Antonello Venturino

Sylvain Bertrand

Cristina Stoica Maniu

Teodoro Alamo

Eduardo F Camacho

A new L-step neighbourhood distributed moving horizon estimator

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Distributed algorithms have pervaded, in the last few years, many aspects of control engineering with applications for multi-robot systems, sensor networks, and others, covering topics such as control [START_REF] Segovia | Distributed model predictive control using optimality condition decomposition and community detection[END_REF]- [START_REF] Conte | Distributed synthesis and stability of cooperative distributed model predictive control for linear systems[END_REF], state estimation [START_REF] Simonetto | A distributed moving horizon estimator for mobile robot localization problems[END_REF]- [START_REF] Battistelli | Distributed moving-horizon estimation with arrival-cost consensus[END_REF], fault detection and mitigation [START_REF] Wu | Distributed fault detection for nonlinear multi-agent systems under fixed-time observer[END_REF], cyber-attack detection and mitigation on cyber-physical systems [START_REF] Gheitasi | A novel networked control scheme with safety guarantees for detection and mitigation of cyberattacks[END_REF], [START_REF] Gallo | A distributed cyber-attack detection scheme with application to DC microgrids[END_REF], etc. Despite their different purposes, these topics share common characteristics as a consequence of the development in distributed schemes. Indeed, they face problems like scalability and communications between agents. If, on one hand, sharing more data could lead to have better performance in terms of accuracy, on the other hand the complexity could raise as well as the communication burden. In centralised frameworks, with the same purpose, usually a central unit manages all the resources involved in the network; nevertheless it can hardly deal with scalability issues due to physical and computational limitations. Moreover it is not robust with respect to the loss of the central unit.

For centralised state estimation problems, Moving Horizon Estimation (MHE) techniques have been investigated over the past years due to their capability to take into account constraints in the finite horizon "least-square" optimisation problem on which their formulation is based [START_REF] Muske | Receding horizon recursive state estimation[END_REF].

Within the context of distributed state estimation by sensor networks, each sensor is delegated to estimate (at least partially) the state of the system on the basis of local measurements and information received from its neighbours. Several works on Distributed MHE (DMHE) methods have A. Venturino and C. Stoica Maniu are with Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des signaux et systèmes, 91190, Gif-sur-Yvette, France (e-mail: {antonello.venturino; cristina.stoica}@l2s.centralesupelec.fr).

A. Venturino and S. Bertrand are with Université Paris-Saclay, ONERA, Traitement de l'information et systèmes, 91123, Palaiseau, France (e-mail: {antonello.venturino; sylvain.bertrand}@onera.fr).

T. Alamo and E.F. Camacho are with Department of Ingeniería de Sistemas y Automática, Universidad de Sevilla, Camino de los Descubrimientos, 41092 Sevilla, Spain (e-mail: talamo@us.es, eduardo@esi.us.es).

been dedicated to guarantee stability of the estimation error dynamics, e.g. [START_REF] Farina | Distributed moving horizon estimation for linear constrained systems[END_REF], [START_REF] Farina | Distributed moving horizon estimation for nonlinear constrained systems[END_REF], [START_REF] Battistelli | Distributed moving-horizon estimation with arrival-cost consensus[END_REF]. In [START_REF] Farina | Distributed moving horizon estimation for linear constrained systems[END_REF] the authors have proposed a DMHE algorithm proving that it is stable even under weak observability conditions (due to consensus on estimates and a consensus weight term in the DMHE formulation). More recently, [START_REF] Battistelli | Distributed moving-horizon estimation with arrival-cost consensus[END_REF] introduced another consensusbased mechanism in a DMHE approach to fuse local arrival costs and guarantee stability of the estimation errors in a fully distributed way. The main drawback of the MHE paradigm is its computational load, since an optimisation problem must be solved online, at each instant, to compute the state estimate. This may be problematic when dealing with limited-computational resources, especially in the distributed case with low-computational sensors. Although adapted optimisation methods have been developed and can be used to reduce the computation time, an additional way of improvement concerns the structure of the optimisation problem itself. One idea, inter alia, is to introduce in the MHE formulation the use of an observer, that would enable to decrease the number of optimisation parameters. This has been first introduced in [START_REF] Sui | Linear constrained moving horizon estimator with pre-estimating observer[END_REF] for centralised linear MHE, considering a Luenberger observer, and then extended to the centralised non-linear case in [START_REF] Suwantong | Stability of a nonlinear moving horizon estimator with pre-estimation[END_REF], and to the distributed linear case in [START_REF] Venturino | Distributed moving horizon estimation with pre-estimating observer[END_REF] as an extension of the DMHE formulated in [START_REF] Farina | Distributed moving horizon estimation for linear constrained systems[END_REF].

This paper extends the approach proposed by the authors in [START_REF] Venturino | Distributed moving horizon estimation with pre-estimating observer[END_REF] to the DMHE formulation of [START_REF] Battistelli | Distributed moving-horizon estimation with arrival-cost consensus[END_REF] which has proven to obtain more general stability results as well as enhanced performance compared to [START_REF] Farina | Distributed moving horizon estimation for linear constrained systems[END_REF]. The current paper leads to a reduced computation time due to a pre-estimating observer. Another contribution concerns the improvement of the convergence of the estimation error by mitigating unobservability issues. This situation could arise in sensor networks when some nodes may have no sensing capacities (inactive sensors), or are able to only measure some parts of the state of the system that would make it non observable using only these sensors. For this purpose, the new proposed DMHE technique exploits the exchanges of information amongst local nodes based on an -step neighbourhood information spreading mechanism.

The paper is structured as follows. Section II introduces the problem formulation and proposed communication protocol. The proposed DMHE algorithm is presented in Section III. Before concluding remarks, simulations examples are presented and analysed in Section IV.

II. PROBLEM STATEMENT

This section describes the state estimation problem of a system over a sensor network. The dynamical system is described as a discrete-time linear time-invariant (LTI) system

x t+1 = Ax t + w t , (1)
with x t ∈ X ⊆ R nx representing the state and w t ∈ W ⊆ R nx the disturbance or unknown input. The measurements are performed by heterogeneous sensors, and can be modelled by

y i t = C i x t + v i t , i = 1, . . . , M (2)
where

y i t ∈ R n i y is the measurement vector, v i t ∈ V i ⊆ R n i y
the measurement noise and M > 1 the number of the sensors composing the network. The superscript i refers to the sensor i. The sets X , W and V i are assumed to be convex sets.

In the distributed architecture, each sensor also shares data with its neighbours. The sensor network is described by a directed graph G = (N , E), where N = {1, 2, . . . , M } is the set of all nodes (sensors) and E ⊆ N × N is the set of all edges (communication links). Specifically, the pair (i, j) ∈ E is defined if and only if the sensor j can receive information from the sensor i. The neighbourhood N i of the sensor i is denoted by N i = {j ∈ N : (i, j) ∈ E} and its cardinality

M i = card(N i ).
In addition to the shared data coming from the neighbour sensors in N i , each sensor i could exploit past information from other sensors j / ∈ N i , if there exists a path that connects these sensors to sensor i.

For this reason, denoting d(i, j) the distance, in terms of number of edges, between node i and j, we define the -step neighbourhood N i = {j = i ∈ N : d(i, j) }, i.e. the set of sensors j ∈ N for which there is a path of length at most to node i. Notice that N i 1 = N i .

A. Communication protocol

As mentioned above, the network is composed by possibly different types of nodes, some of them with no sensing capabilities, i.e. C i = 0, or at least partially, meaning that a sensor may observe only some part of the state vector of the system, i.e. the pair (A, C i ) is possibly not detectable. Moreover, the network could be deployed such that some neighbourhoods are composed only by nodes resulting in weak local or regional observability properties [START_REF] Farina | Distributed moving horizon estimation for linear constrained systems[END_REF], meaning that the pair (A, Ci ) could be not detectable, where Ci is the regional output matrix, i.e. Ci = [(C i ) , (C j1 ) , . . . , (C j M i ) ] , {j 1 , . . . , j M i } ∈ N i . Therefore, with the aim to enhance collective observability [START_REF] Farina | Distributed moving horizon estimation for linear constrained systems[END_REF] by the network, it is proposed that each node i ∈ N exploits measurements received from itsstep neighbourhood N i . This section details how information coming from N i will be considered in the formulation of the DMHE, by choosing = N , where N is the length of the horizon of past information considered for state estimation by the algorithm.

The communication network uses a single-hop routing protocol, in which it is assumed that there is no delay or packet losses. Moreover, a time synchronisation is required for all sensors in the network, to make them able to exchange data with their neighbours at each time instant.

Each node i ∈ N keeps information received from each of its in-neighbours nodes in a time-sliding batch of size N and relays this information to out-neighbours nodes at the next time instant. Old information (i.e. received from time instant t r < t -N ) is removed from the batch. Therefore, at time t each node disposes of past measurements from nodes in its -step neighbourhood over the time window [t -N, t]. Since dealing with a single-hop routing protocol, each sensor i ∈ N receives information only from its neighbours j ∈ N i at each time t. Let ȳi denote the measurements collected by sensor i from all the nodes j ∈ N i . Then, at the time instant t, these collected measurements from nodes j are

ȳi [t-N,...,t] =    y j1 t-N . . . y j1 t . . . . . . . . . y j M i t-N . . . y j M i t    , {j 1 , . . . , j M i } ∈ N i .
At the same time, each sensor j ∈ N i has data collected from its own neighbours z ∈ N j , with z = i, from the previous time step t -1, i.e.

ȳj [t-N,...,t-1] =    y z1 t-N . . . y z1 t-1 . . . . . . . . . y z M j t-N . . . y z M j t-1    , {z 1 , . . . , z M j } ∈ N j .
that they can share with sensor i. This philosophy can be reiterated back in time, and so along the communication links in N i t-k , ∀k = t -N, . . . , t -1, with a maximum of N = back steps.

To summarise, the node i has the collection of data ȳi [t-N,...,t] , ȳj [t-N,...,t-1] , . . . , ȳz t-N , with j, . . . , z ∈ N i , which is useful in the local MHE optimisation problem to improve the accuracy of the estimates.

B. Problem formulation

We define as a Poorly-Observing Sensor Network a network containing at least one node i ∈ N having any of the following characteristics

• it has no sensing capabilities, i.e. C i = 0;

• it has sensing capabilities and can provide a measurement on the state of the system, i.e. C i = 0, but the pair (A, C i ) still remains non detectable; • nodes in its neighbourhood are such that the pair (A, Ci ) is non detectable. The problem addressed in this paper, namely Distributed State Estimation over a Poorly-Observing Sensor Network can be stated as follows.

Given the discrete-time LTI system (1), the sensor network G with linear sensors as in [START_REF] Bono | A distributed model predictive control strategy for vehicle teams in uncertain narrowed environments[END_REF], under the assumptions that:

• the pair (A, C) is observable, where C = col(C i ) with i ∈ N is the collective output matrix, i.e. C = [(C 1 ) , . . . , (C M ) ] ; • the graph G = (N , E) is strongly connected, i.e. every
node is reachable from every other node. The role of each sensor i ∈ N , at each time t, is to (possibly) get measurement on the system, to exchange information among neighbour nodes N i and to process locally available information in order to determine a local estimate xi t of the real state of the system x t .

III. PROPOSED DMHE TECHNIQUE

This section presents the proposed DMHE approach. It extends the one of [START_REF] Battistelli | Distributed moving-horizon estimation with arrival-cost consensus[END_REF] with consensus on the arrival costs, by accounting for information from -step neighbourhoods and taking advantage of a pre-estimating observer to reduce computation time.

A. Local optimisation problem

At time t, let xi t-N |t , . . . , xi t|t be the sequence of estimates of the state of system (1) to be computed by each sensor i ∈ N over a given past horizon of length N 1. The estimate of the state x t to be provided by each sensor at time t corresponds to xi t = xi t|t . To do so, a local minimisation problem can be formulated for each sensor i as follows

xi t-N |t = arg min xi t-N J i t (3) s.t. xi k+1 = Ax i k + L i vi k + j∈N i t-k L j vj k , (4)
vj k = C j xi k -y j k , j ∈ {i} ∪ N i t-k , (5)
xi k ∈ X , (6)
vj k ∈ V j , j ∈ {i} ∪ N i t-k , ( 7
) ∀k = t -N, . . . , t
The sequence of state estimates xi t-N +1|t , . . . , xi t|t is then computed from the optimal solution xi t-N |t and using (4). The main difference in this formulation is that a Luenberger observer is used in (4) instead of the state equation of the system, as classically used in MHE formulations and in [START_REF] Battistelli | Distributed moving-horizon estimation with arrival-cost consensus[END_REF], which requires to consider the disturbance sequence over the past horizon as additional optimisation parameters. This reduces the computation cost, while simultaneously preserving the accuracy of the state estimate. Under the assumption that the gain L i is computed such that

Φ i = A -L i C i , ∀i ∈ N (8)
is Schur stable, then, in order to mitigate the effects on the estimation errors at certain frequencies or to increase robustness for each frequency, the gain L i can be computed off-line according to some criteria, for example H 2 , H ∞ [17, p. 293]. Note that the assumption of (8) being Schur can be satisfied only when the pair (A, C i ) is observable, otherwise, as extrema ratio, it is sufficient to design L i so as to keep spectrum radius of Φ i as low as possible. Another difference w.r.t. [START_REF] Battistelli | Distributed moving-horizon estimation with arrival-cost consensus[END_REF] is that the optimisation problem (3) uses the set N i instead of N i , leading to improve the estimation accuracy. In fact, N i appears also in the objective function J i t defined as

J i t = Γ i t (x i t-N |t ) + t k=t-N y i k -C i xi k 2 R i + t k=t-N j∈N i t-k y j k -C j xi k 2 R j (9)
where the weight matrices R i (resp. R j ) can be chosen as the inverse of the covariance matrix of the measurement noise. The first term is the so called initial penalty function Γ i t (•), known in the MHE environment as arrival cost. It is assumed to be non negative and it summarises the effect of the past measurements, before time t -N . Further details on the arrival cost are provided in Section III-B, because it plays a major role in the convergence and the performance of the algorithm.

Note that, when the current time instant t is t N then the horizon length N is set to N = t.

Remark 1: The local optimisation problem can be formulated using information coming only from neighbour sensors, in other words = 1. For later comparisons, we denote by DMHE pre the minimisation problem (3) having N i , with = N , and by DMHE 1 pre the one with N i equal to N i , i.e. = 1. Indeed, DMHE 1 pre could be a combination of the methods of [START_REF] Battistelli | Distributed moving-horizon estimation with arrival-cost consensus[END_REF] and [START_REF] Venturino | Distributed moving horizon estimation with pre-estimating observer[END_REF].

B. Fused arrival cost

The objective function (9) contains the arrival cost term Γ i t (•). In the MHE approach, it usually penalizes deviations from some a priori information xi t-N on the state at the beginning of the horizon as detailed in [START_REF] Muske | Receding horizon recursive state estimation[END_REF] and can be formulated as in [START_REF] Battistelli | Distributed moving-horizon estimation with arrival-cost consensus[END_REF] by

Γ i t (x) = x -xi t-N 2 P i t-N , (10)
with P i t-N a positive definite weight matrix. The a priori state xi t-N can be computed as a one step prediction from the solution of the optimisation problem at the previous instant. In classical (D)MHE approaches, this prediction is done using the state equation. In the proposed algorithm with pre-estimation formulation, this prediction is computed as

xi t-N = (A -L i C i )x i t-N -1|t-1 . (11)
The matrix P i t-N is defined as the inverse of the covariance matrix of the prediction xi t-N and can be computed recursively, as we explain later on, initialised as P i 0 in order to quantify the confidence on the initial a priori information xi 0 . In a distributed setting, this cost has an essential role to propagate information amongst the sensors in the network in order to ensure convergence of the state estimations to the real state of the system, since the local observability for one node or observability amongst the neighbourhood depends on the network topology and sensing capabilities. As in [START_REF] Battistelli | Distributed moving-horizon estimation with arrival-cost consensus[END_REF], the idea is then to fuse the arrival costs of the neighbourhood

N i in a convex combination Γ i t (x) = π i,i x -xi t-N 2 P i t-N + j∈N i π i,j x -xj t-N 2 P j t-N , (12)
where all the weights π i,i and π i,j are strictly positive and fulfil the condition

π i,i + j∈N i π i,j = 1, ∀i ∈ N . (13)
Consequently, the initial penalty function is defined as a consensus on the arrival costs by means of relation [START_REF] Gheitasi | A novel networked control scheme with safety guarantees for detection and mitigation of cyberattacks[END_REF] ensuring that the local arrival cost is a weighted average of the local arrival costs from neighbours.

In the following, the covariance matrix P i t-N is recursively updated using only local information available at time t to sensor i. Consider the observability matrix F i associated to the pair (A -L i C i , C i ) along the horizon length N and its relative collective output weight matrix

F i =      C i C i (A -L i C i ) . . . C i (A -L i C i ) N      , Ψ i = diag(R i , . . . , R i N +1 times
).

Then a preliminary consensus weight matrix can be computed by using only data locally available at node i from the previous time instant

Ω i t-N -1 = π i,i P i t-N -1 + j∈N i π i,j P j t-N -1 + F i Ψ i F i .
(14) Following [START_REF] Battistelli | Distributed moving-horizon estimation with arrival-cost consensus[END_REF], we now introduce a scalar α such that 0 < α < 1 and a positive definite matrix S for any t, and adapt the equations to the proposed algorithm with pre-estimation. Then the updated consensus covariance matrix is defined by

P i t-N = α 8 A i L Ω i t-N -1 -1 A i L + S -1 -1 , (15)
where

A i L = A -L i C i .

C. DMHE algorithm

Finally, we can describe the modus operandi of the proposed distributed algorithm. First of all, it is worth to mention that the steps of the algorithm could be run in a parallel scheme by each sensor i ∈ N , after they have sent and received the information from the neighbours at each time t, with the assumptions on the network and communication protocol provided in Section II-A.

The steps of the DMHE pre procedure are described in Algorithm 1. To get the one-step DMHE 1 pre procedure, it is sufficient to remove the step 10 and to use N i instead of N i .

Remark 2: All nodes are synchronised at the step 12, since each sensor i needs the data from its neighbours j ∈ N i . This is the only communication step.

Remark 3: The path length of the -step neighbourhood N i can also be chosen lower than the horizon length N of the receive from the neighbours j ∈ N i : L j , C j , R j , V j 5: Initialization: ∀i ∈ N , at the first time step t = 0 compute the prediction xi t-N and the consensus weight matrix P i t-N according to ( 11) and ( 15), resp.

12:

receive from the neighbours j ∈ N i the collected, gathered and computed data in the steps 9, 10 and 11 13: compute the fused arrival cost Γ i t according to [START_REF] Gheitasi | A novel networked control scheme with safety guarantees for detection and mitigation of cyberattacks[END_REF] 14:

solve the local MHE, minimising J i t as in ( 9) subject to the constraints ( 4)- [START_REF] Farina | Distributed moving horizon estimation for nonlinear constrained systems[END_REF] 15:

store the solution xi t-N |t and the estimate xi t|t DMHE, i.e. 1 N . One can design then the DMHE pre in order to have a good trade-off between the accuracy of the estimation and the amount of data exchanged in the network. This trade-off depends on the observability conditions and the network topology.

IV. SIMULATIONS

In this section an evaluation of the proposed DMHE algorithm is provided via simulations examples. To compare it with existing results in literature, the scenario in [START_REF] Battistelli | Distributed moving-horizon estimation with arrival-cost consensus[END_REF] is considered. The goal is to track a 2D moving target using a sensor network, that could model, for example, a distributed camera network. As illustrated in Fig. 1, the network is composed of 100 sensors randomly disposed with a uniform distribution on a plane of [-500, 500] × [-500, 500] m, in which only the green nodes are active sensors (i.e. with sensing capabilities), while the white nodes are inactive sensors (i.e. null output matrix). In the following it is considered that a communication link between two nodes exists if the distance between them is less than a given communication radius equal to 160 m.

The 2D moving target is modelled as a double integrator system. The state of the system is represented by x = [p x p y v x v y ] which corresponds to the Cartesian coordinates of its position and velocity vectors. The dynamics of the target are described by model [START_REF] Segovia | Distributed model predictive control using optimality condition decomposition and community detection[END_REF]

with A = I 2 T s I 2 0 I 2 ,
where T s = 1 s is the sampling time used for discretization of the continuous-time dynamics of the target. The input disturbance w t in (1) is a four dimensional vector and is assumed to be modelled by a noise vector with uniform distribution Two simulation cases are further analysed. Case 1. A first simulation considers a horizon length N = 4; the a priori information about the state xi 0 is set equal to [0 0 0 0] for each node; the initial arrival cost weight matrix is P i 0 = diag(10 -5 , 10 -5 , 1, 1), taking into account different magnitudes of the states; the matrices Q and R i are set for each sensor i as the inverse of the covariance matrices of w t and v i t , respectively. Q is used in [START_REF] Farina | Distributed moving horizon estimation for linear constrained systems[END_REF], [START_REF] Venturino | Distributed moving horizon estimation with pre-estimating observer[END_REF] and [START_REF] Battistelli | Distributed moving-horizon estimation with arrival-cost consensus[END_REF] for (D)MHE algorithms without preestimation. This weight matrix penalises the norm on the sequence of disturbance input terms in the cost function of these algorithms. All these parameters are identically set in all the considered algorithms. The consensus weights π i,j are chosen to be equal among the neighbourhood, satisfying [START_REF] Gallo | A distributed cyber-attack detection scheme with application to DC microgrids[END_REF].

In order to compare the proposed DMHE 1 pre (namely Distributed Moving Horizon Estimation with pre-estimation) and DMHE pre (namely DMHE with -neighbourhood diffusion and pre-estimation) algorithms with existing techniques, the simulation has been run also for the centralised MHE of [START_REF] Muske | Receding horizon recursive state estimation[END_REF] and the DMHE of [START_REF] Farina | Distributed moving horizon estimation for linear constrained systems[END_REF], [START_REF] Venturino | Distributed moving horizon estimation with pre-estimating observer[END_REF] and [START_REF] Battistelli | Distributed moving-horizon estimation with arrival-cost consensus[END_REF]. To avoid confusion, the DHME in [START_REF] Venturino | Distributed moving horizon estimation with pre-estimating observer[END_REF] is an extension of the one of [START_REF] Farina | Distributed moving horizon estimation for linear constrained systems[END_REF] using the pre-estimation strategy and it is called DMHE pre in [START_REF] Venturino | Distributed moving horizon estimation with pre-estimating observer[END_REF].

The performance metrics that have been taken into account are the Position Root Mean Square Error (PRMSE) averaged over the M =100 nodes of the network, denoted by PRMSE(t) = 1 M i∈N C i x t -xi t|t , and the computation time τ (t) averaged also over the entire network.

Figure 2 shows the time behaviour of the PRMSE of all considered algorithms. The proposed DMHE 1 pre technique offers similar results as the DMHE of [START_REF] Battistelli | Distributed moving-horizon estimation with arrival-cost consensus[END_REF], with a faster convergence (about 4 seconds) with respect to the DMHE of [START_REF] Farina | Distributed moving horizon estimation for linear constrained systems[END_REF] and [START_REF] Venturino | Distributed moving horizon estimation with pre-estimating observer[END_REF] (about 18 seconds to converge). Further, we can notice that DMHE pre ensures improved performance in terms of convergence time among the considered distributed algorithms. Evaluating the computation times in Fig. 3 shows that the algorithms with pre-estimation (red, purple and blue dots) are always less computationally demanding compared to their respective version without pre-estimation (yellow and green dots). Indeed, adding the pre-estimation reduces the computation time of about 30%. In particular, the proposed DMHE 1 pre technique (purple dots) converges faster and has comparable performances with the DMHE of [START_REF] Venturino | Distributed moving horizon estimation with pre-estimating observer[END_REF]. Moreover, it is worth to notice that DMHE pre has the best convergence time from all the considered approaches (see Fig. 2) and needs almost the same computation time as DMHE 1 pre and DMHE of [START_REF] Venturino | Distributed moving horizon estimation with pre-estimating observer[END_REF].

Case 2. A second simulation of nine trials has been performed using the same parameters but changing the fixed window size N = {2, 3, . . . , 10}, to the end of evaluating how the horizon length affects the performance of the considered DMHE algorithms. In addition, the initial state of the system x 0 is randomly generated with uniform distribution over the plan [-500, 500] × [-500, 500] m and in velocity [-1, 1] × [-1, 1] m/s. The simulation duration is chosen to be t f = 20s.

To emphasise the influence of the horizon length N on the estimations, Fig. 4 shows the evolution of the sum of the PRMSE, i.e. t∈(0,t f ] PRMSE(t), of each algorithm with respect to N . As for N = 4, the proposed technique DMHE pre has always the best performance with respect to the distributed algorithms. Moreover, even considering information belonging to neighbours, i.e. N i = N i , the DMHE 1 pre method has comparable results in terms of PRMSE with the DMHE of [START_REF] Battistelli | Distributed moving-horizon estimation with arrival-cost consensus[END_REF]. In fact, this is noticeable in the zoom part on Fig. 4 because the PRMSEs are one above the other. Finally, the Fig. 5 points out the differences amongst the sum of the computation time τ , i.e.

t∈(0,t f ] τ (t), of all algorithms when changing the horizon length N . As expected, the algorithms with pre-estimation are less computation demanding for every N since their local optimisation problems involves less optimisation parameters. Another significant aspect to observe in Fig. 5 is that the difference on τ amongst algorithms with and without preestimation increases with N . In addition, the Fig. 5 shows also the bounds representing the minimum and maximum computation time of the DMHE algorithms. It can be noticed that these bounds are tighter and less varying w.r.t. N for the algorithms with pre-estimation.

To summarise, the numerical simulations have shown that the proposed DMHE algorithm, DMHE pre , with preestimation and -step neighbourhood information diffusion, is able to solve the considered distributed estimation problem while, at the same time, it turns out to be lower computation demanding and gives better estimation accuracy w.r.t. other existing methods [START_REF] Farina | Distributed moving horizon estimation for linear constrained systems[END_REF], [START_REF] Battistelli | Distributed moving-horizon estimation with arrival-cost consensus[END_REF].

V. CONCLUSIONS AND PERSPECTIVES

The proposed -step neighbourhood Distributed Moving Horizon Estimation (DMHE) algorithm is able to solve the Distributed State Estimation problem for a linear system over a poorly-observing sensor network. In particular, the simulation results have shown that the proposed DMHE technique with pre-estimation DMHE 1 pre is able to converge with analogous performance with respect to the DMHE of [START_REF] Battistelli | Distributed moving-horizon estimation with arrival-cost consensus[END_REF] and, simultaneously, to reduce by a significant factor the computation time. The best result comes from the -step neighbourhood DMHE algorithm DMHE pre that, spreading out information from neighbourhood to neighbourhood, both improves accuracy (in terms of the Position Root Mean Square Error) and reduces computation time.

Current work concerns stability and robustness analysis of the proposed approach. Further developments focus on implementing the proposed -step neighbourhood Distributed Moving Horizon Estimation on a real application with multisensor system.

Algorithm 1 DMHE pre procedure 1 : 2 : 3 :

 123 Off-line: ∀i ∈ N compute the Luenberger gain L i store the a priori initial estimation xi 0|0 = x0 of x 0 and the covariance matrix P i 0 = P 0 of x 0 4:

6 : 7 : 8 : 9 : 10 :

 678910 collect a first local measurement y i 0 receive from the neighbours j ∈ N i their measurements y j 0 Online: ∀i ∈ N , ∀t > 0 collect the local measurement y i t gather past information received at time t -1 from j ∈ N i , as in Section II-A 11:

Fig. 1 .

 1 Fig. 1. Topology of the sensor network composed by 100 nodes.

Fig. 2 .

 2 Fig. 2. PRMSE time behaviour comparison.

Fig. 3 .

 3 Fig. 3. Computation time comparison.

Fig. 4 .

 4 Fig. 4. Comparison of the sum of PRMSE for a different horizon length N .

Fig. 5 .

 5 Fig. 5. Comparison of the sum of computation time for a different horizon length N .


Charles Laidi
email: charleslaidi@gmail.com

Tomas Hajek

Filip Spaniel

Marian Kolenic

Marc - Antoine D'albis

Samuel Sarrazin

Jean-François Mangin

Edouard Duchesnay

Paolo Brambilla

Michele Wessa

Julia Linke

Mircea Polosan

Pauline Favre

Amelia L Versace

Mary L Phillips

Jose V Manjon

Jose E Romero

Franz Hozer

Marion Leboyer

Pierrick Coupe

Josselin Houenou

Cerebellar parcellation in schizophrenia and bipolar disorder Running title: Cerebellum in psychosis

Keywords: Cerebellum, psychosis, parcellation, lithium Word count

Objective. Cerebellum plays a crucial role in the regulation of cognition and emotion processing. Cerebellar alterations could explain symptoms of schizophrenia spectrum disorder (SZ) and bipolar disorder (BD). In addition, literature suggests that lithium might influence cerebellar anatomy. Our aim was to study cerebellar anatomy in SZ and BD, and investigate the effect of lithium. Methods. Participants from 7 centers worldwide underwent a 3T MRI.

We included 182 patients with SZ, 144 patients with BD and 322 controls. We automatically segmented the cerebellum using the CERES pipeline. All outputs were visually inspected. Results. Patients with schizophrenia SZ showed a smaller global cerebellar grey matter volume compared to controls, with most of the changes located to the cognitive part of the cerebellum (Crus II and lobule VIIb). This decrease was present in the subgroup of patients with recent-onset SZ. We did not find any alterations in the cerebellum in patients with BD. However, patients medicated with lithium had a larger size of the anterior cerebellum, compared to patients not treated with lithium.

Conclusion.

Our multicenter study supports a distinct pattern of cerebellar alterations in SZ and BD.

Significant outcomes:

• Cortical grey matter volume in the cognitive part of the cerebellum is smaller in patients with schizophrenia compared to controls.

• Cortical grey matter volume in the cerebellum was no different in patients with bipolar disorders compared to controls.

• Lithium increases the volume of the anterior part of the cerebellum Limitations:

• The design of the study was cross-sectional

Introduction

The cerebellum represents 10% of the brain volume but contains more than 50% of its neurons [START_REF] Li | Using a million cell simulation of the cerebellum: network scaling and task generality[END_REF] . In humans, the cerebellum and the prefrontal cortex are the two most developed parts in comparison with other non-human primates [START_REF] Barton | Rapid evolution of the cerebellum in humans and other great apes[END_REF] .

Following the seminal work of Leiner et al. [START_REF] Leiner | Does the cerebellum contribute to mental skills?[END_REF] suggesting that the cerebellum was involved in non-motor function, Schmahmann et al. described a decade later the cognitive and affective cerebellar syndrome [START_REF] Schmahmann | The cerebellar cognitive affective syndrome[END_REF] . It is now admitted that the cerebellum is involved in a broad range of cognitive function, including working memory, emotion processing or social cognition [START_REF] Guell | Triple representation of language, working memory, social and emotion processing in the cerebellum: Convergent evidence from task and seed-based resting-state fMRI analyses in a single large cohort[END_REF] .

Functional magnetic resonance imaging (fMRI) [START_REF] Buckner | The cerebellum and cognitive function: 25 years of insight from anatomy and neuroimaging[END_REF] and lesion studies [START_REF] Adamaszek | Impairment of emotional facial expression and prosody discrimination due to ischemic cerebellar lesions[END_REF] suggested that the cerebellum can be functionally divided in regions based on their connectivity with the cerebrum. The posterior lobe of the cerebellum is involved in cognition and connected to associative regions such as the prefrontal cortex, whereas the anterior cerebellum likely modulates sensorymotor cortical activity. As the cerebellar architecture is considered to be homogeneous through the cerebellar lobules [START_REF] Zhou | Functional Connectivity of the Caudal Anterior Cingulate Cortex Is Decreased in Autism[END_REF] , it has been proposed that the cerebellum regulates in a similar manner the activity of the sensory-motor and the associative cortex. Similarly to the sensorimotor regions of the cerebellum in which damages lead to motor dysmetria [START_REF] Schmahmann | Disorders of the cerebellum: ataxia, dysmetria of thought, and the cerebellar cognitive affective syndrome[END_REF] , abnormalities in cerebellar regions involved in executive functions and affective regulation may have implications for cognitive dysmetria. This concept has been developed to explain symptoms of psychiatric disorders such as schizophrenia [START_REF] Andreasen | The Role of the Cerebellum in Schizophrenia[END_REF] and affective symptoms [START_REF] Schmahmann | Disorders of the cerebellum: ataxia, dysmetria of thought, and the cerebellar cognitive affective syndrome[END_REF] .

In schizophrenia, Moberget et al. [START_REF] Moberget | Cerebellar volume and cerebellocerebral structural covariance in schizophrenia: a multisite megaanalysis of 983 patients and 1349 healthy controls[END_REF] found in a large sample of 2332 patients and controls a robust reduction of the cerebellar grey matter volume. A more refined analysis within the cerebellar lobules showed that this reduction had a maximum effect size in regions of interest located in the posterior cerebellum.

In contrary to schizophrenia, cerebellar anatomy in BD has been investigated to a lesser extent. In a sample of 115 patients, we did not find any global cerebellar size reduction [START_REF] Laidi | Cerebellar volume in schizophrenia and bipolar I disorder with and without psychotic features[END_REF] . However we were not able to perform our analysis at the level of the cerebellar lobules.

Understanding the exact location of the cerebellar alterations in schizophrenia and bipolar disorder is important because of the regional specialization of the cerebellum [START_REF] Buckner | The organization of the human cerebellum estimated by intrinsic functional connectivity[END_REF] . Since the first descriptions of Kraepelin, it has been proposed that schizophrenia and bipolar disorder may share common pathophysiological features. Whereas some studies reported overlapping alterations between schizophrenia and bipolar disorder It has been proposed that lithium may have a neurotrophic effect in the brain (see Hajek et al. for full review [START_REF] Hajek | Neuroprotective Effects of Lithium in Human Brain? Food for Thought[END_REF] ). In addition, Johnson et al. 16 17 reported, using quantitative T1p mapping, cerebellar abnormalities in patients with BD.

T1p mapping abnormalities were normalized in patients treated with lithium.

Last, lithium is known to affect the cerebellum [START_REF] Banwari | Persistent cerebellar dysfunction following acute lithium toxicity: A report of two cases[END_REF] and cerebellar tremor is a common side effect of this type of medication.

The goals of our study were threefold. First, we aimed to replicate previous results on cerebellar anatomy in SZ spectrum disorders. Second, we wanted to understand the specificity of these findings within the psychosis spectrum by studying a sample of patients with BD. Last, we aimed to understand how the cerebellar lobules volumes were influenced by lithium.

Material and Methods:

Participants. Patients with SZ spectrum disorder and controls were recruited in 3 different sites located in France (Créteil), Czech Republic (Prague) and the USA (New Mexico). Patients with BD and controls were recruited in 5 different centers in USA (Pittsburgh), Germany (Mannheim), Italy (Udine) and 2 sites in France (Créteil and Grenoble). All details on inclusion criteria are described in Supplementary Material 1. Demographic characteristics are reported in Table 1. Only one center (Créteil, France) recruited patients with SZ, BD and controls. All other centers recruited either SZ and controls or BD and controls.

Data acquisition and data processing. All participants underwent a T1

weighted MRI (T1) with a full coverage of the cerebellum. Details on acquisition sequences are reported in Supplementary Material 1. All T1 images were processed using the CERES pipeline that performs a fully automated segmentation and parcellation of the cerebellum [START_REF] Romero | CERES: A new cerebellum lobule segmentation method[END_REF] following the protocol described in Park et al. paper [START_REF] Park | Derivation of high-resolution MRI atlases of the human cerebellum at 3 T and segmentation using multiple automatically generated templates[END_REF] . Afterwards, we divided the cerebellar grey matter in 6 regions of interest: lobule I-V, lobule V, Crus I, Crus II, lobule VIIb and the postero-inferior lobe (lobules VIII to X). We chose these regions of interest because the resolution on standard T1 images do not allow to visually differentiate the five first (lobules I to V) and the three last lobules (lobule VIII to X) of the cerebellum. We extracted the grey matter volume of these 6 regions of interest (ROIs) and the global cerebellar grey matter volume.

Quality control. All segmentations were visually inspected by an examiner (CL) blind of the diagnoses. Each segmented cerebellum (3D volume) was inspected in axial, coronal and sagittal views using Brainvisa/Anatomist software (http://brainvisa.info) (Figure 1). First, the examiner checked the quality of the segmentation, to ensure that no extra cerebellar tissue was labeled as cerebellar tissue and vice-versa. Second, the examiner inspected the quality of the lobular parcellation. Last, CL inspected the global brain segmentation to ensure the validity of the total intracranial measure (ICV), included as a covariate in the statistical analysis. Details on the included subjects and examples of rejected subjects are provided in Supplementary Material 2. All images not meeting our quality criteria were excluded of the study.

Statistical analyses.

We used a Chi-square test to test for significant differences in the proportion of males and females and a Student t-test to compare ages between patients and controls (Table 1).

To compare the size of the global cerebellar volume and ROIs in the cerebellum between patients and controls, we performed a linear model with (i) age and ICV as covariates and (ii) sex, diagnosis and site as cofactors.

Before performing pairwise comparisons (t-tests) between patients and controls, we ensured that the standardized residuals were normally distributed, as assessed by the Shapiro-Wilk test (p > 0.05) and a QQ-plot.

We applied a false discover rate correction (FDR -Benjamini Hochberg correction) to control multiple testing. We present both uncorrected (presented as "p" value) and corrected p values (presented as "p corrected"). Results that survived correction for multiple comparisons were considered as significant.

We conducted our statistical analyses separately on the SZ and the BD sample. Similarly, we compared the size of the global cerebellar volume and ROIs in patients with BD medicated vs not medicated with lithium, with a linear model including (i) age and ICV as covariates and (ii) sex, lithium status and site as cofactors.

We conducted several supplementary analyses. Because of the exploratory nature of theses analyses, we did not correct our results for multiple comparisons.

Changes in cerebellar volume could be related to the lifetime dose of medication or to the duration of the illness. We thus repeated our analysis on the significant regions previously identified in the subsample of first episode patients with SZ spectrum disorder and controls recruited in Czech Republic (n = 174).

We studied the association between the severity of schizophrenia (assessed with the Positive and Negative Symptoms Schizophrenia (PANSS) scale) and the regions of interest in the cerebellum. The PANSS scale was available in patients recruited in France (n = 35) and Czech Republic (n = 79). Tests were conducted with the Z-score of PANSS total, negative, positive and general psychopathology scores. We performed linear models with (i) age, ICV and PANSS Z scores as covariates and (ii) sex and site as cofactors.

Clinical, genetic and neuroimaging studies [START_REF] Sarrazin | Corpus callosum area in patients with bipolar disorder with and without psychotic features: an international multicentre study[END_REF] suggested that there might be distinct patterns between patients with BD (n = 53) with and without psychotic features (n = 56). We thus repeated our analysis on patients with BD with and without psychotic features. In this analysis, patients were recruited in Créteil, France (n = 19), Germany (n = 34), USA (n = 39), Grenoble, France (n = 13) and Italy (n = 4).

Alcohol dependence is known to have a damaging effect on the cerebellum: cerebellar ataxia is frequent in alcoholics and prenatal exposure to alcohol affects the cerebellum [START_REF] Luo | Effects of ethanol on the cerebellum: advances and prospects[END_REF] . Thus, alcohol dependence might be a confounding factor in our study. We repeated our analysis to ensure that our results remained significant when considering only patients without alcohol dependence.

Likewise, it has been proposed that cannabis could affect the structure of the cerebellum [START_REF] Battistella | Long-Term Effects of Cannabis on Brain Structure[END_REF] . Thus, we conducted an analysis in the sample recruited in Czech Republic to study the effect of cannabis consumption, during the month prior to the inclusion, on the cerebellum. We performed a linear model, considering age and intracranial volume as covariates and sex and cannabis status as cofactors.

Antipsychotics are known to affect brain structures and could be also a potential confounding factor, explaining the structure differences in the cerebellum between patients with schizophrenia and controls. To address this question, we repeated our analyses in a sample of first episode patients with schizophrenia, with limited exposure to antipsychotics. In addition we conducted a linear model, in our population of patients with schizophrenia recruited in Czech Republic (Supplementary Material 4) to study the effect of medication load (computed in chlorpromazine equivalent dose) on our regions of interest in the cerebellum. We considered age, ICV and medication load as covariates and sex as a cofactor.

We conducted all statistical analyses with the python statsmodels open source library [START_REF] Kubilius | A framework for streamlining research workflow in neuroscience and psychology[END_REF] .

Results

Total cerebellar volume and lobular analysis in schizophrenia

One hundred and eighty-two patients with SZ and 198 controls were included in this analysis. Demographic characteristics are reported in Table 1.A.

The global cerebellar volume, Crus II and lobule VIIb were significantly smaller in patients compared to controls (Table 2 and Figure 2).

Total cerebellar volume and lobular analysis in bipolar disorder

One hundred and forty four patients with type I BD and 176 controls were included in this analysis. Demographic characteristics are reported in Table 1.B.

We did not find any significant difference between patients and controls for any of the regions of interest. Results are reported in Table 2.

BD participants treated with Li at the time of scanning (n=56) had significantly larger volume of the left anterior cerebellum grey matter volume than participants not treated with Li (n = 86), see Table 3.

Exploratory analyses

Cerebellar volume in the sample of first episode patients with schizophrenia

We found a decreased volume in patients compared to controls in the total cerebellar volume (p= 0.0005), Crus II (p = 0.0006) and lobule VIIb (p = 0.0036). This suggests that the size reduction in the cerebellum is present since the beginning of the illness and is not related to its evolution.

Influence of PANSS score in patients with schizophrenia

We found a negative correlation between the volume of Crus I and the PANSS general psychopathology score (p = 0.007). In addition, we found a negative correlation between the volume of Crus I and the PANSS total score (p = 0.047) (Supplementary Material 3).

Influence of psychotic features in patients with bipolar disorder

We compared patients with BD with and without psychotic features. We did not find any significant difference between the two groups for any of our regions of interest.

Influence of alcohol dependence

Patients recruited in France and Czech Republic did not suffer from alcohol dependence (Supplementary Material 4). However this information was lacking in patients recruited in the USA. Thus, we repeated our analyses in patients with schizophrenia recruited in France and Czech Republic. Our results remained significant in the total cerebellar (p = 0.0001; t-value = -3.9), Crus II (p = 0.0002; t-value = -3.8) and lobule VIIb (p = 0.0004; t-value = -3.6) volumes.

Influence of cannabis use

Sixteen patients with schizophrenia recruited in Czech Republic had a history of cannabis consumption in the month prior to the inclusion (Supplementary Material 4). However, there was no effect of cannabis history in the last month prior to the inclusion on the volume of the total cerebellar (p-value = 0.60; tvalue = -0.57), Crus II (p-value = 0.12; t-value = -1.57) and lobule VIIb (pvalue =0.41; t-value = -0.81) volumes.

Influence of medication load in patients with schizophrenia

There was no significant effect of medication load on the volume of the total cerebellar (p = 0.41; t-value = -0.82), Crus II (p=0.79; t-value = -0.26) and lobule VIIb (p = 0.19; t-value = 1.33) volumes.

Discussion

We conducted a volumetric MRI study in 648 participants to study the cerebellar anatomy of patients with SZ and BD.

We found a decreased volume of the total cerebellar volume, Crus II and the lobule VIIb in patients with SZ compared to controls. There was no significant difference between patients with BD and controls per se, but BD patients treated with lithium had a larger anterior cerebellar volume compared to controls. To date, this is the first multicenter study probing cerebellar differences at a lobular level in both patients with SZ and BD.

Our results are in line with findings from the recent Moberget et al. study [START_REF] Moberget | Cerebellar volume and cerebellocerebral structural covariance in schizophrenia: a multisite megaanalysis of 983 patients and 1349 healthy controls[END_REF] and with our previous study [START_REF] Laidi | Cerebellar volume in schizophrenia and bipolar I disorder with and without psychotic features[END_REF] . Moberget rare in a field which is known for preponderance of false positive findings [START_REF] Ioannidis | Excess significance bias in the literature on brain volume abnormalities[END_REF] .

Jointly, these findings suggest that smaller volumes of cerebellum are among the most robust findings in schizophrenia, replicable with different methodological approaches. These findings may be specific to schizophrenia, as we did not find size reduction in bipolar disorder. Similarly, Traut et al., [START_REF] Traut | Cerebellar Volume in Autism: Literature Meta-analysis and Analysis of the Autism Brain Imaging Data Exchange Cohort[END_REF] did not reported cerebellar alterations in autism in a large scale study.

Moreover, we found a decreased volume in two cerebellar lobules (Crus II and lobule VIIb, adjacent to Crus II) located in the cognitive part of the cerebellum. The Crus II region is connected to the prefrontal cortex, a region that has been linked to schizophrenia. Our result supports the hypothesis of a cognitive dysmetria in schizophrenia, where the cortical-subcortical circuit between the cerebellum and the prefrontal cortex might be altered [START_REF] Andreasen | The Role of the Cerebellum in Schizophrenia[END_REF] . A disruption in this circuit may lead to difficulty in prioritizing, processing and responding to information, which could account for the wide range of symptoms.

This result may also have therapeutic implications. The cerebellum is connected to almost every region of the brain, with the exception of the visual cortex and might represent a potential therapeutic target. For example, trans cranial direct current stimulation (tDCS) has been applied in patients with nonclinical psychosis to improve their skills in procedural learning [START_REF] Gupta | Cerebellar Transcranial Direct Current Stimulation Improves Procedural Learning in Nonclinical Psychosis: A Double-Blind Crossover Study[END_REF] .

Understanding the exact location of the cerebellar alterations in schizophrenia and bipolar disorder is important since the location of the cerebellar stimulation influence the effect of brain stimulation in the cerebellum and distant cortical regions [START_REF] Farzan | Enhancing the Temporal Complexity of Distributed Brain Networks with Patterned Cerebellar Stimulation[END_REF] .

We did not find difference in the cerebellar anatomy when comparing patients with bipolar disorder and controls. Previous key studies from Hibar et al. 28 29 investigated the cortical and subcortical anatomy of patients with bipolar disorder in multicentric samples from the ENIGMA consortium. However the anatomy of the cerebellum was not investigated in these studies.

Lithium is the gold standard treatment for BD. We investigated the effect of Lithium on cerebellar morphology for several reasons. First, Lithium shows neuroprotective effects (see [START_REF] Hajek | Neuroprotective Effects of Lithium in Human Brain? Food for Thought[END_REF] for full review) and is much more often used in BD than in SZ. Consequently, it is possible that the absence of cerebellar changes in BD and their presence in SZ could be related to differential exposure to Lithium. Second, Johnson et al. 16 17 reported, using quantitative T1p mapping, cerebellar abnormalities in patients with BD. T1p mapping abnormalities were normalized in patients treated with lithium. Last, 5 cases of a lithium induced long-lasting cerebellar toxicity 30 18 have been reported and intention tremor, also known as cerebellar tremor, is a common side effect of lithium.

We found an increased volume in the anterior cerebellum in patients with BD medicated with lithium compared to BD patients not medicated with lithium.

The anterior cerebellum is connected to the sensori-motor cortex [START_REF] O'reilly | Distinct and overlapping functional zones in the cerebellum defined by resting state functional connectivity[END_REF] , which could explain the cerebellar motor syndrome described in patients medicated with lithium. However, we did not see effects of Lithium on the posterior cerebellum, which showed abnormalities in SZ. Therefore, it is unlikely that the absence of cerebellar changes in BD was related to neuroprotective effects of Lithium.

We investigated if confounding factors could explain our results.

Chronic consumption of alcohol in well known to affect the cerebellum [START_REF] Cardenas | Deformation based morphometry of brain changes in alcohol dependence and abstinence[END_REF] and can lead to its size reduction 32 33 . Thus, our results might have been related

to alcohol dependence in patients with schizophrenia. However, when repeating our analyses in a sub-sample of patients with schizophrenia without alcohol dependence, our results remained significant.

Likewise, there is evidence that cannabis could affect the cerebellar structure [START_REF] Battistella | Long-Term Effects of Cannabis on Brain Structure[END_REF] . However, our analyses suggested that there was no effect of cannabis on our results.

Antipsychotics are known to affect brain structures and could be also a potential confounding factor, explaining the structure differences in the cerebellum between patients with schizophrenia and controls. However, when repeating our analyses in a sample of first episode patients, with limited exposure to antipsychotics, our results remained significant. In addition we did not find any association between the regions differing between patients with schizophrenia and control and the medication load.

Our study comes with several strengths. First, we processed a large sample of 648 patients with SZ, BD and controls using the same state of the art method. Second, we visually inspected every cerebellar parcellation to ensure the quality of the segmentation and we compared cerebellar volumes in the subjects' native space. We used the CERES pipeline [START_REF] Romero | CERES: A new cerebellum lobule segmentation method[END_REF] based on a multi-atlas framework using the segmentation protocol defined in Park et al. paper [START_REF] Park | Derivation of high-resolution MRI atlases of the human cerebellum at 3 T and segmentation using multiple automatically generated templates[END_REF] . A recent publication [START_REF]Comparing fully automated state-of-the-art cerebellum parcellation from magnetic resonance images[END_REF] stated that the CERES pipeline outperformed various automated segmentation methods, including a voxel-based morphometry based pipeline. The manual segmentations of the used atlas have been carefully validated and automated segmentation based on these atlases have been compared to manual tracing [START_REF] Cardenas | Deformation based morphometry of brain changes in alcohol dependence and abstinence[END_REF] . Third, we conducted several additional analyses to assess the effect of potential confounding factors, such as the treatment with antipsychotics, alcohol dependence and cannabis. Last, our results remained significant in a population of patients with a first episode of schizophrenia, suggesting that cerebellar alterations are present at the beginning of the illness.

Several limitations should be considered before interpreting our results. Our study is cross sectional and subjects were recruited in centers using different acquisitions parameters. For the most part, individual centers recruited participants only with schizophrenia or only with BD, thus making a direct transdiagnostic comparison difficult. There is an overlap with our previous study conducted in a sample of patient with SZ, BD and controls [START_REF] Laidi | Cerebellar volume in schizophrenia and bipolar I disorder with and without psychotic features[END_REF] . However our current study was conducted in a larger sample, with different software, allowing us to investigate the morphometry of the cerebellum at a lobular level.

Our results suggest that Lithium has an effect on the size of the anterior cerebellum in patients with bipolar disorder. However, due to the crosssectional nature of this study, it is not possible to draw any firm conclusion on the effect of lithium on the cerebellum. Further studies with a longitudinal design are warranted to better understand the effect of lithium on the cerebellum.

We did not find size reduction in the cerebellum in patients with bipolar disorder compared to controls. Although we investigated the cerebellar anatomy in a multicenter sample of 144 patients, we cannot exclude that this negative finding might be related to a lack of power. In addition, the number of inclusion centers and the variability in the imaging acquisition protocols might as well explain some of our negative findings. We conducted several sensitivity analyses in sub-samples of our study. Because of the lack of power, our results (in particular negative findings) must be interpreted with caution.

Because this study was focused on the cerebellum, we did not investigate the relations between cerebellar abnormalities and distant cerebral abnormalities.

Further studies are warranted to better understand how cerebellar abnormalities could be linked to abnormalities in functionally connected regions such as the prefrontal cortex.

In conclusion, our study confirms that the size of the total cerebellar grey matter volume is decreased in SZ and that this size reduction is located within the cognitive, posterior parts of the cerebellum including Crus II and lobule

VIIb. Association between schizophrenia and smaller cerebellar volumes is one of the most robust and replicated brain imaging findings in this disorder.

In line with other studies, we did not find any alterations in the cerebellum in bipolar patients. However BD patients medicated with Lithium had an increased size of the anterior part of the cerebellum, compared to BD patients not treated with Lithium. Our study supports a distinct pattern of cerebellar alterations in SZ and BD.

Tables Table 1. Demographic characteristics of participants

Figure 2 :

 2 Figure 2: Grey matter volume of the cerebellum, Crus II and lobule VIIb in

Table 1 .

 1 A: Patients with schizophrenia and controls

		Patients	Controls	Statistical test	Statistics, p-value
		(n = 182)	(n = 198)
	Mean age (std)	32 (11)	33 (11)	Student t-test	t = -0.16, p = 0.87
	Sex: Male / Female 122 / 60	96 / 102	Chi-2 test	q = 13.3, p < 0.001
	Site of Inclusion	COB = 57	COB = 58	-	-
		PRA = 86	PRA = 88
		CRE = 39*	CRE = 52
	COB = COBRE dataset
	PRA = Prague, Czech Republic
	CRE = Créteil, France
	*: including 3 patients with schizo-affective disorder

Table 1 .

 1 B: Patients with bipolar disorder and controls

		Patients	Controls	Statistical test	Statistics, p-value
		(n = 144)	(n = 176)
	Mean age (std)	38 (11)	35 (10)	Student t-test	t = 2.41, p = 0.02
	Sex: Male / Female 53 / 91	68 / 108	Chi-2 test	q = 0.11, p = 0.73
	Site of Inclusion	CRE = 31	CRE = 52	-	-
		MAN = 37	MAN = 36
		PIT = 51	PIT = 23
		GRE = 13	GRE = 10
		UDI = 12	UDI = 55
	CRE = Créteil, France
	MAN = Mannheim, Germany
	PIT = Pittsburgh, USA
	GRE = Grenoble, France
	UDI = Udine, Italy

Table 2 . Effect of diagnosis on cerebellar anatomyTable 2 .

 22 A: Effect of diagnosis in patients with SZ compared to controls

	Grey matter volumes (cm3)	pval	FDR corr. P value	tval	[95% conf. inter.]	dof	Effect size for diagnosis ; sex ; age ; site ; ICV
							(η 2 effect size)
	Total cerebellum	0.0003	0.001 *	-3.62	[-4.16 ; -1.23]	6	0.034 ; 0.071 ; 0.079 ; 0.025 ; 0.312
	Lobule I-V	0.0828	0.097	-1.74	[-0.48 ; 0.03]	6	0.008 ; 0.043 ; 0.011 ; 0.018 ; 0.147
	Lobule VI	0.0459	0.074	-2.00	[-0.81 ; -0.01]	6	0.011 ; 0.002 ; 0.017 ; 0.024 ; 0.225
	Crus I	0.1928	0.19	-1.30	[-1.03 ; 0.21]	6	0.005 ; 0.020 ; 0.050 ; 0.020 ; 0.139
	Crus II	0.0008	0.002 *	-3.38	[-1.06 ; -0.28]	6	0.030 ; 0.022 ; 0.065 ; 0.048 ; 0.060
	Lobule VIIb	0.0000	0.0001 *	-4.33	[-0.64 ; -0.24]	6	0.048 ; 0.025 ; 0.029 ; 0.021 ; 0.065
	Lobule VIII -X	0.0535	0.074	-1.94	[-1.09 ; 0.01]	6	0.010 ; 0.073 ; 0.016 ; 0.002 ; 0.183
	*, significant results after FDR correction
	pval: p -value
	tval: t -value
	conf. inter.: confidence interval
	dof: degrees of freedom

Table 2 .

 2 B: Effect of diagnosis in patients with BD compared to controls

	Grey matter volumes (cm3)	pval	FDR corr. P value	tval	[95% conf. inter.]	dof	Effect size for diagnosis ; sex ; age ; site ; ICV
							(η 2 effect size)
	Total cerebellum	0.2273	0.5754	1.21	[-0.66 ; 2.76]	8	0.005 ; 0.016 ; 0.124 ; 0.115 ; 0.304
	Lobule I-V	0.3288	0.5754	0.98	[-0.15 ; 0.45]	8	0.003 ; 0.016 ; 0.042 ; 0.025 ; 0.157
	Lobule VI	0.6841	0.6841	0.41	[-0.37 ; 0.57]	8	0.001 ; 0.000 ; 0.034 ; 0.012 ; 0.135
	Crus I	0.4217	0.5904	0.80	[-0.37 ; 0.89]	8	0.002 ; 0.021 ; 0.089 ; 0.212 ; 0.095
	Crus II	0.3044	0.5754	1.03	[-0.22 ; 0.71]	8	0.003 ; 0.001 ; 0.045 ; 0.171 ; 0.150
	Lobule VIIb	0.1756	0.5754	1.36	[-0.07 ; 0.40]	8	0.006 ; 0.000 ; 0.025 ; 0.048 ; 0.200
	Lobule VIII -X	0.5801	0.6768	0.55	[-0.42 ; 0.74]	8	0.001 ; 0.025 ; 0.068 ; 0.031 ; 0.202
	*, significant results after FDR correction
	pval: p -value
	tval: t -value
	conf. inter.: confidence interval
	dof: degrees of freedom

Table 3 . Effect of Lithium on cerebellar morphology in patients with BD

 3

	Grey matter volumes	pval	FDR	tval	[95% conf.		Effect size for diagnosis ; sex ; age ; site ; ICV
	(cm3)		corr. p		inter.]	dof	(η 2 effect size)
			value
	Global cerebellar volume	0.0322	0.095	2.17	[0.25 ; 5.43]	8	0.034 ; 0.001 ; 0.103 ; 0.070 ; 0.367
	Lobule I-V	0.0042	0.029*	2.92	[0.21 ; 1.11]	8	0.060 ; 0.000 ; 0.033 ; 0.029 ; 0.190
	Lobule VI	0.0668	0.12	1.85	[-0.05 ; 1.42]	8	0.025 ; 0.021 ; 0.019 ; 0.022 ; 0.223
	Crus I	0.5434	0.63	0.61	[-0.64 ; 1.20]	8	0.003 ; 0.023 ; 0.112 ; 0.053 ; 0.118
	Crus II	0.8163	0.81	0.23	[-0.63 ; 0.80]	8	0.000 ; 0.004 ; 0.047 ; 0.104 ; 0.151
	Lobule VIIb	0.2955	0.41	1.05	[-0.18 ; 0.58]	8	0.008 ; 0.004 ; 0.023 ; 0.064 ; 0.212
	Lobule VIII -X	0.0408	0.10	2.07	[0.04 ; 1.68]	8	0.031 ; 0.000 ; 0.031 ; 0.109 ; 0.290
	*, significant results after FDR correction
	pval: p -value
	tval: t -value
	conf. inter.: confidence interval
	dof: degrees of freedom

Figures

Figure

1:

Cerebellar segmentation with the CERES pipeline

Acknowledgments:

We thank all subjects for their participation in this study. CL wrote the main manuscript text and prepared Figures and Tables. PC processed the anatomical images. All authors reviewed the manuscript

Financial support

This study was supported by public funding from Agence Nationale de la Recherche (ANR-11-IDEX-0004, ANR-11-INBS-006, ANR-14-CE35-0035, ANR MNP VIP 2008), Deutsche Forschungsgemeinschaft (SFB636/C6 and We3638/3-1), Ministerstvo Zdravotnictví Ceské Republiky (NV16-32696A), National Institute of Mental Health (R01 MH076971) and Regione Veneto Italy (159/03, DGRV n. 4087)

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.