Colin Cros
email: colin.cros@gipsa-lab.fr

Pierre-Olivier Amblard
email: pierre-olivier.amblard@cnrs.fr.

Christophe Prieur
email: christophe.prieur@gipsa-lab.fr

Jean-François Da Rocha
email: jeanfrancois.darocha@telespazio.com.

P.-O Amblard

† C Cros

J.-F Da Rocha

Fusion of Distance Measurements between Agents with Unknown Correlations

Keywords: Filtering, sensor network, cooperative control

Cooperative localization is a promising solution to improve the accuracy and overcome the shortcomings of GNSS. Cooperation is often achieved by measuring the distance between users. To optimally integrate a distance measurement between two users into a navigation filter, the correlation between the errors of their estimates must be known. Unfortunately, in large scale networks the agents cannot compute these correlations and must use consistent filters. A consistent filter provides an upper bound on the covariance of the error of the estimator taking into account all the possible correlations. In this paper, a consistent linear filter for integrating a distance measurement is derived using Split Covariance Intersection. Its analysis shows that a distance measurement between two agents can only benefit one of them, i.e., only one of the two can use the distance measurement to improve its estimator. Furthermore, in some cases, none can. A necessary condition for an agent to benefit from the measurement is given for a general class of objective functions. When the objective function is the trace or the determinant, necessary and sufficient conditions are given.

Introduction

Accurate positioning is a key challenge for numerous strategic applications. Global Navigation Satellite Systems (GNSSs) provide a low-cost and effective solution for achieving satisfactory accuracy in open-sky environments. However, in GNSS-denied environments such as urban canyons, indoors or underwater, other methods must be used. Cooperative localization is a promising alternative because it can slow down the loss of accuracy in the absence of GNSS signal [START_REF] Roumeliotis | Analysis of multirobot localization uncertainty propagation[END_REF]. A simple form of cooperation is the measurement of distances between the users, hereafter called agents. Inter-agent distances can be inexpensively determined by measuring the Received Signal Strength or the Time-of-Flight of a signal. For example, cooperative localization has been used for terrestrial vehicles [START_REF] Hossain | Cooperative vehicular positioning with vanet in urban environments[END_REF] and UAVs [START_REF] Causa | Improving navigation in gnss-challenging environments: Multi-uas cooperation and generalized dilution of precision[END_REF][START_REF] Chen | Distributed and collaborative localization for swarming uavs[END_REF]. Recent research has demonstrated the effectiveness of cooperation in significantly improving positioning accuracy while using simple algorithms [START_REF] Minetto | On the trade-off between computational complexity and collaborative gnss hybridization[END_REF].

One of the challenges in cooperative systems when integrating inter-agent distance measurements is dealing with the correlations between the errors of the agents' estimators. The use of inter-agent measurements tends to correlate these errors. If the cross-covariances are not taken into account, they lead to an underestimation of the errors and a potential divergence of the filter [START_REF] Arambel | Covariance intersection algorithm for distributed spacecraft state estimation[END_REF]. These problems can be avoided by calculating and storing the cross-covariances between each pair of estimators. In centralized systems, the estimators of all agents are stacked into a global estimator whose covariance includes all the cross-covariances. This global estimator is then updated in a centralized manner (at a computing station), or in a distributed manner [START_REF] Roumeliotis | Distributed multirobot localization[END_REF]. However, such solutions are difficult to implement for networks with a large number of agents due to the computational and transmission costs. Another solution to avoid correlation problems involves ensuring that no redundant information is used. In [START_REF] Bahr | Consistent cooperative localization[END_REF], the authors propose to keep a table of all past interactions to ensure that only independent information is used in the filter. This requires storing the history of past interactions and may be prohibitive for large networks.

In order to use two-way cooperation when some correlation information is unknown, covariance consistent filters should be used. The literature also refers to these as "conservative filters", see, e.g., [START_REF] Forsling | Conservative linear unbiased estimation under partially known covariances[END_REF]. Covariance consistency ensures that the estimator is not overconfident by providing an upperbound on the estimator error. Several methods have been developed to fuse different estimators with unknown covariances. A recent paper [START_REF] Forsling | Conservative linear unbiased estimation under partially known covariances[END_REF] unifies some of them in a general framework called the Conservative Linear Unbiased Estimator (CLUE). Among these methods, Covariance Intersection (CI) [START_REF] Julier | A non-divergent estimation algorithm in the presence of unknown correlations[END_REF] and its extension the Split Covariance Intersection (SCI) [START_REF] Julier | General decentralized data fusion with covariance intersection (CI). Handbook of Multisensor Data Fusion[END_REF] are particularly efficient: several papers, see, e.g., [START_REF] Chen | Estimation under unknown correlation: covariance intersection revisited[END_REF], have described the efficiency of CI, and more recently CI has been shown to be the best method to fuse two estimators under unknown correlations [START_REF] Reinhardt | Minimum covariance bounds for the fusion under unknown correlations[END_REF]. Even if SCI has not been shown to be optimal, its simplicity makes it a widely used method for decentralized cooperative positioning [START_REF] Li | Cooperative multi-vehicle localization using split covariance intersection filter[END_REF][START_REF] Lima | Data fusion with split covariance intersection for cooperative perception[END_REF][START_REF] Pierre | Range-only based cooperative localization for mobile robots[END_REF]. The integration of distance measurements using SCI have already been considered by [START_REF] Julier | Using covariance intersection for slam[END_REF][START_REF] Noack | Automatic exploitation of independencies for covariance bounding in fully decentralized estimation[END_REF] in the context of SLAM. The authors take benefit from the fact that the observation noise is assumed independent and therefore provides new information. However, to the best of our knowledge, no specific study on the usefulness of the integration of a distance measurement between two agents with an unknown correlation using SCI has been performed in the literature.

In this paper, we consider cooperative networks where agents estimate their positions and store their own estimators. Here, cooperation occurs through distance measurements between agents. We address the problem of integrating, through linear filtering, a distance measurement between two agents who have covariance consistent estimators of their states but the cross-covariance between these estimators is unknown. Throughout the paper, we only consider the integration of a distance measurement. In particular, the dynamics of the system is beyond the scope of this paper and is the subject of further work. Our first contribution is the derivation of the optimal SCI filter for this problem, optimal w.r.t. a given increasing cost function. In practice, this function is often the trace or the determinant of the covariance of the estimate. Our second contribution is the analysis of the SCI filter which shows whether a distance measurement can be used to improve the estimation. We provide a necessary condition in the general case, and necessary and sufficient conditions for the cases where the cost function is the trace or the determinant.

The paper is organized as follows. Section 2 introduces the problem of optimal filtering, then Section 3 derives a candidate: the SCI filter. Section 4 investigates the usefulness of such a filter, and Section 5 discusses the results, in particular in simulations. Finally, Section 6 provides some perspectives and future directions.

Notation. In the sequel, vectors are underlined e.g., x ∈ R n , random variables are denoted in lowercase boldface letters e.g., z for a scalar or x x x for a vector, and matrices are denoted in uppercase boldface variables e.g., M ∈ R n×n . The notation E[•] denotes the expected value of a random variable and • the Euclidean norm of a vector. The trace, the determinant, the inverse and the transpose of a matrix M and the identity matrix are denoted as tr M , |M |, M -1 , M ⊺ and I respectively. For two matrices A and B, A B means that the difference B -A is positive semidefinite. A positive definite matrix P is represented in the figures by the ellipsoid

EP = x | x ⊺ P -1 x ≤ 1 .

Problem statement

Let us first recall the definition of a covariance consistent estimator. Definition 1. An estimator (x x x, P ) of a random variable x x x is said to be covariance consistent (or shortly consistent in the sequel) if E [x x x] = 0 and P P where x x x = x x x -x x x denotes the error and P = E [x x xx x x⊺ ] the mean-squared error (MSE).

In other words, a consistent estimator is unbiased and does not underestimate the covariance of the error.

Consider two agents, denoted A and B, characterized by their states x x x A and x x x B in R n . For the sake of simplicity and without loss of generality, the states are assumed to contain only the positions of the agents; in practice, they may also contain their orientations or their velocities, for example. The two agents have consistent estimators of their states denoted (x x xA , PA) and (x x xB , PB). Notice that the true (centralized) covariances of the errors, defined as,

P = E x x xA x x xB x x xA x x xB ⊺ = PA PAB P ⊺ AB PB , (1)
is unknown, i.e., PA, PB, and PAB are unknown. However, consistency restrains the set of possible P to:

P = PA PAB P ⊺ AB PB 0 | PA PA, PB PB . (2)
Furthermore, consider z = x x x A -x x x B + z a measurement of the distance between A and B where z denotes the error. The error z is assumed centered, E [z] = 0, with variance E z2 = σ 2 m , and independent of the errors of the estimators x x xA and x x xB . Finally, consider that the measurement can be linearized around the means of the estimators xA and xB . Introducing the unit-length director vectors

u BA = xA -x B xA -x B
, the linearized observation writes:

z = u ⊺ BA (x x x A -x x x B ) + z, (3)
where z is still assumed independent from the errors of the estimators. This assumption is necessary to use SCI which is a linear fusion. It is quite idealistic and causes a loss of precision as the distance is a nonlinear function. It is reasonable if the agents are sufficiently far from each other, as the second-order terms become negligible. Without this assumption, no linear filter can be used.

The objective is to improve the estimator of A using the estimator of B and the distance measurement to create a better estimator (x x xF , PF ) of the state of A. Throughout this paper, the estimators are compared w.r.t. an increasing cost function J (increasing in the sense of the Loewner ordering i.e., P ≺ Q =⇒ J(P ) < J(Q)). For two consistent estimators (x x x1 , P1) and (x x x2 , P2) of the same random variable x x x, (x x x1 , P1) is said to be better than (x x x2 , P2) if J(P1) < J(P2). The estimator (x x xF , PF ) is designed as an unbiased1 linear filter defined as follows.

Definition 2. A linear filter for the state of A is an estimator (x x xF , PF ) of x x x A where x x xF is a linear combination of x x xA , x x xB and z. It is defined by two matrices KF , LF and a vector w F such that:

x x xF = KF x x xA + LF x x xB + w F z. (4)
To have x x xF unbiased, since

E[z] = u ⊺ BA (E [x x x A ] -E [x x x B ]
) the gains must be dependent and satisfy:

KF = I -w F u ⊺ BA , LF = w F u ⊺ BA . (5)
An unbiased linear filter is then only defined by w F as:

x x xF = (I -w F u ⊺ BA ) x x xA + w F u ⊺ BA x x xB + w F z. (6)
For a possible P ∈ P, the covariance of the error x x xF = x x x A -x x xF is:

PF = PA + σ2 A + σ2 B -2γ + σ 2 m w F w ⊺ F -PA -PAB u BA w ⊺ F -w F u ⊺ BA PA -P ⊺ AB (7)
where

σ2 A = u ⊺ BA PAu BA , σ2 B = u ⊺ BA PBu BA , (8) γ
= u ⊺ BA PABu BA . (9)
If the true covariance P is known, the optimal linear filter is calculable. It is derived from the classical Kalman Filter equations [START_REF] Anderson | Optimal filtering[END_REF]:

x x x * F = x x xA + z -u ⊺ BA (x x xA -x x xB ) σ2 A + σ2 B -2γ + σ 2 m ( PA -PAB)u BA , (10a)
P * F = PA - ( PA -PAB)u BA u ⊺ BA ( PA -P ⊺ AB ) σ2 A + σ2 B -2γ + σ 2 m . (10b)
In this case, P * F is the minimum (in the Loewner ordering) of the PF , and thus is optimal w.r.t. any increasing function J.

However, as P is not known, such a consideration is not possible, and the linear filter must be consistent, i.e., it must satisfy PF PF for every possible covariance. We are now in a position to state the main problem.

Problem 1. Find a gain w F and a covariance PF such that the estimator (x x xF , PF ) defined by ( 6) is consistent and optimal w.r.t. J. In other words:

minimize w F ,P F J(PF ) subject to ∀ P ∈ P, PF PF (P1)
where PF is given by [START_REF] Chen | Distributed and collaborative localization for swarming uavs[END_REF].

The main result of this paper is the design of a candidate filter solving Problem 1 using SCI and its analysis.

SCI for a distance measurement

As a preliminary remark, (6) can be rewritten as:

x x xF = x x xA + w F [z -u ⊺ BA (x x xA -x x xB )] (11a) = (I -w F u ⊺ BA ) x x xA + w F u ⊺ BA (x x xB + zu BA ) . ( 11b
)
These two expressions highlight the equivalence between linear filtering and fusion. Equation (11a) represents the usual form of the Kalman correction step: w F is a gain and z -u ⊺ BA (x x xA -x x xB ) is the innovation on the measurement. Equation (11b) on the other hand represents a fusion: the term x x xB + zu BA is another estimator of the state of A.

Let x x x′

A denote this estimator. Equation (11b) corresponds to the fusion of x x xA and the observation of x x x′ A through u BA . Since the errors x x xA and x x xB are correlated to an unknown degree and z is independent of x x xA , the fusion of x x xA and x x x′

A respects the assumptions of SCI [START_REF] Julier | General decentralized data fusion with covariance intersection (CI). Handbook of Multisensor Data Fusion[END_REF]. SCI fuses two estimators using a linear combination whose weights depend on their covariances. For any ω ∈ [0, 1), SCI provides a consistent estimator (x x xSCI (ω), PSCI(ω)) for the state of A, hereafter called an SCI filter. It is obtained by the SCI equations [12, Eq. (12.24)-(12.25)]. The covariance is:

PSCI(ω) = (1 -ω)P -1 A + ω u BA u ⊺ BA σ 2 B + ωσ 2 m -1 = 1 1 -ω PA - ωPAu BA u ⊺ BA PA ωσ 2 A + (1 -ω) (σ 2 B + ωσ 2 m ) , (12a)
where

σ 2 A = u ⊺ BA PAu BA and σ 2 B = u ⊺ BA PBu BA .
The corresponding estimator is x x xSCI (ω) = x x xF given by ( 11) with a gain w F = w SCI (ω):

w SCI (ω) = ω ωσ 2 A + (1 -ω) (σ 2 B + ωσ 2 m ) PAu BA . (12b)
The optimal filter w.r.t. J within the family of SCI filter is called the optimal SCI filter. To find the optimal SCI filter, the parameter ω is chosen to minimize the cost function J:

ω * = arg min 0≤ω≤1 J(PSCI(ω)). ( 13
)
We have proved the following result.

Theorem 1. The optimal SCI filter for the state of A is (x x xSCI (ω * ), PSCI(ω * )) given by [START_REF] Julier | General decentralized data fusion with covariance intersection (CI). Handbook of Multisensor Data Fusion[END_REF] where the parameter ω * is defined by [START_REF] Julier | Unscented filtering and nonlinear estimation[END_REF].

In the sequel, this optimal SCI filter is denoted (x x x * SCI , P * SCI ). SCI has not been shown to be optimal for the fusion, therefore the optimal SCI filter (x x x * SCI , P * SCI ) may be suboptimal for Problem 1. However, since it is very simple to implement, it is widely used in practice. Furthermore, when the measurement is very accurate, i.e., when σ 2 m tends to 0, SCI becomes CI which is optimal (only when σ 2 m = 0) [START_REF] Reinhardt | Minimum covariance bounds for the fusion under unknown correlations[END_REF]. These two reasons lead us to consider this filter and analyze its usefulness.

Usefulness of the filtering 4.1 General increasing cost function

Theorem 1 gives the expression of the optimal SCI filter. Setting ω = 0 in the fusion ensures that the estimator (x x x * SCI , P * SCI ) is at least as good as the original estimator (x x xA , PA). In fact, setting ω = 0 corresponds to keeping the original estimator and ignoring the measurement. However, there is no reason why there should be a better estimator, the optimal parameter (13) could be ω * = 0. In such a case, SCI filters cannot improve the estimator of A. We call pertinent a linear filter for the state of A that provides a better estimator than (x x xA , PA). This section characterizes the pertinence of SCI filters. Definition 3. A pertinent linear filter for the state of A is a linear filter (x x xF , PF ) such that J(PF ) < J(PA).

By definition of the optimal SCI filter, there is a pertinent SCI filter if and only if the optimal SCI filter is pertinent. The underlying question is therefore: Is the optimal SCI filter (x x x * SCI , P * SCI ) pertinent? Let us start with two corollaries of Theorem 1 that provide a necessary condition for the existence of a pertinent SCI filter.

Corollary 1. If σ 2

A ≤ σ 2 B , there is not any pertinent SCI filter for the state of A.

Proof. Assume that σ 2

A ≤ σ 2 B and let ω ∈ [0, 1). We will show that PSCI(ω) PA which is sufficient since J is increasing. In (12a), P -1 SCI (ω) is expressed as a convex combination of P -1

A and Q =

u BA u ⊺ BA σ 2 B +ωσ 2 m . Therefore, if Q P -1 A , then P -1 SCI (ω) P -1 A . Let us prove that Q P -1 A , by proving that ∀v, v ⊺ (P -1 A -Q)v ≥ 0. Write v = αu BA + w with α ∈ R and w ⊺ u BA = 0. Then, v ⊺ Qv = α 2 /(σ 2 B + ωσ 2 m ) ≤ α 2 /σ 2 A . Finally, let us prove that v ⊺ P -1 A v ≥ α 2 /σ 2 A .
Consider an orthonormal basis B = (u BA , u 2 , . . . , u n ) containing u BA and the orthogonal matrix R = u BA u 2 • • • u n . In this basis, the covariance of the error of the estimator of A and the vector v become:

PA = R σ 2 A b ⊺ b C R ⊺ , v = R α w ′
where b, w ′ ∈ R n-1 and C ∈ R (n-1)×(n-1) . Let S = C -bb ⊺ /σ 2 A be the Schur complement of the first entry. S is invertible and S -1 0, see, e.g., [START_REF] Horn | Matrix analysis[END_REF]Chap. 7]. The inverse of PA and v ⊺ P -1

A v become:

P -1 A = R σ -2 A + σ -4 A b ⊺ S -1 b -σ -2 A b ⊺ S -1 -σ -2 A S -1 b S -1 R ⊺ , v ⊺ P -1 A v = α 2 σ 2 A + α σ 2 A b -w ′ ⊺ S -1 α σ 2 A b -w ′ . Hence, v ⊺ P -1 A v ≥ α 2 σ 2 A . Corollary 2.
If there is a pertinent SCI filter for the state of A, then there is not any pertinent SCI filter for the state of B.

Proof. If there is a pertinent SCI filter for the state of A, Corollary 1 implies σ 2 B < σ 2 A . Similarly, if there is a pertinent SCI filter for the state of B, then σ 2 A < σ 2 B . Both inequalities cannot hold simultaneously.

Corollary 1 provides only a necessary condition for the existence of pertinent SCI filters. In the following, we extend the property to provide a necessary and sufficient condition for the two most commonly used cost functions: the trace and the determinant. It is based on the following result.

Lemma 1. Assume that the function f : ω → J(PSCI(ω)), with PSCI(ω) given by (12a), is convex on [0, 1). Then, the optimal SCI filter (x x x * SCI , P * SCI ) is pertinent if and only if f ′ (0) < 0.

Proof. By definition, (x x x * SCI , P * SCI ) is pertinent if and only if f (ω * ) < f (0) = J(PA). If f ′ (0) < 0, then there is a ω0 > 0 such that f (ω0) < f (0) and therefore f (ω * ) < f (0). If f ′ (0) ≥ 0, since f is convex on [0, 1), ∀ω ∈ [0, 1), f (ω) ≥ f (0) and therefore, f (ω * ) = f (0).

Particular case of the trace

In this paragraph, consider J(•) = tr • and let g : ω → tr PSCI(ω) be the cost function to be optimized. According to (12a), the cost function is:

g(ω) = tr PA 1 -ω 1 - ωrAσ 2 A ωσ 2 A + (1 -ω)(σ 2 B + ωσ 2 m ) (14)
where:

rA = 1 σ 2 A PAu BA 2 tr PA . ( 15
)
The meaning of rA is discussed in Section 5. In order to apply Lemma 1, let us prove the following.

Lemma 2. The cost function g is convex on [0, 1).

Proof. Let us first prove that 0 < rA ≤ 1. Consider again the orthonormal basis B = (u BA , u 2 , . . . , u n ) containing u BA and the orthogonal matrix

R = u BA u 2 • • • u n .
In this basis, the covariance of the estimator of A becomes:

R ⊺ PAR =      σ 2 A ρ2σAσ2 • • • ρnσAσn ρ2σAσ2 σ 2 2 * * . . . * . . . * ρnσAσn * * σ 2 n      ( 16
)
where only the diagonal coefficients and the correlations with the first component have been labeled. With these notations, rA is developed as:

rA = 1 σ 2 A PAu BA 2 tr PA = σ 2 A + n i=2 ρ 2 i σ 2 i σ 2 A + n i=2 σ 2 i . ( 17
)
Since the correlations satisfy |ρi| ≤ 1, the ratio satisfies 0 < rA ≤ 1.

To prove the convexity of g, let us first assume that rA < 1. By putting all the terms in the same fraction, the cost function g is a rational function:

g(ω) = tr PA 1 -ω (1 -rA)ωσ 2 A + (1 -ω)(σ 2 B + ωσ 2 m ) ωσ 2 A + (1 -ω)(σ 2 B + ωσ 2 m ) = tr PA 1 -ω σ 2 B + [(1 -rA)σ 2 A -σ 2 B + σ 2 m ]ω -σ 2 m ω 2 σ 2 B + (σ 2 A -σ 2 B + σ 2 m )ω -σ 2 m ω 2
.

Let P : ω → σ 2 B +(σ 2 A -σ 2 B +σ 2 m )ω -σ 2 m ω 2 and Q : ω → σ 2 B +[(1-rA)σ 2 A -σ 2 B +σ 2 m ]ω - σ 2
m ω 2 be the polynomials of degree 2 at the denominator and numerator. Since P (0) = σ 2 B and P (1) = σ 2

A , P has two roots b and d which satisfy

: b < 0 < 1 < d. Similarly, by noting that Q(ω) = P (ω) -rAσ 2 A ω, Q(b) = -rAbσ 2 A > 0, Q(1) = (1 -rA)σ 2 A > 0 and Q(d) = -rAdσ 2
A < 0, Q has two roots a and c which satisfy:

a < b < 0 < 1 < c < d.
The cost function is therefore:

g(ω) = - tr PA(ω -a)(ω -c) (ω -1)(ω -b)(ω -d) .
Then, using partial fraction decomposition:

g(ω) = tr PA A ω -1 + B ω -b + C ω -d (18)
with:

A = - (1 -a)(1 -c) (1 -b)(1 -d) < 0, B = - (b -a)(b -c) (b -1)(b -d) > 0, C = - (d -a)(d -c) (d -1)(d -b) < 0.
Since tr PA > 0, the three terms in [START_REF] Noack | Automatic exploitation of independencies for covariance bounding in fully decentralized estimation[END_REF] are convex on (b, 1). Therefore, since b < 0, the cost function g is convex on [0, 1). Finally, if rA = 1, then c = 1 which simplifies the expression of g. By the same logic, g is convex on (b, d), and thus on [0, 1]. Note that rA = 1 corresponds to the one-dimensional case.

We are now in a position to apply Lemma 1 to the trace. Theorem 2. The optimal SCI filter for the state of A w.r.t. to the trace is pertinent if and only if:

σ 2 B < rAσ 2 A . (19)
Proof. Since g is convex on [0, 1) as stated in Lemma 2, by Lemma 1, the optimal SCI filter is pertinent if and only if g ′ (0) < 0. The claim then follows from g ′ (0

) = tr PA 1 -rAσ 2 A /σ 2 B .

Particular case of the determinant

In this paragraph, consider J(•) = |•| and let h : ω → |PSCI(ω)| be the cost function to be optimized. According to (12a), the cost function is:

h(ω) = |PA| (1 -ω) n I - ωP 1/2 A u BA u ⊺ BA P 1/2 A ωσ 2 A + (1 -ω)(σ 2 B + ωσ 2 m ) (20)
where

P 1/2 A
denotes the square root of PA and n is the dimension of the state. Using that |I + uv ⊺ | = 1 + u ⊺ v, (20) becomes:

h(ω) = |PA| (1 -ω) n-1 σ 2 B + ωσ 2 m ωσ 2 A + (1 -ω)[σ 2 B + ωσ 2 m ] . (21)
As for the trace, the convexity of h is proven in the following result.

Lemma 3. The cost function h is convex on [0, 1).

Proof. This proof is very similar to the proof of Lemma 2. First, let P :

ω → σ 2 B + (σ 2 A -σ 2 B + σ 2 m )ω -σ 2 m ω 2 .
As P (0) > 0, P (1) > 0 and P (-σ 2 B /σ 2 m ) < 0, the cost function h is a rational function which can be expressed as follows.

h(ω) = - |PA| (ω + σ 2 B /σ 2 m ) (1 -ω) n-1 (ω -a)(ω -b)
where the zeros of P , denoted a and b, satisfy:

-σ 2 B /σ 2 m < a < 0 < 1 < b.
Then, using partial fraction decomposition:

h(ω) = |PA| A (ω -a)(1 -ω) n-1 + |PA| B (ω -b)(1 -ω) n-1 (22)
with:

A = -(a + σ 2 B /σ 2 m ) (a -b) > 0, B = -(b + σ 2 B /σ 2 m ) (b -a) < 0.
Both terms in [START_REF] Roumeliotis | Analysis of multirobot localization uncertainty propagation[END_REF] are convex on [0, 1).

We are now in a position to apply Lemma 1 to the determinant.

Theorem 3. The optimal SCI filter for the state of A w.r.t. to the determinant is pertinent if and only if:

σ 2 B < 1 n σ 2 A . ( 23
)
Proof. Since h is convex on [0, 1) as stated in Lemma 3, the optimal SCI filter is pertinent if and only if h ′ (0) < 0. The claim then follows from h ′ (0

) = |PA| n -σ 2 A /σ 2 B . 9 -4 -2 0 2 4 20 -2 0 2 A B (a) J(•) = tr • -4 -2 0 2 4 20 -2 0 2 A B (b) J(•) = |•|
Figure 1: Example of the optimal SCI filters for two different cost functions J. The black solid ellipses represent the covariance of the estimators P A and P B . The dashed ellipse represents the covariance of the optimal SCI filter P * SCI . The grey solid ellipses represent possible covariances PF obtained by [START_REF] Chen | Distributed and collaborative localization for swarming uavs[END_REF] with w F = w SCI (ω * ) and different P ∈ P, and the grey area their convex hull. In both figures: P A = [16, 8; 8, 9], P B = [1, 1; 1, 4], and σ 2 m = 1. The optimal parameters are ω * ≈ 0.28 for the trace and ω * ≈ 0.36 for the determinant.

Discussion and numerical simulations

Illustrations of the optimal SCI filters for the trace and the determinant are shown in Fig. 1. In this figure, the consistency of the filter has been illustrated by generating several possible P ∈ P and plotting the resulting MSE given by [START_REF] Chen | Distributed and collaborative localization for swarming uavs[END_REF]. As observed, the two optimal SCI filters are different. This means that they depend on the cost function which is a first difference with the usual case where P is known.

The main implication of Theorem 1 and Corollary 1 is that it is not always possible to improve an estimator using SCI. Improvements require that the helping agent, here Agent B, has a sufficiently good precision (w.r.t. Agent A) in the direction of the measure, i.e., that σ 2 B < σ 2 A as stated in Corollary 1. Furthermore, this condition is only necessary, harder constraints should be satisfied in practice, they depend on the cost function J. It can happen that neither agent can improve its estimator, for example when considering the trace, if rAσ 2 A < σ 2 B < σ 2 A as shown in Theorem 2. This is an important difference from the usual case. If P is known, it is almost always possible to improve the estimator: the optimal linear filter was recalled in (10) (Section 2). In this case, even extremely poor precision on the helping agent provides (tiny) improvements.

The expression of the SCI filter [START_REF] Julier | General decentralized data fusion with covariance intersection (CI). Handbook of Multisensor Data Fusion[END_REF] involves the whole statistic of the estimate of Agent A but only the variance of the estimate of Agent B in the direction of the measurement. This asymmetry reduces the communication cost of the filter: the agents only need to send their estimate and their variance in the direction of the measurement (but not their full covariance). The asymmetry is due to the linearization assumption.

Moreover, for the two commonly used cost functions, the trace and the determinant, the necessary and sufficient conditions given in Theorem 2 and Theorem 3 allow to check efficiently the pertinence of the SCI filter before doing the measurement. In practice, this criterion can save energy by avoiding useless measurements. Furthermore, the precision of the measurement σ 2 m does not appear in these conditions ( 19) or ( 23). This means that if the property is not satisfied, even a perfect measurement cannot improve the estimate. However, the variance σ 2 m does affect the improvement (if it occurs). As [START_REF] Julier | General decentralized data fusion with covariance intersection (CI). Handbook of Multisensor Data Fusion[END_REF] suggests, the larger σ 2 m , the smaller the improvement. In addition, it seems that the condition for the existence of a pertinent filter becomes more difficult to satisfy as the dimension increases: the right-hand sides of ( 23) and [START_REF] Pierre | Range-only based cooperative localization for mobile robots[END_REF] tend to 0 as the dimension increases (except for the trace when the errors are perfectly correlated in almost all directions).

Finally, it is worth mentioning that the optimal parameter ω * can be calculated analytically for the cases of the trace and the determinant. Since the cost functions g and h are convex, the minimum is reached either at 0, at 1 (only possible when rA = 1), or when g ′ (ω) = 0 or h ′ (ω) = 0. From their decompositions [START_REF] Noack | Automatic exploitation of independencies for covariance bounding in fully decentralized estimation[END_REF] and [START_REF] Roumeliotis | Analysis of multirobot localization uncertainty propagation[END_REF], solving g ′ (ω) = 0 requires finding the roots of a polynomial of degree 4, and solving h ′ (ω) = 0 requires finding the roots of a polynomial of degree 3, both of which can be done analytically.

Concluding remarks

The SCI filter is a candidate for the optimal filtering problem. It is based on the linearization of the observation. Under this linearization assumption, we have been able to fully characterize the usefulness of the filter. The validity of this assumption may be evaluated by computing the second-order terms. The extension of this work to nonlinear observation remains open. To better fit the nonlinearities, a possible improvement could be the use of the Unscented Transform [START_REF] Julier | Unscented filtering and nonlinear estimation[END_REF]. Furthermore, the optimal SCI filter proposed here may not be the solution to Problem 1: a better linear filter may exist outside the family of SCI filters. We conjecture, after numerous simulations, that the optimal SCI filter is indeed the solution to Problem 1. The conjecture is based on the facts that SCI is the natural extension of CI when the errors of the estimators have independent components, and that CI is optimal for the problem it considers [START_REF] Reinhardt | Minimum covariance bounds for the fusion under unknown correlations[END_REF].

Finally, we only studied the integration of one distance measurement. Future works will focus on the simultaneous integration of distance measurements with multiple helping agents. They will consider filters such as: Most of the concepts introduced in this paper can be adapted, but as [START_REF] Ajgl | Fusion of multiple estimates by covariance intersection: Why and how it is suboptimal[END_REF] explains, CI is not optimal for more than two estimators, therefore SCI should also be suboptimal.

  x x xF = Kx x xA + L1x x x1 + z1w 1 + • • • + Lmx x xm + zmw m .

A bias would increase the MSE PF in the Loewner ordering sense.


Alice Michel

Chervin Hassel

Hortense Petat

Marie Leoz

Meriadeg Le Gouil

Alice Moisan

Olivier Join-Lambert

Astrid Vabret

Christophe Marguet

Jean-Christophe Plantier

Upper and lower respiratory tract microbiomes analysis in infants with severe asthma

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Upper and lower respiratory tract microbiomes analysis in infants with severe asthma

The human respiratory tract is colonized by microorganisms which constitute the respiratory microbiota. This microbiota is involved in the protection of the host against microbial infections. Its dynamics is regulated by mechanisms such as respiratory air flows, contributing to the inhalation of microorganisms, which is balanced by the host's immune system. An imbalance of these microbial communities, called dysbiosis, seems to be involved in the occurrence of respiratory pathologies such as asthma. This pathology affects about 25% of infants and its pathophysiology suggests a microbiota damage, associated with an inflammation. Major aims of this project are to characterize the global composition of the respiratory microbiome of infants with severe asthma, to follow its evolution over time and to determine the inflammatory profile of these patients. The cohort will include 100 patients from University Hospitals of Rouen and Caen. Nasopharyngeal and bronchoalveolar samples from these patients will be sequenced by metatranscriptomic approach and will allow to describe the respiratory microbiome composition. Inflammatory markers will also be analyzed through flow cytometry and ELISA to identify links between inflammation profiles involved in severe asthma and microbiome composition. This original project will allow to assess the dynamics of the airway microbiome, to determine the nature of potential dysbiosis promoting severe asthma in infants, potentially correlated with inflammatory markers involved in its pathogenesis. Leading to the better management of patients and to the discovery of new therapeutic targets as long-term aims. P-17

1

  o p h a r y n g e a l s w a b C l i n i c a l d a t a B r o n c h o a l v e o l a r l a v a g e f l u i d B l o o d s a m p l e INCLUSION 1 YEAR CHECK-UP EVERY 3 MONTH N a s o p h a r y n g e a l s w a b C l i n i c a l d a t a 1 , C. Hassel 1 , H. Petat 2 , M. Leoz 1 , M. Le Gouil 3 , A. Moisan 4 , O. Join-Lambert 5 , A. Vabret 3 , C. Marguet 2 , JC. Plantier 4 Univ Rouen Normandie, Univ de Caen Normandie, INSERM, DYNAMICURE UMR 1311 F-76000 Rouen, France ; 2 Univ Rouen Normandie, Univ de Caen Normandie, INSERM, DYNAMICURE UMR 1311, CHU Rouen, Department of Pediatrics, F-76000 Rouen, France ; 3 Univ de Caen Normandie, Univ Rouen Normandie, INSERM, DYNAMICURE UMR 1311, CHU Caen, Department of Virology, F-14000 Caen, France ; 4Univ Rouen Normandie, Univ de Caen Normandie, INSERM, DYNAMICURE UMR 1311, CHU Rouen, Department of Virology, F-76000 Rouen, France ; 5Univ de Caen Normandie, Univ Rouen Normandie, INSERM, DYNAMICURE UMR 1311, CHU Caen, Department of Microbiology, F-14000 Caen, France alice.michel@inserm.fr

CONCLUSIONS 1 ]

 1 Rajagopala, Seesandra V et al. "Metatranscriptomics to characterize respiratory virome,microbiome, and host response directly from clinical samples." Cell reports methods vol. 1,6 (2021): 100091. doi:10.1016/j.crmeth.2021.100091 2] Thorsen, Jonathan et al. "Infant airway microbiota and topical immune perturbations in the origins of childhood asthma." Nature communications vol. 10,1 5001. 1 Nov. 2019, doi:10.1038/s41467-019-12989-7 3] Teo SM, Mok D, Pham K, Kusel M, Serralha M, Troy N, et al. The Infant Nasopharyngeal Microbiome Impacts Severity of Lower Respiratory Infection and Risk of Asthma Development. Cell Host Microbe. 2015 May 13;17(5):704-15. 4] Kumpitsch, C.; Koskinen, K.; Schöpf, V.; Moissl-Eichinger, C. The Microbiome of the Upper Respiratory Tract in Health and Disease. BMC Biol. 2019, 17 (1), 87.


Adopting an urban political economy approach, she examines housing sub-markets triggered by tenure inertia and tenure insecurity in the legal urban fabric of certain Lebanese cities. Her previous research has addressed urban violence, urban stigmatization and its correlation to the housing sector and public space in marginalized neighborhoods.

More Than a Squat

The al-Khatib Buildings in Khandaq al-Ghamiq Aside from the social and the political conflict, the squat has its origin in land tenure, a history yet to be told. During my many field visits to Khandaq al-Ghamiq, I was always intrigued by the al-Khatib buildings, known to be a centre of squatting in the area (figure 01, and 02 for location). These buildings form a variegated mosaic of spatial decay: windowless facades contrasting with well-maintained ones, while precarious commercial activities occupy the dozen shops on the ground floor. The footprint and the density of the 1970s architecture stand out morphologically from their surroundings, mostly built before the 1960s. The building is nine stories high, offering a housing stock of about 64 flats (figure 01).

The wear and tear of time, urban violence or the multiple conflicts that happened in Beirut remain insufficient reasons to explain the current state of the al-Khatib buildings. My experience in the land and property ownership aspects of urban decay led me quickly to assume that a precarious land tenure situation lies behind these deteriorated façades and squatted floors. My first field observations did indeed confirm this hypothesis. Residents have various forms of tenure that are at odds with the blanket reputation of these buildings as a squatting centre. Formal owners and tenants live side by side with squatters. How can this complex tenure system within a single building be explained?

Squatting is "defined as inhabiting-or otherwise using-a dwelling [or property] without the consent of the owner" (Vasudevan, 2017, p. 22). In the context of Middle Eastern cities, studies on squatting tend to focus on irregular neighbourhoods where illegal occupation of land has enabled the construction of housing (Ababsa et al., 2012;[START_REF] Clerc | Les quartiers irréguliers de Beyrouth : Une histoire des enjeux fonciers et urbanistiques dans la banlieue sud[END_REF]. In this case, the squat is primarily land-based. This dynamic is different from that observed in regular neighbourhoods. In the latter, squatting occurs in old buildings, which comply with architectural standards and town planning rules and the property titles of which are legal.

Urban studies have examined the mechanisms, actors and consequences of tenure regularization in irregular districts [START_REF] Clerc | Reconquérir ou reconfigurer les marges de la ville ? La concurrence des politiques de résorption des quartiers informels à Damas[END_REF][START_REF] Denis | The Commodification of the Ashwaiyyat: Urban Land, Housing Market Unification, and De Soto's Interventions in Egypt[END_REF][START_REF] Lutzoni | In-formalized urban space design. Rethinking the relationship between formal and informal[END_REF]. However, processes of informalization or those falling in a formal-informal continuum [START_REF] Acuto | Transcending (in) formal urbanism[END_REF][START_REF] Issar | Conceptualizing the connections of formal and informal housing markets in low-and middleincome countries[END_REF] in regular neighbourhoods are yet to be analysed. How can the variegated tenure system in the al-Khatib buildings explain the formal-informal continuum of land tenure, including squatting in a legal building fabric? And how does this continuum interact with the formal land market in a neighbourhood with high potential land rent such as Khandaq al-Ghamiq?

In response to these questions, I adopted a land history approach. My two main sources were the land register for this property and qualitative interviews conducted in 2019 and 2021. I carried out two in-depth interviews, the first with the Mukhtar whose office is on the ground floor, and the second with a local figure living nearby. I also met four of the building's residents.

An Unresolved Legal History

The history of these buildings since their construction clarifies the how and why of the informalformal continuum of land tenure. The buildings were constructed in 1973, just before the outbreak of the civil war in Lebanon, by the Syrian al-Khatib family, who left the country before completing the work. Their hasty departure impeded the legalization of the land titles for the buyers who had paid for their flats in advance. Faced with a fait accompli, the buyers moved in and put the finishing touches to their property, each according to their own capacity and taste. This explains the inconsistency of the materials and finishes of the façades. Moreover, Mr al-Khatib had not sold all the units, notably the ground floor ones dedicated to commercial activities. Thus, when the residents moved in, the parts of the buildings that were still under construction were abandoned.

The civil war broke out in 1975 and resulted in massive population movements. In this context, the practice of squatting increased throughout the city. The abandoned spaces of the al-Khatib buildings were then occupied and remained so until after the end of the civil war. In 1994, the National Fund for Displaced Population compensated the migrants, who gradually began to vacate the occupied buildings. A decade later, a short temporary second wave of collective squatting occurred during the 2006 war, when displaced persons from South Lebanon found refuge in the empty spaces of al-Khatib buildings.

At the end of the 2010s, and in response to the growing demand by workers, refugees and others for affordable housing in the neighbourhood, squatting reoccurred in these same parts of the buildings, but this time by one individual. The squatter, a local figure of urban violence, proceeds by forcing entry, breaking down a door or a wall, and then renting out the property or selling it on the informal housing market. Squatting in this case is described as commercial in the sense that it sets up a rental market. The squatter and his tenants/buyers thus constitute what can be described as the informal and illegal tenure of these buildings.

The second group in this tenure continuum is the owners. The al-Khatib family and their partners legally own all the land rights of these buildings. They are the de jure owners despite the multiple sales they made at the time of construction. The buyers, however, are de facto owners insofar as their land rights are still not legalized. Therefore, while their tenure is certainly illegal, nevertheless it remains formal due to the purchase procedure carried out. Legal tenure stands by the and formal tenure is a tenure normalized by market practices. Conversely, the de jure owners' tenure is legal, but informal since it is subject to judicial reservations, as the scrutiny of the land register will show. 1973-1978 "legal pressure" 1978-2010 "tenure inertia" 2010-2018 "tenure tension." The first period is characterized by a hope among the new buyers of putting pressure on the de jure owners to complete the registration procedure. They proceeded only by prénotations, which is a classic way to block titles temporarily. However, these appeals were unsuccessful, as the de jure owners were unresponsive and completely uninterested in the property, or the country for that matter. The indifference of the de jure owners towards their property and the frozen land rights of the de facto owners effectively excluded the property from the formal land market. This defines and initiates a state of tenure inertia, a second phase lasting from 1978 to 2010. This long period favoured the emergence of a parallel economic system of tenure. During these years, tenure evolved according to the needs for housing in the neighbourhood and was shaped by informal mechanisms of housing production. Two activities describe the economic system in place. First, facing their exclusion from the formal real-estate market, some of the de facto owners sold or rented out their property through informal transactions. Second, squatting started in the buildings, creating an informal housing sub-market. The squatter, a kind of real-estate manager, rents out or sells the abandoned spaces, as single rooms or as flats, without any contractual document. Hence, the transaction is illegal and insecure for the tenants or the purchasers.

The tenure inertia phase concludes with the spike of real estate speculation in the district at the end of the 2000s. This marks the beginning of the third period, tenure tension, in which legal disputes between de facto and de jure owners escalate. Indeed, between 2007 and 2014 a real estate consortium, constituted mainly of Zein, Alyah and Amlak, acquired almost the entire block between Fouad Chehab Avenue and Daoud Ammoun Street where the al-Khatib buildings are located (figure 02). In response to this pressure, thirteen out of the eighteen lawsuits noted on the folio were launched against the de jure owner. This clearly describes a change in the behaviour of the de facto owners who have seen their level of tenure insecurity increase with the real estate pressure.

The de jure owners, driven by the potential ground rent of their property, also changed their behaviour during this period. In 2014, after the death of one of the de jure owners, these latter proceeded to register a lien to secure their land rights. Thus, the de jure owners are now motivated to remain present on the formal land market, especially with the rising speculation. This contradicts the discourse of the neighbourhood inhabitants and those profiting from the squatting activities, still describing the property as a case of land abandonment.

In conclusion, the mechanisms of squatting and dilapidation of the al-Khatib buildings are rooted in its land history. The insecurity of tenure of the de facto owners caused by the passivity of the de jure owners with regard to their property has translated into a situation of general tenure inertia, excluding the property from the formal land market. As a result, an informal economic tenure system has developed over the years. Commercial squatting in this legally constructed building is only possible as a consequence of the existing tenure inertia. Nevertheless, this inertia had mitigated the insecurity of tenure for the de facto owners until the beginning of a heightened real estate pressure in the neighbourhood at the end of the 2000s. Thus, the resulting increase in land disputes is a way for the de facto owners to secure their land rights against any sale by the de jure owners. Conversely, this could also be viewed as an attempt to prolong the state of inertia. As long as disputes accumulate, the property remains unsaleable and the rights of the de facto owners are controlled and maintained, albeit in a state of insecurity. The relationship between tenure inertia and tenure insecurity on the formal market is dialectical, constantly redefining the boundaries between the formal market and the informal housing sub-market in the central districts of Beirut. It thus constitutes one of the mechanisms of tenure informalization in a legally built fabric.

1Figure 01 -

 01 Figure 01-The al-Khatib buildings, photo taken from Chidiac Street, see figure 02. @Rouba Wehbe (2018) Keywords: Beirut, housing, land tenure, formal-informal continuum, squat, Khandaq el-Ghamiq.

Figure 02 -

 02 Figure 02-Khandaq el-Ghamiq location in administrative Beirut (a) Al-Khatib Buildings site in the neighbourhood; real-estate activity since 2014 (b). @ Wehbe (2022)


Listen to a three minutes walk in area in the city of Reze (France) Wich sound structure can be noticed?

Wich emerging shapes?

[...] A streetcar bell (irruption effect), we hear it well before the whistle of the progressive arrival of the streetcar (crescendo effect). There follows a spiral of familiar elements (chain effect): the light screeching of the brakes, the sound of the doors opening, and then a puff of human sound that escapes.

Barely passed the front of the tramway, the rumor of the market increases, we are not yet inside.

Then one penetrates in its heart, the difference is immediate (cut effect), it is not any more a continuous rumour, a uniform hubbub, but a very alive and significant dynamics (effect of metabole). We are inside. So much inside that we don't remember having heard the streetcar leave. Here, all is proximity and all is changing, all disappears and reappears, nothing remains and all is there. We no longer distinguish anything continuous, there are no longer elements in the background and others that remain in the foreground. We go from one sound to another, without necessarily realizing it. It is at the same time all the time different and it is at the same time all the time identical (effect of metabole).

A streetcar arrives (effect of crescendo). It completely covers the sound environment of the market (mask effect). The noise increases gradually, to become deafening. Then, curiously, it lets it reappear more quickly than it had been masked. But in spite of the strong activity of the market, in spite of the passage of the tramway, a street singer with his accordion is heard (effect of irruption). He finds a place in the already busy sound space (niche effect). Located with his back to the tramway, at the edge of the market, the competition of decibels around him does not seem as harsh as that. We listen to him (attraction effect). We only hear him (synecdoche effect). All the rest is erased (asyndeton effect). Whereas one left the market one always hears it. One moves away, he moves away. The car rumor is there, gradually found (fade out effect), always, in the background. [...] Verification of interdisciplinarity in the investigation.

Proposition

The Sonic Effect is an interdisciplinary tool in order to process the interactions between :

1) the acoustics sources 2) the layout or constructs surroundings 3) the sonic perception 4) the sonic actions.

What is the logical status of the effect as we understand it?

It is the manifestation of a phenomenon that accompanies the existence of the object.

The perceptible "effect" is immediately linked to the whole of the conditions surrounding the existence of the object.

Every sound phenomenon is linked to an effect.

Structural character of the sound effect.

Any sound in situ is inseparable from an "effect" bound to the perceptio and to the culture of the listeners.

Any physically situated sound is linked to one or more effects. The pure sounds and without effects exist only in vitro.

Interdisciplinary value of this effect

1) Measurable: it is the ratio of the variation of intensity, or of timbre or of time of reverberation time as a function of the duration of the phenomenon.

2) Reproducible in the built environment: the morphological characteristics of the space where it propagates being essential components of its audible structure.

3) Perceptible: it invites to further investigations in physiology and psychology of the perception.

4) Structuring element of the perception of the territories: it invites to approach under a new day the question the question of the forms of sociability.

5) As a universal instrument of musical composition: it invites us to work on an aesthetic of the sound experience in general.

A guide to Everyday Sounds

• For twelve years the Cresson Researchers have been patiently working on detailed investigations in many fields.

• They have registered and described more than 80 of Sound Effects. These ones have been explored and distributed in five categories belonging to their predominant description.

• The published book (in French, English, Italian) constitute a many directions Repertory. It contains the sufficiently well known effects those clearly described and easy to find or "in situ" or in the hearing. It gathers the classical effects of environment and urban culture.

The elementary effects

They deal with sonic material and the acoustic modality (crenel, attraction, phonotonic effect... ) Niche

Semantic effects

Playing on the variations of sense between the situation and the emergent meaning (imitation, repetition) Sharawadji

  effects…. …connected to the perceptive organization, to the culture memory (Cocktail-party,Ubiquity, asyndeton, synecdoche, phonomnesis....) Metamorphosis4. Psychomotor effects……which necessarily involve the existence of an action.

A l'écoute de l'environnement. Répertoire des effets sonores. Marseille, Ed. Parenthèses. 2006 Sonic experience. Montreal, McGill Univ Press . 2003 Repertorio degli effetti sonori. Lucca, Ed. L.I. M. https://www.cartophonies.fr/


Alice Michel

Jean-Christophe Plantier

Chervin Hassel

Chervin Hassel " Microbiosthme

"Microbiosthme" : Upper and lower airway microbiomes analysis in infants with severe asthma