Valérie Pommier-Budinger
email: valerie.budinger@isae.fr

Yannick Bury
email: yannick.bury@isae.fr.researcher

Guilhem Michon
email: guilhem.michon@isae.fr.engineer

Gael Napias
email: gael.napias@isae.fr

In-flight piv for cror flight test demonstration

Designing innovative solutions for future aircraft is one the concerns of aeronautical engineers. Among the studied technologies, Counter Rotating Open Rotor (CROR) propeller technologies are appearing as a promising, though highly challenging, alternative solution to Ultra High Bypass Ratio (UHBR) engines. Amongst these challenges, the interaction of the wake of the CROR engine pylon with the counter rotating blades, positioned downstream of the pylon (pusher configuration), are responsible for both airframe noise and vibrations that penalize aircraft certification. 'Erasing' the pylon wake such as to recover a strictly uniform flow upstream of the rotating blades would suppress this major source of airframe noise and vibrations. The current study is put forward in that context and is part of a project funded by the European Commission through the Cleansky/FP7/SFWA (Smart Fixed Wing Aircraft) Demonstrator program. In the first part of the project, an optimal design of the pylon, associated with advanced flow control strategies, has been proposed. The current study aims at developing an advanced experimental methodology, based on vibration-controlled stereoscopic Particle Image Velocimetry (3C-PIV), able to be flight-operated and that will serve the validation of the above mentioned active flow control system when operated on the Flying Test Bench. This article deals particularly with the characterization of the limits of the vibration spectrum acceptable by the on-board PIV subsystems and with the design of a system to lessen potentially harmful vibrations.

I. Context of the study

The study proposed in this article is carried out in the framework of Counter Rotating Open Rotor (CROR) propellers development. It is part of the Cleansky/FP7/SFWA program, funded by the European Commission. Strong pressure fluctuations are induced by the interaction of the pylon wake with the counter rotating blades located downstream the pylon. Such fluctuations are responsible for both airframe noise and vibration issues that penalize aircraft certification. The thorough analysis of the complex physics driving the flow is required in order to reduce the wake/blades interaction thanks to an optimal design of the pylon, associated with advanced flow control strategies.

The understanding of the highly non-linear and unsteady phenomena driving the complex flow physics involved by the interaction of the pylon wake with the blades impose to implement:  advanced measurement techniques, such as three Components Particle Image Velocimetry (3C-PIV). Such technique should allow deciphering the flow structure in the wake of the pylon, and quantifying the efficiency of the active flow control strategy in mitigating the pylon wake. However, this technique is highly sensitive to vibrations and must be suited to harsh vibratory environment representative of flight conditions,  advanced post processing methods for the quantification of the flow in terms of, for instance, vortex dynamics, unsteady wake topology or modal signature of the flow.

The proposed study focuses on in-flight PIV for CROR flight test demonstration. The issues that will have to be tackled to implement successfully in-flight PIV are:  the definition of a vibratory environment representative of a CROR-propelled aircraft. This task is not obvious since CROR-propelled aircraft do not exist yet and the vibratory environment spectrum can only be estimated.  the characterization of the limits of the vibration spectrum acceptable by the PIV subsystems (cameras, laser and laser sheet forming lenses). More precisely, from the above mentioned vibratory environment spectrum, determination of the limits beyond which vibration-induced errors on PIV measurements impose attenuation of vibrations transmitted to PIV subsystems.  the design of a system for vibration control of PIV subsystems.

II. Definition of a vibratory environment representative of a CROR-propelled aircraft

The spectrum of the vibratory environment representative of a CROR-propelled aircraft can only be assumed since no measurement exists and Finite Element analyses on full aircraft model are not accurate enough to confidently predict the dynamics of an airplane structure. The reference documents used to define this spectrum are the standard DO-160 on Environmental Conditions and Test Procedures for Airborne Equipment 1 and the articles [START_REF] Catherines | Vibrations measured in the passenger cabins of two jet transport Aircraft[END_REF]3,[START_REF] Rivin | Passive vibration isolation[END_REF] that give some measurements on aircraft such as jet transport aircraft and a DASH 8-300 with turbopropellers (Figure 1). This former generation of aircraft is assumed to be more representative of a CROR-propelled aircraft than the new generation of aircraft that is optimized in terms of vibratory confort in cabin. In the end, Figure 2(left) gives the spectrum in the vertical direction (Z-axis) chosen for the vibratory environment representative of CRORpropelled aircraft. This spectrum is a compromise between curve C of the RTCA/DO-160 and measurement data available in the literature. The global shape of this spectrum complies with the RTCA/DO-160 curve and ensures a high safety factor in comparison with effective vibration levels measured in standard existing aircraft [START_REF] Catherines | Vibrations measured in the passenger cabins of two jet transport Aircraft[END_REF]3 . Tests have been performed with a shaker to measure the behavior of a camera fixed on a Scheimpflug board and subject to such a spectrum. In the Z-direction, the RMS acceleration is 1.2g and the maximal displacement is 1.02mm (Figure 2(right)). These are not extreme conditions but future flight tests with on-board PIV measurements will not to be performed in harsh conditions. For the X-axis and Y-axis, the vibration spectrums are assumed to have the same shape as Z-axis spectrum but with significantly lower amplitudes (approximatively 10% of the Z-axis amplitude) since measurements of the vibrations in the X and Y directions show that the coupling with the Z direction is relatively weak.

III. Characterization of the limits of the vibration spectrum acceptable by the PIV subsystemss

The spectrum of the vibratory environment representative of a CROR-propelled aircraft defined in the previous section will be used to define the limits of the vibration spectrum acceptable by the PIV subsystems for in-flight measurements. More particularly, the limits beyond which vibrations propagating on PIV subsystems induce displacements which noticeably affect the measurements will be determined from this spectrum. It should also be noted that the deformation of the aircraft cabin, even if it induces small displacements on the PIV system, may also alter the measurements. These static displacements should also be considered for the definition of a vibration control system dedicated to the PIV subsystems.

A. In-Flight to Indoor PIV Measurements

First of all, it is important to imagine the 3D-PIV system implemented in the aircraft. One configuration is proposed in Figure 3 in which the laser sheet as well as two windows for the cameras are represented. A specific test bench has been developed to represent this configuration and to allow defining the limit of the displacements and the vibration spectrum acceptable by the PIV subsystems.

Figure 3 -Schematic of the on-board 3D-PIV system

This test bench is based on the use of a target which simulates the images of the seeding particles that should be obtained in the pylon wake, when illuminated by the laser sheet. The target is obtained by applying an appropriate coating over a plane structure. A good result was obtained by using a sand with a specific size fixed with a black resin (Figure 4 (left)). This target can be moved compared to the cameras in order to simulate the relative displacements of the seeding particles in the laser sheet imposed by the mean flow past the CROR-propelled aircraft. Figure 5 shows the relative displacements in translation of the laser sheet compared to the cameras. The target has been designed to simulate typical flow past an aircraft. In order to simulate local vortical and shear flows superimposing with a mean uniform flow, the target can be moved in translation along the out-of-target-plane direction and two moving parts actuated by micrometric platforms are included into the target (Figure 4 (right)). They allow simulating a vortex (rotation of a moving part of the target around X (𝜃 𝑋 )) and a shear flow (translation of a moving part of the target along Y (𝑇 𝑌 ) superimposing with the mean uniform flow (translation along X of the target itself (𝑇 𝑋 )).

The test bench also comprises two cameras for 3C-PIV measurements. In order to simulate the effect of the onboard vibrations (which cause relative displacements of a given PIV subsystem, e.g. one camera, compared to the other elements of the PIV system), one camera is fixed on top of an hexapod which is used to generate small and accurate static displacements, while the other PIV subsystems (the 2 nd camera and the target faking the position of the laser sheet) do not move (Figure 5).

Note that the static approach using solid target made of sand and epoxy resin in place of a lasersheet illuminating a volume of fluid makes sense due to the time scale involved for PIV measurements. Indeed, for PIV, a pair of pictures used for the determination of an "instantaneous" velocity field is acquired in less than a few tens of micro seconds. Within this time, the displacements of the PIV subsystems induced by the CROR-generated vibrations, of the order of a few micrometers, can faithfully be neglected. Consequently, time independant measurements may be used. In order to simulate the in-flight acquisition of pairs of particle images used for the computation of instantaneous velocity fields, successive pairs of images are acquired by imposing a given displacement of the target and of its moving parts between the first and the second image of each pair. The vibrations experienced in the cabin of the CROR-propelled aircraft cause the relative displacements of the PIV subsystems along the 3 translation and the 3 rotation axes. The latter are modeled by imposing, from one pair of images to the consecutive one the displacement of the mobile camera relative to the other camera along each of the 6 axes. In order to determine the maximal acceptable displacement allowing the reconstruction of the velocity/displacement field along each of the 6 axes, a step-by-step approach is applied for the displacement of the mobile camera, from the reference position of the camera to a maximum value, chosen on the basis of the maximum displacements imposed by the assumed vibration spectrum (Fig. 2). Performing all the required PIV measurements manually would take a long time due to the total number of tests that must be done, thus the test bench is run in an automatic way thanks to controllable displacement platforms.

B. Determination of the 3C-PIV Test bench Accuracy

The objective of this part is to describe the methodology developed for the determination of the 3C-PIV test bench accuracy, for a reference configuration simulating a typical flow past an aircraft. The measurement errors in the PIV test bench mainly come from optical and algorithm limitations: magnification factor, optical quality of the lenses, observation angle and associated Scheimpflug correction angles, PIV cross correlation algorithms. To perform the calibration, the two cameras are first fixed in a given reference position. Then the moving parts of the target are displaced and the displacement of the target measured by the PIV system is compared to the reference displacement measured by an external sensor that has itself an accuracy of 6µm.

Three kinds of tests are performed: -Tests to measure the standard deviation of the PIV measurement system when the target simulates the mean uniform flow. Only translations along X are carried out; the 2 others movements of the target are set to zero.

The micrometric platform generates a step to move the target for 4 reference displacements: 0.5mm, 1mm, 2mm and 5mm. The movement for each step value is repeated 300 times to compute the standard deviation between the actual displacement imposed to the target and the one measured by the 3C-PIV system. -Tests to measure the standard deviation of the PIV measurement system when the target simulates a shear flow.

Only translations along Y are carried out. The micrometric platform generates a step to move the target for 3 reference displacements: 0.5mm, 1mm and 3mm. As for the mean uniform flow, the movement for each step value is repeated 300 times. -Tests to measure the standard deviation of the PIV measurement system when the target simulates a vortical flow. Only rotations along X are carried out. The micrometric platform generates a step to move the target for 3 reference rotations: 0.5°, 1° and 2°. The movement for each step value is also repeated 300 times.

Table 1 to Table 3 summarize the results obtained for these tests. For all three kinds of movement, the standard deviation is reasonably constant whatever the values of the step. The error is always very small: less than a few percents. The conclusions are that the PIV measurement system is very accurate, and the standard deviations are evaluated for the three movements that allow simulating the typical flow of an aircraft.

Measurement configuration: translation along

C. Simulating the typical flow past an aircraft

The three different possible displacements of the target (Figure 4(right): translation of the target itself in the outof-target-plane direction, rotation of a part of the target and translation of another part to simulate vortical and shear flows respectively), aim at simulating a typical flow past an aircraft at Mach number 0.23 (typical takeoff air speed). This typical flow is represented in our test bench by the three following displacements of the target: -Step of 1mm in translation along X -Step of 1mm in transalation along Y -Step of 1° in rotation around X These steps are carried out simultaneously 300 times and Table 4 gives the reference displacements and the measured displacements. The results show that the test bench allows producing the displacements of the target precisely. The static deviations in a configuration with a combination of the three possible movements of the target are consistent with the results of the test bench calibration. The moving target of the test bench is thus an efficient device to simulate the typical flow of an aircraft. It will be used in the next section to measure the limit of the displacements acceptable by an on-board 3D-PIV measurement system.

Type of displacement Reference displacement

Displacement

D. Limit of the displacements acceptable by on-board 3D-PIV measurement systems

The objective of this section is to measure the limit of the displacements acceptable by on-board 3C-PIV measurement systems, i.e. the maximal displacements allowing the reconstruction of the velocity/displacement field along each of the 6 axes. Remember that the test bench comprises two moving devices: the target that simulates the typical airflow of the aircraft and the camera on the hexapod that simulates the vibrating environment of the aircraft. The camera acquires pairs of images used for the computation of instantaneous velocity fields. As previously mentioned since during a real PIV test campaign each pair of pictures is acquired in less than a few tens of micro seconds, thus the camera can be considered as fixed between the two images of a pair in the PIV test bench considered in this study.The two images of each pair are taken for two positions of the main target and/or of its moving parts. For determining the maximal acceptable displacements, a step-by-step approach is performed and consists in moving of the mobile camera along each of the axes, from a reference position of the camera to a maximum value given in Table 5, while between each image of each pair, the target moves according to values given by Table 4. The number of measures to perform this step-by-step approach is very important (750) and tests have been automated.

Type of displacement Displacement

Translation along X 0 to 1.5mm (step 10µm)

Translation along Y 0 to 1.5mm (step 10µm)

Translation along Z 0 to 1.5mm (step 10µm)

Rotation around X 0 to 0.4° (step 0.004°) Rotation around Y 0 to 0.4° (step 0.004°)

Rotation around Z 0 to 0.4° (step 0.004°)

Table 5 -Camera displacements on the hexapod for determining the limit of the displacements acceptable by on-board 3C-PIV measurement systems

For the chosen configuration in our test bench (Table 5), test results reveal that the most critical parameter for the computation of the velocity field is the rotation around Z. Figure 5 shows the target displacement measured by the PIV (for a target step reference of 1mm) as a function of the rotation angle  z of the camera. It can be observed that if  z exceeds 0.17°, the PIV measurement system is no more able to reconstruct the velocity field and to compute the displacement. Note that this 0.17° camera rotation around Z corresponds to a 11.8mm shift on target's z-direction. Given the magnification factor of the images imposed by the PIV test bench, this value corresponds to the shifting of more than a cross correlation windows (classically fixed to 32*32 pixels 2 ) in the 3 components rebuild process. Thanks to the distance between cameras and the target (i.e. 4m), this high displacement tolerance is possible contrary to classical short distance PIV measurement systems. For rotations around X and Y axes, as the configuration in aircraft is not yet well-defined, we consider the same limits, even if it is pessimistic in the case of our test bench.

At this stage of the work, the study has shown that the displacements resulting from the rotations imposed by vibrations are more critical than the linear displacements. However this study is yet limited to a configuration where only one Scheimpflug platform is considered, since only one angle of observation is imposed between the PIV cameras and the laser sheet. For the final 'on board' configuration, the PIV cameras will be supposed to observe the scene, i.e. the laser sheet, from two angles (see Fig. 3), imposing the implementation of two Scheimpflug platforms. The admissible shift of the target should then be computed taking into account this more complex implementing configuration.

IV. Design of a system for vibration control of PIV subsystems

A stabilization platform must be implemented to attenuate the vibrations imposed by the CROR aircraft environment to the PIV subsystems, which induce larger displacements than acceptable displacements for the PIV measurements system (large displacements that may prevent the reconstruction of the velocity/displacement field or damage the cameras). Several technologies are under study for this platform: -Passive dampers: viscoelastic dampers are well-known to introduce damping in a structure. Their efficiency mainly depends on the type of physical properties. Of course, their location is also an important key-driver for the design of a vibration isolation system. Optimization procedures for the choice and the locations of dampers have been developed [START_REF] Rivin | Passive vibration isolation[END_REF] . The main drawbacks of this isolation technology are the poor performance at low frequency and the nonlinear behavior of viscoelastic material 5. -Tunes mass-damper vibrations absorbers: this technology allows attenuating vibrations in a narrow bandwidth in a very efficient way. The drawback of this technology is the narrowness of the efficiency. To prevent this drawback, adaptive tuned mass-dampers are developed to make the frequency range of the absorbers vary with the frequency range of the perturbation [START_REF] Rana | Parametric study and simplified design of tuned mass dampers[END_REF][START_REF] Sun | Passive, adaptive and active tuned vibration absorbers-a survey[END_REF] . -Active and hybrid vibration control systems. Active vibrations control systems are used when the limits of passive devices are reached. Many studies on active vibration control concern automotive applications X and civil engineering [START_REF] Soong | Passive and active structural vibration control in civil engineering[END_REF][START_REF] Sun | Adaptive robust vibration control of full-car active suspensions with electrohydraulic actuators[END_REF] . These systems are based on the use of actuators that can be of many kinds (electromagnetic, piezoelectric, hydraulic, pneumatic). When passive dampers are combined with an active vibration control system, we get a two-stage hybrid system. The passive stage aims at filtering high frequency vibrations while the active stage will be controlled to attenuate low frequency vibrations.

For our application, two technologies are assessed (in simulation for the moment): passive dampers and hybrid vibration control systems. The performance is computed for displacements along the Z axis and compared to the vibrations induced by the vibration spectrum when no isolation system is implemented.

A. Passive system

Elastomer dampers are chosen to attenuate vibration frequencies above 15Hz. These passive dampers need to be characterized. Indeed, passive dampers filter high frequencies but to the detriment of an amplification of low frequency vibrations. This amplification, which depends on the load supported by the dampers, is not well characterized in datasheets. Thus, experiments are necessary to measure the spectrum of the dampers along 3 axes at low and high frequency with a load representative of the future on-board PIV system (Figure 7). Figure 7 highlights a resonance frequency at low frequency, close to 8 Hz. 1-dof simulations in the Z-axis are carried out in order to estimate the influence of the dampers. Figure 8 shows the results of the simulation: the damper attenuates the high frequencies but is less efficient regarding low frequencies, specially around 8 Hz which corresponds to the frequency amplified by the damper. Figure 10 shows the simulation results along the Z-axis: the remaining low frequencies are attenuated by a factor 5 compared to the solution with the dampers only. This solution allows increasing the performance of the vibration attenuation system but is costly in terms of weight and complexity. Further tests on the real system should help us evaluating which solution is the most appropriate for implementing on-board 3C-PIV measurement systems.

V. Conclusion

This article has presented the results of a study concerning Counter Rotating Open Rotor (CROR) propeller technologies, and specially the development of an advanced experimental methodology, based on vibrationcontrolled stereoscopic Particle Image Velocimetry (3C-PIV), able to be implemented on board to measure typical flow of this kind of aircraft when it will be operated for flight test validation. This study is made difficult by the fact that this kind of aircraft does not exist yet. For our work, assumptions must be made on the vibrating environment in cabin of such aircraft. In this context a test bench reproducing this environment has been developed and allowed establishing the limits of the vibration spectrum acceptable by the on-board PIV subsystems. The results of this study is that the displacements in rotation of the cameras used for the 3C-PIV are the most critical displacements that prevent the reconstruction of the velocity/displacement field. This article ends with the proposal of a hybrid vibration control system to lessen potentially harmful vibrations.

Figure 1 -

 1 Figure 1 -Acceleration power spectral density measured in the vertical direction during 5 different descents in a bi-motor aircraft of years 70 2

Figure 2 -

 2 Figure 2 -(left) Assumed spectrum in the Z-axis (vertical direction) for the vibratory environment representative of CROR aircraft (right) Acceleration, velocity and displacement in the Z direction

Figure 4 -

 4 Figure 4 -(left) Target with the coating used to simulate particle images (right) Schematic of the target's calibrated displacements and nature of the displacements in the (X,Y,Z) reference frame.

Figure 5 -

 5 Figure 5 -Scheme of the test bench for In-Flight to Indoor 3C-PIV Measurement.

Table 4 -

 4 Target displacements representing a typical airflow: references imposed to the target and corresponding displacements measured by the 3C-PIV system

Figure 6 -

 6 Figure 6 -Target displacement measured by the PIV (for a target step reference of 1mm) as a function of the rotation angle  z of the camera.

Figure 7 -

 7 Figure 7 -(left) Passive dampers with a load on the shaker, (right) Spectrum of dampers with a load along 3 axes.

Figure 8 -

 8 Figure 8 -Vibration amplitudes (top) without any isolation vibration system, (bottom) with passive dampers

Figure 9 -

 9 Figure 9 -Scheme of the hybrid solutionThe model of the hexapod along the Z-axis is identified thanks to step and sinus tests. The hexapod is a closedloop system whose open-loop can be modeled by a second-order system controlled by a PI controller:

Figure 10 -

 10 Figure 10 -Vibration amplitudes as a function of time and amplitude spectrum (top) without any isolation vibration system (middle) with dampers, (bottom) with passive dampers and hexapod

Table 1 -Calibration of the PIV measurement system for the measure of the simulated mean uniform flow

 1 X of the target

	Reference displacement (mm)	Displacement measured by the PIV system (mm)	Standard deviation (µm)	Error (%) (reference displacement -measured displacement)
				/reference displacement
	0.5	0.489	11.3	2.25
	1	0.981	10.9	1.94
	2	2.024	11.7	1.19
	5	4.999	11.6	0.02
	Measurement configuration: translation along Y of one moving part of the target
	Reference (mm) displacement	system (mm) measured by the PIV Displacement	deviation (µm) Standard	Error (%)
	0.5	0.502	4.2	0.4
	1	1.006	5.7	0.6
	3	3.001	5	0.03

Table 2 -Calibration of the PIV measurement system for the measure of a simulated shear flow

 2

	Measurement configuration: rotation along X of one moving part of the target
	Reference rotation	Rotation measured by	Standard deviation (°)	Error (%)
	(°)	the PIV system (°)
	0.5	0.485	0.0015	3.1
	1	0.991	0.0089	0.91
	2	2.044	0.0457	2.25

Table 3 -Calibration of the PIV measurement system for the measure of a simulated vortical flow

 3

Acknowledgments

The authors acknowledge financial support by the European commission through the research project ACcTIOM, part of the european research program CleanSky.


Ankit Chiplunkar

Emmanuel Rachelson

Michele Colombo

Joseph Morlier

Sparse Physics-based Gaussian Process for Multi-output Regression using Variational Inference

Keywords: Gaussian Process, Kernel Methods, Variational Inference, Multi-output Regression, Flight-test data

In this paper a sparse approximation of inference for multi-output Gaussian Process models based on a Variational Inference approach is presented. In Gaussian Processes a multi-output kernel is a covariance function over correlated outputs. Using a general framework for constructing auto-and cross-covariance functions that are consistent with the physical laws, physical relationships among several outputs can be imposed. One major issue with Gaussian Processes is efficient inference, when scaling up-to large datasets. The issue of scaling becomes even more important when dealing with multiple outputs, since the cost of inference increases rapidly with the number of outputs. In this paper we combine the use of variational inference for efficient inference with multi-output kernels enforcing relationships between outputs. Results of the proposed methodology for synthetic data and real world applications are presented. The main contribution of this paper is the application and validation of our methodology on a dataset of real aircraft flight tests, while imposing knowledge of aircraft physics into the model.

INTRODUCTION

In this work we consider the problem of modelling multiple output Gaussian Process (GP) regression [START_REF] Rasmussen | Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)[END_REF] correlated through physical laws of the system, while in presence of large number of inputs. In the literature inference on multiple output data is also known as co-kriging [START_REF] Stein | Interpolation of Spatial Data: Some Theory for Kriging[END_REF] or multi-kriging [START_REF] Boyle | Dependent gaussian processes[END_REF]. Modelling multi-output kernels is particularly difficult because we need to construct auto-and crosscovariance functions between different outputs. We turn to a general framework [START_REF] Constantinescu | Physicsbased covariance models for gaussian processes with multiple outputs[END_REF] to calculate these covariance functions while imposing prior information of the physical processes. While a joint model developed using correlated covariance functions gives better predictions, it incurs a huge cost on memory occupied and computational time. The main contribution of this paper is to apply variational inference on these models of large datasets (of the order O(10 5 ))and reduce the heavy computational costs incurred.

Let us start by defining a P dimensional input space and a D dimensional output space. Such that {(x j i , y j i )} for j ∈ [1; n i ] are the training datasets for the i th output. Here n i is the number of measurement points for the i th output, while x j i ∈ R P and y j i ∈ R. We next define x i = {x 1 i ; x 2 i ; . . . ; x n i i } and y i = {y 1 i ; y 2 i ; . . . ; y n i i } as the full matrices containing all the training points for the i th output such that x i ∈ R n i ×P and y i ∈ R n i . Henceforth we define the joint output vector Y = [y 1 ; y 2 ; y 3 ; . . . ; y D ] such that all the output values are stacked one after the other. Similarly, we define the joint input matrix as X = [x 1 , x 2 , x 3 , . . . , For simplicity take the case of an explicit relationship between two outputs y 1 and y 2 . Suppose we measure two outputs with some error, while the true physical process is defined by latent variables f 1 and f 2 . Then the relation between the output function, measurement error and true physical process can be written as follows.

y 1 = f 1 + ε n1 y 2 = f 2 + ε n2 (1)
Where, ε n1 and ε n2 are measurement error sampled from a white noise gaussian N (0, σ n1 ) and N (0, σ n2 ). While the physics based relation can be expressed as,

f 1 = g ( f 2 , x 1 )
(2) Here g is an operator defining the relation between f 1 and an independent latent output f 2 . A GP prior in such a setting with 2 output variables is:

y 1 y 2 ∼ GP m 1 m 2 , K 11 + σ 2 n1 K 12 K 21 K 22 + σ 2 n2
(3) K 12 and K 21 are cross-covariances between the two inputs x 1 and x 2 . K 22 is the covariance function of independent output, σ 2 n1 and σ 2 n2 are the variance of measurement error, while K 11 is the auto-covariance of the dependent output variable. m 1 and m 2 are the mean of the prior for outputs 1 and 2. The full covariance matrix is also called the joint kernel, henceforth we will denote the joint-kernel as K XX .

K XX = K 11 K 12 K 21 K 22 (4)
while the joint error matrix will be denoted by Σ;

Σ = σ 2 n1 0 0 σ 2 n2 (5)
Full joint-kernel of a multi-output GP has huge memory and computational costs. For a multi-output GP as defined earlier the covariance matrix is of size N, needing O N 3 calculations for inference and O N 2 for storage. [START_REF] Snelson | Sparse gaussian processes using pseudo-inputs[END_REF] introduced "Fully independent training conditional" (FITC) and [START_REF] Quionero-Candela | A unifying view of sparse approximate gaussian process regression[END_REF] introduced and "Partially Independent Training Conditional" (PITC) approximations on inference of a GP using inducing inputs. Later [START_REF] Alvarez | Sparse convolved gaussian processes for multi-output regression[END_REF] extended the application of FITC and PITC to approximate the inference of multi-output GP's constructed through convolution processes. One problem with FITC and PITC approximation is their tendency to over-fit. In this work we extend the use of a variational approach [START_REF] Titsias | Variational learning of inducing variables in sparse gaussian processes[END_REF] to approximate the inference in a joint-kernel for both linear and non-linear relationships. We observe that the current approximation reduces the computational complexity to O N(MD) 2 and storage to O (NMD), where M denotes the number of inducing points in the input space.

In Section 2, we start with an introduction to multi-output GP and later derive the multi-output GP regression in presence of correlated covariances. In Section 3 we discuss various methods of approximating inference of a GP and later derive application of variational inference on the problem of multi-output kernels. Finally, in Section 4 we demonstrate the approach on both theoretical and flight-test data.

MULTI-OUTPUT GAUSSIAN PROCESS

Choosing covariance kernels for GP regression with multiple outputs can be roughly classified in three categories. In the first case, the outputs are known to be mutually independent, and thus the system can be decoupled and solved separately as two or more unrelated problems. In the second case, we assume that the processes are correlated, but we have no information about the correlation function. In this case, a model can be proposed, or nonparametric inferences can be carried out. In the third situation, we assume that the outputs have a known relationship among them, such as equation 2. The last point forms the scope of this section.

Related Work

Earlier work developing such joint covariance functions [START_REF] Bonilla | Multitask gaussian process prediction[END_REF] have focused on building different outputs as a combination of a set of latent functions. GP priors are placed independently over all the latent functions thereby inducing a correlated covariance function. More recently [START_REF] Alvarez | Sparse convolved gaussian processes for multi-output regression[END_REF] have shown how convolution processes [START_REF] Boyle | Dependent gaussian processes[END_REF] can be used to develop joint-covariance functions for differential equations. In a convolution process framework output functions are generated by convolving several latent functions with a smoothing kernel function. In the current paper we assume one output function to be independent and evaluate the remaining auto-and crosscovariance functions exactly if the physical relation between them is linear [START_REF] Solak | Derivative observations in gaussian process models of dynamic systems[END_REF] or use approximate joint-covariance for non-linear physicsbased relationships between the outputs (Constantinescu and Anitescu, 2013).

Multi-output Joint-covariance Kernels

If the two outputs y 1 and y 2 satisfy a physical relationship given by equation 2 with g (.) ∈ C 2 , and known covariance matrix K 22 . Then the joint-covariance matrix for a linear operator g (.) can be derived analytically as [START_REF] Stein | Interpolation of Spatial Data: Some Theory for Kriging[END_REF]:

K XX = g(g(K 22 , x 2 ), x 1 ) g(K 22 , x 1 ) g(K 22 , x 2 ) K 22 (6)
Since a non-linear operation on a GP does not result in a GP, for the case of non-linear g (.) the above joint-covariance matrix as derived in equation 6 is not positive semi-definite. Therefore we will use an approximate joint-covariance as developed by [START_REF] Constantinescu | Physicsbased covariance models for gaussian processes with multiple outputs[END_REF] for imposing non-linear relations:

K XX = LK 22 L T LK 22 K 22 L T K 22 + O δy 3 2 (7)
Where L = ∂g ∂y y 2 = ȳ2

is the Jacobian matrix of g (.)

evaluated at the mean of independent output y 2 . δy 2 is the amplitude of small variations of y 2 , introduced by the Taylor series expansion of g(K XX ) with respect to y 2 .

The above kernel takes a parametric form that depends on the mean value process of the independent variable. Equation 7 is basically a Taylor series expansion for approximating related kernels. Since a Taylor series expansion is constructed from derivatives of a function which are linear operations the resulting approximated joint kernel is a gaussian kernel with the non-gaussian part as the error. Higher-order closures can be derived with higher order derivatives of the relation g(.). For simplicity we will restrict ourselves to first order approximation of the auto-and cross-covariance functions leading to an error of the order O δy 3 2 . [START_REF] Constantinescu | Physicsbased covariance models for gaussian processes with multiple outputs[END_REF] provide a more detailed derivation of equation 7.

GP Regression using Joint-covariance

We start with defining a zero-mean prior for our observations and make predictions for y 1 (x * ) = y * 1 and y 2 (x * ) = y * 2 . The corresponding prior according to equation 3 and 4 will be:

Y (X)) Y (X * )) = GP 0 0 , K XX + Σ K XX * K X * X K X * X * + Σ (8)
The predictive distribution is then given as a normal distribution with expectation and covariance matrix given by [START_REF] Rasmussen | Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)[END_REF])

Y * | X, X * ,Y = K X * X (K XX ) -1 Y (9) Cov (Y * | X, X * ,Y ) = K X * X * -K X * X (K XX ) -1 K XX * (10)
Here, the elements K XX , K X * X and K X * X * are block covariances derived from equations 6 or 7.

The joint-covariance matrix depends on several hyperparameters θ. They define a basic shape of the GP prior. To end up with good predictions it is important to start with a good GP prior. We minimize the negative log-marginal likelihood to find a set of good hyperparameters. This leads to an optimization problem where the objective function is given by equation 11

log(P(y | X, θ)) = log[GP(Y |0, K XX + Σ)] (11)
With its gradient given by equation 12

∂ ∂θ log(P(y | X, θ)) = 1 2 Y T K -1 XX ∂K XX ∂θ K -1 XX Y - 1 2 tr(K -1 XX ∂K XX ∂θ ) (12)
Here the hyperparameters of the prior are θ = l 2 , σ 2 2 , σ 2 n1 , σ 2 n2 . These correspond to the hyperparameters of the independent covariance function K 22 and errors in the measurements σ 2 n1 and σ 2 n2 . Calculating the negative log-marginal likelihood involves inverting the matrix K XX + Σ. The size of the K XX + Σ matrix depends on total number of input points N, hence inverting the matrix becomes intractable for large number of input points.

In the next section we describe how to solve the problem of inverting huge K XX + Σ matrices using sparse GP regression. We also elaborate on how variational approximation overcomes the problem of overfitting by providing a distance measure between two approximate models.

SPARSE GP REGRESSION

The above GP approach is intractable for large datasets. For a multi-output GP as defined in section 2.2 the covariance matrix is of size N, where O N 3 time is needed for inference and O N 2 memory for storage. Thus, we need to consider approximate methods in order to deal with large datasets. Sparse methods use a small set of m function points as support or inducing variables.

Suppose we use m inducing variables to construct our sparse GP. The inducing variables are the latent function values evaluated at inputs x M . Learning x M and the hyperparameters θ is the problem we need to solve in order to obtain a sparse GP method. An approximation to the true log marginal likelihood in equation 11 can allow us to infer these quantities.

Related Work

Before explaining variational inference method, we review FITC method proposed by [START_REF] Snelson | Sparse gaussian processes using pseudo-inputs[END_REF]. The approximate log-marginal likelihoods have the form

F = log[GP(y|0, Q nn + Σ)] (13)
Sparse Physics-based Process for Multi-output Regression using Variational Inference where Q nn is an approximation to the true covariance K nn . In FITC approach, the Q nn takes the form,

Q nn = diag[K nn -K nm K -1 mm K mn ] + K nm K -1 mm K mn (14)
Here, K mm is a m × m covariance matrix on inducing points x m , K nm is a n × m cross-covariance matrix between the training and inducing points.

Hence the position of x M now defines the approximated marginal likelihood. The maximization of the marginal likelihood in equation 13 with respect to (x M ; θ), is prone to over-fitting especially when the number of variables in x M is large. Fitting a modified sparse GP model implies that the full GP model is not approximated in a systematic way since there is no distance measure between the two models that is minimized.

Variational Approximation

During variational approximation, we seek to apply an approximate variational inference procedure where we introduce a variational distribution q(x) to approximate the true posterior distribution p(x|y). We take the variational distribution to have a factorized Gaussian form as given in equation 15

q(x) = N (x|µ, A) (15)
Here, µ and A are parameters of the variational distribution. Using this variational distribution we can express a Jensens lower bound on the logP(y) that takes the form:

F(q) = q(x)log p(y|x)p(x) q(x) dx = q(x)logp(y|x)dx -q(x)log q(x) p(x) dx = Fq -KL(q||p) (16)
Where the second term is the negative Kullback Leibler divergence between the variational posterior distribution q(x) and the prior distribution p(x). To determine the variational quantities (x M , θ), we minimize the KL divergence KL(q||p). This minimization is equivalently expressed as the maximization of the variational lower bound of the true log marginal likelihood

F V = log(N [y|0, σ 2 I + Q nn ]) - 1 2σ 2 Tr( K) (17)
where Q nn = K nm K -1 mm K mn and K = K nn -K nm K -1 mm K mn . The novelty of the above objective function is that it contains a regularization trace term: -1 2σ 2 Tr( K). Thus, F V attempts to maximize the log likelihood Q nn and simultaneously minimize the trace Tr( K). Tr( K) represents the total variance of the conditional prior. When Tr( K) = 0, K nn = K mp K -1 mm K mn , which means that the inducing variables can exactly reproduce the full GP prediction.

Variational Approximation on

Multi-output GP ( Álvarez et al., 2010) derived a variational approximation to inference of multi-output regression using convolution processes. They introduce the concept of variational inducing kernels that allows them to efficiently sparsify multi-output GP models having white noise latent functions. White noise latent functions are needed while modelling partial differential equations. These variational inducing kernels are later used to parametrize the variational posterior q(X).

For our expression of the joint kernel, the autoand cross-covariance functions are dependent on the independent covariance function K 22 . Hence the independent output function in our case behaves like a latent function for the case of ( Álvarez et al., 2010). Moreover, the latent function or independent output is a physical process and hence will almost never be white noise. Henceforth we will place the inducing points on the input space. We now extend the variational inference method to deal with multiple outputs.

We try to approximate the joint-posterior distribution p(X|Y ) by introducing a variational distribution q(X). In the case of varying number of inputs for different outputs, we place the inducing points over the input space and extend the derivation of [START_REF] Titsias | Variational learning of inducing variables in sparse gaussian processes[END_REF] to multi-output case.

q(X) = N (X|µ, A) (18)
Here µ and A are parameters of the variational distribution. We follow the derivation provided in section 3.2 and obtain the lower bound of true marginal likelihood.

F V = log(N [Y |0, σ 2 I + Q XX ]) - 1 2σ 2 Tr( K) ( 19
)
where

Q XX = K XX M K -1 X M X M K X M X and K = K XX - K XX M K -1 X M X M K X M X .
K XX is the joint-covariance matrix derived using equation 6 or 7 using the input vector X defined in section 1. K X M X M is the joint covariance function on the inducing points X M , such that X M = [x M1 , x M2 , ..., x M2 ]. We assume that the inducing points x Mi will be same for all the outputs, hence

x M1 = x M2 = ... = x M2 = x M .
While K XX M is the crosscovariance matrix between X and X M .

Note that this bound consists of two parts. The first part is the log of a GP prior with the only difference that now the covariance matrix has a lower rank of MD. This form allows the inversion of the covariance matrix to take place in O N(MD) 2 time. The second part as discussed above can be seen as a penalization term that regularizes the estimation of the parameters.

The bound can be maximized with respect to all parameters of the covariance function; both model hyperparameters and variational parameters. The optimization parameters are the inducing inputs x M , the hyperparameters θ of the independent covariance matrix K 22 and the error while measuring the outputs Σ. There is a trade-off between quality of the estimate and amount of time taken for the estimation process. On the one hand the number of inducing points determine the value of optimized negative logmarginal likelihood and hence the quality of the estimate. While, on the other hand there is a computational load of O N(MD) 2 for inference. We increase the number of inducing points until the difference between two successive likelihoods is below a predefined quantity.

NUMERICAL RESULTS

In this section we provide numerical illustration to the theoretical derivations in the earlier sections. We start with a synthetic problem where we try to learn the model over derivative and quadratic relationships. We compare the error values for the variational approximation with respect to the full GP approach. Finally we look at the results in presence of a real world dataset related to flight loads estimation.

The basic toolbox used for this paper is GPML provided with [START_REF] Rasmussen | Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)[END_REF], we generate covariance functions to handle relationships as described in equations 6 and 7 using the "Symbolic Math Toolbox" in MATLAB 2014b. Since variational approximation was not coded in GPML we have wrapped the variational inference provided in gpStuff [START_REF] Vanhatalo | Gpstuff: Bayesian modeling with gaussian processes[END_REF] in the GPML toolbox. All experiments were performed on an Intel quad-core processor with 4Gb RAM.

Numerical Results on Theoretical Data

We begin by considering the relationship between two latent output functions as described in equation 2. Such that f 2 ∼ GP[0, K SE (0.1, 1)]

σ n2 ∼ N [0, 0.2] σ n1 ∼ N [0, 2] (20)
K SE (0.1, 1) means squared exponential kernel with length scale 0.1 and variance as 1.

Differential Relationship

We take the case of a differential relationship g(.), such that g( f , x) = ∂ f ∂x . Since the differential relationship g(.) is linear in nature we use the equation 6 to calculate the auto-and cross-covariance functions as shown in table 1.

Table 1: Auto-and cross-covariance functions for a differential relationship.

Initial Covariance K 22 σ 2 exp( -1 2 d 2 l 2 ) Cross-Covariance K 12 σ 2 d l 2 exp( -1 2 d 2 l 2 ) Auto-covariance K 11 σ 2 d 2 -l 2 l 4 exp( -1 2 d 2 l 2 )
To generate data a single function is drawn from f 2 as described in equation 20 which then is used to calculate y 1 and y 2 . 10,000 training points are generated for both the outputs. Values of y 2 for x ∈ [0, 0.3] are removed from the training points. Next we optimize the lower bound of log-marginal likelihood using variational inference, for independent GP's on y 1 and y 2 as described in 3.2. Later we optimize the same lower bound but with a joint-covariance approach as described in section 3.3 using y 1 , y 2 and g(.). As explained in section 3.3 we settled on using 100 inducing points for this experiment because there was negligible increase in lower bound F V of log-marginal likelihood upon increasing the inducing points.

In figure 1(a) we show the independent (blue shade) and joint fit (red shade) of two GP for the differential relationship. The GP model with joint covariance gives better prediction even in absence of data of y 2 for x ∈ [0, 0.3] because transfer of information is happening from observations of y 1 present at those locations. Nonetheless we see an increase in variance at masked 'x' values even for jointcovariance kernel. The distribution of inducing points may look even in the diagram but are uneven at places where we have removed the data from y 2 . For the second experiment we generate 1000 points and compare the Root Mean Squared Error (RMSE) and log-marginal likelihood between full relationship kernel as described in section 2.3 and variational inference relationship kernel as described in section 3.3, using 10, 20, 50 and 100 inducing points. 10 sets of experiments were run on 75% of the data as training set and 25% of the data as the test set, the training and test sets were chosen randomly. We learn the optimal values of hyper-parameters and inducing points for all the 10 sets of experiments of training data. Finally, RMSE values are evaluated with respect to the test set and negative log-marginal likelihood are evaluated for each learned model. The RMSE values are calculated for only the dependent output y 1 and then plotted in the figures below.

In figure 1(b) the mean and variance of negative log-marginal likelihood for 10 sets of experiments are calculated with varying number of inducing inputs at the top in red. As expected upon increasing the number of inducing points we see that the values of mean negative log-marginal likelihood tends to become an asymptote reaching the value of mean negative log-marginal likelihood of a full multi-output GP. In the bottom figure we have the mean and variance of RMSE for 10 sets of experiments. We see that the mean value of RMSE does not vary drastically upon increasing the inducing points but the value of variance gets reduced. One can observe an almost constant RMSE in figure 1(b) with a sudden decrease for a full GP (as M tends to N). Although this behaviour was expected, providing an explanation for the slow decrease with low values of M requires further investigation.

Quadratic Relationship

Now we take the case of a quadratic relationship g(.), such that g( f , x) = f 2 . Since the quadratic relationship g(.) is non-linear in nature we use the equation 7 to calculate the auto-and cross-covariance functions as shown in table 2. The Jacobin L for this case becomes 2 f2 (x). Here the value of f2 (x) is the mean value of the latent-independent output function f 2 calculated at the input point x.

Initial Covariance K 22 σ 2 exp( -1 2 d 2 l 2 ) Cross-Covariance K 12 2 f2 (x)K 22 Auto-covariance K 11 4( f2 (x) 2 K 22 + 2K 2 22 )
As stated in the earlier section 4.1.1 the output data was generated using the same draw of f 2 as explained in equation 20. 10,000 training points are generated for both the outputs. Values of y 2 for x ∈ [0, 0.3] are removed from the training points. Note that the above calculated auto-and cross-covariances are third order Taylor series approximations as presented in equation 7.

Calculating the Jacobin f2 (x) at the inducing and input points was a requirement of the algorithm. Moreover, during the optimization process the value of x m keeps on changing, since the f2 (x m ) depends on the value of x m . Many a times we don't have value of latent independent output function at these new points. To solve this problem we first learn an independent model of the output y 2 recover an estimate of f2 (x) and use this value to calculate the required auto-and cross-covariances.

In figure 2(a) we show the independent and joint fit of two GP for the quadratic relationship. Even in the presence of the error the joint-covariance GP model gives better prediction because of transfer of information. The GP model with joint covariance gives better prediction even in absence of data of y 2 for x ∈ [0, 0.3] because transfer of information is happening from the observations of y 1 present at those locations.

Upon performing the second experiment as described in section 4.1.1 we get the figure 2(b). The top part in the figure is the mean and variance of negative log-marginal likelihood for varying number of inducing points. We see that the value mean negative log-marginal likelihood reaches the value of full negative log-marginal likelihood. In the lower figure we have mean and variance of RMSE of the dependent output y 1 . As seen in the earlier experiment we observe high amount of variance for lower inducing points due to the approximate nature. In this section we conduct experiments, applying our approach on the flight loads data recovered during flight test campaigns at Airbus. We look at normalized data of a simple longitudinal maneuver. The two outputs in our case are shear load T z and bending moment M x as described in figure 3. The input space is 2-dimensional space with wing span η or point of action of forces as he first input variable and angle of attack α as the second input variable. The maneuver is quasi-static which means that airplane is in equilibrium at all time and there are no dynamic effects observed by the aircraft. The relation between T z and M x can be written as:

Numerical Results on Flight Test Data

M x (η, α) = η edge η T Z (x, α)(x -η)dx. (21)
Note that the above equation is calculated only on the η axis. Here, η edge denotes the edge of the wing span. The above equation is linear in nature and hence we will use equation 6 to calculate the auto-and cross-covariance functions. The forces are measured at 5 points on the wing span and at 8800 points in the second axis. We follow the procedure described in earlier experiments where we compare plots of relationship-imposed multioutput GP and independent GP. Secondly, we compare the measures of negative-log marginal likelihood and RMSE for varying number of inducing points.

In figure 4(a) we see the plot of independent and dependent T Z at the top with the dependent plot in red and independent plot in blue. In the bottom part of figure we show the plot for M X . Only one α is plotted here for better viewing. 100 inducing points in the input space are used to learn and plot the figure. We see that the region in red, has a tighter variance than the one in blue confirming the improvement by our method. The relationship in equation 21 is acting as extra information in tightening the error margins of the loads estimation. This becomes very useful when we need to identify faulty data because the relationship will impose a tighter bound on the variance and push faulty points out of the confidence interval.

In figure 4(b) we see how the negative logmarginal likelihood and RMSE plots improve upon increasing number of inducing points. As expected the likelihood and RMSE improves with more inducing points because of the improvement in the approximate inference. We settle on choosing 100 inducing points for figure 21 because there is not much improvement in RMSE and likelihood upon increasing the inducing points.

CONCLUSIONS AND FUTURE WORK

We have presented a sparse approximation for physics-based multiple output GP's, reducing the computational load and memory requirements for inference. We extend the variational inference as derived by [START_REF] Titsias | Variational learning of inducing variables in sparse gaussian processes[END_REF] and reduce the computational load for inference from O N 3 to O N(MD) 2 and load on memory from O N 2 to O (NMD).

We have demonstrated our strategy to work on both linear and non-linear relationships, in presence of large amount of data. The strategy of imposing relationships is one way of introducing domain knowledge into the GP regression process. We conclude that due to the support of added information provided by the physical relationship we have a tighter confidence interval for the joint-GP approach, eventually leading to better estimates. Additionally, the results in identification of measurements incoherent with the physics of the system and a more accurate estimation of the latent function, which is crusial in aircraft design and modeling. The approach can be extended to larger number of related outputs giving a richer GP model.

Aircraft flight domain can be divided into various cluster of maneuvers. Each cluster of maneuvers has a specific set of features and mathematically modelled behaviour, which can also be seen as domain knowledge. Recent advancements in approximate inference of GP such as "Distributed Gaussian Process" [START_REF] Deisenroth | Distributed gaussian processes[END_REF] distribute the input domain into several clusters. Such kind of approximate inference technique should be explored in the future for our kind of dataset. Future work deals with handling clustered input space with individual features.

  x D ]. If Σn i = N for i ∈ [1, D]. Hence N represents the total number of training points for all the outputs combined. Then Y ∈ R N and X ∈ R N×P .

  (a) Independent fit for two GP's in blue variance is represented by light blue region and mean is represented by solid blue line. Variance and mean of the dependent are represented in red region and solid red line. The dashed black line represents the true latent function values; noisy data is denoted by circles. Experiment was run on 10,000 points but only 100 data points are plotted to increase readibility.Here f 1 = ∂ f 2 ∂x , y 1 = f 1 + σ n1 and y 2 = f 2 + σ n2 . The + points refer to the location of the inducing points in the inference. (b) The figure shows progression of different measures upon increasing number of inducing points from 10, 20, 50 to 100 and finally full Multi-output physics based GP. The top figure in red shows the value of mean and variance of negative log-marginal likelihood, while the bottom figure in blue shows the mean and variance of root mean squared error. 10 sets of experiments were run on 75% of the data as training set and 25% of the data as the test set, the training and test sets were chosen randomly.

Figure 1 :

 1 Figure 1: Experimental results for differential relationship with approximate inference using variational approximation.

  (a) Independent fit for two GP in blue variance is represented by light blue region and mean is represented by solid blue line. Variance and mean of the dependent are represented in red region and solid red line. The dashed black line represents the true latent function values; noisy data is denoted by circles only 100 data points are plotted to increase readibility. Here f 1 = f 2 2 , y 1 = f 1 + σ n1 and y 2 = f 2 + σ n2 . The + points refer to the location of the inducing points in the inference. (b) The figure shows progression of different measures upon increasing number of inducing points from 10, 20, 50 to 100 and finally full Multi-output physics based GP. The top figure in red shows the value of mean and variance of negative log-marginal likelihood, while the bottom figure in blue shows the mean and variance of root mean squared error. 10 sets of experiments were run on 75% of the data as training set and 25% of the data as the test set, the training and test sets were chosen randomly.

Figure 2 :

 2 Figure 2: Experimental results for quadratic relationship with approximate inference using variational approximation.

Figure 3 :

 3 Figure 3: Wing Load Diagram.

  (a) Independent fit for two GP in blue variance is represented by light blue region and mean is represented by solid blue line. Variance and mean of the dependent are represented in red region and solid red line. The dashed black line represents the true latent function values; noisy data is denoted by circles only 1 α step is plotted. (b) The figure shows progression of different measures upon increasing number of inducing points from 10, 20, 50 to 100. The top figure in red shows the value of mean and variance of negative log-marginal likelihood, while the bottom figure in blue shows the mean and variance of root mean squared error.

Figure 4 :

 4 Figure 4: Experimental results for aircraft flight loads data with approximate inference using variational approximation.

Table 2 :

 2 Auto-and cross-covariance functions for a quadratic relationship.

Sparse Physics-based Gaussian Process for Multi-output using Variational Inference

Sparse Physics-based Gaussian Process for Multi-output Regression using Variational Inference


I Goy-Thollotab

Z Djelouadji

M Nennig

G Hazart

M Hugonnard

Screening for Leptospira DNA in blood and urine from 30 apparently healthy dogs Recherche d'ADN de leptospires pathogènes dans le sang et l'urine de 30 chiens apparemment sains☆

Keywords:

Asymptomatic carriage of leptospires in dogs has not been extensively studied. A prospective study was conducted on thirty apparently healthy male dogs to screen for Leptospiral DNA in their blood and urine. All dogs were up to date with a bivalent vaccine against Leptospirosis. Microscopic agglutination test (MAT) and real-time PCR analysis in blood and urine were performed for each dog. Six of 30 (20%) dogs had positive PCR results: two from blood samples (all MAT titres ≤ 160) and four from urine samples (one with maximal MAT titres of 2560 and three with all MAT titres ≤ 160).

Ten months later, PCR analyses of blood and urine were repeated for these six dogs.

Four had negative PCR results from blood and urine. The other two, with positive urine results at the first sampling, had negative PCR results from urine and positive PCR results from blood. In this study, the urinary shedding of leptospires appeared to be transient. As shown previously, serological status was not correlated with PCR results.

Low but detectable amounts of Leptospiral DNA in the blood of some apparently healthy dogs raises questions regarding the pathophysiology of Leptospirosis and the diagnostic consequences of high-sensitivity PCR assays.

Résumé

Le portage asymptomatique des leptospires chez le chien est encore mal connu. Une étude prospective sur 30 chiens mâles apparemment sains a recherché la présence d'ADN de leptospires pathogènes dans leur sang et leur urine. Chaque chien était à jour de ses vaccins contre la leptospirose (vaccins inactivés bivalents sur tous les chiens). Une sérologie par technique de microagglutination (MAT) et une PCR en temps réel sur l'urine et le sang ont été réalisées pour chaque chien. Six chiens sur 30 (20 %) avaient des résultats PCR positifs : deux chiens sur sang (titres MAT ≤ 160 pour ces deux chiens), quatre chiens sur urine (titre MAT maximal de 2560 pour un chien, titres MAT ≤ 160 pour les trois autres). Dix mois plus tard, les analyses PCR ont été répétées pour les six chiens avec un résultat positif. Quatre chiens avaient des résultats négatifs sur sang et urines. Deux chiens, qui avaient initialement une PCR positive sur urines, avaient une PCR positive sur sang et négative sur urines. Dans cette étude, l'excrétion urinaire des leptospires apparaissait discontinue. Comme dans d'autres études, les profils sérologiques n'étaient pas corrélés aux résultats PCR. Trouver dans le sang de chiens apparemment sains des quantités faibles mais détectables d'ADN de leptospires interroge sur la physiopathologie de la leptospirose et les conséquences diagnostiques de tests PCR très sensibles.

INTRODUCTION

Leptospirosis is a re-emerging worldwide zoonotic disease caused by spirochetes that can affect many domestic mammals, including dogs. Humans and dogs are both susceptible species and represent incidental hosts who can develop similar severe acute forms of the disease. Many wild rodents are maintenance hosts for leptospires.

They chronically shed the spirochetes from their renal tubules into their urine without expressing any symptoms [1]. The renal carriage of leptospires that is associated with high-intensity, constant and long-term Leptospiruria is characteristic of maintenance hosts [START_REF] Chernukcha | Zenkovitch Pathogenicity of Leptospires of various serological types for some species of wild rodents[END_REF]. Dogs are the maintenance host for the Canicola serovar and an incidental host for other serovars [1]. Canine Leptospirosisvaccines are reported to provide longterm protection not only against infection but also against the renal carrier state for the vaccine-targeted serogroups [START_REF] Klaasen | Duration of immunity in dogs vaccinated against leptospirosis with a bivalent inactivated vaccine[END_REF], [START_REF] Klaasen | Sutton A novel tetravalent Leptospira bacterin protects against infection and shedding following challenge in dogs[END_REF], [START_REF] Minke | Onset and duration of protective immunity against clinical disease and renal carriage in dogs provided by a bivalent inactivated leptospirosis vaccine[END_REF], [START_REF] Wilson | Duration of immunity of a multivalent (DHPPi/L4R) canine vaccine against four Leptospira serovars[END_REF]. Leptospiruria in incidental hosts is considered a transient, intermittent and low-intensity condition [START_REF] Chernukcha | Zenkovitch Pathogenicity of Leptospires of various serological types for some species of wild rodents[END_REF], [START_REF] Ganoza | Vinetz Asymptomatic renal colonization of humans in the peruvian amazon by Leptospira[END_REF]. The asymptomatic carriage of leptospires in canines has not been extensively studied.

However, many studies have demonstrated that apparently healthy dogs can have high microscopic agglutination test (MAT) titres for different serogroups, indicating that exposure to Leptospira spp. is common in dogs [START_REF] Houwers | Agglutinating antibodies against pathogenic Leptospira in healthy dogs and horses indicate common exposure and regular occurrence of subclinical infections[END_REF], [START_REF] Scanziani | Serological survey of Leptospiral infection in kennelled dogs in Italy[END_REF], [START_REF] Stokes | Prevalence of serum antibodies against six Leptospira serovars in healthy dogs[END_REF]. Some studies have investigated the prevalence of urinary shedding of leptospires based on PCR analyses in apparently healthy dogs or in a general referral hospital population. The percentage of positive samples has been reported to vary between 1.5 and 20% [START_REF] Gay | Goarant Though not reservoirs, dogs might transmit Leptospira in New Caledonia[END_REF], [START_REF] Harkin | Comparison of polymerase chain reaction assay, bacteriologic culture, and serologic testing in assessment of prevalence of urinary shedding of leptospires in dogs[END_REF], [START_REF] Llewellyn | Urinary shedding of leptospires and presence of Leptospira antibodies in healthy dogs from upper Bavaria Bern Munch Tierarztl[END_REF], [START_REF] Oliveira | Exposure to Leptospira spp. in sick dogs, shelter dogs and dogs from an endemic area: points to consider[END_REF], [START_REF] Rojas | Detection and quantification of leptospires in urine of dogs: a maintenance host for the zoonotic disease leptospirosis[END_REF]. In those studies, the renal carrier state was evaluated at a single point in time and the vaccination status was not specifically addressed in two of the studies [START_REF] Harkin | Comparison of polymerase chain reaction assay, bacteriologic culture, and serologic testing in assessment of prevalence of urinary shedding of leptospires in dogs[END_REF], [START_REF] Rojas | Detection and quantification of leptospires in urine of dogs: a maintenance host for the zoonotic disease leptospirosis[END_REF]. The aim of the current study was to screen for Leptospira DNA in urine and blood samples from 30 apparently healthy, privately owned dogs with a well-known vaccination history and behavioural habits that favoured contamination and to recheck those with a positive result (if any) on a second occasion, without administering a course of antibiotics in between tests.

MATERIALS AND METHODS

Selection of the population

Thirty apparently healthy student-owned dogs from a veterinary university were included. The ethical committee of the University approved this study. Recruitment of the population was made on a voluntary basis in June 2011. The study population was selected based on answers to a questionnaire. Male dogs with a lifestyle that favoured exposure to Leptospirosis were eligible for inclusion. The risk of exposure was assessed based on the frequency of walks near water sources and/or bathing in these water sources during the past two months.

Vaccination status

Vaccination histories (date and type of vaccine), which were obtained from valid vaccination certificates, were recorded for each dog.

Epidemiological data

The data recorded for each dog included living place; lifestyle (number of dogs in the household and the main source of activity: work dog, hunting dog, breeder or pet); walking, bathing and drinking habits (ditchwater, running water, lake, swamp, pond or tap water exclusively); and proximity to wild rodents, which was estimated by the owner.

Clinical status evaluation

A thorough medical history was recorded, and each dog underwent a complete physical examination, including a urinary dipstick (Combur 10 Test, Roche) test and specific gravity measurements using refractometer (URC-Ne, Atago). Only apparently healthy dogs were included in the study. Glucosuria was not an exclusion criterion.

Sample collection and DNA isolation

The owners collected urine via the free-catch method in a sterile plastic urine cup during the dog's first morning walk. Subsequently, one millilitre of urine was placed in a dry tube for PCR analysis. Two blood samples were collected at the University Hospital on the same day as the urine sampling: one in a clot tube for MAT serology and the other in a tube containing EDTA for PCR analysis. The urine samples were centrifuged at 3500 g for 10 min prior to DNA extraction to obtain 200 μL of pellet. DNA was isolated (QIAamp DNA Mini Kit, Qiagen, Courtaboeuf, France) from 200 μL of blood and the urine pellet within hours of sampling, following the manufacturer's instructions. The DNA template was stored at -20 °C.

PCR assay on blood and urine

A public laboratory 1 completed the real-time PCR analysis. Blood and urine were handled separately, and the assays were completed using a kit (Taqvet Pathogenic Leptospira Real-Time PCR kit (LEPT/50), LSI, Lissieu, France). The Taqvet kit included a specific internal control to check for PCR inhibitors. Each run included a no template control, a positive control using a reference laboratory strain (L. interrogans Icterohaemorrhagiae strain 9120) as a template and a negative control (PCR mix without the targeted DNA). The no template control was defined as a negative control that undergoes all the process including extraction of nucleic acids.

A final volume of 25 μL, which consisted of 20 μL lepto mix (LSI) and 5 μL of the targeted DNA, was analysed via PCR using a Rotor-Gene Q system (Qiagen) using the following program: 10 min of enzyme activation at 95 °C, followed by 45 cycles of 95 °C for 15 s and 60 °C for 1 min The detection threshold was 10 3 leptospires/mL of blood or urine. Each PCR was performed in duplicate to assess the reproducibility of the results. All samples have been tested in the same run.

Genotyping

The 16S rRNA gene was amplified with the primers LeptA (5′-GGCGGCGCGTCTTAAACATG-3′) and LeptB (5′-TTCCCCCCATTGAGCAAGATT-3′) [START_REF] Merien | Saint Girons Polymerase chain reaction for detection of Leptospira spp. in clinical samples[END_REF]. The amplified fragments were sequenced (see appendix), and the sequences were compared to 16S rRNA sequences available in Genbank 2 . Further characterization was performed using the multispacer sequence typing (MST) method with MST1, MST3 and MST9 primers as described by Zilber et al. [START_REF] Zilber | KodjoHigh-resolution typing of Leptospira interrogans strains by multispacer sequence typing[END_REF]. For one dog (dog N o 19), a variable-number tandem-repeat (VNTR) analysis was subsequently performed as an alternative molecular approach [START_REF] Salaün | Picardeau Application of multilocus variablenumber tandem-repeat analysis for molecular typing of the agent of leptospirosis[END_REF].

Serologic examination

Microscopic agglutination tests were performed by a public laboratory for 19 serovars (Australis, Bratislava, Munchen, Autumnalis, Bim, Canicola, Grippotyphosa, Vanderhoedoni, Icterohaemorrhagiae, Copenhageni, Panama, Mangus, Pomona, Mozdok, Pyrogenes, Sejroe, Saxkoebing, Hardjo, and Wolfii) representing nine serogroups (Australis, Autumnalis, Canicola, Grippotyphosa, Icterohaemorrhagiae, Panama, Pomona, Pyrogenes and Sejroe). The initial dilution used was 1/40.

Follow up

Owners were individually informed of the MAT and PCR results. For dogs that had positive PCR results on the initial test, informed consent of the pet owner was required to recheck the blood and urine ten months later using the same PCR method. The vaccination schedule was continued unchanged. All dogs with at least one positive PCR result either the first or second test were treated with doxycycline (5 mg/kg BID, 14 days) following the second sampling of blood and urine.

RESULTS

The study included 30 male dogs, 14 of which were neutered (47%). In total, two dogs (6%) were small breeds (< 10 kg), 20 (67%) were medium breeds (between 10 and 30 kg), and eight (27%) were large breeds (> 30 kg), including eight mixed-breed dogs, Serological and PCR data are presented in Table 1.

Six out of 30 (20%) dogs had a unique positive PCR result at the first sampling (two from blood, four from urine). At the recheck ten months later, four of those six dogs had negative PCR results for both their blood and urine samples, and two had a unique positive PCR result for their blood sample. These last two dogs had both positive PCR results for their urine at the first sampling. The detailed results are described below. At the first sampling, twenty-four dogs (80%) had negative PCR results for both their blood and urine samples. Among them, [START_REF] Rojas | Detection and quantification of leptospires in urine of dogs: a maintenance host for the zoonotic disease leptospirosis[END_REF] Ten months later, the PCR results for both blood and urine samples were negative for these two dogs.

DISCUSSION

In this study, 4/30 (13.3%) vaccinated, apparently healthy, privately owned, male dogs had positive PCR assay results from urine samples. This indicates that asymptomatic urinary shedding of leptospires in male dogs with behavioural habits that favour contamination is not an anecdotal phenomenon, even when the dogs are vaccinated.

Overall, 4/30 dogs (13.3%, 95% Confidence Interval 5.3-29.6%) had positive PCR results from urine samples. This frequency is among the highest of those previously reported in asymptomatic dogs: 1.5% [START_REF] Llewellyn | Urinary shedding of leptospires and presence of Leptospira antibodies in healthy dogs from upper Bavaria Bern Munch Tierarztl[END_REF], 4.4% [START_REF] Gay | Goarant Though not reservoirs, dogs might transmit Leptospira in New Caledonia[END_REF], 7% [START_REF] Rojas | Detection and quantification of leptospires in urine of dogs: a maintenance host for the zoonotic disease leptospirosis[END_REF], 8% 12] and 20% [START_REF] Oliveira | Exposure to Leptospira spp. in sick dogs, shelter dogs and dogs from an endemic area: points to consider[END_REF].

However, the small number of dogs in the current study and its design, which favoured the recruitment of urinary shedders, precludes further comparisons of the frequency found in this study with those of previous studies. Indeed, only male dogs were included because they have an increased risk of contamination [START_REF] Birnbaum | Simpson Naturally acquired leptospirosis in 36 dogs: serological and clinicopathological features[END_REF], [START_REF] Azocar-Aedo | Monti Meta-Analyses of factors associated with leptospirosis in domestic dogs[END_REF], [START_REF] Ward | Prevalence of and risk factors for leptospirosis among dogs in the United States and Canada: 677 cases (1970-1998[END_REF]. The selected dogs were mainly living in suburban areas and had access to one or more water sources during their walks. Living in an urban area, the proximity of outdoor water sources, swimming or drinking from outdoor water sources and indirect exposure to wild rodents have all been suggested to promote Leptospirosis contamination [START_REF] Azocar-Aedo | Monti Meta-Analyses of factors associated with leptospirosis in domestic dogs[END_REF], [START_REF] Ghneim | Use of a casecontrol study and geographic information systems to determine environmental and demographic risk factors for canine leptospirosis[END_REF].

In our study, all urine samples showing positive PCR results had a quantity of DNA that was notably lower than that reported in the pioneer study describing the MST method [START_REF] Zilber | KodjoHigh-resolution typing of Leptospira interrogans strains by multispacer sequence typing[END_REF]. A positive PCR result does not determine whether the pathogen is viable, and the low Leptospiral burden observed here in the urine of apparently healthy dogs is of questionable biological significance. Ideally, cultures would have been recommended to investigate this issue. However, this option was rejected because of the difficulties in obtaining positive culture results, even in acute cases with high Leptospiral burdens. The sequencing results showed that there were no false positive results. The diagnostic kits used appeared to be very sensitive, which is a technical advantage. According to the manufacturer's notice for the Taqvet PCR kit, the sample is positive if Ct values are < 45. However, a positive test with such a low Leptospiral burden possibly calls into question the biological significance of the results. Limits of detection (LODs) as low as three genome equivalents per 5 μL (6 × 10 2 Leptospires/mL) have been reported for specific real-time PCR assays for detecting Leptospira that have been validated for diagnostic application [START_REF] Villumsen | Krogfelt Novel TaqMan ® for detection of Leptospira species in urine and blood: pit-falls in silico validation[END_REF]. Because of this, the positive PCR results observed in this study reflect those obtained under clinical conditions and should be considered in the same regard.

All four dogs with initial positive PCR results for their urine samples had negative PCR results from urine ten months later, without antibiotic treatment. Possible explanations for this include intermittent shedding, natural cleansing or methodological limits due to the detection threshold of the test and the low infectious burden. Surprisingly, two of these dogs had positive results for their blood samples at the recheck.

At the time of the current study, only bivalent inactivated vaccines were available in genotyping at the serovar level would have been of particular interest here and is absolutely necessary to extend the assessment of vaccination status and renal carrier state. However, the amount of DNA was insufficient for molecular characterization at the serovar level.

In a previous report [START_REF] Harkin | Comparison of polymerase chain reaction assay, bacteriologic culture, and serologic testing in assessment of prevalence of urinary shedding of leptospires in dogs[END_REF], serologic testing was shown to be a poor predictor of urinary shedding. Three of the four dogs with positive PCR results from urine samples had MAT titres that were all ≤ 160. One of them was glucosuric, suggesting tubular lesions, possibly caused by leptospires. The dog with a positive PCR result for the urine sample and multiple MAT titres ≥ 640, including non-vaccinal titres, was the sole dog with isosthenuric urine, but he was not azotemic. At the population scale, a low prevalence of renal carriage (≤ 1%) and a low rate of excretion are expected in incidental hosts, whereas a high prevalence (≥ 10%) and a high sustained excretion rate are expected in maintenance hosts [START_REF] Chernukcha | Zenkovitch Pathogenicity of Leptospires of various serological types for some species of wild rodents[END_REF], [START_REF] Desvars | Epidémiologie d'une zoonose, la leptospirose, dans deux îles de l'Océan Indien, la Réunion et Mayotte. Étude comparée du rôle de différentes espèces sauvages et domestiques Université de la Réunion[END_REF]. Whether dogs can be considered as a maintenance host for serovars other than Canicola has not been determined [START_REF] Gay | Goarant Though not reservoirs, dogs might transmit Leptospira in New Caledonia[END_REF], [START_REF] Goldstein | Canine leptospirosis[END_REF]. A recent publication suggests that, except for Canicola, dogs should be considered as vectors rather than as maintenance hosts [START_REF] Gay | Goarant Though not reservoirs, dogs might transmit Leptospira in New Caledonia[END_REF]. Dogs represent an epidemiological link between the environment, wild fauna and humans, and therefore, they might play a significant role in human contamination in endemic areas [START_REF] Ganoza | Vinetz Asymptomatic renal colonization of humans in the peruvian amazon by Leptospira[END_REF].

The positive PCR results from blood in four dogs (two at the first sampling and two others at the recheck) were unexpected in France, which is a non-endemic region. The leptospiremic phase has been shown to be short in models susceptible to acute clinical infections and in a rat model of persistent renal colonization [START_REF] Alves | De Brito Leptospiral antigens (L. interrogans serogroup ictero-haemorrhagiae) in the kidney of experimentally infected guinea pigs and their relation to the pathogenesis of the renal injury[END_REF], [START_REF] Athanazio | Rattus norvegicus as a model for persistent renal colonization by pathogenic Leptospira interrogans[END_REF], [START_REF] Branger | Polymerase chain reaction assay specific for pathogenic Leptospira based on the gene hap1 encoding the hemolysisassociated protein-1[END_REF]. As a consequence, a positive PCR result from blood should correspond to a recent infection.

Positive PCR results from blood have been previously reported in Brazil in apparently healthy dogs from a control centre for zoonoses (CCZ) and in dogs living in a neighbourhood with a high prevalence of human leptospirosis with respective frequencies of 7.7 and 14.8% [START_REF] Oliveira | Exposure to Leptospira spp. in sick dogs, shelter dogs and dogs from an endemic area: points to consider[END_REF]. The prevalence of Leptospiraemia was higher than that of Leptospiruria in the dogs living in the endemic area for human leptospirosis.

Explanations proposed by the authors included a recent outbreak in this area. In our study, which was conducted in a non-endemic area, it seems unlikely that the four positive results obtained from blood were due to recent asymptomatic infections. Nonspecific amplification was also excluded because all positive results obtained with the PCR diagnostic kit were confirmed via sequencing, and the negative controls were valid. Furthermore, for one dog, the L. interrogans Grippotyphosa Valbuzzi genotype was identified using the VNTR method. Vaccination also did not affect the PCR results because a recent vaccination does not interfere with the use of real-time PCR in the diagnosis of canine leptospirosis [START_REF] Midence | Effects of recent Leptospira vaccination on whole blood real-time PCR testing in healthy client-owned dogs[END_REF]. The leptospiral burden was low for the four dogs with a positive result for their blood samples, which renders the biological significance of these findings questionable. The amplification of residual DNA from a past infection seems very unlikely as DNA is degraded very quickly. The physiopathological process involved might also be more complex than is commonly thought: recurrent phases of Leptospiraemia in asymptomatic chronic carriers could be a possibility. The fact that two dogs with positive PCR results for their urine at the first sampling had positive PCR results from their blood at the recheck could suggest a complex phenomenon of recurrence. However, evidence for this hypothesis has never been documented, and it requires further investigation.

The limitations of the present study were the small population size, the absence of supporting culture results, the lack of PCR monitoring for the 24 dogs with negative PCR results from blood and urine at first sampling and the failure to genotype the Leptospira species identified at the serovar level. It would also have been interesting

to test the blood PCR-positive dogs for antibodies a few weeks after the first sample to explore the assumption of a recent infection.

Despite these limitations, this study supports the fact that asymptomatic urinary shedding of leptospires in apparently healthy vaccinated dogs does occur in nonendemic areas and that it may be transient. Whether dogs can be a reservoir for serovars other than Canicola remains controversial. A larger prospective study with close and repeated monitoring of dogs via PCR would be advisable, with a specific effort to determine the serovars involved and to screen for possible associated renal damage. Serological status was found to be a poor predictor of urinary shedding, as has been shown previously [START_REF] Harkin | Comparison of polymerase chain reaction assay, bacteriologic culture, and serologic testing in assessment of prevalence of urinary shedding of leptospires in dogs[END_REF], [START_REF] Llewellyn | Urinary shedding of leptospires and presence of Leptospira antibodies in healthy dogs from upper Bavaria Bern Munch Tierarztl[END_REF]. The significance of positive PCR results from blood samples for Leptospiral DNA in dogs from a non-endemic area warrants further investigation. The very low but detectable amounts of Leptospiral DNA in the blood and urine of some apparently healthy dogs raises questions regarding the diagnostic consequences of the extremely high sensitivity of PCR assays. It is also potentially important from a public health point of view. Given the potential veterinary and public health implications of these observations, repeated monitoring in a cohort study would be advisable.

ETHICAL APPROVAL

All applicable international, national, and/or institutional guidelines for the care and use of animals were followed. All procedures performed in studies involving animals were in accordance with the ethical standards of the institution or practice at which the studies were conducted.

DISCLOSURE OF INTEREST

The authors declare that they have no competing interest.

France.

  Three of the four dogs with positive PCR results from urine samples were vaccinated with the same vaccine (EURICAN LR, Merial) and one with a different one (ENDURACELL LR, Zoetis). The relative efficiency of vaccines in preventing urinary shedding could not be compared based on the results presented here because of the small size of the study population and because the majority of the study group (25/30) was vaccinated with the same vaccine (EURICAN LR, Merial). The genomospecies identified in all four cases was L. interrogans. L. interrogans contains both pathogenic vaccinal and non-vaccinal serovars associated with canine Leptospirosis. Further

Table 1

 1 to lakes and/or running water (rivers, streams and springs) outside the Veterinary Campus area. During the 15 days preceding the first sampling, 20 (67%) dogs went for a swim, and all dogs drank from a natural water source.Less than one year before the first sampling, all dogs had received a booster vaccine

	or completed a primo-vaccination against Leptospirosis with a two-serovar inactivated
	vaccine, including the Canicola and Icterohaemorrhagiae serovars (25 EURICAN LR
	(Merial), two CANIGEN L (Virbac), two ENDURACELL LR (Zoetis), one NOBIVAC L
	(MSD)) (

four Border Collies, four Australian Shepherds, two Samoyeds, two Labradors, two Golden Retrievers, one flat-coated Retriever, one Entlebuch Mountain Cattle dog, one Collie, one Brittany Spaniel, one Long-Haired Pyrenean Sheepdog, one Beagle, one Bernese Mountain dog and one Manchester Terrier. The age of the dogs varied from eight months to 14.5 years, and the median age was 26.5 months (8-177 months). All dogs primarily lived in suburban areas and had access to one or more water sources during walks. They also had access to the Veterinary Campus and its surrounding areas, which contain many sources of stagnant water (especially ponds, puddles and ruts) and is adjacent to areas where dairy cows and wild rodents lived. Certain dogs also had access

Table 1 .

 1 ). Summary table of vaccinal, serological and PCR data.No dog had history of renal and/or hepatic diseases. Clinical examinations were normal for all dogs. Urine specific gravity varied between 1.012 and 1.050. In total, one dog (dog N o 23) had glucosuria without clinical symptoms. Glycaemia was normal (4.5 mmol/L, 0.8 g/L) for this dog (Accu-Chek Active, Roche). In total, 19 dogs (63%)

	Abbreviations of serovars: 19: Icterohaemorrhagiae; COP: Copenhageni (serogroup Icterohaemorrhagiae); MUN: Munchen; AUS: Australis, BRAT: Bratislava
	(serogroup Australis); AKI: Autumnalis; BIM: Bim (serogroup Autumnalis); CAN: Canicola (serogroup Canicola); GRIP: Grippotyphosa; VAN: Vanderhoedoni
	(serogroup Grippotyphosa); PAN: Panama; MAN: Mangus (serogroup Panama); POM: Pomona; MOZ: Mozdok (serogroup Pomona); PYR: Pyrogenes
	(serogroup Pyrogenes); SJ: Sejroe; SAX: Saxkoebing; HJ: Hardjo; WOLF: Wolfii (serogroup Sejro).

  Ct = 34.55 for dog N o 8 and Ct = 37.78 for dog N o 18) and negative PCR results for their urine samples. For those two dogs, all MAT titres were ≤ 160. The 16S rRNAgene amplification results indicate that the genomospecies was L. borgpetersenii for dog N o 8 and L. interrogans for dog N 18. Further molecular characterization via MST failed because an insufficient amount of DNA was available.

	from blood samples (Ct = 30.20 for dog N o 19, Ct = 32.64 for dog N o 23). The 16S
	rRNA gene	amplification	results	indicate	that	the	genomospecies
	was L. interrogans. Further molecular characterization via MST for these two dogs
	failed. For dog N o 19, who had the most substantial Leptospiral DNA burden, another
	molecular typing method (variable-number tandem-repeat analysis) was completed,
	and the L. interrogansGrippotyphosa Valbuzzi genotype (results for VNTR locus 4, 7,
	10: 2, 1, 1, respectively) was identified [18].
	At the first sampling, two dogs (6.7%) (dogs N o 8 and 18) had positive PCR results for
	their blood samples (
	At the first sampling, four dogs (13.3%) (dogs N o 2, 13, 19 and 23) had positive PCR
	results for their urine samples (cycle threshold (Ct) = 35.62, 37.07, 38.11 and 40.0,
	respectively) and negative PCR results for their blood samples. One dog had
	glucosuria without hyperglycaemia (dog N o 23). One out of four dogs (dog N o 2) had
	multiple MAT titres ≥ 640 (1/2560 for Panama, Australis and Canicola; 1/1280 for
	Icterohaemorrhagiae). The other three dogs, including the dog with glycosuria (dog
	N o 23), had MAT titres that were all ≤ 160. The 16S rRNA gene amplification results
	indicate that all the sequences were similar and that the genomospecies found in these
	four dogs was L. interrogans. Further molecular characterization via MST failed

(62.5%) had MAT titres that were all ≤ 160, and five (20.8%) had MAT titres ≥ 640 for the vaccinal serogroups only (1/640 for Canicola for one dog, 1/1280 for Canicola for one dog, 1/2560 for Canicola for two dogs and 1/640 for Canicola and Icterohaemorrhagiae for one dog). Two of these five dogs (dog N o 1 with a titre of 1/2560 for Canicola and dog N o 5 with a titre of 1/640 for Canicola and Icterohaemorrhagiae) had received their last booster less than three months prior to the study. Four dogs (16.6%) had multiple titres ≥ 640 for at least one non-vaccinal serogroup (1/1280 for Panama and Grippotyphosa and 1/640 for Australis for one dog, 1/1280 for Australis and Autumnalis for one dog, 1/1280 for Australis for one dog and 1/640 for Grippotyphosa for one dog). because an insufficient amount of DNA was available. Ten months later, PCR results for both blood and urine samples were negative for two dogs (dogs N o 2 and 13). For the other two (dogs N o 19 and 23), PCR results were negative from urine and positive

ACKNOWLEDGEMENTS

The authors are grateful to Pr. A. Kodjo (public laboratory) for the free analyses and for the technical support.