Bulletin q[M~Mwmatical Bioloyr Vol. 53, No. 5, pp. 679 684, 1991. Printed in Great Britain.

CONSENSUS

0092 8240/9153.00+0.00 Pergamon Press plc ~? 1991 Society for Mathematical Biology

WEAK HIERARCHIES

F. R. MCMORRIS* and R. C. POWERS Department of Mathematics, University of Louisville, Louisville, KY 40292, U.S.A_ An axiomatic characterization is presented for consensus functions defined on weak hierarchies. These functions are generalizations of the majority rule consensus.

1. Introduction. The notion of consensus has been a very useful one in biology. For example, finding consensus patterns among several D N A sequences is an important step in the comparison of these sequences (Waterman et al., 1984; Waterman, 1986), while finding consensus trees is standard practice among numerical taxonomists (cf. Adams., 1972; D a y and McMorris, 1985; Rohlf, 1982; Shao, 1983). In the second case, hierarchical trees are usually considered. Recently, Bandelt and Dress (1989) introduced a weakening of the tree condition to allow the possibility of overlapping clusters, yet maintain some tree-like properties. They also presented ways to construct these "weak hierarchies" from similarity data. Several different algorithms operating on the same data set, or one algorithm operating on different similarity data sets, will rarely produce the same hierarchy each time so it is important to determine ways in which these output hierarchies can be aggregated into a consensus hierarchy. In this note, we initiate such a study by focusing on those consensus functions that create no new clusters in the output, i.e. functions that put a cluster in the output by virtue of it appearing sufficiently often among the input hierarchies. These rules are frequently referred to as "counting rules". In Bandelt and Dress (1989), a consensus rule for weak hierarchies was given that generalizes the simple majority rule for strong hierarchies found in Margush and McMorris (1981). The next step, as we see it, is to axiomatically characterize this rule. In fact, we will characterize all the counting rules defined on Weak hierarchies in a m a n n e r similar to that found in McMorris and N e u m a n n (1983) and Monjardet (1978). We should point out that Barth~lemy and Janowitz (1990), Bandelt and Meletiou (1989) and Monjardet (1990) have axiomatic characterizations for certain counting rules in very general settings. Indeed, some aspects of our results are contained in their very elegant and * Research s u p p o r t e d by G r a n t N00014-89-J-1643 f r o m the Office of N a v a l Research. 679

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abstract models. However, we feel that by focusing only on weak hierarchiesl we m a k e ourresults more accessible to potential users. For further motivation and examples using weak hierarchies, we urge the reader to consult Bandelt and Dress (1989). 2. Results. Let S be a finite set with n >/3 elements. A collection H of subsets of S is called a weak hierarchy on S if X n Y ~ Z E { X n Y, Y n Z , X n Z } for all X, Y, Z e H, The elements of H are called clusters. If instead of looking at triple intersections, we require Xc~ Y e { X , Y, ~ } for all X, Y e l l , then H is called a strong hierarchy on S. Note that every strong hierarchy on S is.,a w~..al¢ hierarchy but not conversely. If H is a strong hierarchy on S satisfying: S e H; IZiCH; and { x } e H for all x e S , then H is called an n-tree on S. N o t e that, effectively, strong hierarchies are precisely n-trees. Let $ ( , 5 f and 3- denote the sets of all weak hierarchies on S, strong hierarchies on S and n-trees on S, respectively. Thus we have Y __ oug __ ~/(. Let k be a positive integer. A consensus function for weak hierarchies on S is a function C: $¢zk~ ~W, where ~ k is the k-fold cartesian product of ~//. We refer to an element P of ~K~ as a profile. For any profile P we will write CP for the image of P under the function C. Following Monjardet (1978) and McMorris and N e u m a n n (1983), we list some axioms that are reasonable for a consensus function C to satisfy. Let K = { 1 . . . . ,k}. For any two profiles P = (H I . . . . , Hk) and P ' = (Hi, . . . , Hk), we say !

(N) (M) (P)

(coP) (S) (A)

!

C is neutral whenever {i:XeHi} = {j: Yell]} implies X~ CP if and only if Y e C P ' ; C is monotone whenever {i:X~Hi}___{j:XEH~} implies X E C P ' if X~ CP; Cis Pareto if {i:XeHi} = K implies X e CP; C is co-Pareto if CP~_Hlu. . .uHk; C is symmetric if for every permutation a on K, CP~ = CP where P . = (H.0), • . ., Hn(k)); C is autonomous if for every nonempty subset X of S there exists a profile P such that X~ CP.

The next three observations, made first for consensus functions on n- trees by McMorris and N e u m a n n (1983), follow easily from set theoretic arguments: (i) C satisfies (M) and (N). if and only if C satisfies monotone neutrality (MN), where C satisfies ( i N ) if whenever {i:XGHi} _ {j: YeHi}, then X e CP implies Ye CP'. (ii) If C satisfies (MN), then C satisfies (coP). (iii) If C satisfies (MN) and (A), then C satisfies (P). Suppose similarity/dissimilarity data are given for the study collection S and we run k clustering algorithms on these data. Assume L___K and that the

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algorithms indexed by L produce weak hierarchies while those indexed by L c, the complement of L in K, produce strong hierarchies. We would like to capture, in a rigorous way, the c o m m o n agreement of these k output hierarchies. Let ~ *L ~W denote the set of all profiles P = ( H I , . . . , Hk) in ~/¢#k such that H i e ~ for all i e U . We will call a profile P from ~ *L ~K an L-profile. Our central goal in this paper is to characterize, using the previously stated axioms, the counting consensus functions C: ~*Lfg'--*"W. The axioms, of course, must be slightly adjusted in order to account for L-profiles. For (S), only those permutations a are feasible for an L-profile P which maps P onto an L-profile P~. Notice that if L = ~ , then ~ *e ~g'= ~ k and that if L = K, then H,L~#/'=~#/"k. Finally, for any subset V o f K w e let V~= V n L c and V,,= VnL. Let @ be a set of subsets of K. N is an L-weak decisive family if the following two conditions are satisfied: (1) if V e N and V~_ W~_K, then W e N ; (2) if V 1, V 2, V 3 e N , then either V~c~V{¢~ for some iCj in {1, 2, 3} or THEOREM 1. Let N be an L-weak decisivefamily and P = (Hi, . . . , Hk) a profile in ~ *L "/g'" Then

U g H, Ve~

I~V

is a weak hierarchy. Proof. Let X, Y, and Z be subsets of S that belong to U V~

(~Hi i~V

So there exists sets V 1, V 2, V 3 in N such that

XE ("~ Hi, i~v

Ye 0

I

i~v

Hi, 2

ge

('-] H i. iEv

~

Since N is an L-weak decisive family it follows that either V~c~ V~~ ~ for some iCj in {1,2,3} or v l n V ~ n V 3 ¢ ~ . If V ~ c ~ V ~ n V 3 ¢ ~ , l e t j be in this intersection. Then X, Y and Z are in the weak hierarchy H i and so J(c~YnZe{J(c~Y, X n Y , X~Z}. If V~c~VZc~v3=fgJ, then we may assume without loss of generality that V] ~ V2 va ~ . Ifj e V] ~ V2, then the sets J ( a n d Y both belong to the strong hierarchy Hi. Therefore, either Y _ X , J ( ~ Y or J(c~ Y= ~ . In any case it follows that Xc~ Y n Z e { X n Y, Yc~Z, XnZ} and the proof is complete. • For an L-weak decisive family N we define

M~: jvf *g ~tg'~/"

by

M~(P)= U Vc~

n H,. i~V

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F . R . M c M O R R I S A N D R. C. P O W E R S

The proposition above implies that the function M~ is well defined. We will characterize M~ in terms of the axioms listed above. But first we gives some examples of L-weak decisive families. Example 1. Assume that L ¢ ~ let i and j be fixed indices from L c and L, respectively. Define ~ __~(K) by VE@ if and only if i¢ Vs implies jE Vw. Note that if L c = ~ then My is the dictatorial rule for weak hierarchies. Example 2. Let ILl = l and let (k + l)/3 < t (k + 1)/3. Proof. If C = M t then C clearly satisfies (MN), (A) and (S).

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683

F o r the converse, since C satisfies (MN) and (A), it follows from Theorem 2 that C = M e where ~ is some L-weak decisive family. In fact, from the p r o o f of Theorem 2, ~ is the set of all decisive sets with respect to C. Let V~ ~ and let P be a profile in ~ k such that V = {i:X~H~} where X i s some nonempty subset of S. Then X e C P from the definition of Me(P ). N o w suppose a is any permutation on K. Then, using the fact that P, ~ ~ k and symmetry, we get X e CP~. Therefore a V i s a decisive set with respect to C, i.e. o-Vbelongs to 9 , If V ~ K such that ~ (k + l)/3, then there exist permutations o-a and o 2 on K such that the sets V, o-1 V and o-2 V belong to @ and violate (2). Therefore, any set in ~ must have more than (k + 1)/3 elements. Thus, if t = sup{r:r < IV[ for all Ve @}, then t > (k + 1)/3, so that M e ~_M t. But, by the Lemma, we get M t ___M e and the p r o o f is complete. • The next two examples are interesting special cases of the last corollary and generalize the majority rule consensus as defined in Margush a n d M c M o r r i s (1981). Example 3. I f L = ~Z~,then the consensus function C is defined on ~ k and has output in ~/¢F.In this case, the Corollary says that C satisfies (MN), (A) and (S) if and only if, for any profile P = (H l, . . . , Hk) in ~ k,

l vi

CP=Mt(P )= U

~ Hi

V~_K i~V

Ivl>~t where t ~ (k/3, k). If t is the smallest integer greater than k/3, then we can think o f M t as the consensus rule that puts a cluster in the output if it appears in more than ½ of the input hierarchies. This could appropriately be called the 1-rule. Example 4. If L = K, then the consensus function C is defined on ~¢rk and has output in ~/¢F.In this case, the Corollary says that C satisfies (MN), (A) and (S) if and only if, for any profile P = (H l, . . . , Hk) in yc/'k,

CP=Mt(P )= U

(~ Hi-

V~_K i~V

where t e (2k/3, k]. If t is the smallest integer greater than 2k/3, then we can think o f M t as the consensus rule that puts a cluster in the output if it appears in more than 2 of the input hierarchies. This could be appropriately called the

~-rule. LITERATURE Adams, E. N_ Ill. 1972. Consensus techniques and the comparison of taxonomic trees. Syst. Zool_ 21,390-397. Bandelt, H.-J. and A. Dress_ 1989. Weak hierarchies associated with similarity measures--an additive clustering technique. Bull. math. Biol. 51, 133 166.

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Bandelt, H.-J. and G. C. Meletiou. 1989. The algebra of majority consensus. Preprint. Barth~lemy, J. P. and M. F. Janowitz_ 1990. A formal theory of consensus. Preprint. Day, W. H. E. and F. R. McMorris. 1985. A formalization of consensus index methods. Bull. math. Biol. 47, 215-229. Margush, T. and F. R. McMorris. 1981. Consensus n trees. Bull. math. Biol. 43, 239-244. McMorris, F. R. and D. A. Neumann_ 1983. Consensus functions defined on trees. Math. Social Sci. 4, 131-136. Monjardet, B. 1978. An axiomatic theory of tournament aggregation. Math. Oper. Res_ 3, 334~351. Monjardet, B. 1990. Arrowian characterizations of latticial federation consensus functions. Math. Social Sci. (in press). Rohlf, F. J. 1982. Consensus indices for comparing classifications. Math. Biosci. 59, 131-144. Shao, K. 1983. Consensus methods in numerical taxonomy. Ph.D. Thesis, State University of New York at Stony Brook. Waterman, M. S. 1986. Multiple sequence alignment by consensus. Nucl_ Acids Res. 14, 90954102. Waterman, M. S., R. Arratia and D. J. Galas. 1984_ Pattern recognition in several sequences: consensus and alignment. Bull. math. Biol. 46, 515-527.

R e c e i v e d 23 M a r c h R e v i s e d 10 S e p t e m b e r

1990 1990

Consensus weak hierarchies.

An axiomatic characterization is presented for consensus functions defined on weak hierarchies. These functions are generalizations of the majority ru...
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