Document Open Access Logo

Towards Ordinal Data Science

Authors Gerd Stumme , Dominik Dürrschnabel , Tom Hanika

PDF HTML 5

Files

Thumbnail PDF
PDF
TGDK.1.1.6.pdf
  • Filesize: 1.98 MB
  • 39 pages
HTML5 Logo HTML (experimental)
TGDK.1.1.6.html

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Computing methodologies → Ontology engineering
  • Computing methodologies → Nonmonotonic, default reasoning and belief revision
  • Computing methodologies → Semantic networks
  • Computing methodologies → Algebraic algorithms
  • Computing methodologies → Boolean algebra algorithms
  • Computing methodologies → Unsupervised learning
  • Computing methodologies → Inductive logic learning
  • Computing methodologies → Rule learning
Keywords
  • Order relation
  • data science
  • relational theory of measurement
  • metric learning
  • general algebra
  • lattices
  • factorization
  • approximations and heuristics
  • factor analysis
  • visualization
  • browsing
  • explainability

Metrics

Abstract

Order is one of the main instruments to measure the relationship between objects in (empirical) data. However, compared to methods that use numerical properties of objects, the amount of ordinal methods developed is rather small. One reason for this is the limited availability of computational resources in the last century that would have been required for ordinal computations. Another reason - particularly important for this line of research - is that order-based methods are often seen as too mathematically rigorous for applying them to real-world data. In this paper, we will therefore discuss different means for measuring and ‘calculating’ with ordinal structures - a specific class of directed graphs - and show how to infer knowledge from them. Our aim is to establish Ordinal Data Science as a fundamentally new research agenda. Besides cross-fertilization with other cornerstone machine learning and knowledge representation methods, a broad range of disciplines will benefit from this endeavor, including, psychology, sociology, economics, web science, knowledge engineering, scientometrics.

Cite As Get BibTex

Gerd Stumme, Dominik Dürrschnabel, and Tom Hanika. Towards Ordinal Data Science. In Special Issue on Trends in Graph Data and Knowledge. Transactions on Graph Data and Knowledge (TGDK), Volume 1, Issue 1, pp. 6:1-6:39, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/TGDK.1.1.6

Author Details

Gerd Stumme
  • Knowledge & Data Engineering Group, Research Center for Information System Design &, Department of Electrical Engineering and Computer Science, University of Kassel, Germany
Dominik Dürrschnabel
  • Knowledge & Data Engineering Group, Research Center for Information System Design &, Department of Electrical Engineering and Computer Science, University of Kassel, Germany
Tom Hanika
  • Institute of Computer Science, University of Hildesheim, Germany
  • Berlin School of Library and Information Science, Humboldt-Universität zu Berlin, Germany

References

  1. Ittai Abraham, Yair Bartal, and Ofer Neiman. Advances in metric embedding theory. Advances in Mathematics, 228(6):3026-3126, 2011. URL: https://doi.org/10.1016/j.aim.2011.08.003.
  2. Ravi P Agarwal, MA El-Gebeily, and Donal O'Regan. Generalized contractions in partially ordered metric spaces. Applicable Analysis, 87(1):109-116, 2008. URL: https://doi.org/10.1080/00036810701556151.
  3. Rakesh Agrawal, Tomasz Imieliński, and Arun Swami. Mining association rules between sets of items in large databases. SIGMOD Rec., 22(2):207-216, jun 1993. URL: https://doi.org/10.1145/170035.170072.
  4. Alexandre Albano and Bogdan Chornomaz. Why concept lattices are large: extremal theory for generators, concepts, and vc-dimension. International Journal of General Systems, 46(5):440-457, 2017. URL: https://doi.org/10.1080/03081079.2017.1354798.
  5. Simon Andrews. Making use of empty intersections to improve the performance of cbo-type algorithms. In International Conference on Formal Concept Analysis, pages 56-71. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-59271-8_4.
  6. Franz Baader. Computing a minimal representation of the subsumption lattice of all conjunctions of concepts defined in a terminology. In Proceedings of the International Symposium on Knowledge Retrieval, Use, and Storage for Efficiency, KRUSE 95, pages 168-178, Santa Cruz, USA, 1995. Google Scholar
  7. Franz Baader, Bernhard Ganter, Baris Sertkaya, and Ulrike Sattler. Completing description logic knowledge bases using formal concept analysis. In Manuela M. Veloso, editor, IJCAI, pages 230-235, 2007. URL: https://doi.org/10.25368/2022.155.
  8. K. Becker, G. Stumme, R. Wille, U. Wille, and M. Zickwolff. Conceptual information systems discussed through an it-security tool. In R. Dieng and O. Corby, editors, Knowledge Engineering and Knowledge Management. Methods, Models, and Tools., volume 1937 of LNAI, pages 352-365, Heidelberg, 2000. Springer. URL: https://doi.org/10.1007/3-540-39967-4_27.
  9. P. Becker, J. Hereth, and G. Stumme. ToscanaJ: An open source tool for qualitative data analysis. In V. Duquenne, B. Ganter, M. Liquiere, E. M. Nguifo, and G. Stumme, editors, Advances in Formal Concept Analysis for Knowledge Discovery in Databases., pages 1-2, Lyon, France, 7 23 2002. Google Scholar
  10. Peter Becker and Joachim Hereth Correia. The toscanaj suite for implementing conceptual information systems. In Formal Concept Analysis, pages 324-348. Springer, 2005. URL: https://doi.org/10.1007/11528784_17.
  11. Radim Belohlavek. Fuzzy relational systems: foundations and principles, volume 20. Springer Science & Business Media, 2012. URL: https://doi.org/10.1007/978-1-4615-0633-1.
  12. Radim Belohlávek and Vilém Vychodil. Formal concepts as optimal factors in boolean factor analysis: Implications and experiments. In Peter W. Eklund, Jean Diatta, and Michel Liquiere, editors, Proceedings of the Fifth International Conference on Concept Lattices and Their Applications, CLA 2007, Montpellier, France, October 24-26, 2007, volume 331 of CEUR Workshop Proceedings. CEUR-WS.org, 2007. URL: https://ceur-ws.org/Vol-331/Belohlavek1.pdf.
  13. Radim Belohlavek and Vilem Vychodil. Discovery of optimal factors in binary data via a novel method of matrix decomposition. Journal of Computer and System Sciences, 76(1):3-20, 2010. URL: https://doi.org/10.1016/J.JCSS.2009.05.002.
  14. R. Bendaoud, A. Napoli, and Y. Toussaint. Formal concept analysis: A unified framework for building and refining ontologies. In EKAW, 2008. URL: https://doi.org/10.1007/978-3-540-87696-0_16.
  15. Olivier Berne, C. Joblin, Y. Deville, J. D. Smith, M. Rapacioli, J. P. Bernard, J. Thomas, W. Reach, and A. Abergel. Analysis of the emission of very small dust particles from spitzer spectro-imagery data using blind signal separation methods. Astronomy & Astrophysics, 469(2):575-586, 2007. URL: https://doi.org/10.1051/0004-6361:20066282.
  16. J. Bertin and M. Barbut. Sémiologie graphique: les diagrammes, les réseaux, les cartes. Gauthier Villars, 1967. Google Scholar
  17. T Gnana Bhaskar and V Lakshmikantham. Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Analysis: Theory, Methods & Applications, 65(7):1379-1393, 2006. URL: https://doi.org/10.1016/j.na.2005.10.017.
  18. Robi Bhattacharjee and Sanjoy Dasgupta. What relations are reliably embeddable in euclidean space? In Aryeh Kontorovich and Gergely Neu, editors, Proceedings of the 31st International Conference on Algorithmic Learning Theory, volume 117 of Proceedings of Machine Learning Research, pages 174-195, San Diego, California, USA, 08 February-11 February 2020. PMLR. URL: http://proceedings.mlr.press/v117/bhattacharjee20a.html.
  19. G. Birkhoff. On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society, 31(4):433-454, 1935. URL: https://doi.org/10.1017/s0305004100013463.
  20. Garrett Birkhoff. Universal algebra. In Comptes Rendus du Premier Congrès Canadien de Mathématiques, pages 310-326, Toronto, 1946. University of Toronto Press. Google Scholar
  21. Garrett Birkhoff. Lattice theory. Rev. ed, volume 25 of Colloq. Publ., Am. Math. Soc. American Mathematical Society (AMS), 1948. Google Scholar
  22. Vincent D. Blondel, Jean-Loup Guillaume, Renaud Lambiotte, and Etienne Lefebvre. Fast unfolding of communities in large networks. Journal of Statistical Mechanics: Theory and Experiment, 2008:10008, 2008. URL: https://doi.org/10.1088/1742-5468/2008/10/p10008.
  23. George Boole. Investigation of The Laws of Thought On Which Are Founded the Mathematical Theories of Logic and Probabilities. Walton and Maberly, 1853. Also available from Dover, New York 1958, ISBN 0-486-60028-9. Google Scholar
  24. Daniel Borchmann, Tom Hanika, and Sergei Obiedkov. On the usability of probably approximately correct implication bases. In Karell Bertet, Daniel Borchmann, Peggy Cellier, and Sébastien Ferré, editors, ICFCA, volume 10308 of LNCS, pages 72-88. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-59271-8_5.
  25. Daniel Borchmann, Tom Hanika, and Sergei Obiedkov. Probably approximately correct learning of horn envelopes from queries. Discrete Applied Mathematics, 273:30-42, 2020. Advances in Formal Concept Analysis: Traces of CLA 2016. URL: https://doi.org/10.1016/J.DAM.2019.02.036.
  26. Ronald J. Brachman, Peter G. Selfridge, Loren G. Terveen, Boris Altman, Alex Borgida, Fern Halper, Thomas Kirk, Alan Lazar, Deborah L. McGuinness, and Lori Alperin Resnick. Integrated support for data archaeology. In Proceedings of the 2nd International Conference on Knowledge Discovery in Databases, AAAIWS'93, pages 197-211. AAAI Press, 1993. URL: https://doi.org/10.1142/S0218215793000083.
  27. L. Breiman, J. Friedman, C.J. Stone, and R.A. Olshen. Classification and Regression Trees. The Wadsworth and Brooks-Cole statistics-probability series. Taylor & Francis, 1984. Google Scholar
  28. Laura F. Bringmann and Markus I. Eronen. Heating up the measurement debate: What psychologists can learn from the history of physics. Theory & Psychology, 26(1):27-43, 2016. URL: https://doi.org/10.1177/0959354315617253.
  29. Russ Bubley and Martin Dyer. Faster random generation of linear extensions. Discrete mathematics, 201(1-3):81-88, 1999. URL: https://doi.org/10.1016/S0012-365X(98)00333-1.
  30. S. Burris and H. P. Sankappanavar. A Course in Universal Algebra. Springer, New York, 1981. Google Scholar
  31. Xiao Cai, Feiping Nie, and Heng Huang. Multi-view k-means clustering on big data. In Twenty-Third International Joint conference on artificial intelligence. Citeseer, 2013. URL: http://ijcai.org/Abstract/13/383.
  32. Jaime S Cardoso, Joaquim F Pinto da Costa, and Maria J Cardoso. Modelling ordinal relations with svms: An application to objective aesthetic evaluation of breast cancer conservative treatment. Neural Networks, 18(5-6):808-817, 2005. URL: https://doi.org/10.1016/J.NEUNET.2005.06.023.
  33. Claudio Carpineto and Giovanni Romano. Concept Data Analysis: Theory and Applications. John Wiley & Sons, 2004. Google Scholar
  34. N. Caspard, B. Leclerc, and B. Monjardet. Finite Ordered Sets: Concepts, Results and Uses. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2012. URL: https://doi.org/10.1017/cbo9781139005135.
  35. Nicholas R. Chrisman. Rethinking levels of measurement for cartography. Cartography and Geographic Information Systems, 25(4):231-242, 1998. URL: https://doi.org/10.1559/152304098782383043.
  36. Constanze Ciavarella, Laura Fumanelli, Stefano Merler, Ciro Cattuto, and Marco Ajelli. School closure policies at municipality level for mitigating influenza spread: a model-based evaluation. BMC Infectious Diseases, 16(1):576, 2016. URL: https://doi.org/10.1186/s12879-016-1918-z.
  37. Philipp Cimiano, Andreas Hotho, Gerd Stumme, and Julien Tane. Conceptual knowledge processing with formal concept analysis and ontologies. In Peter Eklund, editor, Concept Lattices, volume 2961 of LNAI, pages 189-207, Heidelberg, 2004. Second International Conference on Formal Concept Analysis, ICFCA 2004, Springer. URL: https://doi.org/10.1007/978-3-540-24651-0_18.
  38. C. H. Coombs, L. C. Coombs, and J. C. Lingoes. Stochastic cumulative scales. In S. Shye, editor, Theory construction and data analysis in the behavioral sciences. Jossey-Bass., San Francisco, 1978. Google Scholar
  39. Clyde H. Coombs. A Theory of Data. Wiley, New York, 1964. Google Scholar
  40. Frithjof Dau and Baris Sertkaya. An extension of ToscanaJ for FCA-based data analysis over triple stores. In CUBIST Workshop, 2011. URL: https://ceur-ws.org/Vol-753/paper02_cubistws11.pdf.
  41. Thomas H. Davenport and Jeanne G. Harris. Competing on Analytics: The New Science of Winning. Harvard Business School Press, 2007. URL: https://doi.org/10.5860/choice.44-6322.
  42. B. A. Davey and H. A. Priestley. Introduction to Lattices and Order. Cambridge mathematical textbooks. Cambridge University Press, 2002. URL: https://doi.org/10.1017/CBO9780511809088.
  43. Stephan Doerfel, Tom Hanika, and Gerd Stumme. Clones in graphs. In Michelangelo Ceci, Nathalie Japkowicz, Jiming Liu, George A. Papadopoulos, and Zbigniew W. Ras, editors, ISMIS, volume 11177 of LNCS, pages 56-66. Springer, 2018. URL: https://doi.org/10.1007/978-3-030-01851-1_6.
  44. Stephan Doerfel, Robert Jäschke, and Gerd Stumme. Publication analysis of the formal concept analysis community. In F. Domenach, D.I. Ignatov, and J. Poelmans, editors, ICFCA 2012, volume 7278 of Lecture Notes in Artificial Intelligence, pages 77-95, Berlin/Heidelberg, 2012. Springer. URL: https://doi.org/10.1007/978-3-642-29892-9_12.
  45. Jean-Paul Doignon and Jean-Claude Falmagne. Knowledge Spaces. Imprint: Springer, Berlin, Heidelberg, 1999. URL: https://doi.org/10.1007/978-3-642-58625-5.
  46. Michel Dumontier and Tobias Kuhn. Data science – methods, infrastructure, and applications. Data Science, 1:1-5, 2017. URL: https://doi.org/10.3233/DS-170013.
  47. Dominik Dürrschnabel, Tom Hanika, and Gerd Stumme. Drawing order diagrams through two-dimension extension, 2019. URL: https://arxiv.org/abs/1906.06208.
  48. Dominik Dürrschnabel, Tom Hanika, and Maximilian Stubbemann. FCA2VEC: Embedding techniques for formal concept analysis. In Rokia Missaoui, Léonard Kwuida, and Talel Abdessalem, editors, Complex Data Analytics with Formal Concept Analysis, pages 47-74. Springer International Publishing, 2022. URL: https://doi.org/10.1007/978-3-030-93278-7_3.
  49. Dominik Dürrschnabel, Tom Hanika, and Gerd Stumme. Dimdraw - a novel tool for drawing concept lattices. In Diana Cristea, Florence Le Ber, Rokia Missaoui, Léonard Kwuida, and Baris Sertkaya, editors, ICFCA (Supplements), volume 2378 of CEUR Workshop Proceedings, pages 60-64. CEUR-WS.org, 2019. URL: https://ceur-ws.org/Vol-2378/shortAT6.pdf.
  50. Dominik Dürrschnabel, Maren Koyda, and Gerd Stumme. Attribute selection using contranominal scales. In Tanya Braun, Marcel Gehrke, Tom Hanika, and Nathalie Hernandez, editors, Graph-Based Representation and Reasoning - 26th International Conference on Conceptual Structures, ICCS 2021, Virtual Event, September 20-22, 2021, Proceedings, volume 12879 of Lecture Notes in Computer Science, pages 127-141. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-86982-3_10.
  51. Dominik Dürrschnabel and Gerd Stumme. Force-directed layout of order diagrams using dimensional reduction. In Agnès Braud, Aleksey Buzmakov, Tom Hanika, and Florence Le Ber, editors, Formal Concept Analysis - 16th International Conference, ICFCA 2021, Strasbourg, France, June 29 - July 2, 2021, Proceedings, volume 12733 of Lecture Notes in Computer Science, pages 224-240. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-77867-5_14.
  52. Dominik Dürrschnabel and Gerd Stumme. Greedy discovery of ordinal factors. CoRR, abs/2302.11554, 2023. URL: https://doi.org/10.48550/ARXIV.2302.11554.
  53. Dominik Dürrschnabel and Gerd Stumme. Maximal ordinal two-factorizations. In Proceedings of the 27th International Conference on Conceptual Structures, ICCS 2022, 2023. URL: https://doi.org/10.1007/978-3-031-40960-8_5.
  54. Peter Eades. A heuristic for graph drawing. Congressus numerantium, 42:149-160, 1984. Google Scholar
  55. David A Edwards. The structure of superspace. In Studies in topology, pages 121-133. Elsevier, 1975. URL: https://doi.org/10.1016/b978-0-12-663450-1.50017-7.
  56. P. Eklund, B. Groh, G. Stumme, and R. Wille. Contextual-logic extension of toscana. In B. Ganter and G. W. Mineau, editors, Conceptual Structures: Logical, Linguistic, and Computational, volume 1867 of LNAI, pages 453-467, Heidelberg, 2000. Springer. URL: https://doi.org/10.1007/10722280_31.
  57. Peter W. Eklund and Jean Villerd. A survey of hybrid representations of concept lattices in conceptual knowledge processing. In Léonard Kwuida and Baris Sertkaya, editors, ICFCA, volume 5986 of LNCS, pages 296-311. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-11928-6_21.
  58. Yves Escoufier, Bernard Fichet, Edwin Diday, Ludovic Lebart, Chikio Hayashi, Noburu Ohsumi, and Yasumasa Baba, editors. Data Science and its Applications. Acadamic Press/Harcourt Brace, Tokyo, 1995. Google Scholar
  59. Martin Ester, Hans-Peter Kriegel, Jörg Sander, and Xiaowei Xu. A density-based algorithm for discovering clusters in large spatial databases with noise. In Evangelos Simoudis, Jiawei Han, and Usama M. Fayyad, editors, Proceedings of the Second International Conference on Knowledge Discovery and Data Mining (KDD-96), Portland, Oregon, USA, pages 226-231. AAAI Press, 1996. URL: http://www.aaai.org/Library/KDD/1996/kdd96-037.php.
  60. Ernesto Estrada. The communicability distance in graphs. Linear Algebra and its Applications, 436(11):4317-4328, 2012. URL: https://doi.org/10.1016/j.laa.2012.01.017.
  61. Maximilian Felde and Tom Hanika. Formal context generation using dirichlet distributions. In Dominik Endres, Mehwish Alam, and Diana Sotropa, editors, ICCS, volume 11530 of LNCS, pages 57-71. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-23182-8_5.
  62. Maximilian Felde, Tom Hanika, and Gerd Stumme. Null models for formal contexts. J. Information, 11(3, 135), 2020. URL: https://doi.org/10.3390/INFO11030135.
  63. A. Ferguson, C. S. Myers, R. J. Bartlett, H. Banister, F. C. Bartlett, W. Brown, N. R. Campbell, K. J. W. Craik, J. Drever, J. Guild, R. A. Houstoun, J. O. Irwin, G. W. C. Kaye, S. J. F. Philpott, L. F. Richardson, J. H. Shaxby, T. Smith, R. H. Thouless, and W. S. Tucker. Quantitative estimates of sensory events. Advancement of Science, 2:331-349, 1940. (Final report of a committee appointed by the British Association for the Advancement of Science in 1932 to consider the possibility of measuring intensities of human sensation.). URL: https://doi.org/10.1038/130334a0.
  64. Sébastien Ferré, Marianne Huchard, Mehdi Kaytoue, Sergei O Kuznetsov, and Amedeo Napoli. Formal concept analysis: from knowledge discovery to knowledge processing. A Guided Tour of Artificial Intelligence Research: Volume II: AI Algorithms, pages 411-445, 2020. URL: https://doi.org/10.1007/978-3-030-06167-8_13.
  65. Ralph Freese. Automated lattice drawing. In International Conference on Formal Concept Analysis, pages 112-127. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-24651-0_12.
  66. Thomas MJ Fruchterman and Edward M Reingold. Graph drawing by force-directed placement. Software: Practice and experience, 21(11):1129-1164, 1991. URL: https://doi.org/10.1002/SPE.4380211102.
  67. Simon Funk. Netflix update: Try this at home, 2006. [Online; accessed 11-August-2020]. Google Scholar
  68. B. Ganter, G. Stumme, and R. Wille, editors. Formal Concept Analysis: Foundations and Applications, volume 3626 of LNAI, Heidelberg, 2005. Springer. URL: https://doi.org/10.1007/978-3-540-31881-1.
  69. Bernhard Ganter. Composition and decomposition in formal concept analysis. In Hans-Herrmann Bock, editor, Classification and related methods of data analysis, pages 561-566, Amsterdam, 1988. North-Holland. Google Scholar
  70. Bernhard Ganter. Diskrete mathematik: Geordnete mengen. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-37500-2.
  71. Bernhard Ganter and Cynthia Vera Glodeanu. Ordinal factor analysis. In International Conference on Formal Concept Analysis, pages 128-139. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-29892-9_15.
  72. Bernhard Ganter and Cynthia Vera Glodeanu. Factors and skills. In Cynthia Vera Glodeanu, Mehdi Kaytoue, and Christian Sacarea, editors, Formal Concept Analysis - 12th International Conference, ICFCA 2014, Cluj-Napoca, Romania, June 10-13, 2014. Proceedings, volume 8478 of Lecture Notes in Computer Science, pages 173-187. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-07248-7_13.
  73. Bernhard Ganter and Sergei O. Kuznetsov. Stepwise construction of the dedekind-macneille completion. In Marie-Laure Mugnier and Michel Chein, editors, Conceptual Structures: Theory, Tools and Applications, pages 295-302, Berlin, Heidelberg, 1998. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/BFB0054922.
  74. Bernhard Ganter and Sergei A. Obiedkov. Conceptual Exploration. Springer, 2016. URL: https://doi.org/10.1007/978-3-662-49291-8.
  75. Bernhard Ganter, Jürgen Stahl, and Rudolf Wille. Conceptual measurement and many-valued contexts. In W. Gaul and Martin Schader, editors, Classification as a tool of research, pages 169-176, Amsterdam, 1986. North-Holland. Google Scholar
  76. Bernhard Ganter and Gerd Stumme. Creation and merging of ontology top-levels. In Aldo de Moor, Wilfried Lex, and Bernhard Ganter, editors, Conceptual Structures for Knowledge Creation and Communication., volume 2746 of LNAI, pages 131-145, Heidelberg, 2003. Springer. URL: https://doi.org/10.1007/978-3-540-45091-7_9.
  77. Bernhard Ganter and Rudolf Wille. Conceptual scaling. In Frank Roberts, editor, Applications of combinatorics and graph theory to the biological and social sciences, pages 139-167. Springer-Verlag, New York, 1989. URL: https://doi.org/10.1007/978-1-4684-6381-1_6.
  78. Bernhard Ganter and Rudolf Wille. Formale Begriffsanalyse: Mathematische Grundlagen. Springer, Heidelberg, 1996. URL: https://doi.org/10.1007/978-3-642-61450-7.
  79. Bernhard Ganter and Rudolf Wille. Formal Concept Analysis: Mathematical Foundations. Springer, Heidelberg, 1999. URL: https://doi.org/10.1007/978-3-642-59830-2.
  80. J. C. Maxwell Garnett. General ability, cleverness and purpose. British Journal of Psychology, 9(3):345, 1919. Google Scholar
  81. Cynthia Vera Glodeanu. Tri-ordinal factor analysis. In International Conference on Formal Concept Analysis, pages 125-140. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-38317-5_8.
  82. Cynthia Vera Glodeanu and Bernhard Ganter. Applications of ordinal factor analysis. In International Conference on Formal Concept Analysis, pages 109-124. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-38317-5_7.
  83. Leo A. Goodman and William H. Kruskal. Measures of Association for Cross Classifications. Springer Verlag, New York, 1979. URL: https://doi.org/10.1007/978-1-4612-9995-0.
  84. S. Goto, H. Bono, H. Ogata, W. Fujibuchi, T. Nishioka, K. Sato, and M. Kanehisa. Organizing and computing metabolic pathway data in terms of binary relations. Pac Symp Biocomput, pages 175-186, 1997. Google Scholar
  85. Olivier Gouvert, Thomas Oberlin, and Cédric Févotte. Ordinal non-negative matrix factorization for recommendation. In Proceedings of the 37th International Conference on Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event, volume 119 of Proceedings of Machine Learning Research, pages 3680-3689. PMLR, 2020. URL: http://proceedings.mlr.press/v119/gouvert20a.html.
  86. Mikhael Gromov. Groups of polynomial growth and expanding maps. Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 53(1):53-78, 1981. URL: https://doi.org/10.1007/bf02698687.
  87. George Grätzer. Universal Algebra. Springer, 1968. URL: https://doi.org/10.1007/978-0-387-77487-9.
  88. George Grätzer. General Lattice Theory. Birkhäuser Verlag, 2. edition, 1998. URL: https://doi.org/10.1016/s0079-8169(08)x6085-4.
  89. L. Guttman. A basis for scaling qualitative data. American Sociological Review, 9:139-150, 1944. URL: https://doi.org/10.2307/2086306.
  90. Paul R Halmos. Measure theory, volume 18. Springer, 2013. URL: https://doi.org/10.1007/978-1-4684-9440-2.
  91. Tom Hanika and Johannes Hirth. Conexp-clj - a research tool for fca. In Diana Cristea, Florence Le Ber, Rokia Missaoui, Léonard Kwuida, and Baris Sertkaya, editors, ICFCA (Supplements), volume 2378 of CEUR Workshop Proceedings, pages 70-75. CEUR-WS.org, 2019. URL: https://ceur-ws.org/Vol-2378/shortAT8.pdf.
  92. Tom Hanika and Johannes Hirth. Quantifying the conceptual error in dimensionality reduction. In Tanya Braun, Marcel Gehrke, Tom Hanika, and Nathalie Hernandez, editors, Graph-Based Representation and Reasoning - 26th International Conference on Conceptual Structures, ICCS 2021, Virtual Event, September 20-22, 2021, Proceedings, volume 12879 of Lecture Notes in Computer Science, pages 105-118. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-86982-3_8.
  93. Tom Hanika and Johannes Hirth. Knowledge cores in large formal contexts. Ann. Math. Artif. Intell., 90(6):537-567, 2022. URL: https://doi.org/10.1007/S10472-022-09790-6.
  94. Tom Hanika, Maren Koyda, and Gerd Stumme. Relevant attributes in formal contexts. In Dominik Endres, Mehwish Alam, and Diana Sotropa, editors, ICCS, volume 11530 of LNCS, pages 102-116. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-23182-8_8.
  95. Tom Hanika, Friedrich Martin Schneider, and Gerd Stumme. Intrinsic dimension of geometric data sets. Tohoku Mathematical Journal, 74(1):23-52, 2022. URL: https://doi.org/10.2748/tmj.20201015a.
  96. D. Harrington. Confirmatory Factor Analysis. Oxford scholarship online: Social Work module. Oxford University Press, USA, 2009. Google Scholar
  97. Egbert Harzheim. Ordered Sets. Springer, 2005. URL: https://doi.org/10.1007/b104891.
  98. Xiangnan He, Lizi Liao, Hanwang Zhang, Liqiang Nie, Xia Hu, and Tat-Seng Chua. Neural collaborative filtering. In Proceedings of the 26th international conference on world wide web, pages 173-182, 2017. URL: https://doi.org/10.1145/3038912.3052569.
  99. Rajneesh Hegde and Kamal Jain. The hardness of approximating poset dimension. Electronic Notes in Discrete Mathematics, 29:435-443, 2007. European Conference on Combinatorics, Graph Theory and Applications. URL: https://doi.org/10.1016/J.ENDM.2007.07.084.
  100. J. Hereth and G. Stumme. Reverse pivoting in conceptual information systems. In H. Delugach and G. Stumme, editors, Conceptual Structures: Broadening the Base., volume 2120 of LNAI, pages 202-215, Heidelberg, 2001. Springer. URL: https://doi.org/10.1007/3-540-44583-8_15.
  101. Joachim Hereth, Gerd Stumme, Rudolf Wille, and Uta Wille. Conceptual knowledge discovery - a human-centered approach. Journal of Applied Artificial Intelligence (AAI), 17(3):281-301, 2003. URL: https://doi.org/10.1080/713827122.
  102. Seok-Hee Hong, Peter Eades, and Sang-Ho Lee. Drawing series parallel digraphs symmetrically. Computational Geometry, 17(3-4):165-188, 2000. URL: https://doi.org/10.1016/S0925-7721(00)00020-1.
  103. John Hopcroft and Robert Tarjan. Efficient planarity testing. Journal of the ACM (JACM), 21(4):549-568, 1974. URL: https://doi.org/10.1145/321850.321852.
  104. Andreas Hotho, Steffen Staab, and Gerd Stumme. Explaining text clustering results using semantic structures. In Nada Lavrač, Dragan Gamberger, and Hendrik BlockeelLjupco Todorovski, editors, Knowledge Discovery in Databases: PKDD 2003, 7th European Conference on Principles and Practice of Knowledge Discovery in Databases, volume 2838 of LNAI, pages 217-228, Heidelberg, 2003. Springer. URL: https://doi.org/10.1007/978-3-540-39804-2_21.
  105. Zhexue Huang. Extensions to the k-means algorithm for clustering large data sets with categorical values. Data Min. Knowl. Discov., 2(3):283-304, 1998. URL: https://doi.org/10.1023/A:1009769707641.
  106. Eyke Hüllermeier, Johannes Fürnkranz, Weiwei Cheng, and Klaus Brinker. Label ranking by learning pairwise preferences. Artificial Intelligence, 172(16-17):1897-1916, 2008. URL: https://doi.org/10.1016/J.ARTINT.2008.08.002.
  107. IT Joliffe and BJT Morgan. Principal component analysis and exploratory factor analysis. Statistical methods in medical research, 1(1):69-95, 1992. URL: https://doi.org/10.1177/096228029200100105.
  108. Toshihiro Kamishima and Jun Fujiki. Clustering orders. In Gunter Grieser, Yuzuru Tanaka, and Akihiro Yamamoto, editors, Discovery Science, pages 194-207, Berlin, Heidelberg, 2003. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-540-39644-4_17.
  109. K Keller and E Petrov. Ordinal spaces. Acta Mathematica Hungarica, 160(1):119-152, 2020. URL: https://doi.org/10.1007/s10474-019-00972-z.
  110. Matthias Keller. Intrinsic metrics on graphs: A survey. In Delio Mugnolo, editor, Mathematical Technology of Networks, pages 81-119, Cham, 2015. Springer International Publishing. URL: https://doi.org/10.1007/978-3-319-16619-3_7.
  111. Maurice G. Kendall. Rank correlation methods. C. Griffin, 1948. Google Scholar
  112. Ales Keprt and Václav Snásel. Binary factor analysis with help of formal concepts. In CLA, volume 110, pages 90-101, 2004. URL: https://ceur-ws.org/Vol-110/paper10.pdf.
  113. Aleš Keprt. Algorithms for Binary Factor Analysis. PhD thesis, PhD thesis, 2006. Google Scholar
  114. S.C. Kleene. Representation of events in nerve nets and finite automata. Annals of Mathematics Studies, 34:3-41, 1956. URL: https://doi.org/10.1515/9781400882618-002.
  115. Matthäus Kleindessner and Ulrike Luxburg. Uniqueness of ordinal embedding. In Maria Florina Balcan, Vitaly Feldman, and Csaba Szepesvári, editors, Proceedings of The 27th Conference on Learning Theory, volume 35 of Proceedings of Machine Learning Research, pages 40-67, Barcelona, Spain, 13-15 June 2014. PMLR. URL: http://proceedings.mlr.press/v35/kleindessner14.html.
  116. Matthäus Kleindessner and Ulrike Luxburg. Dimensionality estimation without distances. In Artificial Intelligence and Statistics, pages 471-479, 2015. URL: http://proceedings.mlr.press/v38/kleindessner15.html.
  117. Sigrid Knecht and Rudolf Wille. Congruence lattices of finite lattices as concept lattices. In J. Almeida, G. H. Bordalo, and P. Dwinger, editors, Lattices, semigroups, and universal algebra, pages 323-325. Plenum Press, New York-London, 1990. URL: https://doi.org/10.1007/978-1-4899-2608-1_32.
  118. Maren Koyda and Gerd Stumme. Factorizing lattices by interval relations. International Journal of Approximate Reasoning, 157:70-87, 2023. URL: https://doi.org/10.1016/J.IJAR.2023.03.003.
  119. Patryk Kozieł and Małgorzata Sulkowska. Uniform random posets. Information Sciences, 515:294-301, 2020. URL: https://doi.org/10.1016/J.INS.2019.12.018.
  120. Petr Krajca, Jan Outrata, and Vilem Vychodil. Parallel algorithm for computing fixpoints of galois connections. Annals of Mathematics and Artificial Intelligence, 59(2):257-272, 2010. URL: https://doi.org/10.1007/S10472-010-9199-5.
  121. David H. Krantz, R. Duncan Luce, Patrick Suppes, and Amos Tversky. Foundations of Measurement - Additive and Polynomial Representations, volume 1. Academic Press, 1971. URL: https://doi.org/10.1016/C2013-0-11004-5.
  122. Francesco Kriegel and Daniel Borchmann. Nextclosures: parallel computation of the canonical base with background knowledge. International Journal of General Systems, 46(5):490-510, 2017. URL: https://doi.org/10.1080/03081079.2017.1349570.
  123. Rajeev Kumar, BK Verma, and Shyam Sunder Rastogi. Social popularity based svd++ recommender system. International Journal of Computer Applications, 87(14), 2014. URL: https://doi.org/10.5120/15279-4033.
  124. Léonard Kwuida and Stefan E. Schmidt. Valuations and closure operators on finite lattices. Discrete Applied Mathematics, 159(10):990-1001, 2011. URL: https://doi.org/10.1016/J.DAM.2010.11.022.
  125. Jens Kötters and Peter W. Eklund. Conjunctive query pattern structures: A relational database model for formal concept analysis. Discrete Applied Mathematics, 273:144-171, 2020. Advances in Formal Concept Analysis: Traces of CLA 2016. URL: https://doi.org/10.1016/J.DAM.2019.08.019.
  126. Abdul Latif, Antonio-Francisco Roldán-López-de Hierro, and Wutiphol Sintunaravat. On common α-fuzzy fixed points with applications. Fixed Point Theory Appl., 2014:22, 2014. Id/No 234. URL: https://doi.org/10.1186/1687-1812-2014-234.
  127. Deutsche Fußball Liga. Spielordnung, 2019. URL: https://media.dfl.de/sites/2/2019/09/Spielordnung-SpOL-2019-08-22-Stand.pdf.
  128. Nathan Linial, Eran London, and Yuri Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15(2):215-245, 1995. URL: https://doi.org/10.1007/BF01200757.
  129. Tie-Yan Liu. Learning to rank for information retrieval. Foundations and Trends® in Information Retrieval, 3(3):225-331, 2009. URL: https://doi.org/10.1561/1500000016.
  130. F. M. Lord. On the statistical treatment of football numbers. American Psychologist, 8(12):750-751, 1953. URL: https://doi.org/10.1037/h0063675.
  131. R. Duncan Luce, David H. Krantz, Patrick Suppes, and Amos Tversky. Foundations of Measurement - Representation, Axiomatization, and Invariance, volume 3. Academic Press, 1990. URL: https://doi.org/10.1016/C2009-0-21635-7.
  132. R. Duncan Luce and John W. Tukey. Simultaneous conjoint measurement: a new type of fundamental measurement. Journal of Mathematical Psychology, page 27, 1964. URL: https://doi.org/10.1016/0022-2496(64)90015-x.
  133. Peter Luksch and Rudolf Wille. Substitution decomposition of concept lattices. In G. Eigenthaler, H. K. Kaiser, W. B. Müller, and W. Nöbauer, editors, Contributions to general algebra, volume 5, pages 213-220, Wien, 1987. Hölder-Pichler-Tempsky. Google Scholar
  134. Laurens van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of machine learning research, 9(Nov):2579-2605, 2008. URL: http://jmlr.org/papers/v9/vandermaaten08a.html.
  135. H. M. MacNeille. Partially ordered sets. Trans. Am. Math. Soc., 42:416-460, 1937. URL: https://doi.org/10.1090/s0002-9947-1937-1501929-x.
  136. Hiroshi Maehara. Euclidean embeddings of finite metric spaces. Discrete Mathematics, 313(23):2848-2856, 2013. URL: https://doi.org/10.1016/J.DISC.2013.08.029.
  137. K. Meinke and J. V. Tucker. Universal Algebra, pages 189-368. Oxford University Press, Inc., USA, 1993. URL: https://dl.acm.org/doi/10.5555/162573.162534.
  138. Facundo Memoli. On the use of gromov-hausdorff distances for shape comparison. In M. Botsch, R. Pajarola, B. Chen, and M. Zwicker, editors, Eurographics Symposium on Point-Based Graphics. The Eurographics Association, 2007. URL: https://doi.org/10.2312/SPBG/SPBG07/081-090.
  139. Joel Michell. Quantitative science and the definition of measurement in psychology. British Journal of Psychology, 88(3):355-383, 1997. URL: https://doi.org/10.1111/j.2044-8295.1997.tb02641.x.
  140. Charles Minard. Carte figurative des pertes successives en hommes de l'armée française dans la campagne de russie 1812-1813, 1869. URL: https://commons.wikimedia.org/wiki/File:Minard.png.
  141. Ann Mische. Relational sociology, culture, and agency. The SAGE handbook of social network analysis, pages 80-97, 2011. URL: https://doi.org/10.4135/9781446294413.n7.
  142. Rokia Missaoui and Kevin Emamirad. Lattice miner-a formal concept analysis tool. In 14th International Conference on Formal Concept Analysis, Supplementary Proceedings, page 91, 2017. Google Scholar
  143. B. Monjardet. Metrics on partially ordered sets—a survey. Discrete Mathematics, 35(1):173-184, 1981. Special Volume on Ordered Sets. URL: https://doi.org/10.1016/0012-365X(81)90206-5.
  144. Frederick Mosteller and John W Tukey. Data analysis and regression. a second course in statistics. Addison-Wesley series in behavioral science: quantitative methods, 1977. Google Scholar
  145. Kevin Musgrave, Serge Belongie, and Ser-Nam Lim. A metric learning reality check, 2020. URL: https://doi.org/10.1007/978-3-030-58595-2_41.
  146. Facundo Mémoli. Metric structures on datasets: stability and classification of algorithms. In International Conference on Computer Analysis of Images and Patterns, pages 1-33. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-23678-5_1.
  147. Facundo Mémoli. Some properties of gromov-hausdorff distances. Discrete & Computational Geometry, 48(2):416-440, 2012. URL: https://doi.org/10.1007/S00454-012-9406-8.
  148. Takao Nishizeki and Md Saidur Rahman. Planar graph drawing, volume 12. World Scientific Publishing Company, 2004. URL: https://doi.org/10.1142/5648.
  149. Megan Norris and Luc Lecavalier. Evaluating the use of exploratory factor analysis in developmental disability psychological research. Journal of autism and developmental disorders, 40(1):8-20, 2010. URL: https://doi.org/10.1007/s10803-009-0816-2.
  150. Charles S. Peirce. Collected Papers. Harvard Universit Press, Cambridge, 1931-1935. URL: https://doi.org/10.5860/choice.37-1485.
  151. J. Pfanzagl. Theory of Measurement. Physica, Heidelberg, 1971. URL: https://doi.org/10.1007/978-3-662-41488-0.
  152. Gregory Piatetsky-Shapiro. Knowledge discovery in real databases: A report on the ijcai-89 workshop. AI Magazine, 11(5):68-70, 1991. URL: https://doi.org/10.1609/AIMAG.V11I4.873.
  153. Jonas Poelmans, Sergei O. Kuznetsov, Dmitry I. Ignatov, and Guido Dedene. Formal concept analysis in knowledge processing: A survey on models and techniques. Expert Syst. Appl., 40(16):6601-6623, 2013. URL: https://doi.org/10.1016/J.ESWA.2013.05.007.
  154. Silke Pollandt and Rudolf Wille. Functorial scaling of ordinal data. Discret. Appl. Math., 147(1):101-111, 2005. URL: https://doi.org/10.1016/J.DAM.2004.06.024.
  155. Klaus Reuter. A note on the number of irreducibles of subdirect products. Algebra Universalis, 22:304-305, 1986. URL: https://doi.org/10.1007/bf01224035.
  156. Klaus Reuter. Matchings of linearly indecomposable modular lattices. Discrete Mathematics, 63:245-247, 1987. URL: https://doi.org/10.1016/0012-365X(87)90013-6.
  157. Klaus Reuter and Rudolf Wille. Complete congruence relations of concept lattices. Acta Sci. Math., 51:319-327, 1987. URL: http://acta.bibl.u-szeged.hu/id/eprint/15144.
  158. Fred S. Roberts. Measurement Theory. Cambridge University Press, 1984. URL: https://doi.org/10.1017/CBO9780511759871.
  159. Ryan A Rossi, Luke K McDowell, David William Aha, and Jennifer Neville. Transforming graph data for statistical relational learning. Journal of Artificial Intelligence Research, 45:363-441, 2012. URL: https://doi.org/10.1613/JAIR.3659.
  160. Camille Roth and Paul Bourgine. Lattice-based dynamic and overlapping taxonomies: The case of epistemic communities. Scientometrics, 69(2):429-447, 2006. URL: https://doi.org/10.1007/S11192-006-0161-6.
  161. Andreas Schmidt and Gerd Stumme. Prominence and dominance in networks. In Catherine Faron Zucker, Chiara Ghidini, Amedeo Napoli, and Toussaint Yannick, editors, Proceedings of the 21th International Conference on Knowledge Engineering and Knowledge Management (EKAW), LNCS, pages 370-385. Springer, 2018. URL: https://doi.org/10.1007/978-3-030-03667-6_24.
  162. Annemarie Zand Scholten and Denny Borsboom. A reanalysis of lord’s statistical treatment of football numbers. Journal of Mathematical Psychology, 53(2):69-75, 2009. URL: https://doi.org/10.1016/j.jmp.2009.01.002.
  163. Bernd Schröder. Ordered sets. An introduction with connections from combinatorics to topology. Basel: Birkhäuser/Springer, 2nd edition edition, 2016. URL: https://doi.org/10.1007/978-3-319-29788-0.
  164. Ernst Schröder. Algebra der Logik I, II, III. 1890, 1891, 1895. Thoemmes Press, Bristol, 2001. Google Scholar
  165. Yuan Shi, Wenzhe Li, and Fei Sha. Metric learning for ordinal data. In Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, pages 2030-2036, 2016. URL: https://doi.org/10.1609/AAAI.V30I1.10280.
  166. Dan A. Simovici and Chabane Djeraba. Mathematical Tools for Data Mining - Set Theory, Partial Orders, Combinatorics. Second Edition. Advanced Information and Knowledge Processing. Springer, 2014. URL: https://doi.org/10.1007/978-1-4471-6407-4.
  167. Gregor Snelting and Frank Tip. Understanding class hierarchies using concept analysis. ACM Trans. Program. Lang. Syst., 22(3):540-582, 2000. URL: https://doi.org/10.1145/353926.353940.
  168. John Sowa. Conceptual Structures: Information Processing in Mind and Machine. The Systems Programming Series. Addison-Wesley, 1984. Google Scholar
  169. N Spangenberg and KE Wolff. Conceptual grid evaluation, classification and related methods of data analysis, 1988. Google Scholar
  170. C. Spearman. The proof and measurement of association between two things. The American Journal of Psychology, 15(1):72-101, 1904. URL: https://doi.org/10.2307/1422689.
  171. Suvrit Sra and Inderjit S. Dhillon. Generalized nonnegative matrix approximations with bregman divergences. In Advances in neural information processing systems, pages 283-290, 2006. URL: https://proceedings.neurips.cc/paper/2005/hash/d58e2f077670f4de9cd7963c857f2534-Abstract.html.
  172. Jörg Stephan. Substitution decomposition of lattices. PhD thesis, TH Darmstadt, 1991. Google Scholar
  173. Jörg Stephan. Substitution products of lattices. In D. Dorninger, G. Eigenthaler, H. K. Kaiser, and W. B. Müller, editors, Contributions to general algebra, volume 7, pages 321-336, Wien, 1991. Hölder-Pichler-Tempsky. Google Scholar
  174. S. S. Stevens. On the theory of scales of measurement. Science, 103(2684):677-680, 1946. URL: https://doi.org/10.1126/science.103.2684.677.
  175. Selma Strahringer. Direct products of convex-ordinal scales. Order, 11:361-383, 1994. URL: https://doi.org/10.1007/bf01108768.
  176. Selma Strahringer and Rudolf Wille. Towards a structure theory for ordinal data. In M. Schader, editor, Analysing and modelling data and knowledge, pages 129-139, Heidelberg, 1992. Springer-Verlag. URL: https://doi.org/10.1007/978-3-642-46757-8_14.
  177. Selma Strahringer, Rudolf Wille, and Uta Wille. Mathematical support for empirical theory building. In Harry S. Delugach and Gerd Stumme, editors, Conceptual Structures: Broadening the Base, 9th International Conference on Conceptual Structures, ICCS 2001, Stanford, CA, USA, July 30-August 3, 2001, Proceedings, volume 2120 of LNCS, pages 169-186. Springer, 2001. URL: https://doi.org/10.1007/3-540-44583-8_13.
  178. Maximilian Stubbemann, Tom Hanika, and Friedrich Martin Schneider. Intrinsic dimension for large-scale geometric learning. Transactions on Machine Learning Research, 2023. URL: https://openreview.net/forum?id=85BfDdYMBY.
  179. Maximilian Stubbemann, Tom Hanika, and Gerd Stumme. Orometric methods in bounded metric data. In Michael R. Berthold, Ad Feelders, and Georg Krempl, editors, Advances in Intelligent Data Analysis XVIII - 18th International Symposium on Intelligent Data Analysis, IDA 2020, Konstanz, Germany, April 27-29, 2020, Proceedings, volume 12080 of LNCS, pages 496-508. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-44584-3_39.
  180. Maximilian Stubbemann and Gerd Stumme. The mont blanc of twitter: Identifying hierarchies of outstanding peaks in social networks. In Proc. 34th European Conference on Machine Learning / 27th Intl. Conference on Prinicples of Knowledge Discovery in Databases, 2023. URL: https://doi.org/10.1007/978-3-031-43418-1_11.
  181. Milan Studeny. Probabilistic Conditional Independence Structures. Springer Publishing Company, Incorporated, 1st edition, 2010. URL: https://doi.org/10.1007/b138557.
  182. G. Stumme. Hierarchies of conceptual scales. In T. B. Gaines, R. Kremer, and M. Musen, editors, Proc.Workshop on Knowledge Acquisition, Modeling and Management (KAW'99), volume 2, pages 78-95. Banff, October 16-22 1999. Google Scholar
  183. G. Stumme. Conceptual on-line analytical processing. In K. Tanaka, S. Ghandeharizadeh, and Y. Kambayashi, editors, Information Organization and Databases, chapter 14, pages 191-203. Kluwer, Boston-Dordrecht-London, 2000. URL: https://doi.org/10.1007/978-1-4615-1379-7_14.
  184. G. Stumme. Off to new shores - conceptual knowledge discovery and processing. Intl. J. Human-Comuter Studies (IJHCS), 59(3):287-325, sep 2003. URL: https://doi.org/10.1016/S1071-5819(03)00044-2.
  185. G. Stumme and A. Maedche. Fca-merge: Bottom-up merging of ontologies. In B. Nebel, editor, Proc. 17th Intl. Conf. on Artificial Intelligence (IJCAI '01), pages 225-230, Seattle, WA, USA, 2001. URL: https://www.ijcai.org/Proceedings/01/IJCAI-2001-b.pdf.
  186. Gerd Stumme. Knowledge acquisition by distributive concept exploration. In G. Ellis, R. Levinson, W. Rich, and J. F. Sowa, editors, Conceptual structures: applications, implementation and theory, number 954 in Lecture Notes in Artificial Intelligence, Berlin-Heidelberg-New York, 1995. Springer-Verlag. (supplementary proceedings). URL: https://doi.org/10.1007/3-540-60161-9.
  187. Gerd Stumme. Attribute exploration with background implications and exceptions. In H.-H. Bock and W. Polasek, editors, Data analysis and information systems, pages 457-469, Berlin-Heidelberg-New York, 1996. Springer-Verlag. URL: https://doi.org/10.1007/978-3-642-80098-6_39.
  188. Gerd Stumme. Local scaling in conceptual data systems. In P. W. Eklund, G. Ellis, and G. Mann, editors, Conceptual Structures: Knowledge Representation as Interlingua. Proc. ICCS'96, volume 1115 of LNAI, pages 308-320, Heidelberg, 1996. Springer. URL: https://doi.org/10.1007/3-540-61534-2_20.
  189. Gerd Stumme. Concept exploration - a tool for creating and exploring > conceptual hierarchies. In D. Lukose, H. Delugach, M. Keeler, L. Searle, and J. F. Sowa, editors, Conceptual structures: Fulfilling Peirce’s dream, volume 1257 of Lecture Notes in Artificial Intelligence, Berlin-Heidelberg-New York, 1997. Springer-Verlag. URL: https://doi.org/10.1007/BFB0027880.
  190. Gerd Stumme. Free distributive completions of partial complete lattices. In: Order, 14:179-189, 1998. URL: https://doi.org/10.1023/A:1006002507294.
  191. Gerd Stumme. On-line analytical processing with conceptual information systems. In K. Tanaka and S. Ghandeharizadeh, editors, Proc. 5th Intl. Conf. on Foundations of Data Organization (FODO'98), pages 117-126, November 12-13 1998. Google Scholar
  192. Gerd Stumme. Acquiring expert knowledge for the design of conceptual information systems. In D. Fensel and R. Studer, editors, Knowledge Acquisition, Modeling, and Management. Proc. 11th European, volume 1621 of LNAI, pages 275-290, Heidelberg, 1999. Springer. URL: https://doi.org/10.1007/3-540-48775-1_17.
  193. Gerd Stumme. A finite state model for on-line analytical processing in triadic contexts. In Bernhard Ganter and Robert Godin, editors, Proc. 3rd Intl. Conf. on Formal Concept Analysis, volume 3403 of LNCS, pages 315-328, Heidelberg, 2005. Springer. URL: https://doi.org/10.1007/978-3-540-32262-7_22.
  194. Gerd Stumme and Rudolf Wille. A geometrical heuristic for drawing concept lattices. In R. Tamassia and I.G. Tollis, editors, Graph Drawing, volume 894 of LNCS, pages 452-459, Heidelberg, 1995. Springer. URL: https://doi.org/10.1007/3-540-58950-3_399.
  195. Gerd Stumme and Rudolf Wille, editors. Begriffliche Wissensverarbeitung - Methoden und Anwendungen, Heidelberg, 2000. Springer. URL: https://doi.org/10.1007/978-3-642-57217-3.
  196. Karl-Theodor Sturm et al. On the geometry of metric measure spaces. Acta mathematica, 196(1):65-131, 2006. URL: https://doi.org/10.1007/s11511-006-0002-8.
  197. Kozo Sugiyama, Shojiro Tagawa, and Mitsuhiko Toda. Methods for visual understanding of hierarchical system structures. IEEE Transactions on Systems, Man, and Cybernetics, 11(2):109-125, 1981. URL: https://doi.org/10.1109/TSMC.1981.4308636.
  198. Patrick Suppes, David H. Krantz, R. Duncan Luce, and Amos Tversky. Foundations of Measurement - Geometrical, Threshold, and Probabilistic Representations, volume 2. Academic Press, 1989. URL: https://doi.org/10.1016/C2009-0-21665-5.
  199. Atsushi Suzuki, Atsushi Nitanda, Jing Wang, Linchuan Xu, Kenji Yamanishi, and Marc Cavazza. Generalization error bound for hyperbolic ordinal embedding. In Marina Meila and Tong Zhang, editors, Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pages 10011-10021. PMLR, 18-24 July 2021. URL: http://proceedings.mlr.press/v139/suzuki21a.html.
  200. Viola Takàcs. Two applications of Galois graphs in pedagogical research. Manuscript of a lecture given at TH Darmstadt, 1984. Google Scholar
  201. Eran Tal. Measurement in science. In Edward N. Zalta, editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, fall 2017 edition, 2017. URL: https://plato.stanford.edu/Entries/measurement-science/.
  202. M. Thomas. Mathematization, not measurement: A critique of stevens’ scales of measurement. Journal of Methods and Measurement in the Social Sciences, 10(2):76-94, 2020. URL: https://doi.org/10.2458/v10i2.23785.
  203. Ioannis Tollis, Giuseppe Di Battista, Peter Eades, and Roberto Tamassia. Graph drawing. Algorithms for the visualization of graphs. Upper Saddle River, NJ: Prentice Hall, 1999. Google Scholar
  204. Günter Trendler. Measurement theory, psychology and the revolution that cannot happen. Theory & Psychology, 19:579-599, oct 2009. URL: https://doi.org/10.1177/0959354309341926.
  205. Günter Trendler. Conjoint measurement undone. Theory & Psychology, 29(1):100-128, 2019. URL: https://doi.org/10.1177/0959354318788729.
  206. William T. Trotter. Combinatorics and Partially Ordered Sets: Dimension Theory. Johns Hopkins University Press, 2001. Google Scholar
  207. William T. Trotter and Bartosz Walczak. Boolean dimension and local dimension. Electron. Notes Discret. Math., 61:1047-1053, 2017. URL: https://doi.org/10.1016/J.ENDM.2017.07.071.
  208. John W. Tukey. The future of data analysis. Annals of Mathematical Statistics, 33:1-67, 1962. URL: https://doi.org/10.1214/aoms/1177704711.
  209. Alexey A. Tuzhilin. Who invented the gromov-hausdorff distance?, 2016. cite arxiv:1612.00728Comment: 7 pages, 10 references. URL: https://doi.org/10.48550/arXiv.1612.00728.
  210. Leena Chennuru Vankadara and Ulrike von Luxburg. Measures of distortion for machine learning. In Samy Bengio, Hanna M. Wallach, Hugo Larochelle, Kristen Grauman, Nicolò Cesa-Bianchi, and Roman Garnett, editors, Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, 3-8 December 2018, Montréal, Canada, pages 4891-4900, 2018. URL: https://proceedings.neurips.cc/paper/2018/hash/4c5bcfec8584af0d967f1ab10179ca4b-Abstract.html.
  211. Fei Wang and Jimeng Sun. Survey on distance metric learning and dimensionality reduction in data mining. Data Min. Knowl. Discov., 29(2):534-564, 2015. URL: https://doi.org/10.1007/S10618-014-0356-Z.
  212. Joe H. Ward. Hierarchical grouping to optimize an objective function. Journal of the American Statistical Association, 58(301):236-244, 1963. URL: https://doi.org/10.1080/01621459.1963.10500845.
  213. Frank Wilcoxon. Individual comparisons by ranking methods. Biometrics Bulletin, 1(6):80-83, 1945. URL: https://doi.org/10.1007/978-1-4612-4380-9_16.
  214. Marcel Wild. A theory of finite closure spaces based on implications. Advances in Mathematics, 108(1):118-139, 1994. URL: https://doi.org/10.1006/aima.1994.1069.
  215. R. Wille. Concept lattices and conceptual knowledge systems. Computers and Mathematics with Applications, 23:493-515, 1992. URL: https://doi.org/10.1016/0898-1221(92)90120-7.
  216. Rudolf Wille. Restructuring lattice theory: an approach based on hierarchies of concepts. In Ivan Rival, editor, Ordered sets, pages 445-470. Reidel, 1982. URL: https://doi.org/10.1007/978-94-009-7798-3_15.
  217. Rudolf Wille. Complete tolerance relations of concept lattices. In G. Eigenthaler, H. K. Kaiser, W. B. Müller, and W. Nöbauer, editors, Contributions to general algebra, volume 3, pages 397-415. Hölder-Pichler-Tempsky, Wien, 1985. Google Scholar
  218. Rudolf Wille. Finite distributive lattices as concept lattices. Atti Inc. Logica Mathematica, 2:635-648, 1985. Google Scholar
  219. Rudolf Wille. Tensorial decomposition of concept lattices. Order, 2:81-95, 1985. URL: https://doi.org/10.1007/bf00337926.
  220. Rudolf Wille. Subdirect product construction of concept lattices. Discrete Mathematics, 63:305-313, 1987. URL: https://doi.org/10.1016/0012-365X(87)90019-7.
  221. Rudolf Wille. Lattices in data analysis: how to draw them with a computer. In Ivan Rival, editor, Algorithms and order, pages 33-58, Dordrecht-Boston, 1989. Kluwer. URL: https://doi.org/10.1007/978-94-009-2639-4_2.
  222. Rudolf Wille. Tensor products of complete lattices as closure systems. In D. Dorninger, G. Eigenthaler, H. K. Kaiser, and W. B. Müller, editors, Contributions to general algebra, volume 7, pages 381-385. Hölder-Pichler-Tempsky, Wien, 1991. Google Scholar
  223. Rudolf Wille and Uta Wille. Uniqueness of coordinatizations of ordinal structures. In Contributions to general algebra, volume 9, pages 321-324, Wien, 1995. Hölder-Pichler-Tempsky. Google Scholar
  224. Rudolf Wille and Uta Wille. Coordinatization of ordinal structures. Order, 13, 1996. URL: https://doi.org/10.1007/bf00338747.
  225. Uta Wille. Representation of ordinal contexts by ordered n-quasigroups. Eur. J. Comb., 17(2-3):317-333, 1996. URL: https://doi.org/10.1006/EUJC.1996.0027.
  226. Uta Wille. Linear measurement models - axiomatizations and axiomatizability. Journal of Mathematical Psychology, 44(4):617-650, 2000. URL: https://doi.org/10.1006/jmps.2000.1326.
  227. Svante Wold, Kim Esbensen, and Paul Geladi. Principal component analysis. Chemometrics and intelligent laboratory systems, 2(1-3):37-52, 1987. URL: https://doi.org/10.1016/0169-7439(87)80084-9.
  228. Mihalis Yannakakis. The complexity of the partial order dimension problem. SIAM Journal on Algebraic Discrete Methods, 3(3):351-358, 1982. URL: https://doi.org/10.1137/0603036.
  229. Javier Yáñez and Javier Montero. A poset dimension algorithm. J. Algorithms, 30(1):185-208, 1999. URL: https://doi.org/10.1006/JAGM.1998.0974.
  230. Alexander Zap, Tobias Joppen, and Johannes Fürnkranz. Deep ordinal reinforcement learning. In Ulf Brefeld, Élisa Fromont, Andreas Hotho, Arno J. Knobbe, Marloes H. Maathuis, and Céline Robardet, editors, Machine Learning and Knowledge Discovery in Databases - European Conference, ECML PKDD 2019, Würzburg, Germany, September 16-20, 2019, Proceedings, Part III, volume 11908 of LNCS, pages 3-18. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-46133-1_1.
  231. Yiqun Zhang and Yiu ming Cheung. An ordinal data clustering algorithm with automated distance learning. In The Thirty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2020, The Thirty-Second Innovative Applications of Artificial Intelligence Conference, IAAI 2020, The Tenth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2020, New York, NY, USA, February 7-12, 2020, pages 6869-6876. AAAI Press, 2020. URL: https://doi.org/10.1609/AAAI.V34I04.6168.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact