On the Maximization of Quadratic Weighted Kappa
An analytical expression for the optimal estimation of the numerical dependence by the criterion of a quadratic weighted kappa and also the expression for the optimal value of this criterion were obtained. It is shown that the optimal decision function is obtained from the regression function by a linear transformation. The coefficients of this transformation can be found from the condition of equality of mathematical expectations and variances of the predicted value and its estimate. The quadratic weighted kappa coefficient was originally proposed as an alternative to the correlation coefficient to reflect the strength of dependence between two characteristics, but recently it has been widely used as a criterion for the quality of the forecast in the problem of recovery of dependencies (regression analysis). At the same time, the properties of this coefficient in this context are still poorly understood. The properties of the quadratic weighted kappa criterion revealed in the work allow us to conclude that the expediency of using it as a criterion for the quality of the decision function in most cases raises doubts. This criterion provides a solution that is actually based on the regression function, but the variance of the forecast is artificially made equal to the variance of the original value. This distorts the forecast without improving the statistical properties of the decision function.
About the Authors
Victor M. Nedel’ko, Cand. Sci. (Phys.–Math.), Assoc. Prof., Sobolev Institute of Mathematics SB RAS, 4, Acad. Koptyug Ave., Novosibirsk, 630090, Russian Federation Novosibirsk State University, 2, Pirogov st., Novosibirsk, 630090, Russian Federation, e-mail: firstname.lastname@example.org
1. Berikov V.B., Lbov G.S. Bayes estimates for recognition quality on finite sets ofevents Doklady Mathematics, 2005, vol. 71, no 3, pp. 327-330.
2. Vityaev E.E. Semantic Probabilistic Inference of Predictions Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika. [The Bulletin of Irkutsk State University. Series Mathematics], 2017, vol. 21, pp. 33-50. (In Russian) https://doi.org/10.26516/1997-7670.2017.21.33
3. Genrikhov I.E., Djukova E.V., Zhuravlyov V.I. About full regression decisiontrees Machine learning and data analysis, 2016, vol. 2, no 1. pp. 116-126. (In Russian). https//doi.org/10.21469/22233792.2.1.09
4. Kovalevskii A.P., Shatalin E.V. Vybor regressionnoy modeli zavisimosti massy telaot rosta s pomoschju empiricheskogo mosta [The Choice of a Regression Modelof the Body Weight on the Height via an Empirical Bridge]. Vestnik Tomskogo gosudarstvennogo universiteta [Tomsk State University Journal of Mathematicsand Mechanics], 2015, vol. 37, no 5, pp. 35-47. (In Russian). https://doi.org/10.17223/19988621/37/3
5. Linke Yu.Yu. Asymptotic properties of one-step weighted M-estimators withapplication to some regression problems. Teor. Veroyatnost. i Primenen., 2017, vol. 62, no 3, pp. 468-498. (in Russian). https://doi.org/10.4213/tvp5122
6. Nedelko V.M. Some aspects of estimating a quality of decision functions construction methods. Tomsk state university. Journal of control and computerscience, 2013, vol. 24, no 3, pp.123-132. (in Russian)
7. Nedel’ko V.M. Estimation of feature importance for quantile regression Machinelearning and data analysis, 2017, vol. 3, no 2, pp. 151–159. https://doi.org/10.21469/22233792.3.2.05
8. Nedel’ko V.M. Regressionnye modeli v zadache klassifikacii [Regression models inthe classification problem]. Sibirskij zhurnal industrial’noj matematiki [SiberianJournal of Industrial Mathematics], 2014, vol. XVII, no 1, pp. 86-98. (In Russian)
9. Brenner, Hermann, and Ulrike Kliebsch. Dependence of Weighted Kappa Coefficients on the Number of Categories Epidemiology, vol. 7, no 2, 1996, pp.199–202.
10. Cohen Jacob. A coefficient of agreement for nominal scales Educational and Psychological Measurement, 1960, vol. 20, no 1, pp. 37–46.
11. Alvan R. Feinstein, Domenic V. Cicchetti. High agreement but low Kappa: I. theproblems of two paradoxes Journal of Clinical Epidemiology, 1990, vol. 43, issue 6, pp. 543–549.
12. Kilem L. Gwet. Kappa Statistic is not Satisfactory for Assessing the Extentof Agreement Between Raters Statistical Methods For Inter-Rater Reliability Assessment, 2002, no. 1.
13. Ludbrook J. Statistical Techniques For Comparing Measurers And MethodsOf Measurement: A Critical Review Clinical and Experimental Pharmacologyand Physiology, 2002, vol. 29, pp. 527-536. https://doi.org/10.1046/j.1440-1681.2002.03686.x
14. S. Vanbelle, A. Albert. A note on the linearly weighted kappa coefficient for ordinalscales Statistical Methodology, 2009, vol. 6, issue 2, pp. 157-163.
15. David Vaughn, Derek Justice. On The Direct Maximization of Quadratic Weighted Kappa, 2015, arXiv:1509.07107v1.