Lei Mo

Qi Zhou

Angeliki Kritikakou
email: angeliki.kritikakou@irisa.fr

Ji Liu

Energy Efficient, Real-time and Reliable Task Deployment on NoC-based Multicores with DVFS

Task deployment plays an important role in the overall system performance, especially for complex architectures, including several cores with Dynamic Voltage and Frequency Scaling (DVFS) and Network-on-Chips (NoC). Task deployment affects not only the energy consumption but also the real-time response and reliability of the system. In this work, a task deployment approach is proposed to optimize the overall system energy consumption, including computation of the cores and communication of the NoC, under task reliability and real-time constraints. More precisely, the task deployment approach combines task allocation and scheduling, frequency assignment, task duplication, and multipath data routing. The task deployment problem is formulated using mixed-integer non-linear programming. To find the optimal solution, the original problem is equivalently transformed to mixed-integer linear programming, and solved by state-of-theart solvers. Furthermore, a decomposition-based heuristic, with low computational complexity, is proposed to deal with scalability. Finally, extended simulations evaluate the proposed methods.

I. INTRODUCTION

Multicore architectures integrate multiple processors on a single chip, leading to platforms with low supply frequency, high data throughput and better energy efficiency [START_REF] Ali | A survey on system level energy optimisation for MPSoCs in IoT and consumer electronics[END_REF]. With the development of nanoscale technologies, various processors communicate via the Network-on-Chips (NoC), instead of traditional, non-scalable, data buses [START_REF] Bhanu | Fault-tolerant network-onchip design with flexible spare core placement[END_REF]. The NoC uses its underlying micro-network components, i.e., routers, for the communication among the processors. The routers are usually connected by a mesh network, one of the most effective NoC topologies, due to its regularity, high bandwidth and short interconnections [START_REF] Han | Contention-aware energy management scheme for NoC-based multicore real-time systems[END_REF].

The inter-processor communication cost (time and energy) over NoC is not negligible compared to the cost of computation, occurring at the processors [START_REF] He | UNISM: Unified scheduling and mapping for general networks on chip[END_REF]. Furthermore, the communication cost depends on both task mapping and routing path decision. When dependent tasks are allocated and executed on different processors, data must be transmitted. Furthermore, multiple routing paths for the data transmission can exist in NoC, such as in the mesh network [START_REF] Bhanu | Fault-tolerant network-onchip design with flexible spare core placement[END_REF]. To improve the overall system performance, it is necessary to consider inter-processor communication, task mapping and routing path selection as an integral task deployment process.

Meanwhile, critical applications have the real-time and reliability constraints [START_REF] Namazi | A majoritybased reliability-aware task mapping in high-performance homogenous NoC architectures[END_REF], whereas system energy consumption is essential. Although energy efficiency, real-time and reliable task execution are both important design objectives, they are usually conflicting. Enhancing real-time execution and reliability often requires more energy consumption. Dynamic Voltage and Frequency Scaling (DVFS) [START_REF] Li | Minimizing energy consumption for frame-based tasks on heterogeneous multiprocessor platforms[END_REF] has been introduced into modern multicore platforms to improve the energy efficiency, as it adjusts the time and the energy required to execute the tasks. However, DVFS has a negative impact on task execution and reliability. Lower frequencies lead to longer execution times and increased transient fault rates [START_REF] Gou | Reliability-aware energy optimization for throughput-constrained applications on MPSoCs[END_REF]. To increase the reliability, task duplication is applied [START_REF] Xie | Energy-efficient fault-tolerant scheduling of reliable parallel applications on heterogeneous distributed embedded systems[END_REF]. However, task duplication has a significant impact both on energy and time; more tasks are executed, thus, more energy and time is required for the task computation and their communication over the NoC. Therefore, task duplication and frequency assignment should be also considered during the deployment process [START_REF] Radetzki | Methods for fault tolerance in networks-on-chip[END_REF]. The task deployment problem has already been studied in embedded systems. Table I categories representative papers from the literature, performing task allocation (All.), scheduling (Sch.) and duplication (Dup.) on multicores with DVFS (VF.), considering multi-path routing (MP.), task reliability (Reli.) and communication cost (Com.). The solutions are given by optimal (Opt.) and heuristic (Heur.) algorithms. Several approaches exist to map dependent/independent tasks on the multi-core/singlecore embedded platforms under multiple constraints, such as energy, real-time, and reliability [START_REF] Li | Minimizing energy consumption for frame-based tasks on heterogeneous multiprocessor platforms[END_REF], [START_REF] Xie | Energy-efficient fault-tolerant scheduling of reliable parallel applications on heterogeneous distributed embedded systems[END_REF], [START_REF] Mo | Controllable QoS for imprecise computation tasks on DVFS multicores with time and energy constraints[END_REF], [START_REF] Haque | On reliability management of energy-aware real-time systems through task replication[END_REF]. These works consider processors typically connected with a highspeed data bus. Thus, the communication cost between any two processors is usually ignored, as it's much smaller compared to task execution cost. However, for the platforms based on NoC, the communication cost becomes important. To reduce the communication cost on NoC-based platforms, the common methods include mapping tasks to processors [START_REF] He | UNISM: Unified scheduling and mapping for general networks on chip[END_REF], [START_REF] Sayuti | Real-time low-power task mapping in networks-on-chip[END_REF] and adjusting the operating voltage of the routers [START_REF] Han | Contention-aware energy management scheme for NoC-based multicore real-time systems[END_REF], [START_REF] Ishak | Energy-aware task scheduling on heterogeneous NoC-based MPSoCs[END_REF]. However, task reliability is not taken into account, especially when DVFS is available. Existing approaches, to enhance NoC reliability, include spare processors [START_REF] Bhanu | Fault-tolerant network-onchip design with flexible spare core placement[END_REF] and task duplication [START_REF] Namazi | A majoritybased reliability-aware task mapping in high-performance homogenous NoC architectures[END_REF], [START_REF] Chatterjee | Fault-tolerant dynamic task mapping and scheduling for network-on-chip-based multicore platform[END_REF]. However, no DVFS is considered [START_REF] Namazi | A majoritybased reliability-aware task mapping in high-performance homogenous NoC architectures[END_REF], [START_REF] Chatterjee | Fault-tolerant dynamic task mapping and scheduling for network-on-chip-based multicore platform[END_REF].

Compared with the state-of-the-art, we propose a novel task deployment process that includes simultaneous optimization of task allocation and scheduling, frequency assignment, task duplication and path selection, in order to balance the energy consumption of NoC-based multicore platforms, while meeting the system requirements regarding real-time execution and reliability. The main contributions of this work are: 1) An MINLP formulation for energy efficient task deployment problem on NoC-based multicores. The original problem is equivalently converted to an MILP problem, by adding auxiliary variables and constraints, to be optimally solved. 2) A decomposition-based heuristic to solve the above problem, which divides the original problem into three subproblems, having a simpler structure with less constraints and variables. 3) Extensive experimental results to demonstrate the advantages of the proposed approach, and evaluate the impact of parameters on task deployment. Compared with the optimal method, the proposed heuristic has negligible computation time, with an average cost of 26.5% in energy savings.

Next, Section II introduces the system model and the problem formulation. Section III designs the heuristic method. Section IV presents the evaluation and Section V the conclusion.

II. SYSTEM MODEL AND PROBLEM FORMULATION

A. System Model 1) Task Set: The task set consists of M periodic tasks {τ 1 , . . . , τ M }, released at time 0, sharing a common scheduling horizon H. Each task τ i is described by a tuple {C i , D i , p ij , s ij }, where C i is the Worst Case Execution Cycle (WCEC), D i is the relative deadline, and p ij is the (i, j) th element of task dependency matrix p = [p ij ] M ×M . If tasks τ i and τ j are dependent, and τ i is a predecessor of τ j , p ij = 1, else, p ij = 0. When τ i finishes, it will generate a set of data with sizes s ij for its successor τ j (p ij = 1).

2) Platform: The target platform consists of N processors {θ 1 , . . . θ N } and N routers (R), connected through a 2D-mesh network, as a 2 × 2 example shown in Fig. 1(a). Each router communicate with the neighbor routers through a pair of links. Data can be transmitted through multiple paths.

The processors support DVFS and have the same Instruction Set Architecture (ISA). A processor has L different Voltage/Frequency (V/F) levels {(v 1 , f 1 ), . . . , (v L , f L )}. A typical power model [START_REF] Ishak | Energy-aware task scheduling on heterogeneous NoC-based MPSoCs[END_REF] is considered: the processor power with (v l , f l ) is P l = P s l + P d l . The static power is

P s l = L g (v l K 1 e K2v l e K3v b + |v b |I b ). The dynamic power is P d l = C e v 2
l f l . C e is the average switched capacitance. L g is the number of logic gates. K 1 , K 2 and K 3 are parameters depending on the processor type. v b and I b are the body-bias voltage and body junction leakage current. The computation energy of task τ i is e comp i = P l t comp i , where t comp i its computation time. The NoC topology is described by a directed graph G(V, E), where the vertexes represent each processor θ k ∈ V, while the edge l ij ∈ E represents a direct communication link between the routers of processors θ i and θ j . Based on the NoC communication model and the Manhattan distance between

𝜃 $ (0,0) R 1 𝜃 % (0,1) R 2 𝜃 & (1,0) R 3 𝜃 * (1,1) R 4 (a) Platform 𝜃 ! 𝜃 " 𝜃 # 𝜃 $ 𝜃 % 𝜃 & 𝜃 ' 𝜃 ' 𝜃 ( 𝑤 )*

Energy-oriented path

Time-oriented path (b) Multi-path routing the source and destination processors, a positive weight w ij is associated with the edge l ij . The goal of the task deployment is energy reduction under real-time constraints, we consider both energy-oriented and time-oriented paths as available options to transmit data, as shown in Fig. 1(b). Note that, the path with minimal energy can be different from the path with minimal latency [START_REF] He | UNISM: Unified scheduling and mapping for general networks on chip[END_REF]. Our approach explores these different paths.

𝝉 𝟏 𝝉 𝟐 𝝉 𝟑 𝝉 𝟒 𝝉 𝟓 𝝉 𝟔 𝑠 $% 𝑠 *% 𝐿 $ 𝐿 % 𝐿 & (c) Task duplication
For the energy (time)-oriented path, the weight w ij represents the energy (time) required for transmitting and receiving a unit of data between processors θ i and θ j . Hence, the aim of energy (time)-oriented routing is to find the shortest path, e.g., according to Dijkstra's algorithm. Based on the graph G(V, E), we obtain an energy matrix e = [e βγkρ ] N ×N ×N ×P and a time matrix t = [t βγρ ] N ×N ×P , where e βγkρ represents the energy consumed at processor θ k , if a unit of data is routed from θ β to θ γ through the ρ th path, while t βγρ denotes the time required to transmit a unit of data from θ β to θ γ through the ρ th path. When two dependent tasks are mapped on the same processor, the communication cost (time and energy) is zero [START_REF] Sayuti | Real-time low-power task mapping in networks-on-chip[END_REF]. The communication energy of a router, including the communications between this router and its neighbor routers and associated processor, is incorporated into the energy consumption of the processor for convenience.

3) Reliability: This work focuses on transient faults and adopts the Poisson fault probability model [START_REF] Xie | Energy-efficient fault-tolerant scheduling of reliable parallel applications on heterogeneous distributed embedded systems[END_REF]. If a processor uses (v l , f l ) to execute the task τ i with C i cycles, the task reli-

ability is r il = e -λ×10 d(fmax-f l ) fmax-f min × C i f l
, where λ is the maximum failure, d is a constant that indicates the sensitivity of the failure rate corresponding to frequency scaling, f max = max ∀l {f l } and f min = min ∀l {f l }. When the reliability of task τ i is lower than its threshold R th , to enhance the task reliability, task τ i is duplicated (see ( 4) and ( 5)), considering that it is unlikely to have faults occurring concurrently in both copies [START_REF] Gou | Reliability-aware energy optimization for throughput-constrained applications on MPSoCs[END_REF].

Note that the task duplication affects the task model, and thus, the computation and communication cost. For instance, tasks τ 1 , τ 2 and τ 3 are the original tasks, and tasks τ 4 , τ 5 and τ 6 are their copies in Fig. 1(c). By task duplication, the task dependencies change, e.g., due to the dependency of τ 1 and τ 2 , τ 4 and τ 2 become dependent and τ 4 generates data to τ 2 .

B. Problem Formulation

The goal is to deploy tasks to balance the energy consumption of the overall system under real-time and reliability constraints. To achieve that, the following decisions are taken: 1) task frequency assignment, 2) task duplication, 3) path selection, 4) task allocation and 5) task scheduling. To formulate the task deployment problem, we introduce the following variables: 1) binary variable y il = 1 if task τ i is executed with frequency f l , otherwise, y il = 0; 2) binary variable h i = 1 if task τ i exists, otherwise, h i = 0; 3) binary variable c βγρ = 1 if data is transmitted from θ β to θ γ through the ρ th path, otherwise, c βγρ = 0; 4) binary variable x ik = 1 if task τ i is allocated to processor θ k , otherwise, x ik = 0; 5) binary variable u ij = 1 if task τ i proceeds τ j , otherwise, u ij = 0; 6) continuous variable t s i represents the start time of task τ i . For the sake of paper presentation, let N = {1, . . . , N }, M = {1, . . . , M }, M = {1, . . . , 2M }, L = {1, . . . , L} and P = {1, 2}.

1) Task Allocation Constraints: Each task is executed on a single processor [START_REF] Li | Minimizing energy consumption for frame-based tasks on heterogeneous multiprocessor platforms[END_REF], without task migration:

k∈N x ik = 1, ∀i ∈ M .
(1)

2) Path Selection Constraints: One data routing path is selected between two processors [START_REF] Bhanu | Fault-tolerant network-onchip design with flexible spare core placement[END_REF]:

ρ∈P c βγρ = 1, ∀β = γ ∈ N .
(2)

3) Frequency Assignment Constraints: The task is executed with one V/F level [START_REF] Mo | Controllable QoS for imprecise computation tasks on DVFS multicores with time and energy constraints[END_REF]:

l∈L y il = 1, ∀i ∈ M . (3)
4) Task Reliability Constraints: Let τ i and τ i+M (∀i ∈ M) denote the original and duplicated tasks, respectively, where τ i and τ i+M have the same execution cycles. Taking r il and y il into account, the reliability of task τ i (∀i ∈ M), without duplication, is r i = l y il r il . Note that whether a duplicated task τ i+M exists, it dependents on the task reliability r i . To indicate that, we introduce a binary variable h i . Since the original tasks always exist, h i = 1 (∀i ∈ M). If r i ≥ R th , there is no need to duplicate task τ i (h i+M = 0); otherwise, task τ i is duplicated (h i+M = 1). We alternatively rewrite the above comparison in a linearly manner by the following lemma.

Lemma 2.1: 1 Assume that b is a binary variable, while x is a variable bounded by 0 ≤ x ≤ s. The comparison: 1)

x ≥ s 1 ⇒ b = 0; 2) x < s 1 ⇒ b = 1 can be described by x-(s1-σ) s ≤ 1 -b ≤ x s1
, where s 1 ≤ 1 is a constant, and σ is a positive small enough value.

Based on Lemma 2.1, let σ = min ∀i,l {|r il -R th |}. The relationship between r i and h i+M can be described as:

r i -(R th -σ) max ∀i,l {r il } ≤ 1 -h i+M ≤ r i R th , ∀i ∈ M. (4)
By duplicating the task τ i (∀i ∈ M), its reliability is given by

r i = 1 -(1 -h i r i )(1 -h i+M r i+M ).
Therefore, to satisfy the reliability constraint, we have

r i ≥ R th , ∀i ∈ M.
(5)

5) Task Sequence Constraint:

A task τ i cannot start execution until the input data from all its predecessors has arrived. When a task completes execution, its output data is available for transmission to all its successors. A router propagates the received task data, from other routers, towards its processor in sequence, and thus, the time spent for receiving the data required for the task execution is

t comm i = j β γ ρ p ji h i h j x iβ x jγ c βγρ t βγρ .
1 Due to the page limit, the proofs of the lemmas are omitted.

Let t s i and t e i denote the start time and the end time of task τ i , respectively, where 0 ≤ t s i ≤ t e i ≤ H. The task execution time t comp i = t e i -t s i = h i l y il Ci f l . For the dependent tasks τ i and τ j , their start time and end time are bounded by

t s j + (1 -p ij )H ≥ t s i + p ij t comp i + t comm j , ∀i = j ∈ M . (6)
If p ij = 1, τ i precedes τ j and τ j is the closest task of τ i , we get t s j ≥ t e i + t comp j , else, ( 6) is always satisfied. 6) Task Non-Overlapping Constraint: When independent tasks τ i and τ j (i.e., p ij = 0), are allocated to the same processor, their execution sequence must be determined, since the processor executes only one task at a time instance:

t e i ≤ t s j +(2-x ik -x jk )H +(1-u ij )H, ∀i = j ∈ M , ∀k ∈ N . (7)
If τ i and τ j are assigned to the same processor (e.g., x ik = x jk = 1), ( 7) is meaningful, else, ( 7) is always true. With x ik = x jk = 1, if u ij = 1 (i.e., τ i precedes τ j ), we have t e i ≤ t s j , else (i.e., u ij = 0), ( 7) is always satisfied.

7) Real-Time Constraints: Since task τ i should be finished within the scheduling horizon H and its execution time should be smaller than the relative deadline D i , we have

t comp i ≤ D i , ∀i ∈ M , (8)
t e i ≤ H, ∀i ∈ M . (9)
8) Objective Function: Regarding the communication energy, if dependent tasks τ i and τ j are allocated to different processors, e.g., θ β and θ γ , the energy consumed on processor θ k to transmit task data, with size s ij from θ β to θ γ , through the ρ th path, is e comm ijβγkρ = p ij s ij x iβ x jγ e βγkρ . On this basis, taking task duplication h i into account, the communication energy consumed by processor

θ k is E comm k = i j β γ ρ h i h j e comm ijβγkρ .
Regarding the computation energy, to execute task τ i with (v l , f l ), the required energy is e comp i = h i l y il Ci f l P l . Taking task allocation x ik into account, the computation energy consumed by processor

θ k is E comp k = i x ik e comp i .
To balance the energy consumption of the processors [START_REF] Joven | QoS-driven reconfigurable parallel computing for NoC-based clustered MPSoCs[END_REF], the task deployment problem is formulated as follows:

P1 : min x,y,h,c,u,t s (max ∀k {E comm k + E comp k }) (10)
s.t. ( 1) -(8).

9) Problem Linearization: Since the nonlinear items h i y il , x ik h i y il and h i h j x iβ x jγ are included in the constraints ( 5)- [START_REF] Xie | Energy-efficient fault-tolerant scheduling of reliable parallel applications on heterogeneous distributed embedded systems[END_REF], problem [START_REF] Mo | Controllable QoS for imprecise computation tasks on DVFS multicores with time and energy constraints[END_REF] is an MINLP, which is difficult to solve directly. To simplify the structure of the problem, we propose a linearization method to deal with nonlinear items.

Lemma 2.2: Assume that x, y and z are the binary variables. The nonlinear item z = xy can be replaced by the following constraints: z -x ≤ 0, z -y ≤ 0 and x + y -z ≤ 1.

The objective function with the constraints completes the MILP formulation, which can be optimally solved by the existing methods, such as branch-and-bound or Gurobi solver. However, significantly large amount of CPU time and resources is required for the optimal solution, especially with large problem sizes. This limitation reduces the applicability of complex, but more realistic, task deployments approaches in real systems.

eM axComp = max ∀i { C i f S[i] P S[i] } (S[i] = -

21:

Calculate y (i+M )l according to (5); 22: end for III. HEURISTIC ALGORITHM Therefore, to improve the scalability of the proposed task deployment approach, we propose a novel heuristic to efficiently solve problem [START_REF] Mo | Controllable QoS for imprecise computation tasks on DVFS multicores with time and energy constraints[END_REF], which contains the following three phases.

1) Frequency Assignment and Task Duplication: The processors have the same ISA and V/F levels. Therefore, if a frequency is assigned to each task, and then, the tasks are allocated to the processors or different paths are selected for data transmission, the decision regarding the frequency assignment is still valid. In addition, as task duplication h i is determined by frequency assignment y il , according to (4), y il and h i should be still jointly optimized. Since y il and h i mainly influence the time t comp i and the energy e comp i of task computation, to balance the computation energy, the frequency assignment and task duplication problem is formulated as:

P2 : min y,h (max ∀i {e comp i }) (11)
s.t. ( 3), ( 4), ( 5), [START_REF] Xie | Energy-efficient fault-tolerant scheduling of reliable parallel applications on heterogeneous distributed embedded systems[END_REF].

Minimizing the maximum energy consumption of each task execution helps to balance the energy consumption of the processors during task allocation occurring in phase 2. Since task allocation x ik and path selection c βγρ are currently unknown, the communication cost (time and energy) is not considered in [START_REF] Haque | On reliability management of energy-aware real-time systems through task replication[END_REF]. Note that P2 is an INLP problem, as the binary variables y il and h i are coupled nonlinearly in ( 5) and [START_REF] Xie | Energy-efficient fault-tolerant scheduling of reliable parallel applications on heterogeneous distributed embedded systems[END_REF]. To solve this problem, a heuristic is proposed (described in Algorithm 1) based on the Greedy Algorithm (GA) [START_REF] Pathak | Energy-efficient task mapping for datadriven sensor network macroprogramming[END_REF]. The steps are: a. An index S[i] is introduced for frequency assignment per task τ i (∀i ∈ M ), and it is initialized as

S[i] = -1 (Line 3). If (v l , f l ) is used to execute task τ i , then S[i] = l. b.
Algorithm 1 follows the sequence {τ 1 , . . . , τ M } to iteratively assign the frequencies {f 1 , . . . , f L } for each task τ i , with the aim to minimize the increase of energy consumption among the tasks that have already been assigned a frequency, i.e., min(max ∀i {e comp i }) (S[i] = -1) (Line 10). If the real-time constraint (8) cannot be satisfied, the frequency assignment y il = 1 is excluded (Line 7). c. When the frequency assignment y il regarding the original task τ i is known, the existence of the duplicated task h i+M Algorithm 2 Task Allocation and Scheduling 19: end for can be computed through [START_REF] He | UNISM: Unified scheduling and mapping for general networks on chip[END_REF]. A similar method is used to assign a frequency to the duplicated tasks. The main difference is that, when assigning frequency to the duplicated tasks, the selected frequency should satisfy the reliability constraint ( 5) while providing the minimum energy increase. 2) Task Allocation and Scheduling: With the frequency assignment y il and task duplication h i , the computation time and energy of task τ i are t comp i = h i l y il Ci f l and e comp i = h i l y il Ci f l P l , respectively. The next step is to determine the task allocation x ik , task sequence u ij and task start time t s i . To balance the energy consumption of the processors under the task sequence and the task non-overlapping constraints, the task allocation and scheduling problem is given by: P3 : min

x,u,t s (max

∀k {E comm k + E comp k }) (12)
s.t. ( 1), ( 6), [START_REF] Gou | Reliability-aware energy optimization for throughput-constrained applications on MPSoCs[END_REF].

Note that the communication energy E comm k and communication time t comm i are influenced by the path selection c βγρ . However, c βγρ is unknown at the current step. To formulate the problem [START_REF] Sayuti | Real-time low-power task mapping in networks-on-chip[END_REF], we fix the values of E comm k and t comm i to the average communication time of task τ i and average communication energy of processor θ k , respectively, i.e.,

t comm i = M 1 (max ∀β,γ,ρ {t βγρ } + min ∀β,γ,ρ {t βγρ })/2 and E comm k = M 2 (max ∀β,γ {e βγk1 } + min ∀β,γ {e βγk2 })/2. M 1 is
the number of predecessors of τ i , while M 2 is the number of original and duplicated tasks. Once the path selection c βγρ is determined, the value of t comm i and E comm k is updated accordingly. Based on the structure of MINLP problem (12), we design Algorithm 2: a. A task allocation index O[i] is introduced for each task τ i (h i = 1) and it is initialized as

O[i] = -1. If task τ i is allocated to processor θ k , we have O[i] = k. b.
The in-and out-degrees of all tasks are calculated and the tasks are divided into layers. For instance, in Fig. 1(c), tasks (τ 1 , τ 4 ), (τ 2 , τ 5 ) and (τ 3 , τ 6 ) are assigned to the first (L 1 ), the second (L 2 ) and the third (L 3 ) layers, respectively. Tasks in the same layer are sorted in a descending order based on their execution cycles. If the tasks in same layer have same execution cycles, they are ordered randomly. c. Algorithm 2 follows the sequence provided in the previous step to perform task allocation. With this sequence, Algorithm 3 Path selection 1: Input: y il , hi, x ik , uij and t s i ;

2: Output: c βγρ and t s i ;

3:

Initialize: Q[β][γ] = -1 (∀β, γ ∈ N ); 4: for ∀(β, γ) ∈ N do 5:
M inM axCost = ∞;

6:

for ∀ρ ∈ P do 7: 14:

Q[β][γ] = ρ;
M inM axCost = max ∀k {E comm k + E comp k };
15:

if M axCost < M inM axCost then

16:

M inM axCost = M axCost;

17:

f lag = ρ;
18:

21:

end if

22:

end for 23:

Q[β][γ] = f lag;
24: end for the task sequence constraint ( 6) and task non-overlapping constraint ( 7) are handled at the same time. d. Algorithm 2 iteratively allocates a task τ i (h i = 1) to a processor θ k in order to balance the energy consumption of processors, i.e., minimize the maximum computation and communication energy of the processors (Lines 8-15).

[β][γ] = -1. Q[β][γ] = ρ
is the data transmission from θ β to θ γ , through the ρ th path. b. The routing path is determined iteratively for each pair of processors (θ β , θ γ ). The aim is to find a path for θ β and θ γ , that causes the minimum increase of communication and computation energy among these processors, i.e., min(max [START_REF] Ishak | Energy-aware task scheduling on heterogeneous NoC-based MPSoCs[END_REF][START_REF] Chatterjee | Fault-tolerant dynamic task mapping and scheduling for network-on-chip-based multicore platform[END_REF][START_REF] Joven | QoS-driven reconfigurable parallel computing for NoC-based clustered MPSoCs[END_REF][START_REF] Pathak | Energy-efficient task mapping for datadriven sensor network macroprogramming[END_REF][17][18][19][20]. During this process, the real-time constraint (9) should be satisfied.

∀k {E comp k + E comm k }) (Lines

IV. EVALUATION

The evaluation is performed considering a 4 × 4 2D-mesh NoC. The modeling of the energy consumed by processors and routers is based on [START_REF] Han | Contention-aware energy management scheme for NoC-based multicore real-time systems[END_REF]. The following parameters are considered in our experiments: the number of processors (N ), the number of the tasks (M ), the number of V/F levels (L).

Fig. 2(a) compares the energy consumption and the feasibility, considering multi-path routing (described by [START_REF] Mo | Controllable QoS for imprecise computation tasks on DVFS multicores with time and energy constraints[END_REF]) and single-path routing. Compared to multi-path routing, the path selection c βγρ is fixed in single-path routing. Both task deployment problems are optimally solved by Gurobi. We set N = 16, M = 20, L = 6 and H = α i∈C (t comp i,ave + t comm i,ave ),

where C is the set of tasks belonging to the critical path. t comp i,ave = (max ∀l { Ci f l P l } + min ∀l { Ci f l P l })/2 and t comm i,ave = M 1 (max ∀β,γ,ρ {t βγρ } + min ∀β,γ,ρ {t βγρ })/2 are the average computation time and communication time of task τ i , respectively. Fig. 2(a) shows that with small α, e.g., α = 0.1 or α = 0.2, the problem is infeasible, since the constraints are hard to satisfy. The problem feasibility increases with α; the larger the value of α, the smaller the energy consumption. This is because the feasibility region of the problem enlarges with α, and the task deployment is a minimization problem. Fig. 2(a) also shows that under the same value of α, multi-path routing has a higher problem feasibility than single-path routing. Multipath routing achieves a lower energy consumption because the path selection c βγρ is considered in the optimization. Thus, it can find a better path selection, further improving the energy efficiency, compared with single-path routing. , where e comm k = max ∀β,γ,k,ρ {e βγkρ } and e comp k = max ∀i,l { Ci f l P l } are the processor parameters regarding communication and computation energy. The larger the value of µ, the more energy is consumed in data transmission than task execution. Fig. 2(b) shows that M max increases with µ; when tasks have a large communication energy, the dependent tasks are allocated to the same processor to reduce the cost.

Fig. 2(c) explores the influence of processor parameters on task duplication decision h i , where M d represents the number of duplicated tasks for M original tasks, and = max ∀l { P l f l }/ min ∀l { P l f l } is an index that represents the gap regarding task execution energy. Note that t comp i = l∈L y il Ci f l P l is the energy required to execute task τ i . The larger the value of , the more energy is consumed because of task execution with high frequency, compared to task execution with low frequency. Fig. 2(c) shows that M d increases with . When is small, the execution of one task (original task) with high frequency is more energy efficient than executing two tasks (original and duplicated tasks) with low frequency. With increasing, executing two tasks with low frequency becomes more efficient than executing one task with high frequency.

= E comm k + E comp k
is the total energy of processor θ k . To evaluate the performances of these schemes, we introduce an index φ = max ∀k {E all k }/ min ∀k {E all k }, where E all k = 0. The smaller the value of φ, the more balance is achieved regarding energy consumption of all processors. Fig. 2(d) and Fig. 2(e) show that the total energy consumption of ME is lower than BE (average 13.62%). However, the value of φ for BE is smaller than ME. This is because ME allocates the tasks to the same processors to reduce communication energy. Therefore, some processors will consume more energy than others. However, with BE this trend can be avoided in order to achieve energy balance. achieved by the proposed heuristic and the optimal solution, obtained by the Gurobi solver. From Fig. 2(f), we observe that the algorithm computation time increases with task number M , as more variables and constraints are involved. On the contrary, the proposed heuristic has a negligible computation time, since it divides the problem into three subproblems, i.e., P2, P3 and P4, which are solved in sequence. From Fig. 2(g), we observe that the solution of the heuristic has a higher, but still acceptable, energy consumption (average 26.05%) than the optimal solution, since it only provides a feasible solution.

Fig. 2(h) shows the feasibility in solving the task deployment problem [START_REF] Mo | Controllable QoS for imprecise computation tasks on DVFS multicores with time and energy constraints[END_REF] for the optimal and the heuristic methods. The experiments have been repeated n a = 30 times with different task graphs. The used metric is the problem feasible ratio δ = n f /n a , where n f is the number of experiments with feasible solutions. Fig. 2(h) shows that δ increases with α. The constraints are relaxed with α, thus the feasible region of problem is enlarged. The optimal feasibility is higher than the heuristic, since it optimizes the variables concurrently, whereas the heuristic optimizes the variables step by step.

V. CONCLUSION

This work proposes a task deployment process for NoC-based multicore platforms with DVFS, that balances the overall energy consumption, under reliability and real-time constraints. The deployment process is formulated as an MINLP problem and equivalently transformed to an MILP problem. Our formulation jointly optimizes frequency assignment, task allocation, task scheduling, task duplication and path selection. Moreover, a novel heuristic method is proposed to enhance scalability, achieving good solutions with low computation time.

Fig. 1 .

 1 Fig. 1. An example of NoC-based multicore system.

Algorithm 1 Frequency

 1 Assignment and Task Duplication 1: Input: (v l , f l ) (∀l ∈ L), Ci, Di (∀i ∈ M ); 2: Output: y il and hi; 3: Initialize: S[i] = -1 (∀i ∈ M ), hi = 1, h i+M = -1 (∀i ∈ M); 4: for ∀i ∈ M do 5: eM inM axComp = ∞;

3 )

 3 Multi-path Selection: The final phase determines the path selection c βγρ . Note that c βγρ does not influence the computation energy E comp k and time T comp i , only the communication energy E comm k and time T comm i . To balance the energy consumption of the processors, while meeting the realtime constraints, the path selection problem is formulated as: above ILP problem, we propose Algorithm 3. a. A path selection index Q[β][γ] is used for each processor pair (θ β , θ γ ), and initialized as Q

Fig. 2 (

 2 b) shows the influence of processor parameters on task allocation decision x ik . We introduce M max = max ∀k {M k }, where M k is the number of tasks allocated to processor θ k , and µ = e comm k /e comp k

Fig. 2 (

 2 d) and Fig.2(e) compare the energy consumption of the proposed task deployment scheme, with the goal of Balancing the Energy consumption (BE), and the task deployment scheme with the goal of Minimizing the Energy consumption (ME), i.e., min k∈N E all k , where E all k

Fig. 2 (Fig. 2 .

 22 Fig. 2. System performance with different parameters varying.

  Input: y il and hi; 2: Output: x ik , uij and t s

	i ; 3: Initialize: O[i] = -1 (∀i ∈ M );
	4: Sort tasks according to their in-degree and out-degree;
	5: for ∀i ∈ M (hi = 1) do
	6:	M inM axEng = ∞;
	7: 8: 9: 10:	for ∀k ∈ N do Calculate E comm k M axEng = max ∀k {E comm and E comp k ; k if M axEng < M inM axEng then + E comp k	};
	11:	M inM axEng = M axEng;
	12:	O[i] = k;
	13:	else
	14:	Continue;
	15:	end if
	16:	end for
	17: 18:	Calculate x ik according to O[i] = k; Calculate uij and t s i according to x ik , t comm i	and t comp i	;

1:

√ √ √ √

ACKNOWLEDGMENT

This work was partially supported by Fundamental Research Funds for the Central Universities (Grants 2242021R10113 and MCCSE2021B02), and by Southeast University "Zhishan Scholars" (Grant 2242021R40003).


Pierre Hankach

Pascal Gastineau

Pierre Olivier

Pierre-Olivier Vandanjon

Multi

Keywords: distance-based Moran's eigenvector maps, car ownership, spatial structure, spatial vectors, spatial scales, multi-scale spatial analysis, sociodemographic factors, variation partitioning

published or not. The documents may come

Introduction

Transportation is highly correlated with spatial organization at the local, regional and global levels. Transportation data may display recognizable spatial structures at many scales. For our purposes herein, scale refers to the unit of measurement, e.g. broad scale is used to describe phenomena with large extents, while fine scale for those with small extents. Analyzing such spatial structures at each scale could play a very important role in understanding the specific transportation-related context and can therefore be of prime interest to policymakers for transportation planning and governance.

Multi-scale spatial analysis can prove to be an efficient exploratory tool for: 1) identifying those scales at which transportation data vary spatially, ranging from the broadest, encompassing the entire studied area, down to the finest scale; 2) modeling the spatial structures at each scale; and 3) determining, at each scale, the processes at work to create these spatial structures, i.e. in explaining the share of transportation data variation at each scale that can be attributed to certain explanatory variables (e.g. sociodemographic) available for the analysis. Different processes are often at work at these various scales to shape spatial structures.

In recent years, Moran's eigenvector maps (MEM) have been developed and applied to analyse spatial data. MEM were developed in two distinct fields: statistical geography [START_REF] Griffith | Spatial autocorrelation and eigenfunctions of the geographic weights matrix accompanying geo-referenced data[END_REF], 2000a[START_REF] Boots | Global and local spatial autocorrelation in bounded regular tessellations[END_REF], and quantitative community ecology [START_REF] Borcard | All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices[END_REF][START_REF] Dray | Spatial modelling: a comprehensive framework for principal coordinate analysis of neighbour matrices (pcnm)[END_REF][START_REF] Legendre | Numerical ecology[END_REF]. They were applied 1) to filter the effect of spatial autocorrelation out of model residuals; 2) as a multi-scale spatial analysis method to analyze scale-dependent spatial structures. Multi-scale spatial analysis using Moran's eigenvector maps has yet to be applied more widely to transportation problems [START_REF] Wang | Understanding spatial filtering for analysis of land use-transport data[END_REF].

This paper describes and demonstrates how distance-based Moran's eigenvector maps [START_REF] Borcard | All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices[END_REF][START_REF] Dray | Spatial modelling: a comprehensive framework for principal coordinate analysis of neighbour matrices (pcnm)[END_REF]), a multi-scale spatial analysis method, can be applied to study the spatial layout of household car ownership patterns. Distance-based Moran's eigenvector maps analysis uses space as an explicit predictor. It is based on sets of variables describing spatial structures explicitly, derived from the coordinates of the geographic sites or from neighbourhood relationships among sites. This approach offers a detailed and precise description of spatial structures. Since space is fundamental to transportation planning and governance, analyzing transportation data with this method can prove to be determinant.

Car ownership has received considerable attention in recent years due to its important role in transportation and land use planning. [START_REF] Song | Measuring the spatial dimension of automobile ownership and its associations with household characteristics and land use patterns: A case study in three counties, south florida (usa)[END_REF] noted the necessity of understanding the spatial patterns of household car ownership. Various studies carried out have found that geography influences car ownership and moreover that the number of vehicles depends on household location. For example, suburbanization and car ownership have been shown to be closely correlated [START_REF] Auran | Aire urbaine de nantes : Mémento des modes de déplacements[END_REF].

Most previous studies on the determinants of household car ownership rates (see Section 2) were conducted at the household level (i.e. disaggregated data). The advan-tage of this approach is to place the vehicle acquisition decisions made by households in their historical context, e.g. vehicle acquisition or disposal may be tied to changes in household characteristics. However, these studies targeted a sample of the population and were unable to provide a comprehensive view of the study territory. Since disaggregated data are costly to collect, the geographic coverage of surveys remains uneven, making local variations sometimes difficult to identify.

Few works have thus far studied household car ownership by relying on statistical areas as their basic units. A statistical area is a polygon, representing a proportion of a 2-dimensional map, with which one or more aggregated variables can be associated, e.g. car ownership rate, sociodemographic characteristics. This paper analyzes the spatial structures of car ownership rates by using small statistical areas from census data. Patterns of statistical areas are identified at various scales and attributed to a number of explanatory variables. A major advantage of this approach is the lower cost required to construct the database compared with a disaggregated approach. Data can easily be obtained from a number of sources, either public or commercial. For example, periodic population censuses are common and readily available sources for such aggregated data.

In order to represent spatial structures in dbMEM (distance-based Moran's eigenvector maps) analysis, spatial variables are constructed. For this step, the dbMEM method computes a matrix of geographic distances between statistical areas. After truncation (as explained in Section 3), a principal coordinate analysis of the resulting matrix is performed, yielding eigenvalues and eigenvectors (also called spatial eigenfunctions). Taken together, these eigenfunctions depict the multi-scale distance relationships between statistical areas; they model spatial relationships in decreasing order of spatial scale (i.e. the first dbMEM eigenvector with the largest eigenvalue corresponds to the broadest spatial scale). Furthermore, the eigenvectors possess some interesting properties (e.g. orthogonality) that increase their desirability as spatial explanatory variables.

Following this construction step, the variation in car ownership rates is analyzed with respect to the spatial structures represented by the eigenvectors. For this purpose, the eigenvectors are used in generalized linear models as explanatory variables for the car ownership variable. Each significant eigenvector identified by regression explains part of the variation in car ownership. Since the eigenvectors are orthogonal by design, significant ones identified by regression can be grouped into sub-models representing several categories of spatial scales: broad, medium, etc. The subsequent step of the analysis consists of studying, at each scale, the influence of the sociodemographic variables on variation of car ownership rate. The last step of multi-scale spatial analysis consists in partitioning the deviance respectively explained by the sociodemographic variables and the spatial variables through variation partitioning.

The remainder of this paper will be organized as follows. Section 2 presents the existing literature, while Section 3 describes distance-based Moran's eigenvector maps for spatial analysis and other theoretical aspects. Section 4 displays the results of the car ownership study conducted in France's Loire-Atlantique administrative region. These results are then discussed in Section 5, and the final section offers concluding remarks.

Literature review

Car ownership has received a good deal of attention in recent years by virtue of its important role played in transportation and land use planning, as well as its relationship with energy, environmental and health issues [START_REF] Whelan | Modelling car ownership in great britain[END_REF]. Moreover, car ownership is considered by some authors as a key mediating link between residential location and choice of travel mode [START_REF] Ding | Joint analysis of the spatial impacts of built environment on car ownership and travel mode choice[END_REF]. [START_REF] Lavery | Driving out of choices: An investigation of transport modality in a university sample[END_REF] reports that car owners tend to have lower modality, which helps explain why their use of other modes is low. As such, it constitutes a key variable for urban planning policy-making. According to [START_REF] Van Acker | Car ownership as a mediating variable in car travel behaviour research using a structural equation modelling approach to identify its dual relationship[END_REF], urban planners should focus not only on influencing car use directly but also on indirect measures through car ownership, given that once a car has been purchased, it tends to be used more often.

The main determinants of vehicle ownership usually include variables affecting travel demand (e.g. individual and household attributes) and variables affecting transportation supply (e.g. accessibility and characteristics of the built residential environment [START_REF] Zegras | The built environment and motor vehicle ownership and use: Evidence from santiago de chile[END_REF][START_REF] Cao | The impacts of lrt, neighbourhood characteristics, and selfselection on auto ownership: Evidence from minneapolis-st. paul[END_REF]. Table 1, based on [START_REF] Anowar | Alternative modeling approaches used for examining automobile ownership: a comprehensive review[END_REF] comprehensive survey of car ownership in addition to more recent papers [START_REF] Cao | Examining the effects of the built environment on auto ownership in two norwegian urban regions[END_REF][START_REF] Moeckel | Car ownership and the impact of built environment, demographics and transport systems[END_REF][START_REF] Jiang | Influence of land use and street characteristics on car ownership and use: Evidence from jinan, china[END_REF][START_REF] Ding | How does the built environment at residential and work locations affect car ownership? an application of cross-classified multilevel model[END_REF][START_REF] Ma | Development of multivariate ordered probit model to understand household vehicle ownership behavior in xiaoshan district of hangzhou, china[END_REF]), provides the main variables used in previous studies to explain car ownership. These variables are classified into four categories: household demographics, individual attributes, environment and accessibility attributes.

Household

Individual Built Accessibility Environment

• family • age • density • transport speed composition • bicycle possession • urban form • public transport • income • socio-professional availability • education category • distance to city • number of • income center driving licences • education • travel time • number of children • residence type

Table 1: Main determinants of vehicle ownership

Most studies in the literature investigate the determinants of car ownership at a disaggregated (individual or household) level; they rely on surveying a sample of individuals in order to correlate the acquisition or disposal of a vehicle with changes in the characteristics of the household, environment or life events [START_REF] Clark | Changes in level of household car ownership: the role of life events and spatial context[END_REF]. Disaggregated level studies include [START_REF] Bhat | A comparison of two alternative behavioral choice mechanisms for household auto ownership decisions[END_REF], [START_REF] Whelan | Modelling car ownership in great britain[END_REF], [START_REF] Potoglou | Modelling car ownership in urban areas: a case study of hamilton, canada[END_REF], [START_REF] Anastasopoulos | Household automobile and motorcycle ownership analyzed with random parameters bivariate ordered probit model[END_REF] and [START_REF] Oakil | Determinants of car ownership among young households in the netherlands: The role of urbanisation and demographic and economic characteristics[END_REF]. Only a small number of these studies have actually taken into account the spatial dimension, e.g. [START_REF] Adjemian | Estimating spatial interdependence in automobile type choice with survey data[END_REF], [START_REF] Paleti | Modeling of household vehicle type choice accommodating spatial dependence effects[END_REF] and [START_REF] Clark | Changes in level of household car ownership: the role of life events and spatial context[END_REF]. Of these, the study of [START_REF] Páez | Development of an indicator to assess the spatial fit of discrete choice models[END_REF] is the only one to provide evidence that all spatial variability has been properly accounted for by the model.

Few studies have thus far examined vehicle ownership rates at the statistical area level. However, as explained above, a major advantage of this approach is its reliance on more easily accessible data, i.e. obtained from readily available governmental sources and less expensive to collect than survey data.

Geographically Weighted Regression (GWR) [START_REF] Brunsdon | Geographically weighted regression[END_REF][START_REF] Clark | Estimating local car ownership models[END_REF][START_REF] Clark | Spatial modelling of car ownership data: a case study from the united kingdom[END_REF] is one of the approaches employed to study car ownership at the statistical area level. This approach integrates the spatial autocorrelation of regression coefficients in analyzing the spatial distribution of car ownership. GWR fits a series of local models to the data, thus providing a series of locally varying parameter estimates (rather than explicitly accounting for correlations in the residuals). In their seminal paper, using census data for the County of Tyne and Wear in North East England, [START_REF] Brunsdon | Geographically weighted regression[END_REF] showed that car ownership rates are correlated with both social class and male unemployment. [START_REF] Clark | Estimating local car ownership models[END_REF] estimated the spatially varying coefficient of the income parameter and hence produced the local income elasticity. [START_REF] Clark | Spatial modelling of car ownership data: a case study from the united kingdom[END_REF] highlighted the influence of income and population density on the UK car ownership rate at the electoral ward scale.

In another approach, [START_REF] Clark | Estimating local car ownership models[END_REF] and [START_REF] Clark | Spatial modelling of car ownership data: a case study from the united kingdom[END_REF] estimated a spatial error model (SEM) that accounts for spatial dependence in the error terms rather than in the dependent variable. In addition, in [START_REF] Clark | Spatial modelling of car ownership data: a case study from the united kingdom[END_REF], a hierarchical Bayesian spatial model was proposed to handle residual spatial autocorrelation. [START_REF] Morton | Fuel price differentials and car ownership: A spatial analysis of diesel cars in northern ireland[END_REF] investigated the determinants of car fleet composition (i.e. the dieselization rate) recorded in Northern Ireland through a Spatial Durbin Error Model (SDEM), which accounts for spatial dependence among the error terms as well as the exogenous interaction effect. In considering 890 contiguous statistical areas containing a mean of 2,000 residents, these authors found that the availability of relatively inexpensive diesel fuel in the Republic of Ireland affected the diesel car ownership rate in Northern Ireland. [START_REF] Clark | Temporal dynamics in local vehicle ownership for great britain[END_REF] explored the local dynamics of household vehicle ownership in the UK, by comparing a classical Markov chain approach with a spatial Markov chain analysis. They showed that the transitions in relative vehicle ownership are influenced by the neighborhood context.

While the above approaches take neighborhood influence into account in car ownership models, none is well suited for an exhaustive identification of spatial structures at various scales. It is crucial to understand however the spatial patterns (including global and local spatial clustering) of car ownership, as [START_REF] Song | Measuring the spatial dimension of automobile ownership and its associations with household characteristics and land use patterns: A case study in three counties, south florida (usa)[END_REF] pointed out. This exploratory process can be of major interest to decision-makers while also being valuable to modelers in generating new hypotheses on explanatory variables.

Few methods have been proposed to model spatial structures at all scales using space as an explicit variable. Moran eigenvector spatial filtering (MESF) is one such method that has been proposed for this purpose. This method computes spatial eigenvectors as explicit spatial predictors. Interestingly, MESF was developed independently and nearly simultaneously in two distinct fields, i.e. statistical geography [START_REF] Griffith | Spatial autocorrelation and eigenfunctions of the geographic weights matrix accompanying geo-referenced data[END_REF](Griffith , 2000b) ) and quantitative community ecology [START_REF] Borcard | All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices[END_REF]. [START_REF] Griffith | Spatial autocorrelation and eigenfunctions of the geographic weights matrix accompanying geo-referenced data[END_REF]Griffith ( , 2000b) ) set the goal of filtering the effect of spatial autocorrelation out of model residuals, in transferring this component to a model's conditional mean (i.e. intercept), whereas [START_REF] Borcard | All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices[END_REF] sought to explicitly model the multi-scale nature of univariate or multivariate response data. [START_REF] Dray | Spatial modelling: a comprehensive framework for principal coordinate analysis of neighbour matrices (pcnm)[END_REF] formalized the theory originally proposed in [START_REF] Borcard | All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices[END_REF] as distance based Moran's eigenvector maps (dbMEM). [START_REF] Griffith | Spatial modeling in ecology: the flexibility of eigenfunction spatial analyses[END_REF] showed equivalencies of and differences between these two implementations. They advocated that the two methods can be unified under the class of spatial eigenfunction maps. They also highlighted that one important advantage of these two approaches over any other spatial approach is that they provide a flexible tool that allows the full range of general and generalized linear modeling theory to be applied to ecological and geographical problems in the presence of nonzero spatial autocorrelation.

Software implementations of MESF method were proposed in both quantitative community ecology and statistical geography fields. Dray et al. ( 2018) implemented Moran's Eigenvector Maps and related methods for the spatial multi-scale analysis of ecological data in the R package adespatial. 1 Murakami & Griffith (2019) proposed efficient algorithms for MESF which are implemented in the R package spmoran (Moran Eigenvector-Based Scalable Spatial Additive Mixed Models).2 A pedagogical synthesis of spatial (geo-)statistical and spatial econometric methods with R is presented in [START_REF] Yoshida | Chapter Seven -Implementation with R language[END_REF].

The dbMEM method has already been successfully used in several ecological applications including: [START_REF] Borcard | Dissecting the spatial structure of ecological data at multiple scales[END_REF] To the best of our knowledge, the MESF method has only very rarely been used to study transportation-related problems. [START_REF] Moniruzzaman | Accessibility to transit, by transit, and mode share: application of a logistic model with spatial filters[END_REF] investigated the transit shares for the city of Hamilton (Canada) by means of a logistic regression model for proportions and using a spatial filtering approach to control for spatial autocorrelation. [START_REF] Wang | Understanding spatial filtering for analysis of land use-transport data[END_REF] applied MESF to land use data and highlighted the relevance of this method for analyzing both land use and transportation data. [START_REF] Griffith | Modeling spatial autocorrelation in spatial interaction data: empirical evidence from 2002 germany journey-to-work flows[END_REF] demonstrated the use of MESF to study 2002 German journey-to-work flows among 439 German administrative units. [START_REF] Griffith | Visualizing analytical spatial autocorrelation components latent in spatial interaction data: an eigenvector spatial filter approach[END_REF] used MESF method for the visualization of spatial autocorrelation in the journey-to-work dataset of Pennsylvania, USA. Yet MESF has never been applied to study car ownership. The following section provides a detailed description of the MESF method as formalized in [START_REF] Dray | Spatial modelling: a comprehensive framework for principal coordinate analysis of neighbour matrices (pcnm)[END_REF], i.e. the dbMEM multi-scale spatial analysis method.

3 Distance-based Moran's eigenvector maps for multi-scale spatial analysis dbMEM analysis identifies spatial patterns across the entire range of perceptible scales with a given dataset. This method is based on a computation of the principal coordinates of a matrix of geographic neighbors among the geographic sites [START_REF] Borcard | All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices[END_REF]; it can be successfully applied on various spatial designs: linear (transect) and two-dimensional (surface), regular or irregular geographic schemes. This section describes both the construction of dbMEM eigenfunctions and multi-scale spatial analysis.

Construction of dbMEM spatial vectors

As stated above, dbMEM vectors are obtained by performing a principal coordinate analysis of a truncated matrix of Euclidean distances between defined geographic units. These units are given sites (e.g. statistical areas) for which population statistics are available. Notably, the share of car ownership and other variables are associated with each of these units. dbMEM spatial vectors are constructed according to the four following steps:

1. A 2-dimensional matrix of Euclidean distances (D) between geographic unit barycenters is built using the geographic coordinates of the relevant points.

2. A truncated connectivity matrix (W) is then constructed. During this operation, geographic distances are considered to belong to one of two groups: small or large distances. The truncation step is performed according to the following rule:

wij = dij if dij ≤ t (1) and wij = 4t if dij > t (2)
The value of threshold, t, is chosen. This parameter derives from the observation that MEM eigenvectors display variation across the full set of sites under study if the sites form a connected graph in the truncated matrix. Furthermore, a large truncation value implies a loss of the finest spatial structures. The most commonly applied solution calls for computing the minimum spanning tree of a single-linkage clustering of the site coordinates and then retaining the largest edge value. Hence, threshold t is computed as follows:

• A minimum spanning tree (MST) linking all points (i.e. site barycenters) in the study is created.

• The length of the largest edge in the chain forming the MST is determined.

• t is set equal to the length of the largest edge in the MST.

Disconnected pairs are identified in the matrix D truncated by distances equal to 4t. The value 4t is chosen here because computer simulations have shown little change to numerical results when using larger values [START_REF] Borcard | All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices[END_REF].

3. The diagonal values of the distance matrix, which were originally zeros, are replaced by the value 4t; this change on the diagonal of the truncated matrix indicates that a site is not connected to itself.

4. The principal coordinate analysis (PCoA) of the truncated matrix is computed, thereby yielding (n -1) non-zero eigenvalues and their corresponding eigenvectors [START_REF] Gower | Some distance properties of latent root and vector methods used in multivariate analysis[END_REF].

dbMEM spatial vector interpretation

Eigenvectors produced by PCoA decomposition are readily available tools for space partitioning. The components of each eigenvector, associated with corresponding geographic units, form an unique spatial pattern; this can apply to any spatial unit design, but spatial vectors are easier to interpret in the case of regular designs. As an illustration, Figure 1, extracted from [START_REF] Legendre | Numerical ecology[END_REF], shows maps of the first six eigenvectors produced after performing a dbMEM principal coordinate decomposition for a regular 12 x 8 grid. Shades of gray represent the values in each eigenvector, from white (largest negative value) to black (largest positive value). Spatial patterns, corresponding to the eigenvectors, can indeed be clearly identified. 2012)) produced after performing a dbMEM principal coordinate decomposition for a regular 12 x 8 grid

Scale representation

dbMEM variables model the spatial relationships among sites in decreasing order of spatial scale [START_REF] Borcard | All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices[END_REF]. The first dbMEM eigenvector (with the largest eigenvalue) corresponds to the broadest spatial scale, indicating the spatial extent of the entire study area (large-scale variation), while the last dbMEM eigenvector (with the smallest eigenvalue) corresponds to the finest spatial scale (small-scale variation).

To illustrate the spatial scales associated with dbMEM variables, let us consider a one-dimensional transect with n equally-spaced sites. Since the design is regular, the dbMEM eigenvectors representing the spatial variation resemble sine waves. Figure 2, extracted from [START_REF] Legendre | Numerical ecology[END_REF], shows the results for a 50-point transect. The complete sine wave of the first eigenvector has a wavelength of 51 units. The following eigenvectors form sine waves of shorter wavelengths, with the last eigenvector having a wavelength of 2.04. With irregularly-spaced designs (whether one-dimensional or two-dimensional), the dbMEM vectors lose the regularity of their shapes (at times complicating the scale assessment), yet the eigenvectors modeling broader-scaled and finer-scaled phenomena can still be distinguished.

Spatial autocorrelation and dbMEM spatial vectors

Spatial autocorrelation is a measure of similarity between the value of a variable at one location and the value(s) of the same variable at one or more proximal locations. A widely used tool for measuring spatial autocorrelation is Moran's coefficient [START_REF] Moran | The interpretation of statistical maps[END_REF]. In matrix form, Moran's coefficient can be formulated as follows:

M C = n i j wij x W x x x (3)
where x is a vector (n × 1) of mean-centered values of a geo-referenced variable, and W a spatial weights matrix of dimensions (n × n) with elements wij. The elements of the spatial weights matrix assume non-zero values if locations i and j are deemed to be spatially proximate in some sense, and 0 otherwise. [START_REF] Dray | Spatial modelling: a comprehensive framework for principal coordinate analysis of neighbour matrices (pcnm)[END_REF] showed that the eigenvalues of the dbMEM spatial eigenvectors are equal to Moran's I coefficients of spatial correlation (Eq. 3) computed for these same eigenvectors, divided by a constant. A positive eigenvalue therefore corresponds to a positive spatial autocorrelation, i.e. the Moran index is positive. On the contrary, a negative eigenvalue corresponds to a negative spatial autocorrelation at short range. In the case of a linear transect with equally spaced points (example shown in Figure 2), roughly half the eigenvectors have a positive Moran's I and model a positive spatial correlation, while the other half has a negative Moran's I and models a negative spatial correlation at short range.

This paper only considers eigenvectors with positive eigenvalues and Moran's I. The focus in fact lies on the positive spatial autocorrelation of car ownership in the spatial analysis. Most previous studies with dbMEM have considered positive spatial autocorrelation [START_REF] Legendre | Numerical ecology[END_REF]. However, eigenvectors with negative Moran I remain available when investigating negative spatial autocorrelation is needed.

Multi-scale spatial analysis

Significant spatial vectors

The number of dbMEM eigenvectors created by the principal coordinate analysis tends to be high. However, not all these vectors are significant and contribute to an explanation of the dependent variable (e.g. car ownership). Therefore, an appropriate selection method is needed to reduce their number [START_REF] Borcard | Dissecting the spatial structure of ecological data at multiple scales[END_REF]). Since dbMEM eigenvectors are orthogonal by design so as not to correlate with each other, they can be used in linear models as explanatory variables of the dependent variable. The model can thus be made more parsimonious, by means of retaining just the significant dbMEM eigenvectors, using an appropriate selection method (e.g. stepwise eigenvectors selection). The dbMEM eigenvectors can be used as explanatory variables in any model of general and generalized linear modeling theory [START_REF] Griffith | Spatial modeling in ecology: the flexibility of eigenfunction spatial analyses[END_REF].

Spatial scales

Given that dbMEM eigenvectors are orthogonal to one another, they can be combined into sub-models, corresponding to different spatial scales. Furthermore, any sub-model containing a subset of dbMEM is also independent of any other sub-model containing another subset. The spatial scales corresponding to these sub-models vary from large to very fine. For example, if dbMEM eigenvectors are grouped into three sub-models, then the sub-model representing the largest spatial scale is to be characterized by eigenvectors with the largest positive eigenvalues. Similarly, the sub-model representing the medium spatial scale is to be characterized by eigenvectors with intermediate positive eigenvalues and the fine scale by eigenvectors with small eigenvalues.

These sub-models are defined arbitrarily, yet a few procedures can still be followed for this selection step [START_REF] Borcard | All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices[END_REF]):

• predefine spatial scale limits, using the sizes of patterns corresponding to the dbMEM variables;

• identify groups of eigenvectors by examining a scalogram showing in the ordinate the regression coefficients (or the Pearson correlations or the absolute values of the t-statistics associated with the regression coefficients) of the dbMEM eigenvectors ordered along the abscissa in decreasing eigenvalues;

• draw maps of the significant dbMEM variables and group them visually according to the scales of the patterns they represent.

Variation of dependent data

Another step of multi-scale spatial analysis consists in identifying factors that explain the variation of dependent data: • variation of dependent data (e.g. car ownership) at each scale can be attributed to the sociodemographic variables available for the analysis (e.g. sociodemographic variables identified in Section 2);

• variation of dependent data can be represented simultaneously with respect to spatial scales and sociodemographic variables.

Variation partitioning analysis [START_REF] Mood | Partitioning variance in multiple regression analyses as a tool for developing learning models[END_REF][START_REF] Borcard | Partialling out the spatial component of ecological variation[END_REF], Peres-Neto & Jackson 2001) constitutes a comprehensive approach to represent the variation of dependent data. It consists of apportioning the variation of a variable, between two or more explanatory datasets pertaining to the different classes. Variation partitioning is commonly used in the field of ecology as an exploratory tool and to give insights into processes structuring communities [START_REF] Gebrehiwot | Partitioning the influence of hydrodynamics-induced physical variables and nutrients on phytoplankton assemblages in a shallow tropical reservoir (koka, ethiopia)[END_REF][START_REF] Checon | Taxonomic sufficiency and the influence of rare species on variation partitioning analysis of a polychaete community[END_REF], Cao et al. 2019[START_REF] Zbinden | Beta diversity of stream fish assemblages: partitioning variation between spatial and environmental factors[END_REF].

In a partitioning by two explanatory matrices (sets) X and W , both explain some variation of the response data. When the explanatory datasets are not orthogonal to one another, some amount of variation is explained jointly by the two sets. Consequently, the variation explained by all variables taken together is less than the sum of the variations explained by the various subsets. Figure 3 illustrates the partitioning of the explanatory power of different explanatory matrices (X and W) in relation to the same response matrix (Y).

Each explanatory matrix uniquely explains a portion of the variation in Y (X (resp. W) explains partition [a] (resp. [c])). In addition, both matrices explain another portion of the variation in Y (partition [b]), i.e. the explanatory matrices are redundant in this partition. The larger this fraction, the more multicollinearity present in the model. The residual variation in matrix Y (partition [d]) is explained by neither X nor W.

Study of car ownership in a French administrative region

This section will demonstrate the use of dbMEM multi-scale spatial analysis on car ownership rate data. The procedure presented in Section 3 is applied to the Loire-Atlantique Department (France).3 The dataset will be described before conducting a dbMEM analysis of the dependent variable (i.e. car ownership), then dividing the significant eigenfunctions into sub-models and interpreting these sub-models using explanatory variables. As a final step, dbMEM eigenfunctions will be applied within the framework of variation partitioning.

Data considerations

For this analysis, we are using data of the Loire-Atlantique Department from the 2016 census and the FiLoSoFi (localised disposable income system) database. These datasets have been collected respectively by the French National Institute of Statistics and Economic Studies (INSEE) and the French fiscal administration. The Loire-Atlantique department is located in northwestern France and has a population of roughly 1,400,000. The main city Nantes is France's 6 th largest (with a population of about 310,000).

The statistical areas used in this study are known as IRIS units (acronym for "aggregated units for statistical information"); they represent the basic units for disseminating sub-municipal data in France. All municipalities with over 10,000 inhabitants and a large proportion of those with populations between 5,000 and 10,000 are divided into several IRIS units. The municipalities not divided into multiple IRIS units constitute an IRIS unit in themselves. Within a residential IRIS, the population lies between 1,500 and 5,000. Both census and FiLoSoFi datasets are associated with IRIS units. Variables in the census database include household characteristics (e.g. family composition and car ownership) and individual ones (e.g. profession, age and gender). In Appendix B, Table B.1 provides a detailed description of census variables. Household median disposable annual income variable is given in the FiLoSoFi database.

Car ownership rate is one of the variables contained in the census database, meaning that the car ownership rate can be computed for each IRIS. The mean value of this variable for the study area is 87.41 percent, with a standard deviation of 10.92 percent and a median value of 91.79 percent. Households are more likely to own a car if they live in the Nantes outskirts. Figure 4 shows the proportion of households in each IRIS possessing at least one car. Based on the literature review (Section 2) that identified the main determinants of car ownership, we selected an initial set of explanatory variables from those available in the census and FiLoSoFi databases. This set of variables includes:

• family composition variables: shares of family categories (single member, couple without children, couple with children, single-parent and other composition);

• age variables: shares of different age groups (0-2, 3-5, 6-10, 11-14, 15-17, 18-24, 25-29, 30-39, 40-54, 55-64, 65-79 and 80/+);

• profession variables: shares of farmers, tradespeople, executives, intermediate occupations, clerical support workers, lower-skilled technical occupations, retiree and unemployed;

• income variable: median disposable annual income.

However, many of these variables are highly correlated (e.g. income and profession).

In order to avoid co-linearity in our model, only a subset of these variables is retained. This subset is constituted of the following variables:

• share of couples with children variable to take into account the effect of larger households;

• share of individuals in the 25-54 age group variable as individuals in this age group tend to be more active;

• median disposable annual income per consumption unit (consumption units of a household are defined as follows (Eurostat 2021): 1 consumption unit (CU) for the first adult in the household, 0.5 CU for every other person in the household aged 14 years or older, 0.3 CU for each child under 14).

In addition to the three selected variables, we also computed a fourth variable, i.e. population density (in inhabitants/km 2 ), to integrate it into our model. This variable has been used in previous studies of car ownership as a proxy to accessibility variables [START_REF] Clark | Estimating local car ownership models[END_REF]. It could be hypothesised that car ownership is likely to be higher in areas of lesser density because there is a poorer provision of public transport and people need to travel further to access shops and services. Therefore, population density of the IRIS is selected as one to help capture this accessibility effect.

Results

Spatial vectors

This analysis begins by defining the IRIS spatial neighborhood by means of a minimum spanning tree (see Section 3.1), which can be represented by an unweighted graph, as seen in Figure 5. For our study area, the longest edge of this tree has a length (t) equal to 7,604.5 meters. Beyond this truncation distance, IRIS pairs are deemed to be disconnected. The distance between disconnected IRIS pairs has been set at 4t in the distance matrix.

In order to compute spatial vectors, a principal coordinate analysis of the truncated distance matrix between the 499 IRIS has been carried out. This procedure resulted in 498 eigenvectors, 176 of which are associated with positive eigenvalues and the remaining 322 with negative eigenvalues. As explained in Section 3.2.2, only those eigenvectors associated with a positive eigenvalue have been taken into account in our analysis.

Significant spatial vectors

As the car ownership variable is expressed as a percentage, the standard techniques of statistical analysis (i.e. linear regression and ANOVA) are not appropriate. Thus, we use a generalized linear model (GLM) regression (with a quasi-binomial distribution to account for overdispersion and using a logit link function) in order to estimate the explanatory power of the 176 dbMEM vectors associated with a positive Moran's I value. Among the 176 vectors, only 23 significant ones are retained. The reduced model is presented in Table 3

Spatial sub-models

Significant dbMEM variables have been grouped into subsets corresponding to distinct spatial scales. They were partitioned into three distinct subsets corresponding to three scales by inspecting their geographical representation which are displayed in Appendix A:

• the first two significant dbMEM variables (MEM1, MEM4) have been grouped

to represent the broad scale (BS);

• the next nine dbMEM variables (MEM 6,8,9,13,14,15,16,17,20) have been grouped to represent the medium scale (MS);

• and the last twelve dbMEM variables (MEM 34,36,37,39,61,69,73,80,96,129,137,167) have been grouped to represent the fine scale (FS).

We define broad, medium and fine scale sub-models as mathematical operations for computing the variation of car ownership at the corresponding scales. In order to compute the variation of car ownership at each scale, the following steps are applied:

1. the sum of the chosen dbMEM vectors at each scale multiplied by their corresponding estimates (displayed in Table 3) added to the intercept is computed;

2. the variation of the car ownership at each scale is obtained by applying the inverse of the link function (used for GLM regression) to the result of the previous operation.

The values of car ownership variation at broad, medium and fine scale are shown respectively in Figures 6, 7 and8. Spatial structure associated with broad scale submodel matches the structure obtained distinguishing between main cities (Nantes and Saint-Nazaire) versus other areas (Figure 6). Spatial structure associated medium scale sub-model matches the structure obtained distinguishing between main city centers and close surrounding areas (Figure 7). Fine scale spatial structure is harder to interpret (Figure 8).

Multi-scale analysis of explanatory variables and variation partitioning

Let us now turn to identifying the sociodemographic characteristics that contribute to the variation in car ownership rate at each identified scale. For this purpose, the following sociodemographic variables have been considered: share of couples with children, share of individuals in the 25-54 year age range, share of households living in single-family dwellings, population density, and income. These sociodemographic variables are used as explanatory variables in regression analysis of the output of each of the three sub-models. Outputs of broad scale, medium scale and fine scale submodels are respectively regressed on these variables. The results of these regressions establish the contribution of sociodemographic variables at each scale.

The results of multi-scale analysis of explanatory variables are displayed in Table 4. The table reports the signs of the significant coefficients (with a p-value < 0.1) associated with each variable of the following GLM regressions: (i) car ownership on the sociodemographic variables (column 1 of the table); (ii) variations of car ownership at each scale (i.e. broad, medium, fine) as displayed by Figures 6,7 and 8 on the sociodemographic variables (columns 2, 3 and 4).

(1)

(2) (3) (4) car ownership Broad scale Medium scale Fine scale Median income

+ - + +
Couples with children

+ + + +

25-54 yrs -

+

Population density --

Table 4: Signs of significant coefficients related to the spatial analysis of the car ownership variable Some important elements should be highlighted as regards regression analysis displayed in Table 4.

Considering the regression of the car ownership on the sociodemographic variables (Table 4, column 1), the signs of the coefficients of the median income and the population density are as expected. In line with previous literature [START_REF] Clark | Estimating local car ownership models[END_REF][START_REF] Clark | Spatial modelling of car ownership data: a case study from the united kingdom[END_REF], the car ownership rate is positively correlated with the median income, in contrast, it is negatively correlated with the population density. Furthermore, the coefficient is positive for the share of couples with children and negative for the share of individuals between 25-54 years.

Considering the three other regressions (Table 4, columns 2, 3 and 4), the multiscale analysis provides a more nuanced view of the effect of the different sociodemographic explanatory variables. The variables population density, median income and couples with children are significant at the broad scale: population density and median income are negatively correlated with the response variable; while couples with children variable is positively correlated. It is interesting to note that median income is negatively correlated with car ownership variation at broad scale (Table 4, column 2) although it is positively correlated to car ownership in the purely sociodemographic model (Table 4, column 1). This derives from the fact that spatial structure at broad scale is closely related to the structure obtained distinguishing between main cities (Nantes and Saint-Nazaire) versus other areas as shown in Figure 6. Furthermore, centers of the two main cities in Loire-Atlantique Department tend to concentrate higher income households (i.e. median income in town centers is higher than other areas). The combination of these two factors yields a negative correlation between median income and car ownership variation at broad scale. At the medium and fine scales, couples with children and median income are significant and positively correlated with the response variable. The share of individuals in the 25-54 year age range is only significant at the medium scale (Table 4,column 3). As the medium scale distinguishes between main city centers and close surrounding areas (Figure 7), this implies a variation in the share of individuals in the 25-54 year age range between these two areas. Additional data are needed to further explore this result. Finally, one can note that the population density is significant at broad scale (Table 4, column 2) but not significant at medium and fine scales (Table 4, columns 3 and 4).

These results suggest that car ownership variation at each scale and sociodemographic variables are indeed correlated; hence, it can be inferred that at least a portion of the spatial variation in car ownership rate (i.e. structure) is induced by the relevant driving (i.e. sociodemographic) factors. In order to apportion car ownership variation between sub-models and sociodemographic variables, a variation partitioning analysis is undertaken. This analysis estimates both the amount of variation that can be attributed exclusively to sub-models (spatial variables) and the amount explained jointly by spatial and sociodemographic variables.

Figure 9 displays the results of this variation partitioning analysis following the method proposed by [START_REF] Randin | Land use improves spatial predictions of mountain plant abundance but not presenceabsence[END_REF]. The explanatory matrices defined contain variables pertaining to sociodemographic determinants and various spatial scales. The first model (i.e. full) considers all significant dbMEM variables while the other three models (i.e. broad, medium and fine) consider only the subset of dbMEM variables associated with each scale. We partition the variation into four identifiable fractions of deviance: (i) pure sociodemographic (black area), (ii) pure spatial (grey area), (iii) shared sociodemographic and spatial (mustard area), and (iv) unexplained variation (yellow area). • The car ownership rate variation is well explained by the model: the proportion of variation explained by the model ranges from 85 % for the full model to 81 % for the model including the fine scale as spatial model.

• As expected, the sociodemographic model explains most of the variation: the sum of the deviance explained by the pure sociodemographic fraction and the shared sociodemographic and spatial fraction represents at least 81 % of the total variation.

• The car ownership rate variation explained by sociodemographic variables is highly spatialized: 63 % of the deviance in the full model is simultaneously explained by sociodemographic and spatial variables. The shared deviance between the socioeconomic and the spatial variables remains important for the broad scale model (i.e. 53 %) but diminishes dramatically for the medium and fine scale models (i.e. 7 % and 4 %, respectively).

• A small part of the car ownership rate variation is exclusively explained by the spatial structure: in the full model, this part is 4 %. This additional explanation by the spatial model is mainly due to the broad scale model: the independent contribution of the spatial variables is very small for the medium and fine scale models.

Discussion

The results presented in this paper underscore the importance of multi-scale spatial analysis as an explanatory tool applicable to transportation data. Since the data are spatially structured, these structures can be correlated with both explanatory factors and governance policies. Let us note that space per se is not considered as an explanation of transportation data variability; rather, spatial variables serve as proxies to quantify and dissect the spatial variation. A portion of this variation can then be attributed to some of the explanatory variables available for analysis, with the remainder being considered as spatial variation yet to be explained.

In this study, the car ownership rate has been shown to have a spatial structure at three distinct scales. At the broad scale, the spatial structure of car ownership rate variation is in line with the structure obtained when distinguishing between the two main cities (Nantes and Saint-Nazaire) versus other areas (Figs. 6 andA.1), which implies that living in one of the study area's two main cities is a relevant predictor of car ownership, as the rate is less in these cities. The broad-scale spatial structure of car ownership rate has been correlated with classical sociodemographic explanatory variables: negatively with population density and median income, and positively with share of couples with children. Population density is closely correlated with the availability of public transit as corresponding investments in denser areas are far greater. Part of the explanation of the negative correlation between median income and car ownership at broad scale model may lie in the fact that a part of the population with higher income lives in areas with better urban amenities which can offer a good substitute for car ownership [START_REF] Mulalic | Does improving public transport decrease car ownership? evidence from a residential sorting model for the copenhagen metropolitan area[END_REF]. This assumption needs to be explored further by social scientists. In future research, it would be interesting to investigate if this finding is specific to the studied area or valid in different areas. This analysis illustrates the additional value to classical sociodemographic modeling brought by taking into Mems in analyzing the car ownership.

At the medium scale, the spatial structure of car ownership rate variation is in line with the structure obtained when distinguishing between large city centers and the nearby surrounding areas (Figs. 7 andA.2). In the case of Nantes city, three areas can be distinguished. The first is the inner core with a lower rate. The second area is constituted of IRIS located between the inner core and the ring road, where median income and individual house share are higher. These areas tend to have higher ownership rates. The third area, located outside the ring road and characterized by lower median income and a higher dwelling share, tends to have lower ownership. At the fine scale, spatial structures are harder to interpret (Figs. 8 andA.3).

A major advantage of distance-based Moran's eigenvector analysis is its explicit identification of scale patterns, which provides critical information in land use and transportation planning [START_REF] Wang | Understanding spatial filtering for analysis of land use-transport data[END_REF]). Knowing these patterns (i.e. spatial structures) can be of major interest for transportation and urban planners and moreover allow decision-makers to act on targeted zones of the territory to influence the variable of interest. As shown by our analysis, issues are not the same throughout the territory considered. For example, since the car ownership rate is particularly high outside city centers, policymakers can increase the supply of public transit in these zones. Similarly, the analysis has identified a higher car ownership rate within the IRIS located in Nantes between the core and the ring road. A more targeted analysis of this area would serve to determine whether any levers could be used to reduce the rate. Another advantage of this analysis is its use as an explanatory tool to refine models for the dependent variable. When model residuals are not randomly distributed, exploring the spatial patterns of these residuals can in fact be very helpful in identifying missing explanatory factors. Supported by expert opinion, the resulting patterns will then guide the choice of omitted relevant explanatory variables [START_REF] Paez | Using spatial filters and exploratory data analysis to enhance regression models of spatial data[END_REF].

dbMEM vectors are obtained by performing a principal coordinate analysis of a truncated matrix of Euclidean distances between centroids of defined statistical areas. As our concern is transportation data analysis, a more sophisticated approach would use other distance measurements between statistical areas (instead of simple Euclidean distances) such as travel time. 4 In fact, travel time provides another relevant transportation-related measure of scale. In this paper, we have kept the classic Euclidean distance as a first approach and in order to facilitate the analysis. However, it will be the purpose of future works to analyze the data using others distances (e.g. travel time).

Conclusion

This study has contributed to the current transportation literature by introducing multi-scale spatial analysis techniques with space as an explicit predictor. Applying these techniques to study transportation variables (e.g. household car ownership) offers many important methodological and operational advantages. They provide transportation planners with a tool for identifying the spatial structures of response variables at various scales. These structures can be correlated with explanatory variables and used to target specific geographical areas in transportation planning policies.

In this paper, distance-based Moran's eigenvector maps have been implemented for car ownership spatial analysis. For this purpose, aggregated census data for small statistical areas in France's Loire-Atlantique Department were used. Significant spatial vectors explaining car ownership variable were selected by performing generalized linear model regression. These spatial vectors were then divided into sub-models representing broad scale, medium scale and fine scale. Finally, variation partitioning analysis was performed with respect to variables pertaining to both sociodemographic determinants and the distinct spatial scales.

In conclusion, distance-based Moran's eigenvectors maps is an appealing method that can be used to study transport data. In this paper, it has been applied to study spatial structure of a single response variable. An avenue for future research is to apply this methodology in different urban areas to check whether the same findings still For example, some counterintuitive results, such as the negative correlation between income and car ownership at broad scale, underline the interest of this approach, but this result needs to be verified in other urban areas. Furthermore, as this first analysis was based on Euclidean distance between statistical areas to explore spatial structures, it could be enriched by considering other distance measurements for transportation data analysis (e.g. travel time). Moreover, opportunities for continued investigation include applying this method to study multivariate response data. In this approach, spatial layout of multiple transport variables (e.g. single car, multiple cars and motorized two-wheelers ownership) can be explored simultaneously. Finally, another investigation opportunity is the study of both spatial and temporal layout of the response variable. In fact, dbMEM variables can represent effectively a spectral decomposition of both the spatial and temporal relationships.

  , Brind'Amour et al. (2005), Legendre et al. (2005), Laliberté et al. (2009), Brind'Amour et al. (2018), Gáspár et al. (2019), Wagner et al. (2017), Cilleros et al. (2017), Brice et al. (2016), Jin et al. (2020), Chen et al. (2020), Santos et al. (2020), Pollice et al. (2020) and Taddeo et al. (2021).

  Note: shades of gray represent the values in each eigenvector, from white (largest negative value) to black (largest positive value).

Figure 1 :

 1 Figure 1: Maps of the first six eigenvectors (extracted from Legendre & Legendre (2012)) produced after performing a dbMEM principal coordinate decomposition for a regular 12 x 8 grid

  Note: Abscissa, from left to right: sites 1 to 50. Ordinates: values along the dbMEM eigenfunctions.

Figure 2 :

 2 Figure 2: Graphs (extracted from Legendre & Legendre (2012)) of 10 of the 49 dbMEM eigenfunctions representing the spatial variation along a transect with 50 equally-spaced points

Figure 3 :

 3 Figure 3: Diagram describing the partitioning of the variation of a response variable Y by two sets of explanatory variables (X and W).

Figure 4 :

 4 Figure 4: Car ownership, by IRIS

Figure 5 :

 5 Figure 5: Spatial neighborhood

Figure 6 :

 6 Figure 6: Variation in car ownership computed with the broad scale sub-model

Figure 9 :

 9 Figure 9: Variation partitioning

Note:

  Each dbMEM vector is plotted with the component of the vector associated to a given statistical area (IRIS) represented by a square. Black and White squares represent values of different signs (positive or negative). Square sizes are proportional to the absolute value of the component.

Figure

  Figure A.2: MEM 6, 8, 16 and 17 are part of the medium scale sub-model. Spatial structure associated with these MEM matches the structure obtained distinguishing between main city centers and close surrounding areas.

Figure

  Figure A.3: MEM 61, 69 and 129 are part of the fine scale sub-model. Fine scale spatial structures are harder to interpret.

  Table 2 lists the descriptive statistics of this set of explanatory variables.

	Variable	Mean	Median	s.d.
	car ownership (%)	87.41	91.79	10.92
	couples with children (%)	28.41	29.21	11.17
	25-54 yrs (%)	38.96	39.93	6.01
	population density (inhabitants/km 2 ) 2362.89	591.80	3399.96
	median income (e)	21,681	21,512	3,118

Table 2 :

 2 Descriptive statistics(IRIS level)

  .

		Estimate	Std. Error t-value
	Intercept	2.168	0.029	72.676
	MEM1	-0.572	0.024 -23.575
	MEM4	-0.409	0.023 -17.077
	MEM6	0.131	0.028	4.606
	MEM8	-0.108	0.031	-3.406
	MEM9	0.077	0.026	2.897
	MEM13	-0.062	0.024	-2.615
	MEM14	-0.090	0.022	-3.959
	MEM15	-0.082	0.027	-3.035
	MEM16	0.074	0.027	2.678
	MEM17	0.087	0.031	2.797
	MEM20	0.069	0.022	3.027
	MEM34	-0.054	0.028	-1.902
	MEM36	-0.076	0.029	-2.577
	MEM37	-0.083	0.032	-2.597
	MEM39	0.094	0.031	2.964
	MEM61	-0.075	0.028	3.383
	MEM69	-0.075	0.025	-2.906
	MEM73	-0.062	0.029	-2.135
	MEM80	-0.050	0.024	-2.052
	MEM96	-0.057	0.027	-2.063
	MEM129	0.059	0.025	2.297
	MEM137	0.048	0.024	1.978
	MEM167	-0.060	0.027	-2.191
	N			499
	φ			0.033
	Moran I stat. (p-value<0.001)		0.090

Table 3 :

 3 GLM regression: significant eigenfunctions

https://cran.r-project.org/web/packages/adespatial/index.html

https://cran.r-project.org/web/packages/spmoran/index.html

In France's administrative divisions, the department is one of the three administrative jurisdictions of government below the national level, positioned between regions and municipalities.

We thank an anonymous reviewer of this paper for suggesting travel time as another relevant transportation-related measure of scale.

Acknowledgments

Spatial analyses were performed using the packages 'adespatial' and 'spdep' developed in R language. For the variation partitioning analysis, the R package 'ecospat' was run.

 ; 3-5; 6-10; 11-14; 15-17; 18-24; 25-29; 30-39; 40-54; 55-64; 65-79;