Statistical Discrimination Analysis Using the Maximum Function

Abstract

The maximum of k functions defined on R ⁿ , n ≥ 1, by f _max (x) = max{f ₁ (x),…, f _k (x)}, ∀ x ∊ R ⁿ , can have important roles in Statistics, particularly in Classification. Through its relation with the Bayes error, which is the reference error in classification, it can serve to compute numerical bounds for errors in other classification schemes. It can also serve to define the joint L¹-distance between more than two densities, which, in turn, will serve as a useful tool in Classification and Cluster Analyses. It has a vast potential application in digital image processing too. Finally, its versatile role can be seen in several numerical examples, related to the analysis of Fisher's classical iris data in multidimensional spaces.

Keywords:

• Bayes error
• Discriminant function
• L¹-norm
• Linear
• Maximum
• Normal distribution
• Pattern classification
• Quadratic
• Space curves

Mathematics Subject Classification:

• 62H30
• 68T10

Acknowledgment

Research partially supported by NSERC grant AC4294 (Canada). The authors wish to thank Prof. Chris Quach, of the Fellars Institute, Arizona, for having contributed to the solution of this problem. Also, thanks to an anonymous referee for some helpful comments that have helped to improve the presentation of the article.

Notes

IRIS VARITIES: (Se) = SETOSA, (Ve) = VERSICOLOR, (Vi) = VIRGINICA X ₁ = SEPAL LENGTH, X ₂ = SEPAL WIDTH, X ₃ = PETAL LENGTH, X ₄ = PETAL WIDTH