# A Trivial Linear Discriminant Function

### Abstract

In this paper, we focus on the new model selection procedure of the discriminant analysis. Combining re-sampling technique with k-fold cross validation, we develop a k-fold cross validation for small sample method. By this breakthrough, we obtain the mean error rate in the validation samples (M2) and the 95\% confidence interval (CI) of discriminant coefficient. Moreover, we propose the model selection procedure in which the model having a minimum M2 was chosen to the best model. We apply this new method and procedure to the pass/ fail determination of exam scores. In this case, we fix the constant =1 for seven linear discriminant functions (LDFs) and several good results were obtained as follows: 1) M2 of Fisher's LDF are over 4.6\% worse than Revised IP-OLDF. 2) A soft-margin SVM for penalty c=1 (SVM1) is worse than another mathematical programming (MP) based LDFs and logistic regression . 3) The 95\% CI of the best discriminant coefficients was obtained. Seven LDFs except for Fisher's LDF are almost the same as a trivial LDF for the linear separable model. Furthermore, if we choose the median of the coefficient of seven LDFs except for Fisher's LDF, those are almost the same as the trivial LDF for the linear separable model.### References

Fisher, R. A., (1936). The Use of Multiple Measurements in Taxonomic Problems. Annals of Eugenics, 7, 179–188.

Friedman, J. H., (1989). Regularized Discriminant Analysis．Journal of the American Statistical Associ-ation，84/405, 165-175．

Goodnight, J.H.(1981). A tutorial on the SWEEP Operator, The American Statistician, 33, 149-158.

Lachenbruch, P. A., Mickey, M. R., (1968). Estimation of error rates in discriminant analysis. Technomet-rics 10, 1-11.

Sall, J. P., Creighton, L., Lehman, A., (2004). JMP Start Statistics, Third Edition. SAS Institute Inc.

Schrage, L., (1991). LINDO –An Optimization Modeling System (Fourth Edition)-. The Scientific Press.

Schrage, L., (2006). Optimization Modeling with LINGO. LINDO Systems Inc.

Shinmura, S., (1998). Optimal Linear Discriminant 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., (2007a). Comparison of Revised IP-OLDF and SVM. ISI2009, 1-4.

Shinmura, S., (2007b). Overviews of Discriminant Function by Mathematical Programming. Journal of the Japanese Society of Computer Statistics, 20/1-2, 59-94.

Shinmura, S., (2009). Practical discriminant analysis by IP-OLDF and IPLP-OLDF. IPSI 2009 Belgrade VIPSI Conference CD-ROM, 1-17.

Shinmura, S., (2010a). The optimal linear discriminant function. Union of Japanese Scientist and Engineer Publishing.

Shinmura, S., (2010b). Improvement of CPU time of Revised IP-OLDF using Linear Programming. Journal of the Japanese Society of Computer Statistics, 22/1, 39-57.

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 -. ISI2011 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, 2, 114-129.

Shinmura, S., (2014c). Comparison of Linear Discriminant Function by K-fold Cross-validation. Data Analytic 2014, 1-6.

Shinmura, S., (2015a). The 95% confidence intervals of error rates and discriminant coefficients. Statistics, Optimization and Information Computing, 3, 66-78.

Shinmura, S., (2015b). Four Serious Problems and New Facts of the Discriminant Analysis. In Pinson, E., Valente, F., Vitoriano, B., (Eds.), Operations Research and Enterprise Systems, 15-30, Springer (ISSN: 1865-0929, ISBN: 978-3-319-17508-9, DOI:10.1007/978-3-319-17509-6).

Stam, A., (1997). Nontraditional approaches to statistical classification: Some perspectives on lp-norm methods. Annals of Operations Research, 74, 1-36.

Taguchi, G., Jugulum, R., (2002). The Mahalanobis-Taguchi Strategy -A Pattern Technology System. John Wiley & Sons.

Vapnik, V., (1995). The Nature of Statistical Learning Theory. Springer-Verlag.

*Statistics, Optimization & Information Computing*,

*3*(4), 322-335. https://doi.org/10.19139/soic.v3i4.151

- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).