Discriminant Analysis

  • Somesh Das Gupta
Part of the Lecture Notes in Statistics book series (LNS, volume 1)


Between 1936 and 1940 Fisher published four articles on statistical discriminant analysis, in the first of which [CP 138] he described and applied the linear discriminant function. Prior to Fisher the main emphasis of research in this, area was on measures of difference between populations based on multiple measurements. In this presentation, I shall briefly summarize the early work from Karl Pearson up to Mahalanobis, and then describe the major contributions associated with Fisher. These include the derivation of the linear discriminant function and its relationship to regression in the two-sample case, extensions to s populations for s 003E 2, and a test of significance associated with the linear discriminant function.


Discriminant Analysis Linear Discriminant Analysis Secular Variation Linear Discriminant Function Dispersion Matrix 
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References and Bibliography

Early Period

  1. Pearson, K. (1916). “On the Probability that Two Independent Distributions of Frequency Are Really from the Same Population,” Biometrika, 8, 250–254.CrossRefGoogle Scholar
  2. Tildesley, M.L. (1921). “A First Study of the Burmese Skull,” Biometrika, 13, 247–251.Google Scholar
  3. Pearson, K. (1926). “On the Coefficient of Racial Likeness,” Biometrika, 18, 105–117.zbMATHGoogle Scholar
  4. Morant, G.M. (1926). “A First Study of Craniology of England and Scotland from Neolithic to Early Historic Times, with Special Reference to Anglo-Saxon Skulls in London Museums,” Biometrika, 18, 56–98.Google Scholar
  5. Mahalanobis, P.C. (1927). “Analysis of Race Mixture in Bengal,” Journal and Proceedings of the Asiatic Society of Bengal, 23, 301–333.Google Scholar
  6. Mahalanobis, P.C. (1928). “A Statistical Study of the Chinese Head,” Man in India, 8, 107–122.Google Scholar
  7. Morant, G.M. (1928). “A Preliminary Classification of European Races Based on Cranial Measurements,” Biometrika, 20, 301–375.Google Scholar
  8. Pearson, K. (1928). “The Application of the Coefficient of Racial Likeness to Test the Character of Samples,” Biometrika, 20, 294–300.Google Scholar
  9. Romanovsky, V. (1928). “On the Criteria that Two Given Samples Belong to the Same Normal Population,” Metron, 7, 3–46.Google Scholar
  10. Mahalanobis, P.C. (1930). “On Tests and Measures of Group Divergence,” Journal and Proceedings of the Asiatic Society of Bengal, 26, 541–588.Google Scholar
  11. Hotelling, H. (1931). “The Generalization of Student’s Ratio,” The Annals of Mathematical Statistics, 2, 360–378.zbMATHCrossRefGoogle Scholar
  12. Woo, T.L. and Morant, G.M. (1932). “A Preliminary Classification of Asiatic Races Based on Cranial Measurements,” Biometrika, 24, 108–134.Google Scholar
  13. Pearson, E.S. and Wilks, S.S. (1933). “Methods of Statistical Analysis Appropriate for Samples of Two Variables,” Biometrika, 25, 353–378.Google Scholar
  14. Meyer, H.A. and Deming, W.E. (1935). “On the Influence of Classification on the Determination of a Measurable Characteristic,” Journal of the American Statistical Association, 30, 671–677.zbMATHCrossRefGoogle Scholar
  15. Mahalanobis, P.C. (1936). “On the Generalized Distance in Statistics,” Proceedings of the National Institute of Sciences of India, 2, 49–55..zbMATHGoogle Scholar

Contemporaries of Fisher

  1. Barnard, M.M. (. 1935 ). “The Secular Variations of Skull Characters in Four Series of Egyptian Skulls,” Annals of Eugenics, 6, 352–371.CrossRefGoogle Scholar
  2. Bose, S.N. (1936). “On the Complete Moment Coefficients of the D2 Statistic”, Sankhya, 2, 385–396.Google Scholar
  3. Bose, R.C. (1936). “On the Exact Distribution and Moment Coefficients of the D2- Statistic,” Sankhya, 2, 143–154. Corrected and modified paper: Sankhya, 2, 379–384.Google Scholar
  4. Martin, E.S. (1936). “A Study of the Egyptian Series of Mandibles with Special Reference to Mathematical Methods of Sexing,” Biometrika., 28, 149–178.Google Scholar
  5. Smith, H.F. (1936). “A Discriminant Function for Plant Selection,” Annals of Eugenics, 7, 240–250.CrossRefGoogle Scholar
  6. Bose, S.N. (1937). “On the Moment-Coefficients of the D2-Statistic, and Certain Integral and Differential Equations Connected with the Multivariate Normal Populations,” Sankhya, 3, 105–124.Google Scholar
  7. Bose, R.C. and Roy, S.N. (1937). “On the Distribution of Fisher’s Taxonomic Coefficient and Studentised D2-Statistic,” Science and Culture, 3, 335.Google Scholar
  8. Bose, R.C. and Roy, S.N. (1938). “The Distribution of the Studentised D2-Statistic” Sankhya, 4, 19–38.MathSciNetGoogle Scholar
  9. Hsu, P.L. (1938). “Notes on Hotellings Generalized T2,” The Annals of Mathematical Statistics, 9, 231–243.zbMATHCrossRefGoogle Scholar
  10. Bartlett, M.S. (1939). “The Standard Errors of Discriminant Function Coefficients,” Supplement to the Journal of the Royal Statistical Society, 6, 169–173.MathSciNetCrossRefGoogle Scholar
  11. Girshick, M.A. (1939). “On the Sampling Theory of Roots of Determinantal Equations” The Annals of Mathematical Statistics, 10, 203–224.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Hsu, P.L. (1939). “On the Distribution of Roots of Certain Determinanta1 Equations” Annals of Eugenics, 9, 250–258.CrossRefGoogle Scholar
  13. Roy, S.N. (1939). “A Note on the Distribution of the Studentised D2-Statistic,” Sankhya, 4, 373–380.Google Scholar
  14. Roy, S.N. (1939). “P-Statistics and Some Generalizations in Analysis of Variance,” Sankhya, 4, 381–396.Google Scholar
  15. Welch, B.L. (1939). “Note on Discriminant Functions,” Biometrika, 31, 218–220.MathSciNetzbMATHGoogle Scholar
  16. Roy, S.N. and Bose, R.C. (1940). “The Use and Distribution of the Studentised D2-Statistic when the Variances and Covariances are Based on Samples,” Sankhya, 4, 535–542.MathSciNetGoogle Scholar
  17. Bhattacharya, D.P. and Narayan, R.D. (1941). “Moments of the D2-Statistic for Populations with Unequal Dispersions,” Sankhya, 5, 401–412.MathSciNetGoogle Scholar
  18. Goodwin, C.N. and Morant, G.M. (1941). “The Human Remains of Iron Age and Other Periods from Maiden Castle, Dorset,” Biometrika, 31, 295–312.Google Scholar
  19. Wald, A. (1944). “On a Statistical Problem Arising in the Classification of an Individual into One of Two Groups,” The Annals of Mathematical Statistics, 15, 145–162.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Von Mises, R. (1954). “On the Classification of Observation Data into Distinct Groups,” The Annals of Mathematical Statistics, 16, 68–73.CrossRefGoogle Scholar

Recent References

  1. Cacoullos, T. and Styan, G.P.A. (1973). “A Bibliography of Discriminant Analysis,” in Discriminant Analysis and Applications edited by T. Cacoullos. New York: Academic Press.Google Scholar
  2. Das Gupta, S. (1973). “Theories and Methods in Classification: A Review,” in Discriminant Analysis and Applications edited by T. Cacoullos. New York: Academic Press.Google Scholar
  3. Rao, C.R. (1973). Linear Statistical Inference and Its Applications. 2nd edition. New York: John Wiley and Sons. [Discussion of discriminant analysis and classification in Chapter 8].zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1980

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  • Somesh Das Gupta

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