Abstract
We propose a method for estimating the posterior probability of a class at a given point by approximating a discriminant function that takes a zero value at this point. The approximation is based on a supervised training set. Posterior probabilities of classes allow the classification problem to be solved simultaneously for different criteria and different costs of classification errors. The method is based on choosing such a ratio of the costs of classification errors in the construction of an approximation to the discriminant function that the approximation takes the zero value at a given point. We give a model example and an example with real data from the field of medical diagnostics.
Similar content being viewed by others
References
Anderson, T.W., An Introduction to Multivariate Statistical Analysis, New York: Wiley, 2003, 3rd ed.
Tsypkin, Ya.Z. and Kel’mans, G.K., Adaptive Bayesian Approach, Probl. Peredachi Inform., 1970, vol. 6, no. 1, pp. 52–59.
Zenkov, V.V., Approximating Discriminant Functions in the Neighborhood of Zero Values, Izv. Ross. Akad. Nauk, Tekh. Kibern., 1973, no. 2, pp. 152–156.
Zenkov, V.V., Using Weighted Least Squares to Approximate the Discriminant Function with a Cylindrical Surface in Classification Problems, Autom. Remote Control, 2017, vol. 78, no. 9, pp. 1662–1673.
Hastie, T., Tibshirani, R., and Friedman, J., The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Berlin: Springer, 2009, 2nd ed.
Niculescu-Mizil, A. and Caruana, R., Predicting Good Probabilities with Supervised Learning, Proc. 22nd Int. Conf. on Machine Learning, ICML’05, Bonn, Germany, August 7–11, 2005, pp. 625–632. http://www.cs.cornell.edu/~alexn/papers/calibration.icml05.crc.rev3.pdf
Bella, A., Ferri, C., Hernández-Orallo, J., and Ramírez-Quintana, M., Calibration of Machine Learning Models, in Handbook of Research on Machine Learning Applications and Trends: Algorithms, Methods, and Techniques, Hershey: IDI, 2010, pp. 19. http://users.dsic.upv.es/~flip/papers/BFHRHandbook2010.pdf
Zenkov, V.V., Machine Learning. A Method of Approximation of Discriminant Functions and Two Methods of Estimation of Posterior Probabilities of Classes in the Problem of Classification, Proc. 10th Int. Conf. “Management of Large-Scale System Development” (MLSD’2017, 2–4 Oct. 2017), IEEE Conf, pp. 1–4. http://ieeexplore.ieee.org/document/8109715/
Vorontsov, K.V., Matematicheskie metody obucheniya po pretsedentam (teoriya obucheniya mashin) (Mathematical Methods of Precedent Learning (Machine Learning Theory)). http://www.machinelearning.ru/wiki/images/6/6d/Voron-ML-1.pdf
Merkov, A.B., Raspoznavanie obrazov. Vvedenie v metody statisticheskogo obucheniya (Pattern Recognition. Introduction to Statistical Learning), Moscow: URSS, 2011. http://www.recognition.mccme.ru/pub/RecognitionLab.html/slbook.pdf
Anufriev, I.E., Smirnov, A.B., and Smirnova, E.N., MATLAB 7, St. Petersburg: BKhV-Peterburg, 2005. http://fileskachat.com/file/31353 4197b1d0a54318bac8271e5daca525b0.html
Zenkov, V.V., Software for Constructing an Approximation of a Discriminant Function by a Training Sample with Higher Accuracy in the Neighborhood of Zero Values, Software Registration Certificate no. 2017660808, 2017.
Zenkov, V.V., Estimating the Posterior Probability of a Class at a Point by an Approximation of One Discriminant Function, Software Registration Certificate no. 2017660807, 2017.
Problem at TMSh 2014. http://www.machinelearning.ru/wiki/images/e/e1/School-VI-2014-task-3.rar
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.V. Zenkov, 2018, published in Avtomatika i Telemekhanika, 2018, No. 9, pp. 46–58.
Rights and permissions
About this article
Cite this article
Zenkov, V.V. Estimating the Probability of a Class at a Point by the Approximation of One Discriminant Function. Autom Remote Control 79, 1582–1592 (2018). https://doi.org/10.1134/S0005117918090047
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117918090047