A tilting approach to ranking influence

Handle URI:
http://hdl.handle.net/10754/552392
Title:
A tilting approach to ranking influence
Authors:
Genton, Marc G. ( 0000-0001-6467-2998 ) ; Hall, Peter
Abstract:
We suggest a new approach, which is applicable for general statistics computed from random samples of univariate or vector-valued or functional data, to assessing the influence that individual data have on the value of a statistic, and to ranking the data in terms of that influence. Our method is based on, first, perturbing the value of the statistic by ‘tilting’, or reweighting, each data value, where the total amount of tilt is constrained to be the least possible, subject to achieving a given small perturbation of the statistic, and, then, taking the ranking of the influence of data values to be that which corresponds to ranking the changes in data weights. It is shown, both theoretically and numerically, that this ranking does not depend on the size of the perturbation, provided that the perturbation is sufficiently small. That simple result leads directly to an elegant geometric interpretation of the ranks; they are the ranks of the lengths of projections of the weights onto a ‘line’ determined by the first empirical principal component function in a generalized measure of covariance. To illustrate the generality of the method we introduce and explore it in the case of functional data, where (for example) it leads to generalized boxplots. The method has the advantage of providing an interpretable ranking that depends on the statistic under consideration. For example, the ranking of data, in terms of their influence on the value of a statistic, is different for a measure of location and for a measure of scale. This is as it should be; a ranking of data in terms of their influence should depend on the manner in which the data are used. Additionally, the ranking recognizes, rather than ignores, sign, and in particular can identify left- and right-hand ‘tails’ of the distribution of a random function or vector.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
A tilting approach to ranking influence 2014:n/a Journal of the Royal Statistical Society: Series B (Statistical Methodology)
Publisher:
Wiley-Blackwell
Journal:
Journal of the Royal Statistical Society: Series B (Statistical Methodology)
Issue Date:
Dec-2014
DOI:
10.1111/rssb.12102
Type:
Article
ISSN:
13697412
Additional Links:
http://doi.wiley.com/10.1111/rssb.12102
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorGenton, Marc G.en
dc.contributor.authorHall, Peteren
dc.date.accessioned2015-05-06T13:32:43Zen
dc.date.available2015-05-06T13:32:43Zen
dc.date.issued2014-12en
dc.identifier.citationA tilting approach to ranking influence 2014:n/a Journal of the Royal Statistical Society: Series B (Statistical Methodology)en
dc.identifier.issn13697412en
dc.identifier.doi10.1111/rssb.12102en
dc.identifier.urihttp://hdl.handle.net/10754/552392en
dc.description.abstractWe suggest a new approach, which is applicable for general statistics computed from random samples of univariate or vector-valued or functional data, to assessing the influence that individual data have on the value of a statistic, and to ranking the data in terms of that influence. Our method is based on, first, perturbing the value of the statistic by ‘tilting’, or reweighting, each data value, where the total amount of tilt is constrained to be the least possible, subject to achieving a given small perturbation of the statistic, and, then, taking the ranking of the influence of data values to be that which corresponds to ranking the changes in data weights. It is shown, both theoretically and numerically, that this ranking does not depend on the size of the perturbation, provided that the perturbation is sufficiently small. That simple result leads directly to an elegant geometric interpretation of the ranks; they are the ranks of the lengths of projections of the weights onto a ‘line’ determined by the first empirical principal component function in a generalized measure of covariance. To illustrate the generality of the method we introduce and explore it in the case of functional data, where (for example) it leads to generalized boxplots. The method has the advantage of providing an interpretable ranking that depends on the statistic under consideration. For example, the ranking of data, in terms of their influence on the value of a statistic, is different for a measure of location and for a measure of scale. This is as it should be; a ranking of data in terms of their influence should depend on the manner in which the data are used. Additionally, the ranking recognizes, rather than ignores, sign, and in particular can identify left- and right-hand ‘tails’ of the distribution of a random function or vector.en
dc.publisherWiley-Blackwellen
dc.relation.urlhttp://doi.wiley.com/10.1111/rssb.12102en
dc.rightsThis is the peer reviewed version of the following article: Genton, M. G. and Hall, P. (2014), A tilting approach to ranking influence. Journal of the Royal Statistical Society: Series B (Statistical Methodology). doi: 10.1111/rssb.12102, which has been published in final form at http://doi.wiley.com/10.1111/rssb.12102. This article may be used for non-commercial purposes in accordance With Wiley Terms and Conditions for self-archiving.en
dc.subjectBand depthen
dc.subjectData weightsen
dc.subjectFunctional boxploten
dc.subjectFunctional dataen
dc.subjectImage dataen
dc.subjectOutlieren
dc.subjectRobustnessen
dc.titleA tilting approach to ranking influenceen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalJournal of the Royal Statistical Society: Series B (Statistical Methodology)en
dc.eprint.versionPost-printen
dc.contributor.institutionUniversity of Melbourne; Australiaen
dc.contributor.institutionUniversity of California at Davis, USAen
kaust.authorGenton, Marc G.en
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