Robust Q-mode principal component analysis in L1

Authors:
V Choulakian
Affiliations:
Départment de Mathématique & Statistique, Facultéé des Sciences, Université de Moncton, Moncton, NB, Canada E1A 3E9
Venue:
Computational Statistics & Data Analysis
Year:
2001

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Abstract

We propose principal component analysis (PCA) of a data set based on the L"1-norm. We distinguish between Q-mode and R-mode analyses. The Q-mode L"1 principal components are sequentially calculated by an enumeration procedure. We show that the Q-mode L"1-norm PCA is a constrained version of R-mode L"2-norm PCA. Two generalizations are proposed, robustification and extension of Thurston's simple structure. Robustification is achieved by replacing the L"2-norm by an efficient robust A-estimator of scale based on Tukey's biweight function. Extended simple structure is used to discard redundant variables. Examples are provided.