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Volume 9, Issue 6, December 2020, Page: 124-128
Annotations on the Relationship Among Discriminant Functions
Awogbemi Clement Adeyeye, Department of Statistics, National Mathematical Centre, Abuja, Nigeria
Received: Jun. 19, 2020;       Accepted: Jul. 20, 2020;       Published: Dec. 16, 2020
DOI: 10.11648/j.pamj.20200906.14      View  61      Downloads  41
Abstract
Different forms of discriminant functions and the essence of their appearances were considered in this study. Various forms of classification problems were also considered, and in each of the cases mentioned, classification from simple functions of the observational vector rather than complicated regions in the higher-dimensional space of the original vector were made. Ever since the emergence of the Linear Discriminant Function (LDF) by Fisher, several other classification statistics have emerged and violation of condition of equal variance covariance matrix for Linear Discriminant Function (LDF) results to Quadratic Discriminant Function (QDF). While the Best Linear Discriminant Function (BLDF) is referred to Best Sample Discriminant Function (BSDF) when the parameters are estimated from a sample and also optimal in the same sense as Quadratic Discriminant Function (QDF), Rao statistic is best for discriminating between options that are close each other. The relationships among the classification statistics examined were established: Among the methods of classification statistics considered, Anderson’s (W) and Rao’s (R) statistics are equivalent when the two sample sizes n1 and n2 are equal, and when a constant is equal to 1, W, R and John-Kudo’s (Z) classification statistics are asymptotically comparable. A linear relationship is also established between W and Z classification.
Keywords
Discriminant Functions, Classification Statistics, Classification Problems, Covariance Matrix, Probability of Misclassification
To cite this article
Awogbemi Clement Adeyeye, Annotations on the Relationship Among Discriminant Functions, Pure and Applied Mathematics Journal. Vol. 9, No. 6, 2020, pp. 124-128. doi: 10.11648/j.pamj.20200906.14
Copyright
Copyright © 2020 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[1]
Alvin, C. R. (2002). Methods of Multivariate Analysis. New York: John Wiley & Sons.
[2]
Anderson, T. W. (1973). Asymptotic Evaluation of the Probabilities of Misclassification by Linear Discriminant Functions (LDF): Discriminant Analysis and Applications. (T. Cacoullos, Ed.), pp. 17-35, Academic Press, New-York.
[3]
Anderson, T. W. (2003). An Introduction to Multivariate Statistical Methods. New York: New ed. John Wiley & Sons.
[4]
Anderson, T. W. and Bahadur, R. R. (1962). Classification into Two Multivariate Normal Distributions with Different Covariance Matrices. Annals of Mathematical Statistics, 33, pp. 420-431.
[5]
Awogbemi, C. A. (2019). Errors of Misclassification Associated with Edgeworth Series Distribution, Unpublished Ph. D. Dissertation, Department of Statistics, Faculty of Physical Sciences, Nnamdi Azikiwe University, Awka, Nigeria.
[6]
Clumies-Ross, C. W. and Riffenburgh, R. H. (1960). Geometry and Linear Discrimination. Biometrika, 47, pp. 185-189.
[7]
Ekezie, D. D. and Onyeagu, S. I. (2013). Comparison of Seven Asymptotic Expansions for the Sample Linear Discriminant Function. Canadian Journal of Computations in Mathematics, Natural Sciences, Engineering and Medicine, 4 (1), pp. 93-115.
[8]
Fisher, F. A. (1936). The Use of Multiple Measurements in Taxonomic Problems. Annals of Eugenics, 7, pp. 179-188.
[9]
Gilbert, E. S. (1969). The Effect of Unequal Variance-Covariance Matrices on Fisher’s Linear Discriminant Function. Biometrics, 25, pp. 424-427.
[10]
John, I. M. (2015). Multivariate Statistics, Department of Statistics, University of Illinois, Urbana Champaign. URL: http://stat.istics.net/Multivariate.
[11]
John, S. (1965): Corrections to: “On Classification by Statistics, R and Z". Ann. Inst. Math., 7, pp. 113.
[12]
Kudo, A. (1959). The Classificatory Problem viewed as a Two-Decision Problem. Unpublished Masters’ Thesis of the Faculty of Science, Kyushu University, 13, pp. 96-125.
[13]
Lachenbruch, P. A., Sneeringer, C. and Revo, L. T. (1973). Robustness of the Linear and Quadratic Discriminant Function to Certain Types of Non-normality. Journal of Communication Statistics, 1, pp. 39-57.
[14]
Morrison, D. F. (2003). Multivariate Statistical Methods. London: McGraw-Hill Publishing.
[15]
Onyeagu, S. I. (2003). A First Course in Multivariate Statistical Analysis. Awka: Mega Concept.
[16]
Rao, C. R. (1954). A General Theory of Discrimination when the Information about Alternative Population Distribution is Based on Samples. Annals of Mathematical Statistics, 25, pp. 651- 670.
[17]
Siotani, M. (1975). Comparison of Two Procedures in Discriminant Analysis based on Anderson's W-Criterion and John-Kudo's Z-Criterion. Tech. Report No. 67, University Of Manitoba, Department of Statistics.
[18]
Wald, A. (1944). On a Statistical Problem Arising in the Classification of an Individual into One of Two Groups. Ann. Math. Stat. 15, pp. 145-162.
[19]
Welch, B. L. (1939). Notes on Discriminant Functions. Biometrika, 31, pp. 218-220.
[20]
William, R. D. and Matthew, G. (1984). Multivariate Analysis, Methods and Applications. New York: John Wiley & Sons Inc.
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