Annotations on the Relationship Among Discriminant Functions

Awogbemi Clement Adeyeye

doi:doi:10.11648/j.pamj.20200906.14

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Annotations on the Relationship Among Discriminant Functions

Awogbemi Clement Adeyeye

Published in Pure and Applied Mathematics Journal (Volume 9, Issue 6)

Received: 19 June 2020 Accepted: 20 July 2020 Published: 16 December 2020

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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.

Published in	Pure and Applied Mathematics Journal (Volume 9, Issue 6)
DOI	10.11648/j.pamj.20200906.14
Page(s)	124-128
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Discriminant Functions, Classification Statistics, Classification Problems, Covariance Matrix, Probability of Misclassification

References

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[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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Awogbemi Clement Adeyeye. (2020). Annotations on the Relationship Among Discriminant Functions. Pure and Applied Mathematics Journal, 9(6), 124-128. https://doi.org/10.11648/j.pamj.20200906.14

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Awogbemi Clement Adeyeye. Annotations on the Relationship Among Discriminant Functions. Pure Appl. Math. J. 2020, 9(6), 124-128. doi: 10.11648/j.pamj.20200906.14

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AMA Style

Awogbemi Clement Adeyeye. Annotations on the Relationship Among Discriminant Functions. Pure Appl Math J. 2020;9(6):124-128. doi: 10.11648/j.pamj.20200906.14

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@article{10.11648/j.pamj.20200906.14,
  author = {Awogbemi Clement Adeyeye},
  title = {Annotations on the Relationship Among Discriminant Functions},
  journal = {Pure and Applied Mathematics Journal},
  volume = {9},
  number = {6},
  pages = {124-128},
  doi = {10.11648/j.pamj.20200906.14},
  url = {https://doi.org/10.11648/j.pamj.20200906.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200906.14},
  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.},
 year = {2020}
}

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TY  - JOUR
T1  - Annotations on the Relationship Among Discriminant Functions
AU  - Awogbemi Clement Adeyeye
Y1  - 2020/12/16
PY  - 2020
N1  - https://doi.org/10.11648/j.pamj.20200906.14
DO  - 10.11648/j.pamj.20200906.14
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 124
EP  - 128
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20200906.14
AB  - 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.
VL  - 9
IS  - 6
ER  -

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Author Information

Awogbemi Clement Adeyeye

Department of Statistics, National Mathematical Centre, Abuja, Nigeria

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APA Style

Awogbemi Clement Adeyeye. (2020). Annotations on the Relationship Among Discriminant Functions. Pure and Applied Mathematics Journal, 9(6), 124-128. https://doi.org/10.11648/j.pamj.20200906.14

Copy | Download

ACS Style

Awogbemi Clement Adeyeye. Annotations on the Relationship Among Discriminant Functions. Pure Appl. Math. J. 2020, 9(6), 124-128. doi: 10.11648/j.pamj.20200906.14

Copy | Download

AMA Style

Awogbemi Clement Adeyeye. Annotations on the Relationship Among Discriminant Functions. Pure Appl Math J. 2020;9(6):124-128. doi: 10.11648/j.pamj.20200906.14

Copy | Download

@article{10.11648/j.pamj.20200906.14,
  author = {Awogbemi Clement Adeyeye},
  title = {Annotations on the Relationship Among Discriminant Functions},
  journal = {Pure and Applied Mathematics Journal},
  volume = {9},
  number = {6},
  pages = {124-128},
  doi = {10.11648/j.pamj.20200906.14},
  url = {https://doi.org/10.11648/j.pamj.20200906.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200906.14},
  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.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Annotations on the Relationship Among Discriminant Functions
AU  - Awogbemi Clement Adeyeye
Y1  - 2020/12/16
PY  - 2020
N1  - https://doi.org/10.11648/j.pamj.20200906.14
DO  - 10.11648/j.pamj.20200906.14
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 124
EP  - 128
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20200906.14
AB  - 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.
VL  - 9
IS  - 6
ER  -

Copy | Download