70 research outputs found
Functional deconvolution in a periodic setting: Uniform case
We extend deconvolution in a periodic setting to deal with functional data.
The resulting functional deconvolution model can be viewed as a generalization
of a multitude of inverse problems in mathematical physics where one needs to
recover initial or boundary conditions on the basis of observations from a
noisy solution of a partial differential equation. In the case when it is
observed at a finite number of distinct points, the proposed functional
deconvolution model can also be viewed as a multichannel deconvolution model.
We derive minimax lower bounds for the -risk in the proposed functional
deconvolution model when is assumed to belong to a Besov ball and
the blurring function is assumed to possess some smoothness properties,
including both regular-smooth and super-smooth convolutions. Furthermore, we
propose an adaptive wavelet estimator of that is asymptotically
optimal (in the minimax sense), or near-optimal within a logarithmic factor, in
a wide range of Besov balls. In addition, we consider a discretization of the
proposed functional deconvolution model and investigate when the availability
of continuous data gives advantages over observations at the asymptotically
large number of points. As an illustration, we discuss particular examples for
both continuous and discrete settings.Comment: Published in at http://dx.doi.org/10.1214/07-AOS552 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Short-Term Load Forecasting: The Similar Shape Functional Time Series Predictor
We introduce a novel functional time series methodology for short-term load
forecasting. The prediction is performed by means of a weighted average of past
daily load segments, the shape of which is similar to the expected shape of the
load segment to be predicted. The past load segments are identified from the
available history of the observed load segments by means of their closeness to
a so-called reference load segment, the later being selected in a manner that
captures the expected qualitative and quantitative characteristics of the load
segment to be predicted. Weak consistency of the suggested functional similar
shape predictor is established. As an illustration, we apply the suggested
functional time series forecasting methodology to historical daily load data in
Cyprus and compare its performance to that of a recently proposed alternative
functional time series methodology for short-term load forecasting.Comment: 22 pages, 6 Figures, 1 Tabl
On convergence rates equivalency and sampling strategies in functional deconvolution models
Using the asymptotical minimax framework, we examine convergence rates
equivalency between a continuous functional deconvolution model and its
real-life discrete counterpart over a wide range of Besov balls and for the
-risk. For this purpose, all possible models are divided into three
groups. For the models in the first group, which we call uniform, the
convergence rates in the discrete and the continuous models coincide no matter
what the sampling scheme is chosen, and hence the replacement of the discrete
model by its continuous counterpart is legitimate. For the models in the second
group, to which we refer as regular, one can point out the best sampling
strategy in the discrete model, but not every sampling scheme leads to the same
convergence rates; there are at least two sampling schemes which deliver
different convergence rates in the discrete model (i.e., at least one of the
discrete models leads to convergence rates that are different from the
convergence rates in the continuous model). The third group consists of models
for which, in general, it is impossible to devise the best sampling strategy;
we call these models irregular. We formulate the conditions when each of these
situations takes place. In the regular case, we not only point out the number
and the selection of sampling points which deliver the fastest convergence
rates in the discrete model but also investigate when, in the case of an
arbitrary sampling scheme, the convergence rates in the continuous model
coincide or do not coincide with the convergence rates in the discrete model.
We also study what happens if one chooses a uniform, or a more general
pseudo-uniform, sampling scheme which can be viewed as an intuitive replacement
of the continuous model.Comment: Published in at http://dx.doi.org/10.1214/09-AOS767 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Wavelet Estimators in Nonparametric Regression: A Comparative Simulation Study
Wavelet analysis has been found to be a powerful tool for the nonparametric estimation of spatially-variable objects. We discuss in detail wavelet methods in nonparametric regression, where the data are modelled as observations of a signal contaminated with additive Gaussian noise, and provide an extensive review of the vast literature of wavelet shrinkage and wavelet thresholding estimators developed to denoise such data. These estimators arise from a wide range of classical and empirical Bayes methods treating either individual or blocks of wavelet coefficients. We compare various estimators in an extensive simulation study on a variety of sample sizes, test functions, signal-to-noise ratios and wavelet filters. Because there is no single criterion that can adequately summarise the behaviour of an estimator, we use various criteria to measure performance in finite sample situations. Insight into the performance of these estimators is obtained from graphical outputs and numerical tables. In order to provide some hints of how these estimators should be used to analyse real data sets, a detailed practical step-by-step illustration of a wavelet denoising analysis on electrical consumption is provided. Matlab codes are provided so that all figures and tables in this paper can be reproduced
A Functional Wavelet-Kernel Approach for Continuous-time Prediction
We consider the prediction problem of a continuous-time stochastic process on
an entire time-interval in terms of its recent past. The approach we adopt is
based on functional kernel nonparametric regression estimation techniques where
observations are segments of the observed process considered as curves. These
curves are assumed to lie within a space of possibly inhomogeneous functions,
and the discretized times series dataset consists of a relatively small,
compared to the number of segments, number of measurements made at regular
times. We thus consider only the case where an asymptotically non-increasing
number of measurements is available for each portion of the times series. We
estimate conditional expectations using appropriate wavelet decompositions of
the segmented sample paths. A notion of similarity, based on wavelet
decompositions, is used in order to calibrate the prediction. Asymptotic
properties when the number of segments grows to infinity are investigated under
mild conditions, and a nonparametric resampling procedure is used to generate,
in a flexible way, valid asymptotic pointwise confidence intervals for the
predicted trajectories. We illustrate the usefulness of the proposed functional
wavelet-kernel methodology in finite sample situations by means of three
real-life datasets that were collected from different arenas
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