SINCE the publication of our communication on the metric properties of the one-complement of Jaccard's similarity coefficient1, we have learned that this result was shown earlier by Marczewski and Steinhaus2. We regret the consequent redundancy of our paper. We tried to ensure its novelty by searching standard references and enquiring among mathematicians and statisticians, but apparently the earlier work, more extensive than ours, is not well known. We hope publication of our paper will have the positive effect of making it more well known. It is interesting that the earlier work also occurred in an ecological context3. A discussion of these and other properties of this useful function occurs in a recent contribution by Holgate4, who refers to it as the Marczewski–Steinhaus coefficient of similarity.

On a certain distance of sets and the corresponding distance of functions

Distance between Sets

Information granulation and concept approximation are some of the fundamental issues of granular computing. Granulation of a universe involves grouping of similar elements into granules to form coarse-grained views of the universe. Approximation of concepts, represented by subsets of the universe, deals with the descriptions of concepts using granules. In the context of rough set theory, this paper examines the two related issues. The granulation structures used by standard rough set theory and the corresponding approximation structures are reviewed. Hierarchical granulation and approximation structures are studied, which results in stratified rough set approximations. A nested sequence of granulations induced by a set of nested equivalence relations leads to a nested sequence of rough set approximations. A multi-level granulation, characterized by a special class of equivalence relations, leads to a more general approximation structure. The notion of neighborhood systems is also explored.

background

/pdf/information-granulation-and-rough-set-approximation-2tshd4yvzp.pdf

Information Granulation and Rough Set Approximation

LetR n
 denote the set of rooted trees withn leaves in which: the leaves are labeled by the integers in {1, ...,n}; and among interior vertices only the root may have degree two. Associated with each interior vertexv in such a tree is the subset, orcluster, of leaf labels in the subtree rooted atv. Cluster {1, ...,n} is calledtrivial. Clusters are used in quantitative measures of similarity, dissimilarity and consensus among trees. For anyk trees inR

 n
 , thestrict consensus tree C(T
1, ...,T

 k
 ) is that tree inR

 n
 containing exactly those clusters common to every one of thek trees. Similarity between treesT
1 andT
2 inR

 n
 is measured by the numberS(T
1,T
2) of nontrivial clusters in bothT
1 andT
2; dissimilarity, by the numberD(T
1,T
2) of clusters inT
1 orT
2 but not in both. Algorithms are known to computeC(T
1, ...,T

 k
 ) inO(kn
2) time, andS(T
1,T
2) andD(T
1,T
2) inO(n
2) time. I propose a special representation of the clusters of any treeT R

 n
 , one that permits testing in constant time whether a given cluster exists inT. I describe algorithms that exploit this representation to computeC(T
1, ...,T

 k
 ) inO(kn) time, andS(T
1,T
2) andD(T
1,T
2) inO(n) time. These algorithms are optimal in a technical sense. They enable well-known indices of consensus between two trees to be computed inO(n) time. All these results apply as well to comparable problems involving unrooted trees with labeled leaves.

Optimal algorithms for comparing trees with labeled leaves

In this paper, we analyze quantitative measures associated with if-then type rules. Basic quantities are identified and many existing measures are examined using the basic quantities. The main objective is to provide a synthesis of existing results in a simple and unified framework. The quantitative measure is viewed as a multi-facet concept, representing the confidence, uncertainty, applicability, quality, accuracy, and interestingness of rules. Roughly, they may be classified as representing one-way and two-way supports.

methods

/pdf/an-analysis-of-quantitative-measures-associated-with-rules-5g85tl2z6r.pdf

An Analysis of Quantitative Measures Associated with Rules

Given a set S of points in a metric space with distance function D, the nearest-neighbor searching problem is to build a data structure for S so that for an input query point q, the point s 2 S that minimizes D(s,q) can be found quickly. We survey approaches to this problem, and its relation to concepts of metric space dimension. Several measures of dimension can be estimated using nearest-neighbor searching, while others can be used to estimate the cost of that searching. In recent years, several data structures have been proposed that are provably good for low-dimensional spaces, for some particular measures of dimension. These and other data structures for nearest-neighbor searching are surveyed.

/pdf/nearest-neighbor-searching-and-metric-space-dimensions-5ctfpevfig.pdf

Nearest-Neighbor Searching and Metric Space Dimensions

/pdf/on-a-certain-distance-of-sets-and-the-corresponding-distance-2qarkjhv3f.pdf

On a certain distance of sets and the corresponding distance of functions

Citations

Distance between Sets

Information Granulation and Rough Set Approximation

Optimal algorithms for comparing trees with labeled leaves

An Analysis of Quantitative Measures Associated with Rules

Nearest-Neighbor Searching and Metric Space Dimensions

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