A Määttänen

F Montmessin

B Gondet

H Hoffmann

F Scholten

E Hauber

F Gonzalez-Galindo

F Forget

J.-P Bibring

J.-L Bertaux

G Neukum

HIGH-ALTITUDE CO2 CLOUDS ON MARS: OMEGA AND HRSC OBSERVATIONS

Introduction:

.

Instruments:

. The HRSC is a camera imaging the Martian surface in stereo and color. It images the surface through several channels that are looking slightly offnadir. When atmospheric features, like clouds, are imaged through two (or more) filters, the apparent parallax of the features enables very accurate determination of the cloud altitude. The possible across-track movement of the cloud in different images, taken at slightly different times, gives an estimate of the local winds.

Results: Mars Express has acquired already more than 2.5 years of data with varying spatial and temporal resolution. The OMEGA dataset has revealed about 60 occurrences of these high-altitude CO2 clouds. These observations show that the clouds are mainly confined to the equatorial region (20°S -

20°N) and to a quite specific longitudinal band (-120°E -30°E, between Tharsis Montes and Terra Meridiani). Only a handful of observations lie outside these limits, most notably two clouds at midlatitudes (one at 50°S, other at 45°N). The seasonal variations are notable, since the clouds appear mainly right after the spring equinox, disappear between Ls=60°-90°, and reappear after the summer solstice. The activity seizes rapidly after Ls=150°, but one cloud has been observed at Ls=250°. When observed from high altitudes (2500 to 4000 km), the clouds show convective morphologies (roundish shapes) in about 10 cases (15%). From lower altitudes (< 500 km), the width of the OMEGA image limits the interpretation: the narrow image reveals only a part of the cloud and any extrapolation would be very difficult. However, simultaneous HRSC observations may shed light on the morphologies. The HRSC has imaged clouds in several occasions and clearly two kinds of clouds can be separated from the HRSC dataset: the low-(< 40 km) and high-altitude clouds. The low altitude clouds are thoughts to be primarily H2O rich, while the high altitudes ones are made of CO2, as confirmed by a simultaneous observation by OMEGA. These clouds are observed close to the equator, show similar seasonal behavior, and the hygropause level should be at significantly lower altitudes thus limiting the amount of water vapor in the upper atmosphere.

The HRSC cloud altitudes vary from 53 km to 84 km (±2 km) and they exhibit mainly cirrus-like, filamented shapes instead of convective, round shapes (see Fig. 2). The East-West cloud speed (related to the local wind speeds) range is 15-105 m/ s but mainly in the range 60-90 m/s (±13 m/s). One cloud, observed in southern midlatitudes, is actually moving from West to East with varying wind speeds of 5-40 m/s. Implications: Formation of CO2 ice clouds is known to affect the CO2 cycle as a whole [3,4]. Although the cloud species reported here does not belong to the class of low-altitude clouds encountered in the polar night, it shares similar formation processes with the latter, and thus offers a unique opportunity to decipher the microphysical and dynamical processes at work. This in turn may help us derive a unified and robust theory for the formation of CO2 clouds in the Martian atmosphere.

Figure 1 :

 1 Figure 1: OMEGA observation of a convectivetype cloud: the surface of Mars in grayscale and the analyzed cloud signal intensity in color.Two shadow observations have been used to analyze the cloud optical thickness and particle size, as already mentioned by Montmessin et al.[7]. These authors also calculated the cloud altitude to be around 80 km. In our study we have also mapped the variations of these parameters inside the cloud by analyzing them pixel-by-pixel. These observations have confirmed the previous results of Montmessin et al.[7] as well as produced size distributions of the

Figure 2 :

 2 Figure 2: An example of a high-altitude cloud observed by HRSC (orbit 0567). The cloud observations have been compared with the LMD-MGCM climatology. The MGCM shows strong diurnal temperature variations caused by the strong thermal tides in the mesosphere of Mars at the cloud altitudes. At the altitudes of our interest (60-80 km), the coldest temperatures are attained in the afternoon (16 local time) and the warmest temperatures in the morning hours (04 LT). Most of the clouds are observed in the afternoon, but in a few

Additional Information: AM wishes to thank the financial support from Centre National d'Etudes Spatiales.

7023.pdf Third International Workshop on Mars Polar Energy Balance and the CO2 Cycle (2009)


Antoine Muller
email: antoine.muller@irisa.fr

Charles Pontonnier

Georges Dumont

Uncertainty propagation in multibody human model dynamics

Keywords: Inverse dynamics, Kinematics, Dynamic residuals, Optimization, BSIP, Monte Carlo method

In biomechanics, calibration of body segment inertial parameters (BSIP) is crucial to take into account subject morphological specificities. To avoid strenuous protocols, identification methods based on rigid body dynamics laws have been proposed. Thanks to a motion capture system and force platforms, these methods optimize BSIP by minimizing errors in the equations of motion. These errors can be defined as the dynamic residuals reflecting inaccuracies arising from estimated BSIP, as well as from kinematics and force plate measurements. The current study aims at evaluating the part of uncertainty on the dynamic residuals directly related to kinematics and force plate measurements. To answer this question, we captured the movements of 10 participants performing a standardized motion. We then applied a Monte Carlo-based approach to introduce variations in the kinematics and force plate measurements, and evaluated the reconstructed difference on the dynamic residuals. Results show that, first, the BSIP evaluation using a regression method seemed to be an acceptable estimate for the studied subjects. Second, the part of uncertainty in the dynamic residuals was significantly higher than the dynamic residuals obtained. In conclusion, a subject-specific calibration of the BSIP based on dynamic residuals, for this model and protocol, seems irrelevant and prone to overfitting of BSIP.

Introduction

Computation of body segment inertial parameters (BSIP) has been shown to be crucial in biomechanical studies [START_REF] Andrews | Methods for investigating the sensitivity of joint resultants to body segment parameter variations[END_REF][START_REF] Rao | Influence of body segments' parameters estimation models on inverse dynamics solutions during gait[END_REF]. Specifically, the personalization of mass, position of center of mass (CoM) and inertia of each body part makes inverse dynamics more accurate and realistic. Several methods have therefore been suggested to estimate subject-specific BSIP.

First, regression methods based on anthropometric databases collected from cadaver studies [START_REF] Chandler | Investigation of inertial properties of the human body[END_REF][START_REF] Dempster | Space requirements of the seated operator: geometrical, kinematic, and mechanical aspects of the body, with special reference to the limbs[END_REF] or from in vivo body scanning [START_REF] Dumas | Adjustments to McConville et al. and Young et al. body segment inertial parameters[END_REF][START_REF] Leva | Adjustments to Zatsiorsky-Seluyanov's segment inertia parameters[END_REF][START_REF] Zatsiorsky | The mass and inertia characteristics of the main segments of the human body[END_REF][START_REF] Zatsiorsky | In vivo body segment inertial parameters determination using a gamme-scanner method[END_REF] have been proposed to estimate BSIP. These methods aim at scaling BSIP based on subject's height and weight. The use of regression rules is simple and useful to get a generic or average representation of human inertial characteristics. However, such methods, based on averages from a limited sample of subjects, do not take into account subject morphological specificities, e.g. obesity, amputated limb...

A second approach consists in using in vivo geometrical measurements [START_REF] Jensen | Estimation of the biomechanical properties of three body types using a photogrammetric method[END_REF][START_REF] Pearsall | Inertial properties of the human trunk of males determined from magnetic resonance imaging[END_REF][START_REF] Davidson | Estimating subject-specific body segment parameters using a 3-dimensional modeller program[END_REF]. These methods use 3D scanning or magnetic resonance imaging to reconstruct segmental geometries. BSIP are then estimated by assigning density values to the reconstructed volumes. These methods are accurate but expensive and can be invasive (radiations).

More recently, a third family of methods has been proposed to estimate BSIP in vivo. It consists in using motion capture and external force measurements to reproduce the motion dynamics and find the best BSIP that fit with these equations [START_REF] Vaughan | Selection of body segment parameters by optimization methods[END_REF]. This problem was solved in two ways. [START_REF] Venture | Identification of human mass properties from motion[END_REF] and [START_REF] Hansen | An individual and dynamic Body Segment Inertial Parameter validation method using ground reaction forces[END_REF] suggested to write the inverse dynamics problem under a linear form with respect to the inertial parameters. Then, a least squares method solved the optimization problem. Meanwhile, [START_REF] Reinbolt | Are patientspecific joint and inertial parameters necessary for accurate inverse dynamics analyses of gait[END_REF] and [START_REF] Zhao | Estimating human body segment parameters using motion capture data[END_REF] focused on the 6 degrees of freedom (DoF) joint between the floating-base system and the global reference frame as a measure of the simulation accuracy. The optimization problem consisted in minimizing the generalized forces at this virtual joint, that corresponds to the dynamic residuals.

In their work, [START_REF] Ojeda | The effect of kinematic constraints in the inverse dynamics problem in biomechanics[END_REF] stated that "the magnitude of the vector of residuals gives an idea of the accuracy of the simulation, including the kinematic data, the mechanical model and the ground reaction forces' measurements". Thus, errors associated to kinematic data and force platform measurements directly influence optimization results. This may cause overfitting, and distort the BSIP estimates. Indeed, [START_REF] Babyak | What you see may not be what you get: a brief, nontechnical introduction to overfitting in regression-type models[END_REF] defined overfitting as "asking too much from the available data. Given a certain number of observations in a data set, there is an upper limit to the complexity of the model that can be derived with any acceptable degree of uncertainty". Considering this observation is fundamental and requires a rigorous analysis of the influence of these errors on the dynamic residuals.

Some studies have been conducted to analyze the uncertainties introduced in the inverse dynamics calculus. On one hand, during the walking motion, [START_REF] Reinbolt | Are patientspecific joint and inertial parameters necessary for accurate inverse dynamics analyses of gait[END_REF][START_REF] Pàmies-Vilà | Analysis of different uncertainties in the inverse dynamic analysis of human gait[END_REF][START_REF] Riemer | Uncertainties in inverse dynamics solutions: A comprehensive analysis and an application to gait[END_REF] analyzed the variation of the resulting joint torques introduced by the uncertainties of different dynamic inputs. On the other hand, [START_REF] Ojeda | The effect of kinematic constraints in the inverse dynamics problem in biomechanics[END_REF] assessed different inverse kinematics methods -particularly the effect of the kinematic data consistency [START_REF] Silva | Kinematic Data Consistency in the Inverse Dynamic Analysis of Biomechanical Systems[END_REF] -on inverse dynamics consistency. To achieve this, they evaluated inverse dynamics consistency using dynamic residuals.

The current study aims at analyzing the influence of both kinematics and force plate measurements on the dynamic residuals of the equations of motion for a multibody human model. We adopte a Monte Carlo-based approach of error propagation from kinematics and force plate measurements to dynamic residuals, presented in the next section. The study was conducted on 10 subjects performing a standardized whole body motion. Results are presented and discussed in Section 3, and a focus is made on the propensity of such residuals to be used as a criterion to optimize the BSIP of such a model.

Materials and methods

For most biomechanical studies, regression methods are used to estimate BSIP. In identification methods based on optimization, regression approaches are also considered as the initial estimate of the optimization problem. Evaluating the accuracy of regression-based methods in estimating BSIP is therefore important in order to evaluate their validity when used to compute inverse dynamics. Moreover, some studies identify the subject-specific BSIP thanks to the minimization of dynamic residuals. As previously discussed, it then seems important to evaluate if there is a risk of overfitting during the optimization. Therefore the present study proposes to evaluate the following hypotheses:

H1: regression methods compute acceptable body segment inertial parameters (BSIP) estimate for the inverse dynamics problem;

H2: dynamic residuals can be used to achieve a subject-specific BSIP calibration without overfitting.

To answer these questions, movements were performed by a group of subjects. The experimental data were used to drive a multibody human model scaled using a classical regression rule and dynamic residuals were computed and analyzed to investigate hypothesis H1. Then, a Monte Carlo-based method was applied to both kinematics and force plate measurements to determine the uncertainty due to these errors in the dynamic residuals. This approach allowed us to investigate hypothesis H2.

General notations

In this paper, re, pe and dr refer respectively to the reconstruction error, the platform measurement error and the dynamic residuals. Considering x and y as the general notation of these abbreviations, x refers to the accuracy of x. For inputs of Monte Carlo methods, ∆ x indicates the coefficient of dispersion on x-perturbed data. For outputs, ∆ x y indicates the coefficient of dispersion on y introduced by the x-perturbed data.

Reference data

Reference data -data of the analysis pipeline without perturbation -are evaluated using an inverse dynamics framework (Fig. 1) based on the methods described in [START_REF] Winter | Biomechanics and motor control of human movement[END_REF], [START_REF] Damsgaard | Analysis of musculoskeletal systems in the AnyBody Modeling System[END_REF] or [START_REF] Delp | OpenSim: open-source software to create and analyze dynamic simulations of movement[END_REF]. In this study, all reference variables are marked with * . Experimental data used as input of this step are the marker positions X * exp and ground reaction forces of the two platforms F * p (1) (see the experimental protocol in Section 2.3).

F * p = F * p1 F * p2 (1)
The reference data are then computed by the four steps presented below (Fig. 1). First, an inverse kinematics step computes the joint coordinates q * . Joint torques are then computed in the inverse dynamics step by using the experimental ground reaction forces F * p . For the inverse kinematics and inverse dynamics steps, the accuracy of the results are quantified by assessing two indicators: * re and * dr .

Inverse

Inverse kinematics

The inverse kinematics problem consists in finding the vector of generalized joint coordinates that best reconstructs the movement from the experimental marker positions X exp . For each time step i, the sum of squared distances between measured and model-determined marker positions is minimized (2) by using the global optimization method proposed by [START_REF] Lu | Bone position estimation from skin marker co-ordinates using global optimisation with joint constraints[END_REF]. min

q(i) f (q(i)) = nm m=1 X mod,m (q(i)) -X exp,m (i) 2 (2)
q(i) is the vector of generalized coordinates at time step i, X mod,m (q(i)) the coordinates of marker m obtained by forward kinematics, X exp,m (i) the experimental coordinates of marker m. The total number of markers is n m . The joint coordinate trajectories are composed of the set of joint coordinates obtained for each time step. Then, these trajectories are filtered using a 4-th order Butterworth low pass filter with a cut-off frequency of 5 Hz and no phase shift.

Reconstruction error computation Accuracy of the inverse kinematics step is assessed by computing the reconstruction error re . This indicator corresponds to the average distance between real and model markers over the whole recording period, defined as the global reconstruction error in [START_REF] Laitenberger | Refinement of the upper limb joint kinematics and dynamics using a subject-specific closedloop forearm model[END_REF]. Finally, the average error for all markers is computed by Equation [START_REF] Chandler | Investigation of inertial properties of the human body[END_REF].

re = 1 n m n f nm m=1 n f i=1 X mod,m (q(i)) -X exp,m (i) ( 3
)
n f is the total number of time steps i.

Inverse dynamics

The equation of motion of the system can be written in the general form (4).

M (q).q + C(q, q) + G(q) + λ + E = 0

M (q) is the inertia matrix, C(q, q) is the centrifugal and Coriolis force vector, G(q) is the gravity force vector, λ is the generalized internal force vector and E is the generalized external force vector. Ground reaction forces F * p are considered as external forces, included in E. A recursive Newton-Euler algorithm [START_REF] Featherstone | Rigid Body Dynamics Algorithms[END_REF] is used to solve (4), yielding the vector of generalized internal forces λ.

Dynamic residuals computation Equation ( 4) can be developed as a floatingbase dynamics equation [START_REF] Dumas | Adjustments to McConville et al. and Young et al. body segment inertial parameters[END_REF]. The upper part of the equation describes the 6 DoF motion of the base (e.g. position and orientation of the pelvis). The lower part describes the motion of all the bodies in the model.

M 11 (q) M 1 * (q) M * 1 (q) M * * (q) . q1 q * + C 1 (q, q) C * (q, q) + G 1 (q) G * (q) + λ 1 λ * + E 1 E * = 0 (5)
The subscripts 1 and * refer, as proposed in [START_REF] Featherstone | Rigid Body Dynamics Algorithms[END_REF], respectively to the first joint (6 DoF virtual joint) and to all the remaining joints. So λ 1 corresponds to the vector of generalized forces at the 6 DoF joint. It contains 6 components: 3 forces and 3 torques along the 3-axis of the global reference frame [START_REF] Leva | Adjustments to Zatsiorsky-Seluyanov's segment inertia parameters[END_REF].

λ 1 = [F x F y F z M x M y M z ] T (6)
Superscript T means transpose. When performing a motion, F x is the anteroposterior force residual, F y is the lateral-medial force residual, F z is the inferiorsuperior force residual, M x is the abduction moment residual, M y is the flexion moment residual and M z is the internal rotation moment residual.

If the inverse dynamics problem is perfectly solved, all the components of λ 1 would be null. This property is then used to define the following indicator of the accuracy of the inverse dynamics step: the dynamic residuals indicator dr , defined as the root-mean-square (RMS) of λ 1 throughout the motion [START_REF] Zatsiorsky | The mass and inertia characteristics of the main segments of the human body[END_REF]. To allow for inter-subject comparisons, the three effort components of dr are normalized by the body weight of the subject (m s .g) and the three torque components are normalized by the body weight times the height (m s .g.h s ) [START_REF] Reinbolt | Are patientspecific joint and inertial parameters necessary for accurate inverse dynamics analyses of gait[END_REF].

dr = 1 m s .g 1 n f   n f i=1 (F x (i)) 2 n f i=1 (F y (i)) 2 n f i=1 (F z (i)) 2 1 h s n f i=1 (M x (i)) 2 1 h s n f i=1 (M y (i)) 2 1 h s n f i=1 (M z (i)) 2   T (7)

Perturbed data

To identify the uncertainty introduced by the kinematic data and the force plate measurements on the dynamic residuals, we perform a Monte Carlo-based approach, by running Monte Carlo samplings separately on both sources of uncertainty. In Monte Carlo methods, the reference data are perturbed to obtain a set of perturbed samples. For each sample, a known perturbation is applied by adding a noise following a probability distribution. The perturbed inputs are processed through the analysis pipeline. Then, we analyze the distribution of the output error. The following two sections feature the differences in the methods applied for each source of uncertainty.

Reconstruction error First, we aim at determining the influence of the uncertainty introduced by the kinematic data on the dynamic residuals. To apply a Monte Carlo method, a set of perturbed motions in which the reconstruction error follows a known distribution is required. Since reconstruction error is a result of the inverse kinematics step, it can not be directly perturbed. Therefore, we apply perturbations to the reference joint coordinates to generate a database of perturbed motions corresponding to a specified reconstruction error. From a subset of this database, which follows a known distribution, and by fixing the reference ground reaction forces, the distribution of the dynamic residuals is estimated thanks to the inverse dynamics step (Fig. 2). This step corresponds to the inverse dynamics step in the reference pipeline (right part of Fig. 1) with perturbed kinematic data. To distort the reference motion, for each time step i, a Gaussian noise is added on the reference joint coordinates q * (8) to generate all the perturbed sample N (Fig. 3).

∀ i ∈ [[0, n f ]], q N (i) = q * (i) + N nq (0, σ N q ) ( 8
)
n q is the total number of joints, q N is the vector of generalized coordinates at frame i for the sample N and N n (µ, σ) is a n-dimensional vector with random variables normally distributed with mean µ and variance σ. For each sample, σ N q is randomly selected over the interval [0, σ qmax ]. The value of σ qmax is arbitrarily chosen to allow the maximal value of reconstruction error to be higher than the maximal value needed for the subset selection. : Representation of the method used to introduce noise to the inverse kinematics data. This method is used to generate a database which contains a set of perturbed motions linked to their reconstruction error Perturbing kinematics data aims at reproducing uncertainties introduced by the motion capture and global optimization steps. To consider these data as an uncertainty, we apply the exact same processing pipeline used for the reference data construction (Section 2.1), which avoids very large accelerations nonexistent in the standard processing. Thus, joint coordinates trajectories are filtered using a 4-th order Butterworth low pass filter with a cut-off frequency of 5 Hz and no phase shift and finally gathered in a database as shown in Fig. 3.

As the reference reconstruction error * re is the minimal error that can be achieved, we base the subset distribution from this value. We choose a reconstruction error distribution of the subset following a uniform distribution over the interval [ * re , * re + ∆ re ] (∆ re > 0). This distribution is reformulated in the range [0, ∆ re ], with ∆ re being the maximal relative reconstruction error added [START_REF] Jensen | Estimation of the biomechanical properties of three body types using a photogrammetric method[END_REF].

∆ re = ∆ re * re (9)
In order to constitute a subset of perturbed motions where reconstruction errors follow the desired distribution law, we perform a random selection with replacement on the database. It is conducted according to the previously defined uniform distribution.

The dynamic residuals indicator dr is then computed for each perturbed motion in the subset using the reference ground reaction forces. Since the distribution of the input (distribution of the subset reconstruction error) is uniform, the dispersion coefficient should be associated with the range ∆ re . As some outliers may appear, the dispersion coefficients of the dynamic residuals indicators ∆ re dr are not directly computed with the maximum and the minimum values. Instead, standard deviations σ are computed and ∆ re dr is defined in agreement with the range of a uniform distribution [START_REF] Pearsall | Inertial properties of the human trunk of males determined from magnetic resonance imaging[END_REF].

∆ re dr = √ 12 σ (10)
The study we conducted (see Results in Section 3) is performed with different values of ∆ re to analyze the influence of the noise added on the dynamic residuals. To ensure the objective, the maximum relative reconstruction error added is chosen to add once the reference reconstruction error. Thus, the method is repeated five times with ∆ re = {0.2, 0.4, 0.6, 0.8, 1}.

Force plate measurement error Similarly, to assess the influence of the uncertainty introduced by the force plate measurements on dynamic residuals, we use a set of perturbed force plate data generated by adding a Gaussian noise. Inverse dynamics is then computed using both reference motion data and perturbed force plate data to evaluate dynamic residuals (Fig. 4). This pipeline corresponds to the inverse dynamics step in the reference pipeline (right part in Fig. 1), but using perturbed force plate data. Figure 4: Pipeline evaluating the effect of the force plate data measurement uncertainties on te dynamic residuals. This corresponds to the inverse dynamics reference pipeline in Figure 1, using perturbed force plate data In the AMTI specifications (see the reference of the used force plates in Section 2.3), force sensitivities are reported to be less than 0.1% of the applied load, with a minimum of 0,22 N. The accuracy of the center of pressure (CoP) position estimation is reported to be less than 0.2 mm.

Thus, we apply two different Gaussian noise distributions respectively to the forces and the CoP positions to generate all the perturbed samples N [START_REF] Davidson | Estimating subject-specific body segment parameters using a 3-dimensional modeller program[END_REF].

∀ i ∈ [[0, n f ]],      CoP N (i) = CoP * (i) + N 3 (0, σ CoP ) F N P1 (i) = F * P1 (i) + N 3 (0, σ F P ) F N P2 (i) = F * P2 (i) + N 3 (0, σ F P ) (11)
CoP N (i) is the CoP position vector at time step i for the sample N , F N P1 (i) and F N P2 (i) are the ground reaction force vectors for the sample N on platforms 1 and 2 respectively and σ CoP and σ F P are the variances of the Gaussian distribution for the CoP positions and for the forces respectively. The values of σ CoP and σ F P are chosen according to the force plate datasheet. The uncertainties, corresponding to the dispersion coefficients, will then be evaluated using the 3-sigma values.

We generate a set of perturbed force plate data, and compute the dynamic residuals indicator dr for each sample. In the same way as in the force plate data, the dispersion coefficients on the dynamic residuals indicators are associated to the 3-sigma values: ∆ pe dr .

Experimental protocol and biomechanical model

We applied the method previously described on experimental data. For this purpose, 10 male participants (mean age±std: 27.5±5 years old, height: 180±7 cm, mass: 74.1±7.3 kg) participated in this study. All subjects signed an informed consent form before participation. An anonymization protocol was followed for data safeguarding. The experimental setup was composed of 47 motion capture markers placed on standardized anatomical landmarks as recommended by the International Society of Biomechanics (ISB) [START_REF] Wu | ISB recommendation on definitions of joint coordinate system of various joints for the reporting of human joint motion-part I: ankle, hip, and spine[END_REF][START_REF] Wu | ISB recommendation on definitions of joint coordinate systems of various joints for the reporting of human joint motion -Part II: shoulder, elbow, wrist and hand[END_REF]. Motion capture markers were recorded using a Vicon R motion capture system with 16 cameras (100 Hz). Subjects stood over two AMTI R 120x60 cm force platforms (1000 Hz), each supporting one foot. Each subject visualized a video displaying an avatar performing a standardized movement and imitated the displayed movements (Fig. 5). These movements were designed to activate sequentially each degree of freedom of the kinematical model used thereafter. The whole body model used in this study is composed of 16 rigid segments linked by 15 joints and exhibits 35 degrees of freedom (Fig. 6). A virtual 6-DoF joint connects the pelvis to the global reference frame, allowing to convert a floating-base system into an equivalent fixed base system [START_REF] Featherstone | Rigid Body Dynamics Algorithms[END_REF]. To simplify the inverse kinematics and inverse dynamics steps, the model is hierarchically described, allowing the use of recursive functions [START_REF] Muller | Dealing with Modularity of Multibody Models[END_REF]. Conventions for global and local reference frame and for rotation combinations follow ISB recommendations [START_REF] Wu | ISB recommendations for standardization in the reporting of kinematic data[END_REF].

A preliminary subject-specific kinematical calibration was performed using motion capture data [START_REF] Muller | A Simple Method to Calibrate Kinematical Invariants: Application to Overhead Throwing[END_REF]. Using a set of postures, limb lengths and model marker positions were optimized to best fit captured marker positions. From this calibrated model, BSIP were then estimated using the regression method proposed by [START_REF] Dumas | Adjustments to McConville et al. and Young et al. body segment inertial parameters[END_REF]. This regression method, widely used in human multibody dynamics analyses, is the one evaluated with respect to hypothesis H1.

F * p 1 - → Y F * p 2 - → Z - → X - → Y - → Z - → X

Results and discussion

The present study analyzes the uncertainties propagation in the dynamic residuals. Such a study is suitable to assess the relevance of a BSIP calibration achieved by dynamic residuals minimization. In the following sections, we will analyze separately the influence of the different sources of errors using a Monte Carlo-based approach (Section 3.1 and 3.2). We will then evaluate the part of uncertainty introduced by these errors in the dynamic residuals values (Section 3.3).

Reconstruction error

We first computed the dispersion coefficients obtained from reconstruction error propagation (crosses in Fig. 7, where each subject is represented by a distinctive color). To compare results between subjects, the different coefficients are displayed according to the range of the uniform distribution ∆ re . The reference reconstruction error being different for each subject, the abscissa values of ∆ re are different among subjects.

For each component of the dynamic residuals, the propagation of the reconstruction error in the dynamics seems to be linear. To characterize it, we perform a linear regression using a least-squares method on the results of each subject (straight lines in Fig. 7). The regression function is constrained to have an intercept at zero. This corresponds to a zero coefficient of dispersion with no ∆ re (cm)

0 0.5 1 1.5
∆ re (cm)

0 0.5 1 1.5
∆ re (cm)

0 0.5 1 1.5
∆ re (cm)

Figure 7: Components of the dispersion coefficients ∆ re dr of the dynamic residuals indicators according to the range of the uniform distribution ∆ re . The 3 effort components of these residuals are normalized by the weight of the subject and the 3 torque components are normalized by the body weight times the height more reconstruction error added. The linear regressions result in a 6-coefficient slope vector m re ∆ for each subject. Furthermore, the R-squared value is always within the [0.963 0.999] range.

From this result, we can assume that the uncertainty due to the reconstruction error corresponds to the dispersion coefficients obtained with ∆ re = * re ( ∆ re = 1). Therefore, a priori knowledge of m re ∆ for a given subject enables the computation of the uncertainty due to the kinematics only based on the reference reconstruction error. As the reference reconstruction error depends largely on the motion, we compare the consistency between subjects by using m re ∆ . Fig. 8 shows, for each dynamic residuals component, the repartition of the different slopes obtained by including the results of all the subjects. These boxplots show that, for each component, the slopes are consistent while the reference reconstruction errors are slightly different.

Moreover, the slope values are different according to the considered axis. For instance, the uncertainty due to reconstruction error is smaller on the vertical axis (z-axis) for both force and moment components. For the force component on the z-axis, the relative influence of kinematics is relatively lower compared to the weight which is influential in this component. Because the inertia of whole body is lower around the vertical axis, the influence of kinematics is less important on the z-axis moment component. Moreover, the results obtained with the x-axis and with the y-axis are of the same order of magnitude. This is probably due to the fact that the performed motion did not occur in a specified direction unlike, for example, walking motions.

Force plate measurement error

For each subject, we performed a statistical analysis of the repartition of the dynamic residuals indicators after the propagation of the force plate data measurement uncertainties. Fig. 9 represents the resulting dispersion coefficients. The magnitude of the results seems to be consistent for every subject.

Comparison with the reference dynamic residuals

One of the purposes of the study was to evaluate the possibility of refining BSIP based on dynamic residuals minimization without overfitting (H2). Therefore, we compare the reference dynamic residuals to the uncertainties introduced by the reconstruction error and by the force plate measurements. The global uncertainties ∆ dr [START_REF] Vaughan | Selection of body segment parameters by optimization methods[END_REF] are obtained from the different uncertainties computed separately (Sections 3.1 and 3.2). The first part of [START_REF] Vaughan | Selection of body segment parameters by optimization methods[END_REF] corresponds to the uncertainties due to the reconstruction error and the second part corresponds to the uncertainties due to the force plate data measurement.

∆ dr = * re .m re ∆ + ∆ pe dr ( 12
)
First of all, it is easy to see that the part of uncertainty due to the force plates measurement was significantly lower than the one due to the reconstruction error (5 orders of magnitude lower). This source of uncertainty can therefore be neglected in this study. Fig. 10 shows the reference dynamic residuals along with the uncertainties in the dynamic residuals for each subject. For all subjects, the mean value and the standard deviation (mean(std)) of the reference force components dynamic residuals indicators is 2%(0.6%) (percentage of the weight of the subject) and for the reference torque components is 1.3%(0.7%) (percentage of the weight times the height). These orders of magnitude of inverse dynamics inconsistency are similar to the values obtained by [START_REF] Hansen | An individual and dynamic Body Segment Inertial Parameter validation method using ground reaction forces[END_REF] after a subject-specific BSIP calibration. These results seem to indicate that the regression-based BSIP evaluation is an acceptable estimate to calibrate the model, fully supporting hypothesis H1. This might be attributed to the fact that the volunteers were regular people (healthy, no high-level athletes, neither underweight nor obese: Body Mass Index within [18.5 30] range). So we can suppose that, for these subjects, the anthropometric data of [START_REF] Dumas | Adjustments to McConville et al. and Young et al. body segment inertial parameters[END_REF] were close to the subject-specific parameters.

Overall, the uncertainties in the dynamic residuals are higher than the reference dynamic residuals, except for the vertical moment component (z-axis). However, for this component, the difference between the global uncertainty and the reference value is low compared to the other differences.

The study shows that, in the computation of dynamic residuals, uncertainties introduced by kinematic data and by force plate measurements are too high to use these residuals to achieve a subject-specific BSIP calibration. Indeed, the residuals are here widely explained by the reconstruction error and minimizing them to achieve a BSIP calibration would lead to erroneous results. In conclusion, with regard to those results, the experimental protocol and the multibody model used, hypothesis H2 is refuted. This result partly contradicts some recent studies about BSIP calibration [START_REF] Hansen | An individual and dynamic Body Segment Inertial Parameter validation method using ground reaction forces[END_REF][START_REF] Zhao | Estimating human body segment parameters using motion capture data[END_REF]. Some lines of approach are proposed below to explain these contradictions.

First, we showed that reference dynamic residuals in our study were similar those obtained by [START_REF] Hansen | An individual and dynamic Body Segment Inertial Parameter validation method using ground reaction forces[END_REF] after a subject-specific BSIP calibration. That means that the inconsistency of the inverse dynamics problem is small. This makes it difficult to use this inconsistency to achieve a subject-specific BSIP calibration. Therefore, the method would meet a more competitive challenge with nonregular people, for example high-level athletes or amputees.

Moreover, our study showed that uncertainties introduced by the kinematic data explain most of the dynamic residuals. To reduce the influence of the kinematic errors on dynamic results, [START_REF] Kuo | A least-squares estimation approach to improving the precision of inverse dynamics computations[END_REF] and [START_REF] Van Den Bogert | A weighted least squares method for inverse dynamic analysis[END_REF] proposed a global resolution of the inverse dynamics (least-squares method). Unlike the recursive Newton-Euler algorithm which propagates the errors along the kinematical chain, the least-squares method determines joint torques that better respect the global dynamic balance. By considering all the time steps simultaneously, [START_REF] Kuo | A least-squares estimation approach to improving the precision of inverse dynamics computations[END_REF] also proposed to eliminate constant bias as a wrong BSIP estimation. However, even for only one time step, using this method requires the implementation and the resolution of a large and complex system of equations, for which it is often difficult to find a solution and for which an initial guess is required to start the optimization. This problem is then increased tenfold by considering all the time steps simultaneously. This is why such a method asks for further developments to be fully usable as a solution to the BSIP estimation problem.

Finally, the kinematic model used in the current study was relatively simple. For example the complex structure of the shoulder was here modeled as a spherical joint which does not allow to take into account the motion of the clavicle or the sliding of the scapula. Thus, the reference reconstruction error -average of 1.4 cm -is more important than data available in the literature [START_REF] Lund | Scaling of musculoskeletal models from static and dynamic trials[END_REF] for the lower part of the body. This can have two different impacts on the performed study.

On one side, the reconstruction error introduced by the use of this model could directly introduce dynamic residuals. Complementary studies could be explored: a first approach could consist in adaptively modifying the kinematical model depending on the most important reconstruction error. This could for instance take into account the complex closed-loop geometry of the shoulder [START_REF] Van Der Helm | A finite element musculoskeletal model of the shoulder mechanism[END_REF] or a detailed lumbar spine model [START_REF] De Zee | A generic detailed rigid-body lumbar spine model[END_REF]. A second approach could be to select only a set of motion sequences to perform BSIP calibration, by selecting only sequences where the uncertainties introduced by the reconstruction error would be sufficiently low to achieve a subject-specific BSIP calibration. Acting on the same principle, [START_REF] Venture | A numerical method for choosing motions with optimal excitation properties for identification of biped dynamics -An application to human[END_REF] suggested a method to select sequences of motion with optimal excitation properties. These two methods could be then combined.

On the other side, the reconstruction error introduced by the use of this model could introduce an additional uncertainty without adding dynamic residuals. This seems the case here because, despite the important reconstruction error, reference dynamic residuals are similar to those obtained with a subjectspecific calibrated model [START_REF] Hansen | An individual and dynamic Body Segment Inertial Parameter validation method using ground reaction forces[END_REF]. However, the model could be improved by better understanding uncertainties introduced by kinematic data. For this end, we can use soft tissue artefacts (STA) models [START_REF] Cheze | A solidification procedure to facilitate kinematic analyses based on video system data[END_REF]. These artifacts have been recognized to be the major source of error in human motion analysis [START_REF] Andriacchi | Studies of human locomotion: past, present and future[END_REF]. The reconstruction error could also be computed by weighting differently each marker, unlike the global indicator used in this study. As [START_REF] Ojeda | The effect of kinematic constraints in the inverse dynamics problem in biomechanics[END_REF] proposed to assign weights to each marker during the global optimization method -according to STA related to the considered segment -, weights may reflect the impact of the marker reconstruction error on dynamics. Indeed, the impact of marker reconstruction error on dynamic residuals is influenced by its position in the kinematic model. For example, the reconstruction error of markers of the hand (distal segment with low mass) has probably a lower impact on the dynamics than the reconstruction error of markers of the trunk. Moreover, a segment with several markers has a stronger impact on the reconstruction error computation whereas its impact on the dynamics is not necessarily dominant. In this manner, we could use a Monte Carlo approach to add noise on each marker, proportionally to its reference reconstruction error. This would probably add some noise close to the characteristics of the reference reconstruction error.

Conclusion

In this paper, we proposed a method to quantify the uncertainties propagation in the dynamics of multibody human models. In particular, we assessed possibility of minimizing dynamic residuals to calibrate BSIP of the model without overfitting.

In addition to BSIP, two sources of uncertainties are propagated in the inverse dynamics problem: kinematic data and force plate data. We performed a Monte Carlo-based approach, by running Monte Carlo samplings separately on both sources of uncertainty to assess their propagation their propagation in the dynamic residuals. The global uncertainty introduced was then compared to the dynamic residuals obtained with the reference pipeline.

Our study allowed to assess two hypotheses presented at the beginning of this paper. First, estimating BSIP with regression methods seems to provide an acceptable subject-specific approximation, as hypothesized in H1. Second, the introduced uncertainties were significantly higher than the reference dynamic residuals. Therefore, using dynamic residuals for subject-specific BSIP calibration does not guarantee that overfitting can be avoided, thus not supporting hypothesis H2. Some lines of approach were proposed to explain these results and to improve the uncertainty propagation understanding in multibody human model dynamics.

Figure 2 :

 2 Figure 2: Pipeline used to evaluate the propagation of the uncertainties introduced by the reconstruction error in the dynamic residuals. It corresponds to the inverse dynamics reference pipeline in Figure 1, using perturbed motions

Figure 3

 3 Figure3: Representation of the method used to introduce noise to the inverse kinematics data. This method is used to generate a database which contains a set of perturbed motions linked to their reconstruction error

Figure 5 :

 5 Figure 5: Different postures of the video guide. The video was generated using an avatar animated from motion capture data

Figure 6 :

 6 Figure 6: Kinematic model, model marker position and external forces applied the biomechanical model (bones graphics are adapted from the AnyBody Managed Repository: http://www.anybodytech.com/)

1 )Figure 8 :

 18 Figure 8: Boxplots representing the slopes of the linear regressions which evaluate dispersion coefficients on the dynamic residuals indicators according to ∆ re

Figure 9 :

 9 Figure 9: Boxplots representing the dispersion coefficients in the dynamic residuals indicators after the propagation of the force plate data measurement uncertainties. The 3 effort components are normalized by the weight of the subject and the 3 torque components are normalized by the body weight times the height

Figure 10 :

 10 Figure 10: Reference dynamic residuals indicators and global uncertainties in the dynamic residuals indicators for each component. Each graph represents the results with one subject

Acknowledgements

The authors wish to thank Ana Lucia Cruz Ruiz, Diane Haering, Ludovic Hoyet and Anthony Sorel for their kind support. This study was partially funded by the ANR project ENTRACTE (Grant agreement: ANR 13-CORD-002-01).


Crícia Z Felício

Klérisson V R Paixão
email: klerisson@ufu.br

Guilherme Alves
email: guilhermealves@ufu.br

Sandra De Amo
email: deamo@ufu.br

Philippe Preux
email: philippe.preux@inria.fr

Exploiting Social Information in Pairwise Preference Recommender System

Keywords: H.3.3 [Information Storage and Retrieval]: Clustering-Information filtering; J.4 [Computer Applications]: Social and behavioral sciences Pairwise Preferences, Social Network, Social Recommender System

There has been an explosion of social approaches to leverage recommender systems, mainly to deal with cold-start problems. However, most of the approaches are designed to handle explicit user's ratings. We have envisioned Social PrefRec, a social recommender that applies user preference mining and clustering techniques to incorporate social information on the pairwise preference recommenders. Our approach relies on the hypothesis that user's preference is similar to or influenced by their connected friends. This study reports experiments evaluating the recommendation quality of this method to handle the cold-start problem. Moreover, we investigate the effects of several social metrics on pairwise preference recommendations. We also show the effectiveness of our social preference learning approach in contrast to state-of-the-art social recommenders, expanding our understanding of how contextual social information affects pairwise recommenders.

INTRODUCTION

Social recommender (SR) could shatter the barriers for users to consume information. Thus, there has been an explosion of social approaches in this context [START_REF] Tang | Social Recommendation: a Review[END_REF], mainly to deal with cold-start problems. Typical SR systems assume a social network among users and make recommendations based on the ratings of the users who have direct or indirect social relations with the target user [START_REF] Jamali | A Matrix Factorization Technique with Trust Propagation for Recommendation in Social Networks[END_REF]. However, explicit user's ratings may not capture all the user's interests without loss of information [START_REF] Balakrishnan | Two of a Kind or the Ratings Game? Adaptive Pairwise Preferences and Latent Factor Models[END_REF]. Pairwise preference learning shows clear utility to tackle such problem [START_REF] De Amo | Improving Pairwise Preference Mining Algorithms using Preference Degrees[END_REF]. The "marriage" between pairwise preference recommender and social network can be used in a unique manner to enhance the recommender effectiveness.

In such way, we advance earlier work, PrefRec [de Amo and Oliveira 2014], a model-based hybrid recommender system framework based on pairwise preference mining and preferences aggregation techniques. We propose Social PrefRec, an approach to incorporate social networks information in recommendation task to minimize user cold-start problem. Different factors of social relationships have influence on users. Some of these factors contribute or even harm social recommender systems [START_REF] Yuan | How Friends Affect User Behaviors? An Exploration of Social Relation Analysis for Recommendation[END_REF]]. Understanding the extent to which these factors impact SR systems provides valuable insights for building recommenders. We aim to investigate the role of several social metrics on pairwise preference recommendations. Given that user's preference is similar to or influenced by their connected friends [START_REF] Tang | Social Recommendation: a Review[END_REF], we also aim to study how to apply social similarities in a pairwise preference recommender. Social PrefRec is evaluated on two datasets, named Facebook and Flixster, to verify the integrity of our results. Focusing on social pairwise preference recommendation, our study addresses six questions: Q1: How accurately does social information help on item recommendation?

We will assess the accurateness of Social PrefRec by comparing it to PrefRec. This is the key to determine whether a pairwise preference recommender can benefit from social information.

Q2: How relevant are the recommendations made by a social pairwise preference recommender?

One of the main reasons for the relevance of Social PrefRec is to mitigate the cold-start problem for users through social information. To further assess our model, we compare Social PrefRec to three state-of-art social recommenders.

Q3: Which social metrics are the most important for item recommendation?

The previous questions focus on understanding whether pairwise recommenders could benefit from contextual social information. Here, we want to evaluate the overall performance of each social metric: friendship, mutual friends, similarity, centrality and interaction.

Q4: How effective is Social PrefRec to mitigate data sparsity problems?

In social recommender systems there is a common assumption that contextual social information mitigates data sparsity problems. To assess our model in this context, we evaluate the effectiveness of Social PrefRec with regards to PrefRec against five data sparsity levels.

Q5: Does social degree affect Social PrefRec as much as profile length affects PrefRec?

To achieve high-quality personalization, recommender systems must maximize the information gained about users from item ratings. The more ratings a user's profile has, the merrier will be. We want to check whether increasing the number of friends impacts our approach. Q6: Are there major differences between recommendations quality of popular and unpopular users?

Here we further investigate social popularity effects on recommender systems. This question complements Q5, offering valuable insights into when and which social metric impacts the predictions. This article is a follow up to our earlier study of social information on pairwise preference recommendation [START_REF] Felício | Social PrefRec framework: leveraging Recommender Systems Based on Social Information[END_REF], which tackled only Q1 and Q2. Here, we extend this study by revisiting how Q2 is addressed, and introducing Q3, Q4, Q5 and Q6 to give thoughtful discussions about the practical implications of our findings for pairwise recommender systems.

The remainder of this article is structured as follows: Section 2 presents the background knowledge undertaking in this work and review related work. Section 3 describes our proposed framework the Social PrefRec, as well as the applied social metrics and recommender model selection strategies. Section 4 describes our experimental settings and Section 5 presents the results. Finally, Section 6 concludes the article.

BACKGROUND AND LITERATURE REVIEW

In this section, we introduce the main concepts underlying this work. Section 2.1 presents the notion of pairwise preference recommender systems. Please refer to de Amo et al. [2013] for more details on

Pairwise Preference Recommender Systems

Let U = {u 1 , ..., u m } be a set of users and I = {i 1 , ..., i n } be a set of items, RU (A 1 , ..., A r ) be a relational scheme related to users, and RI(A 1 , ..., A t ) be a relational scheme related to items. The user-item rating matrix in a system with m users and n items is represented by R = [r u,i ] m×n , where each entry r u,i represents the rating given by user u on item i. Table I shows a set of 8 items (movies) and their attributes. A user-item rating matrix with 7 users and movies ratings in the range [1,5] is illustrated in Table II.

In traditional recommender systems, the recommendation task is based on the predictions of the missing values in the user-item matrix. Pairwise preference recommender systems predicts the preference between a pair of items with missing values in the user-item matrix. Both types of systems use the predictions to extract a ranking of items and recommend the top-k.

In our work, we focus on the PrefRec framework, a hybrid model-based approach to design pairwise preference recommender systems. Essentially PrefRec works in two phases: (A) construction of the recommendation models, and (B) recommendation.

A) Construction of the recommendation models. The main activities of this phase are Preferences Clustering, Consensus Calculus and Preference Mining.

Preferences Clustering: First, PrefRec clusters users according to their preferences. This process applies a distance function and a clustering algorithm C over the rows of the user-item rating matrix. A preference vector of user u x is defined as θ ux = R ux , where R ux is a row of matrix R. The output of the clustering algorithm is a set of clusters C, where each cluster C s has a set of users with the most similar preference vectors.

Consensus Calculus: For each cluster C s , a consensus operator A is applied to compute θs , the consensual preference vector of C s . θs,j is the average rating for item j in cluster C s . Please note that the θs,j element is computed if and only if more than half of the users in C s rated the item. Otherwise, this position will be empty.

An example of clustering and consensus calculus can be seen in Table III. The users from Table II were clustered in two groups according to their preference vectors, and a consensual preference vector for each cluster was computed using the group average rating per item.

In comparison with the original PrefRec, one main enhancement done in these two activities, Preferences Clustering and Consensus Calculus, was the replacement of the preference matrix and the consensual matrix by vectors. This new representation not only reduces the algorithm complexity and execution time, but remarkably allows a clustering of a better quality. Preferences Mining: Having the consensual preference vector from each cluster, the system could establish the preference relation between pairs of items. Formally, a preference relation is a strict partial order over I, that is, a binary relation P ref ⊆ I × I transitive and not reflexive. We denote by i 1 > i 2 the fact that i 1 is preferred to i 2 . According to the previous example, a preference relation over consensual preference vector θ 1 is presented in Table IV.

A preference miner P builds a recommendation model for each group using item's features. The set of recommendation models is M = {M 0 = ( θ1 , P 1 ), . . . , M K = ( θk , P K )}, where K is the number of clusters, θs is the consensual preference vector, and P s is the preference model extracted from θs , for 1 ≤ s ≤ K.

In this scenario, a recommendation model is a contextual preference model. Thus, each model P s in M is designed as a Bayesian Preference Network (BPN) over a relational schema RI(A 1 , ..., A t ). A BPN is a pair (G, ϕ) where G is a directed acyclic graph in which each node is an attribute, and edges represent attribute dependency; ϕ is a mapping that associates to each node of G a set of conditional probabilities P[E 2 |E 1 ] of the form of probability's rules:

A 1 = a 1 ∧ . . . ∧ A v = a v → B = b 1 > B = b 2
where A 1 , . . . , A v and B are item attributes. The left side of the rule (condition event E 1 in conditional probability) is called the context and the right side (condition event E 2 in conditional probability) is the preference on the values of the attribute B. This rule reads: if the values of the attributes A 1 , . . . , A v are respectively a 1 , . . . , a v then for the attribute B the value b 1 is preferred to b 2 . Please note that the preferences on B depend on the values of the context attributes. A contextual preference model is able to compare items: given two items i 1 and i 2 , the model can predict which is preferred.

The constructing of a BPN comprehends in: (1) the construction of a network structure represented by the graph G and (2) the computation of a set of parameters ϕ representing the conditional probabilities of the model. The preference miner used in this work, CPrefMiner [START_REF] De Amo | Mining User Contextual Preferences[END_REF], uses a genetic algorithm in the first phase to discover dependencies among attributes and then, compute conditional probabilities using the Maximum Likelihood Principle [START_REF] Nielsen | Bayesian Networks and Decision Graphs[END_REF].

Example Overview. Considering the relational schema of movie attributes in Table I and the useritem rating matrix in Table II: PrefRec cluster users, extract preference consensual vector, from θ1 (Table III), and builds the pairwise preference relation (Table IV). Then, CPrefMiner can build the BPN depicted in Figure 1. P N et 1 represents the contextual preference model that is used to compare the set of pairs of items and make the predictions. B) Recommendation. In its second phase, PrefRec aims at using a recommendation model M s to recommend items for a new user. It is executed online, in contrast to the first phase which is offline. The recommendation process is executed according to the following steps:

(1) Given a target user u x and a (small) set of ratings provided by u x over some items of I, the first task consists in obtaining the consensual preference vector θs more similar to u x 's preferences.

We compute the similarity between θ u (the u x 's preference vector) and each consensual preference vector. Let θs be the consensual preference vector, related to cluster C s , the most similar to θ u .

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Table IV. C 1 pairwise preference relation (i 1 > i 2 ) (i 1 > i 3 ) (i 3 > i 6 ) (i 5 > i 6 ) (i 2 > i 6 ) (i 5 > i 3 ) (i 2 > i 4 ) (i 6 > i 7 ) Fig. 1. Bayesian Preference Network PNet 1 over C 1 preferences.
(2) Consider the preference model P s corresponding to θs . P s is used to infer the preference between pairs of items in I which have not been rated by the user u x in the past.

(3) From the set of pairs of items (i j , i k ) indicating that user u x prefers item i j to item i k , a ranking can be built by applying a ranking algorithm adapted from the algorithm Order By Preferences [START_REF] Cohen | Learning to Order Things[END_REF]]. Thus, the output is a ranking (i 1 , i 2 , . . . , i n ) where an item i j is preferred or indifferent to an item i k , for j < k and j, k ∈ {1, ..., n}.

Example: To illustrate how a preference model can be used in a recommendation phase, suppose that the preference vector θ u of a new user u x is most similar to the consensual preference vector of group C 1 , θ1 . Let us consider the BPN PNet 1 built over θ1 and depicted in Figure 1. This BPN allows to infer a preference ordering on items over relational schema RI(Decade, Director, Star, Genre) of data movie setting. For example, according to this ordering, item i 5 = (2010, Scorsese, Di Caprio, Thriller) is preferred than item i 8 = (2010, Spielberg, Tom Hanks, Drama). To conclude that, we execute the following steps:

(1) Let ∆ : I × I → {A i , ..., A l } be the set of attributes for which two items differ. In this example, ∆(i 5 , i 8 ) = {Director, Star, Genre}.

(2) Let min(∆(i 5 , i 8 )) ⊆ ∆(i 5 , i 8 ) such that the attributes in min(∆(i 5 , i 8 )) have no ancestors in ∆(i 5 , i 8 ). According to the PNet 1 structure, directed edge linking Genre and Star implies remove Star, therefore, in this example, min(∆(i 5 , i 8 )) = {Director, Genre}. To have i 5 preferred rather than i 8 is necessary and sufficient that i To compare i 5 and i 8 we focus only on p 1 and p 3 and select the highest one. In this example, p 1 > p 3 so that we infer that i 5 is preferred to i 8 . If p 1 = p 3 was true, we would conclude that i 5 and i 8 are incomparable.

Studies on Pairwise Preference Recommendation

Balakrishnan and Chopra [2012] have proposed an adaptive scheme in which users are explicitly asked for their relative preference between a pair of items. Though it may give an accurate measure of a user's preference, explicitly asking users for their preference may not be feasible for large numbers of users or items, or desirable as a design strategy in certain cases. [START_REF] Park | Pairwise Preference Regression for Cold-start Recommendation[END_REF] proposed a pairwise preference regression model to deal with the user cold-start problem. We corroborate with their idea. They argue that ranking of pairwise users preferences minimize the distance between real rank of items and then could lead to better recommendation for a new user.

In the same direction, [START_REF] Sharma | Pairwise Learning in Recommendation: Experiments with Community Recommendation on Linkedin[END_REF] propose a probabilistic latent semantic indexing model for pairwise learning, which assumes a set of users' latent preferences between pairs of items.

We build on previous work [de Amo and Oliveira 2014] by adapting a pairwise preference recommender to leverage a graph of information, social network.

Studies on Social Recommender Systems

This research field especially started because social media content and recommender systems can mutually benefit from one another. Many social-enhanced recommendation algorithms are proposed to improve recommendation quality of traditional approaches [START_REF] Krohn-Grimberghe | Multi-relational Matrix Factorization Using Bayesian Personalized Ranking for Social Network Data[END_REF][START_REF] Alexandridis | Improving Social Recommendations by Applying a Personalized Item Clustering Policy[END_REF][START_REF] Canamares | Exploring Social Network Effects on Popularity Biases in Recommender Systems[END_REF]. In terms of using user's social relationships to enhance recommender systems effectiveness, there are some common themes between Social PrefRec and the works of [START_REF] Ma | SoRec: Social Recommendation Using Probabilistic Matrix Factorization[END_REF]Ma et al. [ ] [2011Ma et al. [ ] [2011]]. They developed two approaches called SoRec and SoReg. The former is based on probabilistic matrix factorization to better deal with data sparsity and accuracy problems. The latter also relies on a matrix factorization framework, but incorporates social constraints into its built models. Furthermore, TrustMF is an adaption of matrix factorization technique to map users in terms of their trust relationship, aiming to reflect reciprocal users' influence on their own opinions [START_REF] Yang | Social Collaborative Filtering by Trust[END_REF]. SocialMF also explores the concept of trusting among users, but in the sense of propagation into the model [START_REF] Jamali | A Matrix Factorization Technique with Trust Propagation for Recommendation in Social Networks[END_REF]. In comparison, the goal of Social PrefRec is to exploit social networks in pairwise preference fashion. Although we also employ both users' social network information and rating records, we do it in a different way. Instead of embedding social information in recommendation models, we built a loosely coupled approach based on clustering techniques to choose a suitable model for a given user.

SOCIAL PREFREC

Social PrefRec proposes a new approach to address the new user problem through social information. It is a PrefRec framework extension, incorporating social information at recommendation phase. There were no modification on how models are built, but at recommendation phase we propose an alternative based on social information to recommend items for new users.

In a simple way, a recommendation for new users using social information could recommend items well rated by his direct friends. Another option is to leverage the connection weight among friends to provide better recommendations. The challenge here is to determine how much influence or similarities exist among user's relationship. Connect weight among users can be computed through similarities on profiles (profession, age bracket, location, etc.), interaction between users (messaging, photos, etc.) and degree of influence.

To support this feature, we extended PrefRec and devise Social PrefRec. Figure 2(a) presents its new structure. To better understand it, let us consider the set of users U and the set of items I aforementioned in Section 2. The weight function w : U × I → R computes a user preference degree for an item and can be represented by a rating r u,i from a user-item rating matrix R. To represent a social network, let G = (V, E) be a social graph, and u x and u y vertices of this graph (users of a social network). A set of friends (neighbors) of a vertex u x is F (u x ) = {u y |u y ∈ V ∧ (u x , u y ) ∈ E} and a function l : F → R defines connection weight between u x and u y in [0, 1].

An illustrative example of a social graph in Social PrefRec is shown on Figure 2(b). Nodes represent users, and edges are friendship relations. Edges are labeled with the connection weights computed as explained in Section 3.2. Dashed groups are clusters of users, and each cluster is associated with a recommendation model. Suppose that Paty is a new user; so, there were no historical preferences associated to her. However, the system already clustered Paty's friends according to their preferences. As soon as Paty shows up, the connection weight is computed, and a suitable recommendation model is selected.

Social PrefRec Framework

The general architecture of Social PrefRec, the interactions among the five modules, as well as their respective input and output are presented in Figure 3. Modules from 1 to 4 are from PrefRec, but in module 5 (Recommendation), Social PrefRec, unlike its predecessor, chooses proper recommendation model using one social metric according to following steps.

(1) Given a target user u x and a social metric, we will select u x 's friends F (u x ) and the related connection weight, previously computed as described in Section 3.2, between u x and each u y ∈ F (u x ).

(2) Using one of the selection model methods (see Section 3.2), we will select the preference model P s corresponding to the cluster C s with more similar friends.

(3) P s is used to infer the preference between pairs of items in I.

(4) From the set of pairs of items (i j , i k ) indicating that user u x prefers item i j to item i k , a ranking can be built as mentioned in PrefRec approach.

Note that using this strategy, it is possible to recommend to a given user without taking into account any previous ratings, but relying on the user's relations in the cluster set.

Computation of connection weights and recommendation model selection

Given the social graph G for a target user u x and for each u y ∈ F (u x ), we compute user's connection weight according the following metrics: -Friendship: in this metric, connection weight is measured by l(u x , u y ) = 1, where 1(•) is the characteristic function (1 if argument is true, 0 otherwise).

-Interaction level : computed as a(ux,uy) a(ux) , where a(u x , u y ) is the number of times that user u y appears at u x 's time-line, and a(u x ) is the number of all occurrences of users u y at u x 's time-line.

-Mutual friends: Represents the fraction of common friends or Jaccard similarity using l(u x , u y ) = F (ux)∩F (uy) F (ux)∪F (uy) .

-Similarity score: Given by demographic similarity between u x and u y according to function l(u x , u y ) = sims(u x , u y ). We compute this value by the average of individual similarity in each demographic attribute (Age bracket, Sex, Religion, etc), using the binary function similarity(u x , u y , A i ), wich returns 1 if attribute A i is similar for u x and u y , 0 otherwise. -Centrality: Calculated by average of closeness, betweenness and eigenvector centrality measures with l(u x , u y ) = centrality(u y ).

Social PrefRec allows the definition of any strategy to find a recommendation model. To do so, Module 5 provides a function to select a recommendation model. Let select : U → M be a function that selects the proper recommendation model from M for a target user u x . In this work, Social PrefRec uses two strategies for recommendation model selection based on connection weights: minimum threshold and average connection weight. Each strategy has a different type of implementation for function select, as explained in the following definitions:

-Minimum threshold : Let ε ∈ [0, 1] be a minimum threshold for connection weight. The minimum threshold strategy selects the preference model P s (associated with model M s ∈ M ) which has more users who have a connection weight with the target user u x equal or above a minimum threshold according to Eq. (1).

select(u x ) = arg max

Ms∈M

|{u y ∈ F (u x ) ∧ l(u x , u y ) ≥ ε}| (1)
-Average: The average strategy selects the preference model P s with users who have the highest average connection weight with the target user u x according to Eq.(2).

select(u x ) = arg max Ms∈M 1 |F (u x )| (ux,uy)∈F (ux) l(u x , u y ) (2)
4. EXPERIMENTAL SETTING

Datasets

Table V summarizes our datasets. Recall that sparsity is the percent of empty ratings in user-item rating matrix and links are the number of users connections in the dataset. The particularities of each dataset is described next:

Facebook Dataset. We surveyed this dataset through a Facebook web application we developed for this purpose. With volunteers permission, we crawled relationship status, age bracket, gender, born-in, lives-in, religion, study-in, last 25 posts in user's time-line, posts shared and posts' likes, and movies rated before on the Facebook platform. In addition, we asked each volunteer to rate 169 Oscar nominated movies on a 1 to 5 stars scale. We got data from 720 users and 1,454 movies, resulting in 56,903 ratings.

In our experiments, we consider only ratings from the 169 Oscar nominated movies, which represent movies rated by most users. We split Facebook data into two datasets, FB50 and FB100, to represent the set of users who rated at least 50 and 100 movies, respectively. This was done to evaluate the overall system performance under datasets with different sparsity and social information levels. The movie's attributes are: genres, directors, actors, year, languages and countries. In FB50 and FB100, we compute user similarity metric using the attributes: relationship status, age bracket, gender, bornin, lives-in, religion and study-in. We also compute the interaction level considering the last 25 posts in the user time-line, posts shared and likes.

Flixster Dataset. [START_REF] Jamali | A Matrix Factorization Technique with Trust Propagation for Recommendation in Social Networks[END_REF] published this dataset. However, movie information was restricted to its title, then we improved it by adding genres, directors, actors, year, languages and countries information retrieved from IMDB.com public data. We also use two datasets from Flixster with different sparsity level, Flixster 175K and Flixster 811K. Flixster social information includes friend's relationships, mutual friends, friends centrality and users similarities. Similarity between users is computed only through three attributes: gender, age bracket and location. Interaction information is not available on Flixster dataset.

Comparison Methods

In our experiments we compare Social PrefRec with PrefRec and three social matrix factorization based recommender systems. The idea is to evaluate Social PrefRec recommendations compared to PrefRec. Note that the former chooses the prediction model using only social information whereas the latter needs user's first ratings to choose a model. Further, the comparison with matrix factorization methods is used to evaluate Social PrefRec compared to other social approaches, which handle cold-start users.

Social matrix factorization methods combine social information with rating data. They are distinct from Social PrefRec that uses social information only to choose a consensual prediction model between preference clusters. In addition, our method has its recommendation model based on pairwise preferences. The three social matrix factorization methods do not make use of any clustering technique. We take these systems as comparison methods because they achieve high accuracy levels for cold-start user as reported by the authors. The social matrix factorization particularities are reported next:

SoRec [START_REF] Ma | SoRec: Social Recommendation Using Probabilistic Matrix Factorization[END_REF]: it is based on latent factors of items, users, and social network relationship. The influence of one neighbor on the prediction of a rating increases if he is trusted by a lot of users while it decreases if the target user has many connections.

SocialMF [START_REF] Jamali | A Matrix Factorization Technique with Trust Propagation for Recommendation in Social Networks[END_REF]: applies a trust propagation mechanism. More distant users have less influence (weight) in rating prediction than the trust direct contacts.

TrustMF [START_REF] Yang | Social Collaborative Filtering by Trust[END_REF]]: represents the influence of connections to target user preferences in two ways: truster and trustee. This approach provides recommendations to users that usually show influence on others and those who are typically influenced by others.

Experimental Protocol

Each experiment was performed on the datasets split into two parts: training set and test set.

PrefRec and Social PrefRec build clusters (K-Means clustering) of similar users using the training set. For each cluster C s the systems associate a recommendation model M s . Then, to recommend items for a given user u x , it is necessary to select the most similar model (cluster) that fits u x . This process is done during the test phase. However, those approaches take different directions. Since PrefRec is not able to deal with social information, it relies on previous ratings of u x to select its best recommendation model. In contrast, as Social PrefRec requires social information to accomplish this task. We employ the leave-one-out protocol [START_REF] Sammut | Leave-One-Out Cross-Validation[END_REF] to better validate our tests and simulate a realistic cold-start scenario. Thus, for each test iteration, one user is taken for test purpose, and the training set is made of all other users. Each experiment is composed by n iterations, where n is the number of users. Importantly, because PrefRec cannot act in a full cold-start scenario, we give PrefRec a few ratings to bootstrap the system.

PrefRec protocol. The PrefRec recommendation model is built offline. For the test phase y ratings of the current test user u x chosen at random were considered for the choice of the most similar cluster C s . Then, computing the similarity between the preference vector of u x , θ ux , and the consensual preference vector of C s , θs is a matter of computing the Euclidian distance between these two vectors weighted by the number of common ratings (z), where

d E (θ ux , θi ) = 1 z z k=1 (θ ux,i k -θi,i k ) 2
. Please note that this similarity distance was used for preferences clustering (training) and selection models (test) phases. Finally, for validation purpose, the remaining ratings of the current test user u x were used.

Social PrefRec protocol. Building the recommendation model is done as in PrefRec. However, during the test phase, we do not take any rating into account. Social PrefRec requires solely social information to find the most similar cluster, C s , according to a given social metric and a model selection strategy.

Matrix Factorization social approaches protocol. For SoRec [START_REF] Ma | SoRec: Social Recommendation Using Probabilistic Matrix Factorization[END_REF], SocialMF [START_REF] Jamali | A Matrix Factorization Technique with Trust Propagation for Recommendation in Social Networks[END_REF] and TrustMF [START_REF] Yang | Social Collaborative Filtering by Trust[END_REF] the experimental protocol builds a model M x for each user u x using friendship preferences information which includes all friend's item ratings. In contrast to previous protocols, the recommendation model, M x , is not a clustered preference model, but a specific preference model for each user.

Parameter Settings. In our experiments, we use LibRec [START_REF] Guo | LibRec: a Java Library for Recommender Systems[END_REF] which contains an implementation of SoRec, SocialMF and TrustMF methods with default parameters. We executed Matrix factorization approaches with 10 latent factors and the number of interactions set to 100. We use K-means as the clustering algorithm for PrefRec and Social PrefRec. In addition, we experimentally test several numbers of clusters. Then we set the optimal number of clusters for each dataset: 7 for FB50, 6 for FB100, 4 for Flixster 175K, and 2 for Flixter 811K. The minimum threshold has optimal values equal to 0.4 for FB50 and FB100, and 0.1 for Flixster 175K and Flixter 811K. However, we executed experiments related with Q5 and Q6, over FB50 and FB100 with = 0.1 to have more users in the result set to evaluate these two questions.

Evaluation methods

Regarding our evaluation method, we present results using two metrics: (1) nDCG is a standard ranking quality metric to evaluate the ability of the recommender to rank the list of top-k items [START_REF] Shani | Evaluating Recommendation Systems[END_REF]. (2) We also compute the standard F 1 score, based on precision and recall, to evaluate the prediction quality of pairwise preferences [de Amo and Oliveira 2014].

In the nDCG equation (3), r u,1 is the rating (according to the ground truth) of the item at the first ranking position. Accordingly, r u,j is the ground truth rating for the item ranked in position j. M is the number of ranked items. DCG(u) is the discounted cumulative gain of predict ranking for 5.2 How relevant are the recommendations made by a social pairwise preference recommender? (Q2 )

Tables VI, VII, VIII, and IX show the nDCG results for rank size 5, 10, 15, 20, under minimum threshold strategy, Section 3.2, and 0-rating scenario. We apply each approach described in Section 4.3 on each dataset to assess the robustness of each. We observe that Social PrefRec obtains better results for new users compared to the other social recommenders. One of the main reasons for the effective performance of our approach is that it chooses a suitable recommendation model based on a consensual set of friends' preferences. The other approaches not only consider all friends' preferences, but SocialMF, for example, also relies on trust propagation mechanism, which incorporates preferences from friends of friends. Thus, we argue that a specific set of friends (neighbors) might be a better source to give more relevant recommendations.

Another main difference is about how each approach deals with item attributes. Matrix factorization profiles both users and items in a user-item rating matrix and through latent factor models project items and users into the same latent space, thus making them comparable.

According to a Kruskal-Wallis test with 95% confidence, Social PrefRec performance is significantly better than social matrix factorization approaches. Mutual Friends is better than others Social PrefRec metrics in ndcg@5. For ndcg@10, there is no significant difference between Mutual Friends, Centrality, Friendship, Similarity and Interaction. The performance with Centrality achieves an equivalent score as Mutual Friends in ndcg@15 results. Finally, the ndcg@20 values show that Mutual Friends, Centrality, Friendship and Similarity are not significantly different.

Table VI. Resulting nDCG@5, @10, @15, and @20 against FB50.

Approach

Size of Rank @5 @10 @15 @20 SoRec 0.8515 ± .138 0.8412 Figure 5 shows that PrefRec is superior on less sparse datasets. However, the social approaches on sparser dataset, i.e. F B100 50 and F B100 40 , exhibit better recommendations quality, particularly for Mutual connection weight metric. These results complement previous analyses of Social PrefRec.

5.5 Does social degree affect Social PrefRec as much as profile length affects PrefRec? (Q5 ) Traditional recommender systems present better performance when they know more user's preferences. Figure 6(a) shows the prediction performance of PrefRec on two Facebook datasets. We observe that the recommender predictions get better as the user's profile gets longer. For instance, PrefRec achieves F 1 equal to 71.18 on FB100 when we use 123 ratings for recommendation model selection (α = 0.8).

However, with Social PrefRec, we do not note a correlation between social degree and prediction performance. Figures 6(b) to 6(f) show the results for different social degrees. The overall picture is the same on all datasets and all social metrics. So, increasing the number of friends to select a recommendation model do not increase the F 1 score. This leads us to the next question that further evaluates all social metrics for higher and lower social degrees. 5.6 Are there major differences between the quality of recommendations considering popular and unpopular users? (Q6 )

To investigate the effects of social degree on Social PrefRec, we begin by recalling the definition of popular and unpopular users. First, we calculate the average number of friends on the subset F B50 and F B100. Popular users are those that have more than the average number of friends, whereas unpopular users have only half the average number of friends.

Figure 7 shows the (F 1 ) achieved by Social PrefRec for each social metric against each subset. Note that, the overall performance is similar between each subset. Regarding the major differences between popular and unpopular users, the mutual friends social metric achieves the worst results, which shows the need of larger amounts of friends to better select a recommendation model. On the other hand, the centrality social metric performance shows that it is not affected by the number of friends.

CONCLUSION

We have devised and evaluated Social PrefRec, an approach whose goal is to help pairwise preferences recommender systems to deal with 0-rating user's profile. Driven by six research questions, we expand earlier work by analyzing and demonstrating the effectiveness of our proposed social preference learning approach. Our analyses were performed on four real datasets. We also carefully investigate the role of five well-known social metrics in pairwise preference recommendation and proposed a clustering based approach to incorporate social networks into recommender systems. With Social PrefRec approach, we brought novel ways to extend traditional recommenders.

Finally, although focused on social networks, our work could be extended to tackle other networks (graphs) where we can compute similarity scores between nodes, such as scientific networks or inferred networks [START_REF] Felício | Visual Perception Similarities to Improve the Quality of User Cold Start Recommendations[END_REF]. Another interesting direction for future work is the study of how to choose more influential nodes, e.g. find out the friends who have a stronger influence on a user and apply their preferences to tackle cold-start recommendations.

  5 [Director] > i 8 [Director] and i 5 [Genre] > i 8 [Genre]. (3) Computing the probabilities: p 1 = probability that i 5 > i 8 = P[Scorsese > Spielberg] * P[T hriller > Drama] = 0.8 * 0.66 = 0.53; p 3 = probability that i 5 > i 8 = P[Spielberg > Scorsese] * P[Drama > T hriller] = 0.2 * 0.33 = 0.06; p 2 = probability that i 8 and i 5 are incomparable = 1-(p 1 +p 3 ) = 0.41.

  Fig. 2. (a) Social PrefRec structure and (b) Social network example.

Fig. 3 .

 3 Fig. 3. Social PrefRec Framework.

Fig. 4 .

 4 Fig. 4. F 1 scores for Social PrefRec and PrefRec for 2 model selection strategies: (a) Minimum threshold and (b) Average connection weight.

Fig. 5 .

 5 Fig. 5. Social PrefRec and PrefRec metrics across sparse scenarios with minimum threshold of 40%.

Fig. 6 .

 6 Fig.6. Profile length factor effect (α, see Eq. 5) over F 1 measure (PrefRec) in Fig.6(a). Social degree effect variants (β, see Eq. 6) over F 1 measure in Fig.6(a) -6(f).

  Fig. 7. (a) F 1 metric for Popular users and Unpopular users in FB100 and (b) FB50 with minimum threshold of 10%.

Table I .

 I Movie attributes.

	Item	Title	Decade	Director	Star	Genre
	i 1	Gangs of New York	2000	Scorsese	Di Caprio	Drama
	i 2	Catch me If You Can	2000	Spielberg	Di Caprio	Drama
	i 3	The Terminal	2000	Spielberg	Tom Hanks	Drama
	i 4	The Departed	2000	Scorsese	Di Caprio	Thriller
	i 5	Shutter Island	2010	Scorsese	Di Caprio	Thriller
	i 6	Saving Private Ryan	1990	Spielberg	Tom Hanks	Drama
	i 7	Artificial Intelligence	2000	Spielberg Haley J. Osment	Drama
	i 8	Bridge of Spies	2010	Spielberg	Tom Hanks	Drama
	pairwise preference mining. Following, Section 2.2 describes the related work on pairwise systems and
	Section 2.3 reviews the literature concerning social recommenders.

Table II .

 II User-item rating matrix.

										Table III. Clusters of users with consensual preferences.
		i 1	i 2	i 3	i 4	i 5	i 6	i 7	i 8		i 1	i 2	i 3	i 4	i 5	i 6	i 7	i 8
	Ted	5	2	-	1	-	2	1	-	Ted	5	2	-	1	-	2	1	-
	Zoe	5	2	4	1	5	1	-	3	Zoe	5	2	4	1	5	1	-	3
	Fred	4	-	5	-	5	-	1	-	Fred	4	-	5	-	5	-	1	-
	Mary	2	5	3	5	-	-	-	5	θ1	4.7 2.0 4.5 1.0 5.0 1.5 1.0	*
	Rose	1	-	2	-	2	-	-	4	Mary	2	5	3	5	-	-	-	5
	Paul	-	-	3	4	1	-	-	5	Rose	1	-	2	-	2	-	-	4
	John	2	-	-	5	2	-	-	-	Paul	-	-	3	4	1	-	-	5
										John	2	-	-	5	2	-	-	-
										θ2	1.3	*	2.7 4.7 1.7	*	*	4.7

Table V .

 V Journal of Information and Data Management, Vol. 7, No. 2, August 2016. Dataset features.

	Exploiting Social Information in Pairwise Preference Recommender System	• 107
	Dataset	Users	Items	Ratings	Sparsity Rates / User	Links	Links / User
					(%)	(Average)		(Average)
	F B50	361	169	44.925	26.36	124.44	2,926	8.6
	F B100	230	169	35,459	8.77	154.16	1,330	6.4
	F lixster 175K	357	625	175,523	26.36	491	706	2.8
	F lixster 811K	1,323 1,175 811,726	47.78	613.54	6,526	5.34

Journal ofInformation and Data Management, Vol. 7, No. 2, August 2016, Pages 99-115.

Journal ofInformation and Data Management, Vol. 7, No. 2, August 2016.

C. Z. Felício would like to thank Federal Institute of Triângulo Mineiro for study leave granted. K. V. R. Paixão and G. Alves are sponsored by scholarships from CAPES. We also thank the Brazilian research agencies CAPES, CNPq and FAPEMIG for supporting this work. Ph. Preux' research is partially funded by Contrat de Plan État Région Data, and the French Ministry of Higher Education and Research, and CNRS; he also wishes to acknowledge the continual support of Inria, and the exciting intellectual environment provided by SequeL.

Exploiting Social Information in Pairwise Preference Recommender System

• 109 a target user u, DCG * (u) is the ground truth and N is the number of users in the result set.

Precision and recall were combined using F 1 score (Eq. ( 4)). The precision of a user u is the percentage of good predictions among all the predictions made for user u. The recall is the percentage of good predictions among the amount of pairs of items in the current iteration. Final precision and recall of the test set are obtained by considering the harmonic mean of average precision and average recall of each user.

Besides those metrics, we further analyze how the user ratings profile length and the number of friends impact the recommendation quality through two other metrics: (1) profile length factor and (2) social degree:

Profile length Factor: Let R be an average number of user ratings and an α coefficient, where α ∈ R. Eq. ( 5) represents the profile length factor calculus. In our experiments (Figure 6(a)) we compute the F 1 score for different profile length factors to determine the number of ratings necessary to better select a recommendation model for a given dataset.

Social Degree: The social degree is given by the average degree of the social network ( S) and a β coefficient where β ∈ R. We compute the social degree according to Eq. ( 6). Using different number of friends to select a recommendation model we evaluate the F 1 results (Figure 6(b) -6(f)).

RESULTS

In this section, we thoroughly assess the effectiveness of our proposed pairwise preference recommender approach, Social PrefRec. First, we analyze the quality of recommendations on the datasets (Q1 ). Then, we measure the relevance of recommendations (Q2 ), focusing on the ranking relevance of Social PrefRec compared to those provided by three social recommender systems, besides the original PrefRec. Furthermore, we measure the performance for each social metric (Q3 ) and under different sparsity levels (Q4 ). We close this section by analyzing how user's profile length versus its social degree (Q5 ) and popular versus unpopular users (Q6 ) influence the quality of the recommendations.

How accurately social information help on pairwise preference recommendation? (Q1 )

F 1 scores are represented in Figure 4, for minimum threshold and average connection weight selection model strategies. Against all datasets with a profile length of 30-ratings for PrefRec versus 0-ratings for Social PrefRec, the social approach achieved better results using Minimum threshold strategy. Rate-15-items baseline is widely used to bootstrap traditional recommender systems [START_REF] Chang | Using Groups of Items for Preference Elicitation in Recommender Systems[END_REF]]. Thus, to make a fair comparison we give 30-ratings for PrefRec, which means that all runs have a good safe margin and should not harm its performance. Nevertheless, our social approach performs at least equivalently to traditional one, as we further discuss on Q3. Note that those results are on cold-start scenarios: under scenarios where a user provides enough ratings, a social approach does not add much value.

• 111

Table VII. Resulting nDCG@5, @10, @15, and @20 against FB100.

Approach Size of Rank @5 @10 @15 @20 SoRec 0.8358 Table IX. Resulting nDCG@5, @10, @15, and @20 against Flixter 811K.

Approach

Size of Rank @5 @10 @15 @20 SoRec 0.8198 (profile length = 30-ratings) result. Thus, the test shows with 95% confidence, that with the first three metrics we can better recommend in social 0-rating profile scenario than 30-rating profile in a traditional recommender approach. Although the others social metrics achieved the same result as the traditional approach, they need none previous rating from a user.

How effective is

Social PrefRec to mitigate data sparsity problems? (Q4 )

As sparsity is a big challenge faced by recommendation systems, we consider five subsets sampled from FB100. The basic idea is to simulate sparse scenarios where input datasets has many items to be rated with very few/sparse ratings per user. For instance, F B100 50 was obtained by eliminating around 50% of the ratings in FB100 in a stratified way, so we keep homogeneous subgroups of the original set. Table X shows the characteristics of the datasets extracted from FB100.