In this paper we show that the so-called array Fr\'echet problem in
Probability/Statistics is (max, +)-linear. The upper bound of Fr\'echet is
obtained using simple arguments from residuation theory and lattice
distributivity. The lower bound is obtained as a loop invariant of a greedy
algorithm. The algorithm is based on the max-plus linearity of the Fr\'echet
problem and the Monge property of bivariate distribution

Truffet, Laurent

English

arXiv

In this paper we show that the so-called array Fréchet problem in Probability/Statistics is (max, +)-linear. The upper bound of Fréchet is obtained using simple arguments from residuation theory and lattice distributivity. The lower bound is obtained as a loop invariant of a greedy algorithm. The algorithm is based on the max-plus linearity of the Fréchet problem and the Monge property of bivariate distribution

HAL Mines Nantes

The Fre´chet Contingency Array Problem is Max-PlusLinearLaurent TruffetTo cite this version:Laurent Truffet. The Fre´chet Contingency Array Problem is Max-Plus Linear. 2009. <hal-00375428>HAL Id: hal-00375428https://hal.archives-ouvertes.fr/hal-00375428Submitted on 14 Apr 2009HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.The Fre´chet Contingency Array Problem isMax-Plus LinearL. TRUFFETEcole des Mines de Nantes4, rue A. Kastler, La ChantrerieBP 20722, Nantes 44307 Cedex 3 France.e-mail: Laurent.Truffet@emn.frApril 14, 2009AbstractIn this paper we show that the so-called array Fre´chet problem inProbability/Statistics is (max,+)-linear. The upper bound of Fre´chetis obtained using simple arguments from residuation theory and latticedistributivity. The lower bound is obtained as a loop invariant of agreedy algorithm. The algorithm is based on the max-plus linearity ofthe Fre´chet problem and the Monge property of bivariate distribution.Keywords: Max-plus algebra, Fre´chet bounds.1 IntroductionAs a preliminary remark the author would like this paper to be a modesttribute to the work of Maurice Fre´chet in Statistics. The work has beenstarted at the occasion of his 130th birthday and the 100th anniversary of hisstay in Nantes (the town the author is living in) as professor in Mathematics.In this paper it is shown that the tropical or max-plus semiring Rmax (i.e.the set R of real numbers with max as addition and + as multiplication, seethe precise definition in subsection 2.2) is the underlying algebraic structurewhich is well suited to the Fre´chet contingency (or correlation) array problem[4]. In other words the Fre´chet problem is a tropical problem which thushas its place in the new trends of idempotent mathematics founded by V. P.Maslov and its collaborators in the 1980s (see e.g.[7] and references therein).From this main result the Fre´chet upper bound is derived by residuationand the distributivity property of Rmax as a lattice. The Fre´chet lower bound1is obtained as a loop invariant of a greedy algorithm. This algorithm is basedon the tropical nature of the Fre´chet problem and the Monge property ofbivariate distribution.1.1 Organization of the paperThe paper is written to be sef-contained. Thus, in Section 2 we introducemain notations used in the paper, we restate the Fre´chet array problem andits bounds. We define the tropical semiring Rmax and recall basic resultson residuation theory and lattices. In Section 3 we prove the main result ofthe paper that is the Fre´chet array problem is max-plus linear in the spaceof cumulative distribution functions (see Theorem 3.1). From this result inSection 4 we derive the Fre´chet bounds using new approaches. The upperbound is derived from residuation theory and the lattice distributivity prop-erty of the max-plus semiring Rmax (see Corollary 4.1). The lower bound isobtained as the loop invariant of a greedy algorithm (see Proposition 4.1).In Section 5 we conclude this work.2 PreliminariesIn this Section we recall basic results concerning Fre´chet array problem andthe max-plus semi-ring Rmax.2.1 The Fre´chet contingency array problem and its solutionThis problem is described in e.g. [4]. Let n be an integer ≥ 1. The setMatnm(R+) denotes the set of n×m matrices which entries are nonnegativereal numbers. We define the partial orderD on Matnn(R+) as follows:A = [ai,j]D B = [bi,j]def⇔ ∀i, j,i∑l=1j∑k=1al,k ≤i∑l=1j∑k=1bl,k. (1)Introducing the fundamental n× n matrix:Ddef= [1{i≤j}], (2)where 1{i≤j} = 1 if i ≤ j and 0 otherwise, the partial orderD can berewritten as follows:AD B ⇔ DADT ≤ DBDT (entrywise) (3)2where ()T denotes the transpose operator.Let p, q ∈ Matn1(R+) such that∑ni=1 pi = σ =∑nj=1 qj. Without loss ofgenerality we can assume:σ = 1.The problem of Fre´chet is then to find (if exist) the maximum and theminimum element w.r.t the partial orderD of the subset of Matnn(R+):H(p, q)def= {F ∈ Matnn(R+)|F satisfies (F1) and (F2)} (4)with:(F1). F1 = p,(F2). 1TF = qT .Where 1 denotes the n-dimensional vector which coordinates are 1’s.Fre´chet proved that there exist a maximum element,∨DH(p, q)not.=Fmax, and a minimum element,∧DH(p, q)not.= Fmin such that:(DFmaxDT )i,j = min((Dp)i, (qTDT )j)not.= (Fmax)i,j , (5a)and(DFminDT )i,j = max(0, (Dp)i + (qTDT )j − σ)not.= (Fmin)i,j (5b)for all i, j = 1, . . . , n.2.2 The max-plus semiring RmaxLet R be the field of real numbers. The max-plus semiring Rmax is the setR ∪ {−∞} equipped with the addition ⊕ : (a, b) 7→ a ⊕ bdef= max(a, b) andthe multiplication ⊙ : (a, b) 7→ a⊙ bdef= a+ b. The neutral element for ⊕ isO := −∞ and the neutral element for ⊙ is 1 := 0. The max-plus semiring issaid to be an idempotent semiring because the addition is idempotent, i.e.a⊕ a = a, ∀a.The idempotent semiring Rmax or idempotent semirings isomorphic toRmax has many applications in discrete mathematics, algebraic geometry,computer science, computer languages, linguistic problems, optimizationtheory, discrete event systems, fuzzy theory (see e.g. [1], [3], [6], [8]).32.3 Order properties of Rmax and residuationLet us consider the max-plus semiring Rmax already defined in the intro-duction. The binary relation R defined by: aRbdef⇔ a ⊕ b = b coincideswith the standard order ≤ on R. We denote Rmax the semiring completedby adjoining to Rmax a ⊤ := +∞ element which satisfies a ⊕ ⊤ = ⊤, ∀a,a ⊙ ⊤ = ⊤, ∀a 6= O, and O ⊙ ⊤ = O. This is mainly motivated by the factthat some of the further results can be stated in a simpler way in Rmax.The completed max-plus semiring Rmax is a complete sup-semilattice, i.e.∀A ⊆ R ⊕Adef= ⊕x∈Ax exists in Rdef= R ∪ {−∞,+∞}. This implies that Rmaxis also a complete inf-semilattice because ∧A = ⊕{x ∈ R|∀a ∈ A, x ≤ a}.Thus, Rmax is a complete lattice. Finally, let us mention that Rmax is adistributive lattice, i.e.:a⊕ (b ∧ c) = (a ∧ b)⊕ (a ∧ c). (6)As we will see in the sequel this property will be of particular importance.Let us define left and right division in Rmax by: b/adef= ⊕{x ∈ R|x⊙ a ≤ b}and a\bdef= ⊕{x ∈ R|a ⊙ x ≤ b}. Left and right division coincide withthe usual subtraction to which we add the following properties: O\a = ⊤,⊤\a = O if a 6= ⊤, ⊤ otherwise (similar formulae for /).We extend operations and binary relations from scalars to matrices asfollows. If A = [ai,j], B = [bi,j ] then: A ≤ B (entrywise) ⇔ ∀i, j, ai,j ≤ bi,j,A⊕B = [ai,j ⊕ bi,j], A∧B = [ai,j ∧ bi,j], and A⊙B denotes the matrix suchthat its entry (i, j), (A⊙B)i,j, is: (A⊙B)i,j = ⊕kai,k ⊙ bk,j. We obviouslyhave: (A ⊙ B)T = BT ⊙ AT . We also extend the divisions to (possiblyrectangular) matrices with suitable dimension:(A\B)i,jdef= (⊕{X|A ⊙X ≤ B})i,j = ∧kak,i bk,j, (7a)(D/C)i,jdef= (⊕{X|X ⊙ C ≤ D})i,j = ∧ldi,l/cj,l. (7b)The ⊕ in the formulae (7a) and (7b) corresponds to the supremum w.r.tentrywise order between matrices. The application Y 7→ A\Y (resp. Y 7→Y/C) is called the residuated mapping of the application X 7→ A⊙X (resp.X 7→ X ⊙ C).For more details on max-plus algebra and residuation theory we referthe reader to e.g [1, Chp. 4 and references therein].43 Main ResultWe begin this Section by the following fundamental lemma.Lemma 3.1 Let (uj)mj=1 bem elements of Matn1(R+). Define sj =∑jk=1 uk.Then,[u1 · · · un] 1 = [s1 · · · sn]⊙ 1 (8)where 1 denotes the n-dimensional vector which all components are 1’s.Proof. Let us remark that: [u1 · · · un] 1 = sn. Because the vectors uj haveall their coordinates nonnegative: s1 ≤ · · · ≤ sn (componentwise), whichis equivalent to: sn = s1 ⊕ · · · ⊕ sn. Now, we just have to remark that:s1 ⊕ · · · ⊕ sn = [s1 · · · sn]⊙ 1 which ends the proof of the result. 2Remark 3.1 In the previous Lemma define U = [u1 · · · un] and S = [s1 · · · sn].We remark that S = UDT , recalling that D is the matrix defined by (2).Thus, relation (8) can be rewritten:U1 = (UDT )⊙ 1. (9)Theorem 3.1 (Main Result) Let us consider a matrix F ∈ Matnn(R+).Let p and q be two elements of Matn1(R+). Then,F ∈ H(p, q)⇐⇒{(DFDT )⊙ 1 = Dp1T ⊙ (DFDT ) = qTDTProof. F ∈ H(p, q)⇔{F1 = p1TF = qT.Because matrix D is invertible so is DT and:{F1 = p1TF = qT⇐⇒{DF1 = Dp (eq 1)1TFDT = qTDT (eq 2).For (eq 1). We apply Lemma 3.1 with uj := (DF ).,j and sj = (DFDT ).,j bethe jth column vectors of matrices DF and DFDT , respectively. We obtainthe following equality: Dp = DF1 = (DFDT )⊙ 1.For (eq 2). We apply Lemma 3.1 with uj := (DF T ).,j and sj := (DF TDT ).,j.We have: DF T1 = (qTDT )T = (DF TDT )⊙1. By definition of ()T we have:qTDT = 1T ⊙ (DFDT ) which ends the proof of the Theorem. 2This result can be reformulated as follows. Let us define the followingsets:Distn1def= D Matn1(R+) = {Dx, x ∈ Matn1(R+)} (10)5andDistnndef= D Matnn(R+) DT = {DXDT , X ∈ Matnn(R+)} (11)and for all P ,Q ∈ Distn1:H(P ,Q)def= {F ∈ Distnn|F ⊙ 1 = P and 1T ⊙ F = QT}. (12)Then, Theorem 3.1 states that ∀p, q ∈ Matn1(R+) and ∀F ∈ Matnn(R+):F ∈ H(p, q)⇔ DFDT ∈ H(Dp,Dq). (13)Or, equivalently:D H(p, q) DT = H(Dp,Dq). (14)4 New approach for the Fre´chet boundsFrom our main result, Theorem 3.1, we obtain Fre´chet bounds by methodswhich seem to be new to the best knowledge of the author.4.1 Upper boundThe Fre´chet upper bound is obtained as a direct consequence of Theorem 3.1.Corollary 4.1 (Fre´chet upper bound) Let p and q be two elements ofMatn1(R+). Under the condition that pT1 = qT 1 the set H(p, q) is not emptyand the upper Fre´chet bound Fmax is such that DFmaxDT not.= Fmax is thegreatest sub-solution of the following max-plus linear system of equations:{F ⊙ 1 = Dp1T ⊙ F = qTDT(15)that is:Fmax = ((Dp)/1) ∧ (1T \(qTDT )).Proof. Let us study (15) when replacing = by ≤. Then, by definitionof / we have: F ⊙ 1 ≤ Dp ⇔ F ≤ (Dp)/1. And by definition of \ wehave: F ≤ 1T \(qTDT ). The two previous inequalities are equivalent to:F ≤ ((Dp)/1) ∧ (1T \(qTDT )) = Fmax. Now, we have to prove that (A):Fmax ⊙ 1 = Dp and (B): 1T ⊙ Fmax = qTDT .6Let us prove (A).For all i = 1, . . . , n we write:(Fmax ⊙ 1)i = ⊕nj=1((Dp)/1) ∧ (1T \(qTDT ))i,j ⊙ 1= ⊕nj=1(Dp)i ∧ (qTDT )j= (Dp)i ∧ (⊕nj=1(qTDT )j)(by lattice distributivity (6))= (Dp)i ∧ (qTDT )n(∀j: (qTDT )j ≤ (qTDT )n)= (Dp)i ∧ (Dp)n((bTDT )n = qT 1 = pT1 = (Dp)n)= (Dp)i(∀i: (Dp)i ≤ (Dp)n).We prove (B) similarly. Hence the result is now achieved. 24.2 Lower boundIn this section we obtain Fre´chet lower bound by a greedy algorithm basedon max-plus linearity of the Fre´chet problem and the well-known Mongeproperty (see e.g. [2]) of elements of the set Distnn defined by (11), that isfor all F ∈ Distnn:(M). ∀i, j = 0, . . . n− 1: F i,j ⊙ F i+1,j+1 ≥ F i,j+1 ⊙ F i+1,j ,with the convention that ∀k F 0,k = F 0,k = 0.Let p, q ∈ Matn1(R+) be two vectors such that pT 1 = qT 1 = 1. Let usconsider the following algorithm.Lower(n,p,q)∀i = 1, . . . , n, F i,n := (Dp)i ; (a)∀j = 1, . . . , n, Fn,j := (qTDT )j ; (b)For j = n− 1 to 1 doFor i = n− 1 to 1 doF i,j := F−1i+1,j+1 ⊙ (F i,j+1 ⊙ F i+1,j)⊕ 1endend.Proposition 4.1 The algorithm Lower computes the Lower bound of theFre´chet contingency array problem.7Proof. The initial conditions (a) and (b) of the algorithm Lower come fromthe max-plus linearity of the Fre´chet problem and that the Monge property(M) implies:∀i ≤ i′, j ≤ j′, F i,j ≤ F i′,j′.The proof is obtained by recurrence (see e.g. the detailed proof of this resultby Fre´chet himself [5, p. 13])Denoting αl = (Dp)l, βk = (qTDT )k we have to prove that the loop in-variant of the algorithm Lower corresponds to the Fre´chet lower bound, i.e.∀l, k: F l,k = 1−1 ⊙ αl ⊙ βk ⊕ 1.It is easy to see that the previous relation is true for l = n with k =1, . . . , n and for l = 1, . . . , n with k = n. Now, let us assume that the loopinvariant is true for (k, l) ≥ (i, j), (k, l) 6= (i, j). We have:F i,j+1 ⊙ F i+1,j = (1−1 ⊙ αi ⊙ βj+1 ⊕ 1)⊙ (1−1 ⊙ αi+1 ⊙ βj ⊕ 1)= 2−1 ⊙ αi ⊙ αi+1 ⊙ βj ⊙ βj+1⊕1−1αi ⊙ βj+1 ⊕ 1−1 ⊙ αi+1 ⊙ βj ⊕ 1.Because (R+,≤) is a totally ordered lattice:F i+1,j+1 = 1−1 ⊙ αi+1 ⊙ βj+1 ⊕ 1 ∈ {1−1 ⊙ αi+1 ⊙ βj+1,1}Thus, we have two cases to study.1rst case: F i+1,j+1 = 1−1 ⊙ αi+1 ⊙ βj+1Let us compute:F−1i+1,j+1 ⊙ F i,j+1 ⊙ F i+1,j = 1−1 ⊙ αi ⊙ βj ⊕ αi ⊙ α−1i+1 ⊕ βj ⊙ β−1j+1⊕1⊙ (αi+1 ⊙ βj+1)−1.Then, we just have to remark that: αi ⊙ α−1i+1 = p−1i+1 ≤ 1, βj ⊙ β−1j+1 =q−1j+1 ≤ 1 and F i+1,j+1 = 1−1 ⊙ αi+1 ⊙ βj+1 ⇔ 1 ⊙ (αi+1 ⊙ βj+1)−1 ≤ 1.Thus,F i,j = F−1i+1,j+1 ⊙ F i,j+1 ⊙ F i+1,j ⊕ 1 = 1−1 ⊙ αi ⊙ βj ⊕ 1.2nd case: F i+1,j+1 = 1F−1i+1,j+1 ⊙ F i,j+1 ⊙ F i+1,j = F i,j+1 ⊙ F i+1,j= 2−1 ⊙ αi ⊙ αi+1 ⊙ βj ⊙ βj+1 ⊕ 1−1αi ⊙ βj+1⊕1−1 ⊙ αi+1 ⊙ βj ⊕ 18which could be rewritten as follows:F i,j = 1−1 ⊙ αi ⊙ βj ⊙ (1−1 ⊙ αi+1 ⊙ βj+1 ⊕ qj+1 ⊕ pi+1)⊕ 1.We remark that qj+1, pi+1 ≥ 1. We also note that F i+1,j+1 = 1 ⇒ 1−1 ⊙αi+1 ⊙ βj+1 ≤ 1. Thus, we deduce because ⊙ is non-decreasing and thedefinition of the standard order ≤ that:F i,j ≥ 1−1 ⊙ αi ⊙ βj ⊕ 1.On the other hand F i+1,j+1 = 1⇒ 1−1⊙αi⊙βj ≤ p−1i+1⊙ q−1j+1. And wededuce that:F i,j ≤ p−1i+1 ⊙ q−1j+1(1−1 ⊙ αi+1 ⊙ βj+1 ⊕ qj+1 ⊕ pi+1)⊕ 1= 1−1 ⊙ αi ⊙ βj ⊕ p−1i+1 ⊕ q−1j+1 ⊕ 1= 1−1 ⊙ αi ⊙ βj ⊕ 1 (because p−1i+1, q−1j+1 ≤ 1)We conclude because ≤ is antisymmetric.25 ConclusionIn this paper we proved that the Fre´chet correlation array problem is max-plus linear in the space of cumulative distribution function Distnn definedby (11). This remark leads to new methods to obtain the Fre´chet bounds.As a further work we would like to extend results of the paper to thecontinuous case based on the remark that :∫Rf(x, y)dy = supz∈R(∫ z−∞f(x, y)dy)for all nonnegative functions f such that∫Rf(x, y)dy exists.References[1] F. Baccelli, G. Cohen, G.J. Olsder, and J-P. Quadrat. Synchronizationand Linearity. John Wiley and Sons, 1992.[2] R.E. Burkard, B. Klinz, and R. Rudolf. Perspectives of Monge Propertiesin Optimization. Discr. Appli. Math., 70, 1996. (95-161).[3] R.A. Cuninghame-Green. Minimax Algebra. Lecture Notes in Economicsand Mathematical Systems No 166, Springer, 1979.9[4] M. Fre´chet. Sur les Tableaux de Corre´lations dont les Marges sontDonne´es. Ann. Univ. Lyon, Sect. A, 14, 1951. (53-77).[5] M. Fre´chet. Sur les Tableaux dont les Marges et des Bornes sont Donne´es.Revue Inst. Int. de Stat., 28(1/2), 1960. (10-32).[6] J.S Golan. Semirings and Their Applications. Kluwer Acad. Publ., 1999.[7] G. L. Litvinov. The Maslov Dequantization, Idempotent and TropicalMathematics: A Very Brief Introduction. Technical report, January2006. arXiv:math.GM/0501038 v4.[8] J. Richter-Gerbert, B. Sturmfels, and T. Theobald. First Steps in Trop-ical Geometry. Technical report, January 2003. arXiv:math.AG0306366.10

The Fréchet Contingency Array Problem is Max-Plus Linear

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The Fr\'echet Contingency Array Problem is Max-Plus Linear

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