cloudberry (Rubus chamaemorus) Phenolics[END_REF][START_REF] Saénz | Microencapsulation by spray drying of bioactive compounds from cactus pear (Opuntia ficus-indica)[END_REF][START_REF] Faria | Characterization and singlet oxygen quenching capacity of spray-dried microcapsules of edible biopolymers containing antioxidant molecules[END_REF][START_REF] Constanza | Value-added processing of peanut skins: antioxidant capacity, total phenolics, and procyanidin content of spray-dried extracts[END_REF], and blends of maltodextrin/gum arabic [START_REF] Zhang | Procyanidins: extraction and micro-encapsulation[END_REF], or maltodextrin/Kcarrageenan [START_REF] Krishnaiah | Microencapsulation of Morinda citrifolia L. extract by spray-drying[END_REF], and for chitosan [START_REF] Desai | Effect of manufacturing parameters on the characteristics of vitamin C encapsulated tripolyphosphatechitosan microspheres prepared by spray-drying[END_REF][START_REF] Kosaraju | Preparation and characterisation of chitosan microspheres for antioxidant delivery[END_REF][START_REF] Kumar | Evaluation of chitosan as a wall material for microencapsulation of squalene by spray drying: characterization and oxidative stability studies[END_REF].

The objectives of this work were to study the antimicrobial and antioxidant capacities of chitosan/fungal extract microparticles obtained by spray-drying for potential applications in the food industry, including the maintenance of wine stability. Indeed, the combination of these two types of activity should prevent both the spoilage and oxidation of the wine. Microparticles of this type will not only combine two biological properties, but also will incorporate into food in solid form, without dissolution, thereby preventing alterations to organoleptic qualities, such as color [START_REF] Kumar | Evaluation of chitosan as a wall material for microencapsulation of squalene by spray drying: characterization and oxidative stability studies[END_REF]. The chitosan/fungal extract microparticles were subjected to physicochemical (particle size, morphology and moisture) and biological (impact on yeasts growth, antioxidant capacity) characterization.

Materials and Methods

Chemicals

Chitosan 1 (NBI)

This chitosan preparation was a powder with particles of less than 50 µm in diameter, and a product of the deacetylation of chitin extracted from A. niger supplied by KitoZyme company (Herstal, Belgium): KiOfine B ® or No Brett Inside ® (commercially available products). This product was less than 30% acetylated and the viscosity of a 1% solution in acetic acid was about 4 mPa.s.

Chitosan 2 (LMW)

The chitosan preparation was obtained from Glentham Life Sciences (Wiltshire, United Kingdom) and used without purification. It had a very low molecular weight (average molecular weight: 30,000 g mol -1 ) and was less than 10% acetylated.

These two chitosans had low molecular weights and acetylation levels, features known to be associated with the antibacterial activity of chitosan [START_REF] Kong | Antimicrobial properties of chitosan and mode of action: a state of the art review[END_REF].

The anhydrous ethanol (Fisher ® ), acetic acid, and other chemicals (Sigma-Aldrich ® ) used were of analytical grade.

Fungal Extracts

Production of the Crude Fungal Extract

Czapek Yeast Broth (CYB-30 g/L Saccharose, 5 g/L Yeast Extract, 2 g/L NaNO 3 , 0.25 g/L KCl, 0.25 g/L MgSO 4 .7H 2 O, 0.005 g/L FeSO 4 .7H 2 O, 0.5 g/L K 2 HPO 4 , 0.001 g/L ZnSO 4 .7H 2 O and 0.0005 g/L CuSO 4 .7H 2 O) was inoculated with 10 7 spores of A. tubingensis G131, and the resulting fungal cultures were incubated for 7 days at 30 °C. The mycelial cake was then separated from the culture medium. The mycelial cake only was then covered with anhydrous ethanol. The mycelium/ethanol mixture was incubated for 20 min at room temperature (RT) and then subjected to sonication for 20 min at 50 Hz. The sonicated mycelium/ethanol mixture was filtered once through 113 V grade Whatman filter paper. The filtered extract obtained was stored in the dark at 4 °C until required, and is referred here as the crude extract (CE).

Concentration by Rotary Evaporation

The crude extract was concentrated (volume to volume) with a rotary evaporator. The conditions for evaporation were as follows : 60 °C, rotation at 150 rpm, and uncontrolled vacuum and cooling with distilled tap water at 15 °C. Crude extracts (CEs) were named on the basis of their concentration yield. In this study, two concentrated extracts were used: CE*3 (3 V/V: 3 volumes were evaporated in 1 remaining volume) and CE*6 (6 V/V: 6 volumes were evaporated in 1 remaining volume).

Solid-Phase Extraction

Solid-phase extraction (SPE) was performed on a C18 Hypersep column (Thermo Fischer Scientific)-Bed weight: 5 g; Sorbent: Hypersep C18; Particle Size: 40-60 μm; and column capacity: 25 mL. Columns were equilibrated with five volumes of acetonitrile and three volumes of MilliQ water. 5 mL of CE*3 was deposited on the column and washed with two volumes of water/acetonitrile (70:30 mixture). Finally, a semi-purified fraction was eluted in one volume of anhydrous ethanol. This semi-purified SPE product was then concentrated by rotary evaporation (2 V/V: 2 volumes were evaporated in 1 remaining volume) and named (PE*2).

Estimation of the Dry Weight of Fungal Extracts

The dry weight of all the fungal extracts obtained was determined with an EM120-HR Moisture Analyzer (Precisa Gravimetrics AG, Dietikon, Switzerland) by drying to constant weight at 70 °C.

HPLC Characterisation of the Fungal Extracts

HPLC analyses were performed as described in Choque et al. [START_REF] Choque | Wholegenome sequencing of Aspergillus tubingensis G131 and overview of its secondary metabolism potential[END_REF].

Microencapsulation Methodology

Preparation of the Solution

Solution of 1% (w/w) chitosan in 1% (v/v) aqueous acetic acid was prepared with mechanical stirring (500 rpm) at RT. Fungal extracts in ethanol (CE*3, CE*6, and PE*2) were then added to the chitosan solution, 80% of chitosan solution and 20% of fungal extract, with mechanical stirring (500 rpm) to mix them before the spray-drying process.

Preparation of Spray-Dried Microparticles

Obtained chitosan/fungal extract solutions were spray-dried in a Mini Spray Dryer B-290 (Büchi, Flawil, Switzerland) in the open mode under the following conditions: inlet air temperature at 120 ± 4 °C and outlet temperature at 75 ± 4 °C, drying air flow rate of 470 L/h, liquid feed flow rate varying from 0.33 to 0.48 L/h and 100% aspiration. Microparticles were collected from the container, sealed hermetically in an opaque packaging, and stored at room temperature.

Spray-drying yield was calculated as follows:

where M P is the dry mass of the microparticles collected and M RM is the initial dry mass of the solids in the solution of chitosan and fungal extract.

Physicochemical Characterization of Microencapsulated Matrices

Moisture Content

The moisture content of all the powders obtained was determined with an EM120-HR Moisture Analyzer (Precisa Gravimetrics AG, Dietikon, Swiss) by drying to constant weight at 105 °C.

Spraydrying yield (%) = M P ∕M RM × 100

Microparticles Size Distribution

The particle size distribution of dried microparticles was determined from the scattering pattern of the transverse laser light with a Scirocco 2000 instrument (Malvern Instruments, Worcestershire, UK). Mean particle diameter ranged from 0.2 to 2000 µm. The following parameters were used: refractive index of 1.52, pressure of air of dispersion of 4 bars, degree of vibration of 70%. The volume-based particle diameter (D 43 or D v ) was calculated as the mean of three measurements per sample.

Microparticle Microstructure

The morphology of microparticles was examined by scanning electron microscopy (SEM). The particles were deposited on conductive double-faced adhesive tape and sputtercoated with silver. The microparticles were frozen in liquid nitrogen and broken up in a mortar, for examination of their internal structure. SEM observations were performed with a LEO435VP scanning electron microscope (LEO Electron Microscopy Ltd., Cambridge, UK) at 8 kV.

Microencapsulated Matrix Suspension

Each microencapsulated matrix was dissolved in acidified MilliQ water (pH 3.0, acetic acid), 12% ethanol (pH3.0, acetic acid) or 100% ethanol to give a 3 g/L (dry weight) solution. The weighted powder was incubated in the appropriate solvent for 24 h at RT and then stored at 4 °C. Each experiment was run in triplicate.

Effect of Microencapsulated Matrices on Enological Yeasts Growth

The effects of spray-drying chitosan and its combination with a fungal antioxidant on growth of two enological strains-Saccharomyces cerevisiae and Brettanomyces bruxellensis-were assessed. Yeasts were precultured in YEPD (10 g/L yeast extract, 20 g/L meat peptone, 20 g/L d-glucose) for 16 h at 30 °C. Two concentrations of chitosan were used, based on the results of Taillandier et al. [START_REF] Taillandier | Effect of a fungal chitosan preparation on Brettanomyces bruxellensis, a wine contaminant[END_REF], describing the mode of action of NBI chitosan on B. bruxellensis inhibition: 40 mg/L and 400 mg/L [START_REF] Taillandier | Effect of a fungal chitosan preparation on Brettanomyces bruxellensis, a wine contaminant[END_REF]. These concentrations were chosen as they correspond to the ends of the standard range of NBI showing an action on B. bruxellensis. The controls were culture media without any supplementation, and culture media supplemented with low molecular weight chitosan or with No Brett Inside chitosan that had not been spray-dried. The supplements to the YEPD medium were added 16 h before inoculation, and the medium was allowed to equilibrate at 20 °C. The YEPD medium was then inoculated to an OD 600nm of 0.1 with yeast preculture. The growth of the yeasts was monitored by spectrophotometry at λ = 600 nm,at times 0, 8 h, 24 h, and 48 h. OD was also estimated before inoculation to ensure the correct estimation of yeast growth. The experiments were carried out in duplicate, at 20 °C, and flasks were stirred just after inoculation and immediately before sampling.

Trolox Equivalent Antioxidant Capacity

The antioxidant activity of the fungal extracts was determined in the TEAC test, according to a protocol adapted from a previous study [START_REF] Re | Antioxidant activity applying an improved ABTS radical cation decolorization assay[END_REF]. ABTS + was produced by the reaction of 7 mM of ABTS (2,2′-azino-bis(3-ethylbenzothiazoline-6-sulfonic acid)) (Sigma-Aldrich, > 99% HPLC grade) with 2.5 mM of K 2 S 2 O 8 (sodium persulfate) (Sigma-Aldrich, > 99% ACS reagent) in distilled water for 16 h at RT, in the dark. The concentration of ABTS + was then adjusted by dilution to an absorbance of 0.7 ± 0.02 (at 734 nm) on a spectrophotometer (Anthelie advanced, SECONAM). A stock solution at 150 mM was prepared for Trolox (Sigma-Aldrich). A 100 μL aliquot of the desired sample, at the appropriate dilution, or the Trolox standard (final concentration of 2.5, 5, 10 or 15 μM) was added to 900 μLl of the diluted ABTS + . Absorbance was measured at 734 nm after incubation in the dark, at RT, for 6 min. Each sample was tested three times in duplicate. TEAC value was estimated in mM as described by Re et al. [START_REF] Re | Antioxidant activity applying an improved ABTS radical cation decolorization assay[END_REF].

Classification of Additive, Synergistic or Antagonistic Effects

As described by Ribeiro et al. [START_REF] Ribeiro | Spray-drying microencapsulation of synergistic antioxidant mushroom extracts and their use as functional food ingredients[END_REF], theoretical values for the antioxidant activity of the assayed extract mixture were calculated as the weighted mean of the experimentally determined TEAC values of the individual extracts. For example, for CE*3, the estimated TEAC was based on the dilution factor applied to both the chitosan and fungal extracts in the microencapsulated matrix suspension. It was therefore calculated as follows:

where M fe-c is the dry mass of fungal extract in the microencapsulated matrix suspension; M fe-i is the dry mass of the initial fungal extract; M c-c is the dry mass of chitosan in the microencapsulated matrix suspension; and M c-i is the dry mass of the initial chitosan solution.

TEAC est = TEAC fungal extract × M fe-c ∕ M fe-i + TEAC chitosan × M c-c ∕ M c-i
Effects were classified as additive (AE), synergistic (SE) or antagonistic (AN) as described by Ribeiro et al. [START_REF] Ribeiro | Spray-drying microencapsulation of synergistic antioxidant mushroom extracts and their use as functional food ingredients[END_REF]. AE: theoretical and experimental values differed by less than 5%. SE: experimental values were at least 5% higher than theoretical values. AN: experimental values were at least 5% lower than theoretical values. Lower TEAC values are associated with lower levels of antioxidant activity.

Statistical Analyses

Data are presented as means ± SD for triplicate experiments. Statistical analyses were performed with jamovi (jamovi project (2017), jamovi (Version 0.8)). One-way and two-way ANOVA tests were used to assess the significance of differences for each variable (p < 0.05). Tukey post hoc test was applied to determine the significance of differences between conditions.

Results

Microencapsulation of Fungal Extracts

Spray-drying was used to prepare chitosan and ethanol extract-loaded chitosan microparticles. Several parameters can affect the characteristics of the obtained microspheres, including polymer concentration, solvent and assay conditions (inlet temperature, spray rate of feed, etc.) [START_REF] Guerrero | Characterization and in vivo evaluation of ketotifen-loaded chitosan microspheres[END_REF].

For these experiments, chitosan concentration was fixed at 1% (w/w) in 1% (v/v) aqueous acetic acid to ensure complete dissolution of the chitosan and solutions with a suitable viscosity for spraying. Ethanol extracts (CE*3, CE*6, PE*2, 100 mL in each case) were then added to give an 80/20 water/ethanol ratio, which maintained suitable levels of solubility of both chitosan and the extracts in the solution, preventing obstruction of the nozzle during the spray-drying process and increasing encapsulation efficiency [START_REF] Estevinho | Microencapsulation with chitosan by spray drying for industry applications-a review[END_REF][START_REF] Drosou | Encapsulation of bioactive compounds through electrospinning/electrospraying and spray drying: a comparative assessment of food-related applications[END_REF].

For the spray-drying parameters, the chosen inlet temperature of 120 °C was sufficient to evaporate both solvents and acetic acid (which has a boiling point is 118 °C), and the feed spray rate was adapted to maintain the outlet temperature around 75 °C, ensuring that the particle dried correctly with no alteration of the properties of the extracts. Appropriate adjustment between the inlet temperature and the feed flow is essential to ensure the evaporation of maximum of the liquid sprayed before the droplets meet the drying chamber walls, to optimize the production yield [START_REF] Drosou | Encapsulation of bioactive compounds through electrospinning/electrospraying and spray drying: a comparative assessment of food-related applications[END_REF].

The incorporation of ethanol extract, even with a high proportion of fungal material (PE*2: 1.39 ± 0.03 g/L; CE*3: 29.4 ± 0.7 g/L and CE*6: 55.1 ± 1.2 g/L (dry weight)), made it possible to obtain production yields of around 80% (Table 1). These results were highly satisfactory and significantly better than published production yields for the use of chitosan as the encapsulation material, which vary from 23 to 73%, and are generally around 50% [START_REF] Estevinho | Microencapsulation with chitosan by spray drying for industry applications-a review[END_REF][START_REF] Sansone | Technological properties and enhancement of antifungal activity of a Paeonia rockii extract encapsulated in a chitosan-based matrix[END_REF].

Mean particle size ranged from 1.17 to 4.35 µm, consistent with published values for chitosan and loaded chitosan microparticles, which are generally between 1 and 5 µm [START_REF] Estevinho | Microencapsulation with chitosan by spray drying for industry applications-a review[END_REF]. The results of the statistical analysis suggested that mean particle size depends on the type of fungal extract used for supplementation, but not of the type of chitosan. These values were confirmed by SEM observations (Fig. 1). This could be linked to the composition of the extract. HPLC analysis of the extract were performed as described in Choque et al. [START_REF] Choque | Wholegenome sequencing of Aspergillus tubingensis G131 and overview of its secondary metabolism potential[END_REF]. Results show that fungal extracts are composed of two main compounds: melanin (black pigment), and Naphtho-Gamma-Pyrones (NGPs). The proportion of NGPs of each extract was calculated with chromatogram analysis. The proportions were 94.4 ± 2.3% of NGPs The more the NGPs are present, the less the melanin is. Thus, it appears that the particle size of the chitosan microparticles is linked to the proportion of melanin in the fungal extract. The more the melanin is present, the higher is the particle size. The microparticles produced were spherical and of regular shape, but with a slightly cracked surface. No significant difference was observed between chitosan and fungal extracts, other than for the mean size of the microparticles produced. Indeed, initial dry weight differed between fungal ethanol extracts and higher initial dry weights of the spraydried sample were associated with larger mean microparticle size. This reflects a higher concentration of matter in the sprayed droplets.

Effect of Microencapsulated Matrices on the Growth of Enological Yeasts

The approach used here aimed to combine the antimicrobial properties of chitosan with the antioxidant properties of the fungal extract. No Brett Inside ® is a commercial chitosan, obtained from the fungus Aspergillus niger, which is already on sale for limiting wine spoilage due to Brettanomyces bruxellensis [START_REF] Taillandier | Effect of a fungal chitosan preparation on Brettanomyces bruxellensis, a wine contaminant[END_REF]. This micro-organism is recognized as a major source of contamination in wine and as the principal cause of a "horse sweat" flavor [START_REF] Kheir | Impact of volatile phenols and their precursors on wine quality and control measures of Brettanomyces yeasts[END_REF]. Brettanomyces bruxellensis can develop in both musts and wine, throughout the entire winemaking process. As described by Taillandier et al., No Brett Inside ® chitosan (NBI) seemed to be generally more effective than low molecular weight chitosan (LMW) in the winemaking context. NBI seemed to have less impact on the growth of S. cerevisiae, which is required for alcoholic fermentation, and a stronger impact on the growth of B. bruxellensis, the chief contaminant, than LMW (Fig. 2).

Conversely, the spray-drying technique has no impact on the slowing of B. bruxellensis development (Fig. 2b), whereas antioxidant supplementation of the chitosan matrix considerably decreased antimicrobial efficacy, at least for CE*3 and CE*6 supplementation. The addition of a too large quantity of antioxidant might, therefore, limit the antimicrobial efficacy of chitosan or strong presence of melanin in those matrices could limit the mode of action of chitosan.

Antioxidant Activity of Microencapsulated Matrices

Regarding the antioxidant capacity of the fungal extracts before microencapsulation, a relation can be found between the quantity of NGPs present in an extract and its antioxidant capacity. Indeed, an antioxidant capacity between 1.2-and 1.6-mM TEAC could be found per gram of NGPs present in the extract (not statistically different). The antioxidant capacity of NGPs have previously been described confirming this hypothesis [START_REF] Rabache | Physiologic effects of the Aspergillus niger pigments. 2-Antioxygen property of the naphthogamma-pyrones estimated in the rat[END_REF]16,[START_REF] Choque | Fungal naphtho-γ-pyrones-secondary metabolites of industrial interest[END_REF].

Three types of microencapsulated matrix suspensions were tested: anhydrous ethanol (the solvent used to obtain the antioxidant fungal extract); acidified ethanol 12% (pH 3) to represent conditions equivalent to those found in wine; and acidified water (pH 3) to mimic grape must. Antioxidant activity was limited in anhydrous ethanol suspensions (Table 2) possibly due to the very low solubility of chitosan in organic solvents. In such conditions, the chitosan envelope would prevent the release of the antioxidant molecules into the medium, thereby limiting antioxidant activity. Alternatively, there may be an antagonistic effect between chitosan and the antioxidant molecule of the fungal extract in this type of solvent.

By contrast, in the suspensions simulating wine (12% ethanol) and must (acidified water), a synergistic effect on antioxidant properties was observed (Table 2).

Discussion

This study shows that the supplementation of chitosan matrices, already commercially available, with an antioxidant extract, produced from a filamentous fungus isolated from a Mediterranean grapevine, could extend the use of this additive to the entire winemaking process as a sulfites alternative. As discussed in the introduction, sulfites are added during the winemaking process for two reasons: to limit the growth of undesirable microorganisms (antimicrobial properties) and to reduce the oxidation of grapes to preserve wine color by preventing juice browning (antioxidant properties). The greatest risk in must is enzymatic browning [START_REF] Wu | Glutathione suppresses the enzymatic and nonenzymatic browning in grape juice[END_REF]. Alternatives to sulfites have already been proposed for preventing browning at this stage: salicylic acid during harvesting, glutathione or patatin, and a fining agent limiting browning [START_REF] Wu | Glutathione suppresses the enzymatic and nonenzymatic browning in grape juice[END_REF][START_REF] Champa | Preharvest salicylic acid treatments to improve quality and postharvest life of table grapes (Vitis vinifera L.) cv flame seedless[END_REF][START_REF] Gambuti | Performance of a protein extracted from potatoes for fining of white musts[END_REF]. In this study, an antioxidant extract produced from a fungus isolated from Mediterranean grapevines was selected as a natural product to supplement both LMW and NBI for use in the winemaking process. The two natural active ingredients were combined by spray-drying microencapsulation to generate a powder that is easy to use and to store. The spray-drying technique seems to decrease the negative impact of chitosan on S. cerevisiae growth, with synergistic effects, increasing the antioxidant capacity of the mixture. Indeed, the growth of S. cerevisiae was significantly slowed in the presence of the commercial chitosan at the concentration recommended for limiting the growth of B. bruxellensis [START_REF] Taillandier | Effect of a fungal chitosan preparation on Brettanomyces bruxellensis, a wine contaminant[END_REF]. This slowing of growth has been reported to be due to a phenomenon of yeast absorption by the polymer [START_REF] Zakrzewska | Cellular processes and pathways that protect Saccharomyces cerevisiae cells against the plasma membrane-perturbing compound chitosan[END_REF]. This slower growth of S. cerevisiae may slow the alcoholic fermentation, prolonging the winemaking process, if chitosan is added to the must stage. It limits the use of the commercial NBI product for the treatment of wines before bottling. Interestingly, the effect of chitosan on S. cerevisiae growth seems to be reduced by spray-drying and/or antioxidant supplementation. This could allow a wider use of chitosan in winemaking processes, including its addition to the must before the initiation of alcoholic fermentation.

The experimental antioxidant capacity of most powders was two to three times higher than the estimated value for a combination of the antioxidant activities of chitosan and the chosen fungal extract [START_REF] Silva | Chitosan delays ripening and ROS production in guava (Psidium guajava L.) fruit[END_REF]. The spray-drying of a combination of two antioxidant fungal extracts (Suillus luteus and Coprinopsis atramentaria) has already been shown to double the antioxidant activity over that expected (Ribeiro et al. [38]). A similar synergistic effect was observed for the chitosan/fungal extract combination, highlighting the potential value of this powder for the use in the winemaking process. PE*2 supplemented matrices had the strongest antioxidant properties, with a synergistic effect of 2.8 to 3.2 times the expected value. These results are consistent with those of Sansone and colleagues (2014), showing small particles size generally increases microparticle dissolution and the release of the encapsulated active ingredient (Sansone et al. [START_REF] Sansone | Technological properties and enhancement of antifungal activity of a Paeonia rockii extract encapsulated in a chitosan-based matrix[END_REF]). Under laboratory conditions, in simple media, the antimicrobial and antioxidant properties were combined, revealing this powder to be a promising alternative to sulfites for use in winemaking. Under the conditions tested, the PE*2supplemented matrix is a good candidate for use in wine applications. Indeed, the spray-drying technique limited the effects of chitosan on S. cerevisiae growth, whereas the effect on B. bruxellensis growth was similar to that of NBI used in the recommended conditions in wine (Control, 400 mg/L). The addition of the NBI PE*2 matrix to the must could therefore be considered in winemaking. However, for confirmation of this potential, scaling up of the fungal antioxidant production has to be shown in order to manufacture the process, and recent advances have been made on this step [START_REF] Lopez De Leon | Influence of the culture conditions on the production of NGPs by Aspergillus tubingensis[END_REF]. Then, both properties (antioxidant and antimicrobial) will need to be tested directly on a must, with evaluations of browning and microbial spoilage throughout the winemaking process and assessments of the release of the active ingredients from the powder into a complex medium such as must or wine. Finally, innocuity of the product should be confirmed. However, previous studies have already shown beneficial effect on rats' health with a daily intake of NGPs [START_REF] Taillandier | Effect of a fungal chitosan preparation on Brettanomyces bruxellensis, a wine contaminant[END_REF][START_REF] Xie | Protective effects of cassia seed ethanol extract against carbon tetrachloride-induced liver injury in mice[END_REF].

Fig. 1

 1 Fig. 1 Scanning electron micrographs of LMW + CE*6 microparticles (a), NBI + CE*6 microparticles (b), and NBI + None microparticles (c)

Fig. 2

 2 Fig. 2 Growth of Saccharomyces cerevisiae (a) and Brettanomyces bruxellensis (b) populations in the presence of different types of chitosan, as assessed by measuring ΔOD 600nm . LMW low molecular weight chitosan, NBI No Brett Inside ® chitosan; None anhydrous ethanol control, PE*2 SPE semi-purified fungal extract, CE*3 fun-

Table 1

 1 Spray-drying yield and mean micro particle diameter LMW low molecular weight chitosan, NBI No Brett Inside ® chitosan; None anhydrous ethanol control, PE*2 SPE semi-purified fungal extract, CE*3 fungal extract concentrated by rotary evaporation (3 V/V), CE*6 fungal extract concentrated by rotary evaporation (6 V/V) Data with the same letter are not significantly different (two-way ANOVA with Tukey post hoc correction, p-value < 0.05)

	Chitosan	Fungal extract	Initial dry	Spray-drying	Mean microparti-	Particle size coef-
			weight (g)	yield (%)	cle size (μm)	ficient of variation
						(%)
	LMW	None	3.998	61	1.52 ± 0.03 a	1.64
		PE*2	4.100	81	2.49 ± 0.01 b	0.40
		CE*3	6.944	80	3.29 ± 0.04 c	1.09
		CE*6	9.497	81	4.35 ± 0.33 d	7.59
	NBI	None	4.054	78	1.17 ± 0.05 a	4.10
		PE*2	4.170	74	2.26 ± 0.01 b	0.35
		CE*3	6.961	80	3.13 ± 0.01 c	0.10
		CE*6	9.492	79	3.37 ± 0.14 c	4.24

Table 2

 2 Antioxidant capacity of microencapsulated matrix suspensions at 3 g/L (dry weight) The mean of a triplicate analysis is indicated (SD < 0.07) LMW low molecular weight chitosan, NBI No Brett Inside ® chitosan, PE*2 SPE semi-purified fungal extract, CE*3 fungal extract concentrated by rotary evaporation (3 V/V), CE*6 fungal extract concentrated by rotary evaporation (6 V/V), TEAC est estimated TEAC, TEAC exp experimental TEAC Synergistic effect (SE) is indicated in bold; Antagonistic effect (AN) is underlined Data with the same letter are not significantly different (two-way ANOVA with Tukey post hoc correction, p-value < 0.05)

		Ethanol 100%		Ethanol 12%		Acidified water
		TEAC est (mM)	TEAC exp (mM)	TEAC est (mM)	TEAC exp (mM)	TEAC est (mM)	TEAC exp (mM)
	Control
	CE*3		11.22 ± 0.53
	CE*6		18.66 ± 0.21
	PE*2		2.2 ± 0.08
	LMW		0.06 ± 0.001		0.22 ± 0.005		0.31 ± 0.025
	NBI		N.D. ± 0		0.11 ± 0.002		0.19 ± 0.005
	LMW
	CE*3	0.52	0.16 d ± 0.017	0.61	1.15 b ± 0.021	0.66	1.56 ± 0.036
	CE*6	0.61	0.38 c ± 0.016	0.68	0.76 e ± 0.005	0.72	0.90 ± 0.018
	PE*2	0.22	0.44 c ± 0.01	0.37	1.09 b ± 0.036	0.45	1.4 a ± 0.04
	NBI
	CE*3	0.48	0.16 d ± 0.01	0.54	1.26 a ± 0.055	0.59	1.4 a ± 0.07
	CE*6	0.59	0.40 c ± 0.009	0.63	1.13 b ± 0.056	0.67	0.97 b ± 0.018
	PE*2	0.16	0.34 c ± 0.022	0.26	0.83 e ± 0.034	0.34	0.94 b ± 0.004

Laboratoire de Chimie Agro-Industrielle (LCA), Université de Toulouse, INRA, INPT, Toulouse, France

Author Contributions EC, VD, JR ,and FM designed the experiment. EC realizes the crude extract production, extract purification, antioxidant, and antimicrobial analyses. VD and IA realize microencapsulation, physicochemical properties of the obtained particles, and scanning electron microscopy. EC and VD wrote the manuscript; IA, JR ,and FM made a critical reading of the manuscript.

Compliance with Ethical Standards

Conflict of interest The authors declare that they have no conflict of interest (financial, non financial).

Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


domain: stat shs

text:
A De La Foye

P Veysset

Functional regression to model the link between livestock farms' performance and their size

Over the last decades, livestock farms' sizes have increased in Western Europe although the benefits obtained by the farms and agricultural workers are far from being obvious nor demonstrated.

The purpose of this paper is to propose the use of functional regression to study the relationship between farm size and farm performance trends in long-term panel data. While a linear Partial Least Squares regression is performed as a reference model, a regression for functional data enrich the modelling with the chronological information contained in panel data, in particular evolution shapes of performance and size.

The proposed methods are applied on a long-term structural, technical and economic database of cattle beef farms in the Charolais region (France). The achieved regression models, whether classical or functional, suggest that, in our case study, the farm size is not sufficient to explain its performance.

We now expect our work to be a starting point for further analysis. In particular, other structural, economical and technical characteristics could contribute to better model livestock farms' evolution.

Introduction

While the sizes of beef cattle farms have continually increased in France for at least twenty years, the technical and economical performances have stagnated over the same period ( [START_REF] Veysset | Productivity and technical efficiency of suckler beef production systems: Trends for the period 1990 to 2012[END_REF], [START_REF] Veysset | Formation et répartition des gains de productivité en élevage bovin viande[END_REF]). These observations challenge two commonly held opinions. The first is that the expansion of farms should benefit their productivity through economies of scale. The second is that technological progress and agricultural policy should benefit agricultural workers.

In fact, there is no consensus about the relationship between farm size and farm performance. Indeed, in his literature review on family farm size and efficiency, [START_REF] Bojnec | Farm Size and Efficiency during Transition: Insights from Slovenian Farms[END_REF] wrote that in business literature, the corresponding correlation was often estimated as positive. However, he added that in agriculture and development economics papers, a negative correlation between farm size and productivity was often shown. The results are anyway often mixed. [START_REF] Deininger | Can Labor-Market Imperfections Explain Changes in the Inverse Farm Size -Productivity Relationship ? Longitudinal Evidence from Rural India[END_REF] found that though the negative correlation observed in 1982-2008 in Nicaragua gradually declined, it still remained. [START_REF] Lefroy | An Examination of Broadacre Farm Size and Performance in Western Australia[END_REF] also challenged the opinion expressed in previous studies that small-to-medium-sized businesses would deteriorate economically. Yet, [START_REF] Lefroy | An Examination of Broadacre Farm Size and Performance in Western Australia[END_REF] noted the good economic health of larger farms over the decade 2002-2011 in Western Australia. [START_REF] Bojnec | Farm Size and Efficiency during Transition: Insights from Slovenian Farms[END_REF] added that the complexity of the mechanisms governing farm behaviour makes it difficult to draw theoretical conclusions. [START_REF] Henderson | Considering Technical and Allocative Efficiency in the Inverse Farm Size -Productivity Relationship[END_REF] suggested that the observed negative correlation between farm size and productivity in developping countries is probably due to the labour market imperfections. In western Europe, to our knowledge, the issue remains unexplored though it seems crucial to ensure the adequacy of agricultural policy.

In order to model the link between farm performance as a response and size as a predictor, long-term observations on a sample of independant farms are the most appropriate. On this type of data, a first idea is to carry out a panel regression analysis as [START_REF] Bojnec | Farm size, agricultural subsidies and farm performance in Slovenia[END_REF], [START_REF] Henderson | Considering Technical and Allocative Efficiency in the Inverse Farm Size -Productivity Relationship[END_REF] et [START_REF] Deininger | Can Labor-Market Imperfections Explain Changes in the Inverse Farm Size -Productivity Relationship ? Longitudinal Evidence from Rural India[END_REF] did. However, [START_REF] Bojnec | Farm income sources , farm size and farm technical efficiency in Slovenia[END_REF] rejected panel regression analysis in favour of quantile regression analysis for 3 reasons : 1) the marginal effects of changes in some of the variables in estimated model can differ according to the technical efficiency scores, 2) the restrictive assumption of the distribution of error terms in Ordinary Least Squares (OLS) and 3) the better robustness of quantile regression to heteroskedasticity of the residuals.

None of these models, though, take into account the chronological link between years in performance and size evolutions when this information may be essential. In contrast, functional models, as studied by [START_REF] Ramsay | Functional Data Analysis with R and Matlab[END_REF], make it possible to include evolution curves as predictors (scalar-on-function regression), as a response (function-on-scalar regression) or both (function-on-function regression). Function-on-function regressions can thus predict evolution curves with other evolution curves. Numerous functional regression methods have been developped as reviewed by [START_REF] Morris | Functional Regression[END_REF]. Yet, the latter noted that there had been comparatively little work on function-on-function regression which interests us in the first place. The objective of this paper is to show how functional regression can be used to analyse the links between the evolution of farm size and the evolution of its performance comparing Partial Least Squares (PLS) regression ( [START_REF] Tenenhaus | La régression PLS : théorie et pratique[END_REF]) and functional regression. One concern in Functional Data Analysis (FDA) is the transformation of usually discrete observed data to functional data, which we address at first. A second concern is the estimation of the functional regression coefficients which are infinite-dimensional by nature as [START_REF] Ramsay | Functional Data Analysis with R and Matlab[END_REF] noted. To deal with the undetermination issue arising from the induced infinite number of solutions, we perform a functional PLS regression, as developped by [START_REF] Preda | Functional PLS regression with functional response : the basis expansion approach[END_REF].

The paper is structured as follows. The second section presents our methodological approach. The third section describes the long-term structural, technical and economic database of beef cattle farms in the French Massif Central to which our methodology is applied. The farms' performance modelling results of our application study are reported in Section 4. The fifth section is devoted to the discussion and conclusion.

Methodology

In the following, we assume that relevant performance and size indicators are observed on a constant sample of N farms over T years.

Two methods are proposed to model farms' performances with their size. On the one hand, a PLS regression models the links between T performance variables and T size variables over the same T years. On the other hand, a more complex model based on evolution curves such as function-on-function regression could be needed to exploit the chronology of developments.

PLS linear regression

PLS regression ( [START_REF] Tenenhaus | La régression PLS : théorie et pratique[END_REF]) was developped to analyse complex linear relationships between a matrix of output correlated variables Y and a matrix of input correlated variables X. It consists in extracting orthogonal linear combinations U from X in the one hand and orthogonal linear combinations V from Y in the other hand so that the covariances between U t and V t columns are iteratively maximized for t ∈ [ [1, T ]]. Matrix U contains the PLS components of the model.

In our case, we aim at modelling N * T matrix Y as a response where Y it is the performance of farm i in year t. The predictor is N * T matrix X where X it is the size of farm i in year t :

Y = A + XB + E
B is a T * T coefficient matrix and E is the N * T error matrix.

U and V are also of dimension N * T , but an aim of PLS regression is to reduce their dimension to N * h, h < T , so that only useful information is retained in a parsimonious model. The selection of the optimal number of PLS components h and the estimate of the predictive quality of the model R 2 Ŷ ( h) are processed via a double Leave-One-Out (LOO) procedure as advised by [START_REF] Hastie | The elements of statistical learning[END_REF] and [START_REF] Varma | Bias in error estimation when using cross-validation for model selection[END_REF].

Thus,

R 2 Ŷ ( h) = 1 - P RESS Ŷ ( h) SST Y
where

• P RESS Ŷ (h) = N i=1 T t=1 (Y it -Ŷ(-i)t (h)) 2
is the Predicted Sum of Squared Residuals after the outer LOO loop for a model with h PLS components • Ŷ(-i)t (h) is the predicted value of Y it after the outer LOO loop, that is on a model with h PLS components estimated with all observations but i.

• h = arg min h P RESS Ŷ( h) the optimal number of PLS components as estimated after the inner LOO loop executed on all farms but i

• SST Y = N i=1 T t=1 (Y it -Ȳt ) 2
is the Sum of Squares Total, expressing the dispersion of variables Y t to their means Ȳt .

By construction, when the prediction error due to the regression model is greater than the dispersion of Y from the mean, R 2 Ŷ can be negative. In this case, the model is useless.

Functional linear regression

In the PLS regression model described above, nor chronological information between observed years, neither any of the specific information contained in the evolution curves for performance and size was taken into account. Assuming these evolution curves were at our disposal, the performance of farm i at any time s ∈ T could in particular be modelled according to farm i's size at any time t ∈ T , where T is the continuous interval of time [1, T ] :

y i (s) = α(s) + T β(s, t)x i (t)dt + it ∀i ∈ [[1, N ]] (1)
This is a functional regression model where both the predictor and response are functional.

Preliminary step: computing curves from discrete data

The number T of observations on a farm i is necessarily finite by construction and thus do not consist of curves. Therefore, as a first step before performing functional regression, following [START_REF] Ramsay | Functional Data Analysis with R and Matlab[END_REF] and [START_REF] Preda | Functional PLS regression with functional response : the basis expansion approach[END_REF], our observed size data X it are first smoothed into curves x i which are L 2 -continuous functions. These functions are further assumed to be the real information in the whole T interval.

The smoothing is achieved using B-splines basis functions φ k (t) which are known for being well adapted to non periodic data ( [START_REF] Ramsay | Functional Data Analysis with R and Matlab[END_REF]). The underlying function x i can then be assumed as a finite sum of parametric functions :

x i (t) = K k=1 ξ ik φ k (t)
ξ ik , k = 1, ..., K are the coefficients of the basis B-spline functions φ k associated with x i . The dimension of functions x i is given by the basis size K.

How do we choose the B-splines basis functions φ k (t) ? Indeed, a B-spline is a piecewise polynomial function which depends in particular on the degree of the polynoms and on the number of polynomials (and sub-intervals) defined. Under the usual conditions in which we operate, the basis size K is one more than the highest power of the polynoms added to the number of polynomials. Furthermore, in order to avoid overfitting, it is recommended to use roughness penalties for the smoothing. This leads to 2 additional parameters to choose from : the penalty as well as the differential operator to be penalised.

As there is no a priori on the degree of smoothness of functions x i which would be the most relevant for the modelling, several functional variables x are to be tested. On the one hand, higher K favor x i estimations approaching smooth interpolations of X i , which can be relevant when the data are not noisy. On the other hand, lower K could allow for more parsimonious and thus robust models. We fit the x i according to each possible basis size in [[2; T ]]. For each fixed basis size K, we compute 1 to 200 x curves, varying, when relevant, the degree of the polynoms d from 1 to 5, the number of polynoms b from 1 to T + 1 -d, the differential operator D to penalize (no penalized differential operator, a penalized first derivative and a penalized second derivative) and 28 different roughness penalties λ (from 0.01 to 1). For each K, we select the best parameters combination (d, b, D, λ) according to the integrated Mean Squared Error of x considering that :

• the Mean Squared Error (M SE) is based on the differences between x i (t) and X it at each observed year t :

M SE x = 1 N N i=1 T t=1 (x i (t) -X it ) 2 ,
• the integrated Mean Squared Error (M SE int ) also takes into account the noise which could be added between observed points due to overfitting :

M SE int x = 1 N N i=1 T (x i (t) -xi (t))
2 dt where x is the linear interpolation of X, that is a curve obtained by joining the observed points by linear segments. Thus, the lower the distance to the adjusted curve to the linear interpolation, the lower the M SE int .

The same smoothing process is achieved for Y , leading to a set of computed y such that :

y(t) = K k=1 υ ik ψ k (t)
with υ ik the coefficients of the basis functions ψ k associated with y i . Thus, T -1 smoothed functions are selected for X and for Y . The smoothing procedures are processed via the fda R package.

PLS functional linear regression

Each of the (T -1) 2 combinations of functional variables couples (x, y) gives rise to a functional linear model as in equation [START_REF] Bojnec | Farm income sources , farm size and farm technical efficiency in Slovenia[END_REF].

For each model, coefficient function β in equation ( 1) is estimated via functional PLS regression as proposed by [START_REF] Preda | Functional PLS regression with functional response : the basis expansion approach[END_REF]. The latter showed the equivalence between the functional PLS regression of y on x and the PLS regression of basis coefficients matrices (υ) ik on basis coefficients (ξ) ik . Consequently, (T -1) 2 combinations of estimated (x, y) result in (T -1) 2 PLS regression models on corresponding basis coefficients matrices ((υ) ik , (ξ) ik ).

Furthermore, for each model induced by (x, y), the number h of PLS components can vary between 0 and T , leading to T + 1 potential sub-models. Thus, a total of (T + 1)(T -1) 2 models are to be evaluated. The selection of the optimal (x, y, h) combination and the estimate of the predictive quality of the model R 2 Ŷ ( (x,y,h))

are processed via a double Leave-One-Out (LOO) procedure as in previous subsection ( [START_REF] Hastie | The elements of statistical learning[END_REF], [START_REF] Varma | Bias in error estimation when using cross-validation for model selection[END_REF]).

Thus,

R 2 Ŷ ( (x,y,h)) = 1 - P RESS Ŷ ( (x,y,h))
SST Y where 2 is the Predicted Sum of Squared Residuals after outer LOO loop for a model with h components and smoothings x and y,

• P RESS Ŷ( x,y,h) = N i=1 T t=1 (Y it -ŷ(-i) (t, x, y, h))
• ŷ(-i) (t, x, y, h) is the predicted value of y i (t) after the outer LOO loop, that is on a model estimated with all observations but i with h components and x and y smoothings,

• (x, y, h) = arg min (x,y,h) P RESS Ŷ(x,y,h) is the optimal combination of (x, y, h) as estimated after the inner LOO loop executed on all farms but i.

By construction, when the prediction error due to the regression model is greater than the dispersion of Y from the mean, R 2 Ŷ ( (x,y,h)) can be negative. In this case, the model is useless.

Data: INRA network of Charolais beef cattle farms

In order to understand the determinants of macrotrends in beef cattle system farms and capture and analyze the technical and economic potentials of their production systems, an Economics team from INRA UMRH has set and sample-surveyed a Charolais-region farm network since the 1970s. Data have been collected on labour, structure, hectarage and land allocation scheme, herd, intermediate inputs, sales, aids and subsidies, investments and borrowing and allow long-term observational statistics (see [START_REF] Veysset | Productivity and technical efficiency of suckler beef production systems: Trends for the period 1990 to 2012[END_REF]). All nominal-value prices internalize currency depreciation using the national consumer price index as deflator.

In the following study, we focus on the farms' income per worker (Y ) and on the farm's UAA per worker (X). The farms' income per worker is a particularly complex performance indicator as it includes all the products, the subsidies, the operational and fixed costs (including depreciation costs). However, this indicator represents what is left to pay for labour and own capital and it is thus crucial to understand the determinants of its evolution.

From more than 50 farms performance-monitored per year over the T = 25 years from 1992 to 2016, we formed a constant subsample population of N = 38 farms, all located in the Charolais-area pool. Over this period, the UAA per worker (annual work unit, AWU) has been gradually rising (+ 62% on average), as can be seen in Figure 1a. By way of contrast, the farm income per worker evolution has been chaotic and even slightly decreasing in the last years (Figure 1b).

Descriptive analysis

As can be seen in Fig. 2, the relationship between income per worker and UAA per worker is positively linear in the first years but no so much after year 2004. We note that the correlation coefficient even becomes slightly negative and the relationship not obviously linear after year 2012 . These results show the need for a more complex model than one only taking into account both characteristics at the same time points. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q R = 0.37 p = 0.022 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = 0.52 p = 0.00092 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = 0.58 p = 0.00013 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = 0.47 p = 0.0027 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = 0.48 p = 0.0026 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = 0.59 p = 9.6e-05 q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q R = 0.57 p = 0.00019 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = 0.54 p = 0.00046 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = 0.16 p = 0.33 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = -0.075 p = 0.65 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = 0.59 p = 9.5e-05 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q R = 0.59 p = 8.8e-05 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = 0.58 p = 0.00012 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = 0.012 p = 0.94 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = -0.077 p = 0.65 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = 0.41 p = 0.011 q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = 0.65 p = 1.2e-05 q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q R = 0.25 p = 0.14 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = 0.35 p = 0.031 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = -0.14 p = 0.42 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = 0.44 p = 0.0056 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = 0.3 p = 0.069 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = 0.25 p = 0.13 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = 0.36 p = 0.026 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q R = -0.34

PLS linear regression

A linear PLS model is fitted on the 25 income per worker variables (1 per year) as responses and 25 UAA per worker variables as predictors.

PLS components 1 and 2 gather 91% of the information generated by the UAA per worker year variables as mentioned in the correlation plot (Fig. 3). The latter also shows that the UAA per worker year-variables are globally positivally intercorrelated and more particularly years 2009 to 2016 on the one hand, and 1992 to 2008 on the other hand. By construction, PLS components 1 and 2 allow the best representation of the income per worker year variables in the UAA per worker space. The most positivally correlated income per worker year variables to PLS component 1 (and thus globally to UAA per worker year variables) are 1999,2004,2000,2012,1993,1997. However, the correlations between both sets of variables are quite low. On the opposite side of PLS component 1, 2015 and 2016 income per worker are shown slightly and negatively correlated to the UAA per worker variables. This is consistent with what was seen in Fig. 2.

Thus, as could be expected, the estimated PLS model predictive performance is very low with a predicted total R 2 of -0.07.

Computing curves from discrete data

4259 curves were computed for the UAA per worker (vs income per worker) evolution, varying according to the number of b-splines basis functions used (2 to 25) as well as according to 4 other parameters as detailed in section 2. For each b-spline basis size, the optimal curve according to M SE int criterion was selected for further analysis. Thus, 24 curves were selected for the UAA per worker as well as for income per worker evolution.

Figures 4 and5 show that, as expected, the adjustment quality according to M SE y and M SE int y is generally better when the function is adjusted with a larger number of B-splines functions. The maximal (vs minimal) basis size, which allows the finest (vs poorest) fitting to the observed data is shown in Figure 6. An intermediate basis size of 12 (vs 15) appears to be a priori a good compromise between parsimony and fitting quality for the UAA per worker (vs income per worker) according to figure 4 (vs 5) and is also represented on Figure 6.

PLS functional regression

A functional regression model was fitted on the income per worker evolutions as a functional response and the UAA per worker evolutions as a functional predictor. 14400 parameters combinations for the modelling were computed, varying according to the UAA curves fitted (K ∈ [[1, 24]]), the income per worker curves fitted (K ∈ [[1, 24]]) and the number of PLS components kept

(h ∈ [[0, 25]]).
The inner LOO loop allowed the selection of 38 optimal parameters combinations (x, y, h), that is one for each excluded farm and are shown in Table 1. In this table, parameters x and y are represented by their degrees of smoothness K and K . The optimal combinations are quite stable as 24 out of 38 are identical, as shown in Table 1. Indeed, the most frequent optimal model includes income per worker curves estimated with K = 22 B-splines functions, UAA per worker curves estimated with K = 12 B-splines functions and 3 PLS components. Furthermore, the other optimal models have quite close parameters values. Thus, on the one hand, 36 of the optimal models use response curves close to linear interpolations of the observed incomes per worker, with K > 21. On the other hand, in these models, all the observed UAA/worker are fitted with K varying from 9 to 17, which stand for a moderate smoothing. Besides, 3 or 4 PLS components are systematically chosen. However, the satisfying consistency of these results are to be put into perspective. Indeed, the predictive R 2 ŷ for optimal h, x, y is estimated after the outer LOO loop at about 0.01. In other words, our functional model ( 1) is useless as it stands and needs further improvement.

K

Discussion and conclusion

In this paper, we propose a method that could be relevant in the modelling of farm performance evolution. In our case study, though, none of the models tested, including functional data regression, make it possible to predict correctly the farms' income per worker.

Concerning the two-steps functional regression method used, one limit raised by [START_REF] Greven | A general framework for functional regression modelling[END_REF] is that since the fitted continuous curves are further considered as if they were observed curves, the measurement and fitting errors are not taken into account in the functional regression model. In contrast, [START_REF] Greven | A general framework for functional regression modelling[END_REF] proposes a framework which allows direct modelling of the observed data and expands all model terms in suitable basis expansions. In our case, though our models did not, our final prediction quality criterion has been estimated against the real observed values. This paper introduces the application of functional regression in agricultural economics and in particular to the income per worker modelling. As mentioned in the literature, the link between farms' performances and their size is complex. Moreover, the income per worker is also a complex performance indicator including subsidies. Indeed, the evolution of farms' UAA per worker is found to be insufficient to explain the evolution of the farms' income per worker, as was expected. It will therefore be necessary to improve the model. A first obvious option would be to introduce other explanatory functions corresponding to the evolution of other structural, technical or economic characteristics -like beef price. A second option would be to make the model more parsimonious by restricting the time interval on which performance is linked by predictor functions. At last, evolutions' derivatives could be relevant to predict the performances evolution.

More research is thus needed to understand the unsatisfactory evolution of farms' income per worker evolution and more widely farms' performances evolution. Functional regression is promising in the new perspectives it can brings in this field and will hopefully contribute to the political debate on the relevance of increasing farms' size.

Figure 1 :

 1 Figure 1: Evolution of observed (a) UAA per worker and (b) income according to the year The observed yearly points are linearly interpolated

Figure 2 :

 2 Figure 2: Income per worker as a function of UAA per worker and the year R : Pearson correlation coefficient

Figure 3 :

 3 Figure 3: PLS regression model plot Correlation plot of income per worker and UAA per worker on PLS components 1 and 2

Figure 4 :Figure 5 :

 45 Figure 4: Fitting error of the UAA per worker estimated evolution according to the number of B-spline basis functions K MSE stands for means squared error -MSEint stands for means squared integrated error

Figure 6 :

 6 Figure 6: Fitted x (column 1) and y (column 2) according to the time for 3 parameter combinations (rows 1 to 3) (Row 1) Basis sizes K and K' of 25 -(Row 2) Basis sizes K and K' of 12 and 15 -(Row 3) Basis sizes K and K' of 2

Table 1 :

 1 Optimal parameters estimated from the inner LOO loop for functional PLS regression

		K h Frequency
	15 14 4	1
	17	9 3	1
	22 12 3	1
	22 12 4	1
	22 14 3	24
	25 12 4	9
	25 17 4	1