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Studies in Classification, Data Analysis, and Knowledge Organization Managing Editors H.-H. Bock, Aachen W. Gaul, Karlsruhe M. Vichi, Rome Editorial Board Ph. Arabie, Newark D. Baier, Cottbus F. Critchley, Milton Keynes R. Decker, Bielefeld E. Diday, Paris M. Greenacre, Barcelona C. Lauro, Naples J. Meulman, Leiden P. Monari, Bologna S. Nishisato, Toronto N. Ohsumi, Tokyo O. Opitz, Augsburg G. Ritter, Passau M. Schader, Mannheim C. Weihs, Dortmund Titles in the Series: E. Diday, Y. Lechevallier, M. Schader, P. Bertrand, and B. Burtschy (Eds.) New Approaches in Classification and Data Analysis. 1994 (out of print) S. Borra, R. Rocci, M. Vichi, and M. Schader (Eds.) Advances in Classification and Data Analysis. 2001 W. Gaul and D. Pfeifer (Eds.) From Data to Knowledge. 1995 W. Gaul and G. Ritter (Eds.) Classification, Automation, and New Media. 2002 H.-H. Bock and W. Polasek (Eds.) Data Analysis and Information Systems. 1996 E. Diday, Y. Lechevallier, and O. Opitz (Eds.) Ordinal and Symbolic Data Analysis. 1996 R. Klar and O. Opitz (Eds.) Classification and Knowledge Organization. 1997 C. Hayashi, N. Ohsumi, K. Yajima, Y. Tanaka, H.-H. Bock, and Y. Baba (Eds.) Data Science, Classification, and Related Methods. 1998 I. Balderjahn, R. Mathar, and M. Schader (Eds.) Classification, Data Analysis, and Data Highways. 1998 A. Rizzi, M. Vichi, and H.-H. Bock (Eds.) Advances in Data Science and Classification. 1998 M. Vichi and O. Opitz (Eds.) Classification and Data Analysis. 1999 W. Gaul and H. Locarek-Junge (Eds.) Classification in the Information Age. 1999 H.-H. Bock and E. Diday (Eds.) Analysis of Symbolic Data. 2000 H. A. L. Kiers, J.-P. Rasson, P. J.F. Groenen, and M. Schader (Eds.) Data Analysis, Classification, and Related Methods. 2000 W. Gaul, O. Opitz, and M. Schader (Eds.) Data Analysis. 2000 R. Decker and W. Gaul (Eds.) Classification and Information Processing at the Turn of the Millenium. 2000 K. Jajuga, A. Sokołowski, and H.-H. Bock (Eds.) Classification, Clustering and Data Analysis. 2002 M. Schwaiger and O. Opitz (Eds.) Exploratory Data Analysis in Empirical Research. 2003 M. Schader, W. Gaul, and M. Vichi (Eds.) Between Data Science and Applied Data Analysis. 2003 H.-H. Bock, M. Chiodi, and A. Mineo (Eds.) Advances in Multivariate Data Analysis. 2004 D. Banks, L. House, F.R. McMorris, P. Arabie, and W. Gaul (Eds.) Classification, Clustering, and Data Mining Applications. 2004 D. Baier and K.-D. Wernecke (Eds.) Innovations in Classification, Data Science, and Information Systems. 2005 M. Vichi, P. Monari, S. Mignani and A. Montanari (Eds.) New Developments in Classification and Data Analysis. 2005 D. Baier, R. Decker, and L. SchmidtThieme (Eds.) Data Analysis and Decision Support. 2005 C. Weihs and W. Gaul (Eds.) Classification the Ubiquitous Challenge. 2005 M. Spiliopoulou, R. Kruse, C. Borgelt, A. Nürnberger and W. Gaul (Eds.) From Data and Information Analysis to Knowledge Engineering. 2006 Vladimir Batagelj · Hans-Hermann Bock Anuška Ferligoj · Aleš Žiberna Editors Data Science and Classification With 67 Figures and 42 Tables 123 Prof. Dr. Vladimir Batagelj Department of Mathematics, FMF University of Ljubljana Jadranska 19 1000 Ljubljana, Slovenia [email protected] Prof. Dr. Anuška Ferligoj Faculty of Social Sciences University of Ljubljana Kardeljeva pl. 5 1000 Ljubljana, Slovenia [email protected] Prof. Dr. Hans-Hermann Bock Institute of Statistics RWTH Aachen University 52056 Aachen, Germany [email protected] Aleš Žiberna Faculty of Social Sciences University of Ljubljana Kardeljeva pl. 5 1000 Ljubljana, Slovenia [email protected] ISSN 1431-8814 ISBN-10 3-540-34415-2 Springer Berlin Heidelberg New York ISBN-13 978-3-540-34415-5 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. Springer · Part of Springer Science+Business Media springer.com © Springer-Verlag Berlin · Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Softcover-Design: Erich Kirchner, Heidelberg SPIN 11759263 43/3100/YL – 5 4 3 2 1 0 – Printed on acid-free paper Preface This volume contains a refereed selection of papers presented during the 10th Jubilee Conference of the International Federation of Classification Societies (IFCS) on Data Science and Classification held at the Faculty of Social Sciences of the University of Ljubljana in Slovenia, July 25-29, 2006. Papers submitted for the conference were subjected to a careful reviewing process involving at least two reviewers per paper. As a result of this reviewing process, 37 papers were selected for publication in this volume. The book presents recent advances in data analysis, classification and clustering from methodological, theoretical, or algorithmic points of view. It shows how data analysis methods can be applied in various subject-specific domains. Areas that receive particular attention in this book are similarity and dissimilarity analysis, discrimination and clustering, network and graph analysis, and the processing of symbolic data. Special sections are devoted to data and web mining and to the application of data analysis methods in quantitative musicology and microbiology. Readers will find a fine selection of recent technical and application-oriented papers that characterize the current developments in data science and classification. The combination of new methodological advances with the wide range of real applications collected in this volume will be of special value for researchers when choosing appropriate newly developed analytical tools for their research problems in classification and data analysis. The editors are grateful to the authors of the papers in this volume for their contributions and for their willingness to respond so positively to the time constraints in preparing the final versions of their papers. Without their work there would be no book. We are especially grateful to the referees – listed at the end of this book – who reviewed the submitted papers. Their careful reviews helped us greatly in selecting the papers included in this volume. We also thank Dr. Martina Bihn and the staff of Springer-Verlag, Heidelberg for their support and dedication for publishing this volume in the series Studies in Classification, Data Analysis, and Knowledge Organization. May 2006 Vladimir Batagelj, Ljubljana Hans–Hermann Bock, Aachen Anuška Ferligoj, Ljubljana Aleš Žiberna, Ljubljana The 10th IFCS Conference – a Jubilee The International Federation of Classification Societies (IFCS) was founded July 4, 1985 in Cambridge (UK) at a time when classification problems were frequently being encountered in such diverse fields as biology, marketing, social sciences, pattern recognition, picture processing, information retrieval, and library science. These often massive problems had to be solved by quantitative or computerized methods based on data and measurements. In fact, the IFCS founding agreement paved the way for an intensive bilateral and multilateral cooperation and scientific as well as personal contacts among the members of the six ‘national’ classification societies from United Kingdom (BCS), North America (CSNA), Germany (GfKl), Japan (JCS), France (SFC), and Italy (SIS) that formed the IFCS in those times. A main activity of IFCS is the organization of a biennial conference series. The first one with the title ‘Classification, Data Analysis, and Related Methods’ was held in Aachen (Germany) from June 29 to July 1, 1987 with about 300 participants from 25 countries and more than 180 scientific papers. Since that time, eight more IFCS conferences have taken place at different locations around the world (see the table below), always with a broad international participation from inside and outside the IFCS. Typically, a selection of scientific papers was published in a Proceedings volume in the Springer series ‘Studies in Classification, Data Analysis, and Knowledge Organization’ so that the results became available worldwide. The biennial IFCS conferences Year 1987 1989 1991 1993 1996 1998 2000 2002 2004 2006 Place Aachen (Germany) Charlottesville (USA) Edinburgh (UK) Paris (France) Kobe (Japan) Rome (Italy) Namur (Belgium) Cracow (Poland) Chicago (USA) Ljubljana (Slovenia) Hosting Society Organizer GfKl CSNA BCS SFC JCS SIS SFC, VOC SKAD CSNA SDS H.-H. Bock H. Bozdogan D. Wishart, A. Gordon E. Diday Ch. Hayashi A. Rizzi, M. Vichi J.-P. Rasson, H. Kiers A. Sokolowski F.R. McMorris A. Ferligoj, V. Batagelj As a consequence, more and more national groups or societies working in the classification and data analysis field joined the IFCS: the VOC from Belgium/Netherlands, the CLAD from Portugal, the Polish society (SKAD), VIII The 10th IFCS Conference the KCS from Korea, the Irish Pattern Recognition and Classification Society (IPRCS), and finally the Central American and Caribbean society (SoCCAD). The 10th IFCS conference is being hosted by the Statistical Society of Slovenia (SDS) at the University of Ljublajana (Slovenia), in July 2006, with Anuska Ferligoj and Vladimir Batagelj chairing. It will, without any doubt, be a new highlight in the history of IFCS, provide a challenging marketplace for scientific and applied research results, and foster further cooperation and contacts within the worldwide classification community. This Jubilee Conference is certainly an occasion for recalling the history and achievements of the 20 years of IFCS’s life. But it also marks the beginning of another decade of tasks and activities for IFCS: with new challenges for research and application, with interesting scientific conferences, with an intensive cooperation among IFCS members, and hopefully also a large impact on the worldwide development of our favorite domains: data analysis and classification. May 2006 Hans-Hermann Bock Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V The 10th IFCS Conference – a Jubilee . . . . . . . . . . . . . . . . . . . . . . . . VII Hans-Hermann Bock Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX Part I. Similarity and Dissimilarity A Tree-Based Similarity for Evaluating Concept Proximities in an Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emmanuel Blanchard, Pascale Kuntz, Mounira Harzallah, Henri Briand 3 Improved Fréchet Distance for Time Series . . . . . . . . . . . . . . . . . . . 13 Ahlame Chouakria-Douzal, Panduranga Naidu Nagabhushan Comparison of Distance Indices Between Partitions . . . . . . . . . . . 21 Lucile Denœud, Alain Guénoche Design of Dissimilarity Measures: A New Dissimilarity Between Species Distribution Areas . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Christian Hennig, Bernhard Hausdorf Dissimilarities for Web Usage Mining . . . . . . . . . . . . . . . . . . . . . . . . . 39 Fabrice Rossi, Francisco De Carvalho, Yves Lechevallier, Alzennyr Da Silva Properties and Performance of Shape Similarity Measures . . . . 47 Remco C. Veltkamp, Longin Jan Latecki Part II. Classification and Clustering Hierarchical Clustering for Boxplot Variables . . . . . . . . . . . . . . . . . 59 Javier Arroyo, Carlos Maté, Antonio Muñoz-San Roque Evaluation of Allocation Rules Under Some Cost Constraints . 67 Farid Beninel, Michel Grun Rehomme Crisp Partitions Induced by a Fuzzy Set . . . . . . . . . . . . . . . . . . . . . . 75 Slavka Bodjanova X Contents Empirical Comparison of a Monothetic Divisive Clustering Method with the Ward and the k-means Clustering Methods . 83 Marie Chavent, Yves Lechevallier Model Selection for the Binary Latent Class Model: A Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 José G. Dias Finding Meaningful and Stable Clusters Using Local Cluster Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Hans-Joachim Mucha Comparing Optimal Individual and Collective Assessment Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Hans J. Vos, Ruth Ben-Yashar, Shmuel Nitzan Part III. Network and Graph Analysis Some Open Problem Sets for Generalized Blockmodeling . . . . . 119 Patrick Doreian Spectral Clustering and Multidimensional Scaling: A Unified View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 François Bavaud Analyzing the Structure of U.S. Patents Network . . . . . . . . . . . . . 141 Vladimir Batagelj, Nataša Kejžar, Simona Korenjak-Černe, Matjaž Zaveršnik Identifying and Classifying Social Groups: A Machine Learning Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Matteo Roffilli, Alessandro Lomi Part IV. Analysis of Symbolic Data Multidimensional Scaling of Histogram Dissimilarities . . . . . . . . 161 Patrick J.F. Groenen, Suzanne Winsberg Dependence and Interdependence Analysis for Interval-Valued Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Carlo Lauro, Federica Gioia A New Wasserstein Based Distance for the Hierarchical Clustering Of Histogram Symbolic Data . . . . . . . . . . . . . . . . . . . . . . 185 Antonio Irpino, Rosanna Verde Contents XI Symbolic Clustering of Large Datasets . . . . . . . . . . . . . . . . . . . . . . . . 193 Yves Lechevallier, Rosanna Verde, Francisco de A.T. de Carvalho A Dynamic Clustering Method for Mixed Feature-Type Symbolic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Renata M.C.R. de Souza, Francisco de A.T. de Carvalho, Daniel Ferrari Pizzato Part V. General Data Analysis Methods Iterated Boosting for Outlier Detection . . . . . . . . . . . . . . . . . . . . . . . 213 Nathalie Cheze, Jean-Michel Poggi Sub-species of Homopus Areolatus? Biplots and Small Class Inference with Analysis of Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Sugnet Gardner, Niël J. le Roux Revised Boxplot Based Discretization as the Kernel of Automatic Interpretation of Classes Using Numerical Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Karina Gibert, Alejandra Pérez-Bonilla Part VI. Data and Web Mining Comparison of Two Methods for Detecting and Correcting Systematic Errors in High-throughput Screening Data . . . . . . . . 241 Andrei Gagarin, Dmytro Kevorkov, Vladimir Makarenkov, Pablo Zentilli kNN Versus SVM in the Collaborative Filtering Framework . . 251 Miha Grčar, Blaž Fortuna, Dunja Mladenič, Marko Grobelnik Mining Association Rules in Folksonomies . . . . . . . . . . . . . . . . . . . . 261 Christoph Schmitz, Andreas Hotho, Robert Jäschke, Gerd Stumme Empirical Analysis of Attribute-Aware Recommendation Algorithms with Variable Synthetic Data . . . . . . . . . . . . . . . . . . . . . 271 Karen H. L. Tso, Lars Schmidt-Thieme Patterns of Associations in Finite Sets of Items . . . . . . . . . . . . . . . 279 Ralf Wagner XII Contents Part VII. Analysis of Music Data Generalized N-gram Measures for Melodic Similarity . . . . . . . . . 289 Klaus Frieler Evaluating Different Approaches to Measuring the Similarity of Melodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Daniel Müllensiefen, Klaus Frieler Using MCMC as a Stochastic Optimization Procedure for Musical Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Katrin Sommer, Claus Weihs Local Models in Register Classification by Timbre . . . . . . . . . . . . 315 Claus Weihs, Gero Szepannek, Uwe Ligges, Karsten Luebke, Nils Raabe Part VIII. Gene and Microarray Analysis Improving the Performance of Principal Components for Classification of Gene Expression Data Through Feature Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Edgar Acuña, Jaime Porras A New Efficient Method for Assessing Missing Nucleotides in DNA Sequences in the Framework of a Generic Evolutionary Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Abdoulaye Baniré Diallo, Vladimir Makarenkov, Mathieu Blanchette, François-Joseph Lapointe New Efficient Algorithm for Modeling Partial and Complete Gene Transfer Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Vladimir Makarenkov, Alix Boc, Charles F. Delwiche, Alpha Boubacar Diallo, Hervé Philippe List of Reviewers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 Part I Similarity and Dissimilarity A Tree-Based Similarity for Evaluating Concept Proximities in an Ontology Emmanuel Blanchard, Pascale Kuntz, Mounira Harzallah, and Henri Briand Laboratoire d’informatique de Nantes Atlantique Site École polytechnique de l’université de Nantes rue Christian Pauc BP 50609 - 44306 Nantes Cedex 3 [email protected] Abstract. The problem of evaluating semantic similarity in a network structure knows a noticeable renewal of interest linked to the importance of the ontologies in the semantic Web. Different semantic measures have been proposed in the literature to evaluate the strength of the semantic link between two concepts or two groups of concepts within either two different ontologies or the same ontology. This paper presents a theoretical study synthesis of some semantic measures based on an ontology restricted to subsumption links. We outline some limitations of these measures and introduce a new one: the Proportion of Shared Specificity. This measure which does not depend on an external corpus, takes into account the density of links in the graph between two concepts. A numerical comparison of the different measures has been made on different large size samples from WordNet. 1 Introduction With a long history in psychology (Tversky (1977), Rosch (1975)), the problem of evaluating semantic similarity in a network structure knows a noticeable renewed interest linked to the development of the semantic Web. In the 70’s many researches on categorization were influenced by a theory which states that, from an external point of view, the categories on a set of objects were organized in a taxonomy according to an abstraction process. Describing proximity relationships between domain concepts by a hierarchy, or more generally a graph, remains a common principle of the current knowledge representation systems, namely the ontologies associated with the new languages of the semantic Web –in particular OWL (Bechhofer et al. (2004)). As defined by Gruber (1993), “an ontology is a formal, explicit specification of a shared conceptualization” and “a conceptualization is an abstract, simplified view of the world that we wish to represent for some purpose”. From an operational point of view, the development and the exploitation of an ontology remains a complex task in a global process of knowledge engineering. Upstream, extracting and structuring large sets of concepts with increasing sizes represents one of the major difficulties. Downstream, retrieving subsets of concepts requires approaches that are not time-consuming and are efficient in terms of semantic relevance of the results. To make these 4 Blanchard et al. tasks easier, some proposals resort to an estimation of a semantic similarity between the pairs of concepts. Generally speaking, a “semantic similarity” σ(oi , oj ) between two objects oi and oj is related to their commonalities and sometimes their differences. Most of the definitions considered in the literature suppose that the objects are associated with their extension –a collection of elements or a set of descriptors. In this case, the semantic similarities can be roughly classified in three main classes depending on the object description: 1. The extensions are simply subsets of elements. Then, the similarity σ(oi , oj ) between two objects oi and oj is a function of the subsets of the elements common (set intersection) to and the elements different (set symmetric difference) for oi and oj . This class includes similarities wellknown in the taxonomic literature such as the Jaccard’s coefficient and the Dice’s coefficient. 2. Each element in the domain is taken to be a dimension in a vector space (Salton and McGill (1983)), and each object oi can be described by a vector whose components describe the elements in the collection. This representation popular in the information retrieval domain makes use of the usual Cosine-Similarity, and many adaptations have been proposed (Salton and McGill (1983)). 3. A hierarchical domain structure is given. The leaves are the elements of the collection and nodes represents an organization of concepts. Different similarities –often generalizations of case 2– have been proposed to exploit the hierarchical structure (see Ganesan et al. (2003) for a recent review). Very recently, the problem has been extended to graphs (Maguitman et al. (2005)). When considering ontologies, the concepts are not necessarily described by their extension ; the internal organization of a domain ontology is often the product of a consensus of experts (Guarino (1995)). Hence, defining a semantic similarity σ(ci , cj ) between a pair ci , cj of concepts sets specific problems depending on the information at our disposal. Some measures only depend on the concept structuring –often hierarchical– ; others, in addition, require textual corpus of the domain. This paper presents a review of the main measures of the literature available for comparing pairs of concepts within a domain ontology. Our description in a unified framework allows to highlight their commonalities and their differences. In particular, we show that they can be defined as functions of a restricted set of parameters, and that only one of the measures based on a corpus exploits all the parameters. To overcome this limitation, we propose a new measure, the Proportion of Shared Specificity (PSS) which is an adaptation of the Dice’s coefficient to a hierarchical structure. We show that the Wu and Palmer’s coefficient is a particular case of our measure. And, we present experimental comparisons on different large size samples extracted from the semantic network WordNet 2.0. A Similarity in an Ontology 2 5 Graph-based similarities for domain ontologies Formally, an ontology can be modeled by a graph where nodes represent concepts and arcs represent labeled relationships. Here, like often in the literature, we restrict ourselves to the hyperonymy and hyponymy relationships associated to the relationship of subsumption (is-a). For instance: “a dog is an animal” implies that dog is an hyponym of animal and animal is an hyperonym of dog. This relationship is common to every ontology, and different researches have been confirmed that it is the more structuring one (e.g. Rada et al. (1989)). Let O be a domain ontology with a set C = {c1 , c2 , . . . , cn } of concepts. The relationships “hyperonymy” (generalization) and “hyponymy” (specialization) are dual : for any pair (ci , cj ) ∈ C × C if hyperonymy(ci , cj ) then hyponymy(cj , ci ), and vice-versa. And, each concept ci has no more than one hyperonym. Moreover, to maintain the connectivity, it is common to add a virtual concept c0 (“thing” or “entity”). Consequently, O can be modeled by a rooted tree TO (C, A) with the root c0 and so that each arc (ci , cj ) ∈ A represents an hyponymy relationship between ci and cj (figure 1). By construction of an ontology, the deeper is a concept in TO (C, A), the more specific it is. We adopt this restricted framework of a rooted tree in the following of this paper. entity object artifact instrumentality device musical_instrument music_box wind_instrument electronic_instrument percussion_instrument Fig. 1. Example extracted from WordNet. The graph-based measures –also called structural measures– can be regrouped in two main classes : the functions of combinatorial properties, and the measures which incorporate an additional knowledge source from a corpus analysis. Note that, depending on the authors, the definitions have been originally given on the form of a dissimilarity δ or a similarity σ ; as the transformation of σ in δ is not unique, we here retain this diversity. 6 2.1 Blanchard et al. Approaches using combinatorial properties A comparative study of the different measures based on combinatorial properties shows that their definitions depends on four main components: (i) the length (edge number) len(ci , cj ) of the path between two concepts ci , cj in TO (C, A), (ii) the most specific subsumer mscs(ci , cj ) (the lowest common ancestor in the tree) of ci and cj , (iii) the depth h(ci ) (length from the root to ci ) of a concept ci in TO (C, A). The simplest dissimilarity, proposed by Rada et al. (1989), is δr (ci , cj ) = len(ci , cj ). Despite its simplicity, experiments in information retrieval have shown that, when the paths are restricted to is-a links like here, this measure could contribute to good results. A normalization has been introduced by len(c ,cj ) Leacock and Chodorow (1998): σlc (ci , cj ) = − log 2·maxc∈Ci [h(c . 0 )] The Wu and Palmer similarity takes into account the specificity of the 2·h(mscs(ci ,cj )) . subsumer of ci and cj : σwp (ci , cj ) = len(ci ,cj )+2·h(mscs(c i ,cj )) Based on linguistic properties, Sussna (1993) have introduced a weight function for the relationships : wr (ci , cj ) for the hyperonymy (resp. wo (cj , ci ) for the hyponymy). When the concepts ci , cj are adjacent in TO (C, A), the wr (ci ,cj )+wo (ci ,cj ) . dissimilarity is a scaled sum of the weights : δs (ci , cj ) = 2·max(h(c i ),h(cj )) For two arbitrary concepts, it is computed as the sum of the dissimilarities between the pairs of adjacent concepts along the path connecting them. 2.2 Approaches using a corpus These approaches introduce a measure of the information shared by a pair of concepts. One criterion to evaluate the similarity between two concepts is the extent to which they share information in common. The required additional notion is the probability P (ci ) ∈ [0, 1] of encountering an occurrence of ci . In practice, this probability is estimated from a text corpus S by the occurrence frequency of ci in S num(ci )/num(c0 ). To compute num(ci ), Resnik (1995) proposes to count not only the number of occurrences of ci , but also the number of occurrences of the concepts which are subsumed by ci . For Resnik, the information shared by two concepts ci and cj is the “information content” of their most specific common subsumer : σr (ci , cj ) = − log P (mscs(ci , cj )). Lin (1998) and Jiang and Conrath (1997) moderate this value by the quantity of information which distinguishes the two con2·log P (mscs(c ,c )) cepts : σl (ci , cj ) = log P (ci )+log Pi (cjj ) and δjc (ci , cj ) = 2 · log P (mscs(ci , cj )) − (log P (ci ) + log P (cj )). 3 A new measure: the Proportion of Shared Specificity Each of the previous measures attempts to exploit the information contained in the ontology at best to evaluate the similarity between the pairs of concepts A Similarity in an Ontology 7 (ci , cj ). The Rada’s coefficient is the simplest ; it takes only into account the length of the paths joining ci and cj . The Lin’s coefficient is the more complex ; it takes into account the common information shared by the concepts and, via the estimation of P (ci ), the density of concepts between the root c0 and ci and cj . When the density is high, this means that the hyponyms are more specific, and consequently the Lin’s similarity is higher. However, the computation of the Lin’s coefficient requires a corpus in addition to the ontology. And this latter should be significant enough to provide a “good” estimation of P (ci ). As this condition may be very restrictive for numerous real-life applications, we have developed a new measure which shares some properties of the Lin’s coefficient but which only depends on TO (C, A). When each concept ci is described by a set of unitary characteristics cha(ci ), one of the most commonly used measure is the Dice coefficient (Ganesan et al. (2003)): 2 · |cha(ci ) cha(cj )| σd (ci , cj ) = (1) |cha(ci )| + |cha(cj )| In this model, two concepts are all the more similar since they share numerous common characteristics and few different ones. We here propose to adapt this formula to measure the (dis)similarity between a pair of concepts in an ontology. Intuitively, |cha(ci )| is an indicator of a known part of the information carried by the concept ci . In the approaches using a corpus (2.2) the quantity of information carried by a concept is evaluated by − log(P (ci )). Let us remark, that when replacing in (1) |cha(ci )| (resp. |cha(cj )|) by − log(P (ci )) (resp. − log(P (cj ))) and |cha(ci ) ∩ cha(cj )| by − log(P (mscs(ci , cj ))) we rediscover the Lin’s formula. When no additional information is available, we can exploit the hierarchical structure of the ontology TO (C, A). Our reasoning rests on an analogy with the case where we dispose of a set of instances. In an ontological model, each concept is not necessarily described by a collection of characteristics, and the associated instances are not explicitly given. Nevertheless, by construction, we can consider that the probability of an instance to be associated with the more general concept c0 (the root) is equal to 1. If c0 specializes k in k hyponyms c11 , c21 , . . . , ck1 , then i=1 P (ci1 ) = 1. And, if the distribution of the instance number is supposed to be uniform for each concept, then P (ci1 ) = k1 . Then, when this assumption is available for all the levels of TO (C, A), for any concept ci deeper in TO , we have a series of specialisations 1 in k1 , k2 , . . . , kh(ci ) concepts. Then P (ci ) = k1 ·k2 ·····k . Consequently, the h(ci ) quantity of information associated with the specificity of ci is measured by log(k1 ) + log(k2 ) + · · · + log(kh(ci ) ). Moreover, the quantity of information shared by two concepts ci and cj can be measured, like in the Lin’s coefficient, by the quantity of information of their most specific subsumer. Hence, the Proportion of Shared Specificity 8 Blanchard et al. is defined by: 2 · log P (mscs(ci , cj )) σpss (ci , cj ) = log P (ci ) + log P (cj ) (2) Let us remark, that when we consider the simplest case where all the nodes have the same degree k, then P (ci ) = k −h(ci ) for any concept ci at depth h(ci ). Consequently, the quantity of information associated with the specificity of ci is measured by h(ci )·log k and we obtain the Wu and Palmer’s measure. 4 Experimental results This section presents an analysis of the behavior of the previous coefficients on samples of concepts drawn from the semantic network WordNet 2.0 (Fellbaum (1998)). Let us briefly recall that, inspired from the psycholinguistic theories on the lexical human memory, WordNet was created as an attempt to model the lexical knowledge of a native English speaker. The lexical entities (e.g. nouns, verbs, adjectives) are organized into synonym sets that are interlinked with different relations. We here restrict ourselves to nouns and to the subsumption hierarchy (hyperonymy/hyponymy). This hierarchy, which contains 146690 nodes, constitutes the backbone of the noun subnetwork accounting for close to 80% of the links (Budanitsky (1999)). It can be properly represented by the tree model TO (C, A) described in section 2, and consequently, for our experiments we do not enter into the discussion between experts concerning the ontological nature of WordNet. The computations have been performed with the Perl modules of Pedersen et al. (2004). We have randomly drawn a sample of 1000 concepts from TO , and computed the coefficient values for each pair. Figure 2 shows the distributions associated with the different measures. Due to the random draw and the size of TO , numerous pairs contain concepts without semantic links, and it is not surprising to find numerous null values (around 60%). Contrary to other combinatorial measures, the PSS coefficient allows a more subtle differentiation of the concepts. Due to the theoretical relationships between σpss , σl (Lin) and σwp (Wu & Palmer) we have analyzed the values of the rank correlation of Spearman on the pair set of the sample : ρ(wp, l) = 0.813, ρ(l, pss) = 0.835 and ρ(pss, wp) = 0.970. Obviously, it is interesting to find a strong correlation between the PSS coefficient and the Lin’s coefficient which requires an additional corpus. However, a part of these high values can be explained by the great number of distant pairs. Nevertheless, when we restrict ourselves to the subset which contains 1% of the most similar pairs both for σpss and σl , the correlation is still significant (ρ(pss, l) = 0.53). To go deeper in the analysis, we have computed the correlations on different subtrees of TO associated with identified sets of themes (e.g. insect, tree, 0.2 0.4 0.6 0.8 20000 0.0 0.2 0.4 0.6 0.8 1.0 Resnik similarity Proportion of Shared Specificity 40000 4 6 8 10 12 measure values 0 8000 measure values value numbers measure values 2 0.0 0.2 0.4 0.6 0.8 1.0 measure values 6000 Wu and Palmer similarity with length in number of edges 0 value numbers 0 1.0 0 value numbers 0.0 9 Lin similarity value numbers 100000 Inverse of the Rada distance beforehand raised by one 0 value numbers A Similarity in an Ontology 0.0 0.2 0.4 0.6 0.8 1.0 measure values Fig. 2. histograms of value numbers. musical instrument). Table 1 gives the different correlations. As WordNet is homogeneous (the number of hyponyms is barely constant), the correlation between Wu Palmer and PSS measures is naturally strong. It is important to note that ρ(pss, l) > ρ(w, l) for all our experiments. Table 1. Correlation on three WordNet subsets root insect tree musical instrument number of concepts 157 454 1013 number of pairs 12246 102831 512578 median number of hyponyms 3 2 3 ρ(pss, l) 0.65 0.53 0.30 ρ(pss, wp) 0.91 0.85 0.90 ρ(w, l) 0.63 0.52 0.27 10 5 Blanchard et al. Conclusion This paper presents a new similarity for evaluating the strength of the semantic links between pairs of concepts in an ontology. Its computation does not require an external corpus like the well-known Lin’s coefficient. At the origin, our objective was guided by real-life applications, in particular in knowledge management (Berio and Harzallah (2005), Laukkanen and Helin (2005)), where additional corpuses are rarely available. From the Dice’s coefficient, we have built a measure which exploits the structural properties of the ontology. Our numerical experiments on WordNet have confirmed its discriminant behavior and highlighted its links with other coefficients of the literature. The main arguments for the choice of this experimental framework are its size –which allows statistical analysis– and its computational accessibility. However, to assess the semantic significance of the results obtained with the PSS coefficient, we plan to apply it in the near future to a professional environment ontology. Moreover, we have here restricted ourselves to a hierarchical structure deduced from the “is-a” link. Although this structure is known to be the most structuring of a real-life ontology, we now attempt to generalize our approach to a graph structure to simultaneously take other links into account. References BECHHOFER, S., VAN HARMELEN, F., HENDLER, J., HORROCKS, I., MCGUINNESS, D. L., PATEL-SCHNEIDER, P. F., AND STEIN, L. A. (2004): Owl web ontology language reference. http://www.w3.org/TR/2004/REC-owl-ref-20040210/. BERIO, G. AND HARZALLAH, M. (2005): Knowledge management for competence management. Universal Knowledge Management, 0. BUDANITSKY, A. (1999): Lexical semantic relatedness and its application in natural language processing. Technical report, Computer Systems Research Group – University of Toronto. FELLBAUM, C., editor (1998): WordNet: An electronic lexical database. MIT Press. GANESAN, P., GARCIA-MOLINA, H., AND WIDOM, J. (2003): Exploiting hierarchical domain structure to compute similarity. ACM Trans. Inf. Syst., 21(1):64–93. GRUBER, T. R. (1993): A translation approach to portable ontology specifications. Knowledge Acquisition, 5(2):199–220. GUARINO, N. (1995): Formal ontology, conceptual analysis and knowledge representation. Human-Computer Studies, 43(5/6):625–640. JIANG, J. J. AND CONRATH, D. W. (1997): Semantic similarity based on corpus statistics and lexical taxonomy. In Proc. of Int. Conf. on Research in Comp. Linguistics. LAUKKANEN, M. AND HELIN, H. (2005): Competence management within and between organizations. In Proc. of 2nd Interop-EMOI Workshop on Enterprise A Similarity in an Ontology 11 Models and Ontologies for Interoperability at the 17th Conf. on Advanced Information Systems Engineering, pages 359–362. Springer. LEACOCK, C. AND CHODOROW, M. (1998): Combining local context and wordnet similarity for word sense identification. In Fellbaum, C., editor, WordNet: An electronic lexical database, pages 265–283. MIT Press. LIN, D. (1998): An information-theoretic definition of similarity. In Proc. of the 15th Int. Conf. on Machine Learning, pages 296–304. Morgan Kaufmann. MAGUITMAN, A. G., MENCZER, F., ROINESTAD, H., AND VESPIGNANI, A. (2005): Algorithmic detection of semantic similarity. In Proc. of the 14th int. conf. on World Wide Web, pages 107–116. ACM Press. PEDERSEN, T., PATWARDHAN, S., AND MICHELIZZI, J. (2004): Wordnet::similarity – measuring the relatedness of concepts. In Proc. of the Fifth Annual Meeting of the North American Chapter of the Association for Comp. Linguistics, pages 38–41. RADA, R., MILI, H., BICKNELL, E., AND BLETTNER, M. (1989): Development and application of a metric on semantic nets. IEEE Transactions on Systems, Man, and Cybernetics, 19(1):17–30. RESNIK, P. (1995): Using information content to evaluate semantic similarity in a taxonomy. In Proc. of the 14th Int. Joint Conf. on Artificial Intelligence, volume 1, pages 448–453. ROSCH, E. (1975): Cognitive representations of semantic categories. Experimental Psychology: Human Perception and Performance, 1:303–322. SALTON, G. AND MCGILL, M. J. (1983): Introduction to modern information retrieval. McGraw-Hill. SUSSNA, M. (1993): Word sense disambiguation for free-text indexing using a massive semantic network. In Proc. of the Sec. Int. Conf. on Information and Knowledge Management, pages 67–74. TVERSKY, A. (1977): Features of similarity. Psychological Review, 84(4):327–352. Improved Fréchet Distance for Time Series Ahlame Chouakria-Douzal1 and Panduranga Naidu Nagabhushan2 1 2 TIMC-IMAG, Université Joseph Fourier Grenoble 1, F-38706 LA TRONCHE Cedex, France [email protected] Dept. of Studies in Computer Science, University of Mysore Manasagangothri, Mysore, Karnataka- 570 006, India [email protected] Abstract. This paper focuses on the Fréchet distance introduced by Maurice Fréchet in 1906 to account for the proximity between curves (Fréchet (1906)). The major limitation of this proximity measure is that it is based on the closeness of the values independently of the local trends. To alleviate this set back, we propose a dissimilarity index extending the above estimates to include the information of dependency between local trends. A synthetic dataset is generated to reproduce and show the limited conditions for the Fréchet distance. The proposed dissimilarity index is then compared with the Fréchet estimate and results illustrating its efficiency are reported. 1 Introduction Time series differ from ”non-temporal” data due to the interdependence between measurements. This work focuses on the distances between time series, an important concept for time series clustering and pattern recognition tasks. The Fréchet distance is one of the most widely used proximity measure between time series. Fréchet distance uses time distortion by acceleration or deceleration transformations to look for a mapping that minimizes the distance between two time series. We show in section 4, that the Fréchet distance ignores the interdependence among the occurring values; proximity is only based on the closeness of the values; which can lead to irrelevant results. For this reason, we propose a dissimilarity index extending this classical distance to include the information of dependency between local trends. The rest of this paper is organized as follows: the next section presents the definitions and properties of the conventional Fréchet distance. Section 3, discusses the major limitations of such proximity estimate, then gives the definition and properties of the new dissimilarity index. Section 4, presents a synthetic dataset reproducing limited conditions for this widely used time series proximity measure, then perform a comparison between the proposed dissimilarity index and the Fréchet distance before concluding. 14 2 Chouakria-Douzal and Nagabhushan The Fréchet distance between time series The success of a distance, intended to distinguish the events of a time series that are similar from those that are different, depends on its adequacy with respect to the proximity concept underlying the application domain or the experimental context. The Fréchet distance was introduced by Maurice Fréchet in 1906 (Fréchet (1906)) to estimate the proximity between continuous curves. We present a discrete variant of this distance. An in-depth study of the Fréchet distance is provided by Alt (Alt and Godau (1992)) and an interesting comparison of the different distance theories can be found in Eiter and Mannila (1994). The popular and highly intuitive Fréchet distance definition is: ”A man is walking a dog on a leash. The man can move on one curve, the dog on another. Both may vary their speed independently, but are not allowed to go backwards. The Fréchet distance corresponds to the shortest leash that is necessary”. Let’s provide a more formal definition. We define a mapping r ∈ M between two time series S1 = (u1 , ..., up ) and S2 = (v1 , ..., vq ) as the sequence of m pairs preserving the observation order: r = ((ua1 , vb1 ), (ua2 , vb2 ), ..., (uam , vbm )) with ai ∈ {1, .., p}, bj ∈ {1, .., q} and satisfying for i ∈ {1, .., m − 1} the following constraints: a1 = 1, am = p ai+1 = ai or ai + 1 b1 = 1, bm = q bi+1 = bi or bi + 1 (1) (2) We note |r| = maxi=1,..,m |uai −vbi | the mapping length representing the maximum span between two coupled observations. The Fréchet distance δF (S1 , S2 ) is then defined as: δF (S1 , S2 ) = min |r| = min ( max |uai − vbi |) r∈M r∈M i=1,..,m (3) Graphically, a mapping between two time series S1 = (u1 , ..., up ) and S2 = (v1 , ..., vq ) can be represented by a path starting from the corner (1, 1) and reaching the corner (p, q) of a grid of dimension (p, q). The value of the square (i, j) is the span between the coupled observations (ui , vj ). The path length corresponds to the maximum span reached through the path. Then, the Fréchet distance between S1 and S2 is the minimum length through all the possible grid paths. We can easily check that δF is a metric verifying the identity, symmetry and triangular inequality properties (a proof can be found in Eiter and Mannila (1994)). According to δF two time series are similar if there exists a mapping between their observations, expressing an acceleration or a deceleration of the occurring observation times so that the maximum span between all coupled observations is close. Improved Fréchet Distance for Time Series 15 Note that the Fréchet distance is very useful when only the occurring events, not their occurring times, are determinant for the proximity evaluation. This explains the great success of Fréchet distance in the particular domain of voice processing where only the occurring syllables are used to identify words; the flow rate being specific to each person. 3 Fréchet distance extension for time series proximity estimation Generally, the interdependence among the occurring values, characterizing the local trends in the time series, is determinant for the time series proximity estimation. Thus, Fréchet distance fails as it ignores such main information. Section 4 illustrates two major constraints in the Fréchet measure: ignorance of the temporal structure and the sensitivity to global trends. To alleviate these drawbacks in the classical Fréchet estimate we propose a dissimilarity index extending Fréchet distance to include the information of dependency between the time series local trends. The dissimilarity index consists of two components. The first one estimates the closeness of values and is based on a normalized form of the conventional proximity measure. The second component, based on the temporal correlation Von Neumann (1941-1942)), Geary (1954) and (Chouakria-Douzal (2003), estimates the dependency between the local trends. 3.1 Temporal correlation Let’s first recall the definition of the temporal correlation between two time series S1 = (u1 , ..., up ) and S2 = (v1 , ..., vp ): p−1 (u(i+1) − ui )(v(i+1) − vi ) cort(S1 , S2 ) = i=1 p−1 p−1 2 2 i=1 (u(i+1) − ui ) i=1 (v(i+1) − vi ) The temporal correlation coefficient cort ∈ [−1, 1] estimates how much the local trends observed simultaneously on both times series, are positively/negatively dependent. By dependence between time series we mean a stochastic linear dependence: if we know at a given time t the growth of the first time series then we can predict, through a linear relationship, the growth of the second time series at that time t. Similar to the classical correlation coefficient, a value of cort = 1 means that, at a given time t, the trends observed on both time series are similar in direction and rate of growth, a value of -1 means that, at a given time t, the trends observed on both time series are similar in rate but opposite in direction and finally, a value of 0 expresses that the trends observed on both time series are stochastically linearly independent. 16 Chouakria-Douzal and Nagabhushan Contrary to the classical correlation coefficient, the temporal correlation estimates locally not globally the dependency between trends; indeed, two time series may be highly dependent through the classical correlation and linearly independent through the temporal correlation (illustrated in section 4). Finally, contrary to classical correlation, the temporal correlation is global trend effect free. Let’s now present the new dissimilarity index as an extension of the Fréchet distance. 3.2 The dissimilarity index The proposed dissimilarity index consists in the combination of two components. The first one, estimates the closeness of values and is based on a normalized form of the Fréchet distance. The second one is based on the temporal correlation introduced above. Many functions could be explored for such combination function. To illustrate well the additive value of the temporal correlation to account for local trends dependency, we limit this work to a linear combination function. Let’s note DisF the dissimilarity index extending δF : DisF (S1 , S2 ) = α δF (S1 , S2 ) maxSi ,Sj ∈ΩS δF (Si , Sj ) + (1 − α) 1 − cort(S1 , S2 ) 2 where DisF (S1 , S2 ) ∈ [0, 1], ΩS is the set of the observed time series, α ∈ [0, 1] determines the weight of each component in the dissimilarity evaluation and cort the temporal correlation defined above. Note that for α = 1, DisF corresponds to the normalized δF and the proximity between two time series is only based on taken values, considered as independent observations. For α = 0, DisF corresponds to cort and the proximity is based solely on the dependency between local trends. Finally for 0 < α < 1, DisF implies a weighted mean of the normalized δF and cort, the proximity between time series includes then, according to their weights, both the proximity between occurring values and the dependency between local trends. 4 Applications and results In this section, we first present the time series synthetic dataset which reproduces the limited conditions for Fréchet distance. Then we explore and compare the distribution of the temporal and classical correlations between the synthetic dataset time series. Finally, the proposed dissimilarity index is compared to the conventional estimate. 4.1 Synthetic dataset To reproduce the limited conditions for the widely used conventional distances, we consider a synthetic dataset of 15 time series divided into three Improved Fréchet Distance for Time Series 17 classes of functions. The first five time series are of class F1 , the next five are of class F2 and the last five are of the class F3 ; where, F1 , F2 and F3 are defined as follows: F1 = {f1 (t) | f1 (t) = f (t) + 2t + 3 + } F2 = {f2 (t) | f2 (t) = µ − f (t) + 2t + 3 + } F3 = {f3 (t) | f3 (t) = 4f (t) − 3 + } 40 50 f (t) is a given discrete function, µ = E(f (t)) is the mean of f (t) through the observation period, ; N (0, 1) is a zero mean gaussian distribution and 2t + 3 describes a linear upward trend tainting F1 and F2 classes. Figure 1 represents simultaneously these three classes through 15 synthetic time series. Note that F1 and F3 show similar local tendencies, they increase (respectively F1(x) 10 20 F(x) 30 F3(x) 0 F2(x) 2 4 6 8 10 Time Fig. 1. Three classes of synthetic time series decrease) simultaneously. On the contrary, F2 shows local tendencies opposite to those of F1 and F3 , when F2 increases (respectively decreases) F1 and F3 decreases (respectively increases). Finally, F1 and F2 are the closest in values. 4.2 Time series temporal correlation vs classical correlation Let’s explore in figure 2 the distribution of the temporal and classical correlations among the times series into F1 , F2 and F3 classes. On the one hand, the Chouakria-Douzal and Nagabhushan (c) (b) −0.88 −0.90 0.87 −0.56 0.35 0.30 0.20 0.25 0.65 0.55 0.45 (f) (e) (d) −0.60 −0.75 −0.70 0.89 −0.65 0.91 −0.60 −0.86 (a) −0.64 18 Fig. 2. (a) cort(F1 (x), F2 (x)) (b) cort(F1 (x), F3 (x)) (c) cort(F2 (x), F3 (x)) (d)cor(F1 (x), F2 (x)) (e) cor(F1 (x), F3 (x)) (f ) cor(F2 (x), F3 (x)) temporal correlation distribution cort(F1 , F3 ) ∈ [0.87, 0.92], cort(F1 , F2 ) ∈ [−0.73, −0.60] and cort(F2 , F3 ) ∈ [−0.91, −0.86]) reveal a high positive dependency between F1 and F3 classes and a high negative dependency between F2 and the two remaining classes. These results supported well the dependencies illustrated above in figure 1. On the other hand, the classical correlation distribution cor(F1 , F3 ) ∈ [0.15, 0.35], cor(F1 , F2 ) ∈ [0.45, 0.70] and cor(F2 , F3 ) ∈ [−0.66, −0.56]) indicates a weak (nearly independence) positive dependency between F1 and F3 classes and a high positive dependency between F1 and F2 classes. These results illustrate well that the classical correlation estimates globally (not locally) the dependency between tendencies of time series. Indeed, F1 and F2 which are not locally but globally dependent, due to the linear upward trend tainting them, are considered as highly dependent; whereas F1 and F3 which are dependent locally not globally are considered as very weakly dependent. Note that contrary to classical correlation, the temporal correlation is global-trend effect free. 4.3 Comparative analysis To compare the above proximity measures, we estimate first the proximity matrices between the 15 synthetic time series, according to DisF and δF . DisF is evaluated with α = 0.5 and α = 0. For α = 1, results are Improved Fréchet Distance for Time Series 19 0.8 0.6 0.4 0.2 5 1 2 3 4 14 13 15 9 12 7 11 8 10 0.0 6 5 1 4 2 DisF (α = 0.5) 3 15 12 14 13 9 7 11 8 6 10 9 8 6 7 5 Fig. 3. δF 10 4 1 2 3 14 15 12 13 11 0.0 0.0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1.0 1.0 similar to those obtained from δF . A hierarchical cluster analysis is then performed on the obtained proximity matrices. Figure 3 illustrates the obtained dendograms. Note first that the three above proximity measures (δF , DisF (α = 0) DisF (α = 0.5) and DisF (α = 0)) divide the 15 time series on the well expected three classes F1 (from 1 to 5), F2 (from 6 to 10) and F3 (from 11 to 15). In addition, on the one hand, δF dendogram works out the time series of the classes F1 and F2 as the closest. Indeed, for δF , after stretching each class to match well an other class, the proximity evaluation is based solely on the taken values, which are close on F1 and F2 . On the other hand, DisF for α = 0.5 and α = 0 determines successfully the classes F1 and F3 as the closest. Note particularly that for α = 0.5 DisF still provides three classes with a high proximity between F1 and F3 ; whereas for α = 0 F1 and F3 are nearly merged and the respective dendogram comes out with only two main classes. Indeed, for α = 0 the proximity evaluation are based solely on the dependency between time series which is very high between F1 and F3 . 5 Discussion and conclusion This paper focuses on the Fréchet distance between time series. We have provided the definitions and properties of this conventional measure. Then we illustrated the limits of this distance. To alleviate these limits, we propose a new dissimilarity index based on the temporal correlation to include the information of dependency between the local trends. Note that, as this paper introduces the benefits of the temporal correlation for time series proximity estimation, and mainly for clarity reasons, we limit our work on two points. First we restrict the combination function to 20 Chouakria-Douzal and Nagabhushan a linear function to show clearly, by varying the parameter α, the additive value of the temporal correlation. Secondly, we restrict the illustration of the proposed index to a synthetic dataset which reproduces the limited conditions for the conventional Fréchet distance. Future works, on the one hand, will study other combination functions. For instance, if we consider the two dimensional space defined by the components cort and a normalized form of δF , then we can define a new euclidean distance between time series as their norm vector in such two dimensional space. On the second hand, these combination functions will be compared to the conventional Fréchet distance through a wide range of a real datasets. Finally, let’s remark that the proposed dissimilarity index DisF could be very useful for time series classification problem, where the aim consists in determining the most adaptable DisF by looking for the optimal value of α maximizing a classification rate. This is an interesting direction to study through a priori time series classification. References GEARY, R.C. (1954): The contiguity ratio and statistical mapping. The Incorporated Statistician, 5/3, 115-145. VON NEUMANN, J. (1941): Distribution of the ratio of the mean square successive difference to the variance. The Annals of Mathematical Statistics, 12/4. VON NEUMANN, J., KENT, R.H., BELLINSON, H.R. and HART, B.I. (1942): The mean square successive difference to the variance. The Annals of Mathematical Statistics. 153-162. FRÉCHET, M. (1906): Sur quelques points du calcul fonctionnel. Rendiconti del Circolo Mathematico di Palermo, 22, 1-74. GODAU, M. (1991): A natural metric for curves - computing the distance for polygonal chains and approximation algorithms. In Proc. 8th Sympos. Theor. Aspects of Comp. STACS, LNCS 480, 127-136. ALT, H. and GODAU, M. (1992): Measuring the resemblance of polygonal curves. In Proc. 8th Annu. ACM Sympos. Comput. Geom. 102-109. EITER T. and MANNILA, H. (1994): Computing Discrete Fréchet distance, Technical Report CD-TR 94/64, Christian Doppler Laboratory for Expert Systems. TU Vienna, Austria. CHOUAKRIA-DOUZAL, A. (2003): Compression Technique Preserving Correlations of a Multivariate Temporal Sequence. In: M.R. Berthold, H-J Lenz, E. Bradley, R. Kruse, C. Borgelt (eds.). Advances in Intelligent Data Analysis, 5, Springer, 566-577. Comparison of Distance Indices Between Partitions Lucile Denœud12 and Alain Guénoche3 1 2 3 École nationale supérieure des télécommunications, 46, rue Barrault, 75634 Paris cedex 13 (e-mail: [email protected]) CERMSEM CNRS-UMR 8095, MSE, Université Paris 1 Panthéon-Sorbonne, 106-112, boulevard de l’Hôpital, 75647 Paris cedex 13 Institut de Mathématiques de Luminy, 163, avenue de Luminy, 13009 Marseille (e-mail: [email protected]

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KS20-3* 8 str. - 4 sol. 14 str.
KS22** 6 str. - 2 sol. 14 str.
KS22-3* 6 str. - 2 sol. 14 str. 275
KS23** 6 - 2 str. 14 str.
KS25** 4 - 1/0 str. 14 str.
385
KS26** 2 - 2/0 str. 14 str.
KS27** 1 - 3/0 str. 8 sol. 500
KS29** 1 str. - 250 kcmil 8 str.
650
KS31** 1/0 str. - 350 kcmil 1/0 str.
KS34** 2/0 str. - 500 kcmil 2/0 str. 825
KS39 4/0 str. - 750 kcmil 4/0 str. 1000
KS44 300 - 1000 kcmil 4/0 str. 1100
* Not UL Listed or CSA Certified. Consult UL486 tables 7-4, 7-5, 7-6 for smaller
** UL Rated for Direct Burial conductor combinations.
▲ Listed torque values are for maximum
conductor combinations accommodated.

TYPE KSU
Universal SERVIT®

486A
Copper only

Tin plated service connector with spacer bar for all combinations of copper,
aluminum, ACSR, 6201, 5005, and steel.

Run Tap Recommended

Catalog Copper & Run ACSR Copper & Tap ACSR* Tightening
Number Aluminum AAAC, 5005 Aluminum AAC, 5005 Torque in-lb
KSU17 12 - 6 sol. 8 (6-1) 12 sol. - 6 sol. 8 (6-1)
165
KSU20 10 - 4 sol. 6 (6-1) 10 sol. - 4 sol. 6 (6-1)
6 (6-1) - 6 (6-1) -
KSU22 10 - 2 sol. 10 sol. - 2 sol.
4 (7-1) 4 (7-1)
275
3 (6-1) - 6 (6-1) -
KSU23 8 - 2 str. 8 sol. - 2 str.
2 (6-1) 2 (6-1)
3 (6-1) - 6 (6-1) -
KSU25 2 - 1/0 str. 10 str. - 1/0 str.
1 (6-1) 1 (6-1)
385
2 (6-1) - 6(6-1) -
KSU26 2 - 2/0 str. 8 str. - 2/0 str.
1/0 ( 6-1) 1/0 (6-1)
1 (6-1) - 6 (6-1) -
KSU27 1 - 3/0 str. 8 sol. - 3/0 str. 500
2/0 (6-1) 2/0 (6-1)
1 str. - 2/0 (6-1) - 6( 6-1) -
KSU29 8 str. - 250
250 kcmil 4/0 (6-1) 4/0 (6-1)
650
4/0 str. - 3/0 (6-1) - 4 (6-1) -
KSU31 4 str. - 350
350 kcmil 4/0 ( 6-1) 4/0 (6-1)
400 - 336 (30-7) - 2 (6-1) -
KSU34 2 str. - 500 825
500 kcmil 477 (18-1) 477 (18-1)
* Accommodates compressed conductors within conductor ranges.

1
 Mechanical BURNDY® EC&M BURNDY® EC&M Mechanical

TYPE KVS TYPE KSA

OKLIP™ TRITAP™ SERVIT® Tap Connector

486B
Mechanical Connector for Copper and Copperweld. Compact, two-piece,
An aluminum tin-plated universal tap connector for all combinations of alu-
high strength, high copper alloy BURNDY® OKLIP™ recommended for heavy
minum to aluminum, aluminum to copper and copper to copper applications.
duty connections. Neoprene rings hold DURIUM™ bolts in place during
UL486B Listed. 90° C rated.
installation. Installed with ordinary wrench.
Recommended
Catalog Copper Recommended Catalog Run Run Tap Tap Tightening
Number Run Tap Tightening Torque in-lb Number Min. Max. Min. Max. Torque in-lb ▲
KVS26 2 - 2/0 str. 6 sol. - 2/0 str. 180 KSA6 10 sol. 6 str. 10 sol. 6 str.
KVS28 1/0 - 4/0 str. 10 - 4/0 str. 250 165
KSA4 8 sol. 4 str. 10 sol. 4 str.
KVS31 250 - 350 kcmil 10 str. - 350 kcmil 325 KSA2 6 sol. 2 str. 8 str. 2 str. 275
KVS34 400 - 500 kcmil 10 str. - 500 kcmil 375 KSA1/0 2 compact 1/0 str. 8 sol. 1/0 str.
KVS40 400 - 800 kcmil 3/0 str. - 800 kcmil 385
500 KSA2/0 2 compact 2/0 str. 8 str. 2/0 str.
KVS44 500 - 1000 kcmil 3/0 str. - 1000 kcmil KSA4/0 2 compact 4/0 str. 6 str. 4/0 str. 500
KSA350* 1/0 compact 350 kcmil 4 str. 350 kcmil 650
KSA500* 400 compact 500 kcmil 2 compact 500 kcmil 825
*Not CSA Listed.
TYPE KVSU ▲ Listed torque values are for maximum conductor combinations accommodated.
Universal OKLIP™

TYPE SC
SERVIT® Cover
Mechanical Connector for all combinations of Copper, Aluminum, ACSR,
AAAC and 5005. Compact, high strength, tin plated copper alloy two-piece
connector with spacer and tin-plated silicon bronze DURIUM™ hardware.
Recommended for heavy duty connections. Spacer separates dissimilar con-
ductors and provides long contact length. Neoprene ring prevents loss of Range-taking, insulating cover for SERVIT® and equivalent split-bolt
shorter bolt during installation. Longer peened bolt permits swivel action for connectors.
easier installation.
Catalog
Run Tap Number For Use With
ACSR, ACSR, Recommended SC4 KS17, KS17-3, KS20, KSU17, KSU20
Catalog Copper & AAAC, & Copper & AAAC, & Tightening SC2 KS22, KS20-3, KS23, KS22-3, KSA6, KSA4, KSU22, KSU23
Number Aluminum 5005 Aluminum 5005 Torque in-lb SC2/0 KS25, KS26, KSA2, KSA 1/0, KSU25, KSU26
KVSU26 2 - 2/0 str. 3 - 2/0 6 - 2/0 str. 6 - 2/0 180
KVSU28 1/0 - 4/0 str. 1/0 - 4/0 6 - 4/0 str. 6 - 4/0 250
250 - 350 6 str. - 350
KVSU31 4/0 - 300 6 - 300 325
kcmil kcmil
KVSU34
400 - 500 336.4- 4 str. - 500
5 - 397.5 375
TYPE QPX
kcmil 397.5 kcmil VERSITAP™ Parallel Clamp
400 - 800 4/0 str. - 800
KVSU40 4/0 - 800 3/0 - 715.5
kcmil kcmil
500
500 -1000 4/0 str. -1000
KVSU44 4/0 -1000 4/0 - 900
kcmil kcmil
Parallel clamp for wide range of copper cable. Makes parallel tap, tees,
crosses and end-to-end connections. Rounded edges for easy taping.

Cable Range Recommended

Catalog Run Run Tap Tap Tightening
Number Min. Max. Min. Max. Torque in-lb
QPX2C2C 6 str. 2 str. 6 str. 2 str. 150
QPX282C 1 str. 4/0 str. 6 str. 2 str.
250
QPX2828 1 str. 4/0 str. 1 str. 4/0 str.
QPX342C 250 kcmil 500 kcmil 6 str. 2 str.
QPX3428 250 kcmil 500 kcmil 1 str. 4/0 str. 375
QPX3434 250 kcmil 500 kcmil 250 kcmil 500 kcmil
QPX442C 500 kcmil 1000 kcmil 6 str. 2 str.
QPX4428 500 kcmil 1000 kcmil 1 str. 4/0 str.
500
QPX4434 500 kcmil 1000 kcmil 250 kcmil 500 kcmil
QPX4444 500 kcmil 1000 kcmil 500 kcmil 1000 kcmil

2 3
 Mechanical BURNDY® EC&M BURNDY® EC&M Mechanical

TYPE QPX-Y TYPE KA

Universal VERSITAP™ KA-LUG™ Terminal
Parallel Clamp For Copper Cable

Universal parallel clamp for copper and aluminum. Tin-plated. Makes paral-
lel taps, tees, crosses or end-to-end connections. Compact, economical, high copper alloy terminal for joining a wide range of
Conductor Range cable to equipment pads or terminal blocks.
Catalog Aluminum or Copper Catalog Conductor
Number Run Tap Number Range
QPX2C2C-Y 6 - 2 str. 6 - 2 str. KA8C 14 sol. - 8 str.
QPX282C-Y 1 - 4/0 str. 6 - 2 str. KA4C 14sol. - 4 str.
QPX2828-Y 1/0 - 4/0 str. 1/0 - 4/0 str. KA25 4 - 1/0 str.
QPX342C-Y 250 - 500 kcmil 6 - 2 str. KA25-2TC38 4 - 1/0 str.
QPX3428-Y 250 - 500 kcmil 1/0 - 4/0 str. KA28 1 - 4/0 str.
QPX3434-Y 250 - 500 kcmil 250 - 500 kcmil KA34 4/0 str. - 500 kcmil
QPX4444-Y 750 - 1000 kcmil 750 - 1000 kcmil

TYPE QA
TYPES KPA, KPA-UP QIKLUG™ Terminal
SCRULUG™ Terminal
For Copper Cable

Heavy-duty terminal for a wide range of copper cables. One-hole tongue.

Figure 2
Figure 1 Catalog Conductor Stud
High copper alloy terminal for joining a wide range of cable to equipment Number Range Size
pads or terminal blocks. Plain copper finish. QA8C-B 14 sol. - 8 str. 10
Catalog Catalog QA4C-B 8 - 4 str. 1/4
Number Number Conductor QA1C-B 4 - 1 str. 1/4
Fig. 1 Fig. 2 Range QA26-B 1/0 - 2/0 str. 3/8
QA28-B 3/0 - 4/0 str. 3/8
KPA8C KPA8CUP* 14 sol. - 6 str.
QA31-B 250 - 350 kcmil 1/2
KPA4C KPA4CUP* 14 sol. - 4 str.
QA34-B 400 - 500 kcmil 1/2
KPA25 — 6 - 1/0 str.
QA40-B 600 - 800 kcmil 5/8
KPA28 — 6 - 4/0 str.
QA44-B 850 - 1000 kcmil 5/8
KPA34 — 2/0 str. - 500 kcmil
QA46-B 1100 - 1500 kcmil 3/4
* For tin plating drop “UP” suffix and add “TP” suffix.

TYPES QA-2, QA-4

TYPE KLU QIKLUG™ Terminal
SCRULUG™ Terminal
For Copper Cable
Offset Tongue Non-plated
Heavy-duty terminal for a wide range of copper cables. Two-hole and four-
hole tongues.
Catalog Conductor Stud Stud
High copper alloy terminal with offset tongue for joining a wide range of cable Number Range Hole Size Hole Centers
to equipment pads or bar. Easy to install with screwdriver or wrench. Con- QA8C-2B 14 sol. - 8 str. 10 5/8
nector is reusable. Plain copper finish. QA4C-2B 8 - 4 str. 1/4 5/8
QA1C-2B 4 - 1 str. 5/16 7/8
Catalog Conductor
QA26-2B 1/0 - 2/0 str. 3/8 1
Number Range QA28-2B 3/0 - 4/0 str. 3/8 1
KLU25 14 - 10 sol. QA31-2B 250 - 350 kcmil 3/8 1
KLU35 14 sol. - 6 str. QA26-2N 1/0 - 2/0 str. 1/2 1-3/4
KLU70 8 sol. - 2 str. QA28-2N 3/0 - 4/0 str. 1/2 1 -3/4
KLU125 2 - 1/0 str. QA31-2N 250 - 350 kcmil 1/2 1-3/4
KLU175 4 - 3/0 str. QA34-2B 400 - 500 kcmil 3/8 1
KLU225 2 - 4/0 str. QA34-2N 400 - 500 kcmil 1/2 1-3/4
KLU300 1/0 str. - 350 kcmil QA34-4B 400 - 500 kcmil 3/8 1
KLU400 1/0 str. - 500 kcmil QA40-2N 600 - 800 kcmil
QA44-2N 850 - 1000 kcmil
1/2 1-3/4
QA44-4N 850 - 1000 kcmil
QA46-2N 1100 - 1500 kcmil

4 5
 Mechanical BURNDY® EC&M BURNDY® EC&M Mechanical

TYPE QDA TYPE KA-U

QIKLUG™ Terminal Universal Terminal
(1 Conductor)
For Aluminum and Copper Conductors
Fig. 1

Heavy-duty terminal for a wide range of cable to equipment studs. One-hole Fig. 2
tongue finished on both sides.
Catalog Conductor Stud These dual-rated one conductor lugs are constructed from high strength alu-
Number Range Size minum alloy and electro tin-plated to provide low contact resistance.
QDA8C 14 sol. - 8 str. 3/8 Catalog Figure Recommended Tightening
QDA4C 8 - 4 str. 3/8 Number Conductor Range Number Torque (in-lb) ▲
QDA1C 4 - 1 str. 3/8
KA6U 14 - 6 str. 1 45
QDA26 1/0 - 2/0 str. 1/2
KA2U 14 - 2 str. 1 50
QDA28 3/0 - 4/0 str. 1/2
KA25U 14 - 1/0 str. 1 50
QDA31 250 - 500 kcmil 5/8
KA26U 6 - 2/0 str. 1 120
QDA34 400 - 500 kcmil 3/4
KA29U 6 str. - 250 kcmil 2 275
QDA40 600 - 800 kcmil 1
KA30U 6 str. - 300 kcmil 2 275
KA31U 6 str. - 350 kcmil 2 275
KA34U 4 str. - 500 kcmil 2 500
KA36U 2 str. - 600 kcmil 2 500
TYPES Q2A-2, Q2A-2N, Q2A-4N KA40U 300 - 800 kcmil 2 550
QIKLUG™ Terminal KA44U 500 - 1000 kcmil 2 550
▲ Listed torque values are for maximum conductor size accommodated.

Heavy-duty terminal for a wide range of copper cable. Joins two cables to
equipment pads or bars. Two hole and four hole tongue.
TYPE K2A-U
Catalog Conductor Stud Stud
Number Range Hole Size Hole Centers Universal Terminal
Q2A1C-2 4 - 1 str. 3/8 1 (2 Conductors) Fig. 1
Q2A26-2N 1/0 - 2/0 str. For Aluminum and Copper Conductors
Q2A28-2N 3/0 - 4/0 str.
Q2A31-2N 250 - 350 kcmil
Q2A34-2N 400 - 500 kcmil Fig. 2
Q2A40-2N 600 - 800 kcmil Fig. 3
Q2A28-4N 3/0 - 4/0 str. 1/2 1-3/4 Compact, wide-range, tin-plated aluminum terminal for use with two copper
Q2A31-4N 250 - 350 kcmil or aluminum cables.
Q2A34-4N 400 - 500 kcmil
Stud Stud Recommended
Q2A40-4N 600 - 800 kcmil
Q2A44-4N 850 - 1000 kcmil
Catalog Conductor Hole Hole Fig. Tightening
Q2A46-4N 1100 - 1500 kcmil Number Range Size Spacing No. Torque (in-lb) ▲
K2A25U Two: 14 - 1/0 str. 1/4 — 1 50
K2A26U Two: 14 - 2/0 str. 1/4 — 1 120
K2A29U Two: 6 str. - 250 kcmil 3/8 — 2 275
TYPES Q3A-2N, Q3A-4N K2A31U Two: 4 str. - 350 kcmil 1/2 — 2 275
K2A36U Two: 2 str. - 600 kcmil 1/2 — 2 375
QIKLUG™ Terminal
K2A40U Two: 300 - 800 kcmil 5/8 — 2 550
K2A44U Two: 500 - 1000 kcmil 5/8 — 2 550
K2A31U-2N* Two: 6 str. - 350 kcmil 1/2 1-3/4 3 275
K2A36U-2N* Two: 2 str. - 600 kcmil 1/2 1-3/4 3 375
K2A40U-2N* Two: 300 - 800 kcmil 1/2 1-3/4 3 375
Heavy-duty terminal for a wide range of copper cable. Joins three cables to K2A44U-2N* Two: 500 - 1000 kcmil 1/2 1-3/4 3 375
equipment pads or bars.
* Tongue holes drilled per NEMA standards.
Catalog Conductor Stud Stud ▲ Listed torque values are for maximum conductor size accommodated.
Number Range Hole Size Hole Centers
Q3A28-2N 3/0 - 4/0 str.
Q3A31-2N 250 - 350 kcmil
Q3A34-2N 400 - 500 kcmil
Q3A28-4N 3/0 - 4/0 str.
Q3A31-4N 250 - 350 kcmil
1/2 1-3/4
Q3A34-4N 400 - 500 kcmil
Q3A40-4N 600 - 800 kcmil
Q3A44-4N 850 - 1000 kcmil
Q3A46-4N 1100 - 1500 kcmil
Q3A48-4N 1600 - 2000 kcmil

6 7
 Mechanical BURNDY® EC&M BURNDY® EC&M Mechanical

TYPES K3A-U, KK3A-U TYPE CL50-1

Universal Terminal Copper Lay-In QIKLUG™
(3 Conductor) For Copper
For Aluminum & Copper Conductors Fig. 1

Fig. 2
The Lay-In QIKLUG™ is manufactured from high strength pure electrolytic
Dual-rated three conductor lugs are constructed from high strength aluminum copper to ensure maximum strength and conductivity. UL467 Listed for Direct
alloy and electro tin-plated to provide low contact resistance. Burial in earth or concrete. The open-faced design allows for fast lay-in of
the conductor without the need for cutting or breaking.
Catalog Conductor Stud Fig. Recommended Tight-
Number Range Hole Size No. ening Torque (in-lb) ▲ Catalog Number Conductor Range Copper Stud Hole
K3A2U-2* Three: 14 - 2 str. 5/16 1 50 CL50-1 14 - 4 AWG 10
K3A25U-2* Three: 14 - 1/0 str. 3/8 1 50 NOTE: Stainless Steel Screws.
K3A26U-2N Three: 14 - 2/0 str. 1/2 1 50
K3A27U-2N Three: 6 - 3/0 str. 1/2 1 275
K3A29U-2N Three: 6 str. - 250 kcmil 1/2 1 275
K3A31U-2N Three: 6 str. - 350 kcmil 1/2 1 275
K3A36U-2N Three: 2 str. - 600 kcmil 1/2 1 375
KK3A36U-2N Three: 2 str. - 600 kcmil 1/2 2 375
TYPE KPB
KK3A40U-2N Three: 300 - 800 kcmil 1/2 2 375 For Copper
KK3A44U-2N Three: 500 - 1000 kcmil 1/2 2 375
* Slotted screw.
“N” indicates NEMA Standard holes.
▲ Listed torque values are for maximum conductor size accommodated.
UL467 Listed for direct burial application in earth or concrete.
Catalog Number Cable Range Stud Hole
TYPE K4A-U KPB4CG1 10 - 4 str. 10*
* To be assembled with TMH322 stainless steel hardware kit, ordered separately.
Universal Terminal
(4 Conductor)
For Aluminum and Copper Conductors
Fig. 1

TYPE K11A-U
Fig. 2
Universal Terminal Fig. 2 Fig. 3
These dual-rated four conductor lugs are constructed from high strength For Aluminum and
aluminum alloy and electro tin-plated to provide low contact resistance. Fig. 1
Copper Conductors
Catalog Conductor Stud Fig. Recommended Tight-
Number Range Hole Size No. ening Torque (in-lb) ▲
K4A29U-4N Four: 6 str. - 250 kcmil 1/2 1 275 Fig. 4 Fig. 5
K4A31U-4N Four: 6 str. - 350 kcmil 1/2 1 275
Dual-rated panelboard lugs are constructed from high strength extruded alu-
KK4A36U-4N Four: 2 str. - 600 kcmil 1/2 2 375
minum alloy and electro tin-plated to provide low contact resistance.
KK4A40U-4N Four: 300 - 800 kcmil 1/2 2 375
“N” indicates NEMA Standard holes.
Catalog Conductor Stud Fig. Recommended Tight-
▲ Listed torque values are for maximum conductor size accommodated. Number Range Hole Size No. ening Torque (in-lb) ▲
K11A30U* Two: 6 str. - 300 kcmil 5/16 1 275
K11A34U-2 Two: 4/0 str. - 500 kcmil 1/4 2
K11A36U-2 Two: 2 str. - 600 kcmil 3/8 3
K21A36U-2 Three: 2 str. - 600 kcmil 3/8 4
TYPE BGBL K22A36U-2 Four: 2 str. - 600 kcmil 3/8 5
375
Lay-In QIKLUG™ K11A39U-2 Two: 1/0 str. - 750 kcmil 3/8 3
*UL Listed 90°C, 600 V K22A39U-2 Four: 1/0 str. - 750 kcmil 3/8 5
* Not CSA listed.
▲ Listed torque values are for maximum conductor size accommodated.

The Lay-In QIKLUG™, type BGBL is manufactured from high strength 6061-
T6 aluminum, and is ideally suited for grounding and bonding applications
accommodating both copper and aluminum conductor sizes 14 AWG to 250
kcmil.
Catalog Number Conductor Range Hex Size
BGBL-4 14 - 4 str. Slot
BGBL-1/0 14 - 1/0 str. Slot
BGBL-250 6 str. - 250 kcmil 7/32

8 9
 Mechanical BURNDY® EC&M BURNDY® EC&M Mechanical

TYPE KAU-KIT TYPE AMS

Transformer Lug Kit Dual Rated Splicer/Reducer
For Copper and Aluminum Cable

Each kit contains the UL Listed and CSA certified AL/CU rated aluminum set
screw connectors and tongue mounting hardware needed to terminate alu- All splicer/reducers are dual rated for use with aluminum and copper
minum or copper cables in “dry type” transformers. The KVA rating gives an conductors and are constructed from high strength, tin-plated aluminum.
approximate cross reference to the appropriate kit. PENETROX™ oxide inhibiting joint compounds are recommended for all alu-
minum applications.
Terminals
Catalog Transformer Catalog Catalog Wire Range
Number KVA Rating Qty Number Conductor Range Number Min. Max.
AMS-2* 14 AWG 2 AWG
15-37.5 1Ø 8 KA2U 14 - 2 str.
KAU-KIT1 AMS-0* 8 AWG 1/0 AWG
15-45 3Ø 4 KA29U 6 str. - 250 kcmil
AMS-4/0 6 AWG 4/0 AWG
50-75 1Ø
KAU-KIT2 12 KA29U 6 str. - 250 kcmil AMS-250 6 AWG 250 kcmil
75-112.5 3Ø
AMS-350 6 AWG 350 kcmil
100-167 1Ø 6 K2A31U 6 str. - 350 kcmil
KAU-KIT3 AMS-500 3/0 AWG 500 kcmil
150-300 3Ø 7 K2A40U 300 - 800 kcmil
AMS-750 250 kcmil 750 kcmil
KAU-KIT4 400-500 3Ø 15 K2A40U 300 - 800 kcmil
AMS-1000 500 kcmil 1000 kcmil
* Slotted Screws

Catalog Hardware
Number Qty. Bolt Size Qty. Nut Qty. Washer
Captive
KAU-KIT1 8 1/4-20 x 3/4 HH 8 1/4 x 20 HN —
To Nut
8 1/4-20 X 3/4 HH Captive
TYPE UGSKIT
KAU-KIT2 16 1/4 x 20 HN — Watertight/Underground Splice Kit
8 1/4-20 x 2 HH To Nut
KAU-KIT3
5 1/2-13 x 2 HH
11 1/2-13 HN
22 1/2 FW For all Aluminum or
6 1/2-13 x 2-1/2 HH 11 1/2 Belleville Copper/Aluminum Combinations
7 1/-13 x 2 22 1/2 FW
KAU-KIT4 11 1/2-13 HN
4 1/2-13 x 2-1/2 11 1/2 Belleville Fig. 1 Fig. 2

Type UGS Watertight Underground Splice Kit consists of a standard AMS

splicer/reducer and two heavy wall heat-shrink sleeves.

Catalog Figure Conductor Range

Number Number Minimum Maximum
UGSKIT2* 1 8 AWG 2 AWG
UGSKIT250* 2 1 AWG 250 kcmil
* UL486D Listed for Direct Burial.

TYPE UGSKIT8
UF Direct Burial Splice Kit
For All Aluminum or
Copper/Aluminum Combinations

Type UGS UF Splice Kit consists of a UF splice connector and a heavy wall
heatshrink sleeve.
Catalog Number Conductor Range Copper
UGSKIT8* 14 - 8 AWG
* UL486D Listed for Direct Burial.

10 11
 Mechanical BURNDY® EC&M BURNDY® EC&M Mechanical

U-BLOK™ POWER SPEC-BLOK™

DISTRIBUTION BLOCKS POWER DISTRIBUTION
For Junction Box Applications CONNECTORS
The U-BLOK™ system is a modern, state-of-the-art FT4B500 Unique, modular, made-to-order, power-distribution
approach to multi-load power distribution assemblies accommodate any number of supply
applications. Among typical uses are multi-story or and load conductors in any number of poles.
multi-unit buildings, HVAC, refrigeration, control Capacity matches the conductors accommodated
panels, motor control, switch gear, elevator systems and SPEC-BLOK™ assures uniform loading.
and materials-handling equipment. U-BLOK™ is UL
Adjacent poles are separated by easy-to-handle,
Listed for Copper or Aluminum conductors and rated FT3B4/0 wrap-around insulating covers which eliminate
for 600-volt applications. U-BLOK™ can be
taping and reduce heat build-up by allowing air to
mounted on bases for use in troughs or bolted
flow freely around connectors. SPEC-BLOK™ is
directly to junction boxes. AL9CU rated.
UL Listed for copper or aluminum conductors for
600 volts (AL9CU). Assemblies are mounted on
platforms suitable for easy installation in wireway
3S or junction box.

*
3S 3U

3 WIRE POWER DISTRIBUTION BLOCKS

Maximum Features and Benefits
• Accommodate unlimited conductors.
Number of Al or Cu Allen Tor-
〫 Fits wide range of applications.
Catalog Wires per Wire Range * Weight Wrench que Strip
• Connector elements tin-plated.
Number Phase Run Tap W L H Each Size (in-lb) Length 〫 Provides high reliability, low-resistance connections.
FT3B4/0 3/0 - 4/0 6 - 4/0 1-1/4 lbs. 1/4 200 1-1/2 • User friendly, space-saving design.
2 3-7/8 5-7/8 4-1/8
FT3B500 400 - 500 6 - 500 2-1/8 lbs. 3/8 375 2-5/16 〫 Easy to install. Saves labor.
3S 2 250-350 6 - 350 3-5/8 9-3/4 4-7/8 3 lbs. 1/4 200 1-3/4 • 94-VO rated insulation folds into place insulating the components.
Run Tap Run Tap 〫 Saves time and material. Allows easy installation.
3U 1 8 3/0-500 6 - 1/0 4-3/4 9-3/4 5-1/2 3 lbs. 5/32 110 2-5/161-5/32 • Connector caps removable for easy cable lay-in.
〫 Saves labor. Makes installation easier. Allows installation or straight-
4 WIRE POWER DISTRIBUTION BLOCKS through conductors. Eases retrofit.
Maximum • Belleville washers built-in on pressure screw assemblies, except in
assemblies installed with a 5/32 Allen wrench.
Number of Al or Cu Allen Tor-
〫 Provides high-integrity connections.
Catalog Wires per Wire Range * Weight Wrench que Strip
• Conductors can be cut or fed straight through.
Number Phase Run Tap W L H Each Size (in-lb) Length 〫 Straight through installation ideal for riser applications.
FT4B4/0 3/0 - 4/0 6 - 4/0 2 lbs. 1/4 200 1-1/2
2 3-7/8 7-7/8 4-1/8
FT4B500 400 - 500 6 - 500 2-3/4 lbs. 3/8 375 2-5/16 * SPEC-BLOK™ catalog assembly not UL Listed, individual SPEC-BLOK™ connectors are UL Listed.
4S 2 250-350 6 - 350 3-5/8 11-3/4 4-7/8 3-1/2 lbs. 1/4 200 1-3/4 See next page.
Run Tap Run Tap
4U 1 8 3/0-500 6 - 1/0 4-3/4 11-3/4 5-1/2 3-1/2 lbs. 5/32 110 2-5/161-5/32
PENETROX™ inhibitor is recommended for all aluminum wire connections. Contact BURNDY® Technical Services: 1-800-451-4956,
For two wire tap range is 8 through 1/0.
* Aluminum and copper conductors cannot be assembled under the same pressure plate or t-bar. or BURNDY® Customer Service: 1-800-346-4175

U-BLOK™ MOUNTING PLATFORMS

For Trough Applications

Catalog Gutter Weight

Number W L H Size Each
TBPT-6* 4-1/4 5-7/8 1-1/2 6 3/4 lb.
TBPT-8 4-1/4 7-7/8 3-1/2 8 1-1/4 lb.
TBPT-10 4-1/4 9-7/8 4 10 1-1/2 lb.
TBPT-12 4-1/4 11-7/8 4 12 1-3/4 lb.
Hole pattern shown is for reference only.
* Supercedes TBPT4/0-6 and TBPT350/500-6.

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 Mechanical BURNDY® EC&M BURNDY® EC&M Mechanical

SPEC-BLOK™ POWER SPEC-BLOK™

DISTRIBUTION CONNECTORS Mounting Platforms
* SPEC-BLOK™ Mounting Platforms are rigid steel construction with a black
finish. They can be supplied for junction box mounting or wireway construc-
tion allowing trough conductors to pass underneath the assemblies.
Features and Benefits
• The SPEC-BLOK™ system includes 12 connector elements