Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Slope heuristics: overview and implementation

Abstract

Model selection is a general paradigm which includes many statistical problems. One of the most fruitful and popular approaches to carry it out is the minimization of a penalized criterion. Birgé and Massart (Probab. Theory Relat. Fields 138:33–73, 2006) have proposed a promising data-driven method to calibrate such criteria whose penalties are known up to a multiplicative factor: the “slope heuristics”. Theoretical works validate this heuristic method in some situations and several papers report a promising practical behavior in various frameworks. The purpose of this work is twofold. First, an introduction to the slope heuristics and an overview of the theoretical and practical results about it are presented. Second, we focus on the practical difficulties occurring for applying the slope heuristics. A new practical approach is carried out and compared to the standard dimension jump method. All the practical solutions discussed in this paper in different frameworks are implemented and brought together in a Matlab graphical user interface called capushe. Supplemental Materials containing further information and an additional application, the capushe package and the datasets presented in this paper, are available on the journal Web site.

This is a preview of subscription content, log in to check access.

References

  1. Akaike, H.: Information theory and an extension of the maximum likelihood principle. In: Proceedings, 2nd Internat. Symp. on Information Theory, pp. 267–281 (1973)

  2. Arlot, S.: Model selection by resampling penalization. Electron. J. Stat. 3, 557–624 (2009) (electronic)

  3. Arlot, S., Bach, F.: Data-driven calibration of linear estimators with minimal penalties. In: Advances in Neural Information Processing Systems (NIPS), vol. 22, pp. 46–54 (2009)

  4. Arlot, C., Celisse, A.: A survey of cross-validation procedures for model selection. Preprint arXiv:0907.4728v1 (2009)

  5. Arlot, S., Massart, P.: Data-driven calibration of penalties for least-squares regression. J. Mach. Learn. Res. 10, 245–279 (2009) (electronic). http://www.jmlr.org/papers/volume10/arlot09a/arlot09a.pdf

  6. Baudry, J.P.: Sélection de modèle pour la classification non supervisée. Choix du nombre de classes. PhD thesis, University Paris XI. http://tel.archives-ouvertes.fr/tel-00461550/fr/ (2009)

  7. Baudry, J.P., Celeux, G., Marin, J.M.: Selecting models focussing on the modeller’s purpose. In: COMPSTAT 2008: Proceedings in Computational Statistics, pp. 337–348. Physica, Heidelberg (2008)

  8. Biernacki, C., Celeux, G., Govaert, G., Langrognet, F.: Model-based cluster and discriminant analysis with the mixmod software. Comput. Stat. Data Anal. 51, 587–600 (2006)

  9. Birgé, L., Massart, P.: Gaussian model selection. J. Eur. Math. Soc. 3, 203–268 (2001)

  10. Birgé, L., Massart, P.: Minimal penalties for Gaussian model selection. Probab. Theory Relat. Fields 138, 33–73 (2006)

  11. Burnham, K., Anderson, D.: Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd edn. Springer, New York (2002)

  12. Caillerie, C., Michel, B.: Model selection for simplicial approximation. RR 6981, INRIA. http://hal.inria,.fr/inria-00402091/en/ (2009)

  13. Craven, P., Wahba, G.: Smoothing noisy data with spline functions. Estimating the correct degree of smoothing by the method of generalized cross-validation. Numer. Math. 31, 377–403 (1978)

  14. Denis, M., Molinari, N.: Choix du nombre de noeuds en régression spline par l’heuristique des pentes. In: 41èmes Journées de Statistique, SFdS, Bordeaux, Bordeaux, France (2009). http://hal.inria.fr/inria-00386618/en/

  15. Fraley, C., Raftery, A.E.: Enhanced software for model-based clustering, density estimation, and discriminant analysis: mclust. J. Classif. 20, 263–286 (2003)

  16. Huber, P.J.: Robust Statistics. Wiley, New York (1981)

  17. Keribin: Consistent estimation of the order of mixture models. Sankhya, Ser. A 62, 49–66 (2000)

  18. Lebarbier, E.: Detecting multiple change-points in the mean of Gaussian process by model selection. Signal Process. 85, 717–736 (2005)

  19. Lepez, V.: Some estimation problems related to oil reserves. PhD thesis, University Paris XI. http://tel.archives-ouvertes.fr/tel-00460802/fr/ (2002)

  20. Lerasle, M.: Adaptive density estimation of stationary β-mixing and τ-mixing processes. Math. Methods Stat. 18, 59–83 (2009a)

  21. Lerasle, M.: Optimal model selection in density estimation. Preprint. arXiv:0910.1654 (2009b)

  22. Lerasle, M.: Rééchantillonnage et sélection de modèles optimale pour l’estimation de la densité. PhD thesis, University of Toulouse. http://www-gmm.insa-toulouse.fr/~mlerasle/index2.html (2009c)

  23. Mallows, C.L.: Some comments on CP. Technometrics 15, 661–675 (1973). http://www.jstor.org/stable/1267380

  24. Massart, P.: Concentration inequalities and model selection. In: École d’été de Probabilités de Saint-Flour 2003. Lecture Notes in Mathematics. Springer, Berlin (2007)

  25. Maugis, C., Michel, B.: A non asymptotic penalized criterion for Gaussian mixture model selection. ESAIM Probab. Stat. (2009). doi:10.1051/ps/2009004. http://www.esaim-ps.org/index.php?option=article&access=standard&Itemid=129&url=/articles/ps/pdf/forth/ps0842.pdf

  26. Maugis, C., Michel, B.: Data-driven penalty calibration: a case study for Gaussian mixture model selection. ESAIM Probab. Stat. (2010). doi:10.1051/ps/2010002. http://www.esaim-ps.org/index.php?option=article&access=standard&Itemid=129&url=/articles/ps/pdf/forth/ps0843.pdf

  27. Maugis, C., Celeux, G., Martin-Magniette, M.L.: Variable selection for clustering with Gaussian mixture models. Biometrics 65, 701–709 (2009)

  28. Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6, 461–464 (1978)

  29. Verzelen, N.: Data-driven neighborhood selection of a Gaussian field. RR 6798, INRIA (2009)

  30. Villers, F.: Tests et sélection de modèles pour l’analyse de données protéomiques et transcriptomiques. PhD thesis, University Paris XI. http://www.proba.jussieu.fr/~villers/manuscript.pdf (2007)

Download references

Author information

Correspondence to Jean-Patrick Baudry.

Additional information

This work was supported by the select Project (INRIA Saclay—Île-de-France).

Electronic Supplementary Material

Below is the link to the electronic supplementary material.

(ZIP 830 KB)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Baudry, J., Maugis, C. & Michel, B. Slope heuristics: overview and implementation. Stat Comput 22, 455–470 (2012). https://doi.org/10.1007/s11222-011-9236-1

Download citation

Keywords

  • Data-driven slope estimation
  • Dimension jump
  • Model selection
  • Penalization
  • Slope heuristics