Bayesian Kernel Learning Methods for Parametric Accelerated Life Survival Analysis

  • Gavin C. Cawley
  • Nicola L. C. Talbot
  • Gareth J. Janacek
  • Michael W. Peck
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3635)

Abstract

Survival analysis is a branch of statistics concerned with the time elapsing before “failure”, with diverse applications in medical statistics and the analysis of the reliability of electrical or mechanical components. In this paper we introduce a parametric accelerated life survival analysis model based on kernel learning methods that, at least in principal, is able to learn arbitrary dependencies between a vector of explanatory variables and the scale of the distribution of survival times. The proposed kernel survival analysis method is then used to model the growth domain of Clostridium botulinum, that is the food processing and storage conditions permitting the growth of this foodborne microbial pathogen, leading to the production of the neurotoxin responsible for botulism. A Bayesian training procedure, based on the evidence framework, is used for model selection and to provide a credible interval on model predictions. The kernel survival analysis models are found to be more accurate than models based on more traditional survival analysis techniques, but also suggest a risk assessment of the foodborne botulism hazard would benefit from the collection of additional data.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cox, D.R., Oakes, D.: Analysis of Survival Data. Monographs on Statistics and Applied Probability, vol. 21. Chapman and Hall, Boca Raton (1984)Google Scholar
  2. 2.
    MacKay, D.J.C.: Bayesian interpolation. Neural Computation 4, 415–447 (1992)CrossRefGoogle Scholar
  3. 3.
    MacKay, D.J.C.: A practical Bayesian framework for backprop networks. Neural Computation 4, 448–472 (1992)CrossRefGoogle Scholar
  4. 4.
    MacKay, D.J.C.: The evidence framework applied to classification networks. Neural Computation 4, 720–736 (1992)CrossRefGoogle Scholar
  5. 5.
    Tikhonov, A.N., Arsenin, V.Y.: Solutions of ill-posed problems. John Wiley, New York (1977)MATHGoogle Scholar
  6. 6.
    Geman, S., Bienenstock, E., Doursat, R.: Neural networks and the bias/variance dilemma. Neural Computation 4, 1–58 (1992)CrossRefGoogle Scholar
  7. 7.
    Mercer, J.: Functions of positive and negative type and their connection with the theory of integral equations. Philosophical Transactions of the Royal Society of London, A 209, 415–446 (1909)CrossRefGoogle Scholar
  8. 8.
    Aronszajn, N.: Theory of reproducing kernels. Transactions of the American Mathematical Society 68, 337–404 (1950)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Cristianini, N., Shawe-Taylor, J.: An Introduction to Support Vector Machines (and other kernel-based learning methods). Cambridge University Press, Cambridge (2000)Google Scholar
  10. 10.
    Schölkopf, B., Smola, A.J.: Learning with kernels - support vector machines, regularization, optimization and beyond. MIT Press, Cambridge (2002)Google Scholar
  11. 11.
    Kimeldorf, G.S., Wahba, G.: Some results on Tchebycheffian spline functions. J. Math. Anal. Applic. 33, 82–95 (1971)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Schölkopf, B., Herbrich, R., Smola, A.J.: A generalised representer theorem. In: Proceedings of the Fourteenth International Conference on Computational Learning Theory, Amsterdam, The Netherlands, pp. 416–426 (2001)Google Scholar
  13. 13.
    Fletcher, R.: Practical Methods of Optimization, 2nd edn. John Wiley and Sons, Chichester (2000)Google Scholar
  14. 14.
    Micchelli, C.A.: Interpolation of scattered data: Distance matrices and conditionally positive definite functions. Constructive Approximation 2, 11–22 (1986)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Fine, S., Scheinberg, K.: Efficient SVM training using low-rank kernel representations. Journal of Machine Learning Research 2, 243–264 (2001)CrossRefGoogle Scholar
  16. 16.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)MATHGoogle Scholar
  17. 17.
    Baudat, G., Anouar, F.: Kernel-based methods and function approximation. In: Proc. IJCNN, Washington, DC, pp. 1244–1249 (2001)Google Scholar
  18. 18.
    Stone, M.: Cross-validatory choice and assessment of statistical predictions. Journal of the Royal Statistical Society B 36, 111–147 (1974)MATHGoogle Scholar
  19. 19.
    Luntz, A., Brailovsky, V.: On estimation of characters obtained in statistical procedure of recognition (in Russian). Techicheskaya Kibernetica 3 (1969)Google Scholar
  20. 20.
    Nelder, J.A., Mead, R.: A simplex method for function minimization. Computer Journal 7, 308–313 (1965)MATHGoogle Scholar
  21. 21.
    Buntine, W.L., Weigend, A.S.: Bayesian back-propagation. Complex Systems 5, 603–643 (1991)MATHGoogle Scholar
  22. 22.
    Bishop, C.M.: Neural Networks for Pattern Recognition. Oxford University Press, Oxford (1995)Google Scholar
  23. 23.
    Neal, R.M.: Bayesian learning for neural networks. Lecture Notes in Statistics. Springer, Heidelberg (1996)MATHGoogle Scholar
  24. 24.
    MacKay, D.J.C.: Hyperparameters: optimise or integrate out? In: Heidbreder, G. (ed.) Maximum Entropy and Bayesian Methods. Kluwer, Dordrecht (1994)Google Scholar
  25. 25.
    Lund, B.M., Peck, M.W.: Clostridium botulinum. In: Lund, B.M., Baird-Parker, A.C., Gould, G.W. (eds.) The Microbiological Safety and Quality of Food, Aspen, Gaithersburg, USA, pp. 1057–1109 (2000)Google Scholar
  26. 26.
    Peck, M.W.: Clostridium botulinum and the safety of refrigerated processed foods of extended durability. Trends in Food Science and Technology 8, 186–192 (1997)CrossRefGoogle Scholar
  27. 27.
    Fernández, P.S., Peck, M.W.: A predictive model that describes the effect of prolonged heating at 70 − 80° C and incubation at refrigeration temperatures on growth and toxigenesis by non-proteolytic Clostridium botulinum. Journal of Food Protection 60, 1064–1071 (1997)Google Scholar
  28. 28.
    Peck, M.W., Lund, B.M., Fairbairn, D.A., Kassperson, A.S., Undeland, P.C.: Effect of heat treatment on survival of, and growth from, spores of non-proteolytic Clostridium botulinum at refrigeration temperatures. Applied and Environmental Microbiology 61, 1780–1785 (1995)Google Scholar
  29. 29.
    Stringer, S.C., Haque, N., Peck, M.W.: Growth from spores of non-proteolytic Clostridium botulinum in heat treated vegetable juice. Applied and Environmental Microbiology 65, 2136–2142 (1999)Google Scholar
  30. 30.
    Carlin, F., Peck, M.W.: Growth and toxin production by non-proteolytic and proteolytic Clostridium botulinum in cooked vegetables. Letters in Applied Microbiology 20, 152–156 (1995)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Gavin C. Cawley
    • 1
  • Nicola L. C. Talbot
    • 1
  • Gareth J. Janacek
    • 1
  • Michael W. Peck
    • 2
  1. 1.School of Computing SciencesUniversity of East AngliaNorwichU.K.
  2. 2.Institute of Food Research, Norwich Research ParkNorwichU.K.

Personalised recommendations