Classification Based on Depth Transvariations

Suppose y, a d-dimensional (d ≥ 1) vector, is drawn from a mixture of k (k ≥ 2) populations, given by ∏₁, ∏₂,…,∏ _k . We wish to identify the population that is the most likely source of the point y. To solve this classification problem many classification rules have been proposed in the literature. In this study, a new nonparametric classifier based on the transvariation probabilities of data depth is proposed. We compare the performance of the newly proposed nonparametric classifier with classical and maximum depth classifiers using some benchmark and simulated data sets. The authors thank the editor and referees for comments that led to an improvement of this paper. This work is partially supported by the National Science Foundation under Grant No. DMS-0604726. Published online xx, xx, xxxx.     

Seriation in the Presence of Errors: NP-Hardness of <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">∞</Emphasis></Subscript> -Fitting Robinson Structures to Dissimilarity Matrices

In this paper, we establish that the following fitting problem is NP-hard: given a finite set X and a dissimilarity measure d on X (d is a symmetric function from X ² to the nonnegative real numbers and vanishing on the diagonal), we wish to find a Robinsonian dissimilarity d _R on X minimizing the l _∞ -error ||d − d _R || _∞ = max_x,y ∈X{|d(x, y) − d _R (x, y)|} between d and d _R . Recall that a dissimilarity d _R on X is called monotone (or Robinsonian) if there exists a total order ≺ on X such that x ≺ z ≺ y implies that d(x, y) ≥ max{d(x, z), d(z, y)}. The Robinsonian dissimilarities appear in seriation and clustering problems, in sparse matrix ordering and DNA sequencing.     

Weighted Graphs with Distances in Given Ranges

Let \( \mathcal{G} \) = (G,w) be a weighted simple finite connected graph, that is, let G be a simple finite connected graph endowed with a function w from the set of the edges of G to the set of real numbers. For any subgraph G′ of G, we define w(G′) to be the sum of the weights of the edges of G′. For any i, j vertices of G, we define D _{i,j}(\( \mathcal{G} \)) to be the minimum of the weights of the simple paths of G joining i and j. The D _{i,j}(\( \mathcal{G} \)) are called 2-weights of \( \mathcal{G} \). Weighted graphs and their reconstruction from 2-weights have applications in several disciplines, such as biology and psychology.Let \( {\left\{{m}_I\right\}}_{I\in \left(\frac{\left\{1,\dots, n\right\}}{2}\right)} \) and \( {\left\{{M}_I\right\}}_{I\in \left(\frac{\left\{1,\dots, n\right\}}{2}\right)} \) be two families of positive real numbers parametrized by the 2-subsets of {1, …, n} with m _I ≤ M _I for any I; we study when there exist a positive-weighted graph G and an n-subset {1, …, n} of the set of its vertices such that D _I (\( \mathcal{G} \)) ∈ [m _I ,M _I ] for any \( I\in \left(\frac{\left\{1,\dots, n\right\}}{2}\right) \). Then we study the analogous problem for trees, both in the case of positive weights and in the case of general weights.     

On the Discrepancy Measures for the Optimal Equal Probability Partitioning in Bayesian Multivariate Micro-Aggregation

Data holders, such as statistical institutions and financial organizations, have a very serious and demanding task when producing data for official and public use. It’s about controlling the risk of identity disclosure and protecting sensitive information when they communicate data-sets among themselves, to governmental agencies and to the public. One of the techniques applied is that of micro-aggregation. In a Bayesian setting, micro-aggregation can be viewed as the optimal partitioning of the original data-set based on the minimization of an appropriate measure of discrepancy, or distance, between two posterior distributions, one of which is conditional on the original data-set and the other conditional on the aggregated data-set. Assuming d-variate normal data-sets and using several measures of discrepancy, it is shown that the asymptotically optimal equal probability m-partition of , with m ^1/d ∈ , is the convex one which is provided by hypercubes whose sides are formed by hyperplanes perpendicular to the canonical axes, no matter which discrepancy measure has been used. On the basis of the above result, a method that produces a sub-optimal partition with a very small computational cost is presented. Published online xx, xx, xxxx.     

The Metric Cutpoint Partition Problem

Let G = (V, E,w) be a graph with vertex and edge sets V and E, respectively, and w: E → a function which assigns a positive weight or length to each edge of G. G is called a realization of a finite metric space (M, d), with M = {1, ..., n} if and only if {1, ..., n} ⊆ V and d(i, j) is equal to the length of the shortest chain linking i and j in G ∀i, j = 1, ..., n. A realization G of (M, d), is called optimal if the sum of its weights is minimal among all the realizations of (M, d). A cutpoint in a graph G is a vertex whose removal strictly increases the number of connected components of G. The Metric Cutpoint Partition Problem is to determine if a finite metric space (M, d) has an optimal realization containing a cutpoint. We prove in this paper that this problem is polynomially solvable. We also describe an algorithm that constructs an optimal realization of (M, d) from optimal realizations of subspaces that do not contain any cutpoint. Supported by grant PA002-104974/2 from the Swiss National Science Foundation. Published online xx, xx, xxxx.     

Statistical tests on two characteristics of the shapes of cluster diagrams

A cluster diagram is a rooted planar tree that depicts the hierarchical agglomeration of objects into groups of increasing size. On the null hypothesis that at each stage of the clustering procedure all possible joins are equally probable, we derive the probability distributions for two properties of these diagrams: (1)S, the number of single objects previously ungrouped that are joined in the final stages of clustering, and (2)m _k, the number of groups ofk+1 objects that are formed during the process. Ecological applications of statistical tests for these properties are described and illustrated with data from weed communities of Saskatchewan fields.This work was supported by the Natural Sciences and Engineering Research Council of Canada.     

One-Mode Classification of a Three-Way Data Matrix

X is the automatic hierarchical classification of one mode (units or variables or occasions) of X on the basis of the other two. In this paper the case of OMC of units according to variables and occasions is discussed. OMC is the synthesis of a set of hierarchical classifications Delta obtained from X; e.g., the OMC of units is the consensus (synthesis) among the set of dendograms individually defined by clustering units on the basis of variables, separately for each given occasion of X. However, because Delta is often formed by a large number of classifications, it may be unrealistic that a single synthesis is representative of the entire set. In this case, subsets of similar (homegeneous) dendograms may be found in Delta so that a consensus representative of each subset may be identified. This paper proposes, PARtition and Least Squares Consensus cLassifications Analysis (PARLSCLA) of a set of r hierarchical classifications Delta. PARLSCLA identifies the best least-squares partition of Delta into m (1 <= m <= r) subsets of homogeneous dendograms and simultaneously detects the closest consensus classification (a median classification called Least Squares Consensus Dendogram (LSCD) for each subset. PARLSCLA is a generalization of the problem to find a least-squares consensus dendogram for Delta. PARLSCLA is formalized as a mixed-integer programming problem and solved with an iterative, two-step algorithm. The method proposed is applied to an empirical data set.     

Recognition of Robinsonian dissimilarities

We present an O(n ³)-time, O(n ²)-space algorithm to test whether a dissimilarity d on an n-object set X is Robinsonian, i.e., X admits an ordering such that i≤j≤k implies that d(x _i,x_k)≥max {d(x_i,x_j),d(x_j,x_k)}.     

Thresholded consensus for n-trees

A class of (multiple) consensus methods for n-trees (dendroids, hierarchical classifications) is studied. This class constitutes an extension of the so-called median consensus in the sense that we get two numbersm andm such that: If a clusterX occurs ink n-trees of a profileP, withk m, then it occurs in every consensus n-tree ofP. IfX occurs ink n-trees ofP, withm k <m, then it may, or may not, belong to a consensus n-tree ofP. IfX occurs ink n-trees ofP, withk <m then it cannot occur in any consensus n-tree ofP. If these conditions are satisfied, the multiconsensus function is said to be thresholded by the pair (m,m). Two results are obtained. The first one characterizes the pairs of numbers that can be viewed as thresholds for some consensus function. The second one provides a characterization of thresholded consensus methods. As an application a characterization of the quota rules is provided.

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Resume Cet article traite d'une classe de méthodes de consensus (multiples) entre des classifications hiérarchiques. Cette classe est une généralisation du consensus médian dans las mesure oú elle est constituée des méthodes c pour lesquelles il existe deux nombresm etm tels que: Si une classeX appartient ák hiérarchies d'un profilP, aveck m, alorsX appartient á chaque hiérarchie consensus deP. SiX appartient ák hiérarchies deP, avecm k <m, alorsX, peut, ou non, appartenir à une hiérarchie consensus deP. SiX appartient àk hiérarchies deP, aveck <m, alorsX n'appartient á aucune hiérarchie consensus deP. On dit alors que le couple (m,m) est un seuil pour c. Deux résultats sont obtenus. Le premier caractérise les couples de nombres qui sont des seuils de consensus. Le second caractérise les consensus admettant un seuil. Une caractérisation de la régle des quotas est déduite de ce second résultat.

     

Algorithms for ℓ<Subscript>1</Subscript>-Embeddability and Related Problems

Assouad has shown that a real-valued distance d = (d_ij)_{1 ≤ i < j ≤ n} is isometrically embeddable in ℓ₁space if and only if it belongs to the cut cone on n points. Determining if this condition holds is NP-complete. We use Assouad's result in a constructive column generation algorithm for ℓ₁-embeddability. The subproblem is an unconstrained 0-1 quadratic program, solved by Tabu Search and Variable Neighborhood Search heuristics as well as by an exact enumerative algorithm. Computational results are reported. Several ways to approximate a distance which is not ℓ₁-embeddable by another one which is are also studied.     

Can the Causal Paradoxes of QM be Explained in the Framework of QED?

Attemts to explain causal paradoxes of Quantum Mechanics (QM) have tried to solve the problems within the framework of Quantum Electrodynamics (QED). We will show, that this is impossible. The original theory of QED by Dirac (Proc Roy Soc A117:610, 1928) formulated in its preamble four preliminary requirements that the new theory should meet. The first of these requirements was that the theory must be causal. Causality is not to be derived as a consequence of the theory since it was a precondition for the formulation of the theory; it has been constructed so that it be causal. Therefore, causal paradoxes logically cannot be explained within the framework of QED. To transcend this problem we should consider the following points: Dirac himself stated in his original paper (1928) that his theory was only an approximation. When he returned to improve the theory later (Proc Roy Soc A209, 1951), he noted that the new theory “involves only the ratio e/m, not e and m separately”. This is a sign that although the electromagnetic effects (whose source is e) are magnitudes stronger than the gravitational effects (whose source is m), the two are coupled. Already in 1919, Einstein noted that “the elementary formations which go to make up the atom” are influenced by gravitational forces. Although in that form the statement proved not to be exactly correct, the effects of gravitation on QM phenomena have been established. The conclusion is that we should seek a resolution for the causal paradoxes in the framework of the General Theory of Relativity (GTR)—in contrast to QED, which involves only the Special Theory of Relativity (STR). We show that causality is necessarily violated in GTR. This follows from the curvature of the space-time. Although those effects are very small, one cannot ignore their influence in the case of the so-called “paradox phenomena”.     

A Nonlinear Programming Approach to Metric Unidimensional Scaling

Classical unidimensional scaling provides a difficult combinatorial task. A procedure formulated as a nonlinear programming (NLP) model is proposed to solve this problem. The new method can be implemented with standard mathematical programming software. Unlike the traditional procedures that minimize either the sum of squared error (L ₂ norm) or the sum pf absolute error (L ₁ norm), the proposed method can minimize the error based on any L _p norm for 1 ≤p < ∞. Extensions of the NLP formulation to address a multidimensional scaling problem under the city-block model are also discussed.     

The Metric Bridge Partition Problem: Partitioning of a Metric Space into Two Subspaces Linked by an Edge in Any Optimal Realization

Let G = (V,E,w) be a graph with vertex and edge sets V and E, respectively, and w:E → R ⁺ a function which assigns a positive weight or length to each edge of G. G is called a realization of a finite metric space (M,d), with M = { 1,...,n} if and only if { 1,...,n} ⫅ V and d(i,j) is equal to the length of the shortest chain linking i and j in G ∀ i,j = 1,...,n. A realization G of (M,d), is said optimal if the sum of its weights is minimal among all the realizations of (M,d). Consider a partition of M into two nonempty subsets K and L, and let e be an edge in a realization G of (M,d); we say that e is a bridge linking K with L if e belongs to all chains in G linking a vertex of K with a vertex of L. The Metric Bridge Partition Problem is to determine if the elements of a finite metric space (M,d) can be partitioned into two nonempty subsets K and L such that all optimal realizations of (M,d) contain a bridge linking K with L. We prove in this paper that this problem is polynomially solvable. We also describe an algorithm that constructs an optimal realization of (M,d) from optimal realizations of (K,d|_K) and (L,d|_L).     

Constructing Ultrametric and Additive Trees Based on the L 1 Norm

L ₁) criterion. Examples of ultrametric and additive trees fitted to two extant data sets are given, plus a Monte Carlo analysis to assess the impact of both typical data error and extreme values on fitted trees. Solutions are compared to the least-squares (L ₂) approach of Hubert and Arabie (1995a), with results indicating that (with these data) the L ₁ and L ₂ optimization strategies perform very similarly. A number of observations are made concerning possible uses of an L ₁ approach, the nature and number of identified locally optimal solutions, and metric recovery differences between ultrametrics and additive trees.     

Recognizing Treelike k-Dissimilarities

A k-dissimilarity D on a finite set X, |X|????k, is a map from the set of size k subsets of X to the real numbers. Such maps naturally arise from edgeweighted trees T with leaf-set X: Given a subset Y of X of size k, D(Y ) is defined to be the total length of the smallest subtree of T with leaf-set Y . In case k?=?2, it is well-known that 2-dissimilarities arising in this way can be characterized by the so-called ??4-point condition??. However, in case k?>?2 Pachter and Speyer (2004) recently posed the following question: Given an arbitrary k-dissimilarity, how do we test whether this map comes from a tree? In this paper, we provide an answer to this question, showing that for k????3 a k-dissimilarity on a set X arises from a tree if and only if its restriction to every 2?k-element subset of X arises from some tree, and that 2?k is the least possible subset size to ensure that this is the case. As a corollary, we show that there exists a polynomial-time algorithm to determine when a k-dissimilarity arises from a tree. We also give a 6-point condition for determining when a 3-dissimilarity arises from a tree, that is similar to the aforementioned 4-point condition.     

A note on Cohen's profile similarity coefficientr c

Analytic procedures for classifying objects are commonly based on the product-moment correlation as a measure of object similarity. This statistic, however, generally does not represent an invariant index of similarity between two objects if they are measured along different bipolar variables where the direction of measurement for each variable is arbitrary. A computer simulation study compared Cohen's (1969) proposed solution to the problem, the invariant similarity coefficientr _c , with the mean product-moment correlation based on all possible changes in the measurement direction of individual variables within a profile of scores. The empirical observation thatr _c approaches the mean product-moment correlation with increases in the number of scores in the profiles was interpreted as encouragement for the use ofr _c in classification research. Some cautions regarding its application were noted.This research was supported by the Social Sciences and Humanities Research Council of Canada, Grant no. 410-83-0633, and by the University of Toronto.     

Optimal algorithms for comparing trees with labeled leaves

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.The Natural Sciences and Engineering Research Council of Canada partially supported this work with grant A-4142.     

n-Way Metrics

We study a family of n-way metrics that generalize the usual two-way metric. The n-way metrics are totally symmetric maps from E ⁿ into \mathbbR _{\geqslant 0} {\mathbb{R}_{ \geqslant 0}} . The three-way metrics introduced by Joly and Le Calvé (1995) and Heiser and Bennani (1997) and the n-way metrics studied in Deza and Rosenberg (2000) belong to this family. It is shown how the n-way metrics and n-way distance measures are related to (n − 1)-way metrics, respectively, (n − 1)-way distance measures.     

Unidimensional scaling: A linear programming approach minimizing absolute deviations

A Mixed Integer Programming formulation can be developed for the classical unidimensional scaling problem when the measure of goodness-of-fit is thel ₁ norm of the discrepancies rather than the sum of the squares of the discrepancies.The author wishes to thank the editor and the three anonymous referees for their helpful and constructive comments.     

The median procedure for n-trees

Let (X,d) be a metric space The functionM:X ^k 2 ^x defined by is the minimum } is called themedian procedure and has been found useful in various applications involving the notion of consensus Here we present axioms that characterizeM whenX is a certain class of trees (hierarchical classifications), andd is the symmetric difference metricWe would like to thank the referees and Editor for helpful comments