Joan Sol Roo
email: joan-sol.roo@inria.fr

Martin Hachet
email: martin.hachet@inria.fr

One Reality: Augmenting How the Physical World is Experienced by combining Multiple Mixed Reality Modalities

Keywords: ACM Classification H.5.1 Multimedia Information Systems: Artificial, augmented, and virtual realities, H.5.2 User Interfaces: Input devices and strategies Augmented Reality, Mixed Reality, Spatial Augmented Reality, Virtual Reality, Through-the-lens Technique, Tangible User Interfaces, Organic User Interfaces

Figure 1. The presented system, exemplified here by an augmented volcano mock-up, allows one or more users to use and transition between multiple mixed reality modalities while interacting with augmented artifacts. The increase in instrumentation provides increasing flexibility, while keeping the interaction framed in the physical world.

INTRODUCTION

The physical world in which we live provides rich interactions that can hardly be replaced by digital artifacts. In particular, what we see, touch, or smell, and our social collective experiences with face-to-face Humans constitute a unique experience that cannot (should not?) be totally replaced by simulated environments. On the other hand, the flexibility of digital tools has shown undeniable benefits that allow overtaking the constraints of the physical world. As an example, Virtual Reality (VR) allow engineers to immerse themselves inside a simulation and to navigate virtual mock-ups in a way that would not be possible to do in the real life.

In order to complement the physical and the digital worlds, researchers have studied the creation of mixed spaces where digital information can be used in combination with its physical counterpart. This is notably the case of Augmented Reality (AR), where digital and physical information are co-located in a seamless space. The resulting hybrid spaces benefit from digital information anchored onto the physical world. AR is definitively a powerful medium for a number of fields. On the other hand, AR experiences still remain constrained by the physical environment in which the illusion takes place. As an example, neither See-Through AR, nor Spatial AR allows one to see a scene from a point of view that is different from the current viewpoint of the observer.

In our approach, we propose a conceptual framework and its implementation where the physical world that stands in front of the users can be progressively augmented and distorted, with the final goal of augmenting the users' perception. By providing augmented experiences, anchored in the physical reality, we seek for a hybrid space where users benefit from both the force of the physical sensing, and the flexibility of digital interaction.

Our contributions in this work are i) the design of a conceptual framework, where the user can smoothly travel between the physical and the virtual worlds, ii) the implementation of this framework on a unified system, and iii) the exploration of the interaction possibilities. The result builds towards a unified way to look at the intersection between physical and digital realms, combined on a single reality for the user.

Examples

To better understand this concept, let's consider two concrete scenarios: collaborative learning and hardware maintenance.

Imagine a group of students working around an augmented volcano mock-up (Figure 1). They can discuss and observe the simulated behaviour, while physically touching and moving around and maybe dissecting the mock-up. Some information can be difficult to understand from an egocentric perspective, so they can travel inside the mock-up for more information, using VR. In this case, they can "jump" into the physical model, follow the tubes, and experience the volcano from a different point of view. While immersed, they can still discuss with her classmates, their bodies visible as giants around the mock-up. This simple example, inspired form TV shows like "Cosmos" [START_REF] Sagan | Cosmos: A personal voyage[END_REF], "the Magic School bus" [START_REF] Cole | The Magic School Bus[END_REF] or "Once upon the time... Life" [START_REF] Barille | Once Upon a Time[END_REF], can be generalized to any pedagogical content, where the changing of viewpoint of the physical environment may improve the learning and understanding of the studied phenomena. Using the shared physical space as starting point fosters discussion. Now let's consider a maintenance task for a car engine: the engineer is in a process where she has to work physically with the object standing in front of her. Thanks to situated projection, she can benefit from digital support that will guide her during the process (e.g. highlight a given piece) as done in [START_REF] Zhou | Applying Spatial Augmented Reality to Facilitate In-situ Support for Automotive Spot Welding Inspection[END_REF]. Now, the engineer needs to observe the engine from the back, or have a closer view of a specific part. Because she cannot manipulate or move around the physical engine, she decides to virtually change her point of view. To do so, she puts on an HMD, and navigate in and around the virtual object. She can also observe a virtual engineer performing the required task, and take the viewpoint of this expert. Numerical simulation can be launched, too, in order to observe for instance the flow in and around the engine. These situated interactions, which would not be possible with standard SAR approaches, may help the engineer to better understand the physical engine she is working on.

Before exploring in detail the different levels present in our framework and their implementation, the next section briefly reviews the previous work that enabled their conception.

RELATED WORK

The ideas presented in this work build upon 1) the vision that the digital realm can be integrated into the physical one, 2) and the different technologies that were used in the past to explore this possibility.

Since the dawn of digital technologies, researchers and sci-fi writers envisioned a future where digital and physical will be indistinguishable from each other [START_REF] Sutherland | The ultimate display[END_REF]. Science fiction has been used in the past both as inspiration [START_REF] Komiyama | JackIn Space: Designing a Seamless Transition Between First and Third Person View for Effective Telepresence Collaborations[END_REF][START_REF] Sophia | Out of network: technologies to connect with strangers[END_REF] and research tool [START_REF] Blythe | Research through design fiction: narrative in real and imaginary abstracts[END_REF] to study not just how that future can be, but also about the dangers of not designing consciously. Mark Weiser emphasized that ubiquitous technology [START_REF] Weiser | Some computer science issues in ubiquitous computing[END_REF] must be also calm [START_REF] Weiser | The coming age of calm technology[END_REF]: the digital tools should be available when needed, in a nondisruptive way; for this reason, we propose that the increase in immersion should be progressive and at the user's discretion. Bret Victor with his seminal talk "a humane representation of thought" [START_REF] Victor | Humane Representation of Thought: A Trail Map for the 21st Century[END_REF] reflects on how the representation of information frames our way of thinking, and he argues that we must actively channel the advance of technology so it is empowering, instead of limiting. A common theme arises: when designing technology, we should create interfaces that support and collaborate with the users, to help humanity to flourish. In this direction, Jacob et al. formalized the Reality-Based interfaces [START_REF] Robert | Reality-based interaction: a framework for post-WIMP interfaces[END_REF]: computational interfaces that use our already available skills to understand and interact with the external world and each other.

As a first step towards this vision, Milgram presented physical and digital realms not as a dichotomy, but as a Mixed Reality continuum [START_REF] Milgram | Augmented reality: A class of displays on the reality-virtuality continuum[END_REF]. Ullmer and Ishii presented Tangible User Interfaces (TUIs) [START_REF] Ullmer | Emerging frameworks for tangible user interfaces[END_REF], which provide physical handles to digital information. This concept was later extended with the vision of Radical Atoms [START_REF] Ishii | Radical atoms: beyond tangible bits, toward transformable materials[END_REF]: computational matter that can change its physical properties as needed, an indivisible emulsion of digital and physical. This leads to the question of "what is real?": Eissele et al. [START_REF] Eissele | Transition of mixed, virtual, and augmented reality in smart production environments-an interdisciplinary view[END_REF] reminded us the philosophical distinction between "reality" (realitas, what is constructed in the mind) from "actuality" (actualitas, the invariant truth), a specially relevant separation when studying mixed reality, which is performed by altering the information received by the senses. By using this definition, when digital and physical work together, they are just counterparts of the same reality for the user.

Rekimoto et al. [START_REF] Rekimoto | The world through the computer: Computer augmented interaction with real world environments[END_REF] studied the different ways these realms interact with each other, a classification that could be extended to include spatial augmentation [START_REF] Bimber | Modern approaches to augmented reality[END_REF]. The latter has been particularly explored in the construction of interactive spaces. Wellner et al. created DigitalDesk [START_REF] Wellner | Interacting with Paper on the DigitalDesk[END_REF], a projector-augmented desktop that supports paper based interaction. The Office of the Future [START_REF] Raskar | The office of the future: A unified approach to image-based modeling and spatially immersive displays[END_REF] envisioned the extension of CAVEs [START_REF] Cruz-Neira | Surround-screen projection-based virtual reality: the design and implementation of the CAVE[END_REF] to everyday non-flat surfaces, and soon after was implemented using Spatial Augmented Reality (SAR) [START_REF] Raskar | Spatially augmented reality[END_REF]. Initially, the focus of the interaction was on the digital information (and still required to wear stereo glasses), but then the focus moved towards the physical space [START_REF] Raskar | Shader lamps: Animating real objects with image-based illumination[END_REF]. Since then, it was used to support physical-virtual tools [START_REF] Michael R Marner | Physical-virtual tools for spatial augmented reality user interfaces[END_REF], improvised interfaces using objects [START_REF] Walsh | Ephemeral interaction using everyday objects[END_REF], touch [START_REF] Xiao | a. WorldKit: Rapid and Easy Creation of Ad-hoc Interactive Applications on Everyday Surfaces[END_REF], and Organic User Interfaces [START_REF] Holman | Organic User Interfaces: Designing Computers in Any Way, Shape, or Form[END_REF][START_REF] Piper | Illuminating clay: a 3-D tangible interface for landscape analysis[END_REF], among others.

In order to take advantage of the complementary characteristics of mixed reality technologies, hybrid systems have been studied. For instance, see-through displays and SAR have been combined, notably in order to complement the HMD's high resolution with the projectors' scalable FOV (field of view) [START_REF] Ilie | Combining head-mounted and projector-based displays for surgical training[END_REF][START_REF] Benko | Fovear: Combining an optically see-through near-eye display with projector-based spatial augmented reality[END_REF]. Transitioning between see-through AR and VR have been also explored by Kiyokawa et al. [START_REF] Kiyokawa | SeamlessDesign for 3D object creation[END_REF]. In the context of multi-display environments, the combination of screens and projection has been studied, both with [START_REF] Butz | Enveloping users and computers in a collaborative 3D augmented reality[END_REF] and without see-through technologies [START_REF] Rekimoto | Augmented surfaces: a spatially continuous work space for hybrid computing environments[END_REF][START_REF] Gervais | Tangible Viewports: Getting Out of Flatland in Desktop Environments[END_REF]. Dedual et al. [START_REF] Dedual | Creating hybrid user interfaces with a 2D multi-touch tabletop and a 3D see-through head-worn display[END_REF] proposed a system that combines interactive tabletops with video seethrough AR. Smarter Objects [START_REF] Valentin Heun | Smarter objects: using AR technology to program physical objects and their interactions[END_REF] and exTouch [START_REF] Kasahara | exTouch: Spatially-aware Embodied Manipulation of Actuated Objects Mediated by Augmented Reality[END_REF] use video ST-AR to control embedded systems; even when the physical artifact was the focus of attention, no spatial augmentation was presented, except the electronic behaviour itself. Closer to our work is Metadesk [START_REF] Ullmer | The metaDESK: models and prototypes for tangible user interfaces[END_REF] by Ullmer and Ishii, which combines tabletop interaction, tangible tokens and a see-through AR window. Hybrid systems not just complement technological limitations, but also combine well with physical artifacts.

Mixed Reality was also studied as a mean to provide different perspectives of a scene. Magicbook [START_REF] Billinghurst | The magicbook-moving seamlessly between reality and virtuality[END_REF] augmented a physical book using an HMD, while also supporting transitions between AR and VR. Benko et al. also studied the use of mixed reality to explore digital locations collaboratively [START_REF] Benko | Collaborative mixed reality visualization of an archaeological excavation[END_REF]. Komiyama et al. [START_REF] Komiyama | JackIn Space: Designing a Seamless Transition Between First and Third Person View for Effective Telepresence Collaborations[END_REF] used video see-through to enable the transition between viewpoints (objects, users and spaces) inside the physical world. In all these works, user instrumentation is a requirement to access the augmentation; this can be overcome with always-available viewpoints using screen or projectors. Ibayashi et al. created Dollhouse VR [START_REF] Ibayashi | Dollhouse VR: a multi-view, multi-user collaborative design workspace with VR technology[END_REF], where users can interact with a virtual space, either with a top view (displayed on a tabletop), or from inside (using an HMD-VR), keeping the virtual space as the center of attention. In a preliminary work, we started exploring the use of the virtual interaction with the physical space, supporting not only see-through and immersed modalities, but also spatial augmentation [START_REF] Sol | Towards a hybrid space combining Spatial Augmented Reality and virtual reality[END_REF][START_REF] Sol Roo | Inner Garden: Connecting Inner States to a Mixed Reality Sandbox for Mindfulness[END_REF].

To summarize, the research involving the combination of physical and digital is vast, both focusing on a specific technology or combining complementary alternatives. In this line of research, we propose a smooth transition through progressive immersion, as described in the next section.

ONE REALITY

This work focuses on reducing the gap between physical and digital, without limiting the interaction to neither of them. We propose to provide the users the liberty to increment the degree of immersion when needed, without losing contact with the physical world.

In order to explore this incremental immersion, we created a hybrid working space that supports multiple mixed reality modalities simultaneously, and enables the user to free transition between them (Figure 2). In it, the environment and the users are scanned and tracked in real time, enabling their augmentation, movement logging and digital reproduction. To interact with the system, physical tools are rendered available even in virtual spaces, and vice versa. When designing the progressive transition between the physical and digital realms, we considered 6 incremental levels where digital characteristics are progressively included (Figure 3). The following subsections describe, in an incremental way, the features of each of the levels in combination with their implementation and interaction consequences.

Level Zero: Physical World

The starting point of interaction is the physical world, where participants can interact with each other and with physical artifacts. This level is comprised of both natural elements and technological devices that alter physical properties in a non-cosmetic and persistent way (i.e., the modifications stay even when the artifact is not working, for example mechanical artifacts or pens). In the physical world, objects can be manipulated when they are not too heavy and not too big, and people can move their heads and bodies to change their point of view, as long as nothing is blocking them.

Level One: Augmented Surfaces

At the first level, digital information can be placed at the surface of physical objects in the user's environment, as in the classical work of Rekimoto et al. [START_REF] Rekimoto | Augmented surfaces: a spatially continuous work space for hybrid computing environments[END_REF]. This can be used to display complementary information (e.g., show text, change the object perceived texture or show annotations on its surface), or arbitrary information (i.e., using the surface as a screen).

We implemented the augmentation using Spatial Augmented Reality, which allows us to place the digital information directly onto physical surfaces. When the surface of a given object is not suited to support the digital information, the content is then leaked to adjacent surfaces, such as tables or walls. With SAR, digital information is equally available for all the users and viewpoint-independent, as it is the case with physical information.

By tracking objects of interest in the scene, the system supports various interaction modalities (Figure 4). It is possible to directly manipulate the augmented objects, and to use both direct and indirect interaction using pens or pointers. Given the physically nature of the display supports, non-augmented tools can be used naturally. A widely distributed support of information is paper. When paper is tracked, any digital information displayed on it will follow accordingly. The content of augmented paper can be created using augmented pens; it is also possible to use augmented paper as support for digitally created contenta paper window [START_REF] Holman | Paper windows: interaction techniques for digital paper[END_REF] -, bringing computer screens closer to the physical environment [START_REF] Laviole | PapARt: interactive 3D graphics and multi-touch augmented paper for artistic creation[END_REF]. An additional benefit of using projection is that normal paper can be placed over the augmented counterpart, much like tracing paper. Among the possible digital content, paper windows can display a 3D render of the augmented scene, which can be interacted via ray-casting like traditional displays. As a result, users can interact with their surroundings from a given point of view (Figure 5). We call these windows "interactive pictures".

Level Two: Mid-Air Digital Content

While surface augmentation can give dynamism and interactivity to passive objects, it is unable to directly display content in mid-air. In order to provide support for this it is possible to use see-through devices to create the illusion of floating elements.

We prototyped hand-held see-through displays using the interactive pictures presented in the previous level, by attaching the camera position to the paper position (Figure 6). The user can then interact with the digital elements in their field of view using a tracked pen and ray-casting. Compared to head-worn displays (e.g. Hololens), such an approach has several advantages. First, users do not need to wear specific equipment. Then, the proximity to the physical world is strengthened; it is possible to switch very quickly and easily between the two visual modalities. Both the physicalaugmented object (level 1) and the through-the-lens object (level 2) can also be observed at the same time. Finally, multiple observer can use the same through-the-lens image. This reinforces collaborative and social interaction.

The main drawbacks are that, as with other hand-held technologies, they require the device to be held, and they have a limited field of view. For the cases where comfort is relevant or both hands are needed (e.g., long or precise tasks) it is possible to detach the viewpoint and use indirect interaction, as seen in the previous level.

Back to the Examples

To better understand the first three levels and how the system supports them, let's explore the examples in more detail.

The students (Figure 1) around the volcano mock-up (level 0) can observe superficial simulation (level 1) of its activity and in-place information, while also being able to move around and touch the mock-up, sharing a unified experience. A seethrough screen can help the students to visualize mid air information (level 2) such as steam and lava coming from the top of the volcano, or its interaction with virtual trees. They can use the screen to directly take notes of what they see through it, or place different views of the volcano side-by-side to discuss and make their own drawings.

In the case of the engineer performing maintenance tasks on the engine (level 0), the augmentation can be bi-directional. She can both use previously stored information to guide the process, or create new annotations, creating a tutorial on how to perform the task (level 1). She can also use the see-through approach (level 2) to visualize mid air annotations, measurements and spatial instructions, such as "screw here, then drill there". Once again, this can be bidirectional, either following or creating instructions through graphical annotations or by example.

Level Three: Object Decoupling So far, level 1 and 2 aimed at keeping the augmentation as close as possible to the physical world. Starting at level three, we propose to soften the physical constraints in order to give more flexibility to the users. This can be useful when trying to understand complex processes (simulation), when needing to perform tasks that are not normally possible, such as lifting a heavy object (overcoming physical constraints), or trying to observe actions performed in the past (replay).

To this end, beyond the see-through displays we described in the previous section, we introduced a fully immersive modality that reproduces the physical world standing in front of the users, and that is perceived through a VR Head-Mounted display. Both worlds (physical and virtual) are mapped oneto-one. This means that the users can physically touch the objects they are observing virtually (Figure 7). This gives the immersed user a strong anchorage with the physical world. To increase the immersion, we use hand tracking based on Leap Motion, which provide not only feedback but also the possibility to directly interact with purely digital content.

At this level, the users can now free themselves from strict mapping and perform actions that would not be possible in reality (e.g. manipulate a heavy or fragile object). To do so, they can use virtual controllers. In addition to the features we already described, the users have this new ability consisting in distorting the reality.

Awareness considerations

When physical and digital are decoupled, it is necessary to take additional considerations to help the user to prevent acci-dentally hitting physical objects. For this reason, when users gets near a physical object (either with their hands or tools) we display its wire-frame or bounding box.

It is also important to know where the users are (for themselves, and for others). For this, we display user avatars. The representation depends on the modality: augmented surfaces display the position and orientation in an iconic way (arrow), while see-through and HMD-VR show a 3D reconstruction of the users, enabling them to see themselves and others. Currently the 3D reconstruction is only based on the Microsoft Kinect point-cloud, yet it could be possible to extend it to rigged meshes [START_REF] Shapiro | Rapid avatar capture and simulation using commodity depth sensors[END_REF], animated using Kinect skeleton tracking.

Finally, what the helmet shows can be displayed onto an interactive picture. This way, non-immersed users can see what the immersed user does. Given the behaviour of interactive pictures, users are also able to interact with the scene in front of the immersed user, indicating for example a point of interest, or moving elements through the window. Similar results can be obtained for non immersed users using the estimation of head's position and orientation using skeleton tracking.

Back to the Examples

At this level, the students can virtually modify the geometry of the objects. For example, they can grab the volcano standing in front of them, and stretch it to observe what would happen if the volcano was higher. They can also extend the frontiers of the current physical mock-up by using 3D terrains around the volcano, or by visualizing internal structure above it. They can also copy and paste the volcano to experiment interactions between multi-objects.

In the case of the engineer, she can observe the engine in isolation by wearing an HMD, removing occluding elements or only showing their wire-frame. She can also virtually move a piece mid-air, in order to study it and annotate it. The annotations are reflected onto the physical engine in real time, since both views are synchronized. Finally, she can simulate the proper working behaviour of the engine, to clarify the function of the individual pieces. This level gives users the illusion of changes in the physical world, while they keep their body as frame of reference: for a given user, the physical landmarks (objects and other users) stay still relatively to him or herself. The next level explores the separation between physical and digital body.

Level Four: Body Decoupling

The fourth level explores the immersive virtual navigation of the physical space. The users can then change their perceived position, orientation and scale in space (Figure 8).

For this, HMD-VR helmets are ideal, since the users are presented with a 3D scene overriding their senses, with an arbitrary point of view. In order to control the navigation, we use a teleportation-based interaction using a wand controller, where the resulting scale depends on the target surface (and in our case is configured manually). A simple fade transition was implemented; since both a change of position and scale are involved, travelling could lead to cyber-sickness. Besides being able to teleport, users can also rotate either around a point, or on their current location. Since it is possible to ray-cast through the interactive pictures, users can pick the target position through a window, both before and during the immersive session. The navigation is then not restricted to the current table and their surroundings, but could be used for tele-presence. Indeed, it is possible to transition to distant places (e.g., coworkers office, Mars surface), one step at the time (similar to Google Street View [START_REF] Anguelov | Google street view: Capturing the world at street level[END_REF]), or jumping there through an interactive picture. Once the digital environment is framed in relationship with the physical one, it is possible to then transition to purely virtual spaces (the fifth level).

Back to the Examples

Interacting with the mock-up is enriching, but in the end it is a representation of something bigger. If the students want to experience or study the surface of the volcano, they can do so by wearing an HMD-VRs and teleporting on its surface (Figure 8). The virtual model can then be navigated by jumping from place to place, experiencing the represented scale. While on the surface, the students can see their schoolmates and themselves as giants around the table, allowing to asymmetrically collaborate, not just speaking but also using the aforementioned tools. They can use both the volcano surface and the interactive pictures to communicate with each other.

If the engineer needs help to repair the engine, she can observe another engineer performing the task, either using a recording or tele-presence. She can see her colleague working from his or her point of view, or virtually move around for a better view.

Overview

In this section we described the 6 levels of our conceptual framework, accompanied by the implementation of the 4 central levels that provide augmentation. This framework was conceived to incrementally provide digital tools to interact with the physical world, up to the point where purely digital applications can be framed within the physical space.

The selection of a given modality involves a trade-off (Table 1). As digitality increases, so does the flexibility of interaction and simulation, while the connection with the physical world gets thinner by the instrumentation. We took awareness considerations to keep the user anchored in the physical world as much as possible, yet we argue that immersion should not be increased because it is possible, but only when it is needed.

The result moves towards a space where physical and digital do not compete with each other for the users attention, but instead complement in the construction of a unified experience. This work builds on top of the vast Mixed Reality literature, providing a new lens to look at the diverse projects in the field -such as augmented surfaces [START_REF] Rekimoto | Augmented surfaces: a spatially continuous work space for hybrid computing environments[END_REF], magicbook [START_REF] Billinghurst | The magicbook-moving seamlessly between reality and virtuality[END_REF] and physical-virtual tools [START_REF] Michael R Marner | Physical-virtual tools for spatial augmented reality user interfaces[END_REF]-, while also providing a unified ecosystem where they can coexist.

SYSTEM IMPLEMENTATION

The system runs using two Alienware desktop computers, one worked as a server (Intel i7-3820, 8GiB RAM with an NVIDIA GTX 660 Ti) the second one as a client (Intel i7-3820, 24GiB RAM with dual NVIDIA GTX 690s). The use of multiple computers was necessary given the USB BUS bandwith required by the sensors. The client performs dense acquisition (Kinect) and rendering (HTC Vive, projectors), while the server handles sparse tracking (Optitrack, Leap Motion) and stream it using UDP and OSC. We selected this distribution in order to minimize the intranet usage, yet dense acquisition and rendering could be distributed among several computers.

The whole setup was organized around a rectangular table of 130cm by 80 cm, which was used as the reference for calibration. All the sensors and projectors were placed on an overhead platform (Figure 9). Regarding the displays, we use: 1) an HTC Vive as HMD, and 2) three off-the-shelf projectors, providing 360 • augmentation. The projectors are: an LG PF80G, an LG PF1500 and an Asus B1M; this selection was 1. Each of the augmentation levels and their properties.

based on the available hardware, and the setup could be greatly improved by using 3 projectors of the same model, preferably with higher brightness and frame-rate. The tracking uses 4 Optitrack Flex cameras covering the volume over and around the table. A single Microsoft Kinect v1 provides a partial point-cloud of the table contents and the users, including their skeleton tracking. Given that we used HTC Vive, the tracking lighthouses are also placed on the overhead platform. For hand tracking we attached a Leap Motion to the helmet.

Infrared Interference

Besides the USB bandwith, the main issue of this setup is infrared interference between sensors: all of them -except the Kinect -use the same wavelength. To address this, we synchronized the Optitrack cameras with the lighthouses 1 ; Leap Motion emitters can also create interference, but the impact was diminished with Optitrack parametrization.

The use of additional Kinects would greatly improve the pointcloud coverage, yet additional considerations are required to prevent interference [START_REF] Butler | Shake'n'sense: reducing interference for overlapping structured light depth cameras[END_REF]. Regarding Microsoft Kinect v2, we were unable to include it in the system since it uses the same wavelength as the other sensors, leading to interference that could not be mitigated.

Calibration

The alignment of all the subsystems was performed in stages. Both Optitrack and Kinect were calibrated offline, computing the alignment between coordinate systems (3D to 3D calibration, translation and rotation), using the table as frame of reference. The projectors were calibrated using OpenCV camera calibration (2D to 3D calibration, intrinsic and extrinsic estimation), by matching reference pixels with their 3D position, using the already calibrated Optitrack. The HTC Vive was calibrated online, by placing the controller (most reliable transform) on a previously determined place in the table, and computing the transform between coordinate systems (3D to 3D calibration). Finally, the Leap Motion was manually calibrated, by finding the offset with HMD center. 1 The method used was adapted from http://wiki.optitrack.com/ index.php?title=Sync_Configuration_with_an_HTC_Vive_System

Technical Future Work

This work moves towards the construction of spaces that support mixed reality, similarly on how CAVEs support virtual reality. To this end, further work could take inspiration from literature. First, the interaction space could be extended as with Steerable Augmented Reality [START_REF] Wilson | Steerable augmented reality with the beamatron[END_REF], Roomalive [START_REF] Jones | RoomAlive: magical experiences enabled by scalable, adaptive projector-camera units[END_REF] and CAVE [START_REF] Cruz-Neira | Surround-screen projection-based virtual reality: the design and implementation of the CAVE[END_REF] technologies. In the same line, full nonrigid reconstruction [START_REF] Richard A Newcombe | Dynamicfusion: Reconstruction and tracking of non-rigid scenes in real-time[END_REF] could be use to scan and track physical elements. This could also be used to support touch interaction everywhere [START_REF] Xiao | WorldKit: rapid and easy creation of ad-hoc interactive applications on everyday surfaces[END_REF]. The use of projectors to create paper windows has clear limitations (occlusions, tracking), but we look forward towards the promising technology of Organic User Interfaces [START_REF] Holman | Organic User Interfaces: Designing Computers in Any Way, Shape, or Form[END_REF], which could replace the projectors partially or completely. In the mean time, the SAR augmentation could be improved by preventing overlapping between projectors.

We also consider that the use of embedded components would enrich the system. Light sensors were used in the past to perform precise projector calibration [START_REF] Ross | Invited Paper] Automatic Sub-pixel Projector Calibration[END_REF] or perform projector based tracking [START_REF] Lee | Moveable interactive projected displays using projector based tracking[END_REF]. Another use for embedded systems is actuation and sensing [START_REF] Valentin Heun | Smarter objects: using AR technology to program physical objects and their interactions[END_REF], which could be used to render the physical world (level 0 of our conceptual framework) more dynamic and interactive, as with smarter objects [START_REF] Valentin Heun | Smarter objects: using AR technology to program physical objects and their interactions[END_REF].

CONCLUSION AND DIRECTIONS TO FUTURE WORK

In this paper we presented a hybrid mixed reality conceptual framework, providing incremental augmentation and instrumentation. The framework was implemented using a combination of multiple display technologies. The first one is Spatial augmentation, always available augmentation where only the surfaces' appearance is modified. See-through devices can display mid-air information, and partially override physical information. Finally, HMDs provide a virtual replica of the physical scene, taking advantage of the freedom of virtual spaces without losing connection with the environment.

Even when the different technologies coexist simultaneously in our system, the interaction was designed to perform the tasks while keeping the connection with the physical space and other users; when a task is not possible, then the amount of digital support is increased, in combination with awareness considerations to help the users keep the link with their environment. The resulting system focuses on the smooth transition from a purely physical to a purely digital experience.

An underlying question driving this work is "where are these virtual spaces for the mind?". We consider that the cost of multitasking and traditional computers is that our attention constantly switches between two locations perceived as remote, and by easing the transition we can reduce the cognitive impact. If that is the case, then the dissociation created by modern computers and mobile devices could be greatly reduced by rendering them more aware and in synchrony with their surroundings. The next step, is to test these hypothesis with dedicated user studies using the presented ecosystem, and the answers will in term help the improvement of the system.

Figure 2 .

 2 Figure 2. Example scenes: volcano mock-up made out of sand (top), 3D printed Toyota engine (bottom). Each scene can be interacted with different display technologies: spatial augmentation (left), see-through displays (middle), and opaque head mounted displays (right).

Figure 3 .

 3 Figure 3. Conceptual framework with 6 levels of incremental augmentation, from Physical to Virtual worlds.

Figure 4 .

 4 Figure 4. Using spatial augmentation in combination with object tracking enables us: to use augmented pens to draw directly onto artifacts (A) and their surroundings (B), to manipulate the augmented artifacts to use in combination with traditional tools such as rulers (C), or to use of indirect interaction (D).

Figure 5 .

 5 Figure 5. Interactive pictures can display and support interaction with the augmented scene from a different point of view. They can be used in combination with traditional pen and paper.

Figure 6 .

 6 Figure 6. See-through displays, implemented using interactive pictures, allow to create the illusion of mid-air information (red text and arrow) floating over the engine (left), while also supporting indirect interaction with the scene via tracked pens and ray-casting (right).

Figure 7 .

 7 Figure 7. While immersed, the physical and digital counterparts of augmented objects can be mapped one-to-one (top). This coupling can be relaxed for additional flexibility (bottom).

Figure 8 .

 8 Figure 8. Users can navigate the augmented scene virtually. Left: the user jumped inside the scene and experience the volcano from an egocentric point of view. Right: the engineer rotated to her right, to see herself and her colleague.

Figure 9 .

 9 Figure 9. The setup supports surrounding tracking and augmentation.The tracking is performed using 4 Optitrack Flex 3 infrared cameras (sparse information), a Microsoft Kinect v1 (dense information pointcloud), and 2 HTC lighthouses (off the shelf HTC infrared emitters, used to track the HTC components). The augmentation is performed using three overhead projectors.

ACKNOWLEDGEMENTS

This work was possible thanks to the ISAR project (ANR-14-CE24-0013), and the collaboration from Diota. We would also like to thank our colleagues for their enriching feedback.


F Bertrand
email: fbertran@math.u-strasbg.fr

M Maumy
email: mmaumy@math.u-strasbg.fr

N Kobes

Using Factor Analyses to explore data generated by the National Grapevine Wood Diseases Survey

Keywords: Fisher-Freeman-Halton exact test, Multiple correspondence analysis, Factor analysis of mixed data, Factor analysis of multi-tables

The Grapevine Wood Diseases National Observatory yields a cohesive and large data set which may be dealt with using different approaches. In our study, we have to deal with complex data, as it is composed of quantitative and qualitative variables which evolve with time, since data for three successive years are available. The objective was to produce of the largest possible amount of information from this data set, in order to highlight main trends. To this aim, we used several techniques of data analysis. Our study thus developed in three points. First, relationships between the different variables are identified using bivariate measures of association and tests. Then factorial methods, namely multiple correspondence analysis and factor analysis of mixed data are used to look for multivariate dependencies between the variables of the dataset. Lastly, we used factor analysis of multi-tables, each table representing a year, in order to account for the successive years of data. The factor analysis of mixed data and the factor analysis of multi-tables will be detailed here, as well as all of our results.

Foreword

It is easy to perform the following analyses with any statistical software providing MCA techniques and especially with R (Team, 2007) and the FactoMineR package [START_REF] Husson | FactoMineR: Factor Analysis and Data Mining with R[END_REF]. Indeed this R package is fully integrated in the R Commander interface [START_REF] Fox | Rcmdr: R Commander[END_REF] and therefore all the R code is generated automatically using menus and popup windows. Some of the procedures of FactoMineR package were slightly enhanced to produce the graphical outputs that were included throughout this paper. Yet all the R code lines written by the authors are available upon simple request to the corresponding author 1 .

Background

Since the 8 th of November 2001 and the banning of the use of sodium arsenite in every sector of agriculture, in France then in Europe, there is no longer any authorized mean of curing the grapevine trunk diseases, namely the eutypa dieback (ED), the esca decline (ES) and the black dead arm (BDA). The sodium arsenite is a chemical made of arsenic and its use has been forbidden for the sake of wine growers, since there were hard evidence that this chemical is a carcinogenic one. The wine growers have been deeply concerned by this drastic measure since they feared that it may result in a steady rise of the grapevine trunk diseases rates in the French vineyards. Unfortunately, almost every times, the final stage of any of the grapevine trunk diseases is the death of the grapevine plant. Moreover, we should bear in mind that the scientific community lacks of accurate studies on the epidemiology of these diseases; no other cure could be quickly found and its use recommended.

In order to collect all the data and organize the activities of all the research teams working on the grapevine trunk diseases a national technical group was founded in 2001. This national technical group decided to create the National Grapevine Wood Diseases Survey for several years. We lay the emphasis on the fact that no other survey in the world was carried out with the same scale and with as many teams involved. Indeed, up to now, such surveys were only focused on a small area and on some varieties of vine, whereas the data that were collected by the National Grapevine Wood Diseases Survey deals with all the main French vine varieties and with all French vine-growing region. The duration of the survey was already extended from three years to six years and is to be extended to a total duration of nine years in order to provide a reliable set of data for a longitudinal data analysis. The primary objective 2 of the survey is to collect enough data to decide whether the banning of the use of sodium arsenite will result in a steady increase of the grapevine trunk diseases rates in order to be able to evaluate the economic scope of such an interdiction. The secondary objectives are to identify some of the factors that explain the variability of the grapevine trunk diseases rates and eventually design several experiments in order to validate the hypotheses that were highlighted by the survey's results.

Every French vine-growing region suffers from at least one of the grapevine trunk diseases and hereby is to be included in the survey. Yet, virtually, only 12 areas defined as well as vine-growing regions or administrative regions were sampled: Alsace, Aquitaine, Beaujolais -Rhône-Alpes-, Bourgogne, Centre, Diois -Rhône-Alpes-, Jura -Franche-Comté-, Languedoc-Roussillon, Midi-Pyrénées, Pays de la Loire, Poitou-Charentes, Provence-Alpes-Côte d'Azur. These areas are the widest vine-growing regions: the area covered by the grapevines in these regions accounts for 95,5 % of the total area covered by grapevines in France, see Table 1 andFigure 2. The Corse was not included in the survey since, as an island, it has its own specificity.

At least 25 parcels per vine-growing region and per vine variety were to be surveyed, in order to suitably depict the repartitions, the frequencies and the intensities of the diseases according to the vine-growing region and per vine variety.

For any of the parcels that were randomly chosen among the observation networks 300 grapevines are marked and spotted. These 300 grapevines were divided in 10 locations, randomly chosen in the parcel, of 30 grapevines. The size of the parcels was not taken into account for the selection of a parcel since it has no interest from an epidemiological point of view.

The observations were carried out on the same locations, hence on the same grapevines, during the three years of the survey.

The grapevine trunk diseases are searched for at two particular stages of the vine growth: the blossoming for the eutypa dieback and the veraison for the esca decline and the black dead arm. At this very moment, dead or non producing grapevines are also counted.

The only preset criterion for the choice of a parcel is the vine variety.

The features of the survey were specified by the Department of agriculture and the national technical group, to whom the ITV4 and the INRA5 belong, agreed formally to it. From a statistical point of view, the sample sizes chosen for this survey are of the same magnitude of those that are usually used in that kind of studies.

As to the criterion for the choice of a parcel, the vine variety is a simple way of identifying the parcels and is therefore well suited for such a large scale study since more than 40 different groups are involved in the data collection. From a biological point of view, a wide range of vulnerabilities was observed between the vine varieties even among the same vine-growing region. As a result, the vine variety was guessed to be one of the most interesting factors and the survey was designed to make it possible to have that hypothesis thoroughly investigated.

For any of the parcels that were randomly chosen, additional information about the characteristics and the cropping habits were collected using questionnaires, see Figure 3.

Wine-producing region

Area covered by grapevines (ha) Photos: Jacques Grosman.

FIGURE 2

French vine-growing regions and vine varieties produced there. The tags that were created to identify the vine varieties that were part of the analyses are put within brackets.

Background image from the NASA's Earth Observatory : The Topography of France 8 . 8 This image is available at the following address: http://earthobservatory.nasa.gov/Newsroom/NewImages/images.php?img_id=15360.

Alsace • Gewurztraminer (GWZ) • Pinot Auxerrois (AUX) • Riesling (RIS) Jura • Poulsard (PLS) • Savagnin (SAV) • Trousseau (TRS) Provence-Alpes-Côte d'Azur • Alphonse Lavallée • Cabernet Sauvignon (CBS) • Carignan (CAR) • Cinsault (CIN) • Grenache (GRE) • Italia • Lival • Mourvèdre • Muscat Petits Grains (MPG) • Muscat De Hambourg (MDH) • Syrah (SYR) Languedoc-Roussillon • Carignan (CAR) • Cinsault (CIN) • Muscat Petits Grains (MPG) • Syrah (SYR) Aquitaine • Cabernet Sauvignon (CBS) • Merlot (MER) • Sauvignon (SAU) Poitou-Charentes • Ugni Blanc (UB) Pays de la Loire • Chenin (CHE) • Melon (MEL) Bourgogne • Chardonnay (CHD) • Gamay (GAM) • Pinot Noir (PIN) • Sauvignon (SAU) Beaujolais • Gamay (GAM) Diois • Muscat Petits Grains (MPG) Centre • Cabernet Franc (CBF) • Chenin (CHE) • Sauvignon (SAU)

Preliminary statistical study-Bivariate tests

The esca decline (ES) shows all the same symptoms that black dead arm (BDA), see Figure 1 for a visual proof of that statement. As a consequence, these two diseases are always confounded in the same disease that we call esca/BDA. We have three years of data, 2003, 2004 and 2005, at our disposal for these two diseases. In order to account for the variation of the intensities of the diseases among the population of cropped parcels as well as for the mortality rate of the grapevines the following factors and variables were chosen: the vinegrowing region -factor-, the vine variety -factor-, the age of the parcel 9 -variable-, the density of planting -variable-, the rootstock that was used -factor-, the management of pruning residues -factor-, what happened to the dead wood -factor-, pre-pruning -factor-, the pruning method -factor-, the date of beginning and end of the pruning of the grapevinetwo variables-and the number of times the sodium arsenite was used from 1999 to 2001, just before the use of that cure was banned -factor-. Other data, as the vigour of the grapevinefactor-, the type of soil -factor-, the area of the parcel -variable-, were also collected. Yet, these two factors and this variable were not included in the study because of a low level of response or a low reliability of the answers.

We recall that the incidence rate of a disease in a parcel is the percentage of grapevine trunks that suffer from the disease in a given parcel and that the mortality rate is estimated by summing, in a given parcel, the dead or missing grapevines and the young "complants" which are the recently replaced grapevines still not producing grapes, divided by the total number of the grapevines, about 300, whose state was observed in this parcel.

The dataset is made of 701 parcels surveyed in 12 vine-growing regions. 26 different vine varieties 10 belong to the dataset. These data were collected following a minute protocol during the three years of the study. We removed 4 vine varieties, the Alphonse Lavallée, the Italia, the Lival and the Mourvèdre, from the dataset since they accounted only for a total of 1.28 % of the total sample size. We also removed from the database any of the parcels for which there was an outlier or a missing value.

In order to investigate our primary and secondary objectives, we used several statistical methods. We started with a three-step preliminary data analysis.

First, we checked the dependencies between the variables in our dataset which showed that the influence of the age of the parcel was strong on the incidence and mortality rates especially on the incidence rate of the esca/BDA. This led us to identify mainly four different stages (see Table 2) of relationship between the age of the parcel and the three variables that we wanted to explain. These relationships frankly differed from a stage to another with even sign reversals in the coefficients of correlation between the age of the parcel and the incidence rates of the two diseases and the mortality rate. This is a minute work which is available upon simple request to the corresponding author 11 . 9 The age of the parcel is equal to the age of most of the grapevines that belongs to that parcel. 10 The Alphonse Lavallée, the Pinot Auxerrois, the Cabernet Franc, the Cabernet Sauvignon, the Carignan, the Chardonnay, the Chenin, the Cinsault, the Gamay, the Gewurztraminer, the Grenache, the Italia, the Lival, the Melon, the Merlot, the Mourvèdre, the Muscat de Hambourg, the Muscat à Petits Grains, the Pinot Noir, the Poulsard, the Riesling, the Sauvignon, the Savagnin, the Syrah, the Trousseau and the Ugni Blanc. 11 These analyses are detailed in the report of Lionel Fussler.

Then, we carried out many non parametric Kruskal-Wallis tests, [START_REF] Siegel | Nonparametric Statistics for the Behavioural Sciences[END_REF]. The multiple comparisons made after these bivariate tests between a factor and a variable enabled us, for instance, to design sensibility classes to the two diseases among the different vine varieties. These results confirmed what was known up to now about the relative weakness of some of the vine varieties to the two diseases.

As a consequence of not only Kruskal-Wallis tests but also the many bivariate tests, like the independence χ² test, [START_REF] References | Categorical data analysis[END_REF], and Fisher-Freeman-Halton exact test, [START_REF] Freeman | Note on an exact treatment of contingency, goodness of fit and other problems of significance[END_REF], that we made12 we could reduce the number of variables, from 15 to 8, which were to be part of the statistical analysis and only selected those which were linked to the incidence rate of any of the two diseases or to the mortality rate. These 8 variables are the following: vine-growing region, vine variety, age of the parcel, rootstock, management of pruning residues, pre-pruning, pruning method, number of times the sodium arsenite was used between in 1999 and 2001.

The main aim of this case study is to derive the main trends of evolution of the grapevine trunk diseases from an epidemiological point of view.

Multiple Correspondence Analysis

The dataset is made of three tables, one per year. In this section, we consider the mean table of the three tables of data. As a consequence, the values of the incidences of the ED, the ES/BDA and the rate of mortality used in this part of the study, are the means of the values taken over the three years of the survey.

Multiple Correspondence Analysis: references

For a tutorial on basic and advanced multiple correspondence analysis (MCA), we point out the books of [START_REF] Greenacre | Theory and Applications of Correspondence Analysis[END_REF][START_REF] Greenacre | Correspondence Analysis in Practice[END_REF], [START_REF] Bry | Analyses factorielles simples[END_REF][START_REF] Bry | Analyses factorielles multiples[END_REF] and of [START_REF] Greenacre | Multiple Correspondence Analysis and related methods[END_REF].

Application to the dataset generated by the National Grapevine Wood Diseases Survey

The reason why the MCA has been chosen in first, is that this method allows us the simultaneous handling of the two types of attributes -quantitative and even qualitative through the recoding of the quantitative variables-and does not imply a priori assumptions on the variables, such a linear relationship between them or homoscedasticity.

The MCA was applied in two steps. At the beginning we made a multiple correspondence analysis with only the following active variables: the incidence rate of eutypa dieback, the incidence rate of esca/bda, the mortality rate of the grapevines, the age of the parcel.

The main aim of the MCA was to spotlight the dependencies between the grapevine trunk diseases and the mortality rate of the grapevines. As a consequence of the results of the bivariate tests, we had to include the age of the parcel to account for some of the variation of the mortality rate of the grapevines and of the incidence rates of the grapevine trunk diseases. If the vine variety (22 levels) and vine-growing regions (11 levels) were not selected as active variables for this first analysis it is mainly due to both their too high number of levels and their unbalanced factor levels.

Then we studied the influence of the remaining variables as supplementary qualitative variables: these variables were represented on the very display we built during the first step. Yet for the sake of readability, we virtually plotted each of the supplementary qualitative variables on a different graph:

vine-growing region, vine variety, number of times the sodium arsenite was used between 1999 and 2001, rootstock, pre-pruning, management of pruning residues, pruning method.

In order to achieve these analyses and use the MCA, it was mandatory to derive factor levels from segmenting the range of the continuous variables like the incidence rates of the two diseases and the mortality rate.

These factor levels were designed not only to comply with the results of the bivariate tests but also to set up balanced factor levels for each variable in order to avoid the shrinkage of the axes by the factor levels whose sizes are too low.

We now detail, following the recommendation of [START_REF] Savary | Use of Categorical Information and Correspondence Analysis in Plant Disease Epidemiology[END_REF], how to encode the quantitative variables into classes, i.e. define quantitative boundaries of classes and encode the values of the quantitative variables according to these boundaries. Performing this encoding process with care will allow the investigator to:

define the boundaries such that they represent the (maximum possible) error made in the measurement of each variable (variables with low accuracy would be represented by a few, broad classes, while variables with high accuracy would be represented by a larger number of classes), link the definition of classes with key-values, thresholds, or any information that might be available beforehand.

The process of converting quantitative data into coded data is flexible, different options being available depending on the variable at hand and no statistical restriction made.

In our data set, the three rates and the age of the parcel were four continuous variables. As to the rates, they were categorized in three classes: low, medium and high not only on epidemiological grounds but also according to the depencies between these three variables and the age of the parcel that were spotted during the preliminary analyses. So three grades, from 0 to 2, for every rate, as well as incidence or mortality one, are considered. The range of the age of the parcel was segmented in four intervals following the four stages we identified in Section 3. The Table 2 sums up the designed factor levels for the continuous variables.

The further analysis of the resulted coded data by means of contingency tables depends on the class-filling, and therefore on the number of classes relative to the size of the sample. Therefore we checked the consistency of the former dependencies spotted when the coded variables were continuous and the new dependencies we now spotted using χ² tests and Fisher-Freeman-Halton exact tests between the coded variables and the factors of our dataset. The analyses depicted here are the basis for the display of the supplementary qualitative variables. Since the supplementary variables do not contribute to the construction of the axes of the MCA, we will only provide their coordinates and square cosine. The main results are the following: the high mortality rates -Mort2-are associated with high incidence levels of the two trunk diseases -Esca2 and Euty2-, the high incidence levels of eutypa dieback are associated with old grapevines -from 25 to 40 years old-and esca decline/black dead arm are associated with young grapevines -from 15 to 25 years old-, the incidences seem to evolve in opposite ways: if the level of incidence of one of the two diseases is high then the level of incidence of the other one will be low, some vine-growing regions or vine varieties are linked to high level of vulnerabilities to the two diseases, e.g. Poitou-Charentes and Euty2 or Poulsard and Esca2, the use of sodium arsenite does not seem to account for significant changes of the evolution of the diseases, the association of burning the pruning residues and medium levels of incidence of the two diseases -Euty1 and Esca1-, is most probably due to the link between the vinegrowing regions and the vine varieties and the fact that the use of the burning technique to dispose of pruning residues is mainly used in Bourgogne.

Multiple Factor Analysis for Mixed Data

We just made a MCA analysis on the data and therefore we had to turn the quantitative variables -the age of the parcel, the rates of incidence and the rate of mortality-into qualitative ones by cutting the range of values into three classes or four classes. Since this step was handled with care this should have enabled us to spot even non linear associations between the variables we analysed. Yet, it is also possible to perform a multiple factor analysis for mixed data (FAMD) [START_REF] Escofier | Traitement simultané de variables quantitatives et qualitatives en analyse factorielle[END_REF][START_REF] Pagès | Analyse factorielle multiple appliquée aux variables qualitatives et aux données mixtes[END_REF][START_REF] Pagès | Analyse factorielle de données mixtes[END_REF]), i.e. for a dataset mixing quantitative and qualitative active variables.

There exists two advantages and one possible drawback to apply this method. The first advantage is to set the influence of any of the quantitative (4) or the qualititative (1) variables which participate to the analysis as active variables to the same level. Indeed during the process of construction of the directions of maximal inertia the influence of all the variables, quantitative as well as qualitative, will be balanced (p.97 Pagès 3004) and therefore will not depend on the number of the levels of the qualitative variable. The second advantage, following the recommendations (p.94) of the article written by [START_REF] Pagès | Analyse factorielle de données mixtes[END_REF], is to not encode the continuous variables (here, there are four) as factors with respect one qualitative variable (vine-growing region, vine variety, number of times the sodium arsenite was used between 1999 and 2001, rootstock, pre-pruning, management of pruning residues, pruning method). The drawback is that we will not be able to spot any non linear association between the quantitative variables.

The reader may not be familiar with such techniques which we will now briefly survey following the article written by [START_REF] Pagès | Analyse factorielle de données mixtes[END_REF].

Describing the method

Data and notations

Let us consider I units. Every unit i has a weight i w such that 1 1

I i i w = = ∑
. These units are described by: Q quantitative variables; these variables are always supposed to have zero mean and are scaled to unit variance. The use of standardized variables is mandatory since we use both quantitative and qualitative active variables. J qualitative variables; the j th variable has j K levels, the set of all the levels of the J qualitative is of size The total number, T , of quantitative variables and of indicator functions equals to Q K + . iq x denotes the value of the unit i for the q th variable,

1 q Q = K , ij
x the level of the unit i for the j th variable and j ik

x is equal to 1 if i has the level j k of the variable j and else

j ik
x is equal to zero.

Displaying the variables in R I

Let us consider I R the space of functions defined on I. The matrix D defined by the weights of the units provides this space with a diagonal metric. More precisely we have D equal to:

0 if ( , ) if i i j D i j w i j ≠ ⎧ = ⎨ = ⎩ .
As in a normed principal component analysis (PCA), the variables will be represented by vectors whose lengths are equal to 1.

As in a MCA, the j th variable is represented by the set j N of j K vectors representing the j K centered indicator functions of the levels of the j th variable. This set of vectors spans the vector space j E , whose dimension equals to 1 j K -, of the functions that are centered and constant on every set of the partition defined by the levels of the j th variable. In order that j N inherits of the same properties in matter of inertia than in a MCA, we have to set the weight of the j k th indicator function to 1/ j k w . Virtually this weight should be set to ( )

1/ j k Q w ⋅
to have exactly the same inertia than in a MCA. Indeed in a MCA the inertias are averaged by the number of variables. Yet using these weights would lead to an inadequate property -averaging-since for the factorial analysis of mixed data the qualitative variables are compared to quantitative ones whose inertias are not averaged. Some PCA softwares are not designed to assign weights to columns. As a consequence we prefer to divide the values of the j k th indicator functions by j k w .

Displaying the units in R T

The dimension of the vector space T R equals to the sum of the number of quantitative variables, Q , and of the number of indicator functions, K . We provide this vector space with the usual Euclidean metric. The distance between two units i and j is then given by the following formula:

1 ²( , ) ( )² ( )² j j j j iq lq ik lk q Q j J k K k d i l x x x x w ∈ ∈ ∈ = - + - ∑ ∑∑
, with as a sub case the distance between a unit and the gravity center of the all the units. This gravity centre is located at O , the origin of the vector space T R , as soon as the variables are centered. This is an assumption we made on the quantitative variables at the very beginning of this section. As to the indicator functions that were derived from the qualitative variables and divided by j k w in order to assign adequate weights to the columns, the mean of the j k th column is equal to q k w . These renormalizations lead to the following formulas:

2 ( ) 2 ( ) 1 ²( , ) ² j j j j ik j i iq k iq q Q j J k K q Q j J j i k x w d i O x w x w w ∈ ∈ ∈ ∈ ∈ ⎛ ⎞ - ⎜ ⎟ = + - = + ⎜ ⎟ ⎝ ⎠ ∑ ∑∑ ∑ ∑ , with (
) j i the level that the j th variable takes for the i th unit and ( )

j i
w the corresponding proportion.

Graphical outputs

As in any factor analysis we display:

The units by their projection on the inertia axes. We denote by ( ) s F i the projection of the i th unit on the axis s F of order s . The quantitative variables using their correlation coefficients with the axes s F . The levels of the j th qualitative variable using the gravity centres of the corresponding units. We denote by ( )

s j
F k the projection on the axis s F of order s of the centre of gravity of the units featuring the k th level of the j th qualitative variable.

Applying FAMD to the data generated by the National Grapevine Wood Diseases Survey

In order to be able to put side by side the results we will compute with those we got from the MCA, we will proceed in somewhat a similar way than what we did when using the MCA. We first build the factorial axes using the four main variables. The first four active variables will remain the same and are quantitative ones: the incidence of eutypa dieback, the incidence of esca/BDA, the mortality of grapevine trunks and the age of the parcel. The fifth active variable is one qualitative variable chosen among all those we decided to include in the analysis and that were, up to now, used as supplementary variables: vine-growing region, vine variety, number of use of the sodium arsenite, rootstock, pre-pruning, pruning residues, pruning method.

Vine-growing regions and vine varieties

Since vine-growing regions and vine varieties are two strongly dependant variables we will analyse them together, Figure 9 up and down.

A first point is that the grapevine mortality is close to being proportional to the sum of the vectors associated with the two diseases. A second point is that the diseases are opposed one to each other, which agrees with the results we got with the MCA: the two diseases are likely to not completely -or not at all-evolve the same way. The age of the parcel is still strongly linked with the second factor axis which is what we found out using the MCA.

We spot again, as with the MCA, an association between esca/BDA and the vinegrowing region Jura or between Poitou-Charentes and eutypa. As to the associations between the vine varieties and the diseases, the Poulsard, the Savagnin and the Trousseau are close to the esca/BDA, which are the same results as with the MCA. The association between diseases and mortality is still strong as well as the location of the variable age of the parcel along the second factor axis. There is an opposition between the rise of the mortality of the grapevines and the non-use or use twice of sodium arsenite as well as a very close location between high incidence rates of eutypa dieback and the three times repeated use of sodium arsenite. Yet there are only several parcels that were cured three times during the period from 1999 to 2001 and there is no evidence that the use of sodium arsenite accounts for a decrease of the incidence rate of eutypa dieback. This result is to be confirmed by other experiments. Another way to understand this relationship is that parcels with high incidence rates of eutypa dieback were treated every year by vine growers.

Pre-pruning and management of pruning residues

The pre-pruning and the management of pruning residues, two disease prevention policies, seemed to have no effect on the incidence rates of the two diseases and on the mortality rate of the grapevines when analysed with the MCA. Is it still the case with the FAMD; see Figure 11 First of all, we notice the same association pattern between the incidence of eutypa dieback, the incidence of esca/BDA, the mortality of the grapevines and the age of the parcel as those we spotted with the MCA: the higher the incidence of the eutypa dieback and of the esca/BDA, the higher the mortality rate.

We lay the stress on the fact that although there seems to be an opposition between the mortality of the grapevines or the esca/BDA and the pre-pruning of the wine ("PreP1") which was not spotted by the MCA, this result is not backed up by the plot of the association of the the PreP factor with the two first factor axes since these accounts for almost none of the variability of PreP factor.

Burning the pruning residues is opposed to the incidence of the diseases and to the mortality rate. As to the removal of the pruning residues it is more difficult to state conclusions since the number of parcels where it was used is very small (7 out a sample of 191 parcels).

Multi-tables correspondence analysis

The multi-tables correspondence analysis (CA) -also known as K-tables correspondence analysis- [START_REF] Bry | Analyses factorielles multiples[END_REF][START_REF] Escofier | Analyses factorielles simples et multiples[END_REF][START_REF] Cazes | Quelques méthodes d'analyse factorielle d'une série de tableaux de données[END_REF]) will be used in order to study the yearly evolution of the incidences rates of the two diseases and of the mortality of the grapevines. Indeed up to now, our analyses only focused on the mean values of these three variables over the three years : 2003, 2004 and 2005. As a consequence the time factor was not taken into account. The multi-tables correspondence analysis will enable us to check whether or not one of the years has a high impact on the average association patterns we found before. It was a high concern since the summer 2003 was very hot in France.

Description of the method

This statistical tool was applied to the data tables of the three years 2003, 2004 and 2005 for the same units and variables. We now describe more precisely the way we used the multi-tables correspondence analysis. We begin with the analysis not taking time into account:

First we bound together in the same table the data for the incidence rates of the two diseases and of the mortality rate for the three years 2003, 2004 and 2005. We got a table with 321 rows -one for each of the 3*107 units-and three -EUTY_Tot, ESCA_Tot and MORT_Tot-columns. These three new variables will be the active ones. We then computed the Burt The number of points on the Figure 12 is so high that we decide to split it in several sub-diagrams for the sake of readability, see Figure 13.

For the esca/BDA the levels of the three years are close which means a stable pattern for the incidence rates through the three years of the survey.

For the eutypa dieback and the mortality, the two years 2004 and 2005 are close and thus for all the levels of the variable Euty and Mort whereas the year 2003 lies apart. This seems to highlight a stable expression of the symptoms and of the mortality in 2004 and 2005 after a variation between 2003 and 2004. The mean incidence for eutypa-dieback was: 3.5% in 2003, 3.5% in 2004, 3.6% in 2005. As a consequence the multi-tables correspondence analysis spotted differences in the incidence patterns of eutypa dieback whilst the mean values of incidences where close to be equal. Yet the age of the parcel was not included in this analysis. This may have resulted in misleading results. It is what we will check in the following subsection.

The analysis of the second factor plane highlights a stable opposition through the time for the increase of the two diseases. For all the factors and for any given level of these factors, the annual points of these levels lie close and the pattern they draw remain somewhat similar for the three years of the study 2003, 2004 and 2005. We remark (see Figure 16) that the path of the mortality rate stays stable during the three years while the paths of the two incidence rates change slightly during this period. These variations could not be spotten by only looking at the incidence rates which were almost constant for the three years -2003, 2004 and 2005-; for instance the mean incidences, for the eutypa were 3.5 % in 2003, 3.5 % in 2004 and 3.6 % in 2005. The analysis of the second factor plane highlights a steady opposition through the time for the increase of the two diseases.

Other procedures of data analysis were used to study this dataset, namely the Multiple Factor Analysis (MFA) [START_REF] Escofier | Analyses factorielles simples et multiples[END_REF] and the ACT-STATIS method (Lavit, Escoufier, Sabatier and Traissac 1994), using the FactoMineR [START_REF] Husson | FactoMineR: Factor Analysis and Data Mining with R[END_REF] and the ade4 [START_REF] Chessel | The ade4 package -I-One-table methods[END_REF]) packages for R (Team 2007). The results we got this way were similar to those of the K-tables CA.

Conclusion

This case study highlights the interest of the joint use of several statistical procedures. Its main goal is to provide an example of use of adequate tools to analyse quantitative, qualitative and dynamic data. Moreover from a biological point of view, the associations between the variables of our survey were hardly studied before. We aimed to explore deeply the dataset by laying the stress on the analysis of the consequences of the banning of sodium arsenite on the incidence rates of the diseases and the mortality rate as well as on their evolutions.

The multiple correspondence analysis allowed us to depict the associations between the main variables. The factorial analysis of mixed data confirmed these results and enabled us to spot other associations between the variables. The multi-table correspondence analysis took time into account and therefore provided us an analysis of the evolution of the two diseases and of the mortality rate.

One of the main results of these analyses is that the diseases highly depend on the vine variety as well as on the vine-growing region. We also shed light on the fact that the eutypa dieback is mainly linked with the age of the grapevine whereas the higher esca/BDA incidence rates were associated with young grapevines and the increase of its incidence rate is slowed down by the use of sodium arsenite which is not the case for the eutypa dieback. As to the mortality rate of the grapevines the main factors of influence are the eutypa dieback and the esca/BDA incidence rates as well as the age of the grapevine. The link between the use of sodium arsenite and the mortality rate was not spotted to be significant.

Other statistical methods were used to investigate this dataset from a modelling point of view, which will be the core of a upcoming article. For instance, both binary and ordinal logistic regressions models were fitted and the results they highlighted do not depart from those we got with the factor analyses we made throughout this analysis.

FIGURE 1

 1 FIGURE 1 Grapevines suffering from: • Top left: Eutypa dieback. • Top right: Black dead arm. • Bottom right: Esca.

  FIGURE 4Display of the first and second principal axes of the MCA.

  FIGURE 6Display on the first and second principal axes of the MCA of the two supplementary variables the vine-growing region, on top, and the vine variety, on bottom.

  FIGURE 9Display on the first and second principal axes of the FAMD with the two qualitative variables the vine-growing region, on top, and the vine variety, on bottom.

  .

  FIGURE 11Display on the first and second principal axes of the FAMD with the qualitative variable the pre-pruning, on top, and the management of pruning residues, on bottom.

FIGURE 13 (

 13 FIGURE 13 (On top) and 14 (On bottom) Display on the first, second, third and fourth principal axes of the K-tables CA of the means per year without taking into account the age of the parcel for the eutypa dieback disease only.

TABLE 1 French

 1

administrative regions and the area covered by grapevines. In bold the regions that were sampled for the survey.

of use of the sodium arsenite

	Variable	Symbol	Factor levels	Definition of factor levels	Unit
	Diseases and mortality
	Eutypa Dieback	Euty	Euty0; Euty1; Euty2	Euty0: Euty = 0; Euty1: 0 < Euty ≤ 2; Euty2: 2 < Euty	%
	Esca/BDA	Esca	Esca0; Esca1; Esca2	Esca0: Esca= 0; Esca1: 0 < Esca ≤ 3; Esca2: 3 < Esca	%
	Mortality	Mort	Mort0; Mort1; Mort2	Mort0: 0 ≤ Mort < 3; Mort1: 3 ≤ Mort < 10; Mort2: 10 ≤ Mort	%
	Active variable
	Age of the parcel	Age	Age0; Age1; Age2; Age3	Age0: 0 ≤ Age < 15; Age1: 15 ≤ Age < 25; Age2: 25 ≤ Age < 40; Age3: 40 ≤ Age	years
	Supplementary variables
	Vine-growing region	-
		Ars	Ars0; Ars1; Ars2; Ars3	Ars0: 0 time; Ars1: 1 time; Ars2: 2 times; Ars3: 3 times	none
	Rootstock	-	101-14; 161-49; 3309C; 41B; R110; SO4	Name of the rootstock used	none
	Pre-pruning	PreP	PreP0; PreP1	PreP0: no pre-pruning was done; PreP1: grapevine was pre-pruned	none
	Pruning residues	-	Crushed; Burned; Removed	What happened to the pruning residues	none
	Pruning method	-	Royat cordon; Gobelet; Guyot	Name of the method used to prune the grapevine	none

ALS; AQT; BJL; BRG; CEN; DIO; JUR; LRO; PAC; PCH; PDL ALS: Alsace; AQT: Aquitaine; BJL: Beaujolais; BRG: Bourgogne; CEN: Centre; DIO: Diois; JUR: Jura; LRO: Languedoc-Roussillon; PAC: Provence-Alpes-Côte d'Azur; PCH: Poitou-Charentes; PDL: Pays de la Loire none Vine variety -AUX; CAR; CBF; CBS; CHD; CHE; CIN; GAM; GRE; GWZ; MDH; MEL; MER ; MPG; PIN; PLS; RIS; SAU; SAV; SYR; TRS; UB AUX: Pinot Auxerrois; CAR: Carignan; CBF: Cabernet Franc; CBS: Cabernet Sauvignon; CHD: Chardonnay; CHE: Chenin; CIN: Cinsault; GAM: Gamay; GRE: Grenache; GWZ: Gewurztraminer; MDH: Muscat De Hambourg; MEL: Melon; MER: Merlot; MPG: Muscat Petits Grains; PIN: Pinot Noir; PLS: Poulsard; RIS: Riesling; SAU: Sauvignon; SAV: Savagnin; SYR: Syrah; TRS: Trousseau; UB: Ugni Blanc none Number

TABLE 2

 2 Variables and their levels.

TABLE 3

 3 Eigenvalues and percentages of inertia with respect to the principal axes of the MCA.

		Mass	Quality	Relative Inertia	Coord. Dim.1	Inertia Dim.1	Cosine² Dim.1	Coord. Dim.2	Inertia Dim.2	Cosine² Dim.2
	Euty0	0.086	0.657	0.073	-0.093	0.002	0.005	1.030	0.252	0.553
	Euty1	0.103	0.587	0.065	0.551	0.074	0.214	-0.439	0.055	0.136
	Euty2	0.061	0.756	0.084	-0.802	0.092	0.208	-0.703	0.083	0.159
	Esca0	0.013	0.949	0.105	1.002	0.030	0.054	0.802	0.023	0.035
	Esca1	0.143	0.429	0.048	0.469	0.074	0.293	-0.249	0.025	0.083
	Esca2	0.094	0.622	0.069	-0.844	0.158	0.433	0.268	0.019	0.044
	Mort0	0.030	0.882	0.098	1.053	0.077	0.148	1.326	0.144	0.236
	Mort1	0.098	0.606	0.067	0.704	0.115	0.322	-0.222	0.013	0.032
	Mort2	0.122	0.512	0.057	-0.823	0.194	0.646	-0.142	0.007	0.019
	Age0	0.033	0.866	0.096	0.458	0.016	0.032	1.472	0.201	0.335
	Age1	0.105	0.579	0.064	-0.340	0.029	0.084	0.270	0.021	0.053
	Age2	0.087	0.654	0.073	-0.192	0.007	0.020	-0.617	0.091	0.202
	Age3	0.025	0.902	0.100	1.508	0.132	0.248	-0.986	0.066	0.106
		Mass	Quality	Relative Inertia	Coord. Dim.3	Inertia Dim.3	Cosine² Dim.3	Coord. Dim.4	Inertia Dim.4	Cosine² Dim.4
	Euty0	0.086	0.657	0.073	-0.263	0.020	0.036	-0.103	0.003	0.006
	Euty1	0.103	0.587	0.065	0.124	0.005	0.011	0.667	0.174	0.314
	Euty2	0.061	0.756	0.084	0.159	0.005	0.008	-0.985	0.225	0.313
	Esca0	0.013	0.949	0.105	2.489	0.263	0.334	0.981	0.047	0.052
	Esca1	0.143	0.429	0.048	-0.091	0.004	0.011	-0.584	0.185	0.454
	Esca2	0.094	0.622	0.069	-0.200	0.013	0.024	0.749	0.201	0.341
	Mort0	0.030	0.882	0.098	1.226	0.147	0.201	-0.913	0.093	0.112
	Mort1	0.098	0.606	0.067	-0.653	0.140	0.277	0.119	0.005	0.009
	Mort2	0.122	0.512	0.057	0.230	0.021	0.051	0.125	0.007	0.015
	Age0	0.033	0.866	0.096	0.080	0.001	0.001	0.217	0.006	0.007
	Age1	0.105	0.579	0.064	-0.613	0.131	0.273	-0.238	0.023	0.041
	Age2	0.087	0.654	0.073	0.879	0.222	0.409	0.045	0.001	0.001
	Age3	0.025	0.902	0.100	-0.579	0.027	0.037	0.566	0.030	0.035

TABLE 4

 4 Coordinates, contributions and square cosine of the levels of the active variables for the first four principal axes of the MCA.

TABLE 5

 5 Coordinates and square cosine of the levels of the supplementary variables.

table

  Tot B for this table which is the sum over the three years of the yearly Burt tables2003, 2004 and 2005, 2003

	6.2
	B , 2004 B et 2005 B .
	Each of the yearly table is made of 107 rows and 3 columns -EUTY_0X, ESCA_0X
	and MORT_0X, where is equal to 3 for the year 2003, 4 for the year 2004 and 5 for
	the year 2005-.
	We laid on the right of the table Tot B the three yearly Burt tables 2003 B , 2004 B and 2005 B
	of the incidence of diseases and the mortality rate, each one with 13 rows and 13
	columns. The columns of the three Burt tables 2003 B , 2004 B and 2005 B with be used as
	supplementary variables in the upcoming correspondence analysis.
	We laid on the bottom of the table Tot B the three yearly Burt tables 2003 B , 2004 B and
	2005 B of the incidence of diseases and the mortality rate, each one with 13 rows and

.1 Ignoring the age of the parcel Axis Singular Eigenvalue Percentage of inertia

			Individual Cumulated
	1	0.495	33.326	33.326
	2	0.420	23.944	57.269
	3	0.347	16.387	73.657
	4	0.323	14.178	87.835
	5	0.250	8.498	96.333
	6	0.164	3.667	100.000

TABLE 7

 7 Eigenvalues and percentages of inertia with respect to the principal axes of the K-tables CA.

		Coord.	Coord.	Coord.	Coord.	Inertia	Cosine²	Inertia	Cosine²	Inertia	Cosine²	Inertia	Cosine²
		Dim.1	Dim.2	Dim.3	Dim.4	Dim.1	Dim.1	Dim.2	Dim.2	Dim.3	Dim.3	Dim.4	Dim.4
	EUTY_04.0	-0.029	0.307	-0.280	-0.391		0.001		0.073		0.060		0.118
	EUTY_04.1	0.101	-0.251	-0.324	0.855		0.005		0.033		0.056		0.388
	EUTY_04.2	-0.590	-0.207	0.896	0.017		0.152		0.019		0.350		0.000
	ESCA_04.0	0.809	0.585	0.175	0.381		0.271		0.142		0.013		0.060
	ESCA_04.1	0.030	-0.456	0.122	-0.322		0.001		0.155		0.011		0.077
	ESCA_04.2	-0.780	0.477	-0.356	-0.028		0.308		0.115		0.064		0.000
	MORT_04.0	1.011	0.386	0.218	-0.078		0.456		0.066		0.021		0.003
	MORT_04.1	-0.091	-0.517	-0.431	-0.352		0.005		0.157		0.109		0.073
	MORT_04.2	-0.616	0.418	0.120	0.128		0.238		0.110		0.009		0.010
	EUTY_05.0	-0.091	0.418	-0.254	-0.400		0.006		0.125		0.046		0.114
	EUTY_05.1	0.162	-0.318	-0.308	0.794		0.014		0.054		0.051		0.339
	EUTY_05.2	-0.633	-0.321	0.905	-0.050		0.170		0.044		0.347		0.001
	ESCA_05.0	0.895	0.360	0.020	0.670		0.309		0.050		0.000		0.173
	ESCA_05.1	0.083	-0.401	0.356	-0.132		0.005		0.118		0.093		0.013
	ESCA_05.2	-0.758	0.446	-0.350	0.011		0.298		0.103		0.064		0.000
	MORT_05.0	0.860	0.104	0.166	-0.139		0.367		0.005		0.014		0.010
	MORT_05.1	-0.195	-0.561	-0.381	-0.070		0.022		0.182		0.084		0.003
	MORT_05.2	-0.679	0.299	0.190	0.141		0.284		0.055		0.022		0.012

TABLE 8

 8 Coordinates, contributions and square cosine of the levels of the active variables for the first four principal axes of the K-tables CA without taking the age of the parcel into account.FIGURE 12Display on the first and second principal axes of the K-tables CA of the means per year without taking into account the age of the parcel.

TABLE 9

 9 Eigenvalues and percentages of inertia with respect to the principal axes of the K-tables CA. Display on the first, second, third and fourth principal axes of the K-tables CA of the means per year taking into account the age of the parcel.

		1.0							1.0
	Dim 2 (17.27%)	-0.5 0.0 0.5	AGE.0 ESCA_Tot.0 EUTY_Tot.0 AGE.1 AGE.2 AGE.3 EUTY_Tot.1 UTY_Tot.2 ESCA_Tot.1 ESCA_Tot.2 MORT_Tot.0 MORT_Tot.1 MORT_Tot.2 AGE.0 AGE.1 AGE.2 AGE.3 AGE_03 AGE_03.1 AGE_03.2 AGE_03.3 AGE_04.0 AGE_04.1 AGE_04.2 AGE_04.3 AGE_05.0 AGE_05.1 AGE_05.2 AGE_05.3	Dim 2 (17.27%)	-0.5 0.0 0.5	AGE.0 ESCA_Tot.0 EUTY_Tot.0 AGE.1 AGE.2 UTY_Tot.2 ESCA_Tot.2 MORT_Tot.0 MORT_Tot.2 EUTY_Tot.0 UTY_Tot.2 EUTY_03.0 EUTY_03.2 EUTY_04.0 UTY_04.2 EUTY_05.0 TY_05.2 AGE.0 AGE.1 AGE.2 AGE.3 EUTY_Tot.1 ESCA_Tot.1 MORT_Tot.1 EUTY_04.1 EUTY_05.1 EUTY_Tot.1 EUTY_03.1 AGE.3
			-0.5	0.0	0.5	1.0	1.5			-0.5	0.0	0.5	1.0	1.5
				Dim 1 (30.19%)						Dim 1 (30.19%)
		1.0							1.0
	Dim 2 (17.27%)	-0.5 0.0 0.5	AGE.0 ESCA_Tot.0 EUTY_Tot.0 AGE.1 AGE.2 AGE.3 EUTY_Tot.1 UTY_Tot.2 ESCA_Tot.1 ESCA_Tot.2 MORT_Tot.0 MORT_Tot.1 MORT_Tot.2 ESCA_Tot.0 ESCA_Tot.1 ESCA_Tot.2 ESCA_03.0 ESCA_03.1 ESCA_03.2 ESCA_04.0 ESCA_04.1 ESCA_04.2 ESCA_05.0 ESCA_05.1 ESCA_05.2 AGE.0 AGE.1 AGE.2 AGE.3	Dim 2 (17.27%)	-0.5 0.0 0.5	AGE.0 ESCA_Tot.0 EUTY_Tot.0 AGE.1 AGE.2 UTY_Tot.2 ESCA_Tot.2 MORT_Tot.0 MORT_Tot.2 MORT_Tot.0 MORT_Tot.2 MORT_03.0 MORT_03.2 MORT_04.0 MORT_04.2 MORT_05.0 ORT_05.2 AGE.0 AGE.1 AGE.2 AGE.3 ESCA_Tot.1 MORT_Tot.1 MORT_04.1 MORT_05.1 MORT_Tot.1 EUTY_Tot.1 MORT_03.1 AGE.3
			-0.5	0.0	0.5	1.0	1.5			-0.5	0.0	0.5	1.0	1.5
				Dim 1 (30.19%)						Dim 1 (30.19%)
							FIGURE 16

Institut Technique de la Vigne et du Vin (Technical Institute of Grapevine and Wine).

Institut National de la Recherche Agronomique (French National Institute for Agricultural Research).

The Champagne-Ardennes vine-growing region settled its own local grapevine wood disease survey.

The Midi-Pyrénées vine-growing region was to be sampled in 2003 and 2004 yet it virtually started to send back results in 2005 and therefore could not be included in the analysis for the 2003 to 2005 period.

These analyses are detailed in the report of Lionel Fussler.

Acknowledgements

The authors are grateful to Editor-in-Chief Dominique Haughton and two referees for their helpful and valuable comments.

13 columns. The rows of the three Burt tables 2003 B , 2004 B and 2005 B with be used as supplementary units in the upcoming correspondence analysis. Then we completed the empty cells of this global table B with zeros. Finally we performed the correspondence analysis of the table B as described in Table 6 : the 9 variables EUTY_Tot.0, EUTY_Tot.1, EUTY_Tot.2, ESCA_Tot.0 ESCA_Tot.1, ESCA_Tot.2, MORT_Tot.0, MORT_Tot.1 and MORT_Tot.2 were active variables and the 27 variables EUTY_03.0, EUTY_03.1, EUTY_03.2, EUTY_04.0, EUTY_04.2, EUTY_04.3, EUTY_05.0, EUTY_05.1, EUTY_05. We then added the age of the parcel as an active variable and followed the same scheme of analysis.

Applying multi-tables correspondence analysis to the data generated by the National Grapevine Wood Diseases Survey

We carried out two analyses: the first one does not take into account the age of the parcel and the second one does.

It is legitimate to restrict our analyses to the four principal axes since these axes account for 88 % of the total inertia. The values of the inertia led us again to study only the first four axes that accounts for about 72 % of the total inertia of the whole dataset. We lay the stress on the fact that the fifth axe could be interesting to study too.

Coord

Use of the dataset policy

Permission is granted to use the dataset generated by the National Grapevine Wood Diseases Survey to Academic purposes only. Any scientific research works or results based on the use of the dataset generated by the National Grapevine Wood Diseases Survey must be authorized by the French Agriculture Office 13 .