Martin Bordemann
email: martin.bordemann@uha.fr

AND Jaigyoung Choe
email: choe@kias.re.kr

Jens Hoppe
email: jens.r.hoppe@gmail.com

THE MINIMALITY OF DETERMINANTAL VARIETIES

Keywords: minimal submanifold, determinantal variety, helicoidal MSC: 53A10, 49Q05

The determinantal variety Σpq is defined to be the set of all p × q real matrices with p ≥ q whose ranks are strictly smaller than q. It is proved that Σpq is a minimal cone in R pq and all its strata are regular minimal submanifolds.

introduction

Determinantal varieties are spaces of matrices with a given upper bound on their ranks. Given p and q and r < min(p, q), the determinantal variety Y r is the set of all p × q matrices with rank ≤ r. Y r is naturally an algebraic variety as the rank condition on a matrix is equivalent to the vanishing of all of its (r + 1) × (r + 1) minors which are polynomials of degree r + 1. Let Z r = Y r \ Y r-1 denote the set of all p × q-matrices whose rank is equal to r. By projecting each p × q matrix of rank r on the r-dimensional subspace V of R p generated by its columns and on the rdimensional subspace W of R q generated by its rows it can be seen that Z r is a fibre bundle over the cartesian product of two real Grassmannians G r (R p )×G r (R q ) where the fibre over (V, W ) is the set of all linear bijections V → W (canonical bundles), see e.g. [7, p.8]. Hence it is a regular submanifold of dimension r(pr) + qr. It is well-known that Y r is decomposed into the disjoint union of regular subvarieties Z s , s being an integer 0 ≤ s ≤ r (where of course Y -1 = ∅), see e.g. [7, p.4-6] for more details.

More concretely: let M pq the set of all real p × q matrices with p ≥ q and identify M pq with R pq . Let us define Σ pq = {(x 11 , x 12 , . . . , x pq ) ∈ R pq : det(X T X) = 0, X = (x ij ) ∈ M pq } and call Σ pq the determinantal variety in R pq . It can be identified with Y q-1 . The vector space M pq carries a natural positive definite inner product (1.1) X, Y = tr(X T Y ).

Many studies have been done about determinantal varieties, mostly on their algebraic properties. In this paper we are interested in an analytic property, the minimality of Σ pq in R pq , which we prove in 3 different ways (two of which generalize existing results for p = q, cp. [START_REF] Tkachev | Minimal cubic cones via Clifford algebras[END_REF], [START_REF] Hoppe | New minimal hypersurfaces in R (k+1)(2k+1) and S 2k 2 +3k[END_REF], [START_REF] Choe | Some minimal submanifolds generalizing the Clifford torus[END_REF], while concerning the first, parametric proof, see [START_REF] Hoppe | Lectures on Minimal Surfaces[END_REF] for some low-dimensional p > q examples): Theorem 1.1. Every stratum Z r for 0 ≤ r ≤ q -1 of the determinantal variety Σ pq is a regular minimal submanifold of the inner product space M pq .

Parametric Proof

Consider the real vector space R pr+r(q-r) whose elements are written as pairs a λ where a = ( a 1 a 2 . . . a r ) = (a Js ) J=1...p s=1...r is a real p×r-matrix and λ = (λ ss ′ ) s=1...r s ′ =1...r ′ :=q-r is a real r × (qr)-matrix. Let U pqr denote the open subset of R pr+r(q-r) consisting of those pairs a λ where the matrix a is of maximal rank r. We denote the natural euclidean scalar product on U pqr by γ, i.e.

(2.1)

γ a λ , a ′ λ ′ = tr(a T a ′ ) + tr(λ T λ ′ ).
Then the following smooth map X :

U pqr → M pq ∼ = R pq given by (2.2) X : a λ → X a λ = (a, a • λ)
clearly parametrizes an open subset of the stratum Z r (the regular submanifold of all rank r p × q-matrices) of the determinantal variety Σ pq . It can be seen as a natural chart domain of Z r (with the inverse of X being the chart), the other domains being obtained by permutations of the r column vectors a 1 , a 2 , . . . , a r . Due to

(2.3) D ( a λ ) c µ := d dt X a + tc λ + tµ | t=0 = (c, cλ + aµ)
the Riemannian metric Ĝ = X * , on U pqr that is induced from the inner product , (1.1) on R pq ∼ = M pq (the space of real p × q matrices) by X is computed to be (2.4)

G ( a λ ) c 1 µ 1 , c 2 µ 2 = tr(c T 1 (c 2 + c 2 λλ T + aµ 2 λ T )) + tr(µ T 1 (a T aµ 2 + a T c 2 λ)) = γ c 1 µ 1 , φ ( a λ ) c 2 µ 2 , with (2.5) φ ( a λ ) c µ = c(1 + λλ T ) + aµλ T a T cλ + a T aµ .
Written as a (pr + rr ′ ) × (pr + rr ′ ) dimensional symmetric matrix,

(2.6) G = G B B T D = g βγ b βv b γu d uv = R 1+λλ T L a R λ T L a T R λ L a T a
where R Y resp. L Y denote right-, resp. left-, multiplication with Y .

As both G = (g αβ ) α,β=1...p•r and D = (d uv ) u,v=1...r•r ′ are invertible one may invert G in the form (2.7)

G -1 =   G -1 + G -1 BρB T G -1 -G -1 Bρ -ρB T G -1 ρ   =   g γα + b γ u ρ uv b α v -b γ u ρ uv -ρ uv b α v ρ uv  
with ρ, defined as the inverse of D -B T G -1 B, existing due to the factorization

(2.8) G B B T D = 1 0 B T G -1 1 G B 0 D -B T G -1 B ;
or, following from the factorization

(2.9) G B B T D = G ′ B ′ 0 D ′ 1 0 K 1 in the form (2.10) G -1 =   (G ′ ) -1 -(G ′ ) -1 BD -1 -D -1 B T (G ′ ) -1 D -1 + D -1 B T (G ′ ) -1 BD -1   .
Due to the blocks in G being given in terms of left-and right-multiplication one also has, more explicitly, (2.11)

G -1 =   1 -L Pa R λλ T (1+λλ T ) -1 -L a(a T a) -1 R λ T R λ L (a T a) -1 a T ρ = L (a T a) -1 R 1+λ T λ  
with P a := 1-a(a T a)a T projecting onto the r ′′ = p-r dimensional space orthogonal to the column-space C(a), i.e. the span of the a s . Using

L Y L Y ′ = L Y Y ′ , R Y R Y ′ = R Y ′ Y one trivially verifies G -1 • G = 1 also in the R -L form.
To shown the minimality of Y r in the chart defined by X we have to compute the second order derivatives of X summed over the components of G -1 and prove that the component normal to the tangent space of that vector (the mean curvature vector H) vanishes. A straight-forward, but rather lengthy computation gives (µ = (α, u), ν = (β, v))

(2.12) Ĝµν ∂ 2 X ∂x µ ∂x ν a λ = -2D ( a λ ) X 0 (a T a) -1 λ ,
which obviously is tangential to the surface whence H vanishes. The computation can be shortened by the following simple arguments: note that the only non-vanishing second derivatives of X are (independent of s)

(2.13) ∂ 2 X ∂a Js ∂λ ss ′ = E Js ′
(the matrix that is everywhere zero except =1 in the J th row and s ′ column), and that the r ′ r ′′ = r where s ′ = 1 . . . r ′ = qr, s ′′ = 1 . . . r ′′ = pr, and the { e s ′′ } s ′′ =1...r ′′ being an orthonormal basis of the kernel of a T . The only non-vanishing elements of the second fundamental form are therefore

(2.15) h (t ′ t ′′ ) Js,ss ′ = γ t ′ δ t ′ s ′ ( e t ′′ ) J = h (t ′ t ′′ ) ss ′ ,Js , hence (2.16) H (t ′ t ′′ ) ∼ γ t ′ J,s G Js,st ′ ( e t ′′ ) J
for the components of the mean-curvature vector, -all of which vanish, due to the sum over J(= 1 . . . p), involving (only) scalar products of the e t ′′ with the a s (all of which are = 0 as the e 's are by definition in the kernel of a T ). While (cp. (2.11))

G Js,st ′ = (-L a(a T a) -1 R λ T )
Js,st ′ can easily be calculated explicitely, in analogy with

(2.17) B Jt,ss ′ = λ ts ′ ( a s ) J ,
as -( a t ) J ( a T a) -1 ts λ st ′ , note that this is not really necessary, due to the following (cp. [START_REF] Hoppe | Lectures on Minimal Surfaces[END_REF], p.38): as g αβ = g Js,Kt ∼ δ JK is diagonal in the indices transforming as vectors under O(p), its inverse g αβ (as well as, trivially, ρ uv as the inverse of a matrix of O(p)-invariants) will not touch any O(p) vector-index; implying that the (due to (2.13), (2.15) only relevant) components (2.18) G Js,ss ′ = -g Js,Kt B Kt,u ρ uv must be linear combinations of the B Jt,u , hence carry the O(p) vector index entirely through the columns of a (which is sufficient to conclude the vanishing of the mean curvature vector).

Geometric Proof

Our algebraic proof of the minimality of the zero determinant set Σ in [START_REF] Choe | Some minimal submanifolds generalizing the Clifford torus[END_REF] was based on the property that the hypersurface Σ is helicoidal. In this section we prove the minimality of the determinantal variety by using the fact (Theorem 3.1) that a helicoidal subset with higher codimension is still minimal (we should mention that this fact was first proved by Adrian C. Chu [START_REF] Chu | Minimal submanifolds in Lie groups[END_REF]).

Definition. Let M be a complete Riemannian manifold and Σ a subset of M . Suppose that Σ 0 is the subset of Σ which is a twice differentiable submanifold of M and Σ \ Σ 0 has measure zero on Σ. Suppose also that at any point p of Σ there is an isometry ϕ of M such that

ϕ(p) = p, ϕ(Σ) = Σ, ϕ * (v) = -v for any v ⊥ T pΣ 0 .
Then we say that Σ is helicoidal in M .

Theorem 3.1. Every helicoidal subset Σ m of a Riemannian manifold M n is minimal in M wherever Σ is twice differentiable.

Proof. Let H be the mean curvature vector of Σ at a point p ∈ Σ 0 , that is,

H = m i=1 (∇ e i e i ) ⊥ ,
where ∇ is the Riemannian connection on M and e 1 , . . . , e m are orthonormal vectors on a neighborhood of p in Σ 0 . Since ϕ(Σ) = Σ and p is a fixed point of ϕ, one sees that ϕ * (e 1 ), . . . , ϕ * (e m ) are also orthonormal on Σ near p. Hence

(3.1) ϕ * ( H) = m i=1 (∇ ϕ * (e i ) ϕ * (e i )) ⊥ = m i=1 (∇ e i e i ) ⊥ = H.
On the other hand, by the hypothesis, ϕ * ( H) = -H. Hence H = 0 at p. As p is arbitrarily chosen, one concludes that Σ is minimal.

Proof. (of Theorem 1.1): Recall the inner product , on M pq ,

X, Y = tr(X T Y ), X, Y ∈ M pq .
For any

A ∈ O(p) and X ∈ M pq , define ϕ A (X) = AX. Then ϕ A is an isometry on M pq since ϕ A is invertible and for Y ∈ M pq , ϕ A (X), ϕ A (Y ) = AX, AY = tr(X T A T AY ) = tr(X T Y ) = X, Y .
Let Z r ⊂ Σ pq be the set of all p × q matrices of rank r, 0 ≤ r < q. As ϕ A preserves rank, we have

(3.2) ϕ A (Z r ) = Z r .
Given X ∈ Z r , let C(X) be the column space of X and C(X) ⊥ its orthogonal complement in R p . There exists a p × p orthogonal matrix B such that C(X) is its eigenspace with eigenvalue 1 and C(X) ⊥ is its eigenspace with eigenvalue -1. Then we have

(3.3) ϕ B (X) = X.
Let R(X) be the row space of X. Define a subset N of M pq by

N = {Y ∈ M pq : C(Y ) ⊂ C(X) or R(Y ) ⊂ R(X)}.
Choose Y ∈ N and let σ(t) be the curve of p × q matrices from X to Y defined by σ(t) = X + tY, 0 ≤ t ≤ 1. Clearly σ(t) ⊂ Z r for sufficiently small t. The tangent vector σ ′ (0) at X equals

σ ′ (0) = lim t→0 σ(t) -X t = Y ∈ T X Z r .
Therefore N is a subset of T X Z r as well. Suppose W is a p × q matrix which is perpendicular to T X Z r . Then W ⊥ N . Here we claim

C(W ) ⊂ C(X) ⊥ .
Let v i be the orthogonal projection of the i-th column of W onto C(X) and let W X be the p × q matrix with the i-th column vector v i ,

1 ≤ i ≤ q. Then W X is in N and satisfies W X , W X = i |v i | 2 . Since 0 = W X , W = W X , W X = i |v i | 2 ,
we see that v i = 0 for all i and so the claim follows. Hence ϕ

B (W ) = -W . Note that (ϕ B ) * = ϕ B since ϕ B is linear. Thus (3.4) (ϕ B ) * (v) = -v for any v ⊥ T X Z r .
It follows from (3.2), (3.3), (3.4) that Z r is helicoidal and by Theorem 3.1 it is minimal in R pq .

Level-Set Proof

Let us also give a level-set proof, for simplicity restricting to Σ n , the space of all (n + 1) × n matrices of rank (n -1)

(4.1) A =      x 1 x 2 . . . x n x n+1 . . . x 2n . . . x n 2 +1 . . . x n 2 +n     
for which the first and last rows are not identically zero; Σ n can be characterized by the vanishing of the upper and lower n × n determinant (each of which, alone, 'trivially' defining a minimal surface), (4.2)

χ 1 := x 1 . . . x n . . . x n 2 -n+1 . . . x n 2 ! = 0 ! = x n+1 . . . x n+n . . . x n 2 +1 . . . x n 2 +n =: χ 2 • (-1) n-1 ,
and the minimality-condition for the intersection (χ 1 = 0 and χ 2 = 0) taking the form

(4.3) ∀ A ∈ Σ n : tr(P • ∂ 2 χ α )(A) α=1,2 = 0 with (4.4) P = (P JK := δ JK -∂ J χ α (M -1 ) αβ ∂ K χ β )
projecting onto the tangent-space of Σ n if (4.5) M = (M αβ := ( ∇χ α ) T ( ∇χ β )) (cp. [START_REF] Gee | From First to Second Quantized String Theory[END_REF]) tr(∂ 2 χ α ) = 0 (as all terms in χ 1 and χ 2 contain each of the n 2 , resp. n 2 + n variables x 1 , . . . , x n 2 +n at most linearly), and (4.3) follows from the stronger statement (having to do with the 2 constraints already separately defining minimal surfaces)

(4.6) ∀ A ∈ Σ n : ( ∇χ α ) T (∂ 2 χ β ) ∇χ γ = 0 ∀ 1 ≤ α, β, γ ≤ 2.
While for α = γ (4.6) is easy to prove,

(4.7) ( ∇χ α ) T (∂ 2 χ α ) ∇χ α = 1 2 χ α tr(∂ 2 χ α ) 2 (4.8) ( ∇χ 2 1 ) T (∂ 2 χ 1 2 ) ∇χ 2 1 = 1 2 tr(∂ 2 χ 1 ∂ 2 χ 2 ) • χ 2 1
the corresponding identity for α = γ (s.b.) seemed difficult to prove. The following argument however covers all cases : each non-zero entry of (∂ 2 χ α ) is a (n -2)dimensional determinant, which occurs exactly 4 times in ∂ 2 χ α , and those 4 terms cancel in the contraction with the 2 gradients, as (on χ α = 0).

(4.9)

∂ a+kn χ 1 = -λ k+1 ∂ a χ 1 ∂ a+kn χ 2 = -µ k+1 ∂ a ′ χ 1 a = 1 . . . n, k = 0 . . . n, a ′ := a + n 2
with λ n+1 = 0 = µ 1 , λ 1 = -1 = µ n+1 , and constants λ 2 . . . λ n , µ 2 . . . µ n appearing when writing the first row of A as a linear combination of the last n, resp. the last row in terms of the first n. As mentioned above, (4.7) and (4.8) can easily be proven (without the crucial observation (4.9)), by noting (cp. [START_REF] Hoppe | New minimal hypersurfaces in R (k+1)(2k+1) and S 2k 2 +3k[END_REF], [START_REF] Tkachev | Minimal cubic cones via Clifford algebras[END_REF]) (4.10)

∂χ α ∂A Ji = χ α M iJ α ∂ 2 χ α ∂A Li ∂A Kj = χ α (M iL α M jK α -M jL α M ik α ) J, K, L = 1 . . . n + 1, i, j = 1 . . . n
where M •• 1 is the inverse of (M 1 ).., the upper n × n part of A and M •• 2 is the inverse of the lower one, (M 2 ).., with the understanding that M i,J=n+1

1 = 0 = M i,J=1 2 .
For α = γ the trick (cp. [START_REF] Hoppe | New minimal hypersurfaces in R (k+1)(2k+1) and S 2k 2 +3k[END_REF]) that the [ij] antisymmetry of the second derivative, which antisymmetrizes the products of the 2 gradients (hence giving the trace of 2 Hessians when α = γ, with one determinant surviving) does not (simply) applyalthough almost certainly resulting in

(4.11) ( ∇χ 2 ) T ∂ 2 χ 1 ∇χ 1 = χ 2 4 tr∂ 2 χ 1 ∂ 2 χ 2 + χ 1 4 tr(∂ 2 χ 2 ) 2 ,
in analogy with (4.7), (4.8). Eqn (4.9) however, which in matrix-notation, due to x a+(K-1)n = A Ka simply reads (on χ α = 0) (4.12)

∂χ 1 ∂A ka = -λ k ∂χ 1 ∂A 1a (= λ k λ ′ a ∂ 11 χ 1 ), ∂χ 2 ∂A ka = -µ k ∂χ 2 ∂A n+1,a (= µ k µ ′ a ∂ n+1,n+1 χ 2 )
can be used to easily prove (4.6), for any α, β, γ as (4.13)

∂ 2 χ β ∂A Li ∂A Kj = - ∂ 2 χ β ∂A Lj ∂A Ki ,
combined with (4.12), gives (4.14)

∂ 2 χ β ∂A Li ∂A Kj ( ∂χ α ∂A Li ∂χ γ ∂A Kj + ∂χ α ∂A Kj ∂χ γ ∂A Li - ∂χ α ∂A Lj ∂χ γ ∂A Ki - ∂χ α ∂A Ki ∂χ γ ∂A Lj ) ∼ (λ αL λ γK + λ αK λ γL -λ αL λ γK -λ αK λ γL ) = 0, using that ∂χ 1 ∂A 1a = ∂χ 2 ∂A n+1,a .
Concerning (4.12), note that the following (stronger) statement can be easily proven for any n × n determinant χ: the matrix formed out of the derivatives of χ (w.r.t the n 2 variables) on χ = 0 is always of rank 1; choose any element (say M 11 )for which the corresponding row and column are both not identically zero.

M 1a 1 M 2a 2 . . . M nan ε a 1 ...an gives (4.16) ∂χ ∂M ij = a 1 ...a i-1 a i+1 ...an M 1a 1 . . . M i-1a i-1 M i+1a i+1 .
. . M nan ε a 1 ...j...an ↑at i th position ; using (4.17)

M 1a 1 = k =1 λ k M ka 1
and relabelling a 1 as a i after interchanging a 1 and j in the ε-tensor then gives (4.18)

∂χ ∂M ij = -λ i M 2a 2 . . . M ia i . . . M nan ε j a 2 ...an = -λ i ∂χ ∂M 1j (resp. . . . = -λ i ∂χ 2 ∂M n+1,j
), as all the terms k = j in the sum k = 1 give zero, due to the ε-tensor. Analogously, then writing (on the r.h.s. of (4.18

)) χ = c 1 ...cn M c 1 1 M c 2 2 . . . M cnn ε c 1 .
..cn , and using

M c 1 1 = l =1 ρ l M c 1 l gives (4.19) ∂χ ∂M ij = λ i ρ j ∂χ ∂M 11 onχ = 0.
Similar arguments can be used for the general case, p ≥ q ≥ r.

Generalizations

1. Among the possible generalizations, it is natural to replace real by complex numbers and to consider complex p × q matrices of rank r which also forms a r 2 + r(pr) + r(qr) complex submanifold Z C r of the space of all p × q-matrices, C pq . Putting the sesquilinear inner product Z, Z ′ = tr(Z † Z ′ ) on C pq we can view it as a Kähler manifold (recall that the riemannian metric is given by the real part of the sesquilinear form) and Z C r as a complex submanifold. It is wellknown that arbitrary complex submanifolds of Kähler manifolds are always minimal submanifolds, see e.g. [9, p.380] for a simple argument. Hence Z C r is a minimal submanifold of C pq . Moreover, since Z C r is obviously invariant under multiplication with nonzero complex numbers, C × , the above-mentioned fact also implies that the projectivization Z C r /C × is a complex, hence minimal submanifold of complex projective space CP pq-1 = (C pq \ {0})/C × (in the real case the projectivizations are minimal in spheres).

As concrete examples let us here consider only 2 special cases, p = q = r + 1 and p = 3, q = 2, r = 1, i.e. the 8 dimensional real manifold of complex 3 × 2 matrices

Z =   z 1 z 4 z 2 z 5 z 3 z 6   = ( x + i y, u + i v) ∧ = → z ∈ R 12 of the form (5.1) Z( x, y, λ, µ) = ( x + i y, (λ + iµ)( x + i y)),
x, y ∈ R 3 which is the complex analog of the map X is the first Section. We shall put the inner product Z, Z ′ = Re tr(Z † Z ′ ) on the complex matrix space. With (5.2)

∂ x i → z =     e i 0 λ e i µ e i     , ∂ y i → z =     0 e i -µ e i λ e i     , ∂ λ → z =     0 0 x y     , ∂ µ → z =     0 0 -y x    
one gets the following real symmetric 8 × 8-matrix (written in four blocks of dimension 6 × 6, 6 × 2, 2 × 6, and 2 × 2) for the induced metric:

(5.3) ( ĜAB ) =      (1 + λ 2 + µ 2 )1 6×6 λ x + µ y -λ y + µ x λ y -µ x λ x + µ y λ x T + µ y T λ y T -µ x T -λ y T + µ x T λ x T + µ y T ( x 2 + y 2 )1 2×2     
(note that the two 6-dimensional vectors in the off-diagonal block(s) are orthogonal to each other, and of equal length). As the only non-vanishing second derivatives

∂ 2 AB → z are (5.4) ∂ 2 x i λ → z =     0 0 e i 0     = -∂ 2 y i µ → z , ∂ 2 x i µ → z =     0 0 0 e i     = ∂ 2 y i µ → z ,
and the 4 normal directions being of the form

→ n α T = (. . . . . . N T α ) with N α ∈ R 6 , α = 1, 2,
H α ∼ ĜAB → n α • ∂ 2 AB →
z will vanish (cp. the comments around eqn (2.17), p.38 of [START_REF] Hoppe | Lectures on Minimal Surfaces[END_REF]) due to

N α • E 1 = 0 = N α • E 2 ; in the notation of section 2 : G = (1 + λ 2 + µ 2 )1 6×6 , D = (( x 2 + y 2 ))1 2×2 , (5.5) ρ -1 = ( x 2 + y 2 )1 - 1 1 + λ 2 + µ 2 B T B = x 2 + y 2 1 + λ 2 + µ 2 1,
implying that the relevant off-diagonal part of ĜAB is

(5.6) -G -1 Bρ = - 1 x 2 + y 2 (λE 1 -µE 2 , λE 2 + µE 1 ).
Concerning p = q = r + 1, i.e. the space ζ n of rank (n -1) complex n × n matrices (5.7)

Z =    z 1 . . . z n . . . . . . z n 2    , △ := det Z = u + iv = 0,
x ∈ R n 2 , one can use the known fact [START_REF] Hoppe | [END_REF] that the real and imaginary part of the determinant of a complex matrix Z are 'twin-harmonics', i.e. in particular satisfy (5.8) (∇u) 2 = (∇v) 2 , ∇u • ∇v = 0 as well as (5.9)

1 u u i u j u ij = 1 v v i v j v ij (=: ρ(x)),
with ρ being a homogeneous polynomial of degree 2n -4. In the simplest example, n = 2, (5.10) define minimal surfaces (which in this case are simply S 3 × S 3 cones, each). The interesting fact is that the intersection of the 2 generalized Clifford cones also has zero mean curvature. This is most easily seen by differentiating (5.8), yielding

z 1 z 2 z 3 z 4 = (x 1 + iy 1 )(x 4 + iy 4 ) -(x 2 + iy 2 )(x 3 + iy 3 ) one has (5.11) ∇u =             x 4 -x 3 -x 2 x 1 -y 4 y 3 y 2 -y 1             ∇v =             y 4 -y 3 -y 2 y 1 x 4 -x 3 -x 2 x 1             , ρ (
(5.14)

u i u ij = v i v ij , u i v ij + u ij v i = 0, hence (5.15)
v i v ij u j = u i u ij u j = ρu u i u ij v j = v i v ij v j = ρv u i v ij u j = -v i u ij u j = -ρv v i u ij v j = -u i v ij v j = -ρu,
all vanishing on u = 0 = v; using (4.3) it then immediately follows that (5.7), resp.

ζ n (for general n) is minimal.

For these Kählerian examples it may be interesting to consider its quantization or noncommutative or fuzzy analog: most of all Kähler manifolds admit a quantization by Toeplitz operators or geometric quantization (see e.g. [START_REF] Bordemann | Toeplitz Quantization of Kähler Manifolds and gl(N ), N → ∞-Limits[END_REF]) which leads to approximations of the function algebras by a sequence of finite-dimensional matrix algebras. Since the complex Z C r has a relatively simple structure as a complex holomorphic fibre bundle over a cartesian product of two complex Grassmannians it does not seem to be so hard to compute the Toeplitz quantization by representation theory of the homogeneous space. It may be simpler to do it for the projectivization since one is then working inside complex projective space which is compact. In the particular example of complex rank 1 2×2-matrices, the determinant condition gives Z C r and its projectivization the structure of a complex quadric for which quantization exist, see e.g. [START_REF] Dolan | Fuzzy Complex Quadrics and Spheres[END_REF] (concerning recent work on quantum minimal surfaces, see e.g. [12]). 2. Another possible generalization is the case of pseudo-euclidean spaces: consider again the space R pq of all real p × q-matrices where the positive inner product (1.1) is generalized by the following indefinite scalar product where η (resp. ζ) is a diagonal p × p (resp. q × q) matrix with p 1 ≤ p (resp. q 1 ≤ q) entries +1 and p 2 = pp 1 (resp. q 2 = qq 1 ) entries -1. The signature of (5.16) is easily seen to be (p 1 p 2 + q 1 q 2 , p 1 q 2 + p 2 q 1 ). The submanifold Z r ⊂ R pq of rank r matrices is in general no longer nondegenerate w.r.t. (5.16) which is exemplified by the simplest non-trivial case, p = q = 2, r = 1, ζ = 1 0 0 -1 , η = 1 0 0 1 , (A, B) = (( a 1 a 2 ), ( b 1 b 2 )) = a T 1 b 1a 2 b 2 ;

(5.17) X a λ := ( a, λ a); a = a 1 a 2 ∈ R 2 , λ ∈ R gives det( Ĝ.. ) = a T a(λ 2 -1), meaning that the 3 dimensional Z 1 (in R 2,2 ) defined by (5.17) contains a co-dimension 1 (λ 2 = 1) singular part. In order to get geometrically defined nondegenerate submanifolds consider the following open submanifold Z ′ r of Z r : fix two integers p1 , q1 with 0 ≤ p1 ≤ min{p, r}, 0 ≤ q1 ≤ min{q, r}, and set p2 = r -p1 , q2 = r -q1 . We shall use the short notation V M ⊂ R p (resp. W M ) to be the column space (resp. row space) of the p × q-matrix M ∈ Z r . Moreover for a finite-dimensional vector space equipped with an indefinite nondegenerate symmetric bilinear form g we call a vector subspace P of signature (a, b) if the restriction of g to P × P is nondegenerate and has signature (a, b). With these notations we set (5.18) Z ′ r = {M ∈ Z r | V M of signature (p 1 , p2 ), W M of signature (q 1 , q2 )}. where we have suppressed the obvious dependence of Z ′ r on the choice of integers p1 , q1 . A lengthy computation shows that for each such choice Z ′ r is a nondegenerate submanifold of R pq equipped with (5.16) whose induced metric has signature (5.19) (-p 1 q1 -p 2 q2 +p 1 q1 +p 2 q2 +p 1 q 1 +p 1 q 1 , -p 1 q2 -p 2 q1 +p 1 q2 +p 2 q1 +p 1 q 2 +p 2 q 1 ).

As a drawback of the above indefinite scalar product (5.16) we can deduce -using the preceding equation-that a Minkowski signature (1, s) can only be obtained for rather trivial cases. A generalization of the techniques described in the previous Sections, in particular Section 3, shows that each submanifold Z ′ r is helicoidal and hence minimal.

  3, 4, orthogonal to the plane spanned by x y =: E 1 and E 2 := -

  x) = 2; As, trivially, △u = 0 = △v, (5.9) implies that -separately -(5.12) u(x) := (x 1 x 4x 2 x 3 ) -(y 1 y 4y 2 y 3 ) v(x) := (x 1 y 4 + x 4 y 1 ) -(x 2 y 3 + x 3 y 2 ) ! = 0

( 5 .

 5 16) (A, B) := tr(ζA T ηB)

  N s ′ s ′′ := γ s ′ (λ 1s ′ e s ′′ , λ 2s ′ e s ′′ , . . . , λ rs ′ e s ′′ , 0, . . . ,e s ′′ , 0 . . . 0) ↑ at the s ′ -position

′ (pr) normal directions are

(2.14)


Johannes Van Der Pol
email: johannes.van-der-pol@u-bordeaux.fr

David Virapin

Analysing technological influence and dependance for competitive intelligence

Keywords: Patent analysis, citations analysis, data visualisation, technological dependance, technological influence

In this paper we present a method that can be used for the assessment of the technological influence and dependance of players in a domain. Technological influence identifies the extent to which the technologies of a firm are used by other actors for their own innovation activities. Technological dependence measures how the focal firm uses technologies of other actors in order to innovate. Both dimensions are represented using a specific datavisualisation technique aiming to facilitate the analysis. Measuring and visualising these dimensions helps decision makers with a variety of questions ranging from the identification of other applications for their technologies, the identification of potential partners to an assessment of litigation risk.

Introduction

It is no secret that innovation is vital for players' survival on the market [START_REF] Duysters | Organizational modes of strategic technology partnering[END_REF], [START_REF] Penrose | The theory of the growth of the firm[END_REF]). The knowledge used to innovate is a strategic asset for players and therefor requires careful management and curating. The cumulative nature of knowledge results in the creation of a technological trajectory that is specific to the player. On this trajectory each innovation or invention is the result of a combination of new and old knowledge. The latter can be knowledge created by the company itself or found outside the bounds of the players structure (collaboration, books, publications, patents).

The management of this knowledge requires players to pay close attention the knowledge they use in their innovation process but also who is using the knowledge they created. We will refer to these two dimensions as technological dependance and technological influence respectively. The analysis of influence and dependance are important for any innovating player if she wishes to reduce risk and maximise opportunity. Technological dependence, understood here as the extent with which a player uses other players knowledge reflects a risk. Using knowledge from others exposes a player to risk of litigation and a loss in bargaining power when a licence is needs to be negotiated (especially if the knowledge is already implemented in a sold product). In todays market, litigation is no longer a rare occurrence and some companies have specialised in litigation actions. These companies are called Non practicing entities and they are increasingly active [START_REF] Sterzi | Non-practicing entities and transparency of patent ownership in europe : the case of uk dormant companies[END_REF]. The increased risk for litigation should push players to scan for their level of dependance upon technologies that go beyond the bounds of the company. It it therefor important to have detailed information on the knowledge used in the innovation and indeed, the production process. On the opposite side, the extend to which other players use one's knowledge is a valuable source of strategic information as well. The management of existing knowledge can require decisions related to selling licences, deciding wether or not to maintain patents, finding new avenues for revenue on other markets, keeping track on who might be infringing on one's IP. The analysis of the influence of ones technologies can help provide answers for many and more of these cases. The study of technological influence and technological dependance is an activity that enters the domain of economic intelligence. More precisely it refers to a subdomain which is technology intelligence which aims to provide me-thods and techniques that allow actors of the innovation landscape to better understand their environment. These methods and techniques are required given the amount of information and data that is required to be treated for this type of analysis [START_REF] Schneider | On young highly innovative companies : why they matter and how (not) to policy support them[END_REF]) and should be part of any firms innovation process [START_REF] Dodgson | The evolving nature of taiwan's national innovation system : The case of biotechnology innovation networks[END_REF][START_REF] Flamand | Le déploiement de l'intelligence technologique dans le[END_REF]). However, the number of companies proceeding to this type of analysis is too low [START_REF] Flamand | Le déploiement de l'intelligence technologique dans le[END_REF]. This paper contributes to the tools at the disposal of decision makers by providing a method for the analysis of technological influence and dependance. The method presented in this paper was developed at the VIA Inno platform of the University of Bordeaux. VIA Inno has been using this technique for several years to answer a variety of questions for a variety of decision maker (multinationals, universities, government agencies, start-ups, regional agencies). This paper is organised as follows, first we will present the type of data used for this type of analysis. Next we will show the logic of the method for influence and dependance respectively. The final section will provide a case study.

Data & Methodology

The identification of influence and dependance requires data that contains information on which players are using knowledge of a given firm. Patents have the double advantage of containing both information on the knowledge produced by a player (the claims of the patent) and the knowledge that was used to create it (the citations). Patents have been used for decades as a measure of innovation and for the technological analysis of players [START_REF] Jaffe | Technological opportunity and spillovers of r&d : evidence from firms' patents, profits and market value[END_REF], [START_REF] Verspagen | Measuring intersectoral technology spillovers : estimates from the european and us patent office databases[END_REF], [START_REF] Shapiro | Navigating the patent thicket : Cross licenses, patent pools, and standard setting[END_REF], [START_REF] Tseng | Using patent data to analyze trends and the technological strategies of the amorphous silicon thin-film solar cell industry[END_REF]). Patent citations are come in two main forms, examiner and inventor citations. The main difference between the two is that the examiner cites other documents to contain the claims of the assignee [START_REF] Organization | WIPO intellectual property handbook : Policy, law and use[END_REF]. This implies that examiner citations indicate a certain technological proximity between the citing and the cited patent [START_REF] Benner | Close to you ? bias and precision in patentbased measures of technological proximity[END_REF] [START_REF] Alstott | Mapping technology space by normalizing patent networks[END_REF]). For this reason patent citations are also the a privileged datasource for the analysis of knowledge flows (Alcacer andGittelman (2006), Alcacer et al. (2009), [START_REF] Jaffe | Knowledge spillovers and patent citations : Evidence from a survey of inventors[END_REF]). In fact, patent citations are even used for the purpose of evaluating the technological value of patent portfolios and the player holding them [START_REF] Hall | Market value and patent citations[END_REF]).

For the purpose of this paper will use a citation of an indication that the citing play is building upon the knowledge produced by the cited player. Even thought this hypothesis is widely accepted, we do have to specify that not every citation has a strategic value. Citations come in a variety of categories ranging from citation general knowledge (books, manuals, presentations) to citing knowledge that blocks part of the citing patent. The latter are identified in a patent by the category X or Y to signify that the cited patent claims something that has already been achieved. These citations are therefor the strongest signal of technological proximity and are usually used for strategic analyses [START_REF] Shapiro | Navigating the patent thicket : Cross licenses, patent pools, and standard setting[END_REF]. However, in practice we have noticed that even inventor citations which are generally considered discarded can have strategic value for some specific question especially those related to finding new markets for a technology. The method we will present here will start out including all citations, we will show how to highlight categories in order to answer specific questions.

Patent data for the identification of influence and dependance

Technological Influence

For our purpose we define technological influence as the extent to which players exploit the knowledge of a focal player. The latter can either be the player herself analysing her own portfolio, or another player ( a competitor, potential buyout target, investment target etc.). In order to asses who is using the knowledge we will use patent citations. For this purpose we will consider that a player citing the patent of the focal player will have used at least part of the knowledge held by the focal player. Using this information we aim to be able to answer a variety of use cases that arise in innovation and knowledge management. We show some of these cases in table 1.

Some of these indicators are quite basic but they are extremely powerful when presented in a clear manner. The number of citations is easily computed while patents cited together is more complicated to communicate. Beyond the simple measures of we require a method that has a powerful communication aspect to it. This is why we will present the different indicators visually rather than a list with numbers, patents and players. We aim to provide a visualisation that is capable of instantly highlighting elements that are important for innovation management. The visualisation that we use is a graph that contains all layers of information that we have described, patents, citations and the citing players. In order to be able to highlight important information we build the graph, which we will call the influence graph as 110 follows : The nodes represent either patents or players. The central node will represent the focal player to which we will connect all the relevant patents ( this will be the first link, it will be undirected). Then are added the patents citing the patents of the focal actor. This is where the question of the techno-115 logical influence will be visible at the patent level. We will be able to identify patents receiving many or few citations. In order to see if these citations come from one or multiple player we add the final layer which is the patent owners. The last link is therefore an undirected link to highlight ownership, while the link in the centre is a directed citation link.

Using this data we build a graph connecting the different data sources which can be imported in any network analysis or visualisation tool (Gephi, payek, cytoscape, Tulip). The easiest way to create a dataset is then by a simple source-target list :

Source

Target EVE Rubber US9758627B2 (connecting focal firm to patent) WO2019094551A1 US9758627B2 (connecting citing patent to focal's patent) WO2019094551A1 Cabot (connecting citing firm to it's patent)

This table shows that patent US9758627B2 owned by EVE Rubber is cited by patent WO2019094551A1 which is owned by Cabot. This list is completed for all patent of the focal firm to allow the software to visualise the graph. This is the basic logic of the graph of influence. When this is done for all patents citing a firm the result is a graph that has different layers (as we will show later in this paper in figures 5 and 6 for exemple). For now let us focus on the information that this kind of graph can highlight. In figure 2 we show some the of theoretical cases we can expect to find in an influence graph. In the first of the cases, (a), we have one firms that owns one patent citing two patents of our focal firm. This situation implies that the player has combined parts of both patents to somehow innovate. We deduce from this type of citation that the focal firm has incentive to combine both its patents if it wishes to proceed to licence its patents. If the player wishes to continue its own innovation trajectory this citing player might cause a problem since it combined some ideas and is therefore close from a technological point of) view. A thorough study of the freedom to operate is required for this patent. Case (b) shows more atomised citations, even though the number of citations is the same, they are distributed over multiple patents. The level of influence exerted by the focal company is lower at the patent level because of the distribution of the citations. At the player level however it is complicated to assess of the influence is higher or lower. On the one hand the citing firms cites with more patents which would signify increased influence of the focal player, but the influence is lower at the level of each patent. Only an in-depth analysis of the patents will allow for a conclusion. case (c) highlights that the focal player has an important patent since it receives multiple citations. The technologies that the citing player is protecting with its patents depends upon a single patent of the focal firm. The influence of the focal firm is therefore high. These theoretical situations are not exhaustive. In practise other cases might occur that are of interest for the question that one is trying to answer. Borderline cases can also exist. The force of this method is that it is able to highlight situations of interest and provides an easy-to-read visualisation.

Technological Dependance

Technological dependence analyses the extent to which a firm is dependent upon the knowledge of other players. The aim here is to identify high levels of dependance to reduce the risk of litigation or loss of negotiation power. An analysis of the technological dependance of the firm can be used in a variety of use cases such as :

-Identify firms that can block my technological trajectory -Identify litigating firms to which my patents are close -Identify patents that rely too heavily on technologies of one specific player The method we use for this purpose is the same as the method for the analysis of technological influence, the only modification required is to change the direction of the citations link as shown in figure 3. The focal node remains the player we are interested in. Connected to this player are its patents (or at least those that are relevant to the study) to which we then connect the cited patents. These citations are the first elements to inform us about the dependance of the focal player. To complete the picture we add the assignees of the cites patents so we are able to identify the players holding the cited patents. The interconnection of these layers result in a structure that easily highlights different levels of dependancy as we shall illustrate later in this paper. The particular structure we are searching for are those that highlight some level of concentrated citations from the focal firm to either patents or players. Figure 4 shows the type of structure we are searching for, these are identical the ones we search for in an influence graph, the interpretation is however reversed. In case (a) the player has multiple patents citing one patent of another firm implying that the player is dependant upon one particular patent of another firm. This gives more bargaining power the the cited firm and especially a higher risk of litigation.

Case study : EVE Rubber

To show how to use the method described in this paper we provide an example using EVE Rubber, a player in the rubber industry. We identified this player using a method for the identification of player with a particular collaboration strategy (van der Pol ( 2021)). This method uses dynamic network analysis to identify a players that have a behaviour that is a deviation of the average in addition to providing a classification of player into a topology of strategies. EVE Rubber is particular because it is relatively you while having collaborations with large corporations. With the aim of understanding the technological trajectory of the company we look into its technological dependance and influence using the method presented in this paper. For a more extensive study of this player see [START_REF] Brunet | A competitive intelligence view of china's innovation eco-system : the case of silica[END_REF]. Patent information was obtained from the Questel Orbit database1 and Gephi [START_REF] Bastian | Gephi : an open source software for exploring and manipulating networks[END_REF]) was used for the visualisation of the graphs.

EVE Rubber's Influence Graph

Figure 5 shows the influence graph of EVE Rubber. The different levels of the graph have been coloured to highlight their origin. Green nodes are the patents of EVE Rubber, blue nodes are the patents citing EVE Rubber and the purple nodes the assignees of the citing patents. Within this graph we highlight different cases of interest linked to the theoretical cases we have previously identified. The letters next to the circles relate directly to those theoretical cases. We can start this analysis by highlighting Qingdao Tailan, a Chinese company that cites three patents of EVE Rubber with one of their patents. This situation created negotiation power for EVE Rubber if the have proof that Qindao Tailan uses its patent for production purposes. This also shows that the three patents can be grouped together and used as a bundle of patents. This idea is reinforced by the observation that Shandong Yanggu Huatai Chemical cites the same group of patents with one of its own. In addition, the latter has a second patent citing EVE Rubber increasing it's dependance on EVE Rubber. Other interesting cases, such as Petrochina which cites one patent with two of it's own shows that patent CN103205001 attracts a lot of attention and therefor has potential value for EVE Rubber. Patent numbers CN10523783 and CN103224659B, illustrated by (e) in the graph, is cited by two different companies. This patent can therefore also be of interest for EVE Rubber if it wishes to valorise its patents.

Adding more information to the graph. Citations are very informative but they come in many categories, some of which are of more interest than others. We focus here on two categories : X and Y citations. These citations are added by the patent examiners to indicate that the cited patent blocks at least one claim of the citing patent. These citations therefor indicate a high technological proximity [START_REF] Shapiro | Navigating the patent thicket : Cross licenses, patent pools, and standard setting[END_REF] which is what we are searching for. Using the same graph as previously we add two indicators to the graph, one that indicates whether the citation is blocking (either a category X or Y citation) and whether the patent itself is in the top 5% according to its influence in the domain. This indicator is computed by counting the number of direct and indict citations a patent has received. This is computed at the end of the period according to the year of filing. The top 5% is computed and retains those patents with the most direct and indirect citations when compared to other patents filed the same years in the same domain.

In figure 6 we visualise these two indicators : red links represent a category X or Y citation, green patents indicate patent that are in the top 5% of the domain in terms of indirect and direct citations. This additional information shows some of the patents citing EVE Rubber are important in the domain (CN104962069 and CN104356443on the left of the graph). These patents cite the most cited patents of EVE Rubber and they cite them with blocking citations. This means that the most cited patents of EVE are technologically close to those citing them and present a higher risk. EVE Rubber itself has two patents in the top 5% (CN104277255B and CN103600435), however these patents do not receive any blocking citations.

The group of patents that we identified in this graph appears to be rather strong since it blocks some of the claims made by the two citing players (X or Y citations). The influence exerted by EVE is therefor relatively strong vis-à-vis these two players.

The added value of this approach is that the visualisation makes it easy to identify patents and players that present either risks or opportunities for a player. Any additional information can easily be added by adding colours or images of the technologies behind the patents.

EVE Rubber's Dependance Graph

We now look at the inspiration of EVE Rubber used for its patents. We search for companies and patents that receive many citations to identify what the firm is dependant upon. Figure 7 shows the dependance graph of EVE Rubber. The graph highlights the importance of tire manufacturers as an influence (Michelin, Bridgestone, Yokohama, Sumitomo). The highest number of citations point towards Michelin and Cabot, which can make sense since Cabot is a supplier of Silica for Michelin. A more in detail analysis actually shows that the founder of EVE Rubber as worked for Cabot in the past, explaining this high proximity between the firms.

The structure of the graph also highlights that groups of patents appear to cite particular firms. 7 patent to the north of the graph cite JSR and Bridgestone while patents in the south east tends to focus on Cabot and Michelin. The dependance of EVE Rubber on the technologies of these firms is directly visible and contrasts with the influence graph. While EVE Rubber seeks its technological inspiration mainly through tyre manufacturers, none of them cite the patents of EVE Rubber. Actually, mainly Chine players cite their patents.

Just as we did for the influence graph, we add two indicators in the graph to asses in more detail the type of citation. Figure 8 shows that EVE Rubber cites two of Michelin's patents that are in the top 5% with a blocking citation. Therefore, The patents filed by EVE Rubber are close to those of Michelin since part of the claims made by EVE have been blocked by Michelins Patents, there is hence a high proximity between these documents.

A tool to help decision making

The type of graph presented in this paper was initially developed at the VIA Inno platform where it is now a tool currently used for the strategic analysis of player in a technological domain. It has been used to answer a variety of questions, ranging from finding opportunities for selling a technology in another domain, to finding partners to develop a technology further, identifying litigation treats, identifying patents to sell or maintain. The graph allows the analyst to accentuate the elements that answer the question she/he is trying to answer. Experience has shown that the visual of the graph makes communication with decision makers easier. For the analyst building the graph allows for easy identification of elements of value.

300

It is possible to add additional layers of information relevant to the question asked by the decision maker. We do however recommend to not add too much information in one graph but rather provide multiple graphs with less information.

Conclusion 305

In this paper we have shown a method for the measurement of technological influence and dependance. There are multiple use-cases for this type of analysis, ranging from a simple analysis of the patents and player on which a player is highly dependant to the more complex question of finding new applications for a given technology. past years and is broadly very well received. The visual facilitates the discussion with experts and facilitate the decision making process for decision makers. The method can be used by intellectual property experts to assess 315 the risks of certain patents (for instance a pool of patents highly dependant on one player), by engineers to assess what other applications might exist for a technology, for competitive intelligence analysts to better understand the technological proximity between players and so on. We have shown here a basic version of the graphs, but they can be comple-320 mented with additional information. For instance one can add information relative to the IPC classes, the dates of the patents, the inventors, images from the patents or even add a dynamic perspective.

Figure 1 :

 1 Figure 1: Concept of the influence graph, showing the different layers of information and how they are interconnected

Figure 2

 2 Figure 2

Figure 3 :

 3 Figure 3: Concept of the dependance graph, showing the different layers of information and how they are interconnected

Figure 4 :

 4 Figure 4: Citations structures we look for in a dependancy graph.

Figure 5 :

 5 Figure 5: The influence graph of EVE Rubber (patents filed before June 2019)

Figure 6 :

 6 Figure 6: The influence graph of EVE Rubber completed with information on citations and patent importance. Red links represent a category X or Y citation, green patents indicate patent that are in the top 5% of the domain in terms of indirect and direct citations.

Figure 7 :

 7 Figure 7: Dependance Graph of EVE Rubber for its patents in the domain of silica for rubber (patents filed before June 2019)

Figure 8 :

 8 Figure 8: The dependance graph complemented with Red links representing a category X or Y citation, green patents indicating patents that are in the top 5% of the domain in terms of indirect and direct citations.

Table 1 :

 1 Use cases and indicators used.

	Aim

see https ://www.questel.com for more information on this worldwide patent database

What we present in this paper is the logic of the method. We have used this method in many different settings and for many different questions over the


Frédéric Dambreville

Definition of evidence fusion rules based on referee functions

Keywords: Evidence, Referee Function, Sampling, Dempster-Shafer rule, PCR6. 1 2CHAPTER 1. DEFINITION OF EVIDENCE FUSION

This chapter defines a new concept and framework for constructing fusion rules for evidences. This framework is based on a referee function, which does a decisional arbitrament conditionally to basic decisions provided by the several sources of information. A simple sampling method is derived from this framework. The purpose of this sampling approach is to avoid the combinatorics which are inherent to the definition of fusion rules of evidences. This definition of the fusion rule by the means of a sampling process makes possible the construction of several rules on the basis of an algorithmic implementation of the referee function, instead of a mathematical formulation. Incidentally, it is a versatile and intuitive way for defining rules. The framework is implemented for various well known evidence rules. On the basis of this framework, new rules for combining evidences are proposed, which takes into account a consensual evaluation of the sources of information.

Introduction

Evidence theory [START_REF] Dempster | Upper and Lower probabilities induced by a multivalued mapping[END_REF][START_REF] Shafer | A mathematical theory of evidence[END_REF] has often been promoted as an alternative approach for fusing information, when the hypotheses for a Bayesian approach cannot be precisely stated. While many academic studies have been accomplished, most industrial applications of data fusion still remain based on a probabilistic modeling and fusion of the information. This great success of the Bayesian approach is explained by at least three reasons:

• The underlying logic of the Bayesian inference [START_REF] Dambreville | Deterministic modal Bayesian Logic: derive the Bayesian inference within the modal logic T[END_REF] seems intuitive and obvious at a first glance. It is known however [START_REF] Lewis | Probability of Conditionals and Conditional Probabilities[END_REF] that the logic behinds the Bayesian inference is much more complex,

• The Bayesian rule is entirely compatible with the preeminent theory of Probability and takes advantage of all the probabilistic background,

• Probabilistic computations are tractable, even for reasonably complex problems.

Then, even if evidences allow a more general and subtle manipulation of the information for some case of use, the Bayesian approach still remains the method of choice for most applications. This chapter intends to address the three afore mentioned points, by providing a random set interpretation of the fusion rules. This interpretation is based on a referee function, which does a decisional arbitrament conditionally to basic decisions provided by the several sources of information. This referee function will imply a sampling approach for the definition of the rules. Sampling approach is instrumental for the combinatorics avoidance [START_REF] Papoulis | Probability, random variables and stochastic processes[END_REF].

In the recent literature, there has been a large amount of work devoted to the definition of new fusion rules [START_REF] Denoeux | The cautious rule of combination for belief functions and some extensions[END_REF] to [START_REF] Yager | On the Dempster-Shafer Framework and New Combination Rules[END_REF] . The choice for a rule is often dependent of the applications and there is not a systematic approach for this task. The definition of the fusion rule by the means of a sampling process makes possible the construction of several rules on the basis of an algorithmic implementation of the referee function, instead of a mathematical formulation. Incidentally, it is a versatile and intuitive way for defining rules. Subsequently, our approach is illustrated by implementing two well known evidence rules. On the basis of this framework, new rules for combining evidences are also proposed. Typically, these new rules takes into account a consensual evaluation of the sources, by invalidating irrelevant sources of information on the basis of a majority decision.

Section 1.2 summarize some classical results in the domain of evidence. Section 1.3 introduces the notion of referee function and its application to the definition of fusion rules. A sampling method is obtained as a corollary. Section 1.4 establishes the referee functions for two known rules. Section 1.5 defines new fusion rules. Section 1.6 makes some numerical comparisons. Section 1.7 concludes.

Belief fusion

This section introduces the notion of belief function and some classical rules of fusion.

Lattices

Lattices are algebraic structures which are useful for encoding logical information. In particular, lattices are generalizations of structures like Boolean algebra, sets, hyper-power sets or concept lattice [START_REF] Davey | Introduction to Lattices and Order[END_REF][START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF]. Sets and hyper-power sets are widely used as a framework for defining and manipulating belief functions. Concept lattices are frameworks used in the ontologic domain of formal concept analysis.

Definition 1. A (finite) lattice L is a partially ordered (finite) set, i.e. (finite) poset, which satisfies the following properties:

• For any two elements X, Y ∈ L, there is a greatest lower bound X ∧ Y of the set {X, Y } ,

• For any two elements X, Y ∈ L, there is a least upper bound X ∨ Y of the set {X, Y } .

The notations X ∩ Y (respectively X ∪ Y ) are also used instead of X ∧ Y (respectively X ∨ Y ).

Example. Concept lattices [START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF] (which are not defined here) are lattices. Concept lattices are widely used for deriving ontologies.

Definition 2. A bounded lattice L is a lattice which have a least element ⊥ and a greatest element ⊤.

The notations ∅ or 0 (respectively Ω or 1) are also used instead of ⊥ (respectively ⊤).

Proposition. A finite lattice is a bounded lattice.

Proof is achieved by setting: ⊥ = X∈L X and ⊤ = X∈L X .

Definition 3.

A distributive lattice L is a lattice such that ∧ and ∨ are mutually distributive:

X ∧ (Y ∨ Z) = (X ∧ Y ) ∨ (X ∧ Z) and X ∨ (Y ∧ Z) = (X ∨ Y ) ∧ (X ∨ Z) ,
for any X, Y, Z ∈ L .

Example. Being given a frame of discernment Θ, the hyper-power set D Θ is typically the free distributive lattice generated by Θ.

Definition 4.

A complemented lattice is a bounded lattice L, such that each element X ∈ L has a complement, i.e. an element Y ∈ L verifying:

X ∨ Y = ⊤ and X ∧ Y = ⊥ .
The complement of X is often denoted ¬X or X c .

Actually, the complementation is defined by introducing constraints on the lattice. As a consequence, it is possible to have partial complementation on the lattice.

Definition 5. A Boolean algebra is a complemented distributive lattice.

Examples. Being given a frame of discernment Θ, the power set 2 Θ , the superpower set S Θ , or the free Boolean algebra generated by Θ, are Boolean algebra.

Conclusion.

Bounded lattices (especially, finite lattices) are versatile structures, which are able to address many kind of informational frameworks. Typically, bounded lattices generalize power set and hyper-power set. But since complementation is defined by introducing constraints on the lattice, it is also possible to derive lattice with partial complementation (i.e. with a subset of the complementation constraints) which are intermediate structure between hyperpower set and power set. Since bounded lattices are such generalization, this chapter will define evidence fusion rules within this framework. By using a lattice framework, we are also linking our work to the domain of formal concept analysis.

In the domain of evidence theories, basic concepts are generally modeled by means of a set of proposition, Θ, called frame of discernment. The structure G Θ is any finite lattice constructed from Θ . In particular, G Θ may be D Θ , 2 Θ or S Θ . Notice that G Θ is not a lost of generality, and is able to address any finite lattice. From now on, all notions and results are defined within G Θ , which make them applicable also to any finite lattice.

Belief functions

Belief functions are measures of the uncertainty that could be defined on the propositions of a bounded lattice. In the framework of evidence theory, beliefs are defined so as to manage not only the uncertainty, but also the imprecision encoded within the lattice structure.

Basic belief assignment. A basic belief assignment (bba) on G Θ is a mapping m : G Θ → IR such that:

• m ≥ 0 , • m(∅) = 0 , • X∈G Θ m(X) = 1 .
The bba contains an elementary knowledge (in the form of a basic belief) about the whole propositions of G Θ . The bba, however, does not provide directly the knowledge about an individual proposition. This individual knowledge is imprecise and bounded by the belief and the plausibility.

Belief. The belief bel is constructed from the bba m as follows:

bel(X) = Y ∈G Θ Y ⊂X m(X) .
The belief is a pessimistic interpretation of the bba.

Plausibility. The plausibility pl is constructed from the bba m as follows:

pl(X) = Y ∈G Θ Y ∩X =∅ m(X) .
The plausibility is an optimistic interpretation of the bba.

While fusing beliefs, the essential computations are done by means of the bbas. Our contribution is focused on the bba fusion; belief and plausibility will not be manipulated in this chapter. Notice howover that some properties of the belief functions may change, depending on the structure of the lattice being used.

Fusion rules

Let be given s sources of information characterized by their bbas m 1:s . How could we fuse these information into a single fused bba? There is a variety of rules for fusing bba. This section covers different classical rules.

Dempster-Shafer. The fused bba m DST obtained from m 1:s by means of Dempster-Shafer fusion rule [START_REF] Dempster | Upper and Lower probabilities induced by a multivalued mapping[END_REF][START_REF] Shafer | A mathematical theory of evidence[END_REF] is defined by:

m DST (∅) = 0 and m DST (X) = m ∧ (X) 1 -m ∧ (∅) for any X ∈ G Θ \ {∅} . (1.1)
where m ∧ (•) corresponds to the conjunctive consensus:

m ∧ (X) Y1∩•••∩Ys=X Y1,••• ,Ys∈G Θ s i=1 m i (Y i ) , (1.2)
for any X ∈ G Θ \ {∅} .

The rejection rate z = m ∧ (∅) is a measure of the conflict between the sources. Notice that the conflict is essentially a conjunctive notion, here.

Historically, this is the first rule for fusing evidences. Essentially, this rule provides a cross fusion of the information: it is based on a conjunctive kernel.

However, the conjunctive nature of this rule is altered by the necessary normalization implied by the conflict measurement m ∧ (∅).

Disjunctive rule

The fused bba m ∨ obtained from m 1:s by means of a disjunctive fusion is defined by:

m ∨ (X) = Y1∪•••∪Ys=X Y1,••• ,Ys∈G Θ s i=1 m i (Y i ) , (1.3)
for any X ∈ G Θ .

The disjunctive rule alone is not very useful, but it is interesting when fusing highly conflicting information. When at least one sensor provides the good answer, the disjunctive rule will maintain a minimal knowledge. Typically, this rule may be combined with the conjunctive consensus, in an adaptive way [START_REF] Florea | Adaptative combination rule and proportional conflict redistribution rule for information fusion, COGnitive systems with Interactive Sensors[END_REF][START_REF] Dubois | On the relevance of non-standard theories of uncertainty in modelling and pooling expert opinions[END_REF].

Rule of Dubois & Prade. Dempster-Shafer fusion rule will have some unsatisfactory behavior, when the conflict level is becoming high. If it is assumed that at least one sensor provides the good answer, then it is wiser to replace the possible conflict by a disjunctive repartition of the belief product. This idea is implemented by the rule of Dubois and Prade [START_REF] Dubois | On the relevance of non-standard theories of uncertainty in modelling and pooling expert opinions[END_REF].

The fused bba m D&P obtained from m 1:s by means of this fusion rule is defined by:

m D&P (∅) = 0 , m D&P (X) = Y1∩•••∩Ys=X Y1,••• ,Ys∈G Θ s i=1 m i (Y i ) + Y1∩•••∩Ys=∅ Y1∪•••∪Ys=X Y1,••• ,Ys∈G Θ s i=1 m i (Y i ) , (1.4)
for any X ∈ G Θ \ {∅} .

Averaging rule. Averaging, although quite simple, may provide good results on some applications.

Let be given the averaging parameters α 1:s ≥ 0 such that s i=1 α i = 1 . The averaged bba m µ [α] is obtained from m 1:s by:

m µ [α] = s i=1 α i m i .
PCR6. The proportional conflic redistribution rules (PCRn) have been introduced By Smarandache and Dezert [START_REF] Smarandache | Proportional Conflict Redistribution Rules for Information Fusion, Applications and Advances of DSmT for Information Fusion[END_REF]. The rule PCR6 has been proposed by Martin and Osswald in [11] . PCR rules will typically replace the possible conflict by an adaptive averaging of the belief product.

The fused bba m PCR6 obtained from m 1:s by means of PCR6 is defined by: m PCR6 (∅) = 0 , and, for any X ∈ G Θ \ {∅} , by:

m PCR6 (X) = m ∧ (X) + s i=1 m i (X) 2 T s-1 k=1 Y σ i (k) ∩X=∅ Y σ i (1) ,••• ,Y σ i (s-1) ∈G Θ        s-1 j=1 m σi(j) (Y σi (j) ) m i (X) + s-1 j=1 m σi(j) (Y σi(j) )        , (1.5)
where the function σ i counts from 1 to s avoiding i :

σ i (j) = j × I[j < i] + (j + 1) × I[j ≥ i] . N.B. If
• F (X|Y 1:s ; γ) ≥ 0 , • X∈G Θ F (X|Y 1:s ; γ) = 1 , for any X, Y 1:s ∈ G Θ .
A referee function for s sources of information is also called a s-ary referee function. The quantity F (X|Y 1:s ; γ) is called a conditional arbitrament between Y 1:s in favor of X. Notice that X is not necessary one of the propositions Y 1:s ; typically, it could be a combination of them. The case X = ∅ is called the rejection case.

Fusion rule. Let be given s basic belief assignments (bba) m 1:s and a s-ary referee function F with context m 1:s . Then, the fused bba

m 1 ⊕ • • • ⊕ m s [F ]
based on the referee F is constructed as follows:

m 1 ⊕ • • • ⊕ m s [F ](X) = I[X = ∅] Y1:s∈G Θ F (X|Y 1:s ; m 1:s ) s i=1 m i (Y i ) 1 - Y1:s∈G Θ F (∅|Y 1:s ; m 1:s ) s i=1 m i (Y i ) , (1.6)
for any X ∈ G Θ .

From now on, the notation

⊕[m 1:s |F ] = m 1 ⊕ • • • ⊕ m s [F ] is used. The value z = Y1:s∈G Θ F (∅|Y 1:s ; m 1:s ) s i=1 m i (Y i
) is called the rejection rate. Notice that the rejection rate is derived from the rejection generated by F (∅|Y 1:s ; m 1:s ) As a consequence, the rejection is not exclusively a conjunctive notion in this approach. An example of non conjunctive rejection is proposed in section 1.5.

Examples. Refer to section 1.4 and 1.5.

Properties

Bba status. The function ⊕[m 1:s |F ] defined on G Θ is actually a basic belief assignment.

Proof. It is obvious that ⊕[m 1:s |F ] ≥ 0 . Since I[∅ = ∅] = 0 , it is derived ⊕[m 1:s |F ](∅) = 0 . From X∈G Θ F (X|Y 1:s ; m 1:s ) = 1 , it is derived: X∈G Θ Y1:s∈G Θ F (X|Y 1:s ; m 1:s ) s i=1 m i (Y i ) = Y1:s∈G Θ s i=1 m i (Y i ) X∈G Θ F (X|Y 1:s ; m 1:s ) = Y1:s∈G Θ s i=1 m i (Y i ) = s i=1 Yi∈G Θ m i (Y i ) = 1 .
As a consequence:

X∈G Θ I[X = ∅] Y1:s∈G Θ F (X|Y 1:s ; m 1:s ) s i=1 m i (Y i ) + Y1:s∈G Θ F (∅|Y 1:s ; m 1:s ) s i=1 m i (Y i ) = 1 .
Referee function without rejection. Let be given s basic belief assignments (bba) m 1:s and a s-ary referee function F with context m 1:s . Assume that F does not imply rejection, that is:

F (∅|Y 1:s ; m 1:s ) = 0 for any Y 1:s ∈ G Θ \ {∅} .
Then, the fused bba ⊕[m 1:s |F ] based on the referee F has the simplified definition:

⊕ [m 1:s |F ](X) = Y1:s∈G Θ F (X|Y 1:s ; m 1:s ) s i=1 m i (Y i ) , (1.7)
for any X ∈ G Θ .

Proof. It is a consequence of

Y1:s∈G Θ F (∅|Y 1:s ; m 1:s ) s i=1 m i (Y i ) = 0 .
Separability. Let be given s basic belief assignments (bba) m 1:s and a s-ary referee function F with context m 1:s . Assume that there is r and t such that r + t = s, and two sequences u 1:r and v 1:t which constitute a partition of [ [1, s]] , that is:

{u 1:r } ∪ {v 1:t } = [[1, s]] and {u 1:r } ∩ {v 1:t } = ∅ . 10CHAPTER 1.

DEFINITION OF EVIDENCE FUSION RULES BASED ON REFEREE FUNCTIO

Assume also that there are a r-ary referee function G with context m u1:r , a t-ary referee function H with context m v1:t and a parameter θ ∈ [0, 1] such that:

F (X|Y 1:s ; m 1:s ) = θG(X|Y u1:r ; m u1:r ) + (1 -θ)H(X|Y v1:t ; m v1:t ) .
Then F is said to be separable into the two sub-referee functions G and H. Moreover, the fused bba is simplified as follows:

⊕ [m 1:s |F ] = θ ⊕ [m u1:r |G] + (1 -θ) ⊕ [m v1:t |H] (1.8)
Notice that the fusion is easier for small arity. As a consequence, separability provides possible simplifications to the fusion.

Proof. It is derived:

X∈G Θ Y1:s∈G Θ F (X|Y 1:s ; m 1:s ) s i=1 m i (Y i ) = θ X∈G Θ Y1:s∈G Θ G(X|Y u1:r ; m u1:r ) s i=1 m i (Y i ) + (1 -θ) X∈G Θ Y1:s∈G Θ H(X|Y v1:t ; m v1:t ) s i=1 m i (Y i ) .
Now:

X∈G Θ Y1:s∈G Θ G(X|Y u1:r ; m u1:r ) s i=1 m i (Y i ) =   X∈G Θ Yu 1:r ∈G Θ G(X|Y u1:r ; m u1:r ) r i=1 m ui (Y ui )   t i=1 Yv i ∈G Θ m vi (Y vi ) = ⊕[m u1:r |G] t i=1 1 = ⊕[m u1:r |G] .
It is derived similarly:

X∈G Θ Y1:s∈G Θ H(X|Y v1:t ; m v1:t ) s i=1 m i (Y i ) = ⊕[m v1:t |H] .
Of course, the notion of separability extends easily to more than two sub-referee functions.

Sampling process

The definition (1.6) makes apparent a fusion process which is similar to a probabilistic conditional decision on the set of propositions. Notice that the basic belief assignments are not related, in practice, to physical probabilities. But the implied mathematics are similar, as well as some concepts. In particular, the fusion could be interpreted as a two stages process. In a first stage, the sources of information generate independent entries according to the respective beliefs. Then, a final decision is done by the referee function conditionally to the entries. As a result, an output is produced or not.

This interpretation has two profitable consequences. First at all, it provides an intuitive background for constructing new rules: in our framework, a new rule is just the design of a new referee. Secondly, our interpretation makes possible sampling methods in order to approximate and accelerate complex fusion processes. Notice that the sampling method is used here as a mathematical tool for approximating the belief, not for simulating an individual choice. Indeed, evidence approaches deal with belief on propositions, not with individual propositions.

Sampling algorithm. Samples of the fused basic belief assignment ⊕[m 1:s |F ] are generated by iterating the following processes:

Entries generation: For each i ∈ [[1, s]] , generates Y i ∈ G Θ according
to the basic belief assignment m i , considered as a probabilistic distribution over the set G Θ , Conditional arbitrament:

1. Generate X ∈ G Θ according to referee function F (X|Y 1:s ; m 1:s ), considered as a probabilistic distribution over the set G Θ , 2. In the case X = ∅, reject the sample. Otherwise, keep the sample.

The performance of the sampling algorithm is at least dependent of two factors. First at all, a fast implementation of the arbitrament is necessary. Secondly, low rejection rate is better. Notice however that the rejection rate is not a true handicap. Indeed, high rejection rate means that the incident bbas are not compatible in regard to the fusion rule: these bba should not be fused. By the way, the ratio of rejected samples will provide an empirical estimate of the rejection rate of the law.

The case of separable referee function. Assume that F is separable into G and H, i.e. there are θ ∈ [0, 1] and a partition ({u 1:r }, {v 1:t }) of [ [1, s]] such that:

F (X|Y 1:s ; m 1:s ) = θG(X|Y u1:r ; m u1:r ) + (1 -θ)H(X|Y v1:t ; m v1:t ) .
Then, samples of the fused basic belief assignment ⊕[m 1:s |F ] are generated by means of the sub-arbitraments related to G and H:

12CHAPTER 1.

DEFINITION OF EVIDENCE FUSION RULES BASED ON REFEREE FUNCTIO

Choice of a sampling sub-process:

1. Generate a random number x ∈ [0, 1] according to the uniform distribution, 2. If x < θ then jump to Sampling process related to G, 3. Otherwise jump to Sampling process related to H, Sampling process related to G:

Entries generation: For each i ∈ [[1, r]] , generates Y ui ∈ G Θ ac-
cording to the basic belief assignment m ui , Conditional arbitrament:

1. Generate X ∈ G Θ according to G(X|Y u1:r ; m u1:r ) , 2. In the case X = ∅, reject the sample. Otherwise, return the sample.

Sampling process related to H:

Entries generation: For each i ∈ [[1, t]] , generates Y vi ∈ G Θ ac- cording to the basic belief assignment m vi , Conditional arbitrament: 1. Generate X ∈ G Θ according to H(X|Y v1:t ; m v1:t ) , 2.
In the case X = ∅, reject the sample. Otherwise, return the sample.

This result is a direct consequence of the property of separability (1.8). This algorithm will spare the sampling of useless entries. Therefore, it is more efficient to implement the separability when it is possible.

Algorithmic definition of fusion rules

As seen previously, fusion rules based on referee functions are easily approximated by means of sampling process. This sampling process is double-staged.

The first stage computes the samples related to the entry bbas m 1:s . The second stage computes the fused samples by a conditional arbitrament between the different hypotheses. This arbitrament is formalized by a referee function.

In practice, it is noteworthy that there is no need for a mathematical definition of the referee function. The only important point is to be able to compute the arbitrament. We have here a new approach for defining fusion rules of evidences. Fusion rules may be defined entirely by the means of an algorithm for computing the conditional arbitrament.

Assertion. There are three equivalent approaches for defining fusion rules in the paradigm of referee:

• By defining a formula which maps the entry bbas m 1:s to the fused bba

m 1 ⊕ • • • ⊕ m s (classical approach),
• By defining a referee function F , which makes the conditional arbitrament F (X|Y 1:s ; m 1:s ) ,

• By constructing an algorithm which actually makes the conditional arbitrament between Y 1:s in favor of X .

It is sometime much easier and more powerful to just construct the algorithm for conditional arbitrament.

The following section illustrates the afore theoretical discussion on well known examples.

Example of referee functions

Let be given s sources of information characterized by their bbas m 1:s .

Dempster-shafer rule

The fused bba m DST obtained from m 1:s by means of Dempster-Shafer fusion rule is defined by equation (1.1). It has an immediate interpretation by means of referee functions.

Definition by referee function. The definition of a referee function for Dempster-Shafer is immediate:

m DST = ⊕[m 1:s |F ∧ ] where F ∧ (X|Y 1:s ; m 1:s ) = I X = s k=1 Y k .
Algorithmic definition. The algorithmic implementation of F ∧ is described subsequently and typically implies possible conditional rejections:

Conditional arbitrament:

1. Set X = s k=1 Y k ,
2. If X = ∅ , then reject the sample. Otherwise, keep the sample.

Disjunctive rule

The fused bba m ∨ obtained from m 1:s by means of the disjunctive rule is defined by equation (1.3). It has an immediate interpretation by means of referee functions.

14CHAPTER 1. DEFINITION OF EVIDENCE FUSION RULES BASED ON REFEREE FUNCTIO

Definition by referee function. The definition of a referee function for the disjunctive rule is immediate from (1.3):

m ∨ = ⊕[m 1:s |F ∨ ]
where:

F ∨ (X|Y 1:s ; m 1:s ) = I X = s k=1 Y k .
Algorithmic definition. The algorithmic implementation of F ∨ is described subsequently. It does not imply rejections:

Conditional arbitrament:

1. Set X = s k=1 Y k .

Dubois & Prade rule

The fused bba m D&P obtained from

m D&P = ⊕[m 1:s |F D&P ]
where:

F D&P (X|Y 1:s ; m 1:s ) = I s k=1 Y k = ∅ F ∧ (X|Y 1:s ; m 1:s ) + I s k=1 Y k = ∅ F ∨ (X|Y 1:s ; m 1:s ) = I X = s k=1 Y k = ∅ + I s k=1 Y k = ∅ I X = s k=1 Y k .
The first formulation of F D&P is particularly interesting, since it illustrates how to construct a referee function by means of a conditional branching to already existing referee functions. In the case of Dubois & Prade rule, the rule has a disjunctive behavior when there is a conjunctive conflict, i.e. s k=1 Y k = ∅, and a conjunctive behavior otherwise. Thus, the referee function is obtained as the summation of the exclusive sub-arbitraments:

I s k=1 Y k = ∅ F ∨ (X|Y 1:s ; m 1:s ) (disjunctive case) and I s k=1 Y k = ∅ F ∧ (X|Y 1:s ; m 1:s ) (conjunctive case).
Algorithmic definition. The algorithmic implementation of F D&P is described subsequently. It does not imply rejections:

Conditional arbitrament:

1. Set X = s k=1 Y k , 2. If X = ∅ , then set X = s k=1 Y k .

Averaging rule

Let be given the averaging parameters α 1:s ≥ 0 such that s i=1 α i = 1 . The averaged bba m µ [α] = s i=1 α i m i could be obtained by means of a refereee function.

Averaging by referee function. The definition of a referee function for averaging is immediate:

m µ [α] = ⊕[m 1:s |F µ [α]]
where:

F µ [α](X|Y 1:s ; m 1:s ) = s i=1 α i I [X = Y i ] .
Proof by applying the separability. It is noticed that F is separable:

F µ [α](X|Y 1:s ; m 1:s ) = s i=1 α i id i (X|Y i , m i ) ,
where the referee function id i is defined by:

id i (X|Y i , m i ) = I[X = Y i ] for any X, Y i ∈ G Θ
By applying the separability property (1.8), s -1 times, it comes:

⊕[m 1:s |F µ [α]] = s i=1 α i ⊕ [m i |id i ] .
It happens that:

⊕[m i |id i ] = m i ,
so that:

⊕[m 1:s |F µ [α]] = s i=1 α i m i = m µ [α] . 16CHAPTER 1.

DEFINITION OF EVIDENCE FUSION RULES BASED ON REFEREE FUNCTIO

Algorithmic implementation of averaging. It is interesting here to implement the separability of the referee function. Therefore, the entire sampling algorithm is derived from the separability property:

Choice of a sampling sub-process:

1. Generate a random integer i ∈ [ [1, s]] according to probability α , 2. Jump to Sampling process related to id i , Sampling process related to id i :

Entries generation: Generates Y i ∈ G Θ according to the basic belief assignment m i , Conditional arbitrament: Return X = Y i .

PCR6 rule

The fused bba m PCR6 obtained from m 1:s by means of the PCR6 rule is defined by equation (1.5). It has an interpretation by means of referee functions.

Definition by referee function. Definition (1.5) could be reformulated into:

m PCR6 (X) = m ∧ (X) + s i=1 T s k=1 Y k =∅ Y1,••• ,Ys∈G Θ       I[X = Y i ] m i (Y i ) s j=1 m j (Y j ) s j=1 m j (Y j )      
, and then:

m PCR6 (X) = m ∧ (X) + T s k=1 Y k =∅ Y1,••• ,Ys∈G Θ s i=1 m i (Y i ) s j=1 I[X = Y j ] m j (Y j ) s j=1 m j (Y j ) . (1.9)
At last, it is derived a formulation of PCR6 by means of a referee function:

m PCR6 = ⊕[m 1:s |F PCR6 ] ,
where the referee function F PCR6 is defined by:

F PCR6 (X|Y 1:s ; m 1:s ) = I X = s k=1 Y k = ∅ + I s k=1 Y k = ∅ s j=1 I[X = Y j ] m j (Y j ) s j=1 m j (Y j )
. (1.10) Algorithmic definition. Again, the algorithmic implementation is immediate:

Conditional arbitrament:

1. If s k=1 Y k = ∅ , then set X = s k=1 Y k 2. Otherwise:
(a) Define the probability P over [ [1, s]] by:

P i = m i (Y i ) s j=1 m j (Y j ) for any i ∈ [[1, s]] , (b) Generate a random integer k ∈ [[1, s]] according to P , (c) set X = Y k .
It is noticed that this process does not produce any rejection case X = ∅. As a consequence, the last rejection step has been removed.

Essentially, this algorithm distinguishes two cases:

• there is a consensus; then, answer the consensus,

• there is not a consensus; then choose an entry among all entries proportionally to its belief. It is noteworthy that there is no attempt to transform the entries in this case.

This algorithm is efficient and is not time-consuming. The whole sampling approach should be a good alternative for approximating PCR6, particularly on large frames of discernment.

Non conjunctive rejection

As seen in section 1.3, the rejection, resulting from a fusion based on referee functions, is not necessary a conjunctive conflict: this rejection is the result of the arbitrament rejections F (∅|Y 1:s ; m 1:s ) . An example of rule with disjunctive rejection is proposed now. This example is rather unnatural; is is only constructed for illustration. The context of fusion is the following:

Case a: Entries which union is Ω (i.e. is totally imprecise) are rejected; the idea here, is to reject entries which are insufficiently focused, Case b: An averaging of the bba is done, otherwise.

The referee function is thus obtained by means of a conditional branching to the rejection (case a) or to the averaging (case b). As a consequence, the referee function is obtained as the summation of the following exclusive subarbitraments:

18CHAPTER

DEFINITION OF EVIDENCE FUSION RULES BASED ON REFEREE FUNCTIO

Case a:

I s i=1 Y i = Ω I [X = ∅] ,
Case b:

I s i=1 Y i = Ω s i=1 I [X = Y i ] s .
The referee function for this rule is:

F (X|Y 1:s ; m 1:s ) = I s i=1 Y i = Ω I [X = ∅] + I s i=1 Y i = Ω s i=1 I [X = Y i ] s .
Let us apply this rule to the example of s = 2 bbas m 1 and m 2 defined on 2 {a,b,c} by m 1 ({a}) = 0.1 , m 1 ({a, b}) = 0.9 , m 2 ({b}) = 0.2 and m 2 ({b, c}) = 0.8 . Then, the fused bba m = m 1 ⊕ m 2 [F ] is obtained as:

z = m 1 ({a})m 2 ({b, c}) + m 1 ({a, b})m 2 ({b, c}) = 0.8 , m({a}) = m1({a})m2({b}) 2(1-z) = 0.05 , m({a, b}) = m1({a,b})m2({b}) 2(1-z) = 0.45 , m({b}) = m1({a})m2({b})+m1({a,b})m2({b}) 2(1-z) = 0.5 .
In this example, the rejection rate z is not conjunctive, since it involves {a, b} and {b, c} such that {a, b} ∩ {b, c} = ∅.

Any rule?

Is it possible to construct a referee function for any existing fusion rule?

Actually, the answer to this question is ambiguous. If it is authorized that F depends on m 1:s without restriction, then the theoretical answer is trivially yes.

Property. Let be given the fusion rule m 1 ⊕ • • • ⊕ m s , applying on the bbas m 1:s . Define the referee function F by:

F (X|Y 1:s ; m 1:s ) = m 1 ⊕ • • • ⊕ m s (X) for any X, Y 1:s ∈ G Θ .
Then F is actually a referee function and

⊕[m 1:s |F ] = m 1 ⊕ • • • ⊕ m s .

Proof is immediate.

Of course, this result is useless in practice, since such referee function is inefficient. It is inefficient because it does not provide an intuitive interpretation of the rule, and is as difficult to compute as the fusion rule. Then, it is useless to have a sampling approach with such definition.

As a conclusion, referee functions have to be considered together with their efficiency. The efficiency of referee function is not a topic which is studied in this chapter.

1.5 A new rule: PCR♯

Definition. For any k ∈ [[1, s]],
it is defined:

C[k|s] = {γ ⊂ [[1, s]] /card(γ) = k } , the set of k-combinations of [[1, s]] . Of course, the cardinal of C[k|s] is s k .
For convenience, the undefined object C[s + 1|s] is actually defined by:

C[s + 1|s] = {∅} ,
so as to ensure:

min γ∈C[s+1|s]    I   i∈γ Y i = ∅      = 1 1.5.

Limitations of PCR6

The algorithmic interpretation of PCR6 has shown that PCR6 distinguishes two cases:

• The entry information are compatible; then, the conjunctive consensus is decided,

• The entry information are not compatible; then, a mean decision is decided, weighted by the relative beliefs of the entries.

In other words, PCR6 only considers consensus or no-consensus cases. But for more than 2 sources, there are many cases of intermediate consensus. By construction, PCR6 is not capable to manage intermediate consensus. This is a notable limitation of PCR6.

The new rule PCR♯, which is defined now, extends PCR6 by considering partial consensus in addition to full consensus and absence of consensus. This rule is constructed by specifying the arbitrament algorithm. Then, a referee function is deduced.

Algorithm

The following algorithm tries to reach a maximal consensus. It first tries the full consensus, then consensus of s -1 sources, s -2 sources, and so on, until a consensus is finally found. When several consensus with k sources is possible, the final answer is chosen randomly, proportionally to the beliefs of the consensus. In the following algorithm, comments are included preceded by // (c++ convention).

Conditional arbitrament:

1. Set stop = false and k = s , // k is the size of the consensus, which are searched. At beginning, it is maximal.

P γ = ω γ γ∈C[k|s] ω γ , 5. At last, set X = i∈γ Y i .

Referee function

Historically, PCR♯ has been defined by means of an algorithm, not by means of a formal definition of the referee function. It is however possible to give a formal definition of the referee function which is equivalent to the algorithm:

FPCR♯(X|Y1:s; m1:s) = s X k=1 min γ∈C[k+1|s] ( I " \ i∈γ Yi = ∅ #) × min 8 > > > > > < > > > > > : max γ∈C[k|s] ( I " \ i∈γ Yi = ∅ #) , X γ∈C[k|s] I " X = \ i∈γ Yi = ∅ # Y i∈γ mi(Yi) X γ∈C[k|s] I " \ i∈γ Yi = ∅ # Y i∈γ mi(Yi) 9 > > > > > = > > > > > ;
.

(1.11) Sketch of the proof. The following correspondences are established between the arbitrament algorithm and the referee function:

• The summation s k=1 is a formalization of the loop from k = s down to k = 1 , • At step k, the component: min γ∈C[k+1|s]    I   i∈γ Y i = ∅     
ensures that there is not a functional consensus of larger size j > k .

Typically, the component is 0 if a larger sized functional consensus exists, and 1 otherwise. This component is complementary to the summation, as it formalizes the end of the loop, when a functional consensus is actually found,

• At step k, the component:

Ω = γ∈C[k|s] I   X = i∈γ Y i = ∅   i∈γ m i (Y i ) γ∈C[k|s] I   i∈γ Y i = ∅   i∈γ m i (Y i )
encodes the choice of a functional consensus of size k, proportionally to its belief. The chosen consensus results in the production of the sample X ,

• At step k, the component:

max γ∈C[k|s]    I   i∈γ Y i = ∅      22CHAPTER 1.

DEFINITION OF EVIDENCE FUSION RULES BASED ON REFEREE FUNCTIO

tests if there is a functional consensus of size k. The component answers 1 if such consensus exists, and 0 otherwise. It is combined with a minimization of the form:

min    max γ∈C[k|s]    I   i∈γ Y i = ∅      , Ω    , where Ω ≤ 1 .
This is some kind of "if ... then" : if a functional consensus of size k exists, then the value Ω is computed. Otherwise, it is the value 0 . Since the value Ω encodes a sampling decision, we have here sampling decision, which is conditioned by the fact that a functional consensus exists.

The equivalence is a consequence of these correspondences.

Variants of PCR♯

Actually, card(C[k|s]) = s k increases quickly when s is great and k is not near 1 or s . As a consequence, PCR♯ implies hard combinatorics, when used in its general form. On the other hand, it may be interesting to reject samples, when a consensus is not possible with a minimal quorum. In order to address such problems, a slight extension of PCR♯ is proposed now.

Algorithm.

Let r ∈ [ [1, s]] and let k 1:r ∈ [ [1, s]] be a decreasing sequence such that:

s ≥ k 1 > • • • > k r ≥ 1 .
For convenience, the undefined object k 0 is actually defined by:

k 0 = s + 1 ,
so as to ensure:

min γ∈C[k0|s]    I   i∈γ Y i = ∅      = 1
Then, the rule PCR♯[k 1:r ] is defined by the following algorithm.

Conditional arbitrament:

1. Set stop = false and h = 1 , 2. For each γ ∈ C[k h |s] , do: (a) If i∈γ Y i = ∅ , then set ω γ = i∈γ m i (Y i )
P γ = ω γ γ∈C[k h |s] ω γ , 6. Set X = i∈γ Y i . and end.
Referee function

FPCR♯[k 1:r ] (X|Y1:s; m1:s) = I[X = ∅] min γ∈C[kr |s] ( I " \ i∈γ Yi = ∅ #) + r X h=1 min γ∈C[k h-1 |s] ( I " \ i∈γ Yi = ∅ #) × min 8 > > > > > < > > > > > : max γ∈C[k h |s] ( I " \ i∈γ Yi = ∅ #) , X γ∈C[k h |s] I " X = \ i∈γ Yi = ∅ # Y i∈γ mi(Yi) X γ∈C[k h |s] I " \ i∈γ Yi = ∅ # Y i∈γ mi(Yi) 9 > > > > > = > > > > > ;
.

proof is left to the reader.

PCR6 and PCR♯. Assume that PCR6 is applied to s entries m 1:s . Then:

PCR6=PCR♯[s, 1]
DST and PCR♯. Assume that DST is applied to s entries m 1:s . Then:

DST=PCR♯[s]
Variant with truncation and rejection. Let r ∈ [ [1, s]] . The rule PCR♯[s, s-1, • • • , r] will search for maximally sized functional consensus. If it is not possible to find functional consensus with size greater or equal to r , the algorithm rejects the entries.

Variant with truncation and final mean decision. Let r ∈ [ [1, s]] . The rule PCR♯[s, s-1, • • • , r, 1] will search for maximally sized functional consensus. If it is not possible to find functional consensus with size greater or equal to r , the algorithm choose an entry proportionally to its belief.

Numerical examples

It is assumed: 3. For any X ∈ G Θ , compute m(X), the estimate of m(X), by:

G Θ = {∅,
m(X) = 1 N (1 -z) N n=1 I[X n = X] .
It is known that the accuracy of this estimate is of the magnitude of 1 √ N . Typically, when the rejection rate is zero, i.e. z = 0, the variance σ( m(X)) is given by: σ(m(X)) = m(X) • (1m(X)) N .

Convergence

) If Y 1 ∩ Y 2 = ∅ , then set X n = rejected , (c) Otherwise, set X n = Y 1 ∩ Y 2 , 2. Set z = 1 N N n=1 I[X n = rejected] ,
3. For any X ∈ G Θ , compute m(X) by:

m(X) = 1 N (1 -z) N n=1 I[X n = X] .
The following table compares

) If Y 1 ∩ Y 2 = ∅ , then set X n = Y 1 ∩ Y 2 , (c) Otherwise, do: i. Compute θ = m1(Y1) m1(Y1)+m2(Y2) , ii. Generate a random number x uniformly distributed on [0, 1], iii. If x < θ, set X n = Y 1 ; otherwise, set X n = Y 2 , 2. For any X ∈ G Θ , compute m(X) by: m(X) = 1 N N n=1 I[X n = X] .
The following

m 1 ({a, b}) = m 2 ({a, c}) = m 3 ({c}) = 1 .
The bbas m 1 and m 3 are incompatible. However, m 2 is compatible with both m 1 and m 3 , which implies that a partial consensus is possible between m 1 and m 2 or between m 2 and m 3 . As a consequence, PCR♯ should provide better answers by allowing partial combinations of the bbas. The fusion of the 3 bbas are computed respectively by means of DST, PCR6 and PCR♯ , and the results confirm the intuition:

• zDST = 1 and mDST is undefined,

• mPCR6({a, b}) = mPCR6({a, c}) = mPCR6({c}) = 1 3 , • mPCR♯({a}) = mPCR♯({c}) = 1
2 derived from the consensus {a, b} ∩ {a, c} , {a, c} ∩ {c} and their beliefs m1({a, b})m2({a, c}) , m2({a, c})m3({c}).

Example 5. It is assumed 3 bbas m 1:3 on G Θ defined by:

m 1 ({a}) = 0.6 , m 1 ({a, b}) = 0.4 , m 2 ({a}) = 0.3 , m 2 ({a, c}) = 0.7 , m 3 ({b}) = 0.8 , m 3 ({a, b, c}) = 0.2 .
The computation of PCR♯ is done step by step: Full consensus. Full functional consensus are: Notice that there is never a 2-sized consensus involving the pair (Y 2 , Y 3 ) . As a consequence, the belief ratios for the partial consensus, i.e.: DST produces highly conflicting results, since source 3 conflicts with the other sources. However, there are some partial consensus which allow the answer {b} . DST is blind to these partial consensus. On the other hand, PCR6 is able to handle hypothesis {b} , but is too much optimistic and, still, is unable to fuse partial consensus. Consequently, PCR6 is also unable to diagnose the high inconstancy of belief m 3 ({b}) = 0.8 .

Y 1 {a, b} {a, b} {a} {a} Y 2 {a, c} {a} {a, c} {a} Y 3 {a, b, c}{a, b, c}{a, b, c}{a, b, c} i Y i {a} {a} {a} {a} i m i (Y i ) 0.
ω γ = I   i∈γ Y i = ∅   i∈γ m i (Y i ) γ ′ ∈C[2|3] I   i∈γ ′ Y i = ∅   i∈γ ′ m i (Y i ) for γ ∈ C[2|3] , 28CHAPTER

Conclusion

This chapter has investigated a new framework for the definition and interpretation of fusion rule of evidences. This framework is based on the new concept of referee function. A referee function models an arbitrament process conditionally to the contributions of several independent sources of information. It has been shown that fusion rules based on the concept of referee functions have a straightforward sampling-based implementation. As a consequence, a referee function has a natural algorithmic interpretation. Owing to the algorithmic nature of referee functions, the conception of new rules of fusion is made easier and intuitive. Examples of existing fusion rules have been implemented by means of referee functions. Moreover, an example of rule construction has been provided on the basis of an arbitrament algorithm. The new rule is a quite general extension of both PCR6 and Dempster-Shafer rule. This chapter also addresses the issue of fusion rule approximation. There are cases for which the fusion computation is prohibitive. The sampling process implied by the referee function provides a natural method for the approximation and the computation speed-up. There are still many questions and improvements to be addressed. For example, samples regularization techniques may reduce possible samples degeneracy thus allowing smaller particles clouds. Some theoretical questions are 30CHAPTER 1. DEFINITION OF EVIDENCE FUSION RULES BASED ON REFEREE FUNCTIO also pending; especially, the algebraic properties of the referee functions have almost not been studied. However, this preliminary work is certainly promising for future applications.

  m 1:s by means of the rule of Dubois & Prade is defined by equation (1.4). It has an interpretation by means of referee functions. Definition by referee function. The definition of a referee function for Dubois & Prade rule is deduced from (1.4):

2 . 5 . 3 . 4 .

 2534 For each γ ∈ C[k|s] , do: // All possible consensus of size k are tested. (a) If i∈γ Y i = ∅ , then set ω γ = i∈γ m i (Y i ) and stop = true , // If a consensus of size k is found to be functional, then it is no more necessary to diminish the size of the consensus. This is done by changing the value of Boolean stop. // Moreover, the functional consensus are weighted by their beliefs. (b) Otherwise set ω γ = 0 , // Non-functional consensus are weighted zero. 3. If stop = false , then set k = k -1 and go back to 2 , // If no functional consensus of size k has been found, then it is necessary to test smaller sized consensus. The process is thus repeated for size k -1. 4. Choose γ ∈ C[k|s] randomly, according to the probability: P γ = ω γ γ∈C[k|s] ω γ , // Otherwise, choose a functional consensus. Here, the decision is random and proportional to the consensus belief. At last, set X = i∈γ Y i . // Publish the sample related to the consensus. Algorithm without comment. Conditional arbitrament: 1. Set stop = false and k = s , 2. For each γ ∈ C[k|s] , do: (a) If i∈γ Y i = ∅ , then set ω γ = i∈γ m i (Y i ) and stop = true , (b) Otherwise set ω γ = 0 , If stop = false , then set k = k -1 and go back to 2 , Choose γ ∈ C[k|s] randomly, according to the probability:

4 .

 4 and stop = true , (b) Otherwise set ω γ = 0 , 3. If stop = false , then: (a) set h = h + 1 , (b) If h ≤ r, go back to 2 , If h > r, then reject the entries and end, 5. Otherwise, choose γ ∈ C[k h |s] randomly, according to the probability:

  {a}, {b}, {c}, {b, c}, {c, a}, {a, b}, {a, b, c}} . Various examples of bbas, m 1:s , are considered on G Θ and fused by means of rules based on different referee functions. The fused rule, m = ⊕[m 1:s |F ], is computed both mathematically or by means of the sampler. When the fusion is obtained by sampling a particle cloud, the fused bba estimate is deduced from an empirical averaging. The complete process arises as follows: 1. Repeat from n = 1 to n = N : (a) Generate the particle X n ∈ G Θ by sampling ⊕[m 1:s |F ] , (b) If the sampling process failed, then set X n = rejected , 2. Compute z, the estimate of z, by setting z = 1 N N n=1 I[X n = rejected] ,

Example 1 . 1 . 1 .

 111 The bbas m 1 and m 2 are defined by:m 1 ({a, b}) = 0.2 , m 1 ({a, c}) = 0.4 , m 1 ({b, c}) = 0.3 , m 1 ({a, b, c}) = 0.1 , m 2 ({a, b}) = 0.4 , m 2 ({a, c}) = 0.2 , m 2 ({b, c}) = 0.3 , m 2 ({a, b, c}) = 0.These bbas are fused by means of DST, resulting in m = m DST : z = 0, m({a}) = 0.2, m({b}) = 0.18, m({a, b}) = 0.14, m({c}) = 0.18, m({a, c}) = 0.14, m({b, c}) = 0.15, m({a, b, c}) = 0.01. The estimate m of m is obtained by the following process: Repeat from n = 1 to =n = N : (a) Generate Y 1 and Y 2 by means of m 1 and m 2 respectively, (b

Example 2 . 1 .

 21 This is an example with reject. The bbas m 1 and m 2 are defined by: m 1 ({a}) = 0.4 , m 1 ({a, b}) = 0.5 , m 1 ({a, b, c}) = 0.1 , m 2 ({c}) = 0.4 , m 2 ({b, c}) = 0.5 , m 2 ({a, b, c}) = 0.These bbas are fused by means of DST, resulting in m = m DST : z = 0.56, m({a}) = 0.091, m({b}) = 0.568, m({a, b}) = 0.114, m({c}) = 0.091, m({b, c}) = 0.114, m({a, b, c}) = 0.022.

Example 3 . 4 .

 34 The bbas m 1 and m 2 are defined by:m 1 ({a}) = 0.5 , m 1 ({a, b}) = 0.1 , m 1 ({a, b, c}) = 0.4 , m 2 ({c}) = 0.3 , m 2 ({a, c}) = 0.3 , m 2 ({a, b, c}) = 0.These bbas are fused by means of PCR6, resulting in m = m PCR6 : m({a}) = 0.385, m({b}) = 0.04, m({a, b}) = 0.007, m({c}) = 0.199, m({a, c}) = 0.12, m({b, c}) = 0.249. It is noticed that z = 0 in this case of PCR6. Then, the estimate m is obtained by the following process: 1. Repeat from n = 1 to =n = N : (a) Generate Y 1 and Y 2 by means of m 1 and m 2 respectively, (b

3 : 1 .

 31 1. DEFINITION OF EVIDENCE FUSION RULES BASED ON REFEREE FUNCTIOare simplified as follows: )m2(Y2)+m3(Y3)m1(Y1) = m3(Y3) m2(Y2)+m3(Y3) ,The case γ = {2, 3} does not hold. 1-sized consensus. There is no remaining 1-sized consensus. Belief compilation. The different cases resulted in only two propositions, i.e. {a} and {b}. By combining the entry beliefs i m i (Y i ) and ratio beliefs, the fused bba m = m PCR♯ is then deduced: m({a}) = 0.056 + 0.024 + 0.084 + 0.036 + 0.467 × 0.224 + 0.273 × 0.096 + 1 × 0.336 + 1 × 0.144 = 0.811 , m({b}) = 0.533 × 0.224 + 0.727 × 0.096 = 0.189 .As a conclusion:m PCR♯ ({a}) = 0.811 and m PCR♯ ({b}) = 0.189 .It is noticed that z = 0 for this general case of PCR♯. Then, the estimate m is obtained by the following process, working for any choice of m 1:Repeat from n = 1 to =n = N :(a) Generate Y 1 , Y 2 and Y 3 by means of m 1 , m 2 and m 3 respectively,(b) If Y 1 ∩ Y 2 ∩ Y 3 = ∅ , then set X n = Y 1 ∩ Y 2 ∩ Y 3 and return, (c) If (Y 1 ∩ Y 2 ) ∪ (Y 1 ∩ Y 3 ) ∪ (Y 2 ∩ Y 3 ) = ∅ ,then do: i. For any γ ∈ C[2|3] = {1, 2}, {1, 3}, {2, 3} , do: A. If i∈γ Y i = ∅, then set ω g amma = 0 , B. Otherwise, set ω g amma = i∈γ m i (Y i ) , ii. For any γ ∈ C[2|3], set ω g amma = ωg amma Pγ ′ ∈C[2|3] ω gamma ′ , iii. Choose γ ∈ C[2|3] randomly accordingly to the probability ω , iv. Set X n = i∈γ Y i v. return, (d) Otherwise, do: i. Compute ω i = m i (Y i ) m 1 (Y 1 ) + m 2 (Y 2 ) + m 3 (Y 3 ) ,ii. Choose k ∈ {1, 2, 3} randomly accordingly to the probability ω , iii. Set X n = Y k , iv. return, 2. For any X ∈ G Θ , compute m(X) by: n = X] .The following table compares the empirical estimates of m, computed by means of particle clouds of different sizes N : ) 1 0.77 0.795 0.812 0.812 0.811 0.811 0.811 0.811 m({b}) 0 0.23 0.205 0.188 0.188 0.189 0.189 0.189 0.189These results could be compared to DST and PCR6:• zDST = 0.8 and mDST({a}) = 1,• mPCR6({a}) = 0.391 , mPCR6({b}) = 0.341 , mPCR6({a, b}) = 0.073 , mPCR6({a, c}) = 0.195 ,

  the empirical estimates of m, computed by means of particle clouds of different sizes N : For this choice of m 1 and m 2 , there is no conflict. The theoretical accuracy defined previously thus applies. The results are compliant with the theoretical accuracy.

	log 10 N	1	2	3	4	5	6	7	8	∞
	z	0	0	0	0	0	0	0	0	0
	m({a})	0.2 0.18 0.202 0.201 0.200 0.200 0.200 0.200 0.2
	m({b})	0.1 0.19 0.173 0.182 0.181 0.180 0.180 0.180 0.18
	m({a, b})	0.3 0.14 0.139 0.138 0.139 0.140 0.140 0.140 0.14
	m({c})	0.1 0.15 0.179 0.177 0.181 0.180 0.180 0.180 0.18
	m({a, c})	0.3 0.17 0.141 0.136 0.140 0.140 0.140 0.140 0.14
	m({b, c})	0	0.15 0.153 0.155 0.149 0.150 0.150 0.150 0.15
	m({a, b, c}) 0	0.02 0.013 0.011 0.010 0.010 0.010 0.010 0.01

  The estimate m of m is obtained by the same process as for example 1. The following table compares the empirical estimates of m, computed by means of particle clouds of different sizes N :Notice that the theoretical accuracy should be corrected, since z > 0 .

		26CHAPTER 1. DEFINITION OF EVIDENCE FUSION RULES BASED ON REFEREE FUNCTIO
	log 10 N	1	2	3	4	5	6	7	8	∞
	z	0.7	0.6	0.56 0.558 0.560 0.559 0.560 0.560 0.56
	m({a})	0.667 0.15 0.109 0.088 0.092 0.090 0.091 0.091 0.091
	m({b})	0	0.375 0.573 0.565 0.566 0.569 0.568 0.568 0.568
	m({a, b})	0	0.125 0.107 0.114 0.114 0.114 0.114 0.114 0.114
	m({c})	0.333 0.225 0.091 0.091 0.091 0.091 0.091 0.091 0.091
	m({b, c})	0	0.125 0.107 0.125 0.115 0.114 0.114 0.114 0.114
	m({a, b, c}) 0	0	0.013 0.017 0.022 0.022 0.022 0.022 0.022

  table compares the empirical estimates of m, computed by means of particle clouds of different sizes N : It is assumed 3 bbas m 1:3 on G Θ defined by:

	log 10 N	1	2	3	4	5	6	7	8	∞
	m({a})	0.7 0.41 0.39 0.388 0.382 0.384 0.385 0.385 0.385
	m({b})	0.1 0.08 0.045 0.041 0.040 0.040 0.040 0.040 0.040
	m({a, b}) 0	0.01 0.008 0.008 0.008 0.008 0.007 0.007 0.007
	m({c})	0.2 0.13 0.19 0.198 0.200 0.199 0.199 0.199 0.199
	m({a, c}) 0	0.12 0.108 0.121 0.121 0.120 0.120 0.120 0.12
	m({b, c}) 0	0.25 0.259 0.244 0.249 0.249 0.249 0.249 0.249
	1.6.2 Comparative tests
	Example 4.

  056 0.024 0.084 0.036 Partial consensus sized 2. Then the possible partial consensus are:

	Y 1	{a, b} {a, b} {a}	{a}
	Y 2	{a, c} {a} {a, c} {a}
	Y 3	{b}	{b}	{b}	{b}
	Y 1 ∩ Y 2	{a}	{a}	{a}	{a}
	Y 2 ∩ Y 3	∅	∅	∅	∅
	Y 3 ∩ Y 1	{b}	{b}	∅	∅
	m2(Y2) m2(Y2)+m3(Y3)	0.467 0.273	1	1
	m3(Y3) m2(Y2)+m3(Y3)	0.533 0.727	0	0
	i m i (Y i )	0.224 0.096 0.336 0.144