Is uncorrelated linear discriminant analysis really a new method?

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• Uncorrelated linear discriminant analysis is the same as Fisher's linear discriminant analysis.

Fisher's linear discriminant analysis and linear discriminant analysis (LDA) are powerful methods in multivariate data analysis. Recently, a method called “uncorrelated linear discriminant analysis (ULDA)” has attracted attention in the chemometrics community. It has been stated that the major difference between ULDA and LDA is that the discriminant vectors of ULDA must satisfy an “S-orthogonality” constraint. This has led to the impression that ULDA is a different method from LDA. A number of papers published in the chemometrics field and others have generally accepted this statement. However, it can be shown that the so-called ULDA method is equivalent to Fisher's linear discriminant analysis or one simple case of LDA. There is a need to resolve the confusion surrounding ULDA in the chemometrics community. This work clarifies this confusion from a mathematical perspective and demonstrates equivalence using real experimental data sets.