36,822 research outputs found
Weakly dependent functional data
Functional data often arise from measurements on fine time grids and are
obtained by separating an almost continuous time record into natural
consecutive intervals, for example, days. The functions thus obtained form a
functional time series, and the central issue in the analysis of such data
consists in taking into account the temporal dependence of these functional
observations. Examples include daily curves of financial transaction data and
daily patterns of geophysical and environmental data. For scalar and vector
valued stochastic processes, a large number of dependence notions have been
proposed, mostly involving mixing type distances between -algebras. In
time series analysis, measures of dependence based on moments have proven most
useful (autocovariances and cumulants). We introduce a moment-based notion of
dependence for functional time series which involves -dependence. We show
that it is applicable to linear as well as nonlinear functional time series.
Then we investigate the impact of dependence thus quantified on several
important statistical procedures for functional data. We study the estimation
of the functional principal components, the long-run covariance matrix, change
point detection and the functional linear model. We explain when temporal
dependence affects the results obtained for i.i.d. functional observations and
when these results are robust to weak dependence.Comment: Published in at http://dx.doi.org/10.1214/09-AOS768 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Feature Selection for Functional Data
In this paper we address the problem of feature selection when the data is
functional, we study several statistical procedures including classification,
regression and principal components. One advantage of the blinding procedure is
that it is very flexible since the features are defined by a set of functions,
relevant to the problem being studied, proposed by the user. Our method is
consistent under a set of quite general assumptions, and produces good results
with the real data examples that we analyze.Comment: 22 pages, 4 figure
Statistical inferences for functional data
With modern technology development, functional data are being observed
frequently in many scientific fields. A popular method for analyzing such
functional data is ``smoothing first, then estimation.'' That is, statistical
inference such as estimation and hypothesis testing about functional data is
conducted based on the substitution of the underlying individual functions by
their reconstructions obtained by one smoothing technique or another. However,
little is known about this substitution effect on functional data analysis. In
this paper this problem is investigated when the local polynomial kernel (LPK)
smoothing technique is used for individual function reconstructions. We find
that under some mild conditions, the substitution effect can be ignored
asymptotically. Based on this, we construct LPK reconstruction-based estimators
for the mean, covariance and noise variance functions of a functional data set
and derive their asymptotics. We also propose a GCV rule for selecting good
bandwidths for the LPK reconstructions. When the mean function also depends on
some time-independent covariates, we consider a functional linear model where
the mean function is linearly related to the covariates but the covariate
effects are functions of time. The LPK reconstruction-based estimators for the
covariate effects and the covariance function are also constructed and their
asymptotics are derived. Moreover, we propose a -norm-based global test
statistic for a general hypothesis testing problem about the covariate effects
and derive its asymptotic random expression. The effect of the bandwidths
selected by the proposed GCV rule on the accuracy of the LPK reconstructions
and the mean function estimator is investigated via a simulation study. The
proposed methodologies are illustrated via an application to a real functional
data set collected in climatology.Comment: Published at http://dx.doi.org/10.1214/009053606000001505 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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