J.-F Girard

3-D magnetotelluric inversion with coast effect modeling to assess the geothermal potential of Anses d'Arlet (Martinique, Lesser Antilles) 1

SUMMARY

Within the framework of a global French program towards development of renewable energies, Martinique Island (Lesser Antilles, France) has been extensively investigated (from 2012 to 2013) through an integrated multi-disciplinary approach, with the aim to identify precisely the potential geothermal resources previously highlighted (Gadalia et al., 2014). Among the investigation methods deployed (geological, geochemical and hydrogeological), we carried out three magnetotelluric (MT) surveys at three of the four most promising areas of Martinique, namely the Anses d'Arlet, the Montagne Pelée and the Pitons du Carbet prospects. A total of 32 MT tensors were collected in the Anses d'Arlet area in the frequency range 1000 Hz to 10 The 3-D resistivity model reveals a major resistive body elongated in the NNO-SSE direction, corresponding to the main structural volcanic axis of the area, and interpreted as a deep intrusion almost reaching the surface. Based on geological observations -an alteration zone located between Anses d'Arlet and Petite Ansethe shallow conductive layer identified eastward is interpreted as the remaining of an old cap-rock partly eroded that becomes thicker southwards. The latter could be related with the altered core of the Morne Larcher. Other studies allowed the reconstructing of the geothermal system evolution to its present and moderate apparent activity mainly located south of the resistive anomaly. This intrusion could act as heat source for the geothermal system. This sector is identified as the most interesting for further exploration wells. In order to better understand structures highlighted by the real data distribution and test the strategy to integrate correction coefficients for the coast effect (modeled separately) during the inversion, we designed a forward model using the same bathymetry, topography and MT sounding distribution of the survey. Impedance tensors were calculated for the 32 sites. The geometry of the 3-D structure has been designed quasi-independently of the data distribution to assess the impact of un-appropriate acquisition grid. The 3-D inversion was run with these synthetic data (Hautot et al, 2000(Hautot et al, , 2007)). The results indicate that the general structure is recovered for the first 1000 m. Deeper, and southward, the absence of MT soundings surrounding the resistive body prevented a good lateral constraint, causing its disappearance. To the north, the conductive body "pseudo caprock" is well defined, especially on layer 5 (Figure 2), were the apex of a supposed geothermal reservoir could be identified (sites 3 and 6). Deeper the eastern border remains very well constrained by the eastern MT synthetic soundings.

Keywords

  together with TEM soundings for potential static shift correction. A 3-D MT inversion of the full tensor was carried out including the coast effect.

Figure 1 .

 1 Figure 1. Location map of the investigated area on geological background. (Westercamp et al. 1990) and resistivity models from 351 to 1081 m deep (b, c, d, e). Superimposed elements are indicated in the legend.

Figure 2 .

 2 Figure 2. Left: synthetic model of the Anses d'Arlet province with arbitrary 3-D resistivity structure at 350-480 m depth. 32 MT soundings for which response is computed. Resistivity distribution is indicated. Right: inversion results at layer 5 (depth: 270 to 460 m). Be careful, scale is not equivalent on both plots.

:

  Magnetotelluric, 3-D inversion, Anses d'Arlet, Martinique, Geothermal exploration REFERFENCES Gadalia A, Baltassat JM, Bouchot V, Caritg S, Coppo N, Gal F, Girard JF, Gutierrez A, Jacob T, Martelet G, Rad S, Tailame AL, Traineau H, Vittecoq B, Wawrzyniak P, Zammit C (2014) Compléments d'exploration géothermique en Martinique : synthèse et recommandations pour les zones de la Montagne Pelée, des Anses d'Arlet, des Pitons du Carbet et du Lamentin. Rapport final BRGM/RP-63019 -FR, 227 p, 75 fig., 7 tabl., 4 ann., 1 CD. Hautot S, Tarits P, Whaler K, Le Gall B, Tiercelin JJ, Le Turdu C (2000) The deep structure of the Baringo Rift basin (central Kenya) from 3-D magneto-telluric imaging: Implications for rift evolution. Journal of Geophysical Research 105: 23493-23518. Hautot S, Single R, Watson J, Harrop N, Jerram DA, Tarits P, Whaler KA (2007) 3-D magnetotelluric inversion and model validation with gravity data for the investigation of large igneous provinces. Geophysical Journal International 170(3):1418-1430. Westercamp D, Pelletier B, Thibaut PM, Traineau H, Andreieff P (1990) Carte géologique de la France (1/50 000), feuille Martinique, BRGM, Orléans.


Evidence of the magnetotelluric 3-D inversion ability to account for static shift effects: A comparison between MT and heliborne TEM 1

SUMMARY

Within the framework of a global French program towards development of renewable energies, Martinique Island (Lesser Antilles, France) has been extensively investigated (from 2012 to 2013) through an integrated multi-disciplinary approach, with the aim to identify precisely the potential geothermal resources previously highlighted (Gadalia et al., 2014). Among the investigation methods deployed (geological, geochemical and hydrogeological), we carried out three magnetotelluric (MT) surveys above three of the four most promising areas of Martinique, namely the Anses d'Arlet, the Montagne Pelée and the Pitons du Carbet prospects. A total of about 100 MT stations were collected in the frequency range 1000 Hz to 10 -2 Hz together with TEM soundings for potential static shift correction. 3-D MT inversions of the full tensor were run on each sector which took into account the coast effect. Assuming that the static shift observed is the result of near-surface lateral inhomogeneity causing electric field distortion, it can be accounted for by the inversion scheme used by Hautot et al. (2000Hautot et al. ( , 2007) ) and no prior static correction was applied before inversion. The model results show that most of the soundings displaying static shift effect are accurately reproduced by near-surface resistivity distribution in the model. However it is generally difficult to prove that the shallow structures explaining the static distortion obtained by the MT inversion are real features at the scale of the survey. Here we consider additional high resolution geophysical data to overcome this problem. We demonstrate with the help of the results from the recent heliborne TEM survey covering the whole Martinique (marTEM project) that the sub-surface resistivity distribution obtained from 3-D MT inversion reproduces faithfully the resistivity distribution observed by TEM data (Figure 1). Data of two of the three investigated sectors (Anses d'Arlet and Mount Pelée geothermal provinces) are presented because there is a sufficient spatial density of MT soundings. To compare both datasets, TEM data were first homogenized over the investigated area by krigging the resistivity distribution by layer. The median resistivity of this new grid is then extracted above each cell of the MT model and compared to the MT resistivity. Cells that do not contain TEM information or are located too far away from the fly lines are removed from the statistics. The results show that 82% of the cells considered have a maximum resistivity variation below 400% for the Anses d'Arlet prospect and 79 % for the Montagne Pelée. Thus despite a widely different sampling scale between the two sets of data (53 MT soundings,> 20000 TEM for the Mount Pelée and 32 MT soundings, >6200 TEM), the comparison between the two near surface resistivity models illustrates the ability of our inversion scheme to take into account and reproduce static shift effects.

Keywords

Figure 1 .

 1 Figure 1. MT versus TEM resistivity for superficial layers. Above are represented the data of the Anses d'Arlet (AA) geothermal province and below from the Montagne Pelée (MP) geothermal province. Left plots display resistivity from the 3-D MT models for both sector at the indicated depth. Center plots display resistivity of the TEM data (see text for details). Right plots show variation between the two models (log scale).

:

  Magnetotelluric, static-shift, 3-D inversion, TEM, Martinique REFERFENCES Gadalia A, Baltassat JM, Bouchot V, Caritg S, Coppo N, Gal F, Girard JF, Gutierrez A, Jacob T, Martelet G, Rad S, Tailame AL, Traineau H, Vittecoq B, Wawrzyniak P, Zammit C (2014) Compléments d'exploration géothermique en Martinique : synthèse et recommandations pour les zones de la Montagne Pelée, des Anses d'Arlet, des Pitons du Carbet et du Lamentin. Rapport final BRGM/RP-63019 -FR, 227 p, 75 fig., 7 tabl., 4 ann., 1 CD. Hautot S, Tarits P, Whaler K, Le Gall B, Tiercelin JJ, Le Turdu C (2000) The deep structure of the Baringo Rift basin (central Kenya) from 3-D magneto-telluric imaging: Implications for rift evolution. Journal of Geophysical Research 105: 23493-23518. Hautot S, Single R, Watson J, Harrop N, Jerram DA, Tarits P, Whaler KA (2007) 3-D magnetotelluric inversion and model validation with gravity data for the investigation of large igneous provinces. Geophysical Journal International 170(3):1418-1430.


Sławomir B Ąk

Sofia Zaidenberg

Bernard Boulay

Francois Brémond

Improving Person Re-identification by Viewpoint Cues

Re-identifying people in a network of cameras requires an invariant human representation. State of the art algorithms are likely to fail in real-world scenarios due to serious perspective changes. Most of existing approaches focus on invariant and discriminative features, while ignoring the body alignment issue. In this paper we propose 3 methods for improving the performance of person re-identification. We focus on eliminating perspective distortions by using 3D scene information. Perspective changes are minimized by affine transformations of cropped images containing the target (1). Further we estimate the human pose for (2) clustering data from a video stream and (3) weighting image features. The pose is estimated using 3D scene information and motion of the target. We validated our approach on a publicly available dataset with a network of 8 cameras. The results demonstrated significant increase in the re-identification performance over the state of the art.

Introduction

Person re-identification is a well defined task that requires finding a particular person in a network of cameras with non-overlapping fields of view. As video surveillance usually provides low resolution data, this task is approached by extracting a global appearance of the target using its clothing features. The clothing is not as distinctive as face or iris features, thus we often treat this problem as a retrieval task. However, the low recognition accuracy of existing approaches not only comes from insufficient discriminative power of clothing features, but also from significant appearance changes caused by variations in view angles, lighting conditions and person poses.

Current state of the art approaches concentrate either on feature modeling [START_REF] Bazzani | Multiple-shot person re-identification by hpe signature[END_REF][START_REF] Farenzena | Person re-identification by symmetry-driven accumulation of local features[END_REF][START_REF] Wang | Shape and appearance context modeling[END_REF] producing descriptors invariant to camera changes or on metric learning [START_REF] Dikmen | Pedestrian recognition with a learned metric[END_REF]9,[START_REF] Zheng | Person re-identification by probabilistic relative distance comparison[END_REF] that should boost performance regardless of the representation choice. Designing a new descriptor, we need to look for a trade-off between its discriminative power and invariance through cameras. This task is particularly hard, especially, as this trade-off varies from data to data [START_REF] Varma | Learning the discriminative powerinvariance trade-off[END_REF]. Metric learning approaches use training data to search for strategies that combine given features maximizing inter-class variation whilst minimizing intra-class variation. However, these approaches focus on learning a function which transfers the feature space from the first camera to the second one, introducing the requirement of training ( c 2 ) models for c cameras (e.g. a network of 8 cameras needs to train 56 metrics). Moreover, metric learning based approaches need large training data (hundreds of annotated image pairs with the same individual registered by different cameras) for every camera pair. This alone might make these approaches inapplicable in small camera networks where acquisition of labeled data is unattainable.

Perspective changes

Until now, most of appearance-based approaches were usually evaluated on cropped images that are manually extracted from a few video streams taken at eye level (no significant perspective changes). Slight pose changes can usually be approached by adopting perceptual principles of symmetry and asymmetry of the human body [START_REF] Farenzena | Person re-identification by symmetry-driven accumulation of local features[END_REF]. The extracted features are then weighted using the idea that features closer to the bodies' axes of symmetry are more robust against scene clutter.

In this paper we address the real world scenario, where pose orientation of a person might change due to serious perspective changes. Figure 1(a) illustrates that the change might be significant, thus having noticeable impact on recognition performance. We offer a simple but efficient affine transformation using 3D scene information to improve alignment of two images extracted from different cameras (section 2).

Multiple signature appearance

Further classification of appearance-based techniques distinguishes the single-shot and the multiple-shot approaches. The former class extracts appearance using a single image [START_REF] Gray | Viewpoint invariant pedestrian recognition with an ensemble of localized features[END_REF][START_REF] Park | Vise: Visual search engine using multiple networked cameras[END_REF][START_REF] Wang | Shape and appearance context modeling[END_REF], while the latter employs multiple images of the same object to obtain a robust representation [START_REF] Bazzani | Multiple-shot person re-identification by hpe signature[END_REF][START_REF] Farenzena | Person re-identification by symmetry-driven accumulation of local features[END_REF][START_REF] Gheissari | Person reidentification using spatiotemporal appearance[END_REF][START_REF] Prosser | Person reidentification by support vector ranking[END_REF][START_REF] Zheng | Person re-identification by probabilistic relative distance comparison[END_REF]. The main idea of multiple-shot approaches is to take advantage of several frames for extracting a more reliable representation [START_REF] Bak | Learning to match appearances by correlations in a covariance metric space[END_REF][START_REF] Gheissari | Person reidentification using spatiotemporal appearance[END_REF][START_REF] Oreifej | Human identity recognition in aerial images[END_REF]. Although multiple-shot approaches employ several frames for generating the signature, in the end they usually produce a single (averaged) representation ignoring the possible pose and view-angle changes that can occur while tracking the target in a single camera. Those approaches either select more frequent features or blend the appearance by averaging features. On the contrary, we propose to cluster the trajectory based on estimated pose cues and generate the signature for every significantly different appearance (see figure 1(b) where the appearance of the target is clustered into two appearances). Finally we produce a multiple signature consisting of several signatures ordered by the orientation of their pose. Having pose information, we determine weights of features w.r.t. the distance from the frontal viewpoint for every signature (see figure 1(c)). This improves the alignment and the matching accuracy.

This paper makes the following contributions:

• We eliminate perspective distortions by applying a simple affine transformation (rotation) on cropped images containing a person of interest. This transformation is based on 3D scene information (section 2).

• We offer to employ pose cues for clustering the trajectory with a target. Pose is estimated using 3D scene information and motion of the target. The proposed clustering allows to measure the reliability of the detected orientation in every frame (section 3.2).

• We propose an orientation-driven weighting strategy of image features. Our idea is that different regions of the target appearance ought to be aligned using pose cues, minimizing appearance change and increasing matching performance (section 3.3).

We validate all steps of our approach in section 4 before discussing perspectives and concluding.

Target alignment

The changing viewpoint in a network of cameras is an important issue and might significantly distort the visual appearance of a person. This problem has a direct impact on the re-identification accuracy. Eliminating perspective distortions in an image region containing the target is often called image rectification [START_REF] Li | Human detection by searching in 3d space using camera and scene knowledge[END_REF]. Although employing rectification methods gives satisfactory results in pedestrian detection tasks, we observed that the extracted homography between the image and the ground planes can still produce significant distortions in the texture inside the target appearance. As a result, instead of employing rectification, we propose a simple method that only rotates the cropped image with of target by an angle α. This angle is extracted using 3D scene information (see figure 2(b)), by mapping the vector orthogonal to the ground plane living in real world coordinates to the vertical of a given image. This mapping is achieved by employing the calibration information of the camera (i.e. we employ Tsai calibration). Rotation angle α: Given a detected person (a rough bounding box around the person, see figure 2(b)), we select the center point of the detection (point C). Having a pixel location of this point in the image plane (x i c , y i c ), we compute its corresponding point in the world coordinate system (x r c , y r c ) using calibration information and a fixed height of a person h = 1.7 m. From this point, we can easily compute the orthogonal to the ground plane in the world coordinate system meeting the head point (x r h , y r h ) that has its corresponding location (x i h , y i h ) in the image plane (point H). The rotation angle can be computed by

α = arctan x i h -x i c y i c -y i h . (1)
Figure 2(c) illustrates the result of the rotation.

Multiple target appearance

This section introduces the method for extracting the pose of a person using 3D scene information and the motion of the target (section 3.1). Using pose, we cluster the trajectory into clusters with reliable pose detection (section 3.2) and generate multiple signatures for the single trajectory. For each such generated signature, we determine orientation-driven weights for signature matching (section 3.3).

Pose orientation θ

Given detection results for n frames, we compute a set of central points C r = {(x r c,1 , y r c,1 ), . . . , (x r c,n , y r c,n )} that lie in the real world coordinates system and correspond to the center of the detections in the image plane. Using calibration information, we extract the position of the camera projected on the ground plane (x r cam , y r cam ). For each position k ∈ [2, n] on the trajectory, let m k be the motion vector defined as

m k = x r c,k -x r c,k-1 , y r c,k -y r c,k-1 , (2)
and v k the viewpoint vector defined as

v k = x r cam -x r c,k-1 , y r cam -y r c,k-1 . (3)
We define the pose orientation θ k using dot product between these two vectors

θ k = arccos v k • m k |v k ||m k | . (4)

Orientation-driven clustering

In figure 3(a), we can notice that the θ estimation might be noisy. We minimize noise by smoothing the data:

θ s k = k+z l=k-z θ l 2z + 1 , ( 5
)
where z is a smoothing parameter (we set z = 5 in experiments). This operation provides us more reliable pose orientation cues (figure 3(b)). The key idea of clustering the trajectory is to obtain multiple appearances of the target w.r.t. its orientation. By detecting significant changes in orientation, we believe that we can detect significant changes in the appearance. We estimate the pose changes using the control derivative function δ k defined as

δ k = max l=k-z...k+z dθ s k (t + l) dt . (6)
We use this function to measure the reliability of the orientation θ. We assume that peaks in δ (see figure 3(c)) and their neighborhoods might provide unreliable information. Figure 3(d) illustrates the trajectory and its reliability. We can notice that the trajectory is unreliable during the pose change (the color red indicates low reliability, while blue -the highest). Frames with estimated unreliable orientation (δ > 10) are removed from the trajectory, determining gaps in the trajectory, thus clustering it into the multiple appearances. Each appearance cluster is labeled with its mean orientation (e.g. the person in figure 3(d) was separated into two clusters, labeled with orientation 191 • and 99 • ). The output multiple signature for the target consists of signatures computed from each cluster.

Orientation-driven feature weighting and matching

Each trajectory cluster consists of a set of cropped and rotated images with estimated poses. From such images we We propose a weighting strategy of those patches based on the estimated pose. We eliminate background information by employing an Epanechnikov kernel, while determining the feature weights.

We define the Epanechnikov kernel as a function of the orientation θ s with variable width. Modifying the center of the kernel w.r.t. θ s , we address the problem of appearance changes. Let every image be scaled to a size of W × H. Defining the center of the Epanechnikov kernel, we shift it in the horizontal direction by

∆(θ s ) = sgn(sin(θ s )) W sin 2 (θ s ) 2 (7)
(see figure 5). As a result, we define kernel parameters as

h = H 2 (8) w(θ s ) = W 2 + ∆(θ s ). ( 9
)
Given patch p of the dense feature grid, we define its weight as K x,y (θ s ) expressed by Signature matching for retrieval Let A and B be multiple signatures of two individuals whose similarity we want to measure. Each signature consists of a set of signatures extracted w.r.t. θ s . Before computing the similarity between corresponding features in a dense grid, we address the alignment problem. We align two signatures by minimizing the difference between kernels:

   3 4 1 - x w(θ s ) 2 -y h 2 , | x w(θ s ) 2 + y h 2 | ≤ 1; 0, otherwise, (10
min a∈A,b∈B |K(θ s a ) -K(θ s b )|, (11)
where a and b are sub-signatures of A and B, expressed by weight matrices K. Having aligned signatures, we compute similarity of corresponding features using averaged Epanechnikov weights.

Experimental results

This section focuses on evaluating all steps of our approach in a real world scenario. We illustrate the improvements at each step of the processing chain, discussing its importance. Finally, we show that our approach significantly outperforms state of the art results.

Database and experimental setup

We carry out experiments on the SAIVT-SOFTBIO database [START_REF] Bialkowski | A database for person re-identification in multicamera surveillance networks[END_REF]. This dataset consists of 152 people moving through a network of 8 cameras. Subjects travel in an uncontrolled manner, thus most of subjects appear only in a subset of the camera network. This provides a highly unconstrained environment reflecting a real-world scenario. In average, each subject is registered by 400 frames spanning up to 8 camera views in challenging surveillance conditions (significant illumination, pose and viewpoint changes). Each camera captures data at 25 frames per second at a resolution of 704×576 pixels. Although some cameras overlap, we do not use this information while testing re-identification algorithms. The database provides XML files with annotations given by coarse bounding boxes indicating the location of the subjects in each frame. Using the centers of these bounding boxes, we obtain trajectories of the subjects. Subject images are cropped and aligned using the method described in section 2. After clustering the trajectory, we randomly select N = 10 subsequent images from each cluster to compute the sub-signature of the multiple signature appearance.

Appearance model

Every cropped image is scaled into a fixed size window of 64 × 192 pixels. A set P of rectangular sub-regions is produced by shifting 32 × 32 regions with a 16 pixels step. This operation results in |P| = 33 overlapping rectangular sub-regions. From each sub-region, we extract RGB color histograms. We minimize color dissimilarities caused by camera illumination changes by applying histogram equalization to each color channel.

Evaluation metrics

Re-identification performance is analyzed in terms of recognition rate, using the averaged cumulative matching characteristic (CMC) curve [START_REF] Gray | Evaluating Appearance Models for Recognition, Reacquisition, and Tracking[END_REF]. The CMC represents the expectation of finding the correct match in the top n matches. nAUC is a quantitative scalar of the CMC curve computed by normalizing the area under the CMC curve. Every multiple signature is used as a query to the gallery set of multiple signatures from the other cameras. This procedure has been repeated 10 times to obtain averaged CMC results for each camera pair. Having 8 cameras, we evaluate our approach on 56 pairs. While validating the steps of our approach, we provide CMC curves and nAUC values corresponding to the recognition accuracy averaged among all 56 pairs of cameras at different ranks r.

Results

Figure 6 and table 1 illustrate the impact of each step of our algorithm on the re-identification accuracy. BASE-LINE corresponds to signatures extracted using randomly selected N = 10 subsequent frames. Labels TA, MS and W correspond respectively to the given contributions: Target Alignment (section 2), clustering into Multiple target Signature (section 3.2) and the orientation-driven Weighting (section 3.3). From the results it is apparent that each step of the algorithm has a significant impact on the performance. We consistently increase the recognition for all ranks employing the proposed steps. We can notice that the most important step is related to computation of the multiple signature appearance (i.e. for r = 1 we can notice an increase of about 9% in the recognition accuracy). This step is achieved by our clustering method. Although employing Epanechnikov kernel (step W) improves the recognition, we were expecting more significant improvement. Weighting the features or finding the salience regions [START_REF] Zhao | Unsupervised salience learning for person re-identification[END_REF] in images is still a challenging task and we will address this problem in the future work.

For comparing with state of the art approach, we illustrate the performance of Color-soft [START_REF] Bialkowski | A database for person re-identification in multicamera surveillance networks[END_REF] in figure 7. We followed the experimental setup in accordance with [START_REF] Bialkowski | A database for person re-identification in multicamera surveillance networks[END_REF], thus focusing only either on cameras with similar view (figure 7(a)) or on cameras with a significant viewpoint change (figure 7(b)). The results show clearly that our approach significantly improves state of the art performance.

Conclusion and perspectives

This paper proposed 3 improvements for person reidentification in a real-world scenario. We focused on eliminating perspective distortions by using 3D scene informa- tion. Perspective changes are minimized by rotating the cropped images w.r.t. the orthogonal to the ground plane. Using different pose orientations, we divided the trajectory into clusters capturing multiple target appearances, thus significantly improving the recognition accuracy. Further, we proposed a pose-driven weighting strategy to eliminate background information and improve signature alignment. We demonstrated that all steps of our approach consistently improve the re-identification performance, outperforming state of the art. Future work will focus on improving the weighting strategy. We plan to learn a priori weights for matching different poses using a depth sensor. This offline learned weights could be then applied while matching different poses.

Figure 1 .

 1 Figure 1. Improvements on re-identification using viewpoint cues: (a) target alignment (section 2); (b) multiple target appearance based on clustering (section 3.2); (c) pose orientation-driven weighting (section 3.3). The illustration shows an example of the same person viewed from two different cameras (C5 and C8).

Figure 2 .

 2 Figure 2. Affine transformation of the target image: (a) trajectory of the target (color of the trajectory illustrates the reliability of the detected pose, see section 3.2 for details); (b) the cropped image obtained by the detection algorithm ; (c) the rotated image.

Figure 3 (

 3 Figure 3(a) presents θ values for the given trajectory (figure 3(d)) and figure 5 shows an illustration of orientation angle θ.

Figure 3 .

 3 Figure 3. Orientation driven trajectory clustering: (a) original pose orientation θ estimated from the video; (b) the result of the smoothing operation; (c) our control derivative function; (d) the reliability of the trajectory (red color indicates low reliability, while blue stands for the highest).

Figure 4 .

 4 Figure 4. Sample aligned images and their weight distributions for different orientations. The first row shows aligned images, rotated by α. The second row illustrates weight distributions (looking from the left to the right the orientation angle θ in these examples varies from 16 • to 309 • with a ∼ 13 • step).

Figure 5 .

 5 Figure 5. Computation of the new center of the Epanechnikov kernel. The original kernel is shifted based on the detected orientation of the person w.r.t. to the camera. We shift the kernel by ∆(θ s ) in the horizontal direction. On the right we can see the change of weight distributions w.r.t. the detected orientation angle for θ s = 16 • and θ s = 46 • .

  ) where (x, y) are coordinates of patch p computed relatively to the shifted center of Epanechnikov kernel. The weight represents the patch's contribution to the signature. Figure 4 illustrates sample images and determined weights of image regions w.r.t. to orientation θ s .

Figure 6 .

 6 Figure 6. CMC curves obtained on the SAIVT-SOFTBIO dataset. CMC and nAUC values correspond to the recognition accuracy averaged among all 56 pairs of cameras at different ranks r.

Figure 7 .

 7 Figure 7. Performance comparison using CMC curves, our (TA+MS+W) vs. Color-Soft [3]. We can notice the significant improvement, especially for dissimilar views (b). The example of images from camera 5 and 8 are given in figure 1.

Table 1 .

 1 Validation of the proposed contributions on the SAIVT-SOFTBIO dataset. Values correspond to the recognition accuracy averaged among all 56 pairs of cameras at different ranks r.

		11%	21.09%	35.79%	59.60%
	TA	11.12%	28.09%	42.79%	63.60%
	TA+MS	20.30%	39.62%	52.92%	71.91%
	TA+MS+W 23.02% 42.73% 54.64% 72.89%

Acknowledgements

This work has been supported by PANORAMA European project.


Houssam Arbess
email: harbess@laas.fr

Karine Isoird
email: kisoird@laas.fr

Field Plate Termination For High Voltage Diamond Schottky Diode

New field plate architecture is applied to pseudo vertical diamond Schottky diode. New topology structure has been proposed and simulated using Sentaurus TCAD simulation in order to minimize the maximum electric field in the dielectric at high voltage operation. Firstly and after simple variations in the field plate architecture, the breakdown voltage was improved from 1632 V to 2141 V at 700 K. Concerning E max in the dielectric, we obtained high decreasing of the maximum electric field following the policy of pressure distribution.

I.

INTRODUCTION

Power electronics based on silicon has now almost reached its limit (Operating temperature, high break down voltage limit, thermal conductivity, ...). To resolve this problem, there are two approaches: either by developing new component architectures, or turn to new semiconductor material able to function as desired. The first solution is in its operating limit. Thus we see new semiconductor material appear. The best known and most advanced are the gallium nitride for small and medium power application such as the automobile, or silicon carbide for medium and high power application. These semiconductors are at the industrial stage, the first SiC Schottky diode and JFET transistor are now commercialized. considered to be the best material for power semiconductor devices due to its excellent material properties such as high breakdown field (>10 MV/cm), high mobility (4500 and 3800 cm 2 /Vs for electron and holes, respectively), low dielectric constant (5.7) and the highest thermal conductivity in the materials [START_REF] Sussmann | CVD Diamond for Electronic Devices and Sensors[END_REF]. the interest of diamond obvious when reading Fig. 1. The compromise R ON /V BR , which represents the ratio between the ON-resistance and the breakdown voltage of the device is excellent for diamond.

II. Issues

One of principal problems in diamond power device is the periphery protection. Indeed, the diamond has a high critical electric field, so, the junction termination structure is going to have a major impact in the OFF-state performance of the power devices. Without efficiency junction termination, strong field enhancement at the contact edge cannot be avoided, which causes premature avalanche breakdown of the device. The role of periphery protection is to distribute the electric field at the edge of junction or Schottky contact, thereby, increasing the experimental breakdown voltage. Several junction terminations have been explored for Schottky diodes in wide-bandgap semiconductors. Some of the more common ones are the guard ring [START_REF] Gupta | Design, Fabrication, and Characterization of Ni/4H-SiC (0001) Schottky Diodes Array Equipped With Field Plate and Floating Guard Ring Edge Termination Structures[END_REF], [START_REF] Ueno | The guard-ring termination for the high-voltage SiC Schottky barrier diodes[END_REF] the metal field plates extending over an insulating surface layer [START_REF] Ikeda | Fabrication of a field plate structure for diamond Schottky barrier diodes[END_REF], [START_REF] Tarplee | Design rules for field plate edge termination in SiC Schottky diodes[END_REF], resistive Schottky barrier field plate (RESP) [START_REF] Bhatnagar | Edge terminations for SiC high voltage Schottky rectifiers[END_REF], and junction-termination extension (JTE) [START_REF] Alok | SiC device edge termination using finite area argon implantation[END_REF].

Due to difficulties associated with n-type doping in diamond and destructive effects of ion implantation on the diamond cristal, we focus the study in this paper to the Field plate termination due to the technological problems using other termination types.

Diamond is strong, which imposes severe constraints at the device termination. Since the critical field of the overall deposited dielectric material is less than that of diamond, we aim to find new architectures that reduce the electric field in the termination structure.

The critical field of the properties of diamond are close to the properties of best insulations available for power devices (epsilon, electric field, band gap, ...). It is therefore very difficult to use traditional material to realize junction terminations. Therefore it is a must to develop new protection architectures to reduce the maximum electric field in the termination.

All results shown in this paper were extracted at 700 K due to simulation problems at low temperatures. Other simulations have been made in order to evaluate the variation of maximum electric field with temperature shown a negligible difference.

III. PREVIOUS STUDY

The previous structure studied at LAAS-CNRS is a diamond pseudo vertical Schottky diode with Field plate as termination on the diamond surface (Fig. 2). This structure consists of two diamond layers, the first is a 7 µm P + doped diamond with boron concentration of 3.10 20 cm -3 and the other is a 7 µm P -doped diamond with boron concentration of 8.10 15 cm -3 . To study the breakdown voltage of this structure and to look the optimum desired values of different technological parameters, we use Sentaurus TCAD simulation. The physical model used to calculate the breakdown voltage is the default Van Overstreaten model [START_REF]Sentaurus user guide A[END_REF]. For this component, the field plate was optimized for a length of 10 microns and dielectric thickness of 0.7 microns.

The dielectric used in this optimization is the Silicon oxide SiO 2 [START_REF] Thion | Conception de protections périphériques applicables aux diodes Schottky réalisées sur diamant monocristallin[END_REF]. With these parameters, we obtained 1638 V as a breakdown voltage. However the electric field reached at the end of field plate is higher than 19.9 MV/cm at 700 K.

IV. New topology structure and results

In a previous paper, new field plate architecture has been proposed. It is to keep the whole electrode flat, with no corners, and replace the diamond below the field plate region, with an optimized thickness of dielectric material (Fig. 3). The presence of corners on the path of the potential strengthens the electric field. In the first one (Fig. 2), the dielectric is located on the diamond surface, creating a corner in the electrode. At this corner the electric field is stronger in the diamond and then the impact ionization is greater for the first structure than the second for the same external voltage. Consequently, the breakdown voltage in the structure of Fig. 3 is higher than the structure of Fig. 2. The highest electric field at 1700 V is 19.9 MV/cm for the initial structure and 19.8 MV/cm for the new one. The high mesh refinement in the high electric field zone (end of field plate) to 10 nm (node-to-node distance) shows an increase in the electric field to 49 MV/cm and 48.5 MV/cm for the initial and new architecture respectively. Assuming that the dielectric we used has a high quality, the maximum electric field carried by the silicon oxide is around 3 or 4 MV/cm. In this case, the oxide will break before the voltage across the diode reaches the maximum value, and consequently the field plate doesn't play its desired function.

Looking at the experimental results, the average breakdown voltage was 400 V [START_REF] Thion | Conception de protections périphériques applicables aux diodes Schottky réalisées sur diamant monocristallin[END_REF]. One of the reasons responsible for this failure may be the break of the dielectric. To improve the efficiency of the termination, we can change the dielectric as Ikeda [START_REF] Ikeda | Fabrication of a field plate structure for diamond Schottky barrier diodes[END_REF], who used the Al 2 O 3 as a dielectric for the field plate, but it's not enough. We aim to find a dielectric form for the field plate able to minimize the electric field as much as possible. Our idea is to change the topology of the structure including the field plate form. The maximum electric field is at the corner [START_REF] Bhatnagar | Edge terminations for SiC high voltage Schottky rectifiers[END_REF]. Thus, increasing the number of corners in the potential way, reduces the electric field value at the output of the field plate (distribution of peaks). Two architectures may be able to apply this idea. a-The pillars dielectric form. That means, making the oxide in pillars form until the end of the electrode which increase the number of corners in the path of the potential Fig. 4 (a). b-The graduated dielectric form Fig. 4 (b).

For these two terminations architecture, the efficiency depends on many parameters (field plate length, dielectric thickness, distance between pillars (Fig. 4(a)), distance between graduation (Fig. 4 (b)). Two dimensions simulations have been made for the two architectures with two different dielectrics, silicon oxide and aluminum oxide, with variable thicknesses of dielectric. Using the aluminum oxide (ε ≈ 9) the maximum electric field is smaller than using the silicon oxide as dielectric. E max decreases from 50 MV/cm to around 30 MV/cm. This minimum value is much larger than the maximum value supported by the dielectric (silicon oxide or aluminum oxide).

A new idea consists to associate the two proposed structures described previously in order to increase maximum the number of corners without increasing the steps number. Fig. 6 presents this field plate architecture. A simulation shows that the maximum electric field at 1700 V equal to 22.7 MV/cm using the silicon oxide as dielectric and 16,5 MV/cm using the aluminum oxide as dielectric. Fig. 7 Presents the cross sectional view of the electric field for each graduation and at the oxide surface using aluminum oxide as a dielectric. This figure shows also the distribution of the electrostatic potential lines in the different zones of the structure. oxide) can support this electric field value that is focalized in a very small surface. Based on this assumption, simulations of 3 proposed periphery protections have been realized with new break criteria, which is an electric field equal to 18 MV/cm. The dielectric used for those simulations is the layer of 1.6 µm of silicon oxide Results show a breakdown voltage equal to 860 V and 896 V for components of Fig. 4 (a) and Fig. 4 (b) respectively and 1350 V for component of Fig. 6.

V. Discussion

Each proposed solution allows decreasing the peak of electric field, but not in the target range. Also we must consider other ways.

Replacing the silicon oxide with another dielectric material with higher electric permittivity (such as aluminum oxide Al 2 O 3 ) reduces the value of the maximum electric field. However, the reached value remains above the requirement.

In the final study of this paper, the simulation of the Thion structure [START_REF] Thion | Conception de protections périphériques applicables aux diodes Schottky réalisées sur diamant monocristallin[END_REF] using the same mesh we made in this paper shows that the maximum electric field equal 18MV/cm. It can be say, since the maximum electric field is distributed in very small area (10 to 100 nm) , the critical field can be larger than in capacitor case.

A new study with high permittivity material is in progress.

VI. Conclusion

New field plate architecture for diamond Schottky diode was proposed which helps in increasing the breakdown voltage from 1632 V to 2141 V using the same technological parameters. The maximum electric field in the field plate provokes a new problem since it is larger than the critical field of dielectric used. To resolve this problem, several solutions were proposed. The first is to make the field plate in a pillars form, the second is to change the field plate to graduated form and the last is making a mixed oxide form in order to increase the corners numbers. For all solutions, two dielectric material were used. The first one is the silicon oxide and the second one is the aluminum oxide.

VII.

Fig. 1 :

 1 Fig. 1: Operating limits of silicon through the representation of the on-state resistance as a function of the breakdown voltage. In the non-exploitable area, there is the wide bandgap semiconductors Recently, diamond emerged as one of the best semiconductor. The discovery of diamond synthesis by Chemical Vapor deposition (CVD) in the early 1980s opened a large range of new applications for diamond. It's

Fig. 2 :

 2 Fig. 2: Simulated pseudo vertical diamond Schottky diode

Fig. 3 :

 3 Fig. 3: New field plate architecture for diamond schottky diodeFor the new field plate structure, the simulation with the same technological parameters of the new architecture component give a breakdown voltage equal to 2141 V. The chosen parameters are 10 micron for the field plate length and 1 microns for the dielectric thickness. The highest electric field at 1700 V is 19.9 MV/cm for the initial structure and 19.8 MV/cm for the new one. The high mesh refinement in the high electric field zone (end of field plate) to 10 nm (node-to-node distance) shows an increase in the electric field to 49 MV/cm and 48.5 MV/cm for the initial and new architecture respectively. Assuming that the dielectric we used has a high quality, the maximum electric field carried by the silicon oxide is around 3 or 4 MV/cm. In this case, the oxide will break before the voltage across the diode reaches the maximum value, and consequently the field plate doesn't play its desired function. Looking at the experimental results, the average breakdown voltage was 400 V[START_REF] Thion | Conception de protections périphériques applicables aux diodes Schottky réalisées sur diamant monocristallin[END_REF]. One of the reasons responsible for this failure may be the break of the dielectric. To improve the efficiency of the termination, we can change the dielectric as Ikeda[START_REF] Ikeda | Fabrication of a field plate structure for diamond Schottky barrier diodes[END_REF], who used the Al 2 O 3 as a dielectric for the field plate, but it's not enough. We aim to find a dielectric form for the field plate able to minimize the electric field as much as possible. Our idea is to change the topology of the structure including the field plate form. The maximum electric field is at the corner[START_REF] Bhatnagar | Edge terminations for SiC high voltage Schottky rectifiers[END_REF]. Thus, increasing the number of corners in the potential way, reduces the electric field value at the output of the field plate (distribution of peaks). Two architectures may be able to apply this idea.a-The pillars dielectric form. That means, making the oxide in pillars form until the end of the electrode which increase the number of corners in the path of

Fig. 4 :Fig. 5 :

 45 Fig. 4: Pillars dielectric form (a), graduated dielectric form (b)

Fig. 5

 5 Fig. 5 presents the maximum electric field at 1700 V for the two proposed field plate architectures, in terms of pillars height for the pillars dielectric form (a) and in terms of one

Fig. 6 :Fig. 7 :

 67 Fig. 6: mixed form for Schottky field plate

Acknowledgements

This work has been sponsored by the international associated laboratory (LIA) within the Framework of the project DIAMONIX II.

VIII. references


Ilaine Wang
email: i.wang@u-paris10.fr

Sylvain Kahane
email: sylvain@kahane.fr

Isabelle Tellier
email: isabelle.tellier@univ-paris3.fr

Macrosyntactic Segmenters of a French spoken corpus

Keywords: sentence boundary segmentation, spoken French, machine learning

The aim of this paper is to describe an automated process to segment spoken French transcribed data into macrosyntactic units. While sentences are delimited by punctuation marks for written data, there is no obvious hint nor limit to major units for speech. As a reference, we used the manual annotation of macrosyntactic units based on illocutionary as well as syntactic criteria and developed for the Rhapsodie corpus, a 33.000 words prosodic and syntactic treebank. Our segmenters were built using machine learning methods as supervised classifiers : segmentation is about identifying the boundaries of units, which amounts to classifying each interword space. We trained six different models on Rhapsodie using different sets of features, including prosodic and morphosyntactic cues, on the assumption that their combination would be relevant for the task. Both types of cues could be resulting either from manual annotation/correction or from fully automated processes, which comparison might help determine the cost of manual effort, especially for the 3M words of spoken French of the Orfeo project those experiments are contributing to.

Introduction

The tools we present are developed as part of the Orfeo project, funded by the French National Research Agency (ANR), which aim is to propose syntactic annotations in order to compare the syntax of written and spoken French. We gathered a total of 3 million words of transcription of spoken French. It is well known that the notion of sentence is not easily applicable to spontaneous speech (Pietrandrea et al., 2014 to appear) and the transcriptions we have are not segmented, which is a real problem for parsing. Based on the results of the Rhapsodie project [START_REF] Lacheret | Rhapsodie: a Prosodic-Syntactic Treebank for Spoken French[END_REF], we propose a well formalised segmentation for spoken French called macrosyntactic segmentation and we have at our disposal the 33,000-word Rhapsodie treebank. Our approach consists in using this resource and machine learning methods to classify automatically interword spaces into boundaries or non-boundaries -which comes down to segment into sentence-level units-while trying to figure out which features are the most efficient to identify boundaries. We thus built six different segmenters which we present here.

Macrosyntactic Units for Spoken Language

The segmentation we are aiming at is based on the units defined in the Rhapsodie project and described in this section. However, the number and heterogeneity of the boundaries to retrieve (up to 18) heavily complicates the task. We therefore decided to train the segmenters on a simplified version.

Rhapsodie Syntactic Annotation

In the reference treebank, we distinguished two levels of syntactic cohesion: the first one, called microsyntax, is ensured by government relations and is the traditional level of syntax considered by most syntactic theories and treebanks. The second one, called macrosyntax, is the cohesion of sequences, Illocutionary Unit (henceforth IU) which are wrapped up into a unique illocutionary act, realising one and only one assertion, injunction, interrogation, etc. [START_REF] Blanche-Benveniste | Le franc ¸ais parlé. Etudes grammaticales[END_REF][START_REF] Berrendonner | Pour une macro-syntaxe[END_REF][START_REF] Cresti | Corpus di italiano parlato[END_REF][START_REF] Deulofeu | Depends on what the French say spoken corpus annotation with and beyond syntactic functions[END_REF].

The text is thus segmented into IUs noted by // as in the following example: ˆso when you arrive on place de Verdun <+ the prefecture is on your right // you < you continue straight away // in fact < "er" "well" there < you pass "er" & // "well" in fact < it's the tram line > still > there // An IU may be composed of several segments: the main segment, called the nucleus, bears the illocutionary force; separated from the nucleus by < and > are respectively the pre-nuclei (such as donc quand vous arrivez sur la place de Verdun 'so when you arrive on place de Verdun') and the post-nuclei ; IU introducers (donc 'so') are marked by the symbol ˆwhile discourse markers (ben 'well', euh 'er') are surrounded by double quotation marks " ".

The macrosyntactic annotation of Rhapsodie has been carried out by a dozen of scholars working cumulatively. Each part of the corpus has been checked by at least three annotators and in case of strong disagreement, group sessions have been conducted to discuss the most problematic items. Apart from those debates, no evaluation of the interannotator agreement has been done.

Simplified Annotation for the Task

In order to reduce the complexity of the task, differents types of boundaries have been grouped together. As a result, in the experiments we present, boundaries between IUs are considered major boundaries (//) while those around segments of IUs are considered minor boundaries (noted / for all kinds of segments). Each interword space can have one and only one value : two similar boundaries are merged into one, and major boundaries are prioritised in the event of a conflict with minor boundaries.

The previous example would then be annotated:

Features

As the treebank we worked on was provided with rich annotations, we could use a large set of features. However, although this data is interesting in terms of determining segmentation cues and may be reliable enough to guarantee a good outcome, it is not likely to be available in the corpora that are intended as inputs for macrosyntactic segmenters and we therefore felt the need to use more objective information by means of state-of-the-art tools. This alternative also ensures the possibility of a completely automatic segmentation task. Thus, among all the features described in this section, some were directly taken from the manual contribution of naive and expert annotators which was subsequently validated by linguistics experts. On the other hand, a few more features had to be extracted from TextGrids (the preferred format of transcribed speech, produced via the Praat program [START_REF] Boersma | Praat, a system for doing phonetics by computer[END_REF]) and others were calculated either using Analor 1 , a semi-automatic annotation tool for prosodic structures, or using SEM [START_REF] Tellier | Un segmenteur-étiqueteur et un chunker pour le franc ¸ais[END_REF], a morphosyntactic tagger for French built based on a Conditional Random Field (CRF) algorithm. At the end of the process, using a data architecture specifically built to work at the interface of syntax and prosody, parser Farouch [START_REF] Belião | Farouch : une architecture de données pour l'analyse d'interfaces linguistiques[END_REF] enables all data to be gathered together in a single table and properly aligned on tokens.

The collection of all those cues is shown in Figure 1.

In order to provide classifiers as much information as possible, all features span six positions (three words or three syllables before and after each interword space). To determine 1 http://www.lattice.cnrs.fr/Analor?lang=en the class of an interword space, segmenters are then given the possibility to access and use cues such as whether or not the preceding word is a determiner, as well as whether there is a change of speaker after the first, second or third following word.

Prosodic Features

Two types of prosodic features were available for our experiments: acoustic and perceptual data. Acoustic characteristics we considered as interesting cues were (1) the pitch (both the raw value and the average value calculated on a span of two syllables before and after the current one), (2) syllable and pause duration, (3) the rising rate of the pitch and (4) the slowing coefficient : a normalised measure computed by [START_REF] Gendrot | Comparaison de parole journalistique et de parole spontanée : analyses de séquences entre pauses[END_REF] based on the comparison between the duration of each syllable in the corpus and a typical reference duration. A high coefficient indicates the significant lengthening of a syllable and should not only help identify specific events such as hesitation or emphasis but also help detect the end of syntactic or prosodic units, commonly associated with a rhyme lengthening due to a final accent in spoken French. No speaker normalisation has been applied to acoustic values.

Along with those features, we also used perceptual notions: (5) intonative periods defined by [START_REF] Lacheret | La période intonative comme unité d'analyse pour l'étude du franc ¸ais parlé: modélisation prosodique et enjeux linguistiques[END_REF] as maximal intonational units used in the description of prosodic discourse organisation and characterised by their register (low, normal, high) and by their shape (falling, flat, rising), (6) the strength of prominences (or salient syllables), (7) hesitation, and (8) global contour and local register levels, describing the pitch level compared to the average range of the speaker as a contour or a single value for a given unit. Those four features are available in the Rhapsodie treebank, but periods and prominences were also automatically calculated by Analor [START_REF] Avanzi | Process flowchart of a new corpus to be segmented depending on its nature and[END_REF][START_REF] Avanzi | Toward a continuous modeling of French prosodic structure: Using acoustic features to predict prominence location and prominence degree[END_REF].

Morphosyntactic Features

The corpus was also automatically (9) part-of-speech (POS) tagged and (10) shallowly parsed into chunks. tool which was trained on written data but although the margin of error when SEM2 is applied to oral data is relatively significant (around 17% loss on the accuracy score for POS tagging, and a similar impact on chunking), learning a whole new set of tags adapted to transcribed data has a heavily important cost in time and effort. However, an adapted chunker is easier to train and future experiments could use the same approach as the third one described in [START_REF] Tellier | Adapt a text-oriented chunker for oral data: How much manual effort is necessary?[END_REF].

Token

Tags are produced as shown in Figure 2.

According to [START_REF] Abney | Parsing by Chunks[END_REF], a chunk is the "non-recursive core of an intra-clausal constituent, extending from its beginning to its head". Here, the definition of a chunk extends to pre-head phrases as well but stays within the limits of non-recursive phrases, with chunks being the smallest possible. In the example, the sentence is chunked [I] [was born] [in Cannes] [during the war] using two tags from the BIO (Beginning, In, Out) encoding.

Word Features

Our set of features includes other cues such as (11) speaker diarisation (which can be helpful in dialogs, and especially in overlapping sequences) and ( 12) speech disfluency.

The former was directly extracted from one of the layer of annotation in the Rhapsodie treebank and for that reason, cannot be used in a fully automated segmentation, while the latter simply uses an exhaustive list composed of discursive markers or words that are specific to oral data (for instance, euh 'uh', bon 'well', or backchannel ouais 'yeah', etc.). In addition, if a word is cut off in mid-utterance (most of the time transcribed with a hyphen or a tilde at the end), the phenomenon is to be noted with the value of the speech disfluency feature being in this case amorce (false start).

Experiments

Supervised classification experiments were conducted using the open source software Weka [START_REF] Hall | The WEKA data mining software: an update[END_REF], a data mining tool developed by the University of Waikato. Different categories of algorithms from the variety implemented in Weka were tested (notably NaiveBayes, Rules and Decision Trees) but only results from the J48 algorithm are presented in this paper as it proved to be the best performing on our data set. As with any decision tree algorithm, is possible to read and visualise the models we learned. Using all the features at once could lead to the best results but in order to verify this assumption, we separated the features considering their nature and the way they were obtained -either from a manual annotation or an automated process. This setting also allows us to compare the efficiency of each type of feature for the task.

Six different input files were therefore created by selecting and filtering the features we wanted to be taken into account in each model:

1. All features (mixed): with all the features described above (from 1 to 12).

All features (auto)

: with all the features we could collect automatically, either prosodic (1-6) or part of the morphosyntactic and word features (9, 10, 12).

Prosody only (mixed)

: with all the prosodic features (1-8).

4. Prosody only (auto): with automatically calculated prosodic features only (1-6).

No prosody (mixed)

: with all but the prosodic features (9-12).

6. No prosody (auto): with all automatically calculated features, except for the prosodic ones (9,10,12).

Experiments presented in this paper were limited to classification into three classes only: major syntactic boundaries, minor syntactic boundaries, or non-boundaries. A known problem in classifying such data is the bias caused by nonboundaries outnumbering actual boundaries. As a consequence, a simple algorithm with no other rule than classifying all instances in the majority class makes a high baseline.

In a previous study [START_REF] Wang | Segmentation automatique d'un corpus de franc ¸ais oral en unité macrosyntaxiques[END_REF], an attempt to address the bias was made by under-sampling non-boundaries but this failed to improve the performance of classifiers.

Results and Evaluation

In this section we present the results of the models we trained using a 10-fold cross-validation method.

Accuracy F-measure Weighted F-measure 77.4% 0.87 0.68

Table 2: Majority class baseline results

As expected, the baseline for these experiments (Table 2) is quite high for the proportion of non-boundaries makes a 77,4% accuracy baseline but with only 0.68 overall weighted F-measure. This score was obtained with the algorithm ZeroR provided by Weka, which simply puts every instance in the most common class (in our case, that means classifying each interword space as non-boundaries).

The first observation we can make by comparing it to the results of our models (Table 1) is that all of the experiments we conducted have better results, gaining up to 7% accuracy for the best performing models. What then draws our attention with a closer look at Table 1 is the similarity between the scores of (a) models using all features and (b) models using the automatic ones only, with a margin as slim as 0.1% accuracy, which is not a significant difference considering the size of our data set. Furthermore, using all features or excluding prosodic cues also makes almost no difference. It just seems that "no prosody" models have better precision scores for both major and minor boundaries, whereas models using all types of features have the best recall scores which is quite important because the main problem for classifiers seems to be the coverage of all boundaries, not the precision.

As for classifications based on prosodic cues only, they unexpectedly do merely better than the baseline. As a matter of fact, the Rhapsodie corpus gathers 53 samples of about five minutes each for a total amount of three hours of speech recording. The objective was to maximise the variety of speakers and speech situations.

The resulting heterogeneity might have interfered in the classification process by making it difficult for algorithms to identify reliable values for features, especially for prosodic features. Indeed, as shown in the study of [START_REF] Belião | Characterizing oratory speech through both prosody and syntax. Preproceedings of the ESS-LLI[END_REF] on this corpus, the ratio between IUs and intonative periods may vary highly according to the situation of utterance: in political speeches, for instance, the speaker tends to cut IUs into several periods, while in narrative sequences such as those found in interviews, the speaker is inclined to utter several IUs within a single period. Finding a generic pattern is not possible with those two contradictory patterns.

We can eventually notice that the lowest scores are systematically related to minor boundaries with F-measure scores never reaching 0.5 (especially due to low recall), questioning the uniformity of those boundaries and the choice that was made to group them together to simplify the task. Further analysis of the results and experiments will help determine whether minor boundaries should be split back into different groups or not, and in that case, how they should be categorised.

Conclusion

Macrosyntatic segmentation is the first step to linguistic analyses such as parsing. Many works have studied the use of machine learning methods for speech segmentation on English corpora and have proved successful [START_REF] Stolcke | Automatic linguistic segmentation of conversational speech[END_REF][START_REF] Favre | Efficient sentence segmentation using syntactic features[END_REF]. The study we present in this paper is the application of similar techniques but on French data with similar results, besides the unexpected low scores of models using prosodic cues only. semi-interactive speeches etc.) characterised by prosodic specificities as we mentioned in the previous section.

To address this issue, future experiments will test a new feature to identify automatically the type of speeches by running a preliminary clustering task. Prosodic features should be more valuable cues if combined with that information. This process would then be prepended to the main workflow if it proves to be efficient.

Another noteworthy point in building not only one segmenter but different segmenters is that it facilitates application on new data. The whole process is illustrated in Figure 3, where only one "yes" to one of the three options (acoustic cues and/or morphosyntactic cues and/or manual annotation if available) is necessary to start classifying. Furthermore, Weka offers the possibility to select features used in training models making this method fully adaptable to the available features of a new corpus.

We can also note that with only three features, namely discursive markers, POS and chunking tags (automatic features without prosody so no alignment is required), it is possible to achieve 84.3% accuracy, whereas the best model using all features including manual annotation does only 0.2% better. Considering the efforts put into manual annotation and the slight improvement in the outcome of the best model, the minimal segmenter might be an interesting candidate to process non-tagged corpora.

As part of the Orfeo project, these segmenters are meant to be used as a preprocess of newly acquired large corpora and to accelerate the work of syntactic annotators. Experiments are currently conducted on both tagged and untagged versions of the TCOF corpus [START_REF] Benzitoun | TCOF-POS: un corpus libre de franc ¸ais parlé annoté en morphosyntaxe[END_REF] and will soon be evaluated to determine whether the errors made by the segmenters are significant, and whether the use of fully automatic segmenters is indeed time saving when combined to an additional correction phase and worthwhile over pure manual segmentation. Further examination of decision trees will also help measure the discriminating power of features.

The macrosyntactic segmenters presented in this paper are meant to be open source tools and will thus soon be freely available online 3 . Work is still in progress so please contact authors for further information on the state of advancement of the resource.

Figure 2 :Figure 1 :

 21 Figure 2: Example of morphosyntactic tagging with SEM on an excerpt of [Rhap-D2004, Lacheret corpus]

The tool is freely available on the following page, along with a description and the annotation guide used : www.lattice.cnrs.fr/sites/itellier/SEM.htm

Acknowledgements

This study was made possible by the support of the ANR through funding of the Orfeo project.


Denis Carvin
email: carvin@laas.fr

Philippe Owezarski

Pascal Berthou
email: berthou@laas.fr

A Generalized Distributed Consensus Algorithm for Monitoring and Decision Making in the IoT (Invited Paper)

In this paper, we propose a method to distributively monitor a dynamic mobile network. For this purpose, we take advantage of the consensus theory to provide each node with a common view of the network. More specifically, we give a decentralized algorithm to estimate a time-varying distribution, where each node has a partial information on this distribution. Our algorithm allows a trade-off between the precision of its estimation and its bandwidth consumption. We validate our approach by simulation under NS3, considering the distribution of several network metrics. Simulation results demonstrate how our algorithm can give insights on the network behavior that could be exploitable in a decision making process.

I. INTRODUCTION

The work presented in this paper considers the most general case of the IoT. All nodes are objects that can play three roles: they can produce data and send them to a particular receiver in the network, they can be the receivers of data produced by other objects in the network, and they can also serve as routers by being in charge of forwarding data packets with the required QoS. If these objects are mobile, with a mobility model that depends on their specific task, some of them could adapt their moves in favor of the network quality. Using such an approach, it could be possible for nodes to act for the benefit of the whole network in terms of QoS and resilience. To achieve such a behavior, objects must be able to monitor the local state of the network, and to infer the global status of the network by collaborating with their neighbors. For the seek of coordination, objects need to share the same view of the state of the network, in others words, they need to reach a consensus on this value. As it is often done in network monitoring, we derive the network's state from metrics that are measured at the node level. In particular, in this paper, we define the network's state as the distribution of metrics that have been periodically measured on each node. Our main contribution is to provide a decentralized algorithm to estimate this time-varying distribution when nodes are mobile.

The following section briefly reviews basic notions of descriptive statistic and introduces the related work. In Section III, we describe the consensus framework and present our decentralized density estimation method. We apply this method in the case of a wireless dynamic network of objects in Section IV, where we present our simulation results. Finally, in Section V, we discuss possible applications of inter-controlled mobility and give future directions.

II. THEORETICAL BACKGROUND AND RELATED WORK

This section reviews basic notions of descriptive statistics as well as related work on the decentralized density estimation, which will be used throughout the paper.

A. Density Estimation

When measuring a metric in a network, knowing the average might be useful, but it is often said that the average value of a population is meaningless without its standard deviation. In fact, these parameters are only useful for someone who has prior knowledge on the underlying distribution. For instance, average and standard deviation fully characterize a normal distribution, but if the distribution is unknown and appears to be multi-modal, these parameters might be badly interpreted. Moreover, if one wants to deal with qualitative data, the average is not computable and other approaches must be used. Non-parametric estimation techniques can overcome this problem. We mostly associate them to bar-chart (qualitative data) or histogram (continuous data). Kernel Density Estimation (KDE) encompasses those methods which estimate a probability density function (pdf ) without prior knowledge, in exchange for an additional computational cost.

In the case of continuous data, given a set P ⊂ R d of P ddimensional points, drawn from an unknown pdf, f , the kernel density estimation f of f can be defined as

f (p) = P k=1 α k K ′ k (p), (1)
where α k and K ′ k are the weight and the scaled kernel function associated to the sample (or location) p k , that is,

K ′ k (p) = |H k | -1 2 K k (H -1 2 k (p -p k ))
. The scaled kernel function K ′ k can be seen as the contribution of the sample p k to the estimation of the density at the location p. This contribution is defined by a bandwidth H k which is a d-dimensional matrix and by a kernel function K k which is symmetric and that integrates to one. In many cases, H i and K k are the same for all k. Besides, P k=1 α k = 1 and in general we have α k = 1 P . It is important to note that choosing the bandwidth as well as the Kernel is crucial in the quality of the estimation, nevertheless, these aspects are beyond the scope of this paper. For the sake of simplicity, we consider the Gaussian kernel

G(p) = (2π) -d/2 e -p T p 2
, and each node locally estimates the optimal bandwidth using the algorithm given by Scott in [START_REF] Scott | Multivariate density estimation : theory, practice, and visualization, ser. Wiley series in probability and mathematical statistics[END_REF], in practice this is an open choice. When data are qualitative or categorical, sample points p i belong to a discrete and finite set of labels or categories. In that special case we can use the following kernel:

K ′ k (p) = K k (p) = 1 if p k = p, 0 otherwise. (2)

B. Related work

There exist several lines of work that deal with the decentralized estimation of a distribution. We can classify them in two categories: parametric-estimation and non-parametric estimation. In the first case assumptions are made on the underlying distribution which can be determined by a set of parameters. The objective is to determine the parameters' value that minimize a given criteria, like the Maximum Likelihood. Specific Algorithms to distributively compute a Gaussian Mixture Model are given by Nowak in [START_REF] Nowak | Distributed em algorithms for density estimation and clustering in sensor networks[END_REF] and Jiang et al. in [START_REF] Jiang | Scalable and robust aggregation techniques for extracting statistical information in sensor networks[END_REF]. Other solutions proposed by Kowalczyk, Vlassis and Sfakianakis use a gossip protocol framework [START_REF] Kowalczyk | Newscast em[END_REF], [START_REF] Vlassis | Gossip-based greedy gaussian mixture learning[END_REF]. Regarding non-parametric estimation, a distributed kernel estimation is given by Hu and al. in [START_REF] Hu | Distributed density estimation using non-parametric statistics[END_REF]. This last work also provides a compression method (with information loss) to consider nodes with limited resources. Even if these previous work have proven their efficiency, they are either very specific to the considered problem, not trivial to implement or make strong assumption on the network model that might not stick to the reality of wireless mobile networks. As far as the authors' knowledge, the problem of a time-varying distribution has not been treated yet and the adaptation of previous work to this new assumption could become fairly complex.

We argue that decoupling the diffusion of the information from the information itself facilitates both the understanding and the implementation. Consequently, dealing with the timevarying assumption will become straightforward.

In the next section, we explain how to take advantage of the average consensus to estimate a time-varying pdf in a decentralized way.

III. DECENTRALIZED ESTIMATION OF TIME-VARYING DISTRIBUTIONS

We first introduce the average consensus framework and establish the notation, then we present our contribution.

A. The Average Consensus Framework

The term distributed consensus is often associated with the work of Fischer, Lynch and Paterson [START_REF] Fischer | Impossibility of distributed consensus with one faulty process[END_REF], but the consensus problem also has a strong background in the field of control theory [START_REF] Borkar | Asymptotic agreement in distributed estimation[END_REF]. In fact, a consensus usually refers to the reach of an agreement between several agents that share a protocol. In the case of the average consensus, the agreement to reach is the average of nodes' states. The theoretical framework to study the consensus problem in a dynamic system of networked agents has been introduced by Olfati-Saber and Murray in [START_REF] Saber | Consensus protocols for networks of dynamic agents[END_REF]. It is now applied in various domains such as synchronization of oscillators [START_REF] Preciado | Synchronization in generalized erdsrnyi networks of nonlinear oscillators[END_REF], load balancing [START_REF] Xu | Load Balancing in Parallel Computers: Theory and Practice[END_REF], and sensor fusion [START_REF] Martinez | Distributed representation of spatial fields through an adaptive interpolation scheme[END_REF]. In the scope of this paper, we consider network monitoring. The consensus framework can be used under different assumptions on the nature of the exchange between agents which can be either continuous or discrete. In our case, the exchange between agents is materialized by packets, which are discrete by nature.

Network Model: We place our work under a synchronous communication model in which a node iteratively communicates a consensus value at a discrete time index to its neighbors and updates this value by taking into account the values it has received. A network is modeled by a graph G = (N , E) for a set N = [1, n] of n nodes (or agents) and a set E ∈ N 2 of edges. An edge (i, j) ∈ E materializes the possible reception by the node j of a monitoring packet from node i. The state of node i is s i ∈ S. We denote by s := 1 n n i=1 s i the average value of the state over the network. In the case of dynamic states, the value of s i can change over time, thus s i (t) stands for the sample of i at the discrete time index t ∈ N 0 . Similarly, we define s(t) := 1 n n i=1 s i (t). In the case of a dynamic topology, the communication graph evolves with agents' mobility. Agents that were able to communicate may not reach each other, whereas new communication links may appear. In that case, we consider that the set of edges can change over time whereas the set of nodes remains the same, thus, G(t) = (N , E(t)) stands for the graph of the network at t ∈ {0, 1, . . .} . Note that, even if N is constant, we can still model the arrival of a node by adding edges to an isolated node, and the departure of a node by removing its incident edges.

Linear Averaging Iterations: We now review the known results concerning linear iteration for the discrete-time average consensus problem. Existing results differ mainly according to whether the topology and the states of nodes are static (constant) or dynamic (time-varying). In the case of a static topology with static states the authors of [START_REF] Xiao | Fast linear iterations for distributed averaging[END_REF], [START_REF] Xiao | Distributed average consensus with timevarying metropolis weights[END_REF] rewrite the distributed linear averaging consensus as a product of matrices. At each step, nodes iterate their values using the scheme x i (t + 1) = n j=1 w ij x j (t) where w ij is the weight given by the node i to the node j. In the matrix notation the equation becomes x(t+1) = Wx(t), with x(0) = s. In order to respect the network topology, the values of the weights are constrained by G and ∀(i, j) / ∈ E we have w ij = 0. When the topology evolves over time, the iteration becomes x(t+1) = W(t)x(t), where W(t) is constrained by E(t). It is said that the consensus converges if lim t→∞ x(t) = s which is equivalent to the matrix equation lim t→∞ t k=0 W(k) = 11 T n . In our context, states and topology are both dynamic. This case is treated in [START_REF] Zhu | Discrete-time dynamic average consensus[END_REF]. The authors propose a n-order linear algorithm but we only consider the First Order Dynamic Average Consensus (FODAC) algorithm. The FODAC iteration can be re-written in the following way,

x i (t + 1) = δs i (t) + n j=1 w ij x j (t), (3)
where δs i (t) = s i (t)s i (t -1). The authors in [START_REF] Zhu | Discrete-time dynamic average consensus[END_REF] derive an upper bound on the steady state error which is valid under the following assumptions: (A.0) the weighting matrix W(t) is doubly stochastic, (A.1) the first order dynamic of states is relatively bounded, (A.2) the network has a periodical strong connectivity.

The assumption (A.1) formally means that

∃B ∈ S | ∀t max i (δs i (t)) -min i (δs i (t)) < B.
Regarding the assumption (A.2), a succession of graphs G 1 , G 2 , . . . , G n with the same set of nodes N , has a periodical strong connectivity if

∃P ∈ N * | ∀t G u = (N, ∪ t+P i=t E i ) is strongly connected.
In the remainder of the paper, we use the consistent metropolis weighting policy from [START_REF] Xiao | Distributed average consensus with timevarying metropolis weights[END_REF], which is an heuristic that shows better convergence than its concurrents and verifies assumption (A.0). This policy defines the weights that nodes give to their neighbors (and consequently the weighting matrix W(t)) as follows:

w ij (t) =      1/(1 + max(d i (t), d j (t))) if i ∈ V j (t), 1 -j =i w ij (t) if i = j, 0 otherwise, (4)
where V i (t) is the neighborhood of node i at step t and d i (t) its associated cardinal (degree of node i). Note that this policy reasonably assumes that the graph is undirected, which means that, node i considers the value of node j if and only if it is reciprocal. The assumption (A.1) is respected since the states we consider have bounded norms. Finally, the verification of assumption (A.2) relies on the network under study and will determine the convergence towards the average of states.

B. Taking advantage of the Consensus Framework

As we just explained, the dynamic average consensus framework allows nodes to track the average of their states while the communication graph can evolve over time. Note that Equation (3) only requires states to be part of a vector space. Our approach is to rewrite the decentralized density estimation problem in an average consensus problem. To do so, we define the set of states in such a way that their average is the estimation of the distribution we want to track. It means that we need to provide the state space with an internal addition and a scalar multiplication that respect the axioms of a vector space. The main advantages of this approach are the immediacy of convergence proofs and, regarding practical aspects, the triviality of an oriented object implementation. The following subsection explains the transformation of a density estimation problem into an average consensus problem.

C. Average Consensus over Kernel Functions

When looking at Equation (1), it is clear that the kernel density estimation of a pdf is a weighted average of functions. Furthermore, if we set α i = 1 P , it becomes a simple average. Thus, if we define the state of an agent with the correct function, it results that the FODAC algorithm will track the value f . Hence, it follows that we need to model a function space, its addition and scalar multiplication.

Dictionary Representation: In our case functions are linear combinations of scaled kernels fully characterized by a tuple (K k , H k , p k ). As a result, we represent a function f with a dictionary (or associative array), D f : K f → V f where keys c ∈ K f are tuples characterizing scaled kernels and values v ∈ V f ⊂ R * are the corresponding nonzero real coefficients. In the remainder of the section we consider D f ,D g , D h as the representation of three functions f, g, h. D ∅ stands for the empty dictionary.

Addition: We define the addition between dictionaries as follows.

h = f + g ⇐⇒ K h = K f ∪ K g -{c ∈ K f ∩ K g | D f (c) + D g (c) = 0},
and ∀c ∈ K h we have

D h (c) =      D f (c) if c ∈ K f ∩ K g D g (c) if c ∈ K f ∩ K g D f (c) + D g (c) else.
(5) Dictionaries are merged, and for each common key, values are summed, if the sum is null, the key is removed.

Scalar Multiplication: We define the scalar multiplication as follows for γ ∈ R * :

h = γf ⇐⇒ K h = K f , and ∀c ∈ K h , D h (c)= γD f (c), h = 0f ⇐⇒ K h = ∅, (6)
which means that for each key, the value is multiplied by the scalar, if this scalar is zero, then the dictionary is cleared.

Satisfy Axioms: The way we define addition and scalar multiplications allows to satisfy axioms of vector spaces. We will not detail their proof which are trivial.

Initial Value and Convergence: The state of agents are functions that can vary over time. This variation can be due, for instance, to the evolution of the observed set of sample points, their associated kernel or their associated bandwidth. However, for the seek of clarity, we simplify the notation by using s i , x i , fi and f instead of (s i (t))(p), (x i (t))(p), ( fi (t))(p) and ( f (t))(p), respectively. Let P be the set of P sample points partitioned between agents where each of them has a subset P i ⊂ P of P i samples. We denote K ′ i,r the scaled kernel that agent i associates to its r th sample point and we organize the scaled kernels in a multiset (K ′ , m) where K ′ = i,r {K ′ i,r } and m is the multiplicity function. Thus, the centralized KDE of the distribution given the scaled kernel of all the agents with equal weight is

f = 1 P N i=1 Pi r=1 K ′ i,r = 1 P K ′ ∈K ′ m(K ′ )K ′ , where P = N i=1 P i = K ′ ∈K ′ m(K ′ ). (7)
We now define the state of agent i by s i = Pi k=1 K ′ i,k . Since the consensus value x i of node i tracks the value s, we have

x i ≈ s = 1 N N i=1 s i = 1 N N i=1 Pi k=1 K ′ i,k , ∀i. (8)
By identifying f in the right hand side of (8) we obtain

x i ≈ s = P N f = K ′ ∈K ′ m(K ′ ) N K ′ . (9)
This last equation states that each node participating to the FODAC algorithm asymptotically estimates the value of f with a scaling factor P N , which is the average number of sample points per agent. Each (K ′ , m(K ′ ) N ) is identifiable to a (key,value) couple in the dictionary representation of f . As a result, the sum over the value of this dictionary is

K ′ ∈K ′ m(K ′ ) N = P N .
Hence, each node can compute its own scaling factor by summing all the values of its dictionary. 2 , agents exchange their value and compute the average. At a given time t ∈ {1, 2, . . .}, a and b observe their own unique sample p a (t) and p b (t) in R. We choose the following values p a (t) = -1, -2, -2, -2, ... and p b (t) = 1, 1, 1, 1, .... Figure 1 shows the Gaussian estimation x a (left) and x b (right) of agents for the first four steps and the corresponding dictionaries. In the dictionary representation, a key is illustrated by the position of a row, whereas its value is represented by its length and direction.

Remarks: One can notice that the consensus algorithm given by Equation ( 3) introduces a difference between states (equivalent to the introduction of negative kernels), δs i . It results from this difference that x i is not necessarily positive and consequently is not a pdf (as usually defined). However, this is not a concern since x i is an estimation of f . This effect is mainly due to transient nodes' states, it vanishes with the stabilization of the set of observations, as illustrated for x a (t), for t > 1 in Figure 1.

D. The Precision-Resources Trade-off

Besides computational resources, a consensus algorithm requires memory, and communication bandwidth. The required memory is directly linked to the size of the representation of the consensus value. The required communication bandwidth is evaluated by multiplying this size with the rate at which messages are exchanged. We suggest to adjust these parameters, that, also affect the precision of the estimation. For instance, the reduction of the exchanges' rate, save some communication bandwidth. However, if this rate is too low, the FODAC algorithm faces an undersampling situation and the steady state error evaluated using the upper bound in [START_REF] Zhu | Discrete-time dynamic average consensus[END_REF] will be important. Reducing the size of the consensus representation is another solution to save both communication and memory resources, this reduction implies information loss and also degrade the estimation.

The size of the representation of the pdf follows an O(|K ′ |) progression. Intuitively, we can say that the kernel set of a constant distribution is collected in a number of step equal to the network's diameter, increasing quickly the size of the exchanged messages. This is specifically true in the case of a time-varying distribution where new kernels are regularly introduced, while older might never disappear. To tackle this side-effect, Hu and al suggested in [START_REF] Hu | Distributed density estimation using non-parametric statistics[END_REF] the use of compression algorithm between exchanges. While these methods are applicable for constant distribution, they should be carefully applied in our case. Using a compression algorithm is equivalent to changing the vector space addition, and in the meantime losing its associativity. Consequently, the properties of the FODAC algorithm are not guaranteed anymore, possibly letting nodes progressively derive from the average. Actually, an other way to proceed that conserve the addition's associativity is to limit the size of the kernel set. Indeed, the representation of the consensus only grows with the introduction of a new kernel, thus, using a finite set of kernels will limit this growth. As a result, our approach is to bound and discretize the set of kernel, or equivalently, to use categorical kernels.

IV. MONITORING OF WIRELESS MOBILE NETWORKS

Our method has been designed to monitor wireless network, thus, we evaluate it through NS-3 simulations. This section describes the considered scenario. Then, we present our estimation results.

A. Network Topology and Traffic

We simulate a wireless mobile ad-hoc network during 100 seconds. For this purpose, we created two primary geographical zone and placed 60 nodes in each of them. The size of the zones are set such as nodes from the same zone are (most of the time) in the same graph component. Indeed, in our case, the communication graph can be modeled by a

B. Network Metrics and Consensus

We assumed that each second, a node knows its own position and collects metrics that are derived from its local traffic (and in practice, could be extracted from their Management Information Base). Every consensus period τ , node communicate the information required to update their consensus value (degree and dictionary). The metrics related to the traffic are the number of byte processed by the IP layer from the local UDP sources (sent), the number of bytes processed by the IP layer from other sources (f wd) and the number of bytes dropped by the IP layer (dropped).

To describe the local level of satisfaction, we derived from these three metrics a trivial ratio sat = sent+f wd sent+f wd+dropped , which decreases when a node is not able to process a packet (route problem, overload. . . ). We conducted three dynamic average consensus, the first one tracks the average value of the sat metric, the second tracks its distribution in the whole network. In a last consensus nodes estimate the geographical (multivariate) density of their population.

C. Simulation Results

In this subsection, we successively explain the three consensus and illustrate each of them by a figure .  1) Estimation of the Average Satisfaction: A simple solution for a node to estimate the average value of a metric is to use its local value. In that case the error is given by the mean absolute deviation. In this first consensus each node estimates the average value of their satisfaction. Figure 3 shows how the mean absolute error is reduced by the FODAC algorithm and illustrates the trade-off that exists between its precision and its bandwidth consumption. On the top left sub-figure we plot the satisfaction sat i (t) of each node i every second as well as their average. The average satisfaction decreases progressively due to nodes' dynamic and to the start of UDP sources. The top right and the bottom left sub-figures represent the FODAC estimation x i (t), of this average for the consensus period τ = 0.33s and τ = 0.1s, respectively. We can notice a reduction of the average error with the consensus period. In the last 20s, one node estimated the average satisfaction to zero, it was a disconnected node in the transit zone. On the last subfigure we detail the mean absolute error ||x(t+1)-1.x(t)|| 1 /n for several magnitudes of period consensus. From this last subfigure, we can intuitively say, that at some point, in terms of information, the reduction of the uncertainty does not worth the price of the additional required bandwidth. We let this optimal Shanon trade-off as an open question. 2) Estimation of Satisfaction Distribution: Our main contribution is to estimate a time varying distribution. We illustrate this estimation in Figure 5 with a consensus period τ = 0.1s. The top sub-figure is the estimation of the satisfaction distribution across time (which is similar to the centralized estimation) for a node located in the left zone. The bottom sub-figure is the estimation of the same distribution for the node located in the transit zone that was not well connected with its neighbors during the last 20 seconds. At the beginning of the simulation nodes estimate that all the network is fully satisfied. This distribution evolves over time and both nodes are aware of this phenomena and agree on a new estimation. As expected their estimation differs during the last 20 seconds because the node in the transit zone is disconnected. However, since assumption (A.2) is respected, this node recovers the estimation when it joins the network. Estimation from the transit zone 3) Estimation of The Geographical Density: Finally, we consider the estimation of a time varying multivariate density. For this last case, we consider the geographical repartition of nodes over the map. As we noticed in section III-D the size of messages grows with the cardinal of the kernel set. Since most of compression method could not be used without loosing the addition's associativity, our technique is to limit the kernel set. In this case, all nodes consider a set of 216 kernels whose locations are uniformly distributed on the map (≈ 9 × 10 -4 kernel/m 2 ). Instead of communicating the kernel associated to their own position, node use the nearest kernel in the kernel set. Figure 4 is the geographical density estimation of a node located in the right zone at different instants. During the simulation, transit nodes relay the information between zones, and the node improve its view of the entire map over time, being able to distinguish the different zones. An interesting point is that it has a precise view of its neighborhood while approximations only concerns distant zones.

V. CONCLUSION

Motivated by the distributed monitoring of networks with dynamic topology, we developed in this paper a method to estimate a time varying multivariate distribution in a decentralized way. Based on their local measurement and that of their neighbors, nodes iteratively exchange messages to track a dis-tribution in the network. Our work relies on the f -consensus theory from which we can derive convergence conditions and error bounds. The resulting method allows a trade-off between precision, and resource consumption. Simulations showed that using several consensus, nodes could behave for the benefit of the whole network. In the present paper, by coupling the satisfaction distribution with the geographical density, some nodes could decide to move towards the transit zone and increase the global network quality. Finally, we note that the convergence speed is limited by the fact that all nodes are highly mobile and participate to the consensus. However, if we relax those two assumptions, we could build overlay networks and drastically improve the estimation giving the possibility to efficiently monitor large fixed networks.

Fig. 1 .

 1 Fig. 1. Four First Estimation Step of a 2-Nodes Network

Fig. 2 .

 2 Fig. 2. Considered Network Topology With 3 Transition Nodes

Fig. 3 .

 3 Fig. 3. Estimation of The Average Satisfaction

Fig. 5 .

 5 Fig. 5. Estimation of Satisfaction Distribution

  Fig. 4. Geographical Density Estimation of a Node Moving in the Right Zone at t = 10s, t = 60s and t = 90s (From Left to Right) with 216 kernels

	300m				t =10s						300m				t =60s			300m				t =90s
	200m										200m							200m
	100m	Node position							100m	Node position				100m	Node position
		100m	200m	300m	400m	500m	600m	700m				100m	200m	300m	400m	500m	600m	700m	100m	200m	300m	400m	500m	600m	700m
					Estimation from the left zone
			0	20	Tim e (s) 40	60	80 100	0.0	0.2	S a ti s fa c ti o n 0.4 0.6 0.8 1.0 0 2 4 6 8 10 12 Dens ity
			0	20	Tim e (s) 40	60	80 100	0.0	0.2	S a ti s fa c ti o n 0.4 0.6 0.8 1.0 0 2 4 6 8 10 12 Dens ity