The Annals of Statistics

Remarks on Functional Canonical Variates, Alternating Least Squares Methods and Ace

Andreas Buja

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We discuss properties of some data-analytic methods which are intimately related to each other: alternating least squares (ALS), correspondence analysis and more recently Breiman and Friedman's ACE algorithm. The application of these methods to regression produces nonparametric estimators of nonlinear transformations, both of the response and the predictors. These procedures are among the most powerful tools for data analysis, but missing awareness of some artifacts could lead to inappropriate interpretations. We point out some anomalies as well as some curiosities in the mathematics of these methods, and we relate them to some areas in computer-aided tomography, projection pursuit regression and nonlinear devices in the theory of noise.

Article information

Ann. Statist., Volume 18, Number 3 (1990), 1032-1069.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62J99: None of the above, but in this section
Secondary: 62P15: Applications to psychology

ACE alternating least squares correspondence analysis nonlinear multivariate analysis scaling horseshoe effect functional canonical variates optimal correlation transformation of variables in regression orthogonal polynomials polynomial biorthogonality series expansions of bivariate distributions elliptic distributions


Buja, Andreas. Remarks on Functional Canonical Variates, Alternating Least Squares Methods and Ace. Ann. Statist. 18 (1990), no. 3, 1032--1069. doi:10.1214/aos/1176347739.

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