M Joelson

M C Néel

On alpha stable distribution of wind driven water surface wave slope

Keywords: Wind driven water surface wave slopes, Probability density function, Non-Gaussian distribution, Alpha-stable distribution

. Discussions and conclusions are conducted on the basis of the data fit results and the model analysis comparison.

During the last three decades, remote sensing of ocean surface has been extensively investigated to study physical processes on the sea surface and air sea interaction mechanism. One of the most important quantities for which depends the ocean remote sensing process is the surface wave slope probability distribution that is known to be non-Gausssian. In the past the general way to describe such probability distribution was to use an expansion technique of the Gaussian law called as Gram-Charlier series. In the present paper, we propose a new formulation of the surface wave slope using alpha stable probability distribution. Application to data from laboratory experiments and study comparison of the alpha stable slope model with existing models are carried out. Our results show the potentiality of the alpha stable laws to describe the non-Gaussian variability of water surface wave slope.

A knowledge of the sea surface slope distribution and its related statistical quantities is essential to understanding the study of the sea surface from remote sensing process. The use of radar backscatter to characterize the sea surface has been widely investigated as well as from the experiments than from the theoretical approach. At nearly normal radar beam incidence, the importance of the slope distribution is clearly exhibited by the simplest model of cross section radar backscatter based on geometrical optics approach formulated by Valenzuela [START_REF] Valenzuela | Theories for the interaction of electromagnetic and ocean wave-A review[END_REF] as:

σ 0 (θ) = πsec 4 θf (ζ x , ζ y )|R(0)| 2 (1)
where θ is the radar incidence angle, |R(0)| 2 is the Fresnel reflection coefficient for normal incidence and f (ζ x , ζ y ) is the probability distribution function (PDF) of the sea surface slope at specular point (ζ x , ζ y ). Determination of the surface slope distribution has been carried out in a large part with semi-empirical models derived from data observations. The commonly used approach for such purpose was the truncated Gram-Charlier series under the assumptions of the near-Gaussian case and the weakly nonlinear interaction proposed by Longuet-Higgins [START_REF] Longuet-Higgins | The effect of nonlinearities on statistical distributions in theory of sea waves[END_REF]. The Gram-Charlier series was attempted to include two additional factors: the peakedness and the skewness that are respectively due to second and third order wave-wave interactions. Despite the multiple adaptations made on the Gram-Charlier model, it is now well-known that this approach fails to work in the range of large slope or large surface elevation. In other word, this model is not suitable to represent the reality when large events have higher occurrence. To improve the results from Gram-Charlier series, several works were made by different authors. Among the existing models, those of Liu et al [START_REF] Liu | Probability Density Function of Ocean Surface Slopes and Its Effects on Radar Backscatter[END_REF], differs from its formulation and its mathematical nature. Indeed, Liu model does not appeal to an expansion technique and is well suitable to represent extreme events. However, this model is originally a symmetrical law including the Gaussian probability and the Cauchy law as limit cases. Owing to the symmetry property, the Liu model is not able to reproduce the skewness effect. Liu extended empirically the model for this purpose.

In the present paper we propose to use the α-stable law to describe the PDF of surface slope. The stable law depends on four parameters: the index of stability, the skewness, the scale and the location parameters that allow one to represent a probability law of random variations including asymmetry, peakedness and peak shift. Also, the α-stable law includes as limit cases the Gaussian probability and the Cauchy law. Generally speaking, stable law belongs to the family of heavy tails probability laws that is especially well adapted to take in account the large events of random variations. Such strong random variations can be found in fields or laboratory observations of the water surface roughness (Onorato et al [START_REF] Onorato | Observation of strongly non-Gaussian statistics for random sea surface gravity waves in wave flume experiments[END_REF]).

Evidences from in situ and laboratory experiments on non-Gaussian heavy tails distribution of wind driven water surface waves and their slope may be found in many articles on the literature. The first argument on the non-Gaussian behavior of wind driven water surface waves and their slopes concerns the existence of asymmetry property of wind wavefield that can be quantified by higher order spectral analysis (Leykin et al [START_REF] Leykin | Asymmetry of wind waves studied in a laboratory tank[END_REF],Joelson et al [START_REF] Joelson | Searching for chaotic and deterministic features in laboratory water surface waves[END_REF]).

Another hallmark of this non-Gaussian behavior is based on the power law observed in the spectrum of wind driven water surface waves. We cite here some works in this sense. Experiment observations was conducted by Huang [START_REF] Huang | The local properties of ocean surface waves by the phase-time method[END_REF] on time series of sea surface that suggest a random dynamic as a spectral power law associated to fractal description. Mellen and al [START_REF] Mellen | Nonlinear dynamics of wind waves: multifractal phase/time effects[END_REF] performed the study of random nonlinear dynamics of laboratory wind waves by the Lévy index and the co-dimension parameter that quantify the multifractal property of the water surface. As pointed out by Solow [START_REF] Solow | Power law without complexity[END_REF], power law behaviour can arise solely from random distributions having heavy upper tail. A fundamental consequence is that random rough surfaces with slowly decaying power spectral density can have infinite slope variance (Warnick et al [START_REF] Warnick | Physical and Geometrical Optics for 2-D Rough surfaces with power-law height spectra[END_REF]). Such case of slope behavior is clearly beyond the Gaussianity assumption and is only reached by the family of heavy tail probability density model to which belongs the α-stable law. Heavy tail distributions of wind driven water surface slope obtained from experiment observations can be found in different works (see for example Cox and Munk [START_REF] Cox | Slopes of sea surface deduced from photographs of sun glitter[END_REF],

Lui et al [START_REF] Liu | Probability Density Function of Ocean Surface Slopes and Its Effects on Radar Backscatter[END_REF], Chapron et al [START_REF] Chapron | Importance of peakedness in sea surface slope measurements and applications[END_REF]).

Also direct simulation methods such as a Monte-Carlo simulation based on nonlinear hydrodynamical model are able to reproduce pseudo-random samples that present similar aspects. Recently, Soriano et al [START_REF] Soriano | Doppler spectrum from two-dimensional ocean surface at microwave frequency[END_REF] studied the effect of this non-Gaussian character on the radar Doppler spectrum of the random surface at grazing angle and concluded on the importance of the non-Gaussian character. Despite these results and the fact that stable laws are discovered since several decades, there are only few applications concerning the scattering on random rough surface with α-stable law. In particular, we cite here the work of Guérin [START_REF] Guérin | Scattering on rough surface with α-stable non-Gaussian height distributions[END_REF], in which, he gave an extension of the expression (1) for the α-stable surface roughness.

The present study, in our opinion, would be the first attempt to apply the stable probability law on random water surface slope representation. The paper is organized as follow:

in §2, we recall the main properties of the α-stable law. In §3, the application of stable law to fit instantaneous slope of wind driven water surface is presented. The §4 is devoted to the comparison of the stable PDF with the Gram-Charlier model and the Liu model. In §5, we present our conclusions on the results of the study.

II. STABLE PROBABILITY DISTRIBUTION

The stable laws were introduced by Paul Lévy [START_REF] Lévy | Calcul des probabilités[END_REF] during his investigations of the behavior of the sums of independent identically distributed random variables. Then the theory on stable laws was developed by many authors and laid out in many books and articles. We recall here some basic results without proof. A stable distribution law is determined by four parameters: an index of stability α, a location parameter µ, a skewness parameter β and a scale parameter γ. The most common formulation of such law is given by the characteristic function which represents the inverse Fourier transform of the probability distribution function. A random variable X is said to have a stable distribution (noted as X ∼ S α (β, γ, µ))

if its characteristic function satisfies

Φ X (t) = exp (iµt -γ α |t| α [1 -iβ sign(t) W(α, t)]) (2)
where

W(α, t) =      tan( πα 2 ) if α = 1 -2 π log |t| if α = 1 and sign(t) =            1 if t > 0 0 if t = 0 -1 if t < 0
The stability index also called as the characteristic exponent or the tail index satisfies 0 < α ≤ 2 and the skewness parameter -1 ≤ β ≤ 1. The location parameter µ is defined only when the stability index is α > 1 (µ ∈ R). In this case, the first absolute moment is given by

< |X| >= 2γ π Γ(1 -1/α
) where Γ(.) is the gamma function. If β = 0, then the distribution is symmetric around µ. The scale parameter also called as dispersion parameter has a real positive value γ > 0 that determines the width of the distribution law. The location parameter µ describes the shift of the peak. The tail index α determines the rate at which the tails of the distribution taper off. When α < 2, the variance of the random variable X is infinite and the tails exhibit a power-law behavior. More precisely, using the central limit theorem, it can be shown that:

     lim t→∞ t α Pr(X > t) = C α (1 + β)γ α lim t→∞ t α Pr(X < -t) = C α (1 -β)γ α (3)
where

C α = 1 π Γ(α)sin( πα 2 )
. From (3), it yields that in general, the p th order moment of a stable random variable is finite if and if only p < α.

Probability density function

Excepted from three special cases, the stable distribution do not have closed form expression of its PDF. This lack of closed form formulas for the PDF has a negative consequence on the popularity of the stable distribution. By applying Fourier inversion to (2), an integral form of the PDF is obtained as:

f α,β,γ,µ (x) =                      1 π ∞ 0 exp(-γ α z α ) cos (x -µ)z + βz α tan( απ 2 ) dz if α = 1 1 π ∞ 0 exp(-γz) cos (x -µ)z + βz π 2 log(|z|) dz if α = 1 (4)
Expression (4) can be analytically solved only for the following exceptions cases:

-for α = 2 corresponding to the Gaussian distribution,

f 2 (x) = 1 √ 2πγ exp - (x -µ) 2 2γ 2 (5)
in this case, the skewness parameter is irrelevant. The variable X ∼ S 2 (0, γ, µ) reduces to the Gaussian random variable X ∼ N (µ, √ 2γ)

-for α = 1 and β = 0 which leads to the Cauchy distribution

f 1 (x) = 1 π γ γ 2 + (x -µ) 2 (6)
The Cauchy random variable writes as X ∼ S 1 (0, γ, µ)

-for α = 0.5 and β = 1 giving the Lévy distribution

f 1/2 (x) = γ 2π exp -γ 2x x 3/2 (7)
For the general cases, numerical approximations of (4) was widely used by several authors to represent stable PDF. Different methods exist for this purpose with various level of accuracy. In this paper, we adopt the direct integration method developed by Nolan [START_REF] Nolan | Numerical "Calculation of Stable Densities and Distribution Functions[END_REF] and Weron [START_REF] Weron | Computationally intensive Value at Risk calculations[END_REF] using an other formulation of the expression (4) called as the Zolotarev's formulas [START_REF] Zolotarev | One dimensional Stable Distributions[END_REF]. The main interest of the Zolotarev's formulas is that instead of expression (4), they do not include infinite integral and then are well adapted to numerical computations. In which follow, we give the Zolotarev's formulas of the PDF of a stable random variable X ∼ S α (β, γ, µ). First, we consider the case of a random variable X 0 ∼ S α (β, 1, 0). By

setting ζ = -β tan πα 2 , the PDF of X 0 can be expressed as: if α = 1 f α,β,1,0 (x) =                        α(x -ζ) 1 α-1 π|α -1| π 2 -ζ V(θ, α, β) exp -(x -ζ) α α-1 V(θ, α, β) dθ, if x > ζ Γ(1 + 1 α ) cos(ξ) π , if x = ζ f α,-β,1,0 (-x), if x < ζ (8) if α = 1 f 1,β,1,0 (x) =                1 2|β| exp(-πx 2β ) π 2 -π 2 V(θ, 1, β) exp(-exp(- πx 2β ) V(θ, 1, β))dθ, if β = 0 1 π(1 + x 2 ) , if β = 0 where ξ = 1 α arctan(-ζ) and V(θ, α, β) =                  (cos αξ) 1 α-1 cos θ sin α(ξ + θ) α α-1 cos (αξ + (α -1)θ) cos θ , α = 1 2 π π 2 + βθ cos θ exp 1 β ( π 2 + βθ) tan θ , α = 1
From the probability distribution given by ( 8), one can easily build the PDF of the random variable X with the help of the stability property, X = γX 0 + µ ∼ S α (β, γ, µ).

Parameters estimation

The first step on modelling data with a stable law requires the estimation of the four parameters that determine the distribution properties. The estimation process suffers from the lack of known closed form of the PDF. As a consequence, standard methods based on maximum likelihood principle would not be efficient with regards to the stable PDF dependance.

However, there are other numerical ways that do not depend directly on the PDF knowledge to evaluate stable parameters like the sample quantile methods or also the sample characteristic function methods. A comparative analysis and discussions about these methods may be found in Borak and al [START_REF] Borak | Stable Distribution[END_REF]. Among these existing methods of estimation, we will use the sample characteristic function developed by Koutrovelis [START_REF] Koutrouvelis | Regression-Type Estimation of the Parameters of Stable Laws[END_REF] and performed by Kogon and

Williams [START_REF] Kogon | Characteristic function based estimation of stable parameters[END_REF]. The method is based on regression type on the log-characteristic function. Indeed, the logarithm of the real part and the imaginary part of the log-characteristic function are linear and then, give rise to a regression model with the data.

log (-Re[Ψ(t)]) = α log|t| + α logγ Im[Ψ(t)] = -µt -β sign(t) W(α, t) (9)
where Ψ is the log-characteristic function of X and W(.,.) is defined as in expression [START_REF] Chapron | Importance of peakedness in sea surface slope measurements and applications[END_REF]. A more complete description of the method and an extended study may be found in [START_REF] Kogon | Characteristic function based estimation of stable parameters[END_REF].

In the following section, the stable PDF model will be applied on slope data from laboratory experiments on wind driven surface water waves.

III. MEASUREMENTS AND DATA ANALYSIS

Experiments

The data of interest issue from experiments made in the wind wave facility of the IRPHE-IOA Laboratory. Basically, the facility consists of a water tank of 40 m length, 3 m width and 1 m depth. The facility can be described as a combination of a wind tunnel with a wind-wave tank. A schematic representation of the air-sea interaction simulation facility is given in Fig. 1. The facility is equipped on a submerged wavemaker that allows to produce mechanical waves from capillary ripples to breaking waves. However in the present work, we focus only on wind wave fields. By means of an axial fan, air flow with velocities ranging from 0 m/s to about 15 m/s generates wind wave fields. Different types of surface waves can be produced by the action of the air flow. It is to be stressed that the facility includes particular devices intended to control various parasitical effects which could coexist with the physical process under consideration. In particular, a permeable wave absorber is disposed at the upwind end of the water tank to prevent the wave reflection. Within the range of wave frequency of interest in the experiments on surface waves, the reflection coefficient is estimated to be insignificant. As far as wind generated waves are concerned, a specific design has been adopted for a smooth joining of the air flow and the water surface. The design prevents air flow-separation at the facility entrance test section. This insures a natural development of the turbulent boundary layer over the water surface. In addition, later quays disposed in the water tank favor the three-dimensional evolution of the wind waves. Studies with more details about there specific devices are given in a number of publications (see e.g.

Coantic et al [START_REF] Coantic | Activities and preliminary results of air-sea interaction research at IMST[END_REF]). We concentrate here on the time evolution of the water surface deflection level at a given position. Measurements of water surface height η(t) were performed by means of two capacitance wave gauges separated by a 5cm along wind distance. These probes using a capacitance wire gauge of 0.3 mm outer diameter were set up on a carriage moving along the wave tank. Experiments were carried out under various conditions, whose main parameters are the wind velocity and the distance between the gauge and the facility entrance section (hereafter denoted fetch). Such distance determines the length action of the air flow over the water surface and is well known to be of crucial importance on the rate growth of surface waves and on its nonlinearity. The time derivative dη/dt of the water surface heights were obtained by use of analog derivators. Thus, instantaneous slope s(t) on the upwind direction of the water surface waves could be derived as s(t)=-1/c dη/dt where c corresponds to the phase velocity of the dominant wave frequency. This slope measuring method was tested by comparison with direct measurement performed by a laser slope gauge (see Jhane and Riemer [START_REF] Jähne | Two-dimensional wave number spectra of small scale water surface waves[END_REF]). A good agreement was found particularly for capillary and gravity waves at moderate wind speed.

Data analysis

In this section, we estimate the PDF of time water surface slopes at different wind speed for fetch varying from 2m to 26m. Note that wind speed is estimated at 10m height above the surface(U 10 ). Among the large possibilities of combinations from wind speed and fetch values, we present here some cases that are indicative of the typical behavior of water surface slope. In which follows, we focus on slope behavior of three cases: capillary waves, capillary gravity waves at moderate fetch and low wind speed and gravity waves at large fetch and high wind speed that are typical of fully developed sea surface. In order to compare the different runs of study, the data series are normalized with its rms and mean values. The results of the fit are summarized in Tab.I.

The first line of Tab.I corresponds to the case I of capillary wind wave slopes at moderate wind speed and at a very short fetch. In this case, we find a stability index of 1.72 that is indicative of a strong non-Gaussian character. A very weak value of the skewness parameter is also found traducing a lack of asymmetry on the waveform. The scale parameter is at a relative low value with regard to the normalized Gaussian standard deviation(∼ S 2 (0, 1/ √ 2, 0)).

This would be related to a nonlinear effect involved on the generation of capillary waves by parameter also increases. A striking feature is that the skewness parameter reaches to its negative limit value. This is in good agreement with time evolution of the slope presented in Fig3.a. Observing also that the corresponding surface waves are dominated by a Stokes component, we suggest the possible rôle of the Stokes effect in the asymmetric behavior of the slope and in the limit value obtained for the skewness parameter in the present case.

However, in the general case, relation between asymmetry in the surface and in the slope variation is not easy to establish especially for a random wavefield. The Stokes effect yields to a waveform of more steeper crests increasing locally slopes and more flatter troughs.

Results of run at fetch of 6m and wind speed of 6m/s (case II-1) are shown in Fig. 3. The asymmetry property is clearly visible on the slope samples (Fig. 3.a) and is well represented by the PDF model (Fig. 3.b). Modulational instabilities begin to occur in the wavefield owing to the nonlinear wave-wave interaction process called as the Benjamin-Feir instabilities.

The sample of the wave signal presents wave trains formation that are reminiscent of these instabilities effects (see Fig. 3.a). Such nonlinear effects are known in this case to dominate the effects of randomness. This would explain the reduction of slope magnitude and also the increase of the stability index to the near-Gaussian situation. By increasing the fetch value but taking a low wind speed (case II-2), the rate of input wind energy will be weaker than the rate of the self-nonlinearity of the wavefield. Thus, one can more emphasize the wave modulation effect on the slope distribution and on its PDF. The corresponding PDF is shown in Fig. 4.b with a very similar form to the normalized Gaussian except for the negative tail that is indicative of the presence of Stokes effect. Finally, at high wind speed and large fetch corresponding to the case III, one can find surprisingly that values of the index stability but also the skewness and the scale parameter reduce reaching again to a clearly non-Gaussian situation. In agreement with these values, from the Fig. 5.a, it may be found that the magnitude of the wave slopes grows locally at the crests and the wave signal behaves strongly asymmetrical. From the physical background, it was shown by Trulsen and Dysthe [START_REF] Trulsen | Action of windstress and breaking on the evolution of a wavetrain[END_REF] that such situation may be explained from theoretical model based on a modified nonlinear Schrödinger equation that allows one to predict the stabilization of the Stokes effects by the strong wind such that modulational instabilities are suppressed. The PDF of the case III are given in Fig. 5.b in which it may be found that the stable PDF model reproduce well the distribution of the negative slope values and also the skewness of the slopes. However, at large positive values, the model fails to represent the data distribution.

No clear interpretation was found for this difference but we believe that additional physical process such as breaking wave would be involved.

IV. COMPARISON WITH OTHER MODELS OF SLOPE DISTRIBUTION

In this section, we compare the properties of the stable law with the Gram Charlier and the Liu models. The Gram-Charlier model based on an expansion technique called as Edgworth series was widely used in the past and permitted to develop a large part of the understanding of the non-Gaussian behavior of the sea surface. However, the limit of this model was earlier found by several authors. Many efforts were conducted to develop alternative ways from the statistical expansion technique. The Liu model was built in this context to improve the statistical representation of the sea surface slopes.

The Gram-Charlier series

The traditional method of interpreting the wind wave slopes has been to fit them to a Gram-Charlier series, which can be derived by assuming that surface waves are weakly nonlinear (Longuet-Higgins [START_REF] Longuet-Higgins | The effect of nonlinearities on statistical distributions in theory of sea waves[END_REF]). Basically, in this procedure the PDF of the wave slopes s(t) writes as:

φ GC (s) = G(s) 1 + C 3 6 H 3 (s) + C 4 24 H 4 (s) (10)
where

G(s) = 1 σ √ 2π exp   - s-< s > √ 2σ 2  
and σ is the rms of the slope, when H n is the n th Hermite polynomial. The coefficients C n can be calculated from the statistical moments of φ GC (s) and G(s) as:

C n = 1 σ n s n [φ GC (s) -G(s)] ds
Clearly, the coefficients C 3 and C 4 in expression [START_REF] Joelson | Searching for chaotic and deterministic features in laboratory water surface waves[END_REF] are intended to include the skewness and the kurtosis that are the consequences of nonlinearities in the underlying physical processes.

Of interest is that the PDF model becomes non-Gaussian. Note also that expression [START_REF] Joelson | Searching for chaotic and deterministic features in laboratory water surface waves[END_REF] is not always guaranteed to be positive, and is therefore not a valid probability distribution.

The Gram-Charlier series diverges in many cases of interest. It converges only if φ GC (s) falls off faster than exp(-s 2 /4) at infinity (Cramér[6]). For water waves applications, the skewness and kurtosis coefficients (C 3 and C 4 ) are found to be related to physical parameters in particular to the air flow speed value at 10m altitude (noted as the U 10 ). Here we adopt the values of these coefficients used by Cox and Munk [START_REF] Cox | Statistics of sea surface derived from sun glitter[END_REF] and Plant [START_REF] Plant | A new interpretation of sea-surface slope probability density functions[END_REF] from a range of wind speed varying from 0.72 to 10.2 m/s (Note that [START_REF] Cox | Statistics of sea surface derived from sun glitter[END_REF] uses the wind speed at height of 12.5m instead of the traditional U 10 ). We fit the Gram-Charlier model with stable law using the Nelder-Mead [START_REF] Nelder | A simplex Method for function minimization[END_REF] simplex method. The results of the fit are given in Tab.II. The method is based on the minimization of the euclidian norm of the error ǫ = || f α,β,γ,µφ GC || for 1000 points varying from -5 to +5. From each case, values of the best stable parameters and the error ǫ are shown. As a general remark for all cases, the values of the stability index α are closed to 2 that traduces the near-Gaussian assumption adopted in the Gram-Charlier model. The error ǫ increases with the wind speed values. Rather than a simple difference from the calculation, these error values reveal a more profound departure on the nature of both models. More precisely, at low wind speed, when the occurrence of large values of slope would be rare, a better agreement is found in particular around the peak of the PDF. At high wind speed, the results of the Gram-Charlier series differ clearly from those of the stable law as well as at the tails of the probability than at the middle part of the probability law. The Gram-Charlier model underestimates the extreme events giving a less heavy tails on the probability law (see Fig. 6). At the middle part of probability law, it overestimates the effect of the skewness.

Note that no clear tendency is found between the stable parameter β and the skewness C CM 3 . Finally, the location parameter µ is globally around zero and the scale parameter γ is found at a quasi constant value around 1 √ 2 that are directly related to the normalized Gaussian law of the function G(s) included in the expression [START_REF] Joelson | Searching for chaotic and deterministic features in laboratory water surface waves[END_REF].

The Liu model of slope PDF

A more advanced model is developed by Liu et al [START_REF] Liu | Probability Density Function of Ocean Surface Slopes and Its Effects on Radar Backscatter[END_REF] to fit the joint probability of the surface wave slopes. Liu model is also based on the weakly nonlinear assumption of the surface [START_REF] Longuet-Higgins | The effect of nonlinearities on statistical distributions in theory of sea waves[END_REF]. The basic form of Liu slope PDF writes as: [START_REF] Kogon | Characteristic function based estimation of stable parameters[END_REF] where x,y are normalized slope components and n an integer parameter describing the peakedness of the surface slopes. Note that [START_REF] Kogon | Characteristic function based estimation of stable parameters[END_REF] refers to the case of isotropic sea surface and is not always suitable for anisotropic real surface. An empirical formulation inspired by [START_REF] Cox | Statistics of sea surface derived from sun glitter[END_REF][16] is used by [START_REF] Liu | Probability Density Function of Ocean Surface Slopes and Its Effects on Radar Backscatter[END_REF] to handle the skewness effect. According to [START_REF] Liu | Probability Density Function of Ocean Surface Slopes and Its Effects on Radar Backscatter[END_REF], the expression [START_REF] Kogon | Characteristic function based estimation of stable parameters[END_REF] is an improvement of the Gram-Charlier distribution in particular for the range of large slopes. For convenience, here we consider the expression for y=0, i.e. we focus only on upwind direction. As a first remark, the positivity of the expression ( 11) is obviously guaranteed however it may be found that the expression of f n L (x, 0) does not represent a valid probability density function for any value of n. Indeed, the integral of this expression is not equal to unity. We correct this problem by adopting a normalized form of the Liu PDF as:

f n L (x, y) = n 2π(n -1) 1 + x 2 (n -1) + y 2 (n -1) -(n+2)/2
φ m L (x) = 1 √ π 1 + x 2 (m -1) (-m 2 +1) Γ( m 2 -1) 1 m-1 Γ( m 2 -3 2 ) (12)
The expression ( 12) is defined for m ≥ 4. For m=4, one can easily find that

φ 4 L (x) = 1 π √ 3 3 + x 2 (13)
Expression ( 13) belongs to the Cauchy stable law with X ∼ S 1 (0, √ 3, 0). On the other hand, taking the logarithm of ( 12), the limit for large value of m writes as:

log φ m L (x) ≃ m→∞ - m -2 2 log 1 + x 2 m -1 (14)
which reaches by simple series expansion to the normal distribution of X ∼ N (0, 1) or also the stable law with X ∼ S 2 (0, 1 √ 2 , 0). The limit cases of the Liu formulation belong to the symmetrical stable distribution.

For intermediate values, the Liu PDF differs slightly from the stable distribution. In order to illustrate this relation, we compute a numerical adjustment of the stable parameters α and γ to the expression (12) for m=4.5. The adjustment is based on the minimization of the euclidian norm || f α,0,γ,0 (x)φ m L (x)|| ≤ ǫ for 3000 points from x=-15 to 15. The results are shown in Fig. 7. It may be found that the major difference between the two laws are clearly localized at the medium range values of the probability structure.

By varying the peakedness parameter m, this difference stays quite systematically. Analytically, one can also decompose the modified Liu model to study its property. As an example, taking m=6, the Liu PDF writes as:

φ 6 L (x) = 1 π √ 5 5 + x 2 + 1 π √ 5(5 -x 2 ) (5 + x 2 ) 2 (15)
where the first term is a Cauchy law S 1 (0, √ 5, 0). As shown from the Fig. 8, the second term possess a significant positive values at the middle and the medium parts of the probability law. The magnitude of these positive values are responsible of the height of Liu PDF and the departure from the stable law at the medium range values. The negative values of the second term of (15) explain the similarity of the asymptotic behavior of both laws observed in Fig. 7.

For higher values of m, the same expansion holds with a Cauchy law S 1 (0, √ m -1, 0) as a first term but the second term becomes a sum of rational functions whose integral vanishes to ensure the normalization of the PDF. According to the probability theory, the modified Liu PDF belongs to the family of the Generalized Student or T-distribution whose the expression writes as:

φ m S (x) = Γ(m + 1 2 ) Γ( m 2 ) Γ( 1 2 ) a (x 2 + a 2 ) m+ 1 2 ( 16
)
where m is now called as freedom degrees and a is a scale parameter. The characteristic function of the T-distribution is given by:

ψ m S (t) = 2 |at| m 2 K m 2 (|at|) 2 m 2 Γ( m 2 ) ( 17
)
where K i is the modified Bessel function. From ( 2) and ( 17), one can evaluate the difference between the stable and the T-distribution laws. As said in previous section, the stable laws decay as |x| -α-1 with α < 2. Here, it may be shown that the T-distribution laws have the same asymptotic behavior as |x| -m-1 for all m>0. Note that as a consequence, the Tdistribution laws are more accurate than the stable laws to approximate the near-Gaussian situation. Another significant difference concerns the variance of the laws: for all m>2, the T-distribution laws have finite variance given by σ 2 = a 2 m-2 while the stable laws do not possess finite variance except only for α = 2. From the practical viewpoint, this lack of variance may be of crucial importance when one have to compute statistical values as the mean square slope that is necessary to evaluate the radar backscatter cross section.

Finally, from the expression ( 16), it appears that the T-distribution do not include a skewness property while the stable laws are naturally able to express the skewness effect.

Recalling the importance of the skewness effect on the dynamic of the surface waves, this point may have a significant consequence on the accuracy of the slope distribution model.

The main purpose of this study was to highlight the possible use of α-stable laws for water surface slopes representation. As said in the introduction of this paper, the fundamental argument on the use of α-stable laws stems first from experimental observations of the main property of the wind driven water surface waves: a possible fractal representation, related to a spectral power law, nonlinear characters of the wind wavefield resulting in an asymmetry and a modulational instability of the surface, and a randomness of the slope, such as these abrupt changes shown here in section 3.

Considering these strongly disordered behaviors that may be found on slope variations, α-stable laws seem à priori to be a potential candidate for this objective. Starting from this idea, the study propose to apply the stable laws to represent the probability distribution of the slopes of wind driven water surface from laboratory experiments. It is found that the stable laws are well adapted to model the random characteristic of the slopes data in different cases of experiment configurations. From the proposed cases, the four stable parameters behave in good agreement with the effects of different physical processes observed and expected in the experiments. In particular, the stability index traduces fairly the non-Gaussianity of the slopes in the same sense as the Stokes and/or the modulational instability effects. Also the skewness parameter reproduce well the asymmetry tendency of the slopes under various situations. However, at extreme experiment condition (large fetch and high wind speed), the stable laws fail to reproduce the behavior of the upper part of the slopes.

From more theoretical consideration, comparisons with the Gram-Charlier and the Liu et al [START_REF] Liu | Probability Density Function of Ocean Surface Slopes and Its Effects on Radar Backscatter[END_REF] models are carried out. It appears that the stable law differs completely from the Gram-Charlier model as well as in its mathematical structure than from the characteristics of its results. Indeed, one can recall that the Gram-Charlier model at its usual truncation order (third or fourth order) is part of the near-Gaussian approximation models, while the stable law is totally non-Gaussian except only when the stability index reaches to the maximum value.

The comparison with the Liu model shows more similarities. Recalling that Liu model belongs to the T-distribution family, it is well established that both models are part of the so-called "heavy tail probability laws". They are both infinitely divisible probability laws but the T-distribution is not stable. This means that sum of independent identically T-distributed variables does not follow the same law unlike the stable law that it is always the case.

From mathematical viewpoint such stability property lies to a fundamental characteristic based on the generalization of the central limit theorem (see Samorodnitsky and Taqqu [START_REF] Samorodnitsky | Stable Non-Gaussian Random Processes[END_REF]) and traduces the fact that the α-stable laws constitute limit laws (or attractor laws) for all additive random processes. As a special case, we recall that for α = 2, from the central limit theorem, the Gaussian law is a limit law for all additive finite variance random processes. This is the profound justification of the popularity of the Gaussian model in many physical domains. In the realm of water surface waves, the random wavefield is usually considered as the sum of infinite number of random components. Owing to the weakness of the interactions between components, the corresponding motion of each component may be regarded as independent (see for example Phillips [START_REF] Phillips | The dynamics of the Upper Ocean[END_REF]). Consequently, under the generalisation of the central limit theorem, the choice of stable law is then theoretically justified with a clear mathematical basis.

With regard to the practical viewpoint, the differences between the two laws are more important and have to be stressed here with respect to the study objective. These differences stand on four points concerning the closed form of the PDF, the finite variance value, the skewness representation and the interpretation of the physical processes through the PDF parameters.

As said in previous section, the T-distribution has a closed form of the PDF for all values of its parameters . The stable distribution does not possess a closed form of its PDF except for three special cases. This lack of closed form is apparently an inconvenient property of the stable law. However, as shown in this study, there exists robust techniques to implement the integral form of the stable PDF (see Zolotarev [START_REF] Zolotarev | One dimensional Stable Distributions[END_REF], Weron [START_REF] Weron | Computationally intensive Value at Risk calculations[END_REF], Nolan [START_REF] Nolan | Numerical "Calculation of Stable Densities and Distribution Functions[END_REF]). Other expansion techniques based on the Bergström series are also available to evaluate the stable PDF.

For physical applications, an appeal to statistical quantities associated with the PDF model would be useful. For example, for the sea surface remote sensing problem, it is known that the mean and the variance of the slopes are of importance. This is historically related to the assumption of near-Gaussian state in the development. Recalling that T-distribution has finite variance for all freedom degree greater than 2. This later appears as a well adapted choice for this purpose. However, the T-distribution and the stable distribution constitute fully non-Gaussian alternative and then fails to belong in the category of quasi-Gaussian solutions. In our opinion, an extension of the remote sensing or the hydrodynamical relationships in the sense as developed by Guérin [START_REF] Guérin | Scattering on rough surface with α-stable non-Gaussian height distributions[END_REF] is required for a fully non-Gaussian solution.

As far as non-Gaussian random water surface is concerned, the skewness is undoubtedly one of its most important properties. The stable distribution has asymmetric form that allows one to include naturally the skewness. This is not the case of the T-distribution whose no extension to asymmetric case is previewed. Empirical asymmetric formulation has to be found (see Liu et al [START_REF] Liu | Probability Density Function of Ocean Surface Slopes and Its Effects on Radar Backscatter[END_REF]).

Before ending the paper, we focus on the interpretation of the physical processes through the probability distribution model. As we have shown in this study, the four stable parameters are able to traduce correctly the effects of physical processes as the asymmetry and the kurtosis of the slopes or also its level of non-Gaussianity. Use of α-stable distribution instead of the traditional Gaussian or quasi-Gaussian distributions, combined to hydrodynamical laws will be of great importance on understanding of the physical behavior of the water wave slopes. In this sense, extension of the present work to the space representation of water wave slopes by the use of multivariate α-stable distribution has to be carried out and constitutes our prospective task. An other way to be explored is also the use of truncated stable law that allows one to obtain finite values of statistical moments.

FIG. 1 :

 1 FIG.1:A schematic representation of the large air-sea interactions facility.

FIG. 2 :

 2 FIG. 2: Fetch=2m. U 10 = 6m/s. Part(a): Thick line: Normalized sample of surface wave. Thin line: Normalized instantaneous slope sample. Part(b): 'o': PDF of slope experiment data. Solid line: α stabe PDF. Dashed-dotted line: Normalized Gaussian PDF

FIG. 3 :FIG. 4 :FIG. 5 :

 345 FIG. 3: Fetch=6m. U 10 = 6m/s. Part(a): Thick line: Normalized sample of surface wave. Thin line: Normalized instantaneous slope sample. Part(b): 'o': PDF of slope experiment data. Solid line: α stabe PDF. Dashed-dotted line: Normalized Gaussian PDF

U 12 . 5 (

 125 0.101 0.127 1.951 -0.330 0.695 -0.05 1e-1

FIG. 6 :

 6 FIG.6: Results of the fit for U 12.5 =8m/s. The solid line is the Gram-Charlier PDF. Dashed line represents the stable law of the fit. A fixed value of the probability is drawn with dotted line that allows one to visualize the asymmetry of both laws.

FIG. 7 :

 7 FIG. 7: Semi-log representation of Liu PDF for m=4.5 and the stable law fit with α = 1.221 and γ = 1.3384. The error ǫ = 0.066277. Solid line: Liu PDF model. Dashed line: Stable law.

FIG. 8 :

 8 FIG.8: The corresponding figure of expression[START_REF] Liu | Probability Density Function of Ocean Surface Slopes and Its Effects on Radar Backscatter[END_REF]. The Cauchy law (first term) is on dashed line while solid line represents the second term with negative part. The zero value is drawn with dotted line

TABLE I :

 I Estimation of stable parameters (α, β, γ, µ) from the data series.

	Case Fetch (m) U 10 (m/s) α	β	γ	µ
	I	2	6	1.7219 -0.0816 0.5434 0.0056
	II-1 6	6	1.8688 -1.0000 0.6384 -0.0869
	II-2 13	3	1.9822 -1.0000 0.6972 -0.0117
	III 26	13	1.9178 -0.9488 0.6481 -0.0051

TABLE II :

 II

Estimation of stable parameters (α, β, γ, µ) from the Gram-Charlier series used by Cox and Munk (1954) model

Acknowledgements

The authors are particularly grateful to Y. Stepanyants and to Fred Ramamonjiarisoa for their helpful discussions on the first draft of this manuscript.


Raj Ratn-Pranesh

Mehrdad Farokhnejad
email: mehrdad.farokhnejad@univ-grenoble-alpes.fr

Ambesh Shekhar
email: ambesh.sinha@gmail.com

Raj Ratn Pranesh

Genoveva Vargas-Solar
email: genoveva.vargas-solar@liris.cnrs.fr

Looking for COVID-19 misinformation in multilingual social media texts

Keywords: Misinformation, Multilingual Analysis, Micro-text analysis, COVID-19

Introduction

Since late 2019, the coronavirus disease COVID-19 has spread worldwide to more than 216 countries [START_REF] Organization | Coronavirus disease 2019 ( covid-19)[END_REF]. The COVID-19 pandemic has highlighted the extent to which the world's population is interconnected through the Internet and social media. Indeed, social media is a significant conduit where people share their response, thoughts, news, information related to COVID-19, with one in three individuals worldwide participating in social media, with two-thirds of people utilizing it on the Internet [START_REF] Ortiz-Ospina | The rise of social media[END_REF][START_REF] Wilford | Social media use among parents of young childhood cancer survivors[END_REF]. Social media provides particularly fertile ground for the spread of information and misinformation [START_REF] Frenkel | Surge of virus misinformation stumps facebook and twitter[END_REF]. It can even give direct access to content, which may intensify rumours and dubious information [START_REF] Cinelli | The covid-19 social media infodemic[END_REF]. For people with non-medical experience it is difficult to assess health information's authenticity. Misinformation may have intense implications for public opinion and behaviour, positively or negatively influencing the viewpoint of those who access it [START_REF] Brindha | Social media reigned by information or misinformation about covid-19: a phenomenological study[END_REF][START_REF] Kouzy | Coronavirus goes viral: quantifying the covid-19 misinformation epidemic on twitter[END_REF]. Seeking accurate and valid information is the biggest challenge with Internet health information during the pandemic [START_REF] Eysenbach | Empirical studies assessing the quality of health information for consumers on the world wide web: a systematic review[END_REF].

This paper proposes CMTA, a multilingual tweet analysis and information (misinformation) detection method for observing social media misinformation spread about the COVID-19 pandemic within communities with different languages. CMTA proposes a data science pipeline with tasks that rely on popular artificial intelligence models (e.g., multi-lingual BERT and CNN) to process texts and classifying them according to different misinformation classes ('false', 'partly false' and 'misleading'). The paper describes the experimental setting implemented for validating CMTA that uses datasets of COVID-19 related tweets. The paper presents an illustrative statistical representation of the findings insisting on the insights discovered in our study to discuss results. To assess the performance of CMTA, we compared CMTA with eight monolingual BERT models. The comparison shows that CMTA has surpassed various monolingual models and suggests that it can be used as a general method for detecting misinformation in multilingual micro-texts.

The remainder of the paper is organised as follows. Section 2 introduces works that have addressed misinformation detection about COVID-19 on social media datasets. Section 3 describes the general approach behind the method CMTA that we propose. It describes the experiment setting that we used for validating CMTA. Section 4 compares the performance of CMTA against mono-lingual analytics performed on the same dataset. It also discusses the study results about misinformation spread through micro-texts (i.e., tweets) across different languages. Finally, Section 5 concludes the paper and discusses future work.

Related work

The COVID-19 pandemic has resulted in studies investigating the various types of misinformation arising during the COVID-19 crisis [2, ?,?,13]. Studies investigate a small subset of claims [START_REF] Singh | A first look at covid-19 information and misinformation sharing on twitter[END_REF] or manually annotate Twitter data [START_REF] Kouzy | Coronavirus goes viral: quantifying the covid-19 misinformation epidemic on twitter[END_REF]. In [START_REF] Brennen | Types, sources, and claims of covid-19 misinformation[END_REF] authors analyse different types of sources for looking for COVID-19 misinformation. Pennycook et al. [START_REF] Pennycook | Fighting covid-19 misinformation on social media: Experimental evidence for a scalable accuracynudge intervention[END_REF] introduced an attention-based account of misinformation and observed that people tend to believe false claims about COVID-19 and share false claims when they do not think critically about the accuracy and veracity of the information. Kouzy et al. [START_REF] Kouzy | Coronavirus goes viral: quantifying the covid-19 misinformation epidemic on twitter[END_REF] annotated about 600 messages containing hashtags about COVID-19, they observed that about one-fourth of messages contain some form of misinformation, and about 17% contain some unverifiable information. With such misinformation overload, any decision making procedure based on misinformation has a high likelihood of severely impacting people's health [START_REF] Ingraham | Fact versus science fiction: fighting coronavirus disease 2019 requires the wisdom to know the difference[END_REF]. The work in [START_REF] Huang | Disinformation and misinformation on twitter during the novel coronavirus outbreak[END_REF] examined the global spread of information related to crucial disinformation stories and "fake news" URLs during the early stages of the global pandemic on Twitter. Their study shows that news agencies, government officials, and individual news reporters send messages that spread widely and play critical roles. Tweets citing URLs for "fake news" and reports of propaganda are more likely than news or government pages shared by regular users and bots.

The work in [START_REF] Sharma | Covid-19 on social media: Analyzing misinformation in twitter conversations[END_REF] focused on topic modelling and designed a dashboard to track Twitter's misinformation regarding the COVID-19 pandemic. The dashboard presents a summary of information derived from Twitter posts, including topics, sentiment, false and misleading information shared on social media related to COVID-19. Cinelli et al. [START_REF] Singh | A first look at covid-19 information and misinformation sharing on twitter[END_REF] track (mis)-information flow across 2.7M tweets and compare it with infection rates.They noticed a major Spatio-temporal connection between information flow and new COVID-19 instances, and while there are discussions about myths and connections to low-quality information, their influence is less prominent than other themes specific to the crisis. To find and measure causal relationships between pandemic features (e.g. the number of infections and deaths) and Twitter behaviour and public sentiment, the work in [START_REF] Gencoglu | Causal modeling of twitter activity during covid-19[END_REF] introduced the first example of a causal inference method. Their proposed approach has shown that they can efficiently collect epidemiological domain knowledge and identify factors that influence public interest and attention.

The discussion around the COVID-19 pandemic and the government policies was investigated in [START_REF] Lopez | Understanding the perception of covid-19 policies by mining a multilanguage twitter dataset[END_REF]. They used Twitter data in multiple languages from various countries and found common responses to the pandemic and how they differ across time using text mining. Moreover, they presented insights as to how information and misinformation were transmitted via Twitter. Similarly, to demonstrate the epidemiological effect of COVID-19 on press publications in Bogota, Colombia, [START_REF] Saire | What is the people posting about symptoms related to coronavirus in bogota, colombia?[END_REF] used text mining on Twitter data. They intuitively note a strong correlation between the number of tweets and the number of infected people in the area.

Most of the works described above focus on analyzing tweets related to a single language such as English. Our work has designed a single model leveraging multilingual BERT for the analysis of tweets in multiple languages. Furthermore, we used a large data set to train and analyze the tweets. We aim to provide a system that will not be restricted to any language for analyzing social media data.

The CMTA Method

Both misinformation4 and disinformation 5 , according to the Oxford English Dictionary, are false or misleading information. Misinformation refers to information that is accidentally false and spread without the intent to hurt, whereas disinformation refers to false information that is intentionally produced and shared to cause hurt [START_REF] Hernon | Disinformation and misinformation through the internet: Findings of an exploratory study[END_REF]. Claims do not have to be entirely truthful or incorrect; they can contain a small amount of false or inaccurate information [START_REF] Shahi | Fakecovid-a multilingual cross-domain fact check news dataset for covid-19[END_REF]. This work uses the general notion of misinformation and makes no distinction between misinformation and disinformation as it is practically difficult to determine one's intention computationally.

The data science pipeline phases proposed by CMTA are: tokenizing, text features extraction, linear transformation, and classification. The first phases (tokenizing, text feature extraction, linear transformation) correspond to a substantial data-preparation process intended to build a multi-lingual vectorized representation of texts. The objective is to achieve a numerical pivot representation of texts agnostic of the language. CMTA classification task uses a dense layer and leads to a trained network model that can be used to classify microtexts (e.g. tweets) into three misinformation classes: 'false', 'partly false' and 'misleading'.

Text tokenization Given a multilingual textual dataset consisting of sentences, CMTA uses the BERT multilingual tokeniser to generate tokens that BERT's embedding layer will further process. CMTA uses MBERT6 to extract contextual features, namely word and sentence embedding vectors, from text data7 . In the subsequent CMTA phases that use NLP models, these vectors are used as feature inputs with several advantages. (M)BERT embeddings are word representations that are dynamically informed by the words around them, meaning that the same word's embeddings will change in (M)BERT depending on its related words within two different sentences.

For the non-expert reader, the tokenization process is based on a WordPiece model. It greedily creates a fixed-size vocabulary of individual characters, subwords, and words that best fit a language data (e.g. English) 8 . Each token in a tokenized text must be associated with the sentence's index: sentence 0 (a series of 0s) or sentence 1 (a series of 1s). After breaking the text into tokens, a sentence must be converted from a list of strings to a list of vocabulary indices. The tokenisation result is used as input to apply BERT that produces two outputs, one pooled output with contextual embeddings and hidden-states of each layer. The complete set of hidden states for this model are stored in a structure containing four elements: the layer number (13 layers) 9 , the batch number (number of sentences submitted to the model), the word / token number in a sentence, the hidden unit/feature number (768 features) 10 .

In the case of CMTA, the tokenisation is more complex because it is done for sentences written in different languages. Therefore, it relies on the MBERT model that has been trained for this purpose.

Feature Extraction Phase is intended to exploit the information of hidden-layers produced due to applying BERT to the tokenisation phase result. The objective is to get individual vectors for each token and convert them into a single vector representation of the whole sentence. For each token of our input, we have 13 separate vectors, each of length 768. Thus, to get the individual vectors, it is necessary to combine some of the layer vectors. The challenge is to determine which layer or combination of layers provides the best representation.

Linear convolution

The hidden states from the 12th layer are processed in this phase, applying linear convolution and pooling to get correlation among tokens. We apply a three-layer 1D convolution over the hidden states with consecutive pooling layers. The final convolutional layer's output is passed through a global average pooling layer to get a final sentence representation. This representation holds the relation between contextual embeddings of individual tokens in the sentence.

Classification A linear layer is connected to the model in the end for the CMTA classification task. This classification layer outputs a Softmax value of vector, depending on the output, the index of the highest value in the vector represents the label for the given sequence: 'false', 'partly false' and 'misleading'.

Experiment

To validate CMTA, we designed experiments on Google Colab with 64 GB RAM and 12 GB GPU. For implementing the method, we calibrated pre-trained models provided by hugging face 11 . For our experimental setting, we extracted annotated misinformation data from multiple publicly available open databases. We also collected many multilingual tweets consisting of over 2 million tweets belonging to eight different languages.

Misinformation datasets

We collected data from an online fact-checker website called Poynter [START_REF]The international fact-checking network[END_REF]. Poynter has a specific COVID-19 related misinformation detection program named 'CoronaVirusFacts/DatosCoronaVirus Alliance Database12 '. This database contains thousands of labelled social media information such as news, posts, claims, articles about COVID-19, which were manually verified and annotated by human volunteers (fact-checkers) from all around the globe. The database gathers all the misinformation related to COVID-19 cure, detection, the effect on animals, foods, travel, government policies, crime, lockdown. The misinformation dataset is available in 2 languages-'English' and 'Spanish'.

We crawled through the content of two websites using Beautifulsoup 13 , a Python library for scraping information from web pages. We scraped 8471 English language false news/information belonging to nine classes, namely, 'False', 'Partially false', 'Misleading', 'No evidence', 'Four Pinocchios', 'Incorrect', 'Three Pinocchios', 'Two Pinocchios' and 'Mostly False'. We gathered the article's title, its content, and the fact checker's misinformation-type label for each article.

For Spanish 14 , we collected 531 misinformation articles. The collected data contains the misinformation published on social media platforms such as Facebook, Twitter, Whatsapp, YouTube. Posts were mostly related to politicalbiased news, scientifically dubious information and conspiracy theories, misleading news and rumours about COVID-19. We also used a human-annotated fact-checked tweet dataset [START_REF] Alam | Fighting the covid-19 infodemic: Modeling the perspective of journalists, fact-checkers, social media platforms, policy makers, and the society[END_REF] available at a public repository 15 . The dataset contained true and false labelled tweets in English and Arabic language. We used only false labelled tweets consisting of 500 English. We compiled a total of 9,502 micro-articles distributed across 9 misinformation classes shown in Table 1. Dataset Pre-processing The datasets contained noise such as emojis, symbols, numeric values, hyperlinks to websites, and username mentions that were needed to be removed. Since our dataset was multilingual, we had to be very careful while preprocessing as we did not want to lose any valuable information. To preprocesses the training and inference datasets, we used simple regular expressions to remove URLs, special characters or symbols, blank rows, re-tweets, user mentions. We did not remove the hashtags from the data as hashtags might contain helpful information. For example, in the sentence-'Wear mask to protect yourself from #COVID-19 #corona', only the symbol '#' was removed. We removed stop words using NLTK 16 . NLTK supports multiple languages except for few languages, such as Hindi and Thai. For preprocessing the Hindi dataset, we used CLTK (Classical Language Toolkit) 17 . For removing Thai stop words from Thai tweets, we used PyThaiNLP [START_REF] Phatthiyaphaibun | PyThaiNLP: Thai Natural Language Processing in Python[END_REF]. The emojis were removed using their Unicode.

Attribute engineering of the training dataset Table 2 gives an overview of the training dataset and showcase some misinformation articles. Column 2 shows the original label assigned by the fact-checker, column 3 gives a misinformation example associated with the label present in column 2, and column 4 provides reasoning given by the fact-checker behind assigning a particular label (column 2) to the misinformation (column 3). For example, if we look at the entry number '3' in the table 2, the misinformation is about the adverse effect of 5G radiation over the COVID-19 patients. This entry was labelled 'Incorrect' by the factchecker. After analysing the fact-checker rating and the explanation given, we labelled it as 'False' misinformation. Entry number '5' talks about the COVID-19 test cost. The explanation given by the fact-checker is valid as it is not sure if there is any fee in the USA for the COVID-19 test or not. So because of the lack of evidence and uncertainty, we labelled it as 'Partially false'. Entry number '7' in the table talks about a video showing COVID-19 corpus dumping in the sea.

Based on the explanation, the video was coupled with the wrong information to mislead the audience. So it was labelled as 'Misleading' misinformation. The collected data is unevenly distributed across nine classes: 'No evidence', 'Four Pinocchios 18 ', 'Incorrect', 'Three Pinocchios 19 ', 'Two Pinocchios 20 ' and 'Mostly False' (the smallest group). Most collected articles were labelled either as 'False', 'Partially false' and 'Misleading'. We performed an attribute engineering phase for preparing the dataset. We produced a uniformly distributed dataset reorganised the initial dataset as follows. The classes 'Four Pinocchios' and 'Incorrect' were merged with the class 'False'. The classes 'Three Pinocchios' and 'Two Pinocchios' were merged into the class 'Partially false'. The classes 'No evidence' and 'Mostly False' were merged with the class 'Misleading'. Finally, column 1 (see table 2) corresponds to the label assigned during the attribute engineering phase.

Inference Dataset For building an inference dataset to be used to test the CMTA trained model for analysing the misinformation spread across all over the social media platforms in multiple languages, We collected around 2,137,106 multilingual tweets. The tweets were expressed in eight major languages, namely-'English', 'Spanish', 'Indonesian', 'French', 'Japanese', 'Thai', 'Hindi' and 'German'. Therefore, we used a dataset of tweets IDs associated with the novel coronavirus COVID-19 [START_REF] Chen | Tracking social media discourse about the covid-19 pandemic: Development of a public coronavirus twitter data set[END_REF]. Starting on January 28, 2020, the current dataset contains 212,978,935 tweets divided into groups based on their publishing month. The dataset was collected using multilingual COVID-19 related keywords and con-

No Evidence

Media uses photos of puppets on patient stretchers to scare the public.

There is no evidence that any media outlet used this photo for their reporting about COVID-19. Its origin is unclear, maybe it was shot in Mexico and shows a medical training session.

Mostly False

Coronavirus does not affect people with 'O+' blood type.

The post claiming coronavirus does not affect people with 'O+' blood type is misleading.

tained tweets in more than 30 languages. We used tweepy 21 which is a Python module for accessing Twitter API. We decided to retrieve the tweets using the tweet IDs published in the past five months (February, March, April, May and June) for our analysis. Table 3 shows the total number of collected tweets. The distribution of tweets across eight languages corresponds to most English items, almost 1 and 1/8 of the whole data set, then Spanish (1/4 of the total number of tweets) and the rest for French, Japanese, Indonesian, Thai, and Hindi.

Model Setup and Training

Training Setting For training our model, we divided the data into training, validation and testing datasets in the ratio of 80%/10%10% respectively. The final count for the train, validation and test dataset was 7,602, 950, 950. We fine-tuned the Sequence Classifier from HuggingFace based on the parameters specified in [START_REF] Devlin | Bert: Pre-training of deep bidirectional transformers for language understanding[END_REF]. Thus, we set a batch size of 32, learning rate 1e-4, with Adam Weight Decay as the optimizer. We run the model for training for 10 epochs. Then, we save the model weights of the transformer, helpful for further training. iterations on distinct sets of hyper-parameters based on the model's performance analysis, we adopted the one showing promising results on our dataset.

Hyperparameters' Setting

Experiment and Results

We experimented with the multilingual data with their respective linguisticbased BERT models. We set the model with the same training parameters as the CMTA model and preprocessed the data as stated previously. Each monolingual model was fine-tuned for 10 epochs with a batch size of 32, and it was applied to the classification dataset of their respective language. Our model achieved an accuracy(%) of 82.17 (see figure 1)and F 1 score (%) of 82.54 on the test dataset. The precision and recall reported by the model were 82.07 and 82.30 respectively.

Table 5 shows the model's prediction over few examples from the test dataset along with their actual label. As we have shown in the table, in the entry numbers '1', '2', '3' and '4' our model could predict the correct label. However, in the case of entry number '5', the label predicted by our model was 'False', whereas the actual label is 'Misleading'. If we would look at the misinformation at the entry number '5', which is a Spanish text-'El medicamento contra piojos sirve como tratamiento contra Covid-19.' and the corresponding English translation would be-". This misinformation claims about a COVID-19 medicine, and since this could be 'false' and 'misleading' misinformation at the same time, our model predicted it as 'false' misinformation rather than 'misleading'.

Results Assessment

We used two strategies for assessing CMTA results according to the research questions we wanted to prove. The first research question was to determine (Q 1 ) whether it was possible to develop a method that could provide a general multi-lingual classification pipeline? The second question was (Q 2 ) whether it is possible to build conclusions about how misinformation on COVID-19 spreads in different language speaking communities by analysing micro-texts published in social media. For Q 1 , we conducted a comparative study of CMTA with different independent mono-lingual misinformation classifiers. Thereby, CMTA's classification performance for a given set of micro-texts written in a given language was compared against a classification model targeting only that language. For Q 2 we analysed and plotted CMTA's results, and we proposed intuitive arguments according to our observations.

CMTA vs Monolingual Classification

We conducted a comparative performance study of various monolingual BERT models concerning our proposed multilingual CMTA model for comparing their performance for the misinformation detection task. We investigated eight monolingual BERT model22 , namely, 'English', 'Spanish', 'French', 'Germann', 'Japanese', 'Hindi' 'Thai23 ' and 'Indonesian'. We used the same 9,502 tweets distributed across three misinformation classes for training the monolingual models. Our dataset consisted of English and Spanish tweets; therefore, we translated the tweets into eight languages to train each of the eight monolingual models. We used Google Translator API 24 for converting the tweets into a particular language. Results are shown in Table 6. Based on the experiment results, we strongly suggest that the multilingual CMTA model could generalize smoothly on the dataset because its performance was equivalent to the monolingual models.

Multilingual Misinformation Analysis

We provide a detailed analysis of misinformation distribution across multilingual tweets. This analysis responds to the initial question: how is misinformation about COVID-19 spread in communities speaking different languages. Our survey studied and analyzed the distribution of COVID-19 misinformation across eight significant languages (i.e. 'English', 'Spanish', 'Indonesian', 'French', 'Japanese', 'Thai', 'Hindi' and 'German') for five months (i.e. February, March, April, May and June). We used our trained, multilingual model, CMTA, to predict and categorize the misinformation type present in tweets. We conducted our sequential misinformation analysis on a collection of over 2 million multilingual tweets. Figure 2 shows the month-wise distribution of misinformation types for each language. It showcases the overall (all 5 months together) spread of misinformation types across each language. We could see that German tweets have the highest number of 'Misleading' tweets, whereas French have the least. Spanish tweets beat other language's tweets by becoming the largest source of 'False' misinformation. Germany generated the least number of 'False' tweets. Hindi tweets tend to have the highest number of 'Partially false' tweets, whereas Thai have the least. We could observe that for February, March and June months, our model predicted a large number of tweets as 'False', followed by 'Misleading', which is the second largest and the number of 'Partially false' was the least (see Figure 3). Our model discovered that the number of 'Partially false' tweets are more than 'Misleading' tweets and 'False' tweets were again in the majority for the tweets generated during April and May.

Table 7 and 8 present a detailed count of misinformation classes across all the languages. The following specific observations were made concerning the languages: The misinformation distribution for English data indicates that there is a majority of False tweets during the five months, whereas the distribution of Misleading labelled data is slightly less than as compared to False labelled data. Partially False labelled tweets are moderately distributed, as in the month April, we can see that there is a more significant number concerning other months. According to the language wise-distribution shown in Figure 2, Spanish tweets have a greater frequency of False labelled tweets, whereas the Misleading tweets and Partially False tweets shows the almost identical number of tweet across the five months. There was a surge of Misleading labelled tweets during February, and the count remained the same throughout the five months. There was also an increase in Partially False tweets in March, but it decreased in successive months, leading to minor False labelled tweets. According to the language wise-distribution shown in Figure 2, on average throughout the five months, approx 20% of Japanese tweets are labelled False. Similarly, approx 30% of the Japanese tweets are labelled Partially False, leading to the majority of 50% data are labelled as Misleading. We can also see a considerable increase in Misleading tweets in March, tweeted in the Japanese language. According to the distribution of Indonesian tweets, approximately 10% of tweets are labelled as Misleading, and on the contrary, there is a large distribution of False labelled tweets. Approximately 34% of the Indonesian dialect data is labelled as Partially False throughout the five months. Figure2 shows the misinformation distribution across all five months in the French tweets. The largest majority of the tweets were classified as False misinformation. Among Partially false and Misleading, the least number of tweets were labelled as Misleading. The frequency of Hindi tweets is low in the dataset used in our experiment. However, our model can predict or label Hindi tweets. Tweets in Hindi have low numbers of Misleading tweets, whereas the Partially False tweets class has a great frequency. False labelled tweets are slightly low compared to Partially False tweets in this dialect. The distribution of Thai tweets shows that our model prediction is majorly oriented towards the Misleading tweets. The distribution of Misleading labelled tweets is the greatest among the labelled classes, in contrast to Partially False tweets. False labelled tweets are comparatively moderate in this language.

Conclusion and Future Work

This paper introduced CMTA, a multilingual model for analyzing text applied to classify COVID-19 related multilingual tweets into misinformation categories. We demonstrated that our multilingual CMTA framework performed significantly well compared to the monolingual misinformation detection models used independently. Experimental validation of CMTA detected misinformation distribution across eight significant languages. The paper presented a quantified magnitude of misinformation distributed across different languages in the last five months. Our future work aims to collect more annotated training data and perform analysis of a larger multilingual dataset to gain a deeper understanding of misinformation spread. We are currently improving our model's robustness and contextual understanding for better performance in the classification task. We hope that researchers could gain deeper insights about misinformation spread across major languages and use the information to build more reliable social media platforms through our work.

Fig. 1 :

 1 Fig. 1: Training Accuracy(Upper) and Training loss(Lower)

Fig. 2 :

 2 Fig. 2: Month-wise Disinformation Distribution in Languages.

Table 1 :

 1 Collected Misinformation Data set.

	Classes	Number of tweets
	False [18] (English)	2,869
	Partially False (English) 2,765
	Misleading (English)	2,837
	False (Spanish)	191
	Partially False (Spanish) 161
	Misleading (Spanish)	179
	False [1] (English)	500
	Total	9,502

Table 2 :

 2 Misinformation Dataset

	Our Rating IFCN(Poynter) Rating Misinformation	Explanation
	False	False	The border between France and Belgium will be closed.	French and Belgian authorities denied it.
				There was no "Obama rule," just
		Four pinocchios	Trump's effort to blame Obama for sluggish coronavirus testing.	draft guidance that never took effect and was withdrawn before
				President Trump took office.
			Elisa Granato, the first volunteer	Elisa Granato, the first volunteer
		Inaccurate	in the first Europe human tria	in the first Europe human trial
			of a COVID-19 vaccine, has died.	of a COVID-19 vaccine, has died.
	Partially False	Partially False	Media shows a Florida beach full of people while it's empty.	The different videos were not shot at the same time. The beaches are empty when they are closed.
		Two Pinocchios	The bill for a coronavirus test in the US is $3.000	The CDC is not making people pay the test by now.
				"Consuming fruit juices or gargling
			Salty and sour foods cause	with warm water and salt does not
		Partly False	the "body of the COVID-19 virus"	protect or kill COVID-19," the
			to explode and dissolve.	World Health Organization
				Philippines told VERA Files.
				Misbar's investigation of the video
			A clip from Mexico depicts	revealed that it does not depict the
	Misleading	Misleading	the dumping of coronavirus	dumping of coronavirus patients
			patients corpses into the sea.	corpses in Mexico, but rather paratroopers
				landing from a Russian MI 26 helicopter.

Table 3 :

 3 Table 4 lists every hyperparameter for training and testing our model. All the calculations and selection of hyperparameters were made based on tests and the model's best output. After performing several Language-wise Dataset Distribution

	Language ISO Number of tweets
	English	en	1,472,448
	Spanish	es	353,294
	Indonesian in	80,764
	French	fr	71,722
	Japanese	ja	71,418
	Thai	th	36,824
	Hindi	hi	27,320
	German	de	23,316
	Sum		2137106

Table 4 :

 4 Hyper-parameters for training

	Parameters	Value
	Pool Size of Average Pooling 8
	Pool Size of Max Pooling	8
	Dropout Probability	0.36
	Number of Dense layers	4
	Text Length	128
	Batch Size	32
	Epochs	10
	Optimizer	Adam
	Learning Rate	1 × 10 -4

Table 5 :

 5 Misinformation data examples along with model's prediction and actual label

	Test Data	Actual Label Prediction Accuracy(/)
	Dr. Megha Vyas from India died due to
	COVID-19 while treating COVID patients. False	False
	El plátano bloquea "la entrada celular
	del COVID-19"	False	False
	Asymptomatic people are very rarely
	contagious, said the WHO.	Partially False Partially False
	Patanjali Coronil drops can help cure
	coronavirus.	Misleading	Misleading
	El medicamento contra piojos sirve como
	tratamiento contra Covid-19.	Misleading	False

Table 6 :

 6 Precision, Recall and f-score of CMTA model

	Models	Metrics Precision Recall F1-score
	EnglishBERT	82.03	74.18	77.90
	SpanishBERT	80.9	72.02	76.20
	CamemBERT	81.91	71.45	76.32
	GermanBERT	80.61	71.43	75.74
	JapaneseBERT	79.56	65.36	71.76
	HindiBERT	79.56	65.68	71.95
	ThaiBERT	79.11	66.25	72.11
	IndonesianBERT	78.96	65.66	71.69
	CMTA		81.52	74.40	77.79

Table 7 :

 7 Language-wise predicted misinformation labels of tweets in February, March and April.

https://www.oed.com/view/Entry/119699?redirectedFrom=misinformation

https://www.oed.com/view/Entry/54579?redirectedFrom=disinformation

https://github.com/google-research/bert/blob/master/multilingual.md

Embeddings are helpful for keyword/search expansion, semantic search and information retrieval. They help accurately retrieve results matching a keyword query intent and contextual meaning, even in the absence of keyword or phrase overlap.

[START_REF] Frenkel | Surge of virus misinformation stumps facebook and twitter[END_REF] This vocabulary contains whole words, subwords occurring at the front of a word or in isolation (e.g., "em" as in the word "embeddings" is assigned the same vector as the standalone sequence of characters "em" as in "go get em"), subwords not at the front of a word, which are preceded by '##' to denote this case, and individual characters [

25]9 It is 13 because the first element is the input embeddings, the rest is the outputs of each of BERT's 12 layers.

https://huggingface.co/

https://www.poynter.org/covid-19-poynter-resources/

Pretrained model available at https://huggingface.co/models

ThaiBERT is available at https://github.com/ThAIKeras/bert

Please refer https://cloud.google.com/translate/docs