4,068 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
Fourier analysis of stationary time series in function space
We develop the basic building blocks of a frequency domain framework for
drawing statistical inferences on the second-order structure of a stationary
sequence of functional data. The key element in such a context is the spectral
density operator, which generalises the notion of a spectral density matrix to
the functional setting, and characterises the second-order dynamics of the
process. Our main tool is the functional Discrete Fourier Transform (fDFT). We
derive an asymptotic Gaussian representation of the fDFT, thus allowing the
transformation of the original collection of dependent random functions into a
collection of approximately independent complex-valued Gaussian random
functions. Our results are then employed in order to construct estimators of
the spectral density operator based on smoothed versions of the periodogram
kernel, the functional generalisation of the periodogram matrix. The
consistency and asymptotic law of these estimators are studied in detail. As
immediate consequences, we obtain central limit theorems for the mean and the
long-run covariance operator of a stationary functional time series. Our
results do not depend on structural modelling assumptions, but only functional
versions of classical cumulant mixing conditions, and are shown to be stable
under discrete observation of the individual curves.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1086 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On the prediction of stationary functional time series
This paper addresses the prediction of stationary functional time series.
Existing contributions to this problem have largely focused on the special case
of first-order functional autoregressive processes because of their technical
tractability and the current lack of advanced functional time series
methodology. It is shown here how standard multivariate prediction techniques
can be utilized in this context. The connection between functional and
multivariate predictions is made precise for the important case of vector and
functional autoregressions. The proposed method is easy to implement, making
use of existing statistical software packages, and may therefore be attractive
to a broader, possibly non-academic, audience. Its practical applicability is
enhanced through the introduction of a novel functional final prediction error
model selection criterion that allows for an automatic determination of the lag
structure and the dimensionality of the model. The usefulness of the proposed
methodology is demonstrated in a simulation study and an application to
environmental data, namely the prediction of daily pollution curves describing
the concentration of particulate matter in ambient air. It is found that the
proposed prediction method often significantly outperforms existing methods
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