Assessing the finite dimensionality of functional data
Identifieur interne : 009A41 ( Main/Exploration ); précédent : 009A40; suivant : 009A42Assessing the finite dimensionality of functional data
Auteurs : Peter Hall [Australie] ; Céline Vial [France]Source :
- Journal of the Royal Statistical Society: Series B (Statistical Methodology) [ 1369-7412 ] ; 2006-09.
Descripteurs français
- Pascal (Inist)
- Analyse donnée, Analyse fonctionnelle, Analyse multivariable, Bootstrap, Borne inférieure, Confusion, Degré liberté, Donnée observation, Erreur arrondi, Estimation erreur, Estimation variance, Faible bruit, Fiabilité, Fonction répartition, Intervalle confiance, Méthode statistique, Point critique, Test hypothèse, Valeur propre, Variance bruit, Vecteur propre.
- Wicri :
- topic : Méthode statistique.
English descriptors
- KwdEn :
- Actual number, Asymptotic theory, Average noise variance, Average value, Average variance, Bootstrap, Bootstrap distribution, Bootstrap iterations, Bootstrap methods, Bootstrap simulations, Complete orthonormal sequence, Components analysis, Confidence interval, Confounding, Conventional techniques, Converges, Covariance, Covariance function, Critical point, Data anal, Data analysis, Dimensionality, Distribution function, Eigenvalue, Eigenvector, Error estimation, Error process, Experimental error, Finite dimensionality, Freedom degree, Functional analysis, Functional data, Functional data analysis, Hypothesis test, Hypothesis testing, Linear operator, Low noise, Lower bound, Methods part, Multivariate analysis, Nite, Nite dimensionality, Nite number, Noise settings, Noise variance, Null hypothesis, Observation data, Orthonormal, Possible values, Principal components, Ramsay, Random function, Random functions, Random variables, Real data example, Reliability, Rounding error, Sample replications, Sample size, Satisfactory results, Silverman, Statist, Statistical method, Such cases, Such values, True dimension, Unconfounded, Unconfounded noise variance, Unconfounded part, Variance, Variance estimation, Vial, Vial table.
- Teeft :
- Actual number, Asymptotic theory, Average noise variance, Average value, Average variance, Bootstrap, Bootstrap distribution, Bootstrap iterations, Bootstrap methods, Bootstrap simulations, Complete orthonormal sequence, Components analysis, Conventional techniques, Converges, Covariance, Covariance function, Data anal, Dimensionality, Eigenvalue, Error process, Experimental error, Finite dimensionality, Functional data, Functional data analysis, Hypothesis testing, Linear operator, Methods part, Multivariate analysis, Nite, Nite dimensionality, Nite number, Noise settings, Noise variance, Null hypothesis, Orthonormal, Possible values, Principal components, Ramsay, Random function, Random functions, Random variables, Real data example, Sample replications, Sample size, Satisfactory results, Silverman, Statist, Such cases, Such values, True dimension, Unconfounded, Unconfounded noise variance, Unconfounded part, Variance, Vial, Vial table.
Abstract
Summary. If a problem in functional data analysis is low dimensional then the methodology for its solution can often be reduced to relatively conventional techniques in multivariate analysis. Hence, there is intrinsic interest in assessing the finite dimensionality of functional data. We show that this problem has several unique features. From some viewpoints the problem is trivial, in the sense that continuously distributed functional data which are exactly finite dimensional are immediately recognizable as such, if the sample size is sufficiently large. However, in practice, functional data are almost always observed with noise, for example, resulting from rounding or experimental error. Then the problem is almost insolubly difficult. In such cases a part of the average noise variance is confounded with the true signal and is not identifiable. However, it is possible to define the unconfounded part of the noise variance. This represents the best possible lower bound to all potential values of average noise variance and is estimable in low noise settings. Moreover, bootstrap methods can be used to describe the reliability of estimates of unconfounded noise variance, under the assumption that the signal is finite dimensional. Motivated by these ideas, we suggest techniques for assessing the finiteness of dimensionality. In particular, we show how to construct a critical point such that, if the distribution of our functional data has fewer than q−1 degrees of freedom, then we should be willing to assume that the average variance of the added noise is at least . If this level seems too high then we must conclude that the dimension is at least q−1. We show that simpler, more conventional techniques, based on hypothesis testing, are generally not effective.
Url:
DOI: 10.1111/j.1467-9868.2006.00562.x
Affiliations:
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<front><div type="abstract">Summary. If a problem in functional data analysis is low dimensional then the methodology for its solution can often be reduced to relatively conventional techniques in multivariate analysis. Hence, there is intrinsic interest in assessing the finite dimensionality of functional data. We show that this problem has several unique features. From some viewpoints the problem is trivial, in the sense that continuously distributed functional data which are exactly finite dimensional are immediately recognizable as such, if the sample size is sufficiently large. However, in practice, functional data are almost always observed with noise, for example, resulting from rounding or experimental error. Then the problem is almost insolubly difficult. In such cases a part of the average noise variance is confounded with the true signal and is not identifiable. However, it is possible to define the unconfounded part of the noise variance. This represents the best possible lower bound to all potential values of average noise variance and is estimable in low noise settings. Moreover, bootstrap methods can be used to describe the reliability of estimates of unconfounded noise variance, under the assumption that the signal is finite dimensional. Motivated by these ideas, we suggest techniques for assessing the finiteness of dimensionality. In particular, we show how to construct a critical point such that, if the distribution of our functional data has fewer than q−1 degrees of freedom, then we should be willing to assume that the average variance of the added noise is at least . If this level seems too high then we must conclude that the dimension is at least q−1. We show that simpler, more conventional techniques, based on hypothesis testing, are generally not effective.</div>
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