Методы анализа и визуализации структуры данных о близости
Аннотация
В статье приводится обзор моделей анализа и визуализации матриц близости. Рассматриваются три класса моделей: пространственные, теоретикомножественные и графы. Осуждаются формальные и содержательные аспекты применения моделей к анализу социологических данных.
Ключевые слова:
матрица близости, многомерное шкалирование, классификация, иерархическая классификация, аддитивная кластеризация, качественный факторный анализ, ультраметрические деревья, аддитивные деревья, сети
Литература
Coxon А.Р.М. The User's Guide to Multidimensional Scaling. London: Heinemann Educational Books Ltd, 1982.
Дэйвисон М. Многомерное шкалирование: Методы наглядного представления данных. М.: Финансы и статистика, 1982.
AttneaveF. Dimensions of Similarity//American Journal of Psychology. 1950. No. 63. P 546-554.
Shepard R.N. Representation of Structure in Similarity Data: Problems and Prospects//Psychometrika. 1974. Vol. 39. P 373-421.
Coombs C.H. A Theory of Data. N.Y.: Wiley, 1964.
Shepard R.N. Attention and the Metric Structure of the Stimulus Space//Journal of Mathematical Psychology. 1964. Vol. 1. P 54-87.
Pieszko H. Multidimensional Scaling in Riemannian Space//Journal of Mathematical Psychology. 1975. Vol. 14. P 449-477.
Терехина А.Ю. Анализ данных методами многомерного шкалирования. М.: Наука, 1986.
Cox T.F., Cox M.A.A. Multidimensional Scaling on a Sphere//Communicational Statistics. 1991. No. 20. P 2943-2953.
Torgerson WS Theory and Methods of Scaling. N.Y.: Wiley, 1958.
Young F.W, Householder A.S Discussion of a Set of Points in Terms of Their Mutual Distances//Psychometrika. 1938. Vol. 3. PI 19-22.
Torgerson WS Multidimensional Scaling: Theory and Method//Psychometrika. 1952. Vol. 17.
Messick S.M., Abelson R.P. The Additive Constant Problem in Multidimensional Scaling//Psychometrika. 1956. Vol. 21. P 1-15.
Lingoes J.C. Some Boundary Conditions for Monotone Analysis of Symmetric Matrices//Psychometrika. 1971. Vol. 36. P 195-203.
Cailliez F. The Analytical Solution of Additive Constant Problem//Psychometrika. 1983. Vol. 48. P 305-308.
Restle F. A Metric and an Ordering on Sets//Psychometrika. 1959. Vol. 24. P. 207-220.
Shepard R.N. The Analysis of Proximity Data with Unknown Distance Function//Psychometrika. 1962. Vol. 27.
Kruskal J.B. Multidimensional Scaling by Optimizing Goodness of Fit to a Non-metric Hypothesis//Psychometrika. 1964. Vol. 29.
Kruskal J.B. Non-metric Multidimensional Scaling: a Numerical Method//Psychometrika. 1964. Vol. 29.
Guttman L. A General Nonmetric Technique for Finding the Smallest Coordinate Space for a Configuration of Points//Psychometrika. 1968. Vol. 33. P 469-504.
Lingoes J.C., Roskam E.E. A Mathematical and Empirical Analysis of Two Multidimensional Scaling Algorithms//Psychometrika. 1973. Vol. 38.
Johnson R.M. Pairwise Nonmetric Multidimensional Scaling//Psychometrika. 1973. Vol. 38. P 11-18.
Takane Y., Young F.W., DeLeeuw J. Non-metric Individual Difference Multidimensional Scaling: an Alternating Least Squares Method with Optimal Scaling Features//Psychometrika. 1977. Vol. 42. P 7-67.
Bloxom B. An Alternative Method of Fitting a Model of Individual Differences in Multidimensional Scaling//Psychometrika. 1974. Vol. 39. P 365-367.
Carroll J.D., Chang J.J. Analysis of Individual Differences in Multidimensional Scaling via N-way Generalization of «Eckart-Young» Decomposition//Psychometrika. 1970. Vol. 35. P 283-319.
Tucker L.R. Relation Between Multidimensional Scaling and Three-mode Factor Analysis//Psychometrika. 1972. Vol. 37. P 3-
Carroll J.D. Individual Differences and Multidimensional Scaling//Multidimensional Scaling: Theory and Applications in the Behavioral Sciences/Ed. by R.N. Shepard, A.K. Romney, S.B. Nerlove. N.Y.: Seminar Press, 1972.
Tversky A Features of Similarity//Psychological Review. 1977. No. 84(4). P. 227-252.
Shepard R.N., Arabie P. Additive Clustering Representation of Similarities as Combination of Discrete Overlapping Properties//Psychological Review. 1979. No. 86(2). P 87-123.
Дюран Б., Оделл П. Кластерный анализ. М.: Статистика, 1977.
Куперштох В.Л., Миркин Б.Г., Трофимов В.А. Метод наименьших квадратов в анализе качественных признаков//Проблемы анализа дискретной информации. Новосибирск: ИЭиОПП СО АН СССР, 1976. С. 4-23. Вып. 2.
Миркин Б.Г. Группировки в социально-экономических исследованиях. М.: Финансы и статистика, 1985.
ArarbieP, Carroll J.D. MAPCLUS: A Mathematical Programming Approach to Fitting the ADCLUS Model//Psychometrika. 1980. Vol. 45(2). P 211-235.
Carroll J.D., Arabie P INDCLUS: An Individual Difference Generalization of the ADCLUS Model and the MAPCLUS Algorithm//Psychometrika. 1982. Vol. 48. P 157-169.
Johnson SC. Hierarchical Clustering Schemes//Psychometrika 1967. Vol. 32(3). P 241-254.
Sattath S, Tversky A Additive Similarity Trees//Psychometrika 1977. Vol. 42(3). P 319-345.
Buneman P. The Recovery of Trees from Measures of Dissimilarity//Mathematics in the Archeological and Historical Sciences. Edinburgh, UK: Edinburgh University Press, 1971.
Corter J.E. ADDTREE/P: A PASCAL Program for Fitting Additive Trees Based on Sattath and Tversky's ADDTREE Algorithm//Behavioral Research Methods & Instrumentation. 1982. No. 14. P 353-354.
Corter J.E. An Efficient Metric Combinatorial Algorithm for Fitting Additive Trees//Multivariate Behavioral Research. 1998. No. 33(2). P 249-271.
Carroll J.D., Pruzansky Si Discrete and Hybrid Models for Scaling//Similarity and Choice. Bern: Hans Huber, 1980.
De Soete G. A Least Squares Algorithm for Fitting Additive Trees to Proximity Data//Psychometrika. 1983. Vol. 48(4). P 621-626.
De Soete G. Additive-tree Representation of Incomplete Dissimilarity Data//Quality and Quantity. 1984. No. 18. P 387-393.
Carroll J.D., Clark L., DeSarbo WS The Representation of Three-way Proximities Data by Single and Multiple Tree Structure Models//Journal of Classification. 1984. Vol. 1. P 25-74.
Corter J.E., Tversky A Extended Similarity Trees//Psychometrika. 1986. Vol. 51(3). P 429-451.
Makarenkov V., Legendre P Improving the Additive Tree Representation of a Dissimilarity Matrix Using Reticulations//Data Analysis, Classification, and Related Methods. Berlin: Springer, 2000. P 35-46.
Lapointe F.-J. How to Account for Reticulation Events in Phylogenic Analysis: a Comparison of Distance-based Methods//Journal of Classification. 2000. Vol. 17. P 175-184.
Hutchinson J.W NETSCAL: A Network Scaling Algorithm for Nonsymmetric Proximity Data//Psychometrika. 1989. Vol. 54. P 25-51.
Klauer K.C., Carroll J.D. A Mathematical Programming Approach to Fitting General Graph//Journal of Classification. 1989. Vol. 6. P 247-270.
Краскал Дж. Взаимосвязь между многомерным шкалированием и кластер-анализом//Классификация и кластер. М.: Мир, 1980.
Torgerson WS Multidimensional Scaling of Similarity//Psychometrika. 1965. Vol. 30(4).
Degerman R. Multidimensional Analysis of Complex Structure: Mixture of Class and Quantitative Variation//Psychometrika. 1970. Vol. 35(4). P 475-491.
Navarro D.J., Lee M.D. Combining Dimensions and Features in Similarity-based Representations//Advances in Neural Information Processing Systems/Ed. by S. Becker, S. Thrun, K. Obermayer. N.Y.: MIT Press, 2003. P 67-74. No. 15.
Carroll J.D., Chaturvedi A. A General Approach to Clustering and Multidimensional Scaling of Two-way, Three-way, or Higher-way Data//Geometric Representation of Perceptual Phenomena/Ed. by R.D. Luce M. D'Zmura, D.D. Hoffman, G. Iverson, A.K. Romney. Mahwah, NJ: Erbaum, 1995. P 295-318.
Дэйвисон М. Многомерное шкалирование: Методы наглядного представления данных. М.: Финансы и статистика, 1982.
AttneaveF. Dimensions of Similarity//American Journal of Psychology. 1950. No. 63. P 546-554.
Shepard R.N. Representation of Structure in Similarity Data: Problems and Prospects//Psychometrika. 1974. Vol. 39. P 373-421.
Coombs C.H. A Theory of Data. N.Y.: Wiley, 1964.
Shepard R.N. Attention and the Metric Structure of the Stimulus Space//Journal of Mathematical Psychology. 1964. Vol. 1. P 54-87.
Pieszko H. Multidimensional Scaling in Riemannian Space//Journal of Mathematical Psychology. 1975. Vol. 14. P 449-477.
Терехина А.Ю. Анализ данных методами многомерного шкалирования. М.: Наука, 1986.
Cox T.F., Cox M.A.A. Multidimensional Scaling on a Sphere//Communicational Statistics. 1991. No. 20. P 2943-2953.
Torgerson WS Theory and Methods of Scaling. N.Y.: Wiley, 1958.
Young F.W, Householder A.S Discussion of a Set of Points in Terms of Their Mutual Distances//Psychometrika. 1938. Vol. 3. PI 19-22.
Torgerson WS Multidimensional Scaling: Theory and Method//Psychometrika. 1952. Vol. 17.
Messick S.M., Abelson R.P. The Additive Constant Problem in Multidimensional Scaling//Psychometrika. 1956. Vol. 21. P 1-15.
Lingoes J.C. Some Boundary Conditions for Monotone Analysis of Symmetric Matrices//Psychometrika. 1971. Vol. 36. P 195-203.
Cailliez F. The Analytical Solution of Additive Constant Problem//Psychometrika. 1983. Vol. 48. P 305-308.
Restle F. A Metric and an Ordering on Sets//Psychometrika. 1959. Vol. 24. P. 207-220.
Shepard R.N. The Analysis of Proximity Data with Unknown Distance Function//Psychometrika. 1962. Vol. 27.
Kruskal J.B. Multidimensional Scaling by Optimizing Goodness of Fit to a Non-metric Hypothesis//Psychometrika. 1964. Vol. 29.
Kruskal J.B. Non-metric Multidimensional Scaling: a Numerical Method//Psychometrika. 1964. Vol. 29.
Guttman L. A General Nonmetric Technique for Finding the Smallest Coordinate Space for a Configuration of Points//Psychometrika. 1968. Vol. 33. P 469-504.
Lingoes J.C., Roskam E.E. A Mathematical and Empirical Analysis of Two Multidimensional Scaling Algorithms//Psychometrika. 1973. Vol. 38.
Johnson R.M. Pairwise Nonmetric Multidimensional Scaling//Psychometrika. 1973. Vol. 38. P 11-18.
Takane Y., Young F.W., DeLeeuw J. Non-metric Individual Difference Multidimensional Scaling: an Alternating Least Squares Method with Optimal Scaling Features//Psychometrika. 1977. Vol. 42. P 7-67.
Bloxom B. An Alternative Method of Fitting a Model of Individual Differences in Multidimensional Scaling//Psychometrika. 1974. Vol. 39. P 365-367.
Carroll J.D., Chang J.J. Analysis of Individual Differences in Multidimensional Scaling via N-way Generalization of «Eckart-Young» Decomposition//Psychometrika. 1970. Vol. 35. P 283-319.
Tucker L.R. Relation Between Multidimensional Scaling and Three-mode Factor Analysis//Psychometrika. 1972. Vol. 37. P 3-
Carroll J.D. Individual Differences and Multidimensional Scaling//Multidimensional Scaling: Theory and Applications in the Behavioral Sciences/Ed. by R.N. Shepard, A.K. Romney, S.B. Nerlove. N.Y.: Seminar Press, 1972.
Tversky A Features of Similarity//Psychological Review. 1977. No. 84(4). P. 227-252.
Shepard R.N., Arabie P. Additive Clustering Representation of Similarities as Combination of Discrete Overlapping Properties//Psychological Review. 1979. No. 86(2). P 87-123.
Дюран Б., Оделл П. Кластерный анализ. М.: Статистика, 1977.
Куперштох В.Л., Миркин Б.Г., Трофимов В.А. Метод наименьших квадратов в анализе качественных признаков//Проблемы анализа дискретной информации. Новосибирск: ИЭиОПП СО АН СССР, 1976. С. 4-23. Вып. 2.
Миркин Б.Г. Группировки в социально-экономических исследованиях. М.: Финансы и статистика, 1985.
ArarbieP, Carroll J.D. MAPCLUS: A Mathematical Programming Approach to Fitting the ADCLUS Model//Psychometrika. 1980. Vol. 45(2). P 211-235.
Carroll J.D., Arabie P INDCLUS: An Individual Difference Generalization of the ADCLUS Model and the MAPCLUS Algorithm//Psychometrika. 1982. Vol. 48. P 157-169.
Johnson SC. Hierarchical Clustering Schemes//Psychometrika 1967. Vol. 32(3). P 241-254.
Sattath S, Tversky A Additive Similarity Trees//Psychometrika 1977. Vol. 42(3). P 319-345.
Buneman P. The Recovery of Trees from Measures of Dissimilarity//Mathematics in the Archeological and Historical Sciences. Edinburgh, UK: Edinburgh University Press, 1971.
Corter J.E. ADDTREE/P: A PASCAL Program for Fitting Additive Trees Based on Sattath and Tversky's ADDTREE Algorithm//Behavioral Research Methods & Instrumentation. 1982. No. 14. P 353-354.
Corter J.E. An Efficient Metric Combinatorial Algorithm for Fitting Additive Trees//Multivariate Behavioral Research. 1998. No. 33(2). P 249-271.
Carroll J.D., Pruzansky Si Discrete and Hybrid Models for Scaling//Similarity and Choice. Bern: Hans Huber, 1980.
De Soete G. A Least Squares Algorithm for Fitting Additive Trees to Proximity Data//Psychometrika. 1983. Vol. 48(4). P 621-626.
De Soete G. Additive-tree Representation of Incomplete Dissimilarity Data//Quality and Quantity. 1984. No. 18. P 387-393.
Carroll J.D., Clark L., DeSarbo WS The Representation of Three-way Proximities Data by Single and Multiple Tree Structure Models//Journal of Classification. 1984. Vol. 1. P 25-74.
Corter J.E., Tversky A Extended Similarity Trees//Psychometrika. 1986. Vol. 51(3). P 429-451.
Makarenkov V., Legendre P Improving the Additive Tree Representation of a Dissimilarity Matrix Using Reticulations//Data Analysis, Classification, and Related Methods. Berlin: Springer, 2000. P 35-46.
Lapointe F.-J. How to Account for Reticulation Events in Phylogenic Analysis: a Comparison of Distance-based Methods//Journal of Classification. 2000. Vol. 17. P 175-184.
Hutchinson J.W NETSCAL: A Network Scaling Algorithm for Nonsymmetric Proximity Data//Psychometrika. 1989. Vol. 54. P 25-51.
Klauer K.C., Carroll J.D. A Mathematical Programming Approach to Fitting General Graph//Journal of Classification. 1989. Vol. 6. P 247-270.
Краскал Дж. Взаимосвязь между многомерным шкалированием и кластер-анализом//Классификация и кластер. М.: Мир, 1980.
Torgerson WS Multidimensional Scaling of Similarity//Psychometrika. 1965. Vol. 30(4).
Degerman R. Multidimensional Analysis of Complex Structure: Mixture of Class and Quantitative Variation//Psychometrika. 1970. Vol. 35(4). P 475-491.
Navarro D.J., Lee M.D. Combining Dimensions and Features in Similarity-based Representations//Advances in Neural Information Processing Systems/Ed. by S. Becker, S. Thrun, K. Obermayer. N.Y.: MIT Press, 2003. P 67-74. No. 15.
Carroll J.D., Chaturvedi A. A General Approach to Clustering and Multidimensional Scaling of Two-way, Three-way, or Higher-way Data//Geometric Representation of Perceptual Phenomena/Ed. by R.D. Luce M. D'Zmura, D.D. Hoffman, G. Iverson, A.K. Romney. Mahwah, NJ: Erbaum, 1995. P 295-318.
