Advertisement

Data Mining and Knowledge Discovery

, Volume 28, Issue 2, pp 475–518 | Cite as

Aggregative quantification for regression

  • Antonio Bella
  • Cèsar FerriEmail author
  • José Hernández-Orallo
  • María José Ramírez-Quintana
Article

Abstract

The problem of estimating the class distribution (or prevalence) for a new unlabelled dataset (from a possibly different distribution) is a very common problem which has been addressed in one way or another in the past decades. This problem has been recently reconsidered as a new task in data mining, renamed quantification when the estimation is performed as an aggregation (and possible adjustment) of a single-instance supervised model (e.g., a classifier). However, the study of quantification has been limited to classification, while it is clear that this problem also appears, perhaps even more frequently, with other predictive problems, such as regression. In this case, the goal is to determine a distribution or an aggregated indicator of the output variable for a new unlabelled dataset. In this paper, we introduce a comprehensive new taxonomy of quantification tasks, distinguishing between the estimation of the whole distribution and the estimation of some indicators (summary statistics), for both classification and regression. This distinction is especially useful for regression, since predictions are numerical values that can be aggregated in many different ways, as in multi-dimensional hierarchical data warehouses. We focus on aggregative quantification for regression and see that the approaches borrowed from classification do not work. We present several techniques based on segmentation which are able to produce accurate estimations of the expected value and the distribution of the output variable. We show experimentally that these methods especially excel for the relevant scenarios where training and test distributions dramatically differ.

Keywords

Quantification Regression quantification Probability estimation Segmentation Distribution Aggregation 

Notes

Acknowledgments

We would like to thank the anonymous reviewers for their careful reviews, insightful comments and very useful suggestions. This work was supported by the MEC/MINECO projects CONSOLIDER-INGENIO CSD2007-00022 and TIN 2010-21062-C02-02, GVA project PROMETEO/2008/051, the COST—European Cooperation in the field of Scientific and Technical Research IC0801 AT, and the REFRAME project granted by the European Coordinated Research on Long-term Challenges in Information and Communication Sciences & Technologies ERA-Net (CHIST-ERA), and funded by the Ministerio de Economía y Competitividad in Spain.

References

  1. Alonzo TA, Pepe MS, Lumley T (2003) Estimating disease prevalence in two-phase studies. Biostatistics 4(2):313–326CrossRefzbMATHGoogle Scholar
  2. Anderson T (1962) On the distribution of the two-sample Cramer–von Mises criterion. Ann Math Stat 33(3):1148–1159CrossRefzbMATHGoogle Scholar
  3. Bakar AA, Othman ZA, Shuib NLM (2009) Building a new taxonomy for data discretization techniques. In: Proceedings of 2nd conference on data mining and optimization (DMO’09), pp 132–140Google Scholar
  4. Bella A, Ferri C, Hernández-Orallo J, Ramírez-Quintana MJ (2009a) Calibration of machine learning models. In: Handbook of research on machine learning applications. IGI Global, HersheyGoogle Scholar
  5. Bella A, Ferri C, Hernández-Orallo J, Ramírez-Quintana MJ (2009b) Similarity-binning averaging: a generalisation of binning calibration. In: International conference on intelligent data engineering and automated learning. LNCS, vol 5788. Springer, Berlin, pp 341–349Google Scholar
  6. Bella A, Ferri C, Hernández-Orallo J, Ramírez-Quintana MJ (2010) Quantification via probability estimators. In: International conference on data mining, ICDM2010, pp 737–742Google Scholar
  7. Bella A, Ferri C, Hernández-Orallo J, Ramírez-Quintana MJ (2012) On the effect of calibration in classifier combination. Appl Intell. doi: 10.1007/s10489-012-0388-2
  8. Chan Y, Ng H (2006) Estimating class priors in domain adaptation for word sense disambiguation. In: Proceedings of the 21st international conference on computational linguistics and the 44th annual meeting of the Association for Computational Linguistics, pp 89–96Google Scholar
  9. Chawla N, Japkowicz N, Kotcz A (2004) Editorial: special issue on learning from imbalanced data sets. ACM SIGKDD Explor Newsl 6(1):1–6CrossRefGoogle Scholar
  10. Demsar J (2006) Statistical comparisons of classifiers over multiple data sets. J Mach Learn Res 7:1–30zbMATHMathSciNetGoogle Scholar
  11. Dougherty J, Kohavi R, Sahami M (1995) Supervised and unsupervised discretization of continuous features. In: Prieditis A, Russell S (eds) Proceedings of the twelfth international conference on machine learning. Morgan Kaufmann, San Francisco, pp 194–202Google Scholar
  12. Ferri C, Hernández-Orallo J, Modroiu R (2009) An experimental comparison of performance measures for classification. Pattern Recogn Lett 30(1):27–38CrossRefGoogle Scholar
  13. Flach P (2012) Machine learning: the art and science of algorithms that make sense of data. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  14. Forman G (2005) Counting positives accurately despite inaccurate classification. In: Proceedings of the 16th European conference on machine learning (ECML), pp 564–575Google Scholar
  15. Forman G (2006) Quantifying trends accurately despite classifier error and class imbalance. In: Proceedings of the 12th ACM SIGKDD international conference on knowledge discovery and data mining (KDD), pp 157–166Google Scholar
  16. Forman G (2008) Quantifying counts and costs via classification. Data Min Knowl Discov 17(2):164–206CrossRefMathSciNetGoogle Scholar
  17. Frank A, Asuncion A (2010) UCI machine learning repository. http://archive.ics.uci.edu/ml
  18. González-Castro V, Alaiz-Rodríguez R, Alegre E (2012) Class distribution estimation based on the Hellinger distance. Inf Sci 218(1):146–164Google Scholar
  19. Hastie TJ, Tibshirani R, Friedman J (2009) The elements of statistical learning: data mining, inference, and prediction. Springer, BerlinCrossRefGoogle Scholar
  20. Hernández-Orallo J, Flach P, Ferri C (2012) A unified view of performance metrics: translating threshold choice into expected classification loss. J Mach Learn Res (JMLR) 13:2813–2869Google Scholar
  21. Hodges J, Lehmann E (1963) Estimates of location based on rank tests. Ann Math Stat 34(5):598–611CrossRefzbMATHMathSciNetGoogle Scholar
  22. Hosmer DW, Lemeshow S (2000) Applied logistic regression. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  23. Hwang JN, Lay SR, Lippman A (1994) Nonparametric multivariate density estimation: a comparative study. IEEE Trans Signal Process 42(10):2795–2810CrossRefGoogle Scholar
  24. Hyndman RJ, Bashtannyk DM, Grunwald GK (1996) Estimating and visualizing conditional densities. J Comput Graph Stat 5(4):315–336MathSciNetGoogle Scholar
  25. Moreno-Torres J, Raeder T, Alaiz-Rodríguez R, Chawla N, Herrera F (2012) A unifying view on dataset shift in classification. Pattern Recogn 45(1):521–530CrossRefGoogle Scholar
  26. Neyman J (1938) Contribution to the theory of sampling human populations. J Am Stat Assoc 33(201):101–116CrossRefzbMATHGoogle Scholar
  27. Platt JC (1999) Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods. In: Advances in large margin classifiers. MIT Press, Cambridge, pp 61–74Google Scholar
  28. Raeder T, Forman G, Chawla N (2012) Learning from imbalanced data: evaluation matters. Data Min 23:315–331MathSciNetGoogle Scholar
  29. Sánchez L, González V, Alegre E, Alaiz R (2008) Classification and quantification based on image analysis for sperm samples with uncertain damaged/intact cell proportions. In: Proceedings of the 5th international conference on image analysis and recognition. LNCS, vol 5112. Springer, Heidelberg, pp 827–836Google Scholar
  30. Sturges H (1926) The choice of a class interval. J Am Stat Assoc 21(153):65–66CrossRefGoogle Scholar
  31. Team R et al (2012) R: a language and environment for statistical computing. R Foundation for Statistical Computing, ViennaGoogle Scholar
  32. Tenenbein A (1970) A double sampling scheme for estimating from binomial data with misclassifications. J Am Stat Assoc 65(331):1350–1361CrossRefGoogle Scholar
  33. Weiss G (2004) Mining with rarity: a unifying framework. ACM SIGKDD Explor Newsl 6(1):7–19CrossRefGoogle Scholar
  34. Weiss G, Provost F (2001) The effect of class distribution on classifier learning: an empirical study. Technical Report ML-TR-44Google Scholar
  35. Witten IH, Frank E (2005) Data mining: practical machine learning tools and techniques with Java implementations. Elsevier, AmsterdamGoogle Scholar
  36. Xiao Y, Gordon A, Yakovlev A (2006a) A C++ program for the Cramér–von Mises two-sample test. J Stat Softw 17:1–15Google Scholar
  37. Xiao Y, Gordon A, Yakovlev A (2006b) The L1-version of the Cramér-von Mises test for two-sample comparisons in microarray data analysis. EURASIP J Bioinform Syst Biol 2006:85769Google Scholar
  38. Xue J, Weiss G (2009) Quantification and semi-supervised classification methods for handling changes in class distribution. In: Proceedings of the 15th ACM SIGKDD international conference on knowledge discovery and data mining, pp 897–906Google Scholar
  39. Yang Y (2003) Discretization for naive-bayes learning. PhD thesis, Monash UniversityGoogle Scholar
  40. Zadrozny B, Elkan C (2001) Obtaining calibrated probability estimates from decision trees and naive bayesian classifiers. In: Proceedings of the 8th international conference on machine learning (ICML), pp 609–616Google Scholar
  41. Zadrozny B, Elkan C (2002) Transforming classifier scores into accurate multiclass probability estimates. In: The 8th ACM SIGKDD international conference on knowledge discovery and data mining, pp 694–699Google Scholar

Copyright information

© The Author(s) 2013

Authors and Affiliations

  • Antonio Bella
    • 1
  • Cèsar Ferri
    • 1
    Email author
  • José Hernández-Orallo
    • 1
  • María José Ramírez-Quintana
    • 1
  1. 1.DSIC-ELP, Universitat Politècnica de ValènciaValenciaSpain

Personalised recommendations