Sandy Mahfouz
email: sandy.mahfouz@utt.fr

Farah Mourad-Chehade

Paul Honeine
email: paul.honeine@univ-rouen.fr

Joumana Farah
email: farah.mourad@utt.fr

Hichem Snoussi
email: hichem.snoussi@utt.fr

Non-parametric

Keywords: Distance estimation, Kalman filter, machine learning, particle filter, radio-fingerprints, RSSI, target tracking. I

come

time difference of arrival [START_REF] Wendeberg | Robust tracking of a mobile beacon using time differences of arrival with simultaneous calibration of receiver positions[END_REF] and time-of-arrival [START_REF] Xu | Target tracking and mobile sensor navigation in wireless sensor networks[END_REF]. The RSSIbased techniques exploit the attenuation of the signal strength with the traveled distance to estimate the distances separating the emitters from the receivers. These techniques exhibit favorable properties with respect to power consumption, size and cost. Nevertheless, it is important to highlight that distance estimation using RSSI can be really challenging, since the measurements of signals' powers could be significantly altered by the presence of additive noise, multi-path fading, shadowing, and other interferences. One popular model used to characterize the relationship between the RSSIs and the distances is the log-distance propagation model, also known as the Okumura-Hata model [START_REF] Medeisis | On the use of the universal Okumura-Hata propagation prediction model in rural areas[END_REF], [START_REF] Patwari | Locating the nodes: cooperative localization in wireless sensor networks[END_REF], [START_REF] Zanella | Rss-based ranging by multichannel rss averaging[END_REF]. Even though this model has many limitations, it is still widely used because of its simplicity. However, this model is basically for outdoors, since it predicts the signal strength without taking the surrounding environment into account, such as the walls and the floors. Therefore, it becomes inaccurate in cases where there is no line of sight between sensors. Different models have been proposed to overcome this problem, such as modified versions of the logdistance model [START_REF] Sandeep | Wireless network visualization and indoor empirical propagation model for a campus wi-fi network[END_REF], in which the attenuations due to floors and walls are explicitly included. Several researchers have determined a mathematical relationship between RSSIs and distances, without taking physical properties into consideration [START_REF] Yang | Indoor localization using improved rss-based lateration methods[END_REF]. To this end, an empirical model is estimated with a polynomial regression.

Once the distances are estimated from the RSSIs, they are then combined using for instance triangulation [START_REF] Esteves | Generalized geometric triangulation algorithm for mobile robot absolute self-localization[END_REF] or trilateration [START_REF] Manolakis | Efficient solution and performance analysis of 3-D position estimation by trilateration[END_REF] to find the sensors' positions. One can also use methods based on filtering, since they can help in smoothing the target's trajectory, reducing thus the estimation error. One widely used filter is the Kalman filter [START_REF] Welch | An Introduction to the Kalman Filter[END_REF]. However, it is known that this filter is only reliable for systems which are almost linear. To overcome the non-linear estimation problem, the extended Kalman filter (EKF) and unscented Kalman filter (UKF) have been proposed. A tracking approach using the EKF with the distances and acceleration information is proposed in [START_REF] Shareef | Localization using extended kalman filters in wireless sensor networks[END_REF]. The main disadvantage of such an approach is that the linearization and the approximations made might lead to sub-optimal performance and sometimes divergence of the filter [START_REF] Umamageswari | A comparitive study of Kalman filter, extended Kalman filter and unscented Kalman filter for harmonic analysis of the non-stationary signals[END_REF]. Another tracking approach using the Kalman filter is described in [START_REF] Chan | Using wi-fi signal strength to localize in wireless sensor networks[END_REF], where the authors first find the target's positions using the weighted K-nearest neighbor (WKNN) algorithm. Then, they combine these estimates with acceleration information using the Kalman filter with a secondorder state-space model. The authors of [START_REF] Mahfouz | Target tracking using machine learning and kalman filter in wireless sensor networks[END_REF] also propose a tracking method that uses the Kalman filter with a thirdorder state-space model to enhance the position of the target estimated using a kernel-based model proposed by the authors.

Another well-known filter used for tracking is the particle filter (PF) [START_REF] Branko Ristic | Beyond the Kalman Filter: Particle Filters for Tracking Applications[END_REF]. The tracking methods described in [START_REF] Wang | Wlanbased pedestrian tracking using particle filters and low-cost mems sensors[END_REF], [START_REF] Farmani | A hybrid localization approach in wireless sensor networks using a mobile beacon and internode communication[END_REF] take advantage of such filter, by using the target's motion and the distances as well.

In this paper, we propose two original regression models that estimate the distances separating sensors using their received signal strength indicators (RSSIs), in an indoor wireless sensor network. We then propose to solve the tracking problem using two new methods. Our approach consists first of a collection phase, aiming at building a training database composed of RSSI measures and distances based on the radio-fingerprinting concept [START_REF] Robles | 3D fingerprint-based localization for wireless sensor networks[END_REF]. The collection phase is then followed by an offline training phase where the distance models are defined using the collected database. Defining the distance models is an essential step in the tracking process; in fact, in the following, the estimated distances obtained using one of the proposed distance models will be used to determine the target's position. The first proposed distance model is a non-parametric model without any prior knowledge on the physical system, while the second one is a semi-parametric regression model, combining the well-known log-distance theoretical propagation model with a non-linear fluctuation term, estimated within the framework of kernel-based machines. Each model takes the RSSIs as input and yields the distance to the corresponding sensor as output. To the best of our knowledge, both models have not yet been proposed in the literature. Then, in the online tracking phase, a moving target collects RSSI measures and uses them with the computed kernel-based models to estimate the distances separating it from the sensors having known locations. Based on these distances, the target's position is estimated using two different methods. The first one consists of combining these distance estimates with the acceleration information, by means of a Kalman filter [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF], [START_REF] Welch | An Introduction to the Kalman Filter[END_REF], to determine the target's position. The second one consists of localizing the target using a particle filter [START_REF] Branko Ristic | Beyond the Kalman Filter: Particle Filters for Tracking Applications[END_REF] with the distance estimates and the acceleration information. Compared to [START_REF] Mahfouz | Kernel-based localization using fingerprinting in wireless sensor networks[END_REF] and [START_REF]Decentralized localization using fingerprinting and kernel methods in wireless sensor networks[END_REF], the proposed methods take advantage of the target's mobility to enhance the obtained position estimate. Compared to [START_REF] Mahfouz | Target tracking using machine learning and kalman filter in wireless sensor networks[END_REF], where a kernel-based position model is used, we propose in this paper two kernel-based distance models to find the distances separating the target from the sensors. Because of the logarithmic relationship between the RSSIs and the distances, the use of distance models gives more relevant results than position models. Moreover, the most important reason for switching to distance models is that, by doing so, the sensors are allowed to move in the network, without affecting the tracking as it will be shown throughout this paper.

The rest of the paper is organized as follows. The proposed tracking methods are described in Section II. In Section III, we define the two kernel-based distance models using machine learning. In Section IV, we solve the tracking problem using the distances and acceleration information by means of the Kalman filter, while in Section V, we find the target's positions using the particle filter. In Section VI, we examine the performance of the two proposed distance models, and in Section VII, we evaluate the proposed tracking methods, while providing a comparison to state-of-the-art tracking methods. Finally, Section VIII concludes the paper.

II. DESCRIPTION OF THE TRACKING APPROACH

Consider an environment of δ dimensions, with for instance δ = 2 for a two-dimensional environment, and N s sensors having known locations. The sensors are allowed to move in the network, and their locations are considered to be known at any given time step. For simplicity, we denote the sensors' locations by s i = (s i,1 . . . s i,δ ) ⊤ , i ∈ {1, . . . , N s }. In the collection phase, in order to construct a radio-fingerprints database, N reference positions, denoted by p ℓ = (p ℓ,1 . . . p ℓ,δ ) ⊤ , ℓ ∈ {1, . . . , N }, are either generated on a uniform grid or randomly in the studied environment. The sensors continuously broadcast signals in the network at a fixed initial power, and a sensor is placed consecutively at the reference positions to detect the broadcasted signals and measure their RSSIs. Let ρ si,p ℓ , i ∈ {1, . . . , N s }, denote the power of the signal received from the sensor at position s i by the sensor at reference position p ℓ , and let d s i ,p ℓ , i ∈ {1, . . . , N s }, denote the distance separating these sensors. The distance d s i ,p ℓ is then given by:

d si,p ℓ = δ v=1 s i,v -p ℓ,v 2 . (1)
In this way, N s training sets of N pairs (ρ si,p ℓ , d si,p ℓ ) are obtained. The training phase consists then in finding a set of N s models χ i : IR → IR, i ∈ {1, . . . , N s }; each model takes as input the RSSI received from the sensor s i at a reference position p ℓ , ℓ ∈ {1, . . . , N }, and yields as output the distance d s i ,p ℓ separating s i from p ℓ . These distance models can be non-parametric or semi-parametric; details will be provided in the following section. Note that the database construction and the computation of the models χ i are performed only once at a computation center, before the tracking phase, regardless of the chosen model. The models are then communicated to the target, that performs all the following computations. Let x(k) = (x 1 (k) . . . x δ (k)) ⊤ denote the target's position at a given time step k. Then, in the tracking phase, the moving target collects the RSSIs of the signals received from the N s sensors, at time step k, and stores them into the vector ρ x(k) = (ρ s1,x(k) . . . ρ sN s ,x(k) ) ⊤ . An estimate of the distance d si,x(k) , i ∈ {1, . . . , N s }, separating the sensor at position s i from the target is then obtained using the already-defined distance model χ i , as follows:

d si,x(k) = χ i (ρ si,x(k) ).
(

) 2
The target is assumed to be equipped with a gyroscope and an accelerometer that yield respectively its instantaneous orientation and accelerations. Combining this information leads to the instantaneous accelerations of the target over the δ coordinates in the network coordinates system. We propose then to solve the positioning problem in two ways. The first solution is given in Section IV, where the distance estimates are combined with the accelerometer information to obtain an estimate of the target's positions using the Kalman filter. The second solution is given in Section V, where the target's position estimates are obtained by combining the accelerations and the distance estimates using the particle filter.

III. DEFINITION OF χ i USING KERNEL METHODS

The objective of this section is to define the set of models χ i , i ∈ {1, . . . , N s }, using the constructed training database. Each model χ i is defined in such a way that it associates to each RSSI measure ρ s i ,p ℓ , ℓ ∈ {1, . . . , N }, the corresponding distance d si,p ℓ . To this end, we propose two distance models. In the first subsection, we define the set of models χ i , i ∈ {1, . . . , N s }, as non-parametric models, i.e., they are taken as linear combinations of kernels centered on the training samples. The second models, described in the second paragraph of this section, are semi-parametric regression models that combine the physical log-distance propagation model of [START_REF] Medeisis | On the use of the universal Okumura-Hata propagation prediction model in rural areas[END_REF], [START_REF] Patwari | Locating the nodes: cooperative localization in wireless sensor networks[END_REF] with a non-linear fluctuation term, defined in a reproducing kernel Hilbert space. This non-linear term compensates for the missing factors in the log-distance model, therefore allowing a better modeling of the RSSI/distance relationship.

A. Non-parametric regression models

In this paragraph, the distance model is derived using only the training database. No assumptions on the model or prior knowledge of the model are considered. Therefore, determining χ i requires solving a non-linear regression problem. We take advantage of kernel methods, that have been remarkably successful for solving non-linear regression problems. To this end, we consider the kernel ridge regression [START_REF] Saunders | Ridge regression learning algorithm in dual variables[END_REF] to determine the functions χ i .

Each function χ i is determined by minimizing the mean quadratic error between the model's outputs χ i (ρ s i ,p ℓ ) and the desired outputs d s i ,p ℓ , as follows:

min χi∈H 1 N N ℓ=1 (d s i ,p ℓ -χ i (ρ s i ,p ℓ )) 2 + η i χ i 2 H , (3)
where H is a reproducing kernel Hilbert space, and η i is a regularization parameter that controls the tradeoff between the training error and the complexity of the solution. According to the representer theorem [START_REF] Schölkopf | A generalized representer theorem[END_REF], the optimal function can be written as follows:

χ i (•) = N ℓ=1 α ℓ,i κ(ρ s i ,p ℓ , •), (4)
where " • " is the function's input, κ : IR × IR → IR is a reproducing kernel, and α ℓ,i , ℓ ∈ {1, . . . , N }, are parameters to be determined. Let α i be the N ×1 vector whose ℓ-th entry is α ℓ,i . By injecting (4) in (3), a dual optimization problem in terms of α i is obtained, whose solution is given by taking the derivative of the corresponding cost function with respect to α i and setting it to zero. One can easily find the following form of the solution:

α i = (K i + η i N I N ) -1 d s i , (5)
where

I N is the N × N identity matrix, d s i = (d s i ,p 1 . . . d s i ,p N ) ⊤ , i ∈ {1, .
. . , N s }, and K i is the N × N matrix whose (u, u ′ )-th entry is κ(ρ si,p u , ρ si,p u ′ ), for u, u ′ ∈ {1, . . . , N }. For appropriate values of the regularization parameter η i , the matrix between parentheses is always nonsingular.

Notice that this non-parametric model does not consider any prior knowledge on the physical properties of the relationship between the RSSIs and the distances. Indeed, the model does not take into account the fact that this RSSI/distance relationship is logarithmic. In the following subsection, we propose a semi-parametric regression model in such a way that the prior knowledge of the logarithmic RSSI/distance relationship is taken into consideration.

B. Semi-parametric regression models

As we already stated, the theoretical log-distance signal propagation model alone is insufficient to characterize the RSSI/distance relationship, since it neglects all other factors in the environment, such as physical obstacles, multi-path propagation, additive noises, etc. This model characterizes the RSSI/distance relationship as follows:

ρ s i ,p ℓ = ρ 0i -10 n P log 10 d si,p ℓ d 0i , (6)
where ρ s i ,p ℓ is the power received from the sensor at position s i by the node at position p ℓ , that is the i-th entry of the vector ρ ℓ , ρ 0i is the power at the reference distance d 0i , and n P is the path-loss exponent. In this paragraph, we propose a new propagation model by adding to this model a nonlinear fluctuation term ϕ i , that represents a combination of all unknown factors affecting the RSSI measures, as follows:

ρ si,p ℓ = ρ 0i -10 n P log 10 d s i ,p ℓ d 0i + ϕ i , (7)
This term provides the log-distance physical model with further flexibility, resulting in a more accurate model. Equation ( 7) can be written as follows:

log 10 d s i ,p ℓ = ρ 0i 10 n P + log 10 d 0i - ρ si,p ℓ 10 n P + ϕ i 10 n P . (8)
One can see that ( 8) is a combination of a linear model in terms of ρ si,p ℓ and a non-linear model. Let ψ i (•) denote the model that associates to each RSSI ρ s i ,p ℓ the logarithm of the distance log 10 (d s i ,p ℓ ). This model ψ i (•) can be decomposed into two terms: a linear term and a non-linear fluctuation term, such that:

ψ i (•) = ψ i,lin (•) + ψ i,nlin (•),
where we have the following:

ψ i,lin (ρ si,p ℓ ) = a i ρ si,p ℓ + b i , ψ i,nlin (ρ s i ,p ℓ ) = ϕi 10 nP , (9)
with a i = -1 10 nP and b i = ρ0i 10 nP + log 10 d 0i . We now consider that the non-linear term, given by ψ i,nlin (•), lies in a reproducing kernel Hilbert space denoted by H and induced by a positive definite kernel function κ(•, •). It is estimated by minimizing the regularized mean quadratic error on the training set, given by:

1 N N ℓ=1 ε 2 ℓ + η i ψ i,nlin 2
H , (10)
where the quantity η i is a regularization parameter that controls the tradeoff between the training error and the complexity of the solution, and the cost function ε ℓ,i is given by:

ε ℓ,i = log 10 d ℓ -a i ρ s i ,p ℓ -b i -ψ i,nlin (ρ s i ,p ℓ ),
where ℓ ∈ {1, . . . , N }. From the semi-parametric representer's theorem [START_REF] Schölkopf | A generalized representer theorem[END_REF], ψ i,nlin (•) can be written as a linear combination of kernels:

ψ i,nlin (•) = N ℓ=1 β ℓ,i κ(ρ si,p ℓ , •),
where κ : IR × IR → IR, and β ℓ,i , with ℓ ∈ {1, . . . , N }, are parameters to be estimated. By multiplying [START_REF] Patwari | Locating the nodes: cooperative localization in wireless sensor networks[END_REF] by N for convenience and writing it in matrix form, one gets the following:

Λ ⊤ i Λ i + a 2 i ρ ⊤ i ρ i + N b 2 i + β ⊤ i K ⊤ i K i β i -2a i Λ ⊤ i ρ i -2b i 1 ⊤ Λ i -2Λ ⊤ i K i β i + 2a i b i 1 ⊤ ρ i + 2a i ρ ⊤ i K i β i + 2b i 1 ⊤ β i + η i N β ⊤ i K i β i , (11)
where

β i = (β 1,i . . . β N,i ) ⊤ , K i is the N ×N matrix whose (u, u ′ )-th entry is κ(ρ si,p u , ρ si,p u ′ ), for u, u ′ ∈ {1, ..., N }, ρ i
is the N × 1 vector whose ℓ-th entry is equal to ρ s i ,p ℓ , with ℓ ∈ {1, . . . , N }, 1 is the vector of ones of appropriate size, and Λ i is the N×1 vector whose ℓ-th entry is equal to log 10 d si,p ℓ . By taking the partial derivatives in matrix form of [START_REF] Zanella | Rss-based ranging by multichannel rss averaging[END_REF] with respect to a i , b i and β i and setting them to zero, we get the following linear system having the form U i = W i Y i , where

Y i = a i b i β i , U i =   1 ⊤ Λ i Λ i Λ ⊤ i ρ i   , W i =   1 ⊤ ρ i ρ i ρ ⊤ i ρ i N 1 1 ⊤ ρ i 1 ⊤ K i K i + η i N I N ρ ⊤ i K i   ( 12
)
The solution is then given by:

Y i = (W ⊤ i W i ) -1 W ⊤ i U i . (13)
After computing the model's parameters using [START_REF] Yang | Indoor localization using improved rss-based lateration methods[END_REF], one can find the logarithm of any distance separating a sensor at position s i and the target at position x(k) in the network using only the RSSI information, as follows:

log 10 d si,x(k) = ψ i (ρ si,x(k) ) = a i ρ s i ,p ℓ + b i + N ℓ=1 β ℓ κ(ρ s i ,p ℓ , ρ si,x(k) ).
Then, the model χ i (•) yields the distance as follows:

χ i (ρ si,x(k) ) = 10 ψi(ρ s i ,x(k) ) . (14)
IV. TRACKING USING THE KALMAN FILTER In this section, we propose to find the target's position by combining its instantaneous accelerations with the distances separating it from the sensors in the network, by means of a Kalman filter. These distances are estimated using either one of the kernel-based distance models defined in Section III The first and most challenging part is to find a model that describes the target's motion. To overcome cases where the target does not follow a predictable path, we propose to use a third-order state-space model. This model is based on the assumption that the target's accelerations vary linearly between any two consecutive time steps k -1 and k with a slope equal to γ(k)-γ(k-1) ∆t , where γ(k) = (γ 1 (k) . . . γ δ (k)) ⊤ is the acceleration vector at time step k, and ∆t the time period separating two consecutive time steps. Accordingly, one can estimate recursively the velocity vector ν(k) of the target at time step k as follows:

ν(k) = ν(k-1)+γ(k-1) ∆t+ γ(k) -γ(k -1) ∆t ∆t 2 2 . (15)
Then, by taking the primitive integral of ( 15), the position of the target can be written as follows:

x(k) = x(k -1) + ν(k -1) ∆t + γ(k -1) ∆t 2 2 + γ(k) -γ(k -1) ∆t ∆t 3 6 . (16)
Assuming that the target is at a fixed known position x(0) at the beginning of the tracking and having ν(0) and γ(0) null, one can then find x(k) at any time step k using the measured accelerations. This model yields an accurate approximation of x(k) for small values of ∆t. Now let X(k) = (x(k) ν(k)) ⊤ denote the unknown state of the target at time step k. The considered state-space equation is then given by the following:

X(k) = A X(k -1) + B(k) + ǫ KF (k), ( 17
)
where A is the state transition matrix and B(k) is a noisy control-input vector depending on the accelerations given by:

A = I δ 0 ∆tI δ I δ B(k) = γ(k -1) ⊤ ∆t 2 3 γ(k -1) ⊤ ∆t 2 + γ(k) ⊤ ∆t 2 6 γ(k) ⊤ ∆t 2 ,
where I δ is the δ × δ identity matrix, and 0 is the matrix of zeros of appropriate size. As for the quantity ǫ KF (k), it is the state equation error, whose probability distribution is assumed to be normal, having zero mean and covariance matrix V of size 2δ × 2δ.

Remark. In the previous paragraphs, the target is assumed to be rotationally constrained. However, during its motion in real applications, the target could rotate, leading to changes in the coordinate system, where the accelerations are given. The solution to this problem is to equip each target with a gyroscope, which yields its orientations with respect to the world coordinate system. At each time step, the target measures its acceleration vector in its coordinate system, then finds its orientations using the gyroscope. Its accelerations in the world coordinate system can then be computed using the rotation matrix given in [START_REF] Mahfouz | Target tracking using machine learning and kalman filter in wireless sensor networks[END_REF]. For simplicity, we only consider rotationally constrained targets in our paper. However, as just explained, the computations could be easily modified to consider the targets rotations and find the accelerations in the world coordinate system.

Having defined the state-space model, we now need to define the observation in order to use the Kalman filter. However, by using the equations of the estimated distances separating the target from the sensors, the obtained observation model is nonlinear. To overcome this problem, we propose in this paper an alternative linear observation model based on the distances. Indeed, using (1), one can write the following:

d 2 si,x(k) = δ v=1 (s i,v -x v (k)) 2 = δ v=1 s 2 i,v + δ v=1 x 2 v (k) -2 δ v=1 s i,v x v (k).
By subtracting d 2 sN s ,x(k) from d 2 si,x(k) , i ∈ {1, ..., N s -1}, we get the following:

d 2 si,x(k) -d 2 sN s ,x(k) = δ v=1 s 2 i,v - δ v=1 s 2 Ns,v + 2 δ v=1 x v (k)(s Ns,v -s i,v ).
After rearranging the terms in this expression, it becomes possible to set a linear observation model as follows:

z KF (k) = CX(k) + n(k), (18)
where n(k) ∼ N (0, R) is an additive noise, assumed to have a normal distribution, with zero mean and covariance matrix R, and the observation vector z KF (k) can be written as follows:

   d 2 s 1 ,x(k) -d 2 s Ns ,x(k) + δ v=1 s 2 Ns,v -δ v=1 s 2 1,v . . . d 2 s Ns -1 ,x(k) -d 2 s Ns ,x(k) + δ v=1 s 2 Ns,v -δ v=1 s 2 Ns-1,v   
The observation matrix C is then given by:

2   sN s ,1 -s1,1 . . . sN s ,1 -sN s -1,1 . . . . . . s Ns,δ -s 1,δ . . . s Ns,δ -s Ns-1,δ 0 (Ns -1) × δ   ,
To approximate R, a new set of reference positions is generated, and their distances to the sensors are estimated using χ i . Then, the differences between their observation vectors, using their estimated distances, and the product of the matrix C by their exact positions are computed and stored into a vector, that is the error vector. The matrix R, assumed to be constant over time, is then determined by computing the covariance of the error vector.

Now that we have defined the two linear equations ( 17) and ( 18), we can use the Kalman filter to find the target's position at each time step k. The Kalman filter first predicts the unknown position using the previous estimated position and the statespace equation [START_REF] Shareef | Localization using extended kalman filters in wireless sensor networks[END_REF], as follows:

X-(k) = A X(k -1) + B(k),
where X(0) is assumed to be known. Then, the predicted estimation covariance is updated by the following:

T -(k) = A T (k -1)A ⊤ + Q(k),
where T (k -1) is the covariance estimation at time step k -1, and T (0) is null since the initial state is known. The matrix Q(k) is the covariance matrix of X(k) given X(k-1), namely

Q(k) = Cov(X(k)|X(k -1) = Cov(B(k) + ǫ(k)) = Cov(B(k)) + Cov(ǫ(k)) = Cov(B(k)) + V ,
where Cov(•) computes the covariance matrix of its argument. Each coordinate acceleration noise is assumed to be independent with zero-mean normal distribution, having known variances σ 2 γ,d , d = 1, ..., δ. Their values can be estimated by performing a calibration of the accelerometer before the tracking stage. We also assume that all the δ coordinates are statistically independent. The matrix Q(k) is then given by:

Q(k) = V + 5∆t 4 36 Diag σ 2 γ 0 0 ∆t 2 2 Diag σ 2 γ .
The final step of the filter is the correction of the predicted quantities X -(k) and T -(k) using the observation equation [START_REF] Umamageswari | A comparitive study of Kalman filter, extended Kalman filter and unscented Kalman filter for harmonic analysis of the non-stationary signals[END_REF] as follows:

X(k) = X - (k) + G KF (k) (z KF (k) ⊤ -C X - (k)), T (k) = (I 2δ -G KF (k) C) T -(k),
where G KF (k) is the optimal Kalman gain given by:

G KF (k) = T -(k) C ⊤ (C T -(k) C ⊤ + R) -1 .

V. TRACKING USING THE PARTICLE FILTER

The objective in this section is to find the target's position estimates using the particle filter, the distances estimated by either one of the kernel-based distance models of Section III and the acceleration information of the target. Unlike the Kalman filter, the particle filter makes no restrictive assumption about the dynamics of the state-space or the density function. The observation model can be non-linear, and the initial state and noise distributions can take any required form. For this reason, the observations here are taken equal to the estimated distances separating the target from the sensors, that is:

z PF (k) = d x(k) ,
where d x(k) = (d s1,x(k) . . . d sN s ,x(k) ) ⊤ , obtained using the kernel-based model. Like the Kalman-based method, the tracking problem is also defined by the third-order state-space model. In order to solve the problem using the particle filter, N PF particles are first generated at the initial position of the target x(0), considered to be known. All the particles are given equal weights of 1 NPF . It is important to note here that the higher N PF is, the more computations we need. Let x m (0), m ∈ {1, . . . , N PF }, denote the initial particles positions, and ν m (0) their initial null velocities; then the particles positions at time step k are denoted by x m (k), m ∈ {1, . . . , N PF }, and their velocities by ν m (k). Starting from the initial position, the particles positions and their velocities are recursively predicted using ( 15) and ( 16) as follows:

ν m (k) = ν m ′ (k -1) + γ(k -1) ∆t + γ(k) -γ(k -1) ∆t ∆t 2 2 , x m (k) = x m ′ (k -1) + ν m ′ (k -1) ∆t + γ(k -1) ∆t 2 2 + γ(k) -γ(k -1) ∆t ∆t 3 6 + ǫ PF (k),
where x m ′ (k -1) is one of the particles of time step k -1, selected randomly according to the discrete distribution of their weights, ν m ′ (k -1) is the velocity associated to x m ′ (k -1), and ǫ PF (k) ∼ N (0, H) is the state equation error whose probability distribution is assumed to be normal, having zero mean and covariance matrix H of size δ × δ. The particles' weights are updated according to the observed distances. Indeed, a particle is more valuable if its distances to the sensors are closer to the observed ones [START_REF] Farmani | A hybrid localization approach in wireless sensor networks using a mobile beacon and internode communication[END_REF]. This update rule is formulated as follows: Then, the weights are normalized using the following:

w m (k) = 1 d x m (k) -z PF (k) w m ′ (k -1),
w m (k) = w m (k) NPF m=1 w m (k)
.

Finally, the last step in the filtering process is the resampling. In this step, the particles with negligible weights are replaced by new particles in the proximity of the particles with higher weights. It is applied only when the effective number of particles N eff becomes less than a threshold value N th , with N eff given by:

N eff = 1 NPF m=1 (w m (k)) 2 .
The threshold N th is usually taken equal to 10% of the particles number N PF . Having calculated the weights of the particles, the target's position at time step k is then given using the following:

x(k) = NPF m=1 x m (k) w m (k).
Compared to the Kalman-based method, this algorithm needs more computations, due to the generation and resampling of the particles; however, it is more robust when the distances errors are non Gaussian.

VI. EVALUATION OF THE ACCURACY OF THE PROPOSED DISTANCE MODELS

We now propose to evaluate the accuracy of the proposed distance models, in the case of real data and simulated data. The proposed models are compared to the log-distance propagation model, and to the polynomial model introduced in [START_REF] Yang | Indoor localization using improved rss-based lateration methods[END_REF]. The polynomial regression in [START_REF] Yang | Indoor localization using improved rss-based lateration methods[END_REF] determines a mathematical relation between RSSIs and distances, without taking physical properties into consideration. In such a case, the signal propagation model is the q-th degree polynomial given by the following:

d s i ,p ℓ = a 0 + a 1 ρ s i ,p ℓ + a 2 ρ 2 si,p ℓ + • • • + a q ρ q si,p ℓ , (19)
where a j , j ∈ {0, . . . , q}, are the polynomial's coefficients to be determined. These parameters a j are chosen in a way to fit the q-th degree polynomial through the training set, using the least squares method.

A. Evaluation of the distance models on real data

To evaluate the proposed distance models, we use the set of collected measurements available from [START_REF] Zanca | RSSI measurements in indoor wireless sensor networks[END_REF]. In this work, the measurements are carried out in a room of approximately 10 m × 10 m, where 48 uniformly distributed EyesIFX sensor nodes are deployed. Furniture and people in the room cause multi-path interferences affecting the collected RSSI values. Now, for our evaluation, we consider that 5 sensors over the 48 are denoted as fixed sensors. The 43 remaining are used to collect the measurements for the training phase. Fig. 1 shows the topology of the testbed. Note that it is the collected measurements that are needed in the test phase with the 5 fixed sensors, not the 43 remaining ones. It is also worth noting here that the training data for our algorithm could be collected by using one sensor, as explained in Section II, and this only once at the beginning of the experiments. Indeed, by placing consecutively one sensor over the reference positions, one can collect RSSI/distance pairs and construct the training database.

It is important to note that the average values over time of the RSSIs are used here. In fact, the RSSIs vary significantly with respect to time and movements. These variations are known as short-term or multi-path fading. On the other hand, the local average of the signal varies slowly. These slow fluctuations depend mostly on environmental characteristics, and they are known as long-term fading. Therefore, it is more suitable to use the average values of the RSSIs than to use all the collected values [START_REF] Neskovic | Modern approaches in modeling of mobile radio systems propagation environment[END_REF]. Now the objective is to find the RSSI/distance relationship; in other words, we need to find the three distance models, i.e., one model per sensor. Each model has a different set of training data to be used, and different parameters that need to be found. In fact, the RSSIs of the signals exchanged between each fixed sensor and the 43 other sensors are used along with the distances separating this sensor from the other sensors. This information is then used in the training phase as described in III to compute the models' parameters. In the following, we use the Gaussian kernel given by: where σ i is its bandwidth that controls, together with the regularization parameter η i , the degree of smoothness, noise tolerance, and generalization of the solution. The kernel parameters are chosen in a way to minimize the error on the training set. The value of this error for the computed models is given in Table I, along with the error on the training set for the physical log-distance propagation model used in [START_REF] Medeisis | On the use of the universal Okumura-Hata propagation prediction model in rural areas[END_REF], [START_REF] Patwari | Locating the nodes: cooperative localization in wireless sensor networks[END_REF], [START_REF] Zanella | Rss-based ranging by multichannel rss averaging[END_REF] and for the polynomial model of [START_REF] Wang | Wsn multilateration algorithm based on landweber iteration[END_REF], for several degrees q. We also use the leave-one-out (LOO) technique in order to evaluate the performance of the proposed model in the case of data that are not part of the training set. The LOO technique involves using a single observation from the collected data as the validation data, and the remaining observations as the training data. This is repeated 43 times, such that each observation is used once as the validation data. This technique is interesting because it allows us to compare the proposed models to the log-distance model, even though the set of collected data is not really large. Finally, the mean estimation error (in meters) is stored in Table I for the computed models. For convenience, in the comparison tables, let us denote the non-parametric model of Subsection III-A by NPM, and the semi-parametric model of Subsection III-B by SPM. We also denote the log-distance propagation model, that is known for being a physical model, by PM and the polynomial model by PolyM. One can see from Table I that the proposed models yield better results than the log-distance model, when comparing the mean estimation error. Moreover, the two proposed models yield really close results.

κ(ρ si,p u , ρ si,p u ′ ) = exp - ρ s i ,p u -ρ s i ,p u ′ 2 2σ 2 i , 0

B. Evaluation of the distance models on simulated data

In this subsection, we evaluate the accuracy of the distance models on simulated data, in the case of two different scenarios. In the first paragraph, we present the first scenario, where an area without walls is considered. We then compare the results obtained using the proposed distance models to the results obtained with the log-distance propagation model. In the second paragraph, we consider a different scenario, where the signals are attenuated because of the presence of walls in the proposed topology. As it will be shown in the following, such topology allows a better comparison between models. We start with the first scenario, where a 100 m × 100 m area is considered, with N s = 16 sensors and N p = 100 reference positions distributed over the area. Fig. 2 illustrates the considered topology, where no walls or obstacles are present. The RSSIs for the training phase are obtained using the theoretical log-distance propagation model [START_REF] Medeisis | On the use of the universal Okumura-Hata propagation prediction model in rural areas[END_REF] given in [START_REF] Zhang | A novel angle-of-arrival assisted extended kalman filter tracking algorithm with space-time correlation based motion parameters estimation[END_REF], with n P set to 4 as often given in the literature, and ρ 0 set to 1 dBm. As for the test phase, 100 positions are randomly generated in the studied area, and their RSSIs are also obtained using [START_REF] Zhang | A novel angle-of-arrival assisted extended kalman filter tracking algorithm with space-time correlation based motion parameters estimation[END_REF]. Finally, a zero mean additive white noise ε i,ℓ is added to all the RSSIs, with σ ρ being its standard deviation. Here, we take σ ρ equal to 1 dBm. Based on the study given in Section III, we define N s = 16 non-parametric distance models and N s = 16 semi-parametric distance models. Then, in the test phase, we estimate the distances using these models, the logdistance propagation model, and the polynomial model with q = 2, q = 3 and q = 4. The mean estimation errors for the 16 models (in meters) are stored in Table II. One can see from this table that the results are really close, especially for the test error, which is of much great importance than the training error. This result was expected, since we are generating the RSSIs using the log-distance propagation model, and the noise is a zero mean additive noise. Therefore, the proposed distance models are trying to model the noise, which can not be learned nor modeled, because of its randomness.

We now propose another scenario given in Fig. 3, where we consider a 25 m × 5 m area, 2 fixed sensors and 45 known positions for the training phase. This figure shows that there are 5 rooms in the area, meaning that the signal penetrates a maximum of 4 walls during its propagation. Therefore, we consider the average walls model described in [START_REF] Andrade | IEEE 802.11 WLANs: A comparison on indoor coverage models[END_REF] to generate the RSSI measures. This model is a modified version of the log-distance model that explicitly takes into account the attenuations due to walls. The received signal strength indicator is then given by the following:

ρ si,p ℓ = ρ 0 -10 n P log 10 d si,p ℓ -N wi L wi + ε i,ℓ , (20)
where the quantities L wi and N wi denote respectively the loss due to walls and the number of penetrated walls. The quantity L wi is taken equal to 6.9 dBm, since we consider the case of heavy thick walls [START_REF] Andrade | IEEE 802.11 WLANs: A comparison on indoor coverage models[END_REF]. Now for the test phase, 100 positions are randomly generated in the studied area, and their RSSIs are obtained using [START_REF] Mahfouz | Target tracking using machine learning and kalman filter in wireless sensor networks[END_REF]. Then, the distances are estimated using the obtained models from Section III, and the log-distance propagation model. The estimation errors for the 2 models, i.e. a model for each sensor, are stored in Table III, when the standard deviation of ε i,ℓ is taken equal to 0.5 dBm. Table IV yields the estimation errors when the noise's standard deviation is increased to 1 dBm. One can see from both tables that the two proposed distance models outperforms the log-distance propagation model in terms of accuracy. Moreover, it is the non-parametric model that yields the best results.

VII. EVALUATION OF THE PERFORMANCE OF THE TRACKING METHODS

This section evaluates the performance of the two proposed tracking methods on simulated data. In the first subsection, we evaluate the performance of the proposed methods in terms of accuracy for fixed values of the noises standard deviations σ γ and σ ρ . Then, we study the impact of varying the number of sensors N s on the estimation error. Next, in the third subsection, we study the impact of varying the noises standard deviations σ γ and σ ρ on the estimation error. Then, we compare our results to ones obtained with two recently proposed positioning method; the first one is based on the WKNN algorithm combined with a Kalman filter [START_REF] Liu | Exploit kalman filter to improve fingerprint-based indoor localization[END_REF], while the second one makes use of the ridge regression learning method to find the position estimates [START_REF] Mahfouz | Kernel-based localization using fingerprinting in wireless sensor networks[END_REF]. Finally, we compare the accuracy of the proposed methods to the tracking method proposed in [START_REF] Mahfouz | Target tracking using machine learning and kalman filter in wireless sensor networks[END_REF] when the sensors change their locations. In the following, we consider the setup of Fig. 2, and the RSSI measures are generated using (6), with n P set to 4 and ρ 0 set to 1 dBm. Now for the choice of the distance model, one can see from the results of Section VI that the non-parametric model yields the best results in all scenarios. Therefore, this model will be used in this section. As we already explained, the kernel parameters η i and σ i are chosen in such a way to minimize the error on the training set. As for the application of the particle filter in all our simulations, the number of particles N PF is set to 50.

A. Evaluation of the proposed methods

We consider a moving target in the defined area. The target's accelerations are given in Fig. 4, γ 1 and γ 2 being the first and the second acceleration coordinates respectively. By taking twice the primitive integrals of the accelerations, we compute the coordinates expressions of the target. The trajectory obtained is then of 100 points with ∆t = 1 s, and it is illustrated in Fig. 5. We consider that noises are present in all scenarios, since a noiseless setup is not realistic in a practical environment. To this end, we take σ ρ equal to 10% of standard deviation of the RSSI measures, i.e., σ ρ = 1.08 dBm, and we take σ γ equal to 5% of the standard deviation of the accelerations. Let the estimation error be evaluated by the root mean squared distance between the exact positions and the estimated ones. Fig. 5 shows the estimated when using the two proposed methods. The results are averaged over 50 Monte-Carlo simulations. The mean error obtained when using the Kalman filter of Section IV is equal to 1.03 m. As for the mean error obtained with the particle filter of Section V, it is equal to 0.68 m. One can see that both methods allow an accurate tracking of the target, with a better estimation error when considering the particle filter with this setup.

B. Impact of σ γ and σ ρ

In this subsection, we study the impact of the noises standard deviations σ γ and σ ρ on the estimation error. First, we take a fixed value for σ ρ , equal to 10% of standard deviation of the RSSI measures. Then, different percentages of the standard deviation of the acceleration are considered, ranging from 1% to 10%. The estimation errors are averaged over 50 Monte-Carlo simulations. Fig. 6 shows the impact of the variation of σ γ on the estimation error. This figure shows that both filters have similar performances when the noise on the accelerations is small; however, with higher noise values, the particle filter outperforms the Kalman filter.

Next, we take a fixed value for σ γ equal to 5% of the standard deviation of the accelerations, with several percentages of the standard deviation of the RSSI measures, going from 0% to 50%. Fig. 7 shows the impact of the variation of σ ρ on the estimation error for both methods. One can see here as well that the particle filter yields better results than the Kalman filter. This is due to the distribution of the observation errors, assumed to be Gaussian in the Kalman filter. Indeed, small RSSI errors yield slightly varying observation errors. However, with higher RSSI errors, the noise distribution gets farther from a Gaussian one, and the particle filter performs better. Nevertheless, one can see that both methods have relatively small estimation errors, compared to the space dimensions.

C. Comparison to other methods

The objective now is to first compare the proposed approach to two recently proposed positioning methods [START_REF] Chan | Using wi-fi signal strength to localize in wireless sensor networks[END_REF], [START_REF] Liu | Exploit kalman filter to improve fingerprint-based indoor localization[END_REF], [START_REF] Mahfouz | Kernel-based localization using fingerprinting in wireless sensor networks[END_REF]. We consider the same setup as the one in Subsection VII-A, and the same values for σ γ and σ ρ . Then, taking the same setup, we consider the case where sensors are moving in the surveillance area, and we compare the performance of the proposed methods to the tracking method of [START_REF] Mahfouz | Target tracking using machine learning and kalman filter in wireless sensor networks[END_REF] in such a case.

We first describe briefly the methods in [START_REF] Chan | Using wi-fi signal strength to localize in wireless sensor networks[END_REF], [START_REF] Liu | Exploit kalman filter to improve fingerprint-based indoor localization[END_REF]. It consists of estimating the position using the weighted K-nearest neighbor (WKNN) algorithm, then applying Kalman filter to enhance the estimation. A target's first position estimate using WKNN is given by weighted combinations of the K nearest neighboring positions from the training database, with the nearness indicator being based on the Euclidean distance between RSSIs. The number of neighbors K is taken equal to 8 as in the simulations of [START_REF] Chan | Using wi-fi signal strength to localize in wireless sensor networks[END_REF], where more details can be found. As for the position enhancement using the Kalman filter, a second-order state-space model is used. The estimation errors (in meters) obtained when using this algorithm (WKNN + Kalman) are computed 50 times, and the mean estimation error is stored in Table V. The table also shows the mean estimation error of this algorithm without using the acceleration information (WKNN) and the errors standard deviations as well. One can see that the two proposed methods outperform the WKNN-based methods in terms of accuracy. Fig. 8: Estimation error as a function of the noise on the RSSI, using different methods.

As for the localization method of [START_REF] Mahfouz | Kernel-based localization using fingerprinting in wireless sensor networks[END_REF], a radio-fingerprinting database and the ridge regression learning method are used to define a model that takes as input the RSSI measures and gives as output the position of the target. However, this method does not take into account the acceleration information. Table V also shows the estimation error obtained using this method (RR). The errors obtained with the proposed methods are significantly smaller than the one obtained with the method of [START_REF] Mahfouz | Kernel-based localization using fingerprinting in wireless sensor networks[END_REF].

Next, several percentages of the standard deviation of the RSSI measures are taken using the methods in [START_REF] Chan | Using wi-fi signal strength to localize in wireless sensor networks[END_REF] and [START_REF] Mahfouz | Kernel-based localization using fingerprinting in wireless sensor networks[END_REF]. The value of σ γ is taken equal to 5% of the standard deviation of the accelerations as in the scenario of Fig. 7. The estimation error obtained in this case is shown in Fig. 8. This figure shows that the two proposed methods outperform the other methods for all values of σ ρ , compared to the results obtained in Fig. 7.

We now compare these methods for several values of the number of sensors N s , for a fixed number of reference positions N = 100. σ ρ is taken equal to 10% of the standard deviation of the RSSI measures and σ γ equal to 5% of the standard deviation of the accelerations. Fig. 9 shows the estimation error as a function of N s . One can see that the two proposed methods yield the smallest errors for all values of the number of sensors; moreover, at N s = 4, the other methods do not give such accurate results. Another important thing to notice is that the proposed methods are more robust to the changes in the number of sensors and they present less variations than the other methods. Finally, we consider that the N s = 16 sensors are moving in the surveillance area, and we compare the performance of the proposed tracking methods of this paper to the performance of the method of [START_REF] Mahfouz | Target tracking using machine learning and kalman filter in wireless sensor networks[END_REF]. In [START_REF] Mahfouz | Target tracking using machine learning and kalman filter in wireless sensor networks[END_REF], the target is first localized using a predefined model that takes as input the RSSI measures and gives as output the position of the target such as in [START_REF] Mahfouz | Kernel-based localization using fingerprinting in wireless sensor networks[END_REF]. The estimated positions are then combined with acceleration information using a Kalman filter along with a third-order statespace model. This approach gives accurate position estimates in the case of stationary sensors. However, if the sensors were to move, the results would not be accurate anymore, since the positions are included in the learning process. The advantage of the proposed methods is that we find the RSSI/distance relationship, and thus find directly the distance from the RSSI measures. When the sensors change their positions, this relationship does not change, thus we do not need to reconfigure the model, i.e. a new training phase is not needed. Now we consider that the sensors have circular movements. The trajectory of Fig. 5 is estimated using the three methods for different radii ranging from 0 to 30m and for random angles going from 0 to 2π (radians). The results are then averaged over 50 Monte-Carlo simulations. Fig. 10 shows the mean estimation error as a function of the radius for the three tracking methods. One can see that all three methods yield close results at the beginning, when the radius is equal to zero, that is when the sensors remain stationary. However, when the value of the radius increases, the estimation error also increases for the method of [START_REF] Mahfouz | Target tracking using machine learning and kalman filter in wireless sensor networks[END_REF], while the estimation error almost remains the same for the two proposed methods in this paper. This result proves that the new proposed methods are more robust to changes in the initial configuration of surveillance area than the method of [START_REF] Mahfouz | Target tracking using machine learning and kalman filter in wireless sensor networks[END_REF].

VIII. CONCLUSION

In this paper, we proposed two original regression models that relate the received signal strength indicators (RSSIs) to the distances separating sensors in a wireless sensor network. Then, we solved the tracking problem using the estimated distances and two new methods that take into account the target's motion. We provided a fully comprehensive study of the proposed distance models and their performances. Simulation results show that our models yield accurate distance estimation.

Results also show that our tracking methods allow accurate position estimation, and are proved to be robust in the case of noisy data. Both proposed tracking methods outperform tracking using recently developed methods based on the WKNN method and the Kalman filter or some other learning strategy. Future work will handle further improvements of this work, such as wisely choosing a group of sensors instead of using all the available distance information. Solutions to cases where zones of the surveillance area are not covered by all sensors could also be provided.

  where d x m (k) = (d s1,x m (k) . . . d sN s ,x m (k) ) ⊤ , and d x m (k)z PF (k) is the distance between the vectors z PF (k) and d x m (k) .

Fig. 1 :

 1 Fig. 1: Topology of the testbed (real data), where ∆ represents the sensors and + represents the training positions.

Fig. 2 :

 2 Fig. 2: Topology of the testbed, ∆ represents the sensors and + represents the training positions.

Fig. 3 :

 3 Fig. 3: Topology of the simulated area for the second scenario, ∆ represents the sensors and + represents the training positions.

2 )Fig. 4 :

 24 Fig. 4: Acceleration signals of the target.

Fig. 5 :

 5 Fig. 5: Trajectory estimation.

Fig. 6 :Fig. 7 :

 67 Fig. 6: Estimation error as a function of the noise on the accelerations with σ ρ equal to 10% of the standard deviation of the RSSIs.

Fig. 9 :

 9 Fig. 9: Estimation error as a function of the number of stationary sensors.

Fig. 10 :

 10 Fig. 10: Estimation error as a function of the radius defining the movement of the sensors.

TABLE I :

 I Comparison between models (error in meters) for real data.

	Considered model	Mean training error Mean LOO error
	PM	1.36	1.41
	PolyM, q = 2	1.30	1.40
	PolyM, q = 3	1.29	1.48
	PolyM, q = 4	1.29	1.74
	NPM	1.21	1.33
	SPM	1.17	1.33

TABLE II :

 II Estimation errors for simulated data in the case of the first scenario.

	Considered model	Mean training error Mean test error
	PM	2.26	2.59
	PolyM, q = 2	2.74	2.94
	PolyM, q = 3	2.23	2.57
	PolyM, q = 4	2.22	2.60
	NPM	2.18	2.57
	SPM	2.09	2.59

TABLE III :

 III Estimation errors for simulated data in the case of the second scenario, with σ ρ =0.5 dBm.

	Considered model	Mean training error Mean test error
	PM	1.26	1.37
	PolyM, q = 2	0.56	0.62
	PolyM, q = 3	0.55	0.61
	PolyM, q = 4	0.54	0.61
	NPM	0.20	0.32
	SPM	0.22	0.34

TABLE IV :

 IV Estimation errors for simulated data in the case of the second scenario, with σ ρ =1 dBm.

	Considered model	Mean training error Mean test error
	PM	1.29	1.39
	PolyM, q = 2	0.68	0.66
	PolyM, q = 3	0.67	0.65
	PolyM, q = 4	0.67	0.66
	NPM	0.39	0.51
	SPM	0.43	0.52

TABLE V :

 V Estimation error (in meters) for the different methods.

			Tracking method	Estimation error
			Proposed Kalman filter	1.03
			Proposed particle filter	0.68
			RR		2.98
			WKNN	4.57
			WKNN + Kalman	2.15
		12
			Kalman filter
			Particle filter
		10	Localization using the RR
			WKNN
	Estimation error (m)	4 6 8	WKNN + Kalman
		2
		0% 0	10%	20%	30%	40%	50%
		σρ taken as a percentage of the standard deviation of the RSSI

This work is supported by the Champagne-Ardenne Region in France, grant WiDiD


Albert Marouani

Structural and behavioural asymmetries as the norm of market Economies

Keywords: asymmetry, neoclassical economics, market, equilibrium, econometrics, macroeconomics, classical economics, evolutionary economics, structure

Standard neo-classical economic theory is based on the idea of balance and symmetry between opposing forces (supply and demand). Most situations in the field of macroeconomics, in particular, are of a structurally asymmetrical nature. The same is true about the behavior of all agents because asymmetry of information is a characteristic of any contractual relationship in any market. Classical economists, as well as other heterodox economists, also consider that asymmetry, inequality and the conflicts of power and domination are at the very heart of market relations and explain the historical dynamics of the evolution of market economies. In these circumstances, it should be considered that the asymmetrical relationship must be regarded as the norm of the market economy.

marché. Il conviendrait dans ces conditions de considérer que la relation asymétrique doit constituer la norme de l'économie de marché.

Mots clés : asymétrie, économie néo-classique, économie de marché, équilibre, macroéconomie, économétrie, économie classique, économie évolutionniste, structure.

Introduction.

The notion of asymmetry in economics is often confused with the notions of inequality, dissymmetry, imbalance, inequity and injustice. The dominant standard economic theory known as "neoclassical" is based on the assumption that the pure and perfect competition market is characterized by a perfect symmetry between supply and demand. Asymmetrical or rather dissymmetrical situations arise when the principles of free competition are interfered with, in particular because of the behavior of agents who exercise on the other agents an asymmetrical power such as in particular the State, the large enterprises or the unions. This asymmetrical power is always harmful to the community as a whole. Any asymmetry is a transitory evil that must be corrected by suppressing its causes, ie on the one hand the intervention of the State which must be reduced at best to its sole sovereign functions, and on the other hand, all monopolistic and oligopolistic situations must be banned, if necessary, through the introduction of anti-trust laws and measures that break union power.

We want to go up here that we need to radically change the paradigm and consider, on the contrary, that asymmetry is in general the most common norm of market economies.

Admittedly, there are bad and good asymmetries. Social conventions and democratic institutions are necessary to correct and eliminate the bad asymmetries that are the source of environmental degradation, destructive crises and impoverishment. On the contrary virtuous asymmetries that generate emulation, creativity and improved people's well-being in terms of sustainable and inclusive development must be encouraged.

As economists most often deal with measurable concepts by quantities of monetary units (prices) both at the microeconomic level of an individual or a firm and at the macroeconomic level of an industry, a country, a region and that they have more and more accurate databases, they can resort to increasingly complex facilitated by econometric software and the power of computers. This growing ease of accessing and quantifying statistical data makes it even more necessary to have conceptual and theoretical thinking which alone is able to first determine what we are looking for and then to make sense.

The notion of asymmetry does not escape this tension between its empirical and statistical dimension and its conceptual meaning within different theoretical paradigms. We will first discuss what the notion of asymmetry covers in terms of empirical economic studies and then point out some of the major theoretical debates it raises in the field of economics. The notion of asymmetry in the field of empirical studies is often confused with notions of inequality and imbalance. For example: imbalance between supply and demand (goods and services, labor, capital, etc.) in one or more markets or at the overall level of the economy of a country or group of countries, imbalance of foreign trade, balance of capital, balance of payments, migratory flows, inequalities of income, wealth, natural resources (arable land, fossil fuels, water, etc.), public goods and services, gender, age, ethnicity, etc., etc.

We would not finish enumerating all the imbalances and inequalities which in the actual field of the economy can also be interpreted as so many asymmetrical relations between agents (household, company, banks, insurance, etc.) or groups of agents or institutions, or branches, or sectors, or countries, or groups of countries, etc. Let us take the example of income distribution (wages, profits, rents, etc.). This theoretically very complex question is often reduced to its empirical measure.

The Gini indicator measures the degree of inequality of distribution (income, wealth, wages, etc.) among inhabitants of a given country. International organizations rank countries according to this criterion, which becomes synonymous with "Social justice", "distribution fairness of wealth ", or even" social and political democracy ".

The Gini coefficient is measured by the ratio of the shaded area to the blue area. It varies from 0 (perfect equality on the first bisector) to 1 (total inequality). In France, this coefficient applied to household disposable income is 0.352 in 2011. This indicator reflects unequal distribution and not unequal income level or distribution. Nor does it reflect an asymmetry of income distribution. "For example, if the 50% of the most modest people have no income and the wealthiest 50% equally distribute the remaining income, then the Gini coefficient is equal to 0.5. It also reaches 0.5 when the 75% of the lowest income is 25% and the top 25% have 75% of the income. This indicator says nothing more about the nature of these incomes, the way they are formed, and why and how they change over time.

We will address these theoretical issues in the second part of this paper. To get a better idea of the asymmetry of the distribution of incomes, one should consider incomes as a real random variable and look at their distribution by group and by type of agent or group of countries, or by type of income at a given date.

Two well-known indicators measure precisely the form of a statistical distribution: "kurtosis" and "skewness". There are three types of distribution according to whether they are asymmetrical on the left (positive skewness), symmetrical (kurtosis) or asymmetric on the right (negative skewness). There are two other simpler and more empirical asymmetry coefficients.

1. The empirical coefficient of Pearson: difference between the mean and the mode of the observed distribution. This difference is divided by the standard deviation so that is a number without unit: 1.

2. The empirical coefficient of Yule and Kendall is defined from the three quartiles of the observed distribution:

In general the coefficient of flattening (kurtosis) is calculated at the same time as that of asymmetry. The Jarque-Bera test incorporates these two coefficients. To find out whether skewness is compatible with the normality assumption, we refer to tables that give us for a given series and a number of observations the probability and the confidence interval for it to be considered normal. These financial economy ratios are often used in the distribution of losses for securities portfolios where the asymmetry reflects, for example, a limited potential gain with rare but sometimes very large losses. These asymmetry coefficients are also used to analyze the distribution of wages within a firm and reveal, for example, inequalities of gender or age or ethnic discrimination.

b) the structural asymmetries due to shocks: the models « VAR structurel»

There are many examples of major macroeconomic imbalances that can be interpreted as structural asymmetries :

• Imbalance / asymmetry on « stocks » (real, monetary and financial) are macroeconomic magnitudes such as growth, employment, investment, technical capital, energy (natural, non-renewable and renewable fossils), technology, human capital, natural and environmental capital, productive structures, the distribution of wealth and income, etc.

• Imbalance / asymmetry on « flows » (real, monetary and financial): trade in goods and services, natural disasters, global warming, pollution, biodiversity, population movements and migration flows, international capital movements, etc.

Econometric models in macroeconomics very often refer to the (symmetrical) notions of supply shock and demand shock to designate events that translate , by a significant and unforeseen variation in the conditions of production or demand of one, several or all • The notion of supply shock.It is two countries A and B. One of the countries is affected by an increase in its production costs (oil shock, sudden increase in wages, deterioration of its competitiveness); the other country is affected by an inverse shock (or no shock at all). For a given exchange rate, prices were identical in both countries A and B. An asymmetrical shock of negative supply affects country A (its supply function moves upwards).

• The notion of demand shock Country A produces goods that are suddenly less in demand (shifting the demand curve downwards); while country B produces goods that are more in demand (moving the demand curve upwards).

These shocks are generally asymmetric because they impact with intensities and nonidentical or non-uniform effects, firstly the producers, and then, in turn, all the different other agents (companies, households, branches, sectors, country, group of countries, etc.).

These shocks are generally exogenous and can be economic, social, political or natural.

The econometric identification of supply and demand shocks comes from the VAR literature and in particular from the founding article by Blanchard and Quah (1989). There are countless works, articles and theses that are inspired by it.

It's about understanding how a shock spreads, how and to what degree it impacts the agents and the different sectors of the economy of one or several countries and how it ends up being absorbed by market mechanisms or after the intervention of a public authority (State, Central Bank, International Institution, etc.). The model begins by asking theoretical hypotheses that This involves:

• choosing the statistical series that will be used for the demonstration,

• ensuring that the number of observations is sufficient,

• that the stationarity conditions of each series are verified,

• that there is no cointegration between the series (application of the Johansen test, for example).

• that the assumptions of the covariance variance matrix of the shocks, as well as those concerning the The effects of shocks on the different aggregates represent a sufficient number of equations to determine the coefficients aii of the impulse functions.

• If there are several perturbations, it must be verified that they are independent and not self-correlated (their matrix variances-covariances is diagonal).

• Determine for each type of shock, the common component and the specific component for each type of agent (company, branch, country, etc.), particularly using the Kalman filter • Etc.

According to the nature of the shock (s) studied, the period and the perimeter considered, one draws the results of the econometric model of the recommendations in terms of economic policy. For example shock of the return or the exit of a country of the zone euro, shock of change of the monetary policy accommodating central banks, German reunification shock, shock of sharp rise in the price of energy or raw materials, or entry into a free trade agreement, shock of sudden increase in public spending (announcement by D. Trump), fiscal shock of sudden rise or fall, productivity shock, major innovation shock, etc. In terms of the explanatory factors, issues related to the operation of markets, their rigidity, the volatility of prices or rates, the characteristics of the structures of production, distribution, consumption, demographic, etc. For example, the recent announcement of the introduction of tariff barriers by the US could, if translated into action, lead to asymmetric shocks within and between countries (between regions and between branches) in the euro zone, according to the degree of specialization of their respective productive apparatus and their abso the impact of these shocks, the flexibility of their labor markets, etc. The proposed economic policy measures are also generally to mitigate the effects of asymmetry and to restore the conditions for convergence. For example, many studies have examined the question of whether belonging to a common currency area, the Euro zone, which is clearly not an OMA (Optimal Monetary Area) in the sense of Mundell, has mitigated the asymmetry of productive structures between the member countries of the zoned. This question can also be formulated differently: did belonging to the Eurozone reduce the asymmetry (or strengthen the symmetries) of the external shocks suffered by the member countries? In the absence of adjustment by the exchange rate that has become common to all the countries in the zone, the question is to know on which other variables the adjustment will be based (unemployment, the level of wages, public expenditure, inflation ...). Everything will depend on the degree of flexibility (or rigidity) of these other adjustment variables which itself depends on social, institutional and political factors.

Another recent example could be the analysis of a productivity shock on firms. in the wake of ongoing reforms of the labor market and social protection. The comparative competitiveness benefits that may result from these reforms may result in inter-branch and / or intra-industry specializations, at more or less fine levels in the value chain, which could lead to cumulative effects, sources of growing asymmetries that would accentuate the divergent effects from one country to another or from one region to another. Note that the lack of convergence is often the argument invoked by the proponents of protectionist measures. All these questions lend themselves to numerous econometric studies, the results of which must be assessed according to the chosen period and the variables taken into account (indices activity, specialization index, shock index, regions, countries, sectors, industries, etc.) For example a shock (variation) of activity can be measured by the standard deviation of the change in log difference. The actual output of each pair of countries, or regions. The degree of asymmetry of the shocks can be approximated by an index of similarity (or dissimilarity) of the composition of the export structures, but one could also have taken a growth index, an unemployment rate, a level of employment, a level of real GDP, etc. measured in logarithm (except for the unemployment rate). The degree of reciprocal openness can be measured by the average of bilateral exports relative to GDP. The bilateral intensity of trade could be approximated by the logarithm of the ratio of bilateral exports / imports to the sum of exports / imports of the two countries concerned to all other countries. In some cases the polarization effects associated with Krugman agglomeration effects can attenuate the asymmetries between countries but accentuate those between regions. When one analyzes the effects of convergence (real or nominal), one must also ask the question of the accentuation of the divergences by country or by region. For example, the "EU Structural Funds" policy aims to mitigate regional or sectoral divergences or asymmetries. The diversification of economies is also an important variable to take into account when analyzing the effects of structural supply shocks. Current econometric models make it possible to measure and analyze all the effects of structural shocks. It is also necessary to be very vigilant not only on the data chosen, the period envisaged, the degree of disaggregation of the variables, etc., but also and above all on the explanation of the theoretical framework which makes it possible to formulate hypotheses, to construct the model, to interpret its results and to make proposals for economic policy.

Unfortunately, often in theses and econometrics, the sophistication of statistical and econometric tools and techniques prevails over methodological and conceptual thinking.

Beyond its empirical representation, the notion of asymetry is fundamentally a question of conceptual order that challenges all economic theories.

2°) Behavioral asymmetries, an important source of theoretical controverses.

The nature of agents' behaviors and their relations is often at the heart of the theoretical controversies between the major paradigms of economic thought. . The issue of symmetry / asymmetry is an important dimension of these debates. Having emphasized that symmetry (and certainty) constitutes the market standard in neo-classical orthodox thought, we will first show how asymmetric information disturbs the behavior of agents to a certain extent.

prevents s markets to function properly and fulfill their self-regulating function. We will then ask the often-evaded question of the power and consubstantial power of market economies and conclude on the need for a radical paradigm shift in economic theory that would make the concept of asymmetry the central norm rather than the marginal exception.

a) Symmetry is the standard of the ideological representation of the market economy in the neo-classical thought.

The standard model, to which the neoclassical theory and the liberal economists refer is based on the perfect symmetry of supply and demand based on the following assumptions:

• Economic agents are differentiated only by their subjective preferences and their always symmetrical alternative situations of offerers (sellers) or of claimants (buyers).

• They are all perfectly rational (they know how to optimize and thus make economic calculations of maximization under constraints).

• They have all equally of all market information (price, quality and quantity).

• They are virtuous (no one is looking to (or is able to) withhold information, deceive trading partners or try to impose choices on them by force or rudeness.

The Walraso-Paretian general equilibrium comes on the basis of these heroic assumptions to prove the possibility of the market economy : there is a system of unique balance prices that equalize the supply and demand of the agents and thus realize an optimum social (all agents completely realize their production and consumption plans and are fully satisfied.).

The formal demonstration of existence of a unique price system that makes compatible preferences agents is based on a process of symistration by tattonnement. This process of trial and error is initiated and led by a fictional character (it works without costs, without needs and without any interest), a kind of auctioneer or market secretary (or central computer) who records for each price shouted in the light of all the candidates for the exchange, the quantities of goods and services they are willing to produce or buy on this price basis. The market secretary stops when he finds that there is no longer any dissymmetry between the offerers and the applicants, that is to say that the last shouted prices are "balance prices", that is to say that the offer is equal to the demand and that no more agent is dissatisfied. As soon as he finds that there is no more excess or symmetrically no deficit, the authorizing commissioner gives the order to exchange. In equilibrium all markets are necessarily cleared and no one is in a situation of waiting or dissatisfaction or even arbitrage or transfer to another market, because all markets balance at the same time.The wage which is the equilibrium price on the labor market and the profit on that of the capital are fixed at the level of their marginal productivity within the function of production. They maintain between them a perfectly symmetrical and nonantagonistic relation: profit is the remuneration of the factor of production that is the capital in the same way that the wage is the remuneration of the factor of production that is the work or the rent like remuneration the production factor land or natural resources.

Y= f(K,L,T)

With r = ∂Y/∂K, w= ∂Y/∂L, t=∂Y/∂T And Y.p= r.K+w.L+t.T

The equilibrium on factors markets is identical to that which is determined in the production function, it appears that the market is also perfectly fair: each factor receives exactly its due according to its contribution to the value of the final product Y, since each receives a remuneration equal to its marginal productivity (increase of product Y due to the increase of the last unit of the factor of production). Each factor of production is always guaranteed to perceive exactly its "fair" remuneration. The asymmetry between social classes here does not exist. It is this theoretical construction that is still at the heart of the belief in the self-regulating virtues of the market. The idea of the market 'recall strengths', coupled with such phrases as 'the market is always right', nobody can 'beat the market', it is 'the law of the market', and so on. All this implies that the symmetry between the offer and the demand is a norm, a pole, even a center of gravity which exerts a power of attraction all the more strong as one moves away from it.

The "reality" invented by this theoretical construction is different from this ideal theoretical model in the sense that it departs from it. It is assumed that these "gaps" can be measured or analyzed by calculating "fictive prices" of equilibrium. This is how in the neoclassical theory do the obligatory reference of liberalism, neo-liberalism and ultraliberalism) theories of the "natural wage rate" and the "natural interest rate" have been developed to explain unemployment and the conduct of monetary policies by Central Banks. In other words, the real prices observed in the markets revolve around the equilibrium prices, which are an abstract fiction, a purely theoretical construction.

It is this "mystery" that founds the "religious" and / or "magical" belief, that there are "laws of the economy" against which it is futile to oppose and fight. This representation of an ideal model of symmetry is driven by the idea that the equilibrium is always virtuous whereas conversely the imbalance is perverse and harmful. In economics this equilibrium is expressed in terms of symmetrical forces (supply and demand, flow and credit, losses and profits (or benefits), costs and benefits, restoring balance must be done at all costs as we see today in the austerity policies imposed on indebted countries (Greece, Italy, Portugal, Spain, France, etc.) to fully repay their debts. Neo-classical economic theory has nevertheless integrated asymmetry as an "imperfection" of the market, even as its "failure" without questioning its primary role in the coordination of agents through the exclusive prices as signals. A liberal economist like Hayek, who does not believe in the neo-classical vision of rationality and omniscience of economic agents, nevertheless considers that uncertainty, incompleteness of information, asymmetric circulation and complexity of the world can still be managed by the prices of goods and services that, in spite of their imperfection, reflect in a neutral and objective manner the state of their abundance or scarcity and thereby inform current and potential entrepreneurs about profit opportunities.

Prices translate and express the convergence and coordination of individual decisions and actions and they must not be manipulated and therefore distorted by both private agents and public officials who are no longer able to master and understand the complexity of a world of uncertainties and asymmetries.

b) the existence of asymmetric information disrupts the rules of the market.

Contractual relations between agents are generally asymmetrical from the point of view of their respective informational knowledge and the always uncertain nature of the compliance with the contracts (for a debtor vis-à-vis the creditor) or correspondence with the expected performance of the co-contractor (in the context of a labor contract for example) or the expected qualities of the purchased products, or hidden defects, especially on the second-hand goods market (see Akerlof's study on the "Lemon Market "). This founding article has subsequently given rise to a considerable number of articles and works on the one hand to extend to other types of market (insurance, finance, work, etc.) and on the other hand to theoretical deepenings which have been at the origin of many Nobel prizes in economics, in the field of market finance in particular. Let us summarize Akerlof's demonstration.

Let q be the quality index of used cars, where q is a random variable distributed uniformly over the interval [0, 1]. The average quality of a car is then the mathematical expectation of q, or 1/2. The market is made up of a large number of buyers and sellers. Buyers have a reserve price of (3/2) q for a quality vehicle q, while sellers want to get at least q of their car. It is assumed that the buyers are risk-neutral, that is to say that in the ignorance of the quality of the vehicle they have in front of them, they evaluate it on the basis of expectation or anticipation of its quality, and are therefore willing to pay 3/2 * 1/2 = 3/4. If the quality of each vehicle were observable, each seller could sell his vehicle at a price at least equal to q (because there are buyers willing to pay 3 / 2q), and each car would be sold at a price between q and 3 / 2q. If the quality of the cars is not observable by the buyers, then any buyer is ready to pay 3/4 for any vehicle. In this case, the holders of vehicles of quality higher than 3/4 can not find a taker and exit the market. The quality of the remaining vehicles is then between 0 and 3/4, with an expected quality of 3/8. Knowing this, buyers are willing to pay only 3/2 * 3/8 = 9/16, causing the sellers of cars whose quality is higher than 9/16 to exit. By repeating the reasoning, it is deduced that only zero quality vehicles sold at zero prices remain on the market.

In the presence of asymmetrical information, the market is thus unable to select the best products and determine equilibrium prices. On the other hand, we see that we are facing a process of eliminating the best products by adverse selection.

So there is a real "market failure" that follows, not from its absence, as in the case of public goods, but because of its intrinsic dysfunction. Many other cases of asymmetric information can be found in other markets such as the labor market in which employers (job offerers) are never assured that their expectations, as to the qualifications of the people they will hire, will be satisfied and, on the other hand, employees (jobseekers) are also asymmetrically informed about the working conditions in the company and about their career prospects and compensation. This insurmountable asymmetry makes it possible to account for the phenomena of labor market segmentation, discrimination and adverse selection. Another market, that of credit, also presents a problem of information asymmetry between borrowers (households and businesses) and lenders (banks and credit institutions). Effects similar to those of the labor market are obtained in terms of segmentation, discrimination and adverse selection. Even the stock markets, where access to information is a priori the most symmetrical, the freest and the most egalitarian, there is also a strong asymmetry of information that results in mimetic behaviors that lead to speculative bubbles, which end up bursting into financial crises. In the stock market, which is at the very heart of the capitalist market economy and its symbolic representation the most emblematic, the law of supply and demand, expression of the perfect symmetry of markets, can not exist. Indeed, when the price of a stock increases, its profitability also increases which encourages to buy even more. It is the opposite process that occurs in case of decline without any opposite symmetrical force coming to restore the equilibrium.

Ultimately, no market escapes the informational asymmetry and therefore to adverse selection effects.

But asymmetric information can also provoke, after the signing of a contract or because of a sectoral or macroeconomic policy, another pernicious effect: moral hazard. This effect has mainly been highlighted for insurance contracts in which an insured can be driven, after having taken out an insurance policy against a risk (accident, theft, water damage, fire, etc.) to be adopted. a lax attitude that increases the risks and therefore the level of the indemnities paid by the insurance company. The effects of moral hazard are found in many other areas. Let's take a few examples of recent controversies about unemployment, banking crises or the budget deficit.

• Unemployment insurance can delay job search and aggravate unemployment,

• The guarantee that central banks offer in case of bankruptcy of big banks ("too big to fail") can encourage banks to take high risks.

• The guarantee that the Central Banks offers to cover the budget deficit of a state may incite them to make reckless expenditure.

• Etc.

Contemporary analyzes of behavioral economic psychology confirm the frequency of asymmetric behaviors.

The psychologist Daniel Kahneman, winner of the 2002 Nobel Prize for Economics for his "pioneering and original work on behavior in uncertainty and risk," and economist Richard Thaler, have shown that there is often a bias in asymmetry of behavior under uncertainty.

For example, they show that most people are more likely to suffer losses than gains of the same amount. This is why in the financial markets non-professional investors are always reluctant to part with an impaired financial asset, which leads to inefficient management of their securities portfolio. This "aversion to loss", which is found in animals, should not be confused with risk aversion or dispossession (I tend to think that what I own is always worth more than the market offers me). If I have to take a risky decision on a potential gain and if I have a risk aversion, I a solution that is suboptimal but safer. But if I have to make a risky choice leading to a loss, I am willing to take more risks, provided I have a chance, even minimal, to reduce my loss. Similarly, agents are sensitive to scale effects. I do not react in the same way on a probability between 0.99 and 1 as between 0.2 and 0.21 and in general there are differences of behavior near the extremes of probability (close to 0 or 1). As a result, this has implications for the insurance market or financial markets (in terms of price rigidity of financial assets due, for example, to an under-reaction of agents to bad news) that contradicts neoclassical theory where agents are expected to always make choices that maximize their intended utility. These cognitive biases lead to "normal" behaviors that can be considered "irrational" from the point of view of standard neoclassical theory. Other examples can be found in other markets (labor, raw materials) that show that public policies to act on the markets are perhaps wrong about the causes and factors of these rigidities or viscosities. Overall, it is asymmetry and not neoclassical symmetry that is the norm for the behavior of agents on the markets. c) Asymmetric economic and social relations in Ricardo-Marxist, institutionalist and evolutionist theories.

1. For classical economists from Smith to Ricardo to Malthus, Sismondi and Marx, market prices reflect the prices of production, which are themselves determined by the value of goods and services in terms of "quantity of labor". It can be "incorporated" in goods or it can be "bought", it can be "concrete" (hours of work) or "abstract", as "average social standard" of labor time needed to produce "goods" ". Marx, like Ricardo, argues that "market prices" (the concrete reality) revolved around "production prices" which gravitated around "values" ("abstract social working time" unobservable) which are norms. abstract.

But in marx's theory of value, the "quantum of abstract social work" is determined, not in the sphere of the market but in the secret laboratory of production "where the" social relations of production "are formed which are fundamentally asymmetric between agents belonging to social classes with radically divergent economic interests. The "wage relationship" is not a "symmetrical" relationship between suppliers and job-seekers, but an asymmetrical "operating relationship" that generates "surplus value" which is itself an "abstract norm" (unobservable) around which gravitates the "rate of profit", observable and concrete.It is this fundamental asymmetry between social classes which is the source of the general dynamics of the capitalist market economy, both in the short term (recurrent crises

In the long term (a downward trend in the rate of profit and the proletarian revolution), the very idea of surplus implies the existence of an asymmetrical relationship capable of accounting for the emergence of a surplus product. The profit rate, the salary and the rent are not equilibrium "prices" (on capital, labor or land markets) but indicators or asymmetric distribution variables of a surplus product or an economic surplus, Social asymmetric laws (decreasing returns, land scarcity), demographics (overcrowding laws of Malthus) or "social" (Ricardo, Marx), explain the inevitability of downward trends in the rate of profit and the long-term evolution towards a "stationary state" which is a "stable" state within the meaning of the general theory of systems.is a state of absolute stability and symmetric equilibrium. This is a state of absolute stability and symmetrical equilibrium that results of asymmetrical relations. Indeed, the permanent and maximum search for profit leads to 2. The current of evolutionary economists tries to understand the long-term dynamics of the world economic system from the notion of intrinsic imbalance specific to the market economy. They introduce into the economic analysis dynamic processes, notions of evolution, selection, diffusion, cumulative effects, innovations, accidental causes, irreversibility ... which have unpredictable large-scale effects. and that can not be resorbed easily, quickly or automatically by the game of the only rules of the market. The idea of uniform undifferentiated agents in their behavior must be abandoned and their heterogeneity recognized not only in terms of their own rationalities, their divergent strategies and their respective time horizons (degree of uncertainty), but also and above all of their contradictory and asymmetrical relations in terms of the power struggle. The field of evolutionary analysis then shifts the question of short and long-term balances to the idea of "complex" systems subject to long trajectories, like the evolution of species in biology.

The phenomena of chance and selection then appear in these "trajectories" which can only be understood and analyzed by integrating the long time of history and the non-linear nature of the "paths" traveled. How have some innovations and technologies managed to become dominant? How do some companies impose themselves? How have institutions gradually been shaped? What is the role of "chance and necessity" in these evolutions?

How does the emergence of cultural norms and social norms expand or limit the scope of possibilities in systemic evolutions? All of these issues imply the existence of asymmetrical relationships at all levels.

CONCLUSION

Economics includes currents of thought within or on the margins of neoclassical theory that have sought to relax the most unsustainable assumptions of the idealistic view of the model of pure and perfect competition (see the theory of rational expectations or theories imbalance, for example). But it is outside the neoclassical current, among the heterodox (Keynesian and post-Keynesian, neo-institutionalist, evolutionist, new international political economy, regulationist, etc.) that the notions of inequality, rupture, agglomeration, divergence, power, domination, etc., which implies that asymmetric relations dominate the functioning of market economies, in terms of both structures and behaviors as well as longterm and short-term adjustments. However, these ideas remain scattered and do not yet constitute a solid alternative theoretical corpus that would make asymmetrical relations the norm of the economic theory of the market.

Bibliographie choisie

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  Kurtosis (from the ancient Greek feminine name κύρτωσις, "curvature"), is a flattening coefficient. Given a real random variable X, expectation μ and standard deviation σ, unstandardized kurtosis is the fourth order moment of the reduced centered variable. A high flattening coefficient indicates that the distribution is rather pointed in its mean, and has thick distribution tails. We can distinguish several characteristic types of symmetrical distributions of a real random variable (see the graph below ) Plot of several symmetric unimodal probability densities with unit variance. From highest to lowest peak: red, kurtosis 3, Laplace (D)ouble exponential distribution; orange, kurtosis 2, hyperbolic (S)ecant distribution; green, kurtosis 1.2, (L)ogistic distribution; black, kurtosis 0, (N)ormal distribution; cyan, kurtosis -0.593762…, raised (C)osine distribution; blue, kurtosis -1, (W)igner semicircle distribution; magenta, kurtosis -1.2, (U)niform distribution. The asymmetry coefficient (skewness) measures the asymmetry of the distribution of a real random variable. A positive coefficient indicates a distribution shifted to the left of the median, and therefore a distribution tail spread to the right. A negative coefficient indicates a distribution shifted to the right of the median, and thus a distribution tail spread to the left (Mode> Median> Mean). In general, we distinguish in economy three types of asymmetric coefficients, the main one being the Fisher's asymmetry coefficient: ratio of the 3-point centered moment ( ) to the standard deviation cube ( ) :

  over-accumulation of capital and a tendency to lower the rate of profit, which can only be counteracted by the periodic occurrence of increasingly serious economic, financial and social crises that destroy of capital, momentarily increases the rate of profit and favors the concentration of capital. Class relations are more and more asymmetrical to the breaking point that triggers a revolutionary period. it then belongs, according to Marx, to the social class of workers who are under the yoke of capitalist social exploitation, to propose a new model and system of production that transcends the old, frees the dominated social classes and opens the way to a new quantitative and qualitative growth of the productive forces in the framework of new relations of production which will remain asymmetrical ("dictatorship of the proletariat") for a more or less long time to gradually give way to a society freed from all exploitation and asymmetric alienation . It is interesting to note that this new economic system that Marx had scarcely envisioned is characterized above all by the radical lifting of all the economic uncertainties and asymmetrical relations between social groups. Socialism, as an intermediate step towards communism, opens up to central planning the possibility of ex ante social validation of the products of social work and thus the suppression of the market economy, uncertainty and asymmetrical relations attached to it. The central planner, who is supposed to answer a priori the needs of the whole society, replaces in a certain way the secretary or the auctioneer of the neoclassical theory of the general equilibrium. It is surprising to note that Soviet theorists of socialist planning were mostly mathematicians (Kantorovich, Nemchinov ...) who reflected in a general neoclassical framework in which price signals were calculated ex ante by the central office planning to achieve the perfect co-ordination of all productive activities according to a hierarchy of social needs necessarily rationed by natural resources, technical means and limited labor skills.

  Les articles et ouvrages des auteurs et des thèmes cités dans cette communication sont aujourd'hui directement consultables sur internet) • Akerlof George (1970), « The Market of Lemons: Quality, Uncertainty and the Market Mechanism", QJE. • Kenneth J. Arrow et Gérard Debreu, « The Existence of an Equilibrium for a Competitive Economy », Econometrica, vol. XXII, p. 265-90 • Catherine Bruneau et Olivier De Bandt (janvier 1998), La modelisation var structurel : application à la politique monetaire en France, Banque de France, Direction générale des études. • Nathalie Lazaric, Les théories économiques évolutionnistes, La Découverte, collection « Repères », 2010, 125 p. • Diane-Gabrielle Tremblay, « L'apport des théories institutionnalistes au renouvellement de l'approche d'économie politique », Revue Interventions économiques [En ligne], 28 | 2002, mis en ligne le 01 janvier 2002 • Montmarquette Claude (2008), L'économétrie des données expérimentales : défis et opportunités, Economie & prévision, n°182, pp. 7 à 17