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Numerische Klassifikation

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Klassifikation von Mustern

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Zusammenfassung

Die in den vorangehenden beiden Kapiteln erörterten Verarbeitungsmethoden erlauben es, ein aufgenommenes Muster ρ f(x) in einen Merkmalvektor ρ c zu transformieren. Die grundlegende Voraussetzung ist, daß die erhaltenen Merkmale Postulat 3 aus Abschnitt 1.3 genügen. Es bleibt nun noch die Aufgabe, den Merkmalvektor einer Klasse Ωκ zuzuordnen, also die in (1.6) angegebene Abbildung

$$\rho _c \to \kappa \in \left\{ {1, \ldots ,k} \right\}\,oder\rho _c \to \kappa \in \left\{ {0,1, \ldots ,k} \right\}$$

festzulegen und damit eine Klassifikation durchzuführen. Da die Komponenten ρcv des Vektors ρ c gemäß (3.2) reelle Zahlen sind, wird diese Abbildung als numerische Klassifikation bezeichnet. Es wird sich zeigen, daß zu ihrer Durchführung zum Teil umfangreiche numerische Rechnungen erforderlich sind. Die Klassifikation ist der letzte der in Bild 1.5 angegebenen Verarbeitungsschritte, und damit ist die Klassifikationsaufgabe gelöst.

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Literatur

  1. J.O. Berger: Statistical Decision Theory, Foundations, Concepts, and Methods. Springer, New York, Heidelberg, Berlin 1980.

    MATH  Google Scholar 

  2. C.K. Chow: An Optimum Character Recognition System Using Decision Functions. IEEE Trans. EC-6 (1957) 247–254.

    Google Scholar 

  3. T.W. Anderson: Introduction to Multivariate Statistical Analysis. J. Wiley, New York 1958, Chap. 3.

    MATH  Google Scholar 

  4. H. Niemann: Begründung und Anwendung einer Theorie zur quantitativen Beschreibung und Erkennung von Mustern. Dissertation Techn. Universität Hannover 1969.

    Google Scholar 

  5. D.B. Cooper: Multivariate Extension of Onedimensional Probability Distributions. IEEE Trans. EC-12 (1963) 572–573.

    Google Scholar 

  6. D. Kazakos, T. Cotsidas: A Decision Theory Approach to the Approximation of Discrete Probability Densities. IEEE Trans. PAMI-2 (1980) 61–67.

    Google Scholar 

  7. E. Kreyszig: Statistische Methoden und ihre Anwendungen. Vandenhoeck u. Ruprecht, Göttingen 1967, Kapitel 15.

    Google Scholar 

  8. R.B. Crane, W.A. Malila, W. Richardson: Suitability of the Normal Density Assumption for Processing Multispectral Scanner Data. IEEE Trans. GE-10 (1972) 158–165.

    Google Scholar 

  9. C.R. Rao: Linear Statistical Inference and its Applications. J. Wiley, New York 1973, Sect. 4g.1.

    Book  MATH  Google Scholar 

  10. A. Papoulis: Probability, Random Variables, and Stochastic Processes. Intern. Student Ed., McGraw Hill Kogakusha, Tokyo 1965, Sect. 7.4.

    Google Scholar 

  11. H. Schwarz, H. Rutishauser, E. Stiefel: Numerik symmetrischer Matrizen, B.G. Teubner, Stuttgart 1968.

    MATH  Google Scholar 

  12. J. Schürmann, P. Krause: Vergleich zweier quadratischer Klassifikatoren am gleichen Datenmaterial. El. Rechenanlagen 16 (1974) 132–142.

    MATH  Google Scholar 

  13. J. Todd: Survey of Numerical Analysis. McGraw Hill, New York 1962.

    MATH  Google Scholar 

  14. H. Niemann, J. Weiss: A Fast-Converging Algorithm for Nonlinear Mapping of Highdimensional Data to a Plane. IEEE Trans. C-28 (1979) 142–147.

    MathSciNet  Google Scholar 

  15. M.T. Wasan: Stochastic Approximation. Cambridge University Press 1969.

    Google Scholar 

  16. A.E. Albert, L.A. Gardner: Stochastic Approximation and Nonlinear Regression. MIT Press Res. Monograph 42, Cambridge 1966.

    Google Scholar 

  17. G. Sebestyen, J. Edie: An Algorithm for Nonparametric Pattern Recognition. IEEE Trans. EC-15 (1966) 908–915.

    Google Scholar 

  18. D.O. Loftsgaarden, G.P. Quesenbury: A Nonparametric Estimate of a Multivariable Density function. Ann. Math. Stat. 36 (1965) 1049–1051.

    Article  MATH  Google Scholar 

  19. E. Parzen: On Estimation of a Probability Density and Mode. Ann. Math. Stat. 33 (1962) 1065–1076.

    Article  MATH  MathSciNet  Google Scholar 

  20. V.K. Murthy: Estimation of Probability Density. Ann. Math. Stat. 36 (1965) 1027–1031.

    Article  MATH  MathSciNet  Google Scholar 

  21. R.L. Kashyap, C.C. Blaydon: Estimation of Probability Density and Distribution Functions. IEEE Trans. IT-14 (1968) 549–556.

    Google Scholar 

  22. T.M. Cover, P.E. Hart: Nearest Neighbour Pattern Classification. IEEE Trans. IT-13 (1967) 21–27.

    Google Scholar 

  23. T.M. Cover: Learning in Pattern Recognition. In S. Watanabe (ed.): Methodologies of Pattern Recognition. Academic Press, New York 1969, 111–132.

    Google Scholar 

  24. M.E. Hellman: The Nearest Neighbour Classification Rule With a Reject Option IEEE Trans. SSC-6 (1970) 179–185.

    MathSciNet  Google Scholar 

  25. P.E. Hart: The Condensed Nearest Neighbour Rule. IEEE Trans. IT-14 (1968) 515–516.

    Google Scholar 

  26. G.W. Gates: The Reduced Nearest Neighbour Rule. IEEE Trans. IT-18 (1972) 431–433.

    Google Scholar 

  27. I. Tomek: Two Modifications of CNN. IEEE Trans. SMC-6 (1976) 769–772.

    MathSciNet  Google Scholar 

  28. P.A. Devijver, J. Kittler: On the Edited Nearest Neighbour Rule. Proc. 5. ICPR, Miami, Florida 1980, 72-80.

    Google Scholar 

  29. K. Fukunaga, P.M. Narendra: A Branch and Bound Algorithm for Computing k-Nearest Neighbours. IEEE Trans. C-24 (1975) 750–753.

    MathSciNet  Google Scholar 

  30. R.B. Murphy: Nonparametric Tolerance Limits. Ann. Math. Stat. 17 (1948) 377–408.

    Google Scholar 

  31. C.P. Quesenberry, M.P. Gessaman: Nonparametric Discrimination Using Tolerance Regions. Ann. Math. Stat. 39 (1968) 664–673.

    Article  MATH  MathSciNet  Google Scholar 

  32. M.W. Anderson, R.D. Benning: A Distribution Free Discrimination Procedure Based on Clustering. IEEE Trans. IT-16 (1970) 541–548.

    MathSciNet  Google Scholar 

  33. M. Ichino: A Nonparametric Multiclass Pattern Classifier. IEEE Trans. SMC-9 (1979) 345–352.

    MathSciNet  Google Scholar 

  34. A. Wald: Sequential Analysis. J. Wiley, New York 1957.

    Google Scholar 

  35. Y.T. Chien, K.S. Fu: A Modified Sequential Recognition Machine Using Time-Varying Stopping Boundaries. IEEE Trans. IT-12 (1966) 206–214.

    Google Scholar 

  36. E.G. Henrichon, K.S. Fu: A Nonparametric Partitioning Procedure for Pattern Classification. IEEE Trans. C-18 (1969) 614–624.

    Google Scholar 

  37. W.S. Meisel, D.A. Michalopoulos: A Partitioning Algorithm With Application in Pattern Classification and the Optimization of Decision Trees. IEEE Trans. C-23 (1973) 93–108.

    Google Scholar 

  38. H.J. Payne, W.S. Meisel: An Algorithm for Constructing Optimal Binary Decision Trees. IEEE Trans. C-28 (1977) 905–916.

    MathSciNet  Google Scholar 

  39. R.L.P. Chang, T. Pavlidis: Fuzzy Decision Tree Algorithms. IEEE Trans. SMC-7 (1977) 28–34.

    MathSciNet  Google Scholar 

  40. D.H. Ballard, J. Sklansky: A Ladder Structured Decision Tree for Recognizing Tumors in Chest Radiographs. IEEE Trans. C-25 (1976) 503–513.

    Google Scholar 

  41. W.A. Armstrong, J. Gecsei: Adaptation Algorithms for Binary Tree Networks. IEEE Trans. SMC-9 (1979) 276–285.

    Google Scholar 

  42. A.V. Kulkarni, L.N. Kanal: An Optimization Approach to Hierarchical Classifier Design. Proc. 3. IJCPR, Coronado, Calif. 1976, 459-466.

    Google Scholar 

  43. J.C. Stoffel: A Classifier Design Technique for Discrete Variable Pattern Recognition Problems. IEEE Trans. C-23 (1974) 428–441.

    Google Scholar 

  44. L.R. Rabiner, A.E. Rosenberg, S.E. Levinson: Considerations in Dynamic Time Warping Algorithms for Discrete Word Recognition. IEEE Trans. ASSP-26 (1978) 575–586.

    Google Scholar 

  45. H. Sakoe, S. Chiba: Dynamic Programming Algorithm Optimization for Spoken Word Recognition. IEEE Trans. ASSP-26 (1978) 43–49.

    Google Scholar 

  46. L.R. Rabiner, G.E. Schmidt: Application of Dynamic Time Warping to Connected Digit Recognition. IEEE Trans. ASSP-28 (1980) 377–388.

    Google Scholar 

  47. C. Myers, L.R. Rabiner, A.E. Rosenberg: Performance Tradeoffs in Dynamic Time Warping Algorithms for Isolated Word Recognition. IEEE Trans. ASSP-28 (1980) 623–635.

    Google Scholar 

  48. R.K. Moore: A Dynamic Programming Algorithm for the Distance Between Two Finite Areas. IEEE Trans. PAMI-1 (1979) 86–88.

    Google Scholar 

  49. G.T. Toussaint: The Use of Context in Pattern Recognition. PR-10 (1978) 189–204.

    MathSciNet  Google Scholar 

  50. K. Abend: Compound Decision Procedures for Unknown Distributions and for Dependent States of Nature. In L.N. Kanal (ed.): Pattern Recognition. Thompson, Washington D.C. 1968, 204–249.

    Google Scholar 

  51. A.B.S. Hussain: Compound Sequential Probability Ratio Test for the Classification of Statistically Dependent Patterns. IEEE Trans. C-23 (1974) 398–410.

    MathSciNet  Google Scholar 

  52. J. Raviv: Decision Making in Markov Chains Applied to the Problem of Pattern Recognition. IEEE Trans. IT-13 (1967) 536–551.

    Google Scholar 

  53. G.D. Forney: The Viterbi Algorithm. Proc. IEEE 61 (1973) 268–278.

    Article  MathSciNet  Google Scholar 

  54. L.R. Bahl, F. Jelinek: Decoding for Channels With Insertions, Deletions, and Substitutions With Applications to Speech Recognition. IEEE Trans. IT-21 (1975) 404–411.

    Google Scholar 

  55. D.L. Neuhoff: The Viterbi Algorithm as an Aid in Text Recognition. IEEE Trans. IT-21 (1975) 222–226.

    Google Scholar 

  56. W. Doster: Contextual Postprocessing System for Cooperation With a Multiple-Choice Character-Recognition System. IEEE Trans. C-26 (1977) 1090–1101.

    Google Scholar 

  57. E.M. Riseman, A.R. Hanson: A Contextual Postprocessing System for Error Correction Using Binary n-Grams. IEEE Trans. C-23 (1974) 480–493.

    Google Scholar 

  58. A. Rosenfeld, R.A. Hummel, S.W. Zucker: Scene Labeling by Relaxation Operations. IEEE Trans. SMC-6 (1976) 420–433.

    MathSciNet  Google Scholar 

  59. H. Yamamoto: A Method of Deriving Compatibility Coefficients for Relaxation Operators. CGIP-10 (1979) 256–271.

    Google Scholar 

  60. J.O. Eklundh, H. Yamamoto, A. Rosenfeld: A Relaxation Method for Multispectral Pixel Classification. IEEE Trans. PAMI-2 (1980) 72–75.

    Google Scholar 

  61. N.J. Nilsson: Learning Machines. Mc Graw Hill, New York 1965.

    MATH  Google Scholar 

  62. E.G. Gladyshev: On Stochastic Approximation. Automatika e Telemekanika 10 No. 2 (1965) 275–278.

    Google Scholar 

  63. J.M. Mendel, K.S. Fu (eds.): Adaptive, Learning, and Pattern Recognition Systems. Academic Press, New York 1970.

    MATH  Google Scholar 

  64. R.O. Duda, H. Fossum: Pattern Classification by Iteratively Determined Linear and Piecewise Linear Discriminant Functions. IEEE Trans. EC-15 (1966) 220–232.

    Google Scholar 

  65. R. Takiyama: A General Method for Training the Committee Machine. PR-10 (1978) 255–259.

    Google Scholar 

  66. R. Takiyama: A Two-Level Committee Machine: A Representation and a Learning Procedure for General Piecewise Linear Discriminant Functions. PR-13 (1981) 269–274.

    MathSciNet  Google Scholar 

  67. D.A.S. Frazer: Nonparametric Methods in Statistics. J. Wiley, New York 1957.

    Google Scholar 

  68. E.B. Dynkin: Necessary and Sufficient Statistics for a Class of Probability Distributions. Selected Transl. in Math. Statistics and Probability 1 (1961) 17–40.

    MATH  MathSciNet  Google Scholar 

  69. D.G. Keehn: A Note on Learning for Gaussian Properties. IEEE Trans. IT-11 (1965) 126–132.

    MathSciNet  Google Scholar 

  70. H. Niemann: Unüberwachtes Lernen. In E. Triendl (ed.): Bildverarbeitung und Mustererkennung, Informatik Fachberichte 17. Springer, Berlin, Heidelberg, New York 1978, 3–20.

    Chapter  Google Scholar 

  71. R.C. Tryon, D.E. Bailey: Cluster Analysis. Mc Graw Hill, New York 1970.

    Google Scholar 

  72. H.H. Bock: Automatische Klassifikation. Vandenhoeck und Rupprecht, Göttingen 1974.

    MATH  Google Scholar 

  73. M.R. Anderberg: Cluster Analysis for Applications. Academic Press, New York 1973.

    MATH  Google Scholar 

  74. J.W. Sammon: A Nonlinear Mapping for Data Structure Analysis. IEEE Trans. C-18 (1969) 401–409.

    Google Scholar 

  75. H. Niemann: Linear and Nonlinear Mapping of Patterns. PR-12 (1980) 83–87.

    Google Scholar 

  76. G.H. Ball, J.D. Hall: A Clustering Technique for Summarizing Multivariate Data. Behavioral Sci. 12 (1967) 153–155.

    Article  Google Scholar 

  77. F.R. Fromm, R.A. Northouse: CLASS, a Nonparametric Clustering Algorithm. PR-8 (1976) 107–114.

    Google Scholar 

  78. J.C. Dunn: A Fuzzy Relative of the ISODATA Process and its Use in Detecting Compact Well-Separated Clusters. J. Cybern. 3 No. 3 (1974) 32–57.

    MathSciNet  Google Scholar 

  79. J.C. Bezdek: A Convergence Theorem for the Fuzzy ISODATA Clustering Algorithms. IEEE Trans. PAMI-2 (1980) 1–8.

    Google Scholar 

  80. W.L.G. Koontz, K. Fukunaga: A Nonparametric Valley — Seeking Technique for Cluster Analysis. IEEE Trans. C-21 (1972) 171–178.

    MathSciNet  Google Scholar 

  81. R. Mizoguchi, M. Shimura: Nonparametric Learning Without a Teacher Based on Mode Estimation. IEEE Trans. C-25 (1976) 1109–1117.

    MathSciNet  Google Scholar 

  82. C.T. Zahn: Graph-Theoretical Methods for Detecting and Describing Gestalt Clusters. IEEE Trans. C-20 (1971) 68–86.

    Google Scholar 

  83. W.L.G. Koontz, P.M. Narendra, K. Fukunaga: A Graph-Theoretic Approach to Nonparametric Cluster Analysis. IEEE Trans. C-25 (1976) 936–944.

    MathSciNet  Google Scholar 

  84. S.J. Yakowitz, J. Spragins: On the Identifiability of Finite Mixtures. Ann. Math. Stat. 39 (1968) 209–214.

    Article  MATH  MathSciNet  Google Scholar 

  85. S.J. Yakowitz: Unsupervised Learning and the Identification of Finite Mixtures. IEEE Trans. IT-16 (1970) 330–338.

    Google Scholar 

  86. J.H. Wolfe: NORMIX, Computational Methods for Estimating the Parameters of Multivariate Normal Mixtures of Distributions. Res. Memo. SRM68-2, US Naval Personnel Research Activity, San Diego, Calif. 1967.

    Google Scholar 

  87. J.G. Postaire, C.P.A. Vasseur: An Approximate Solution to Normal Mixture Identification With Application to Unsupervised Pattern Classification. IEEE Trans. PAMI-3 (1981) 163–179.

    Google Scholar 

  88. E.A. Patrick, J.C. Hancock: Nonsupervised Sequential Classification and Recognition of Patterns. IEEE Trans. IT-12 (1966) 362–372.

    Google Scholar 

  89. E.A. Patrick, J.P. Costello: On Unsupervised Estimation Algorithms. IEEE Trans. IT-16 (1970) 556–569.

    MathSciNet  Google Scholar 

  90. H.J. Scudder: Adaptive Communication Receivers. IEEE Trans. IT-11 (1965) 167–174.

    MathSciNet  Google Scholar 

  91. A.K. Agrawala: Learning With a Probabilistic Teacher. IEEE Trans. IT-16 (1970) 373–379.

    MathSciNet  Google Scholar 

  92. T. Imai, M. Shimura: Learning With Probabilistic Labeling. PR-8 (1976) 225–241.

    Google Scholar 

  93. C.B. Chittineni: Learning With Imperfectly Labeled Patterns. PR-12 (1980) 281–291.

    Google Scholar 

  94. H. Niemann, G. Sagerer: An Experimental Study of Some Algorithms for Unsupervised Learning. IEEE Trans. PAMI-4 (1982) 400–405.

    Google Scholar 

  95. L. Kanal, S. Chandrasekaran: On Dimensionality and Sample Size in Statistical Pattern Classification. PR-3 (1971) 225–234.

    Google Scholar 

  96. J.M. Van Campenhout: On the Peaking of the Hughes Mean Recognition Accuracy; the Resolution of an Apparent Paradox. IEEE Trans. SMC-8 (1978) 390–395.

    Google Scholar 

  97. S. Raudys, V. Pikelis: On Dimensionality, Sample Size, Classification Error and Complexity of Classification Algorithm in Pattern Recognition. TEEE Trans. PAMI-2 (1980) 242–252.

    Google Scholar 

  98. D.H. Foley: Considerations of Sample and Feature Size. IEEE Trans. IT-18 (1972) 618–626.

    Google Scholar 

  99. P.A. Lachenbruch, M.R. Mickey: Estimation of Error Rates in Discriminant Analysis. Technometrics 10 (1968) 715–725.

    Article  MathSciNet  Google Scholar 

  100. G.T. Toussaint, R.W. Donaldson: Algorithms for Recognizing Contour-Traced Handprinted Characters. IEEE Trans. C-19 (1970) 541–546.

    Google Scholar 

  101. G.T. Toussaint: Bibliography on Estimation of Misclassification. IEEE Trans. IT-20 (1974) 472–479.

    MathSciNet  Google Scholar 

  102. W.H. Highleyman: The Design and Analysis of Pattern Recognition Experiments. Bell System Techn. Journal (1962) 723-744.

    Google Scholar 

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  1. Lehrstuhl für Informatik 5 (Mustererkennung), Universität Erlangen-Nürnberg, Martensstraße 3, 8520, Erlangen, Deutschland

    Professor Dr.-Ing. Heinrich Niemann

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Niemann, H. (1983). Numerische Klassifikation. In: Klassifikation von Mustern. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-47517-7_4

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