Applied Intelligence

, Volume 29, Issue 3, pp 187–203

Estimation of individual prediction reliability using the local sensitivity analysis



For a given prediction model, some predictions may be reliable while others may be unreliable. The average accuracy of the system cannot provide the reliability estimate for a single particular prediction. The measure of individual prediction reliability can be important information in risk-sensitive applications of machine learning (e.g. medicine, engineering, business). We define empirical measures for estimation of prediction accuracy in regression. Presented measures are based on sensitivity analysis of regression models. They estimate reliability for each individual regression prediction in contrast to the average prediction reliability of the given regression model. We study the empirical sensitivity properties of five regression models (linear regression, locally weighted regression, regression trees, neural networks, and support vector machines) and the relation between reliability measures and distribution of learning examples with prediction errors for all five regression models. We show that the suggested methodology is appropriate only for the three studied models: regression trees, neural networks, and support vector machines, and test the proposed estimates with these three models. The results of our experiments on 48 data sets indicate significant correlations of the proposed measures with the prediction error.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bousquet O, Elisseeff A (2002) Stability and generalization. J Mach Learn Res 2:499–526 MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Crowder MJ, Kimber AC, Smith RL, Sweeting TJ (1991) Statistical concepts in reliability. Statistical analysis of reliability data. Chapman & Hall, London, pp 1–11 Google Scholar
  3. 3.
    Bousquet O, Elisseeff A (2000) Algorithmic stability and generalization performance. In: NIPS, pp 196–202 Google Scholar
  4. 4.
    Bousquet O, Pontil M (2003) Leave-one-out error and stability of learning algorithms with applications. In: Suykens JAK et al, Advances in learning theory: methods, models and applications. IOS Press, Amsterdam Google Scholar
  5. 5.
    Kearns MJ, Ron D (1997) Algorithmic stability and sanity-check bounds for leave-one-out cross-validation. In: Computational learing theory, pp 152–162 Google Scholar
  6. 6.
    Breiman L (1996) Bagging predictors. Mach Learn 24:123–140 MATHMathSciNetGoogle Scholar
  7. 7.
    Schapire RE (1999) A brief introduction to boosting. In: Proceedings of IJCAI, pp 1401–1406 Google Scholar
  8. 8.
    Drucker H (1997) Improving regressors using boosting techniques. In: Machine learning: proceedings of the fourteenth international conference, pp 107–115 Google Scholar
  9. 9.
    Ridgeway G, Madigan D, Richardson T (1999) Boosting methodology for regression problems. In: Proceedings of the artificial intelligence and statistics, pp 152–161 Google Scholar
  10. 10.
    Breiman L (1997) Pasting bites together for prediction in large data sets and on-line. Department of Statistics technical report, University of California, Berkeley Google Scholar
  11. 11.
    Tibshirani R, Knight K (1999) The covariance inflation criterion for adaptive model selection. J Roy Stat Soc Ser B 61:529–546 MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Rosipal R, Girolami M, Trejo L (2000) On kernel principal component regression with covariance in action criterion for model selection. Technical report, University of Paisley Google Scholar
  13. 13.
    Elidan G, Ninio M, Friedman N, Shuurmans D (2002) Data perturbation for escaping local maxima in learning. In: Proceedings of AAAI/IAAI, pp 132–139 Google Scholar
  14. 14.
    Gammerman A, Vovk V, Vapnik V (1998) Learning by transduction. In: Proceedings of the 14th conference on uncertainty in artificial intelligence, Madison, WI, pp 148–155 Google Scholar
  15. 15.
    Saunders C, Gammerman A, Vovk V (1999) Transduction with confidence and credibility. In: Proceedings of IJCAI, vol 2, pp 722–726 Google Scholar
  16. 16.
    Nouretdinov I, Melluish T, Vovk V (2001) Ridge regression confidence machine. In: Proceedings of the 18th international conference on machine learning. Kaufmann, San Francisco, pp 385–392 Google Scholar
  17. 17.
    Vapnik V (1995) The nature of statistical learning theory. Springer, Berlin MATHGoogle Scholar
  18. 18.
    Kukar M, Kononenko I (2002) Reliable classifications with machine learning. In: Proceedings of the machine learning: ECML-2002. Springer, Helsinki, pp 219–231 CrossRefGoogle Scholar
  19. 19.
    Bosnić Z, Kononenko I, Robnik-Šikonja M, Kukar M (2003) Evaluation of prediction reliability in regression using the transduction principle. In Proceedings of Eurocon 2003, Ljubljana, pp 99–103 Google Scholar
  20. 20.
    Mitchell T (1999) The role of unlabelled data in supervised learning. In: Proceedings of the 6th international colloquium of cognitive science, San Sebastian, Spain Google Scholar
  21. 21.
    Blum A, Mitchell T (1998) Combining labeled and unlabeled data with co-training. In: Proceedings of the 11th annual conference on computational learning theory, pp 92–100 Google Scholar
  22. 22.
    Li M, Vitányi P (1993) An introduction to Kolmogorov complexity and its applications. Springer, New York MATHGoogle Scholar
  23. 23.
    Press WH et al. (2002) Numerical recipes in C: the art of scientific computing. Cambridge University Press, Cambridge Google Scholar
  24. 24.
    Newman DJ, Hettich S, Blake CL, Merz CJ (1998) UCI repository of machine learning databases. Department of Information and Computer Sciences, University of California, Irvine Google Scholar
  25. 25.
    Department of Statistics at Carnegie Mellon University (2005) StatLib—data, software and news from the statistics community Google Scholar
  26. 26.
    Cestnik B, Bratko I (1991) On estimating probabilities in tree pruning. In: Proceedings of European working session on learning (EWSL-91), Porto, Portugal, pp 138–150 Google Scholar
  27. 27.
    Chang C, Lin C (2001) LIBSVM: a library for support vector machines.
  28. 28.
    Alpaydin E (2004) Introduction to machine learning. MIT Press, Cambridge Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.University of LjubljanaFaculty of Computer and Information ScienceLjubljanaSlovenia

Personalised recommendations