Robust Singular Value Decompositions: A New Approach to Projection Pursuit

Ammann, Larry P.

doi:10.2307/2290330

Cited by 27 publications

(25 citation statements)

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“…Xie et al [35] experimented with simulated annealing techniques for optimizing (1.2). Ammann [1] used an iterative robust regression scheme for estimating the eigenvectors in reverse order, but this is restricted to a specific class of PP-indices. Also general purpose maximization routines could be used; if the objective function is differentiable, Newton-steps can be carried out.…”

Section: Propositionmentioning

confidence: 99%

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High breakdown estimators for principal components: the projection-pursuit approach revisited

Croux

Ruiz‐Gazen

2005

Journal of Multivariate Analysis

272

186

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Li and Chen (J. Amer. Statist. Assoc. 80 (1985) 759) proposed a method for principal components using projection-pursuit techniques. In classical principal components one searches for directions with maximal variance, and their approach consists of replacing this variance by a robust scale measure. Li and Chen showed that this estimator is consistent, qualitative robust and inherits the breakdown point of the robust scale estimator. We complete their study by deriving the influence function of the estimators for the eigenvectors, eigenvalues and the associated dispersion matrix. Corresponding Gaussian efficiencies are presented as well. Asymptotic normality of the estimators has been treated in a paper of Cui et al. (Biometrika 90 (2003) 953), complementing the results of this paper. Furthermore, a simple explicit version of the projection-pursuit based estimator is proposed and shown to be fast to compute, orthogonally equivariant, and having the maximal finite-sample breakdown point property. We will illustrate the method with a real data example.

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Section: Propositionmentioning

confidence: 99%

“…. , (v Sn,1 ) t x n ). Suppose now that the first k − 1 eigenvectors have already been found (k > 1).…”

Section: Introductionmentioning

confidence: 99%

High breakdown estimators for principal components: the projection-pursuit approach revisited

Croux

Ruiz‐Gazen

2005

Journal of Multivariate Analysis

272

186

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“…One practical consequence is that an association can be masked even by a single outlier. The lack of robustness associated with o has given rise to many alternative measures of association; see, for example, Huber (1981, p. 199), Ammann (1993), Davies (1987), Devlin et al (1981), Goldberg & Iglewicz (1992), Kent & Tyler (1996), Rousseeuw & Leroy (1987), Rocke (1996), Hampel et al (1986, chapter 5), Li & Chen (1985), Lupuhaä (1989), Stahel (1981), Donoho (1982), Marrona (1976), Mosteller & Tukey (1977, p. 211) and Wilcox (1997). For the bulk of these correlation coefficients, virtually nothing is known about how they might be used to test the hypothesis of independence, but two exceptions are the Winsorized and percentage bend correlations summarized in Wilcox (1997).…”

Section: Introductionmentioning

confidence: 99%

Inferences Based on a Skipped Correlation Coefficient

Wilcox

2004

Journal of Applied Statistics

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The most popular method for trying to detect an association between two random variables is to test H0 : ρ=0, the hypothesis that Pearson's correlation is equal to zero. It is well known, however, that Pearson's correlation is not robust, roughly meaning that small changes in any distribution, including any bivariate normal distribution as a special case, can alter its value. Moreover, the usual estimate of ρ, r, is sensitive to only a few outliers which can mask a true association. A simple alternative to testing H0 : ρ =0 is to switch to a measure of association that guards against outliers among the marginal distributions such as Kendall's tau, Spearman's rho, a Winsorized correlation, or a so-called percentage bend correlation. But it is known that these methods fail to take into account the overall structure of the data. Many measures of association that do take into account the overall structure of the data have been proposed, but it seems that nothing is known about how they might be used to detect dependence. One such measure of association is selected, which is designed so that under bivariate normality, its estimator gives a reasonably accurate estimate of ρ. Then methods for testing the hypothesis of a zero correlation are studied.Skipped correlation coefficient, inferences, random variables, Pearson's correlation,

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“…A more reasonable assumption, and the one used as the basis for the analysis reported here, is that there is a set of core stations at each site that may be treated as randomly selected observations from a common distribution, but in addition, there are other observations that are derived from a different distribution. Under these circumstances, robust methods [21,22] are more appropriate than the classical procedures. For this reason the following robust methodology was employed.…”

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confidence: 99%

Assessing the relationship between ambient toxicity and instream biological response

Dickson

Waller

Kennedy

et al. 1992

Environmental Toxicology and Chemistry

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Robust canonical correlation and classification methods were used to evaluate relationships between ambient toxicity and instream biological response. Ten studies of freshwater aquatic systems were used in the analysis. These studies included eight site studies conducted by the U.S. Environmental Protection Agency's Complex Effluent Toxicity Testing Program, one study by the University of Kentucky and one study by the University of North Texas. Results from ambient toxicity tests, including Ceriodaphnia neonate production and Pimephales promelas dry weight and survival, were used to predict impact at sampling stations. Instream biological response variables, including fish and benthic richness, were used to classify whether or not a station was impacted. The strength of the relationship between ambient toxicity and instream impact was determined by statistically evaluating agreement between the predicted and observed impacts. Results of the analysis showed that statistically significant relationships between ambient toxicity and instream impact were found in each of the studies.

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Robust Singular Value Decompositions: A New Approach to Projection Pursuit

Cited by 27 publications

References 0 publications

High breakdown estimators for principal components: the projection-pursuit approach revisited

High breakdown estimators for principal components: the projection-pursuit approach revisited

Inferences Based on a Skipped Correlation Coefficient

Assessing the relationship between ambient toxicity and instream biological response

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