The 95% confidence intervals of error rates and discriminant coefficients
Abstract
Fisher proposed a linear discriminant function (Fisher’s LDF). From 1971, we analysed electrocardiogram (ECG) data in order to develop the diagnostic logic between normal and abnormal symptoms by Fisher’s LDF and a quadratic discriminant function (QDF). Our four years research was inferior to the decision tree logic developed by the medical doctor. After this experience, we discriminated many data and found four problems of the discriminant analysis. A revised Optimal LDF by Integer Programming (Revised IP-OLDF) based on the minimum number of misclassification (minimum NM) criterion resolves three problems entirely [13, 18]. In this research, we discuss fourth problem of the discriminant analysis. There are no standard errors (SEs) of the error rate and discriminant coefficient. We propose a k-fold crossvalidation method. This method offers a model selection technique and a 95% confidence intervals (C.I.) of error rates and discriminant coefficients.References
Efron, B., (1979). Bootstrap Methods -Another Look at the Jackknife-. The Annals of Statics, 7/1, 1–26.
Firth, D., (1993). Bias reduction of maximum likelihood estimates. Biometrika, 80, 27-39.
Fisher, R. A., (1936). The Use of Multiple Measurements in Taxonomic Problems. Annals of Eugenics, 7, 179–188.
Flury, B., Rieduyl, H., (1988). Multivariate Statistics: A Practical Approach. Cambridge University Press.
Friedman, J. H., (1989). Regularized Discriminant Analysis．Journal of the American Statistical Association，84/405, 165-175．
Konishi, S., Honda, M., (1992). Bootstrap Methods for Error Rate Estimation in Discriminant Analysis. Japanese Society of Applied Statistics，21/2，67-100．
Lachenbruch, P. A., Mickey, M. R., (1968). Estimation of error rates in discriminant analysis. Technometrics 10, 1-11.
Sall, J. P., Creighton, L., Lehman, A., (2004). JMP Start Statistics, Third Edition. SAS Institute Inc.
Schrage, L., (2006). Optimization Modeling with LINGO. LINDO Systems Inc.
Shinmura, S., (1998). Optimal Linear Discrimrnant Functions using Mathematical Programming. Journal of the Japanese Society of Computer Statistics, 11 / 2, 89-101.
Shinmura, S., (2000). A new algorithm of the linear discriminant function using integer programming. New Trends in Probability and Statistics, 5, 133-142.
Shinmura, S., (2004). New Algorithm of Discriminant Analysis using Integer Programming. IPSI 2004 Pescara VIP Conference CD-ROM, 1-18.
Shinmura, S., (2007). Overviews of Discriminant Function by Mathematical Programming. Journal of the Japanese Society of Computer Statistics, 20/1-2, 59-94.
Shinmura, S., (2010). The optimal linear discriminant function. Union of Japanese Scientist and Engineer Publishing (in Japanese).
Shinmura, S., (2011a)．Problems of Discriminant Analysis by Mark Sense Test Data．Japanese Society of Applied Statistics，40/3，157-172．
Shinmura, S., (2011b). Beyond Fisher’s Linear Discriminant Analysis - New World of Discriminant Analysis -. ISI CD-ROM, 1-6.
Shinmura, S., (2013). Evaluation of Optimal Linear Discriminant Function by 100-fold cross-validation. 2013 ISI CD-ROM, 1-6.
Shinmura, S., (2014a). End of Discriminant Functions based on Variance Covariance Matrices. ICORES, 5-14, 2014.
Shinmura, S., (2014b). Improvement of CPU time of Linear Discriminant Functions based on MNM criterion by IP. Statistics, Optimization and Information Computing, vol. 2, 114-129.
Shinmura, S., (2014c). Three serious problems and new facts of the discriminant analysis. ICORES, 1-16.
Shinmura, S., (2014d). Comparison of Linear Discriminant Function by K-fold Cross Validation. Data Analytic 2014, 1-6.
Stam, A., (1997). Nontraditional approaches to statistical classification: Some perspectives on Lp-norm methods. Annals of Operations Research, 74, 1-36.
Taguchi, G., Jugulu, R. (2002). The Mahalanobis-Taguchi Strategy – A Pattern Technology System. John Wiley & Sons.
Markowitz, H. M., (1959). Portfolio Selection, Efficient Diversification of Investment. John Wiley & Sons, Inc.
Vapnik, V. (1995). The Nature of Statistical Learning Theory．Springer-Verlag.
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