Frédéric Pichon
email: frederic.pichon@univ-artois.fr

Anne-Laure Jousselme
email: anne-laure.jousselme@cmre.nato.int

Nadia Ben Abdallah
email: nadia.benabdallah@cmre.nato.int

Several shades of conflict $

Keywords: Belief functions, Evidence theory, Consistency, Inconsistency, Conflict, Norm, Distance

. Section 4 is an extension of the results of [1] encompassing a parameterised family of conflict measures, itself presented in Section 3.

Introduction

Intelligent systems need to be able to cope with large amount of information that often displays different dimensions of imperfection. In particular, inconsistent evidence is a challenging problem that may arise when an information source is partially reliable and provides inaccurate or aberrant information. When inconsistent information needs to be merged, a particular attention should be paid to characterising, measuring and understanding the conflict. The conflict can be used as an indicator of a lack of reliability of some source and further used to discount the corresponding piece of information in the overall fusion process or to choose the most appropriate combination rule (see, e.g., [START_REF] Pichon | A consistency-specificity trade-off to select source behavior in information fusion[END_REF]). Besides its importance in deriving relevant overall belief, the conflict can be used to support decision making. For instance in maritime security, inconsistency may reveal maritime anomalies such as vessels deviating from normalcy (e.g., "off-route vessels", "too fast vessels") and those possibly spoofing the Automatic Identification System (AIS) signal to hide suspect behaviour [START_REF] Pallotta | Data-driven detection and context-based classification of maritime anomalies[END_REF][START_REF] Ray | Deais project: Detection of AIS spoofing and resulting risks[END_REF].

Investigating methods to characterise and measure conflict in information fusion has attracted particular attention in the setting of belief functions [START_REF] Dempster | A generalization of Bayesian inference[END_REF][START_REF] Shafer | A mathematical theory of evidence[END_REF]. To clarify the semantics of conflict measures for belief functions, axiomatic foundations have been recently provided in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF] and [START_REF] Martin | About conflict in the theory of belief functions[END_REF] and corresponding measures proposed: Martin's measure combines a degree of inclusion with a distance, whereas Destercke and Burger measure conflict between belief functions as the inconsistency yielded by their conjunctive combination. The uniqueness of these measures with respect to their set of axioms was not addressed though. Besides, although the two sets of axioms of [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF] and [START_REF] Martin | About conflict in the theory of belief functions[END_REF] highly overlap, they differ notably by the identity axiom present in [START_REF] Martin | About conflict in the theory of belief functions[END_REF] and not in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]. Whether a conflict measure between two identical belief functions should be null or not has been discussed in several work [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF][START_REF] Martin | About conflict in the theory of belief functions[END_REF][START_REF] Liu | Analyzing the degree of conflict among belief functions[END_REF][START_REF] Daniel | Conflicts within and between belief functions[END_REF][START_REF] Burger | Geometric views on conflicting mass functions: From distances to angles[END_REF], questioning the appropriateness of the classical conflict measure in belief function theory [START_REF] Shafer | A mathematical theory of evidence[END_REF] to quantify conflict in all situations, and led to alternative considerations of conflict. Liu [START_REF] Liu | Analyzing the degree of conflict among belief functions[END_REF] proposes to complement the classical conflict measure with a distance measure, Daniel [START_REF] Daniel | Conflicts within and between belief functions[END_REF][START_REF] Daniel | Conflict between belief functions: A new measure based on their non-conflicting parts[END_REF] distinguishes between internal, external and total conflict, Destercke and Burger [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF] consider knowledge about source dependence, and Burger [START_REF] Burger | Geometric views on conflicting mass functions: From distances to angles[END_REF] investigates the suitability of distance measures and other geometrical objects for conflict measurement.

In this paper, we follow the axiomatization proposed by Destercke and Burger [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF] for conflict measures, which encompasses the classical measure of conflict, and revisit and extend some of their results. Specifically, we tackle two research questions: [START_REF] Pichon | A norm-based view on conflict[END_REF] the uniqueness of the proposed measures, and (2) whether geometrical objects may be relevant for this kind of measures of conflict.

Our starting point are the two different definitions (one being stronger than the other one) of the notion of nonconflicting belief functions proposed in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]. We define a new parameterised family of conflict measures that captures gradual notions ("shades") of conflict. The proposed family satisfies Destercke and Burger's axiomatic approach, and interestingly, subsumes their definitions of nonconflict as extreme cases. In particular, we show that socalled strong nonconflict can be captured by a sound measure which is an alternative to the original contour-based measure proposed in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]. We also show that this family is compatible with a geometric view and shed some new light on the relation between conflict and distances.

The study in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF] is quite general and considers the whole spectrum from unknown dependence to known dependence between the sources providing the belief functions. We do not tackle these more general situations in this paper, and consider only the case where it can be safely assumed that the sources are independent, although we note that our results can readily be extended to the more general case of known dependence.

This paper is organised as follows. Necessary concepts of belief function theory, as well as Destercke and Burger's axiomatic approach to conflict measurement, are recalled in Section 2. In Section 3, a parameterised family of conflict measures is unveiled and its special cases and properties are discussed. A geometric perspective on the proposed measures is presented in Section 4, before concluding in Section 5.

Preliminaries

In this section, necessary concepts of belief function theory are first recalled. Then, the axiomatic approach to conflict measurement of Destercke and Burger [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF] is presented.

Belief function theory

The theory of belief functions is a framework for uncertainty modeling and reasoning. It was originally introduced by Dempster [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF][START_REF] Dempster | A generalization of Bayesian inference[END_REF] in the context of statistical inference, as a theory of imprecise probabilities. It was extended by Shafer [START_REF] Shafer | A mathematical theory of evidence[END_REF] and then by Smets and Kennes [START_REF] Smets | The transferable belief model[END_REF] to handle subjective uncertainty related to fixed quantities. In this latter interpretation of this framework, called Transferable Belief Model (TBM) and considered in this paper, the beliefs held by an agent about the actual value x taken by a variable defined on a finite domain X = {x 1 , . . . , x K } (called frame), are modeled by a so-called mass function defined as a mapping m : 2 X → [0, 1] verifying A⊆X m (A) = 1 with m(∅) ≥ 0. The mass m(A) represents the amount of belief allocated to the fact of knowing only that x ∈ A. The set of all mass functions on X is denoted by M.

Subsets A of X such that m(A) > 0 are called focal sets of m, and the set of focal sets of m is denoted by F. In order to simplify some expressions, the number |F| of focal sets of m is denoted by F. Several specific cases of mass functions are often distinguished. A mass function m is called:

• categorical if m(A) = 1 for some A ⊆ X , in which case it defines a classical set and will be denoted by m A in the following;

• vacuous if m(X ) = 1 and denoted by m X . It represents total ignorance;

• empty if m(∅) = 1 and denoted by m ∅ . It represents total inconsistency in the agent's beliefs about the set of values that are conceivable for x [START_REF] Smets | The nature of the unnormalized beliefs encountered in the transferable belief model[END_REF];

• normalised if m(∅) = 0.
Equivalent representations of a mass function m are the plausibility function and the belief function, defined respectively as, for all A ⊆ X ,

pl(A) = B∩A =∅ m(B), (1)
and

bel(A) = ∅ =B⊆A m(B). (2)
That is, pl(A) is the amount of belief consistent with x ∈ A, and bel(A) is the amount of belief implying x ∈ A. The plausibility function restricted to the singletons of X is the contour function π : X → [0, 1] such that π(x) = pl({x}), for all x ∈ X . Due to the one-to-one correspondence between functions m, bel and pl, any of these functions may be loosely referred to as "belief function" for simplicity -it should nonetheless always be clear from the context what is meant from a technical point of view. We note that in the imprecise probabilistic interpretation of belief function theory, the belief and plausibility measures bel and pl represent boundaries on an ill-known probability measure P on X and m is associated to the set P(m) of compatible probability measures defined by P(m) = {P |∀A ⊆ X , bel(A) < P (A)}.

The informative contents of two mass functions can be compared using the notion of specialisation [START_REF] Dubois | A set-theoretic view of belief functions: logical operations and approximations by fuzzy sets[END_REF]: a mass function m 1 defined on X is said to be a specialisation of another mass function m 2 defined on X , which is denoted by m 1 m 2 , if and only if there exists a non-negative square matrix

S = [S(A, B)], A, B ∈ 2 X , verifying A⊆X S(A, B) = 1, ∀B ⊆ X , S(A, B) > 0 ⇒ A ⊆ B, A, B ⊆ X , m 1 (A) = B⊆X S(A, B)m 2 (B), ∀A ⊆ X .
The term S(A, B) may be seen as the proportion of the mass m 2 (B) which "flows down" to A. The specialisation relation extends the relation of inclusion between classical sets. Let us also recall that we have [START_REF] Dubois | A set-theoretic view of belief functions: logical operations and approximations by fuzzy sets[END_REF]

m 1 m 2 ⇒ pl 1 (A) ≤ pl 2 (A), ∀A ⊆ X . (3)
Another useful notion is that of refined mass function. Recall that a refinement of a space X to a space Y is formally defined as a function ρ : 2 X → 2 Y such that the set {ρ({x})|x ∈ X } is a partition of Y, and ρ(A) = x∈A ρ({x}), ∀A ⊆ X . If ρ is a refinement function from X into Y, the refined mass function ρ(m i ), denoted by m iρ for short, of some mass function m i is defined such that for any A ∈ F i , with F i the set of focal sets of m i , we have m i (A) = m iρ (ρ(A)).

The TBM is appealing because it makes it possible to combine multiple pieces of information about a variable. One of the most classical combination rule of the theory is Dempster's unnormalised rule [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF], also known as conjunctive rule. Let m 1 and m 2 be two mass functions representing pieces of information about x. Their combination by the conjunctive rule, denoted by ∩ , results in the mass function m 1 ∩ 2 defined by, for all A ⊆ X ,

m 1 ∩ 2 (A) = B∩C=A m 1 (B)m 2 (C). ( 4
)
This combination is appropriate when m 1 and m 2 have been provided by two independent and reliable sources. Furthermore, the rule ∩ is commutative and associative, and admits the vacuous mass function m X as neutral element.

In the TBM, conditioning of a mass function m by some B ⊆ X is equivalent to conjunctive combination of m with the categorical mass function m B . The result is denoted by m[B], with m[B] = m ∩ m B . The conjunctive rule admits a simple expression using conditioning:

m 1 ∩ 2 (A) = B⊆X m 1 (B)m 2 [B](A), ∀A ⊆ X .
(5)

Axiomatic approach to conflict measurement

The axiomatic approach to conflict measurement of [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF] relies on the notion of consistency of a mass function, which is recalled first.

Consistency of a mass function

Although a totally inconsistent information state is uniquely represented as the empty mass function m ∅ [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF][START_REF] Smets | The nature of the unnormalized beliefs encountered in the transferable belief model[END_REF], total consistency of an information state represented by a mass function can be understood differently. Specifically, two different definitions of total consistency are considered in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]: Definition 1 (Logical consistency [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]). A mass function m is logically consistent iff A∈F A = ∅. Definition 2 (Probabilistic consistency [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]). A mass function m is probabilistically consistent iff m(∅) = 0 ( i.e., m is normalised).

Logical consistency of m corresponds to logical consistency between the focal sets of m, hence its name. Probabilistic consistency takes its name from the fact that m(∅) = 0 is equivalent to P(m) = ∅. We note that a logically consistent mass function is normalised and thus is also probabilistically consistent. Hence, logical consistency is a stronger form of consistency than probabilistic consistency.

Based on these notions of total inconsistency and total consistency, two properties that a measure of consistency φ of a mass function should obey have been defined in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]:

Property 1 (Bounded [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]). A measure of consistency should be bounded, i.e., possess minimal and maximal values.

Property 2 (Extreme consistent values [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]). A measure of consistency should reach its maximal value if and only if information is totally consistent (according to the considered definition), and its minimal value if and only if information is totally inconsistent.

Property 2 depends thus on the definition of total consistency considered. Remarking that A∈F A = ∅ ⇔ ∃x ∈ X s.t. π(x) = 1 [7, Lemma 1], two consistency measures φ π and φ m from M to [0, 1] are proposed in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]:

φ π (m) = max x∈X π(x), φ m (m) = 1 -m(∅).
The measure φ π (m) has been proposed originally in [START_REF] Daniel | Conflicts within and between belief functions[END_REF] to quantify the internal conflict of a belief function. As detailed in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF], measure φ π satisfies Property 1 and Property 2, in the case where total consistency is understood according to Definition 1. Measure φ m on the other hand satisfies Property 1 and Property 2, in the case where total consistency is understood according to Definition 2. Furthermore, we have φ m (m) ≥ φ π (m) for any m ∈ M [7, Lemma 2].

Measure φ π agrees with the TBM interpretation of belief functions since a mass function is totally logically consistent if and only if it considers that at least one value in X is totally plausible [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]. Furthermore, it is argued in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF] that φ m is in accordance with the imprecise probabilistic interpretation of belief functions since a mass function is totally probabilistically consistent if and only if its associated set P(m) is not empty. Yet, as reported in [7, Section VII.A], the correlation between φ m and φ π is high enough in some cases, to consider φ m as a good approximation of φ π .

Despite Destercke and Burger [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF] position about measure φ m , we remark that this measure has been justified by Smets [START_REF] Smets | The nature of the unnormalized beliefs encountered in the transferable belief model[END_REF] as a measure of consistency in the TBM interpretation of belief functions, using an argument based on the notion of belief updating. Smets concludes in [START_REF] Smets | The nature of the unnormalized beliefs encountered in the transferable belief model[END_REF] that φ m quantifies the amount of consistency present in the mass function representing the agent's beliefs about the set of values X that are conceivable for x by the agent. This can simply be seen by remarking that pl(X ) = 1 -m(∅) = φ m (m), that is φ m (m) is the amount of belief consistent with the proposition that the actual value x is in X (we also have bel(X ) = φ m (m)). In particular, we have m(∅) = 0 ⇔ pl(X ) = 1 so that we could also name the notion of probabilistic consistency as frame consistency (in contrast, φ π (m) is the maximum amount of belief consistent with x = x for some x ∈ X , and logical consistency corresponds thus to consistency with at least one value in X ). This view on φ m is totally in line with the so-called open-world assumption of the TBM, where m(∅) quantifies the belief that x does not lie in X . Overall, φ m and φ π seem thus both relevant in the TBM interpretation of belief functions (φ m being in addition relevant in the imprecise probabilistic interpretation).

Let us finally remark that besides φ π and φ m , another measure of consistency of a mass function has been proposed by Yager in [START_REF] Yager | On considerations of credibility of evidence[END_REF]:

φ Y (m) = A∩B =∅ m(A)m(B). (6)
This measure reaches its maximum, that is a mass function m is considered totally consistent according to Yager, if and only if the focal sets of m have a pairwise non-empty intersection. This alternative view on total consistency will be called in this paper pairwise consistency:

Definition 3 (Pairwise consistency). A mass function m is pairwise consis- tent iff ∀(A, B) ∈ F 2 , A ∩ B = ∅.
It is easy to check that measure φ Y satisfies Property 1 and Property 2, in the case where total consistency is understood according to Definition 3.

Conflict between mass functions

As rightfully remarked in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF], two mass functions can be considered as totally conflicting if none of their focal sets intersect. Formally: Definition 4 (Total conflict [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]). Let m 1 and m 2 be two mass functions with sets of focal sets F 1 and F 2 respectively. Let D i = ∪ A∈F i A denote the disjunction1 of all focal sets of m i . m 1 and m 2 are totally conflicting when

D 1 ∩ D 2 = ∅.
However, there does not seem to be a unique way to define totally nonconflicting mass functions. As a matter of fact, two2 definitions of nonconflicting mass functions are considered in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]: Definition 5 (Strong nonconflict [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]). Two mass functions m 1 and m 2 are said to be strongly nonconflicting if and only if

A∈{F 1 ∪F 2 } A = ∅.
Definition 6 (nonconflict [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]). m 1 and m 2 are said to be nonconflicting if and only if

∀(A, B) such that A ∈ F 1 , B ∈ F 2 , we have A ∩ B = ∅.
Definition 5 requires that all focal sets of m 1 and m 2 have a non-empty intersection, which is stronger than requiring each focal set of m 1 to have a non-empty intersection with each focal set of m 2 as required by Definition 6.

Based on these notions of totally conflicting and totally nonconflicting mass functions, five properties that a measure of conflict κ between two mass functions defined on X and provided by two independent sources, should satisfy are provided in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]. The five properties are the following (further details on the motivations of these requirements can be found in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]):

Property 3 (Extreme conflict values [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]). κ(m 1 , m 2 ) = 0 if and only if m 1 and m 2 are nonconflicting (according to the considered definition) and κ(m 1 , m 2 ) = 1 if and only if m 1 and m 2 are totally conflicting.

Property 4 (Symmetry [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]). κ(m 1 , m 2 ) = κ(m 2 , m 1 ).

Property 5 (Imprecision monotonicity [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]).

If m 1 m 1 , then κ(m 1 , m 2 ) ≥ κ(m 1 , m 2 ). Property 6 (Ignorance is bliss [7]). If m 2 is vacuous, then κ(m 2 , m 1 ) = 1 -φ(m 1 ). Property 7 (Insensitivity to refinement [7]). If ρ is a refinement function from X into Y, then κ(m 1 , m 2 ) = κ(m 1ρ , m 2ρ ).
As Property 2 for consistency measures, which depends on the definition of total consistency considered, Property 3 for conflict measures depends on the definition of nonconflict considered (i.e., either Definition 5 or Definition 6). Property 5 states that the conflict should not increase as the imprecision (defined in terms of specialisation) of a mass function increases. Property 6 states that a state of ignorance should not conflict with any other state of information represented by a mass function m 1 , whilst accounting for the fact that this state of information m 1 may be partially inconsistent itself -we note however that this property does not specify which kind of consistency (e.g., logical or probabilistic) should consistency measure φ satisfy. Property 7 states that the refinement of a mass function should not change its conflict value with any other mass function refined in the same way.

In [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF], the authors propose to evaluate the conflict between two mass functions as the inconsistency of their conjunctive combination, where inconsistency is the "inverse" of consistency (i.e., 1 -φ(•). More precisely, they propose two measures of conflict satisfying Properties 3-7, induced by consistency measures φ π and φ m :

κ π (m 1 , m 2 ) = 1 -φ π (m 1 ∩ 2 ) = 1 -max x∈X π 1 ∩ 2 (x), κ m (m 1 , m 2 ) = 1 -φ m (m 1 ∩ 2 ) = m 1 ∩ 2 (∅).
Measure κ π satisfies Properties 3-7, when nonconflict in Prop. 3 is understood in terms of Definition 5, whereas κ m satisfies these properties when nonconflict is understood in terms of Definition 6. We note that with κ π , Prop. 6 is satisfied for φ = φ π , whereas with κ m this property is satisfied for φ = φ m . Let us finally stress that κ m (m 1 , m 2 ) is nothing but the classical measure of conflict m 1 ∩ 2 (∅) in the TBM.

N -consistency and induced conflict

The previous section has reviewed the axiomatic approach to conflict measurement proposed in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]. It was recalled that the measures κ π and κ m satisfy this approach. In particular, measure κ π , which relies on the measure φ π of logical consistency, is suitable as a measure for strong nonconflict. In this section, it is shown that there exists an alternative measure of logical consistency (i.e., a measure φ satisfying Property 2 for Definition 1) and that its induced conflict measure (defined as the inconsistency of the conjunctive combination) is an alternative measure for strong nonconflict (i.e., a measure κ satisfying Property 3 for Definition 5). More generally, a parameterised family of consistency measures capturing different forms of consistency including logical, pairwise and probabilistic consistency is introduced and is shown to induce a family of conflict measures satisfying the axiomatic approach of [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF] for a family of definitions of the notion of nonconflicting mass functions subsuming strong nonconflict (Definition 5) and nonconflict (Definition 6).

N -consistency

Let us start by introducing the following definition:

Definition 7 (N -consistency). A mass function m is said to be consistent of order N ( N -consistent for short), with 1 ≤ N ≤ F, iff its focal sets are N -wise consistent, i.e., ∀F ⊆ F s.t. |F | = N , we have

A∈F A = ∅.
In addition, let φ N denote the measure from M to [0, 1] such that, for 1 ≤ N ≤ F and all m ∈ M,

φ N (m) := 1 -m N (∅), (7)
where m N denotes the mass function resulting from the combination of m by itself N times, i.e.

m N := m N -1 ∩ m,
with m 0 := m X . Hence, we have m 1 = m, m 2 = m ∩ m and more generally m N = ∩ N 1 m. Note that m N (∅), i.e., the mass associated to the empty set after N combinations of m by itself, is called auto-conflict of order N of mass function m in [START_REF] Osswald | Understanding the large family of Dempster-Shafer theory's fusion operators -a decision-based measure[END_REF].

We will show in the following that the family φ N measures gradual notions ("shades") of consistency of m as N varies and in particular encompasses the three forms of consistency already defined in the literature and recalled in Section 2.2.1.

1-consistency

Let us first remark that probabilistic consistency (Definition 2) of a mass function m is nothing but 1-consistency, since we have

m(∅) = 0 ⇔ A = ∅, ∀A ∈ F.
In addition, we have φ m (m) = φ 1 (m), for all m ∈ M, since φ 1 (m) = 1 -m(∅), and thus φ 1 (m) is a measure of probabilistic consistency (or 1consistency), in the sense that it satisfies Properties 1 and 2 in the case where total consistency is understood according to Definition 2.

Let

κ 1 (m 1 , m 2 ) := 1-φ 1 (m 1 ∩ 2 ), for all m 1 , m 2 ∈ M. We have κ 1 (m 1 , m 2 ) = κ m (m 1 , m 2 )
, for all m 1 , m 2 ∈ M, and κ 1 is thus a measure for nonconflict (Definition 6) in the sense that it verifies Properties 3-7, when nonconflict in Prop. 3 is understood in terms of Definition 6,.

It will be useful to remark that the notion of nonconflicting mass functions (Definition 6) can be equivalently presented as follows. Let m 1 and m 2 be any two mass functions and let

F 12 := {A ∩ B|A ∈ F 1 , B ∈ F 2 }.
It is clear that: m 1 and m 2 are nonconflicting (Definition 6) ⇔ A = ∅, ∀A ∈ F 12 . In other words, nonconflict is equivalent to each set in F 12 being non-empty, i.e., to the sets in F 12 being "1-wise" consistent. Let us note that

F 12 = F 1 ∩ 2 , with F 1 ∩ 2 the set of focal sets of m 1 ∩ 2 .

F-consistency

Lemma 1 below shows that the notion of logical consistency of a mass function (Definition 1) is a particular case of that of N -consistency, when N is the number of focal sets of m.

Lemma 1. m is logically consistent if and only if m is F-consistent. Proof. m is F-consistent ⇔ ∀F ⊆ F s.t. |F | = F, we have A∈F A = ∅. There is only one set F s.t. |F | = F, which is F = F. Hence m is F-consistent ⇔ A∈F A = ∅.
Moreover, the following results hold for any mass function m ∈ M:

Lemma 2. m is F-consistent if and only if m F (∅) = 0. Proof. Follows from m F (∅) = ∩ F i=1 A i =∅ F i=1 m(A i ). Lemma 3. m is totally inconsistent if and only if m F (∅) = 1.
Proof. ⇒: m(∅) = 1, i.e., m is totally inconsistent, implies clearly m F (∅) = 1.

⇐: assume this is not true, i.e., m F (∅) = 1 and m(∅) = 1. We reach a contradiction since m(∅) = 1 implies that ∃A ⊆ X , A = ∅, s.t. m(A) > 0, in which case we have m F (A) ≥ (m(A)) F > 0, and thus m F (∅) = 1.

Lemmas 1, 2 and 3 suggest to use φ F (m) = 1 -m F (∅) as an alternative measure of logical consistency of a mass function m. Indeed, these results show that similarly to φ π , φ F verifies Property 1 and Property 2, in the case where total consistency is understood according to Definition 1 (logical consistency). This measure appears thus as justified as φ π to evaluate logical consistency of a mass function.

Remark 1. We note that the measures φ π and φ F are not equal. For instance, denoting by

m : (A 1 , m(A 1 ); . . . ; A F , m(A F ))
a mass function m with F focal elements A i with associated masses m(A i ),

then if m is defined on X = {d 1 , d 2 , d 3 } by m : ({d 1 , d 2 }, 0.8; {d 3 }, 0.2) , we have φ π (m) = max x∈X π(x) =0.8 = φ F (m) = 1 -m 2 (∅) =0.68 (since F = 2)
. Further details about the relationships between these measures will be given in Section 3.6.

The measure κ π (m 1 , m 2 ) associated to the notion of strong nonconflict (Definition 5) is derived from φ π (m 1 ∩ 2 ), which is a measure of logical consistency of m 1 ∩ 2 . Let F 12 denote the cardinality of F 12 , i.e.,

F 12 = |F 12 |. Then φ F 12 (m 1 ∩ 2
) is also a measure of logical consistency of m 1 ∩ 2 . This prompts us to consider the following conflict measure from M × M to [0, 1]:

κ F 12 (m 1 , m 2 ) := 1 -φ F 12 (m 1 ∩ 2 ). ( 8
)
Proposition 1. Measure κ F 12 satisfies Properties 3-7, when nonconflict in Property 3 is understood in terms of Definition 5 (strong nonconflict).

Proof. We show each property in turn:

• Property 3 (Extreme conflict values): κ F 12 (m 1 , m 2 ) = 0 ⇔ φ F 12 (m 1 ∩ 2 ) = 1 ⇔ m F 12 1 ∩ 2 (∅) = 0, which is equivalent using Lemma 2 to m 1 ∩ 2 is F 12 - consistent, which in turn is equivalent using Lemma 1 to m 1 ∩ 2 is logi- cally consistent, i.e., ∩ A∈F 12 A = ∅ or equivalently ∃x ∈ X s.t. x ∈ A, ∀A ∈ F 12 .
From the definition of ∩ , we have ∃x ∈ X s.t. x ∈ A, ∀A ∈ F 12 iff m 1 and m 2 are strongly nonconflicting.

κ F 12 (m 1 , m 2 ) = 1 ⇔ φ F 12 (m 1 ∩ 2 ) = 0 ⇔ m F 12 1 ∩ 2 (∅) = 1, which using Lemma 3 is equivalent to m 1 ∩ 2 is totally inconsistent, i.e., m 1 ∩ 2 (∅) = 1.
From the definition of ∩ , m 1 ∩ 2 (∅) = 1 iff m 1 and m 2 are totally conflicting.

• Property 4 (Symmetry): We have

κ F 12 (m 1 , m 2 ) = m F 12 1 ∩ 2 (∅) = ( ∩ F 12 i=1 m 1 ∩ 2 )(∅) = ( ∩ F 12 i=1 (m 1 ∩ m 2 ))(∅) = ( ∩ F 12 i=1 (m 2 ∩ m 1 ))(∅) = ( ∩ F 12 i=1 m 2 ∩ 1 )(∅) = m F 12 2 ∩ 1 (∅) = κ F 12 (m 2 , m 1 ).
• Property 5 (Imprecision monotonicity): ∩ is monotonic with respect to [START_REF] Denoeux | Inner and outer approximation of belief structures using a hierarchical clustering approach[END_REF]Proposition 2] and thus

m 1 m 1 ⇒ m 1 ∩ m 2 m 1 ∩ m 2 , ∀m 2 ∈ M. Using [2, Lemma 3], we obtain ∩ F 12 i=1 (m 1 ∩ m 2 ) ∩ F 12 i=1 (m 1 ∩ m 2 ). That is m F 12 1 ∩ 2 m F 12 1 ∩ 2 , which implies using (3) that pl F 12 1 ∩ 2 (X ) ≤ pl F 12 1 ∩ 2 (X ) ⇔ 1 -m F 12 1 ∩ 2 (∅) ≤ 1 -m F 12 1 ∩ 2 (∅) ⇔ κ F 12 (m 1 , m 2 ) ≥ κ F 12 (m 1 , m 2 ). • Property 6 (Ignorance is bliss): m 2 (X ) = 1 means that m 2 = m X . It
is thus the neutral element of ∩ and we have F 12 = F 1 , from which we obtain

κ F 12 (m 1 , m 2 ) = m F 12 1 ∩ 2 (∅) = ( ∩ F 12 i=1 m 1 ∩ 2 )(∅) = ( ∩ F 12 i=1 m 1 )(∅) = m F 12 1 (∅) = m F 1 1 (∅) = 1 -φ F 1 (m 1 ).
• Property 7 (Insensitivity to refinement): We have

κ F 12 (m 1 , m 2 ) = m F 12 1 ∩ 2 (∅) = ∩ F 12 i=1 (A i ∩B i )=∅ F 12 i=1 m 1 (A i )m 2 (B i ),
and

κ F 12 (m 1ρ , m 2ρ ) = m F 12 1ρ ∩ 2ρ (∅) = ∩ F 12 i=1 (ρ(A i )∩ρ(B i ))=∅ F 12 i=1 m 1ρ (ρ(A i ))m 2ρ (ρ(B i )). ∀(A 1 , B 1 , . . . , A F 12 , B F 12 ) ∈ × F 12 i=1 F 1 × F 2 ,
we have:

-either ∩ F 12 i=1 (ρ(A i )∩ρ(B i )) = ∅, in which case since {ρ({x})|x ∈ X } is a partition of Y, ∃x ∈ X s.t. x ∈ A i and x ∈ B i , i = 1, . . . , F 12 , i.e., ∩ F 12 i=1 (A i ∩ B i ) = ∅; -or ∩ F 12 i=1 (ρ(A i ) ∩ ρ(B i )) = ∅, in which case ∃x ∈ X s.t. x ∈ A i and x ∈ B i , i = 1, . . . , F 12 , i.e., ∩ F 12 i=1 (A i ∩ B i ) = ∅. Hence, ∀(A 1 , B 1 , . . . , A F 12 , B F 12 ), when the mass F 12 i=1 m 1ρ (ρ(A i ))m 2ρ (ρ(B i )) is allocated to ∅, so does the mass F 12 i=1 m 1 (A i )m 2 (B i )
, and when the mass

F 12 i=1 m 1ρ (ρ(A i ))m 2ρ (ρ(B i ))
is not allocated to ∅, so does the mass

F 12 i=1 m 1 (A i )m 2 (B i
). Property 7 is then obtained since

F 12 i=1 m 1ρ (ρ(A i ))m 2ρ (ρ(B i )) = F 12 i=1 m 1 (A i )m 2 (B i ), ∀(A 1 , B 1 , . . . , A F 12 , B F 12 ).
In the proof of Property 6 of Proposition 1, we remark that if m 2 is vacuous then κ F 12 (m 1 , m 2 ) = 1 -φ F 1 (m 1 ), i.e., the consistency of m 1 is evaluated using measure φ F 1 , which is a measure of the logical consistency of m 1 . In other words, we have a similar behaviour to that of κ π , which satisfies Prop.6 for φ = φ π that is another measure of the logical consistency of m 1 .

According to Proposition 1, measure κ F 12 (m 1 , m 2 ), which is nothing but the auto-conflict of order F 12 of the mass function m 1 ∩ 2 , constitutes thus a sound alternative to measure κ π (m 1 , m 2 ) for evaluating the (strong) conflict between two mass functions m 1 and m 2 .

Let us finally note that we have the following equivalence: m 1 and m 2 are strongly nonconflicting (Definition 5) ⇔ A∈F 12 A = ∅. In other words, strong nonconflict is equivalent to the intersection of all sets in F 12 being non-empty, i.e., to the sets in F 12 being F 12 -wise consistent.

2-consistency

It is clear that pairwise consistency (Definition 3) of a mass function m is equivalent to m being 2-consistent.

Furthermore, the measure φ Y of consistency proposed by [START_REF] Yager | On considerations of credibility of evidence[END_REF] (see Eq. ( 6)) is obtained with N = 2 in ( 7):

φ Y (m) = φ 2 (m) = 1 -m 2 (∅), ∀m ∈ M.
Hence, φ 2 is a measure of pairwise consistency of m, since it satisfies Properties 1 and 2 in the case where total consistency is understood according to Definition 3.

We established that probabilistic consistency of a mass function m is equivalent to m being 1-consistent and that logical consistency of m is equivalent to m being F-consistent. Hence, pairwise consistency of m can be situated in between the two extremes that are probabilistic consistency and logical consistency:

• probabilistic consistency is the weakest form -it requires focal sets of m to be "onewise" consistent, i.e., each focal set is non-empty;

• then comes pairwise consistency -focals sets need to be pairwise consistent, i.e., the intersection of any two focal sets must be non-empty;

• and logical consistency is the strongest form -focal sets must be Fconsistent, i.e., the intersection of all focal sets must be non-empty.

Similarly, we have seen that nonconflict (Definition 6) is equivalent to each set in F 12 being non-empty, whereas strong nonconflict is equivalent to the intersection of all sets in F 12 being non-empty. Considering the above discussion about the three possible definitions of total consistency, nonconflict and strong nonconflict appear thus to be two extreme forms of nonconflict, and Yager's definition of consistency suggests then an alternative definition of nonconflict: Definition 8. m 1 and m 2 are said to be nonconflicting of order 2 (2-nonconflicting for short) if and only if ∀A ∈ F 12 , B ∈ F 12 , we have A ∩ B = ∅.

Let us denote by κ 2 the measure of conflict induced by φ 2 , i.e., the measure

κ 2 (m 1 , m 2 ) : M × M → [0, 1] defined by κ 2 (m 1 , m 2 ) := 1 -φ 2 (m 1 ∩ 2 ).
Proposition 2. Measure κ 2 satisfies Properties 3-7, when nonconflict in Property 3 is understood in terms of Definition 8 (2-nonconflict).

Proof. We only show Property 3 (the other properties can be shown using a similar proof to that of Proposition 1 -for Property 6, we obtain

κ 2 (m 1 , m 2 ) = 1 -φ 2 (m 2 )): κ 2 (m 1 , m 2 ) = 0 ⇔ φ 2 (m 1 ∩ 2 ) = 1 ⇔ m 2 1 ∩ 2 (∅) = 0 ⇔ m 1 ∩ 2 is 2- consistent ⇔ ∀A ∈ F 12 , B ∈ F 12 , we have A ∩ B = ∅. κ 2 (m 1 , m 2 ) = 1 ⇔ m 2 1 ∩ 2 (∅) = 1
, which, using a similar proof to that of Lemma 3, is equivalent to m 1 ∩ 2 is totally inconsistent, i.e., m 1 ∩ 2 (∅) = 1. From the definition of ∩ , m 1 ∩ 2 (∅) = 1 iff m 1 and m 2 are totally conflicting.

According to Proposition 2, κ 2 (m 1 , m 2 ) is thus a measure for 2-nonconflict.

A family of conflict measures

The three definitions of nonconflict encountered so far suggest the following generalisation, obtained by applying the notion of N -consistency to the sets in F 12 : Definition 9. m 1 and m 2 are said to be nonconflicting of order N (Nnonconflicting for short), with 1 ≤ N ≤ F 12 , if and only if ∀A i ∈ F 12 , i = 1, . . . , N , we have N i=1 A i = ∅.

N -nonconflict subsumes obviously 2-nonconflict (Definition 8), which is recovered for N = 2, but also nonconflict (Definition 6, recovered for N=1) and strong nonconflict (Definition 5, recovered for N = F 12 ).

φ(m) κ(m 1 , m 2 ) = 1 -φ(m 1 ∩ 2 ) Total inconsistency Total conflict m(∅) = 1 D 1 ∩ D 2 = ∅ Total consistency Total nonconflict Probabilistic consistency ∀A ∈ F, A = ∅ φ 1 = φ m [7] 1-consistency Nonconflict ∀A ∈ F 12 , A = ∅ κ 1 = κ m [7] 1-nonconflict Pairwise consistency ∀(A, B) ∈ F 2 , A ∩ B = ∅ φ 2 [17] 2-consistency Pairwise nonconflict ∀(A, B) ∈ F 2 12 , A ∩ B = ∅ κ 2 2-nonconflict Logical consistency A∈F A = ∅ φ F F-consistency Strong nonconflict A∈F12 A = ∅ κ F 12 F 12 -nonconflict φ π [7] κ π [7]
Table 1: Notions of consistency and associated nonconflict.

Let κ N (m 1 , m 2 ) : M × M → [0, 1] be the measure defined, for 1 ≤ N ≤ F 12 , by κ N (m 1 , m 2 ) := 1 -φ N (m 1 ∩ 2 ). (9)
Proposition 3. Measure κ N satisfies Properties 3-7, when nonconflict in Property 3 is understood in terms of Definition 9 (N -nonconflict).

Proof. The proof is similar to that of Proposition 2.

Proposition 3 is a generalisation of Propositions 1 and 2. The parameterised family of conflict measures κ N includes obviously measures κ F 12 and κ 2 , but also the classical measure of conflict in the TBM, i.e., κ m , which is equivalent to κ 1 .

Remark 2. Let m 1 = m A and m 2 = m B , for some A, B ⊆ X . In this case, we have F 12 = 1 and thus the family of conflict measures κ N reduces to the measure κ 1 . Besides, if A and B are consistent, i.e., A ∩ B = ∅, then κ 1 (m 1 , m 2 ) = 0, and if A and B are inconsistent, i.e., A ∩ B = ∅, then κ 1 (m 1 , m 2 ) = 1. In other words, if m 1 and m 2 are categorical, then the family of conflict measures κ N reduces to classical inconsistency assessment between sets.

Table 1 summarises the notions of consistency and associated conflict together with the associated definitions and measures. The measures defined so far in the literature are mentioned with the proper reference, while the ones proposed in this paper are not assigned any specific reference. Only three values of N are considered for consistency (1, 2 and F) and for conflict (1, 2 and F 12 ) in Table 1.

Lemma 4. ∀m 1 ∈ M, ∀m 2 ∈ M, we have κ N -1 (m 1 , m 2 ) ≤ κ N (m 1 , m 2 ).
Proof. Follows from the fact that auto-conflict verifies m N -1 (∅) ≤ m N (∅) [START_REF] Osswald | Understanding the large family of Dempster-Shafer theory's fusion operators -a decision-based measure[END_REF].

Lemma 4 shows that as N increases, the stronger the conflict measure becomes. In particular, κ F 12 is the strongest conflict measure, while κ 1 is the weakest. An equivalent relationship can straightforwardly be derived between consistency measures φ N .

The following Section 3.6 further studies the relative behaviours of the measures considered so far.

Correlation analysis

In this section, we conduct some correlation analysis of the different measures, the new φ N family for N = 1, . . . , F (including φ m = φ 1 and φ Y = φ 2 ), the existing logical consistency measure φ π and their counter-part conflict measures, to illustrate and better grasp their relative behaviours. In particular, the experiments show that the measures capture different shades of consistency (and conflict). We use the Spearman coefficient which is a rank correlation coefficient quantifying how much two measures are monotonically correlated. It reaches the value 1 if the two measures are linked through some increasing monotonic function.

Conflict measures κ N vs κ π

We first study how the family of conflict measures κ N behaves relatively to the existing (strong) conflict measure κ π , using a similar experiment to that in [7, Section VII.A].

For a given size of the frame of discernment X , we have drawn randomly (following Algorithm 3 of [START_REF] Burger | How to randomly generate mass functions[END_REF]) 5000 couples of normalised mass functions (m 1 , m 2 ) having 2 |X | -1 focal sets . Table 2 shows the Spearman correlation between κ π and κ N for different values of N and of |X |. The first line of Table 2 corresponds to the first line of Table II of [7, Section VII.A] with very similar values, while the other lines differ in the sense that our N does not correspond to the number of sources combined as in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]. We can observe that for |X | = 2 and |X | = 3, for which we have F 12 = 4 and F 12 = 8 respectively, the correlation between κ F 12 and κ π is really high (0.9895 and 0.9941, respectively).

In the experiment of [7, Section VII.A], it was observed that the correlation between κ 1 and κ π is relatively high, and that this correlation increases as the cardinality of the frame decreases and as the number of sources to be combined increases. In our experiment, a similar observation can be made: the correlation between κ N and κ π increases as |X | decreases and as N increases, for the values of |X | and N considered here. This latter result can be better observed in Figure 1, which displays the scatter plots for the specific case of |X | = 3, whose corresponding correlation coefficients can be found in column "|X | = 3" of Table 2.

Our experiment shows that while κ 1 and κ π are quite correlated as both are measures of conflict, the family of measures κ N captures different shades of conflict and the correlation with κ π increases with N . However, it shows that the measures κ F 12 and κ π for strong nonconflict do not have a maximal correlation and hence are different (a fact already known thanks to Remark 1).

Consistency measures φ N

Here we study the correlation between φ N and φ π on the one hand and within the family φ N as N varies on the other hand, by analysing the impact of the number of focal elements F as well as the size of the frame of discernment |X |. We thus prefer Algorithm 7 to Algorithm 3, both of [START_REF] Burger | How to randomly generate mass functions[END_REF], which allows to control the number of focal elements of the mass function generated.

As one can see from Figure 2, the correlation between φ F and φ π , which are two measures of logical consistency, is high (> 0.93) whatever the size of the frame of discernment and the number of focal elements considered here. Furthermore, if we consider a particular value of F, for instance F = 5, it appears that φ π is more correlated to φ F than to any φ N , for all N < F, as shown in Figure 3. We conclude this correlation analysis section with the behaviour of the φ N measures between themselves. Table 3 reports observe that the correlation between successive pairs of measures (φ N , φ N +1 ) increases as N increases. Additionally, we observe that φ N is less correlated to φ N +2 than to φ N +1 . The scatter plots of Figure 4 illustrate the correlation coefficients corresponding to the second diagonal of Table 3. For a given m, the measures φ N are less correlated for low values of N .

We illustrated that the family of consistency measures φ N allows to capture different shades of consistency, and thus that the derived family of conflict measures κ N allows to capture different shades of conflict between two mass functions m 1 and m 2 . In practice, the family of measures κ N offers alternatives to evaluate the validity of the result of the combination by the conjunctive rule ∩ , as it will be illustrated in Section 3.7.

Toy example

We illustrate some of the consistency and conflict measures, as well as some of their relationships, described previously, on a problem of vessel destination estimation and associated anomaly detection, in which we would like to estimate a vessel's destination while detecting any inconsistency given different sources of information.

Let X = {d 1 , d 2 , d 3 , d 4 } = {Savona, Genova, La Spezia, Livorno} denote the set of possible destinations for the vessel.

We consider a source S 1 , as an algorithm analysing the kinematic features of the vessel, compared to some pre-extracted patterns-of-life as maritime routes [START_REF] Pallotta | Vessel pattern knowledge discovery from AIS data -A framework for anomaly detection and route prediction[END_REF]. Once computed, the routes provide contextual information describing some normalcy against which the current positions of vessels can be compared. In particular, assigning a vessel to a route provides information about its final destination [START_REF] Pallotta | Data-driven detection and context-based classification of maritime anomalies[END_REF]. However, because the routes share some portions of the sea, some indeterminacy may occur and a subset of possible destinations may rather be deduced. Let m 1 be the mass function that encodes the information provided by the source S 1 :

m 1 : ({d 1 , d 2 , d 3 }, 0.6; {d 1 , d 2 , d 4 }, 0.3; {d 3 , d 4 }, 0.1).
We have: The mass function m 1 is 1-consistent (probabilistically consistent) and 2consistent (pairwise consistent), but not 3-consistent (i.e., not logical consistent since F 1 = 3). Indeed:

m 2 1 : ({d 1 ,
φ 1 (m 1 ) = φ 2 (m 1 ) = 1, φ 3 (m 1 ) = 1 -m 3 1 (∅) = 0.892. (10)
Consider now a second source S 2 which provides a piece of evidence encoded by the following mass function, corresponding to the subjective assessment of a human operator, knowing the maritime traffic of the area and excluding Genova (d 2 ) as a possible destination for that vessel, but considering as more probable the destination of Savona (d 1 ) or Livorno (d 4 ) than that of La Spezia (d 3 ): The mass functions provided by the sources S 1 and S 2 are 1-nonconflicting since κ 1 (m 1 , m 2 ) = 0. However, they are not nonconflicting of order 2 (and consequently higher orders) since some focal sets of m 1 ∩ 2 are pairwise disjoint (e.g, {d 3 , d 4 } ∩ {d 1 } = ∅):

               κ 1 (m 1 , m 2 ) = 1 -φ 1 (m 1 ∩ 2 ) = m 1 1 ∩ 2 (∅) = m 1 ∩ 2 (∅) = 0, κ 2 (m 1 , m 2 ) = 1 -φ 2 (m 1 ∩ 2 ) = m 2 1 ∩ 2 (∅) = 0.0612, κ 3 (m 1 , m 2 ) = 1 -φ 3 (m 1 ∩ 2 ) = m 3 1 ∩ 2 (∅) = 0.1908, κ 4 (m 1 , m 2 ) = 1 -φ 4 (m 1 ∩ 2 ) = m 4 1 ∩ 2 (∅) = 0.2999, κ 5 (m 1 , m 2 ) = κ F 12 (m 1 , m 2 ) = 1 -φ F 12 (m 1 ∩ 2 ) = m F 12 1 ∩ 2 (∅) = 0.3865. ( 11
)
The function

n p w (v) := ( i w i |v i | p ) 1/p , (12)
with p ≥ 1 and where, for i = 1, . . . , N , finite w i > 0, is an example of a norm; if we allow some w i to equal zero, then n p w is a pseudo-norm [START_REF] Yager | Norms induced from OWA operators[END_REF]. The weights w i actually distort the space of reference by increasing or reducing the importance of some dimensions. The (pseudo-)norm n 1 w will be more simply denoted by n w and we have n w (v) = i w i |v i |. Note also that n ∞ w (v) = max v, ∀w such that i w i = 1 [START_REF] Bullen | Handbook of means and their inequalities[END_REF]. As a consequence n ∞ will denote any norm n ∞ w with w such that i w i = 1. A metric or distance function (called simply distance) is a function d : R N × R N → [0, ∞) that satisfies the following properties: (d.1) Definiteness:

d(v 1 , v 2 ) = 0 ⇔ v 1 = v 2 , (d.2) Symmetry: d(v 1 , v 2 ) = d(v 2 , v 1 ), ∀v 1 , v 2 ∈ R N , (d.3) Triangle inequality: d(v 1 , v 2 ) ≤ d(v 1 , v 3 )+d(v 2 , v 3 ), ∀v 1 , v 2 , v 3 ∈ R N .
A pseudo-distance corresponds to changing condition (d.1) in the definition of the distance to

d(v 1 , v 2 ) = 0 ⇐ v 1 = v 2 [25]. If n is a norm (resp. pseudo- norm) then d n (v 1 , v 2 ) := n(v 1 -v 2
) is a distance (resp. pseudo-distance) and is called the induced distance (resp. pseudo-distance) from n. The distance (resp. pseudo-distance) induced by the norm (resp. pseudo-norm) n p w will be denoted by d p w ; d 1 w will be more simply denoted by d w and the distance induced by n ∞ will be denoted by d ∞ .

Consistency as distance to total inconsistency

Let E X denote the Cartesian space R 2 K spanned by the set of vectors {e A , A ⊆ X }. Any vector v of E X can then be written as v = A⊆X v A e A , with v A ∈ R the coordinate of v along dimension e A . A mass function m may then be represented as the vector m of E X such that v A = m(A). Similarly, a plausibility function pl may be represented by the vector pl = A⊆X pl(A)e A , with its plausibility values pl(A) as coordinates of pl. The vector associated to m A is denoted by m A . Special cases are the empty mass function m ∅ and the vacuous mass function m X .

Let k : 2 X → [0, 1] be the function defined by k(A) := 1 -pl(A), for all A ⊆ X . Function k may be represented by the vector k = A⊆X k(A)e A of E X . Lemma 5 below links function k with other well-known quantities.

Lemma 5. For all A ⊆ X , we have

k(A) = κ 1 (m, m A ) (13) = m[A](∅). (14)
Proof. For all A ⊆ X , we have

1 -pl(A) = 1 - B∩A =∅ m(B) = B∩A=∅ m(B) (15)
= (m A ∩ m)(∅).

Besides, κ 1 (m, m A ) = (m A ∩ m)(∅) and m[A](∅) = (m A ∩ m)(∅) by definition.
From [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF], k(A) is the amount of conflict, according to κ 1 , between m and x ∈ A. From [START_REF] Smets | The nature of the unnormalized beliefs encountered in the transferable belief model[END_REF], k(A) can also be interpreted as the amount of belief inconsistent with x ∈ A. Hereafter, we will refer to k as the inconsistency function associated to m. k is in one-to-one correspondence with m.

We are ready to show the first result of this section:

Proposition 4. For all m ∈ M and 1 ≤ N ≤ F, we have

φ N (m) = n m N -1 (pl), (16) m
N (∅) = n m N -1 (k). (17)
Proof. We have m N = m N -1 ∩ m, which using (5) yields

m N (∅) = A⊆X m N -1 (A)m[A](∅) = A⊆X m N -1 (A)k(A) = n m N -1 (k),
and

φ N (m) = 1 -m N (∅) = 1 - A⊆X m N -1 (A)k(A) = 1 - A⊆X m N -1 (A)(1 -pl(A)) = A⊆X m N -1 (A)pl(A) = n m N -1 (pl).
Proposition 4 shows that the N -consistency of a mass function amounts to the n m N -1 pseudo-norm of the vector pl and its N -inconsistency amounts to the same pseudo-norm but of the vector k. Note that n m N -1 is in general a pseudo-norm, and not a norm, since it is possible that m N -1 (A) = 0 for some A ⊆ X .

Corollary 1. From k = 1 -pl and φ N (m) = 1 -m N (∅), we also have the following equalities:

φ N (m) = 1 -n m N -1 (k) = n m N -1 (1 -k),
and

m N (∅) = 1 -n m N -1 (pl) = n m N -1 (1 -pl). (18)
Corollary 1 shows that it is equivalent to compute the pseudo-norm of the inverse of the inconsistency function (resp. plausibility function), and to compute the inverse of the pseudo-norm of the inconsistency function (resp. plausibility function). Briefly, the pseudo-norm and inverse commute.

Proposition 4 leads to the main result of this section, which expresses the consistency measures φ N in terms of pseudo-distances to the state of total inconsistency: Proposition 5. For all m ∈ M and 1 ≤ N ≤ F, we have

φ N (m) = d m N -1 (pl, pl ∅ ). (19)
Proof. Eq. ( 19) follows from [START_REF] Dubois | A set-theoretic view of belief functions: logical operations and approximations by fuzzy sets[END_REF] and pl ∅ = 0.

In short, Proposition [START_REF] Dempster | A generalization of Bayesian inference[END_REF] shows that the N -consistency of a mass function is nothing but its pseudo-distance, induced from n m N -1 , to the totally inconsistent knowledge state. Accordingly, the farther a mass function from total inconsistency, the more consistent it is, which makes sense. It may be worth noting that this proposition holds because pl ∅ is a null vector in space E X , and coincides thus with the origin of this space.

Particular cases N = 1, 2, F, of Proposition (5) yield This mass function has F 1 = 3 focal sets and we have seen (Eq. (

φ 1 (m) = d m 0 (pl, pl ∅ ) = d m X (pl, pl ∅ ), ( 20
) φ 2 (m) = d m 1 (pl, pl ∅ ) = d m (pl, pl ∅ ), ( 21
) φ F (m) = d m F-1 (pl, pl ∅ ). ( 22
)) that φ 1 (m 1 ) = φ 2 (m 1 ) = 1 and φ F 1 (m 1 ) = φ 3 (m 1 ) = 0.892. 10
Using (20)-( 22), we find, respectively:

φ 1 (m 1 ) = d m X (pl 1 , pl ∅ ) = n m X (pl 1 ) = A⊆X m X (A)pl 1 (A) = m X (X )pl 1 (X ) = 1, φ 2 (m 1 ) = d m 1 (pl 1 , pl ∅ ) = n m 1 (pl 1 ) = m 1 (A)pl 1 (A) = m 1 ({d 1 , d 2 , d 3 })pl 1 ({d 1 , d 2 , d 3 }) + m 1 ({d 1 , d 2 , d 4 })pl 1 ({d 1 , d 2 , d 4 }) +m 1 ({d 3 , d 4 })pl 1 ({d 3 , d 4 }) = 1 and φ F 1 (m 1 ) = d m F 1 -1 1 (pl 1 , pl ∅ ) = d m 2 1 (pl 1 , pl ∅ ) = n m 2 1 (pl 1 ) = A⊆X m 2 1 (A)pl 1 (A) = m 2 1 ({d 1 , d 2 , d 3 })pl 1 ({d 1 , d 2 , d 3 }) + m 2 1 ({d 1 , d 2 , d 4 })pl 1 ({d 1 , d 2 , d 4 }) +m 2 1 ({d 3 , d 4 })pl 1 ({d 3 , d 4 }) + m 2 1 ({d 1 , d 2 })pl 1 ({d 1 , d 2 }) +m 2 1 ({d 4 })pl 1 ({d 4 }) + m 2 1 ({d 3 })pl 1 ({d 3 }) = 0.892.
Remark 3. Similar results to Propositions 4 and 5 exist for φ π . Indeed, let us denote by E x the K-dimensional subspace of E X spanned by the set {e x , x ∈ X } corresponding to singletons. Then, the contour function π associated to a mass function m is represented by the vector π = x∈X π(x)e x of E x . Since φ π (m) = max x∈X π(x) and π ∅ is a null vector in space E x , we have

φ π (m) = n ∞ (π) = d ∞ (π, π ∅ ). ( 23
)
Note also that φ 1 (m) = 1 -m(∅) = pl(X ) = max A⊆X pl(A) and thus we have, besides the equality (20),

φ 1 (m) = n ∞ (pl) = d ∞ (pl, pl ∅ ). ( 24
)
Hence, φ π (m) and φ 1 amount to the n ∞ norms of the contour and plausibility functions, respectively, as well as to the d ∞ distances of these functions to total inconsistency.

Let us denote by d φ the pseudo-distance associated to consistency measure φ, via the equalities ( 19) and ( 23) (i.e., d φ N = d m N -1 and d φπ = d ∞ ). This section has thus shown that the consistency of a mass function amounts, for any consistency measure φ considered in this paper, to its pseudo-distance d φ to total inconsistency. Note that the above results suggest that measure φ p N (m) := n p m N -1 (pl), with p = 1, ∞, i.e., p different from the values considered above, might be interesting to investigate as a candidate consistency measure. This is left for further research.

Geometric perspective on conflict measures

As a natural extension of the preceding results, based on the fact that a given conflict measure is induced by a consistency measure, it is shown below how conflict measures κ N can be expressed in terms of distances to the total inconsistency state. Proposition 6. For any m 1 , m 2 ∈ M, we have

κ N (m 1 , m 2 ) = 1 -n m N -1 1 ∩ 2 (pl 1 ∩ 2 ) = n m N -1 1 ∩ 2 (k 1 ∩ 2 )
Proof. Follows from Eq. ( 9) and Proposition 4.

Proposition 6 shows that the κ N conflict between mass functions amounts to the n m N -1 1 ∩ 2 pseudo-norm of the inconsistency function of their combination. Equivalently, it is equal to one minus the n m N -1

As the norm is simply the distance to the origin, Proposition 5 leads to the following relation between conflict and pseudo-distance: Proposition 7. For any m 1 , m 2 ∈ M and 1 ≤ N ≤ F 12 , we have

κ N (m 1 , m 2 ) = 1 -d m N -1 1 ∩ 2 (pl 1 ∩ 2 , pl ∅ ). Special cases N = 1, 2, F 12 , yield κ 1 (m 1 , m 2 ) = 1 -d m X (pl 1 ∩ 2 , pl ∅ ), κ 2 (m 1 , m 2 ) = 1 -d m 1 ∩ 2 (pl 1 ∩ 2 , pl ∅ ), κ F 12 (m 1 , m 2 ) = 1 -d m F 12 -1 (pl 1 ∩ 2 , pl ∅ ).
In particular, we note that the classical conflict measure m 1 ∩ 2 (∅) between mass functions in the TBM is equal to one minus the pseudo-distance d m X between the plausibility function of their combination and the plausibility function of the empty mass function.

Informally, Proposition 7 shows that the conflict between m 1 and m 2 amounts to one minus the pseudo-distance between their conjunctive combination and the totally inconsistent knowledge state (the counterpart to Remark 3 for conflict measure κ π yields a similar conclusion: κ π (m 1 , m 2 ) = 1 -d ∞ (π 1 ∩ 2 , π ∅ )). This shows that while a distance between m 1 and m 2 is not an appropriate measure of their conflict [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF][START_REF] Burger | Geometric views on conflicting mass functions: From distances to angles[END_REF], distances can be used to express it.

Conversely, conflict measures can be used to express the distance between m 1 and m 2 , as detailed in Remark 4 for a special case. Whether such relationships exist for other distances and conflict measures is an open question. By Lemma 5, we obtain for all m 1 , m 2 ∈ M:

d 2 (pl 1 , pl 2 ) = A⊆X (κ 1 (m 1 , m A ) -κ 1 (m 2 , m A )) 2 . ( 25
)
As can be seen with (25), d 2 (pl 1 , pl 2 ) does not evaluate how much m 1 and m 2 are in conflict with each other, but rather quantifies how much they are in conflict (according to the classical conflict measure κ 1 ) with the same categorical mass functions m A (sets A), A ⊆ X . This gives also some intuition behind the property that d 2 (pl 1 , pl 2 ) = 0 iff m 1 = m 2 whereas we can have κ N (m 1 , m 2 ) = 0 for m 1 = m 2 : if m 1 = m 2 = m then m 1 and m 2 have the same conflict with the same sets, i.e., κ 1 (m 1 , m A ) -κ 1 (m 2 , m A ) = 0 ∀A ⊆ X , whatever m may be, whereas κ N (m 1 , m 2 ) = 0 iff m 1 and m 2 are N -nonconflicting, that is m 2 is N -consistent.

Conclusions

Conflict between belief functions has been defined by Destercke and Burger as the inconsistency of the belief function resulting from their conjunctive combination. The three existing definitions of the consistency of a belief function were shown in this paper to be specific cases, or shades, of a parameterised family of consistency definitions, which we called consistency of order N . We introduced a corresponding family of consistency measures and derived an associated family of conflict measures between belief functions. Each of these conflict measures is shown to verify a list of desirable properties for quantifying conflict. The family of conflict measures encompasses the classical measure of conflict in belief function theory, associated to the weakest ("brightest") definition of consistency, as well as two other conflict measures associated, respectively, to a stronger ("darker") definition of consistency by Yager and to the strongest ("darkest") definition of consistency by Destercke and Burger (called logical consistency). In addition, a geometric view on consistency measures as well as on the associated conflict measures was provided. In particular, we showed that measuring the consistency of a belief function amounts to measuring its distance to the totally inconsistent knowledge state, whatever the definition of consistency considered. A similar result was also obtained for conflict measures.

Our conflict measures are applicable in the case where it can be safely assumed that the sources are independent. However, the measures can readily be extended to the case where the dependence between sources is known and is captured by a joint mass function as in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]. Handling the case where the dependence structure cannot be uniquely identified would need some further work, but it seems possible (in particular, measure κ 1 for nonconflict has already been adapted to this case in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]). Besides, similarly to what has been done in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF] for measure κ π , it would be interesting to find a decomposition of the conflict κ N of two belief functions, in terms of some internal conflict (inconsistency) of each of the belief functions and of some external conflict between them, such that this decomposition satisfies some sensible relations originally introduced in [START_REF] Daniel | Conflicts within and between belief functions[END_REF].

Other interesting perspectives include finding theoretical or practical considerations that could help in selecting in a given application, one specific conflict measure among the κ N measures, or in choosing between the two measures for strong nonconflict κ F 12 and κ π . In addition, it would be interesting to know whether there exist other pseudo-norms of the plausibility vector, besides the family of pseudo-norms we have introduced and presented, which could serve as consistency measures.

Figure 3 (

 3 a) displays how the correlation between φ π and φ N increases with N for different sizes of X , while Figure 3(b) is the

Figure 1 :

 1 Figure 1: Scatter plots of measures κ π and κ N obtained from 5000 couples of randomly generated mass functions over X such that |X | = 3.

Figure 2 :

 2 Figure 2: Spearman correlation between φ π and φ F for 5000 randomly generated mass functions.

  (a) Spearman correlation between φ π and φ N , N varying, for several |X |. (b) Scatter plots of φ π and φ N , |X | = 4.

Figure 3 :

 3 Figure 3: Correlation between φ π and φ N , 5000 randomly generated mass functions over X with F = 5 focal sets.

Table 3 :

 3 the pairwise correlation coefficients between φ N and φ M , for 1 ≤ N, M ≤ F = 5 and |X | = 4. We Correlation between φ N and φ M for 1 ≤ N, M ≤ F = 5 and |X | = 4.

Figure 4 :

 4 Figure 4: Scatter plots of φ N and φ N +1 for 5000 random mass functions such that |X | = 4 and F = 5.

m 2 :

 2 ({d 1 , d 3 , d 4 }, 0.7; {d 1 , d 4 }, 0.3) The conjunctive combination of m 1 and m 2 has F 12 = 5 focal sets: m 1 ∩ 2 : ({d 1 , d 3 }, 0.42; {d 1 , d 4 }, 0.3; {d 3 , d 4 }, 0.07; {d 1 }, 0.18; {d 4 }, 0.03)

Remark 4 .

 4 Let the Euclidean distance between the plausibility functions be denoted d 2 and defined asd 2 (pl 1 , pl 2 ) := A⊆X (pl 1 (A) -pl 2 (A)) 2 .

Table 2 :

 2 Spearman correlation between κ π and κ N according to |X | and N ≤ 8.

				Size of the frame of discernment |X |
			2	3	4	5	6	7
		1 0.8467 0.7747 0.7268 0.6624 0.6116 0.5551
		2 0.9535 0.8960 0.8451 0.7772 0.7187 0.6616
		3 0.9796 0.9452 0.9024 0.8395 0.7713 0.7064
	N	4 0.9895 0.9689 0.9373 0.8853 0.8168 0.7477 5 -0.9811 0.9585 0.9175 0.8546 0.7851
		6	-	0.9877 0.9716 0.9395 0.8847 0.8178
		7	-	0.9916 0.9798 0.9548 0.9081 0.8457
		8	-	0.9941 0.9852 0.9655 0.9260 0.8694

  d 2 , d 3 }, 0.36; {d 1 , d 2 , d 4 }, 0.09; {d 3 , d 4 }, 0.01; {d 1 , d 2 }, 0.36; {d 4 }, 0.06; {d 3 }, 0.12). , d 2 , d 3 }, 0.216; {d 1 , d 2 , d 4 }, 0.027; {d 3 , d 4 }, 0.001; {d 1 , d 2 }, 0.486; {d 4 }, 0.036; {d 3 }, 0.126; ∅, 0.108).

	and
	m 3 1 :	({d 1

)

  Example 1. Let us illustrate Eqs. (20)-(22) with mass function m 1 from Section 3.7, i.e., m 1 : ({d 1 , d 2 , d 3 }, 0.6; {d 1 , d 2 , d 4 }, 0.3; {d 3 , d 4 }, 0.1).

The disjunction of all focal sets is called the core of the belief function by Shafer[START_REF] Shafer | A mathematical theory of evidence[END_REF].

In[START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF], three definitions of nonconflicting mass functions are considered (Definitions 4, 5 and 6 in[START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF]). However, in the case of independent sources, which is assumed in this paper, Definitions 5 and 6 in[START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF] are equivalent.

∩

pseudo-norm of the plausibility function of their combination.

Acknowledgements

The authors wish to thank the NATO STO CMRE and the Université d'Artois for their support via their visiting scientist programs.

Note that, as stated in Lemma 4, the conflict does not decrease with N .

The (1-)nonconflict of the mass functions is a valid justification for considering the result of their conjunctive combination for further reasoning [START_REF] Smets | Analyzing the combination of conflicting belief functions[END_REF] and based on some standard decision procedure (such as the pignistic transform [START_REF] Smets | Decision making in the TBM: the necessity of the pignistic transformation[END_REF]), d 1 =Savona would be a sensible estimated destination.

We note also that while κ 1 is null, κ 2 is very small (0.0612), and κ F 12 , the measure for strong nonconflict, is quite higher (0.3865). A criterion for combination based on κ F 12 instead of κ 1 would lead then possibly to the decision to not combine the sources.

A norm-based view on conflict

Section 3 has brought to light the existence of a family of conflict measures, respecting Destercke and Burger's axiomatization of conflict quantification and encompassing Yager's definition of consistency.

As argued in [START_REF] Daniel | Conflicts within and between belief functions[END_REF][START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF] and further developed in [START_REF] Burger | Geometric views on conflicting mass functions: From distances to angles[END_REF], distances between belief functions [START_REF] Jousselme | Distances in evidence theory: Comprehensive survey and generalizations[END_REF] are questionable to measure their conflict; for instance, they do not satisfy Properties 5 (imprecision monotonicity) and 7 (insensitivity to refinement). Still, this does not mean that geometrical objects are not relevant to conflict quantification as will be shown in this section.

After recalling in Section 4.1 necessary material on norms and distances, we lay bare in Section 4.2 a pseudo-norm based view on consistency measures, which leads us to investigate the relationship between the consistency of a mass function and its distance to the state of total inconsistency. Then, this geometric view with respect to consistency measures is carried over to conflict measures in Section 4.3.

Norms and distances

Any vector v of the Cartesian space R N spanned by the set of vectors {e i , 1 ≤ i ≤ N }, can be written as v = 1≤i≤N v i e i , with v i ∈ R the coordinate of v along dimension e i .

By definition, a norm is a function n : R N → [0, ∞) that satisfies the following properties: