Advertisement

Community Ecology

, Volume 3, Issue 2, pp 191–204 | Cite as

Models, measures and messages: an essay on the role for induction

  • M. B. DaleEmail author
Article

Abstract

In this essay, I examine the role of induction in developing vegetation models. Falsification is a necessary component of model building but is not itself sufficient. Induction provides a necessary complement and one that dethrones the null hypothesis from its privileged state. After examining the role of description and environment, I examine several possible criteria useful for valorising models so that we may choose the ‘best’. These criteria include fit, simplicity, precision and interest. Predictability, which is given overwhelming importance in a falsification approach, is found to be ambiguous. It may be obtained by using multiple models without regard to the processes active in the real system. In addition movement towards a model which does reflect the ‘real’ processes can result in loss of predictivity. Finally, some comments are made on what we can infer and how this relates to our understanding of living systems.

Keywords

Falsification Model classes Model selection Null hypothesis Prediction Valorisation 

Abbreviations

MML

Minimum message length

GUHA

General Unary Hypothesis Automaton

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

I would like to thank Sanyi Bartha and Pat Dale for many helpful comments made after reading earlier drafts.

References

  1. Adomavicius, G. and A. Tuzhilin. 1997. Discovery of actionable patterns in databases: the action hierarchy approach. In: D. Heckerman, H. Mannila, D. Pregibon and R. Uthurusamy (eds.), Proc. Third International Conference on Knowledge Discovery and Data Mining. AAA1, pp. 111–114.Google Scholar
  2. Akaike, H. 1977. On entropy maximization principles. In: P. K. Krishnaiah (ed.), Applications of Statistics North Holland, Amsterdam. pp 27–41.Google Scholar
  3. Anand, M. 1997. The fundamental nature of vegetation dynamics -a chaotic synthesis. Coenoses 12: 55–62.Google Scholar
  4. Anand, M. 2000. Fundamentals of vegetation change: complexity rules. Acta Biotheoretica 48: 1–14.CrossRefGoogle Scholar
  5. Anand, M. and Orlóci, L. 1996. Complexity in plant communities: the notion and quantification. J. theoret. Biol. 179:179–186.CrossRefGoogle Scholar
  6. Anderson, C. W. and G. E. McMaster. 1982. Computer assisted modelling of affective tone in written documents. Comput. Humanit. 16: 1–9.CrossRefGoogle Scholar
  7. Antonelli, P. L. 1990. Applied Volterra-Hamilton systems of the Finsler type: increased species diversity as a non-chemical defense for coral against crown-of-thorns. In: R. H. Bradbury (ed.), Acanthaster and the Coral Reef A Theoretical Perspective, Lecture Notes in Biomathematics 88. Springer-Verlag, Berlin pp. 220–235.CrossRefGoogle Scholar
  8. Abarbanel, H. D. I., R. Brown and M. B. Kennel. 1992. Local Lyapunov Exponents Computed from Observed Data. Journal of Nonlinear Science 2:343–365.CrossRefGoogle Scholar
  9. Austin, M. P. 1970. An applied ecological example of mixed data classification, in: R. S. Anderssen and M. R. Osborne (eds.), Data Representation. University of Queensland Press, Brisbane. pp. 113–117.Google Scholar
  10. Badii, R. and A. Politi. 1997 Hierarchical Structure and Scaling in Physics. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
  11. Barsalou, L. W. 1995. Deriving categories to achieve goals. In:. A. Ram and D. B. Leake (eds.), Goal Directed Learning. MIT Press, Cambridge MA. pp. 121–176.Google Scholar
  12. Beeston, G. R. and M. B. Dale. 1975. Multiple predictive analysis: a management tool. Proceedings of the Ecological Society of Australia 9: 172–181.Google Scholar
  13. Boerlijst, M. and P. Hogeweg. 1991. Spiral wave structure in prebiotic evolution: hypercycles stable against parasites. Physica D 48: 17–28.Google Scholar
  14. Brokaw, N. and R. T. Busing. 2000. Niche versus chance in tree diversity in forest gaps. TREE 15: 183–188.Google Scholar
  15. Carley, K. and M. Palmquist. 1992. Extracting, representing and analyzing mental models. Social Forces 70: 601–636.CrossRefGoogle Scholar
  16. Chambers, W. V. 1991. Inferring formal causation from corresponding regressions. The Journal of Mind and Behavior 12:49–70.Google Scholar
  17. Crutchfield, J. P. and C. R. Shalizi. 1999. Thermodynamic depth of causal states: when paddling around in Occam’s pool, shallowness is a virtue. Physical Review E 59: 275–283.CrossRefGoogle Scholar
  18. Crutchfield, J. P. and K. Young. 1989. Inferring statistical complexity, Physical Review Letters 63: 105–108.CrossRefGoogle Scholar
  19. Dale, M. B. 1970. Systems analysis and ecology. Ecology 51: 2–16.CrossRefGoogle Scholar
  20. Dale, M. B. 1980. A syntactic basis for classification. Vegetatio 42: 93–98.CrossRefGoogle Scholar
  21. Dale, M. B. 1988. Some fuzzy approaches to phytosociology: ideals and instances. Folia Geobotanica Phytotaxonomica 23: 239–274.CrossRefGoogle Scholar
  22. Dale, M. B. 1994. Straightening the horseshoe: a Riemannian resolution? Coenoses 9: 43–53.Google Scholar
  23. Dale, M. B. 1999. The dynamics of diversity: mixed strategy systems. Coenoses 13:105–113Google Scholar
  24. Dale, M. B. 2000. Mt Glorious revisited: Secondary succession in subtropical rainforest Community Ecol. 1: 181–193CrossRefGoogle Scholar
  25. Dale, M. B. and Barson, M. M. 1989. Grammars in vegetation analysis. Vegetatio 81: 79–94.CrossRefGoogle Scholar
  26. Dale, M. B., R. Courts and P. E. R. Dale. 1988. Landscape classification by sequences: a study of Toohey Forest. Vegetatio 29: 113–129.Google Scholar
  27. Dale, M. B., P. E. R. Dale and T. Edgoose. 2002a. Markov models for incorporating temporal dependence Acta Oecologica 23: 261–269.CrossRefGoogle Scholar
  28. Dale, M. B., P. E. R. Dale, C. Li and G. Biswas. 2002b. Assessing impacts of small perturbations using a model-based approach Ecological Modelling 156: 185–199.CrossRefGoogle Scholar
  29. Dale, M. B. and Hogeweg, P. 1998. The dynamics of diversity: a cellular automaton approach. Coenoses 13:3–15.Google Scholar
  30. Dale, P., K. Hulsman, B. R. Jahnke and M. B. Dale. 1984. Vegetation and nesting preferences of Black Noddies at Masthead Island, Great Barrier Ree. Part 1. Patterns at the macro scale. Australian Journal of Ecology 9: 335–341.CrossRefGoogle Scholar
  31. Dale, M. B., L. Salmina and L. Mucina. 2001. Minimum message length clustering: an explication and some applications to vegetation data. Community Ecol. 2:231–247.CrossRefGoogle Scholar
  32. Dennett, D. 1991. Real patterns. J. Philosophy 88:27–51.CrossRefGoogle Scholar
  33. Devaney. R. L. 1985. An Introduction to Chaotic Dynamical Systems. Benjamin/Cummings, Menlo Park.Google Scholar
  34. Domingos, P. 1996. Two-way induction. Intematl. J. Artificial Intelligence Tools 5: 113–125.CrossRefGoogle Scholar
  35. Domingos, P. 1998. When and how to combine predictive and causal learning. Proc NIPS-98 Workshop on Integrating Supervised and Unsupervised Learning, Breckonridge, CO. NIPS Foundation.Google Scholar
  36. Domingos P. 1999. The role of Occam’s Razor in knowledge discovery. Data Mining and Knowledge Discovery 3: 409–425CrossRefGoogle Scholar
  37. Eco, U. 1980. II Nome della Rosa. Gruppe Editoriale Fabbri-Bompianni, Sonzogno, Etas S. p. A.Google Scholar
  38. Edgoose, T. and L. Allison. 1999. MML Markov classification of sequential data. Statistics and Computing 9: 269–278.CrossRefGoogle Scholar
  39. Farrands, J. L. 1990. On modelling. In: R. H. Bradbury (ed.), Acan thaster and the Coral Reef: A Theoretical Perspective Lecture Notes in Biomathematics 88. Springer-Verlag, Berlin, pp. 1–5.Google Scholar
  40. Fisher, D. 1992. Pessimistic and optimistic induction. TR CS-92–12 Dept. Comput. Sei., Vanderbilt Univ.Google Scholar
  41. Forster, M. P. and E. Sober. 1994. Key concepts in model selection: performance and generalization. Brit. J. Philosophy Sei. 45:1–35.CrossRefGoogle Scholar
  42. Gell-Mann, M. 1994. The Quark and the Jaguar, W. H. Freeman, San Francisco.Google Scholar
  43. Georgeff, M. P. and C. S. Wallace. 1984. A general criterion for inductive inference. In: T. O’Shea (ed.), Proc. 6th European Conf. Artificial Intelligence, Elsevier, Amsterdam.Google Scholar
  44. Grassberger, P. 1989. Problems in quantifying self-generated complexity. Helvetica Physica Acta 62: 489–511.Google Scholar
  45. Grassberger, P. 1991. Information and complexity measures in dynamical systems. In: H. Atmanspacher and H. Scheingraber (eds.), Information Dynamics. Plenum Press. New York, pp. 15–33.CrossRefGoogle Scholar
  46. Grassberger, P. and F. Procaccia. 1983. Estimation of the Kolmogorov entropy for a chaotic signal. Phys. Rev. A 28: 2591.CrossRefGoogle Scholar
  47. Gunther, R., B. Shapiro and P. Wagner. 1994. Complex systems, complexity measures, grammars and model inferring, Chaos, Solitons and Fractals 4: 635–651.CrossRefGoogle Scholar
  48. Hájek, P., I. Havel and M. Chytil. 1966. GUHA – the method of systematical hypotheses searching. Kybernetika (Prague) 2:31–39.Google Scholar
  49. Hájek, P. and T. Havránek. 1977. On generation of inductive hypotheses. International. J. Man-Mach. Stud. 9: 415–438.CrossRefGoogle Scholar
  50. Herman, G. T. and Rozenberg, G. 1975. Developmental Systems and Languages, North-Holland, American Elsevier, Amsterdam.Google Scholar
  51. Hilderman, R. J. and Hamilton, H.J. 1999. Heuristics for ranking the interestingness of discovered knowledge. Proc. 3rd Pacific-Asia Conf. Knowledge Discovery PKDD’99, Beijing, Springer-Verlag, Berlin, pp. 204–209.Google Scholar
  52. Hoeting, J., D. Madigan, A. E. Raftery and C. T. Volinsky. 1998. Bayesian model averaging: atutorial. Statist. Sei. 14: 382–417.Google Scholar
  53. Hogeweg, P. 2002. Computing an organism: on the interface between informatic and dynamic processes. BioSystems 64: 97–109.CrossRefGoogle Scholar
  54. Howard, E. and N. Oakley. 1994. The application of genetic programming to the investigation of short, noisy, chaotic data series. In: T C. Fogarty (ed.), Evolutionary Computing. Lecture Notes in Computer Science 865, Springer-Verlag, Berlin, pp. 320–332.Google Scholar
  55. Hume, D. 1999. An Enquiry Concerning Human Understanding. Oxford Philosophical Texts, Oxford University Press, Oxford.Google Scholar
  56. Iba, W., Wogulis, J. and Langley, P. 1988. Trading off simplicity and coverage in incremental concept learning. Proc. 5th Internatl. Conf. Machine Learning, Ann Arbor, Morgan Kaufman, CA. pp. 73–86.CrossRefGoogle Scholar
  57. Ivakhnenko, A. G. 1971. Polynomial theory of complex systems I. E. E. E. Trans. Syst. Man Cybern. SMC 1: 364–378.Google Scholar
  58. Jöreskog, K. G. 1966. Some contributions to maximum likelihood factor analysis. Research Bulletin RB-66–41, Educational Testing Service, Princeton, N. J.Google Scholar
  59. Kaufman, S. 2001. Investigations. Oxford University Press, Oxford.Google Scholar
  60. Kelley, H. 1971. Causal schemata and the attribution process. In: E. Jones, D. Kanouse, H. Kelley, N. Nisbett, S. Valins and B. Weiner (eds.). Attribution: Perceiving the causes of behavior. General Learning Press, Morristown, NJ. pp 151–174.Google Scholar
  61. Klemettinen, M., H. Mannila, P. Ronkainen, H. Toivonen and A. I. Verkamo. 1994. Finding interesting rules from large sets of discovered association rules. In: N. R. Adam, B. K. Bhargavaand Y. Yesha (eds.), Third Internatl. Conf. Information and Knowledge Management CIKM’94, ACM Press Association, pp. 401–407,Google Scholar
  62. Kohavi, R. 1995. A study of cross-validation and bootstrap for accuracy estimation and model selection, Proc. International Joint Conference Artificial Intelligence.Google Scholar
  63. Kolmogorov, A. N. 1965. Three approaches to the quantitative description of information. Prob. Inform. Transmission 1: 4–7 (translation)Google Scholar
  64. Kontkainen, P., P. Myllymäki, T. Silander and H. Tirri. 1999. On stochastic complexity approximation. In: A Gammerman (ed.) Causal Systems and Intelligent data Management., Springer-Verlag, Berlin, pp. 120–136.CrossRefGoogle Scholar
  65. Lekkas, G. and N. Avouris. 1994. Case-Based Reasoning in Environmental Monitoring. Applied Artificial Intelligence 8: 359–376.CrossRefGoogle Scholar
  66. Li, C. and G. Biswas. 1999. Temporal pattern generation using hidden Markov model based unsupervised classification. Lecture Notes in Computer Science 1662. pp. 245–257.Google Scholar
  67. Lloyd, S. and H. Pagels. 1988. Complexity as thermodynamic depth. Ann. Physics 188:186–212.CrossRefGoogle Scholar
  68. Löfgren, L. 1974. Complexity of descriptions of systems: A foundational study. International Journal of General Systems 3: 197–214.CrossRefGoogle Scholar
  69. Lux, A. and F. A. Bemmerlein-Lux. 1998. Two vegetation maps of the same island: floristic units versus structural units. Applied Vegetation Science 1: 201–210.CrossRefGoogle Scholar
  70. Mac Nally, R. 2000. Regression and model-building in conservation biology, biogeography and ecology: The distinction between -and reconciliation of -’predictive’ and ‘explanatory’ models Biodiversity and Conservation 9: 655–671.CrossRefGoogle Scholar
  71. Mackay, D. M. 1969. Recognition and action. In: S. Watanabe (ed.), Methodologies of Pattern Recognition, Academic Press, London, pp. 409–416.CrossRefGoogle Scholar
  72. May, R. M 1995. Necessity and chance: deterministic chaos in ecology and evolution. Bull. Amer. Math. Soc. 32: 291–308.CrossRefGoogle Scholar
  73. Murphy, G.L. and P. D. Allopenna. 1994. The locus of knowledge effects in concept learning. Journal of Experimental Psychology: Learning, Memory, and Cognition, 19: 203–222.Google Scholar
  74. Neil, J. R. and K. B. Korb. 1998. The MML evolution of causal models Tech. Rep. 98/17 School of Computer Science and Software Engineering, Monash University, Clayton, Victoria 3168, Australia.Google Scholar
  75. Niven, B. S. 1992. Formalization of some basic concepts of plant ecology. Coenoses 7: 103–113.Google Scholar
  76. Noble, I. R. and R. O. Slatyer. 1980. The use of vital attributes to predict successional changes in plant communities subject to recurrent disturbances. Vegetatio 43: 5–21.CrossRefGoogle Scholar
  77. Oates, T. and D. Jensen 1998. Large datasets lead to overly complex models: an explanation and a solution. KDD-98 Proc. 4th Internatl. Conf. Knowledge Discovery and Datamining. pp. 294–298.Google Scholar
  78. Osledec. V. I. 1968 A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19: 197–231.Google Scholar
  79. Padmanabhan, B. and A. Tuzhilin. 1999. Unexpectedness as a measure of interestingness in knowledge discovery. Decision Support Systems, 27. from http://citeseer.nj.nec.compadmanabhan99unexpectedness.html
  80. Pagie, L. and P. Hogeweg. 1997. Evolutionary consequences of coevolving targets. Evolutionary Computation 5: 401–418.CrossRefGoogle Scholar
  81. Palus, M. 1996a. Detecting nonlinearity in multivariate time series. Physics Letters A 213: 1387.Google Scholar
  82. Palus, M. 1996b. Coarse-grained entropy rates for characterisation of complex time series Physica D 93: 64–77.Google Scholar
  83. Palus, M. 1997. Kolmogorov entropy from time series using information-theoretic functionals. Neural Network World 7: 269–292.Google Scholar
  84. Pazzani, M. J. and D. Kibler. 1992. The utility of knowledge in inductive learning. Machine Learning 9: 57–94.Google Scholar
  85. Pazzani, M., S. Mani and W. R. Shankle. 1997. Comprehensible knowledge discovery in databases. In: M. G. Shafto and P. Langley, (eds.), Proceedings of the Nineteenth Annual Conference of the Cognitive Science Society, Lawrence Erlbaum, pp. 596–601.Google Scholar
  86. Pillar, V. D. 1999a On the identification of optimal plant functional types. J. Veg.Sci. 10:631–640.CrossRefGoogle Scholar
  87. Pillar, V. D. 1999b The bootstrap ordination revisited. J. Veg. Sci. 10:895–902.CrossRefGoogle Scholar
  88. Popper, K. R. 1968. The Logic of Scientific Discovery. Harper, New York.Google Scholar
  89. Posse, C. 1995. Projection pursuit exploratory data analysis. C amputat. Statist. Data Anal. 20: 669–687.CrossRefGoogle Scholar
  90. Provost, F., T. Fawcett and R. Kohavi. 1998. The Case Against Accuracy Estimation for Comparing Induction Algorithms. Presented at ICML-98 (15th Internatl. Conf. on Machine Learning).Google Scholar
  91. Reich, Y. 1993. A model of aesthetic judgment in design. Artif. Intell. in Engineering 8:141–153CrossRefGoogle Scholar
  92. Reichenbach, H. 1950. The Rise of Scientific Philosophy. Univ. California Press, Los Angeles.Google Scholar
  93. Riddle, R. R. and D. J. Hafner. 1999. Species as unit of analysis in ecology and biogeography: time to take the blinkers off. Global Ecology and Biogeography 8: 433–441.CrossRefGoogle Scholar
  94. Rigoutsos, I. and A. Floratos. 1998. Motif discovery without alignment or enumeration. Proc. 2nd. Ann. ACM Internatl. Conf. Computational Molecular Biology (RECOMB 98). New York, NY.Google Scholar
  95. Rissanen, J. 1995. Stochastic complexity in learning. In: P. Vitányi (ed.), Computational Learning Theory Lecture Notes in Computer Science 904. Springer-Verlag, Berlin, pp. 196–201.Google Scholar
  96. Rychlak, J.F. 1988. The Psychology of Rigorous Humanism. New York University Press, New York.Google Scholar
  97. Savill, N. J., P. Rohani and P. Hogeweg. 1997. Self-reinforcing spatial patterns enslave evolution in a host-parasitoid system. J. theoret. Biol. 188:11–20.CrossRefGoogle Scholar
  98. Schmidhuber, J. 1994. Discovering solutions with low Kolmogorov complexity and high generalization ability. Tech. Rep. FKI-194–94 Faculty of Information, Technical University, Munich.Google Scholar
  99. Schwarz, G. 1978. Estimating dimension of a model. Ann. Statist. 6: 461–464.CrossRefGoogle Scholar
  100. Shalizi, C. R. and J. P. Crutchfield. 1999 Computational mechanics: pattern and prediction, structure and simplicity. Sante Fe Institute Working paper 99–07-044.Google Scholar
  101. Shipley, B. and P. A. Keddy. 1987. The individualistic and community-unit concepts as falsifiable hypotheses. Vegetatio 69: 47–55.CrossRefGoogle Scholar
  102. Simberloff, D. 1980. A succession of paradigms in ecology: Essentialism to materialism and probabilism. Synthese 43:3–29.CrossRefGoogle Scholar
  103. van den Bosch A. P. M. 1997. Simplicity and Prediction. Available electronically at http://tcw2.ppsw.rug.nl~vdbosch/simple.ps
  104. Vapnik, V. N. and A. Chervonenkis. 1971. On the uniform convergence of relative frequencies of events to their probabilities. Theory Probability Appl. 16: 264–280.CrossRefGoogle Scholar
  105. Viswanathan, M., C. S. Wallace, D. L. Dowe and K. B. Korb. 1999. Finding outpoints in noisy binary sequences: a revised empirical examination. In: N. Foo (ed.), A1–99 Lecture Notes in Artificial Intelligence 1747. Springer-Verlag, Berlin, pp. 405–416.Google Scholar
  106. Wackerbauer, R., A., Witt, H. Altmanspracher, J. Kurths and H. Scheingraber. 1994. A comparative classification of complexity measures based on distinguishing partitions in phase space as well as structural v. dynamic elements. Chaos, Solitons and Fractals 4: 133–173.CrossRefGoogle Scholar
  107. Wallace, C. S. 1995. Multiple factor analysis by MML estimation. Tech. Rep. 95/218, Dept Computer Science, Monash University, Clayton Victoria 3168, Australia.Google Scholar
  108. Wallace, C. S. 1996. MML Inference of predictive trees, graphs and nets. In: A. Gammerman (ed.), Computational Learning and Probabilistic Reasoning. John Wiley. New York. pp. 43–66.Google Scholar
  109. Wallace, C. S. 1998. Intrinsic classification of spatially-correlated data. Comput. J. 41: 602–611.CrossRefGoogle Scholar
  110. Wallace, C. S. and D. L. Dowe. 2000. MML clustering of multi-state, Poisson, von Mises circular and Gaussian distributions. Statistics and Computing 10: 73–83.CrossRefGoogle Scholar
  111. Wallace, C. S. and P. R. Freeman. 1987. Estimation and inference by compact coding. J. Roy. Statist. Soc. Ser. B 49: 240–252.Google Scholar
  112. Wallace, C. S., K. B. Korb and H. Dai. 1996. Causal discovery via MML. Tech. Rep. 96/254 Dept. Computer Science, Monash University, Clayton, Victoria 3168, Australia.Google Scholar
  113. Watanabe, S. 1969. Knowing and Guessing. Wiley, New York.Google Scholar
  114. Webb, G. I. 1994. Generality is more significant than complexity: Toward alternatives to Occam’s razor. In: C. Zhang, J. Debenham and D. Lukose (eds.), AI’94 - Proceedings of the Seventh Australian Joint Conference on Artificial Intelligence. World Scientific, Armidale. pp. 60–67.Google Scholar
  115. Webb, G. I. 1996. Further experimental evidence against the utility of Occam’s Razor. J. Artific. Intell. Res. 4:387–417.Google Scholar
  116. Webb, L. J., J. G. Tracey and W. T. Williams. 1976. The value of structural features in tropical forest typology. Austral. J. Ecol. 1:3–28.CrossRefGoogle Scholar
  117. Williams, W. T. 1972. The problem of pattern. Austral. Mathem. Teacher 28:103–109.Google Scholar
  118. Williams, W. T., J. M. Lambert and G. N. Lance. 1966. Multivariate methods in plant ecology V. Similarity analysis and information analysis. J. Ecol. 54:427–445.CrossRefGoogle Scholar
  119. Williams, W. T., G. N. Lance, L. J. Webb, J. G. Tracey and M. B. Dale. 1969. Studies in the numerical classification of complex rain-forest communities III. The analysis of successional data. J. Ecol. 57:515–535.CrossRefGoogle Scholar
  120. Wilson, J., A. D. Q. Agnew and T. R. Partridge. 1994. Carr texture in Britain and New Zealand: community convergence compared with a null model. J. Veg. Sci. 5:109–116.CrossRefGoogle Scholar
  121. Wisheu, I. and P. A. Keddy. 1992. Competition and centrifugal organisation of plant communities: theory and tests. J. Veg. Sci. 3: 147–156.CrossRefGoogle Scholar
  122. Wittgenstein, L. 1995. Tractacus Logico-Philosophicus. (trans) 5:3631 Routledge, Keagan & Paul, London.Google Scholar
  123. Wolpert, D. H. and W. G. Macready. 1997. Self-dissimilarity: an empirically observable complexity measure. In: Y. Bar-Yam (ed.), Proc. International. Conf. Complex Systems, New England Complex Systems Inst. pp. 1–8.Google Scholar
  124. Wright, S. 1934. The method of path coefficients. Ann. Mathem. Statist. 5:161–215.CrossRefGoogle Scholar
  125. Yamada, H. and S. Amaroso. 1971. Structural and behavioural equivalences of tessellation automata. Information and Control 18:1–31.CrossRefGoogle Scholar
  126. Yee, C. N. and L. Allison. 1993. Reconstruction of strings past. J. Comp. Appl. BioSci. 9: 1–7.Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest 2002

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Australian School of Environmental StudiesGriffith UniversityNathanAustralia

Personalised recommendations