On numerical evaluation of maximum-likelihood estimates for finite mixtures of distributions

Jiří Grim

Kybernetika (1982)

  • Volume: 18, Issue: 3, page 173-190
  • ISSN: 0023-5954

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Grim, Jiří. "On numerical evaluation of maximum-likelihood estimates for finite mixtures of distributions." Kybernetika 18.3 (1982): 173-190. <http://eudml.org/doc/27287>.

@article{Grim1982,
author = {Grim, Jiří},
journal = {Kybernetika},
keywords = {finite mixtures of distributions; survey of existing methods; maximum- likelihood estimates; iterative procedure; normal distribution; double exponential distribution; uniform distribution; binomial distribution},
language = {eng},
number = {3},
pages = {173-190},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On numerical evaluation of maximum-likelihood estimates for finite mixtures of distributions},
url = {http://eudml.org/doc/27287},
volume = {18},
year = {1982},
}

TY - JOUR
AU - Grim, Jiří
TI - On numerical evaluation of maximum-likelihood estimates for finite mixtures of distributions
JO - Kybernetika
PY - 1982
PB - Institute of Information Theory and Automation AS CR
VL - 18
IS - 3
SP - 173
EP - 190
LA - eng
KW - finite mixtures of distributions; survey of existing methods; maximum- likelihood estimates; iterative procedure; normal distribution; double exponential distribution; uniform distribution; binomial distribution
UR - http://eudml.org/doc/27287
ER -

References

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  1. C. A. Aйвазян З. И. Бежаева O. В. Cтароверов, Kлассификация многомерных наблюдений, (Classification of Multivariate Observations). Статистика, Mocквa 1974. (1974) Zbl0341.10006
  2. H. H. Aпраушева, Алгоритм расщепления смеси нормальных классов, (Algorithm for resolution of a mixture of normal classes). C6. Программы и алгоритмы (1976), 68. (1976) Zbl1079.34527
  3. H. H. Aпраушева, Определение числа классов в задачах классификации I, (Determination of the number of classes in classification problems I). Известия AH CCCP - Teх. kибернетика (1981), 3, 71-77. (1981) Zbl1024.00503MR0691559
  4. J. Behboodian, On a mixture of normal distributions, Biometrika 57 (1970), 1, 215-217. (1970) Zbl0193.18104
  5. C. G. Bhattacharya, A simple method of resolution of a distribution into Gaussian components, Biometrics 23 (1967), 115-137. (1967) 
  6. W. R. Blischke, Moment estimators for the parameters of a mixture of two binomial distributions, Ann. Math. Statist. 33 (1962), 2, 444-454. (1962) Zbl0131.17804MR0137219
  7. W. R. Blischke, Mixtures of distributions, In: Classical and Contagious Discrete Distributions (G. P. Patil, ed.), Pergamon Press, New York 1963. (1963) 
  8. W. R. Blischke, Estimating the parameters of mixtures of binomial distributions, J. Amer. Statist. Assoc. 59 (1964), 306, 510-528. (1964) Zbl0128.13501MR0162310
  9. C. Bürrau, The half-invariants of the sum of two typical laws of errors with an application to the problem of dissecting a frequency curve into components, Scand. Actuar. J. 77 (1934), 1, 1-5. (1934) 
  10. C. V. L. Charlier, Researches into the theory of probability, Meddelanden fran Lunds Astron. Observ. (1906) Sec. 2, Bd. 1. (1906) 
  11. A. C. Cohen, Discussion of "Estimation of parameters for a mixture of normal distributions" by Victor Hasselblad, Technometrics 8 (1966), 3, 445-446. (1966) MR0196842
  12. A. C. Cohen, Estimation in mixtures of two normal distributions, Technometrics 9 (1967), 15-28. (1967) Zbl0147.18104MR0216626
  13. P. W. Cooper, Some topics on nonsupervised adaptive detection for multivariate normal distributions, In: Computer and Information Sciences - II. (J. T. Tou, ed.), Academic Press, New York 1967. (1967) Zbl0214.47205
  14. N. E. Day, Estimating the components of a mixture of normal distributions, Biometrika 56 (1969), 463-474. (1969) Zbl0183.48106MR0254956
  15. N. P. Dick D. C. Bowden, Maximum likelihood estimation for mixtures of two normal distributions, Biometrics 29 (1973), 4, 781 - 790. (1973) 
  16. G. Doetsch, Zerlegung einer Funktion in Gaussche Fehlerkurven und zeitliche Zurückverfolgung eines Temperaturzustandes, Math. Z. 41 (1936), 283 - 318. (1936) MR1545619
  17. R. O. Duda P. E. Hart, Pattern Classification and Scene Analysis, John Wiley, New York-London 1973. (1973) 
  18. E. B. Fowlkes, Some methods for studying the mixture of two normal (lognormal) distributions, J. Amer. Statist. Assoc. 74 (1979), 367, 561-575. (1979) Zbl0434.62024
  19. J. G. Fryer C. A. Roberston, A comparison of some methods of estimating mixed normal distributions, Biometrika 59 (1972), 639-648. (1972) MR0339387
  20. N. T. Gridgeman, A comparison of two methods of analysis of normal distributions, Technometrics 12 (1970), 4, 832-833. (1970) 
  21. J. Grim, Metody shlukové analýzy a jejich využití při zpětnovazebním řízení velkých systému, (Methods of cluster analysis and their application for feedback control of large systems). Dissertation, Institute of Information Theory and Automation, Prague 1979. (1979) 
  22. J. Grim, An algorithm for maximizing a finite sum of positive functions and its application to cluster analysis, Problems of Control and Information Theory 10 (1981), 6, 427-437. (1981) Zbl0476.65100MR0643728
  23. E. J. Gumbel, La dissection d'une repartition, Annales de l'Université de Lyon 3 (1939), 39-51. (1939) Zbl0063.01784
  24. A. K. Gupta T. Miyawaki, On uniform mixture model, Biometrical J. 20 (1978), 631 - 637. (1978) MR0530762
  25. L. F. Guseman J. R. Walton, Methods for estimating proportions of convex combinations of normals using linear feature selection, Comm. Statist. A - Theory Methods A7 (1978), 1439-1450. (1978) 
  26. V. Hasselblad, Estimation of parameters for a mixture of normal distributions, Technometrics 8 (1966), 431-444. (1966) MR0196842
  27. V. Hasselblad, Finite mixtures of distributions from the exponential family, Ph. D. Dissertation University of California, Los Angeles 1967. (1967) 
  28. V. Hasselblad, Estimation of finite mixtures of distributions from the exponential family, J. Amer. Statist. Assoc. 64 (1969), 328, 1459-1471. (1969) 
  29. B. M. Hill, Information for estimating the proportions in mixtures of exponential and normal distributions, J. Amer. Statist. Assoc. 58 (1963), 918-932. (1963) MR0155381
  30. D. W. Hosmer, A comparison of iterative maximum likelihood estimates of the parameters of a mixture of two normal distributions under three different types of samples, Biometrics 29 (1973), 761-770. (1973) 
  31. D. W. Hosmer, A use of mixtures of two normal distributions in a classification problem, J. Statist. Comput. Simulation 6 (1978), 384, 281-294. (1978) Zbl0381.62051
  32. D. W. Hosmer N. P. Dick, Information and mixtures of two normal distributions, J. Statist. Comput. Simulation 6 (1977), 137-148. (1977) 
  33. O. К. Исаенко К. Ю. Ypбax, Разделение смесей распределений вероятностей на их составляюшие, (Decomposition of mixtures of probability distributions into their components). Теория вероятностей и математическая статистика, теор. кибернетика, том 13, 37-58, ВИНИТИ, Mocквa 1976. (1976) 
  34. I. R. James, Estimation of the mixing proportion in a mixture of two normal distributions from simple rapid measurements, Biometrics 34 (1978), 2, 265-275. (1978) Zbl0384.62027
  35. E. John, Bayesian estimation of mixture distributions, Ann. Math. Statist, 39 (1968), 4, 1289-1302. (1968) MR0229334
  36. B. K. Kale, On the solution of likelihood equations by iteration processes: The multiparametric case, Biometrika 49 (1962), 479-486. (1962) Zbl0118.14301MR0156403
  37. R. Kanno, Estimation of parameters for a mixture of two normal distributions, Rep. Statist. Appl. Res. JUSE 22 (1975), 4, 1-15. (1975) Zbl0356.62024MR0420954
  38. S. Kullback, An information-theoretic derivation of certain limit relations for a stationary Markov Chain, SIAM J. Control 4 (1966), 3, 454-459. (1966) Zbl0199.21301MR0203804
  39. S. Kullback, Information Theory and Statistics, Wiley, New York-Dover 1968. (1968) MR0103557
  40. P. D. M. Macdonald, Estimation of finite distribution mixtures, In: Applied Statistics (R. P. Gupta, ed.), North-Holland 1975. (1975) Zbl0303.62023MR0408087
  41. P. Medgyessy, Decomposition of Superpositions of Density Functions and Discrete Distributions, Akadémiai Kiadó, Budapest 1977. (1977) Zbl0363.60013MR0438428
  42. G. Meeden, Bayes estimation of the mixing distributions, the discrete case, Ann. Math. Statist. 43 (1972), 6, 1993-1999. (1972) MR0350943
  43. W. Molenaar, Survey of estimation methods for a mixture of two normal distributions, Statist. Neerlandica 19 (1965), 4, 249-265. (1965) MR0196844
  44. G. D. Murray D. M. Titterington, Estimation problems with data from a mixture, Appl. Statist. 27(1978), 3, 325-334. (1978) 
  45. K. Pearson, Contributions to the mathematical theory of evolution 1: Dissection of frequency curves, Philos. Trans. Roy. Soc. London Ser. A 185 (1894), 71-110. 
  46. B. C. Peters W. A. Coberly, The numerical evaluation of the maximum-likelihood estimate of mixture proportions, Comm. Statist. A - Theory Methods A5 (1976), 12, 1127-1135. (1976) MR0433687
  47. B. C. Peters H. F. Walker, An iterative procedure for obtaining maximum-likelihood estimates of the parameters for a mixture of normal distributions, SIAM J. Appl. Math. 35 (1978), 2, 362-378. (1978) MR0518877
  48. B. C. Peters H. F. Walker, The numerical evaluation of the maximum-likelihood estimate of a subset of mixture proportions, SIAM J. Appl. Math. 35 (1978), 3,447-452. (1978) MR0507946
  49. J. G. Postaire C. P. A. Vasseur, An approximate solution to normal mixture identification with application to unsupervised pattern classification, IEEE Trans, on Pattern Analysis & Machine Intelligence PAMI-3 (1981), 2, 163-179. (1981) 
  50. R. E. Quandt J. B. Ramsey, Estimating mixtures of normal distributions and switching regressions, J. Amer. Statist. Assoc. 73 (1978), 364, 730-752. (1978) MR0521324
  51. C. R. Rao, Advanced Statistical Methods in Biometric Research, John Wiley and Sons, New York 1952. (1952) Zbl0047.38601MR0050824
  52. P. R. Rider, The method of moments applied to a mixture of two exponential distributions, Ann. Math. Statist. 32 (1961), 1, 143-147. (1961) Zbl0106.13101MR0119282
  53. W. Schilling, A frequency distribution represented as the sum of two Poisson distributions, J. Amer. Statist. Assoc. 42 (1947), 407-424. (1947) MR0021280
  54. D. F. Stanat, Unsupervised learning of mixtures of probability functions, In: Pattern Recognition (L. Kanal, ed.), Thompson Book Co., Washington D. C. 1968, 357-389. (1968) 
  55. B. Stromgren, Tables and diagrams for dissecting a frequency curve into components by the half-invariant method, Scand. Actuar. J. 17 (1934), 1, 7-54. (1934) 
  56. M. И. Шлезингер, Взаимосвязъ обучения и самообучения в разпознавании образов, (Relation between learning and self-learning in pattern recognition). Кивернетика (Kиев) (1968), 2, 81-88. (1968) Zbl1099.01025
  57. W. Y. Tan W. C. Chang, Some comparisons of the method of moments and the method of maximum likelihood in estimating parameters of a mixture of two normal densities, J. Amer. Statist. Assoc. (57(1972), 339, 702-708. (1972) 
  58. H. F. Walker, Estimating the proportions of two populations in a mixture using linear maps, Comm. Statist. A - Theory Methods A9 (1980), 8, 837-849. (1980) Zbl0437.62019MR0573116
  59. J. H. Wolfe, A computer program for the maximum likelihood analysis of types, (Technical Bulletin 65-15), U.S. Naval Personnel Research Activity, San Diego 1965. (1965) 
  60. J. H. Wolfe, NORMIX: computational methods for estimating the parameters of multivariate normal mixtures of distributions, (Research Memorandum SRM 68-2), U.S. Naval Personnel Research Activity, San Diego 1967. (1967) 
  61. J. H. Wolfe, Pattern clustering by multivariate mixture analysis, Multivariate Behavioral Research 5 (1970), July, 329-350. (1970) 
  62. S. J. Yakowitz, Unsupervised learning and the identification of finite mixtures, IEEE Trans. Inform. Theory IT- 16 (1970), 5, 330-338. (1970) Zbl0197.45502
  63. T. Y. Young G. Coraluppi, Stochastic estimation of a mixture of normal density functions using an information criterion, IEEE Trans. Inform. Theory IT-16 (1970), 258-263. (1970) 

Citations in EuDML Documents

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  1. Jiří Grim, Multimodal discrete Karhunen-Loève expansion
  2. Jiří Grim, Mixture of experts architectures for neural networks as a special case of conditional expectation formula
  3. Michal Haindl, Vojtěch Havlíček, Jiří Grim, Probabilistic mixture-based image modelling
  4. Jiří Grim, Multivariate statistical pattern recognition with nonreduced dimensionality
  5. Jiří Grim, On structural approximating multivariate discrete probability distributions
  6. Jiří Grim, Neuromorphic features of probabilistic neural networks

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