P. Diaconis, R. Graham, Spearman's Footrule as a Measure of Disarray, Journal of the Royal Society of Statistics, Series B 39 (1977) 262--268.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....are distinct, the authority weights induce a total ranking of the elements in P . If the weights are not all distinct, then we have a partial ranking of the elements in P . We will also refer to total rankings as permutations. The problem of comparing permutations has been studied extensively [30, 16, 18]. One popular distance measure is the Kendall s tau distance which captures the number of disagreements between the rankings. Let denote the set of all pairs of nodes. Let a 1 , a 2 be two LAR vectors. We define the violating set, as follows j) a 1 (i) a 1 (j) a 2 (i) a ....

P. Diaconis and R. Graham. Spearman's footrule as a measure of disarray. Journal of the Royal Statistical Society, 39(2):262 -- 268, 1977.

.... of measures of association for data is an important issue in many quantitative problems arising in diverse disciplines such as political science, psychology and sociology; and in rank statistics, Kendall s and Spearman s ae are commonly used measures of association (or disarray) of data; cf. [11, 22, 29]. In probability theory, the Kolmogorov distance and the total variation distance between two distributions are frequently used measures of closeness. This paper is concerned with problems of the following type: Accepted for publication in Advances in Applied Mathematics. Given a random (under ....

P. Diaconis and R. L. Graham. Spearman's footrule as a measure of disarray. Journal of the Royal Society of Statistics, series B 39 (1977) 262--268.

.... number of runs with ranges of order n is not very useful when handling presorting methods (however useful they may be for other purposes) In general, as in nonparametric inference, a right invariant metric with small variance, say, linear or less than linear, is unsuitable for general use (cf. [2,4]) On the other hand, computations of expectation values in drop of DS : 1jn jj Gamma (j)j or 1jn Gamma1 j(j) Gamma (j 1)j can be made in a similar manner as for I n (but not for generating functions which are intrinsically much harder) For example, applying sw 1; n to a permutation in ....

P. Diaconis and R. L. Graham, Spearman's footrule as a measure of disarray, Journal of the Royal Society of Statistics, Series B, 39 (1977), 262--268.

.... number of runs with ranges of order n is not very useful when handling presorting methods (however useful they may be for other purposes) In general, as in nonparametric inference, a right invariant metric with small variance, say, linear or less than linear, is unsuitable for general use (cf. [2,4]) On the other hand, computations of expectation values in drop of DS : 1#j#n j #(j) or 1#j#n 1 #(j) #(j 1) can be made in a similar manner as for I n (but not for generating functions which are 11 intrinsically much harder) For example, applying sw 1, n to a permutation in ....

P. Diaconis and R. L. Graham, Spearman's footrule as a measure of disarray, Journal of the Royal Society of Statistics, Series B, 39 (1977), 262--268.

.... of measures of association for data is an important issue in many quantitative problems arising in diverse disciplines such as political science, psychology and sociology; and in rank statistics, Kendall s # and Spearman s # are commonly used measures of association (or disarray) of data; cf. [11, 22, 29]. In probability theory, the Kolmogorov distance and the total variation distance between two distributions are frequently used measures of closeness. This paper is concerned with problems of the following type: # Accepted for publication in Advances in Applied Mathematics. Given a random (under ....

P. Diaconis and R. L. Graham. Spearman's footrule as a measure of disarray. Journal of the Royal Society of Statistics, series B 39 (1977) 262--268.

..... The footrule distance between ] 0 , is defined to be : 1 03G C [ 03V 2: D ( objects. A Kendalloptimal ) D ( 0 is minimized; similarly, a footrule optimal aggregation of ( D ] 0 is minimized. It is known [7] that [ b] 0 [ cF . 0 . It follows that if is a footrule optimal aggregation of ( D then the total Kendall distance of from ( D (namely the quantity . 0 ) is within a factor of two of the total Kendall distance of the Kendall optimal ....

P. Diaconis and R. Graham. Spearman's footrule as a measure of disarray. Journal of the Royal Statistical Society, Series B, 39(2):262--268, 1977.

...., #2 , #m denote m permutations of n objects. A Kendalloptimal aggregation of #1 , #m is any permutation # such that i K(#, # i ) is minimized; similarly, a footrule optimal aggregation of #1 , #m is any permutation # such that i F (#, # i ) is minimized. It is known [7] that K(#, #) F (#, #) 2K(#, # ) It follows that if # is a footrule optimal aggregation of #1 , #m , then the total Kendall distance of # from #1 , #m (namely the quantity i K(#, # i ) is within a factor of two of the total Kendall distance of the Kendall optimal ....

P. Diaconis and R. Graham. Spearman's footrule as a measure of disarray. Journal of the Royal Statistical Society, Series B, 39(2):262--268, 1977.

....of equivalence actually gives us an equivalence relation (reflexive, symmetric, and transitive) It follows from Theorem 4.4 that a distance measure is equivalent to a metric if and only if it is a near metric. Our notion of equivalence is inspired by a classical result of Diaconis and Graham [DG77] which states that for every two permutations oe 1 ; oe 2 , we have K(oe 1 ; oe 2 ) F (oe 1 ; oe 2 ) 2K(oe 1 ; oe 2 ) 17) Of course, we are dealing with distances between top k lists, whereas Diaconis and Graham dealt with distances between permutations. Having showed that the notions of ....

P. Diaconis and R. Graham. Spearman's footrule as a measure of disarray. Journal of the Royal Statistical Society, Series B, 39(2):262--268, 1977.

.... Gamma1 log n log log n) space deterministic algorithm for this problem. In all of these algorithms, we make fundamental use of the relationship between Kendall s tau metric (the number of inversions) and Spearman s footrule metric (to be defined in Section 2) proved by Diaconis and Graham [7]. These results are presented in Section 3; the analysis of the most space efficient algorithm is deferred to the Appendix. Despite the latter improvements, we present the e O( p n) and e O(log 2 n) space algorithms because many ideas from their analyses are used in the algorithms for the ....

.... K(oe) is an (1 ffl) approximation to K(oe) if (1 Gamma ffl)K(oe) K(oe) 1 ffl)K(oe) A different metric on S n is the L 1 like Spearman s footrule distance which is given by F (oe; Delta = P n u=1 joe(u) Gamma (u)j. Let F (oe) Delta = F (oe; 1) Diaconis and Graham [7] relate these two metrics via a fundamental inequality. Theorem 1 (Diaconis Graham [7] For any oe; 2 S n , K(oe; F (oe; 2K(oe; These inequalities are fairly tight. In fact, the above inequality automatically gives a logspace algorithm to obtain a 2 approximation to K(oe) since one ....

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P. Diaconis and R. Graham. Spearman's footrule as a measure of disarray. J. of the Royal Statistical Society, Series B, 39(2):262--268, 1977.

....and by F (oe; P j joe(j) Gamma (j)j. This distance is fairly natural for our purpose: it measures the displacement of each page between the two rankings oe and . It turns out that the footrule distance between any two permutations approximates their Kendall tau distance to within factor two [15]: K(oe; F (oe; 2K(oe; Consequently, any algorithm that computes a footrule optimal aggregation is automatically a 2 approximation algorithm for finding Kemeny optimal permutations 2 . We can construct examples where this approximation is tight. On the positive side, we prove: Lemma 11 ....

P. Diaconis and R. Graham. Spearman's footrule as a measure of disarray. J. of the Royal Statistical Society, Series B, 39(2):262--268, 1977.

....the extended Condorcet criterion and fighting search engine spam. Given that Kemeny optimal aggregation is useful, but computationally hard, howdowe compute it The following relation shows that Kendall distance can be approximated very well via the Spearman footrule distance. Proposition 1. [13] For any two full lists oe#, K(oe#) F (oe#) 2K(oe#) This leads us to the problem of footrule optimal aggregation. This is the same as before, except that the optimizing criterion is the footrule distance. In Section 4 we exhibit a polynomial time algorithm to compute optimal footrule ....

P. Diaconis and R. Graham. Spearman's footrule as a measure of disarray. J. of the Royal Statistical Society, Series B, 39(2):262--268, 1977.

....most recently inserted element. The best known upper bound for computing Inv( is O(n log n= log log n) which follows by combining the outlined algorithm with a data structure of Dietz [5] Whether Inv can be computed faster, maybe even in linear time, is an open problem. Diaconis and Graham [3] showed that Inv( D( 2Inv ( where D( P n i=1 j (i) Gamma ij. Hence, since D is trivially computable in linear time, so is a fairly good approximation of Inv . Again, varying ffl such that 0:5= p log log n ffl 1= p log log n and using the above combined data structure, it ....

P. Diaconis and R.L. Graham. Spearman's footrule as a measure of disarray. J. Royal Statistical Society Series B, 39:262--268, 1977.

....is a total order; we shall use the notation X(i) to denote the the ith element of the total order, X . Then Spearman s footrule, d S (X ;Y) 1 = 2 n i=1 jX(i) Gamma Y(i)j. We prove in Appendix A that d S = d. It is well known that d S (X ;Y) bn 2 =4c (see, for example, Diaconis and Graham [6]) and thus D bn 2 =4c. The basic technique we employ is known as coupling. We use the following Coupling Lemma . See, for example, Aldous [1] Lemma 1 (Coupling) Let (X ;Y) be a random process (the coupling) such that, marginally, X and Y are both copies of M f . Moreover, suppose Y 0 is ....

....we mean XY (i) X(Y (i) We defined Spearman s footrule as a metric on total orders: d S (X ;Y) 1 = 2 n i=1 jX(i) Gamma Y(i)j. It should be noted that this definition of Spearman s footrule is in accord with Spearman s usage [18] and that recommended by [7] other authors (e.g. [6]) drop the half from the definition. Theorem 1 Suppose X and Y are distinct total orders. Suppose further that they differ by more than a single transposition (i.e. there is no transposition T , such that X = TY ) and that both X and Y are linear extensions of a partial order, P. Then there ....

Persi Diaconis and R. L. Graham. Spearman's footrule as a measure of disarray. Journal of the Royal Statistical Society, Series B (Methodological), 39(2):262--268, 1977.

....examined briefly. For further details on the various metrics the reader is referred to the monographs of Critchlow [17] and Diaconis [22] Asymptotic properties of the various metrics under the null distribution have been examined by Kendall [42] Alvo, Cabilio and Feigin [2] Diaconis and Graham [24], Feigin and Cohen [29] and Feigin and Alvo [28] Diaconis [22] notes that the full rankings of n items can be viewed as elements of permutation group S n ; the symmetric group on n items. The symmetric group, in turn, can be represented as a collection of invertible matrices in such a way that ....

....with another group of sets of paired comparisons. Spearman s Footrule also extends easily to deal with partial rankings, using the midrank procedure to deal with ties, so that AE(hA; B; Ci Gamma1 ; hA; B; C)i Gamma1 ) AE( 1; 2; 3) 1; 2:5; 2:5) 0 0:5 0:5 = 1: Diaconis and Graham [24] examine the properties of Spearman s Footrule (which they denote D) on full rankings and relate it to Kendall s (the minimal number of transpositions of adjacent elements required to transform one arrangement to another) and Cayley s distance (the minimal number of transpositions required to ....

Diaconis, P. and Graham, R.L. (1977), "Spearman's Footrule as a Measure of Disarray", Journal of the Royal Statistical Society B, 39, 262-268.

....; n is the height of s j with respect to OE i . Then the Spearman s footrule is defined as: Spearman s footrule : ffi S (OE 1 ; OE 2 ) 1 2 n X i=1 jh 1j Gammah 2j j: It is well known that Spearman s footrule is a metric on strict orders and has the range [0; bn 2 =4c] see, for example [4]) The three requirements for a measure to be a metric are the following (for all strict orders OE i ; i = 1; 2; 3) i) Reflexivity. d(OE 1 ; OE 2 ) 0, iff OE 1 and OE 2 are identical. ii) Symmetry. d(OE 1 ; OE 2 ) d(OE 2 ; OE 1 ) iii) Triangle Inequality. d(OE 1 ; OE 3 ) d(OE 1 ; OE ....

P. Diaconis and R.L. Graham. Spearman's footrule as a measure of disarray. Journal of the Royal Statistical Society, Series B (Methodological), 39(2):262--268, 1977.

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P. Diaconis, R. Graham, Spearman's Footrule as a Measure of Disarray, Journal of the Royal Society of Statistics, Series B 39 (1977) 262--268.

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P. Diaconis and R. Graham. Spearman's footrule as a measure of disarray. J. of the Royal Statistical Society, 39(2), pages 262--268, 1997.

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P. Diaconis and R. Graham. Spearman's footrule as a measure of disarray. J. of the Royal Statistical Society, Series B, 39(2):262--268, 1977.

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P. Diaconis and R. Graham. Spearman's footrule as a measure of disarray. J. of the Royal Statistical Society, 39(2), pages 262--268, 1997.

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P. Diaconis and R. Graham. Spearman's footrule as a measure of disarray. Journal of the Royal Statistical Society, Series B, 39(2):262--268, 1977.

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P. Diaconis and R. L. Graham, Spearman's footrule as a measure of disarray, J. Royal. Stat. Soc., Ser. B, 39(1977) 262-268.

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