1,598 research outputs found
Central Limit Theorems for Wavelet Packet Decompositions of Stationary Random Processes
This paper provides central limit theorems for the wavelet packet
decomposition of stationary band-limited random processes. The asymptotic
analysis is performed for the sequences of the wavelet packet coefficients
returned at the nodes of any given path of the -band wavelet packet
decomposition tree. It is shown that if the input process is centred and
strictly stationary, these sequences converge in distribution to white Gaussian
processes when the resolution level increases, provided that the decomposition
filters satisfy a suitable property of regularity. For any given path, the
variance of the limit white Gaussian process directly relates to the value of
the input process power spectral density at a specific frequency.Comment: Submitted to the IEEE Transactions on Signal Processing, October 200
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
Extreme Value Analysis of Empirical Frame Coefficients and Implications for Denoising by Soft-Thresholding
Denoising by frame thresholding is one of the most basic and efficient
methods for recovering a discrete signal or image from data that are corrupted
by additive Gaussian white noise. The basic idea is to select a frame of
analyzing elements that separates the data in few large coefficients due to the
signal and many small coefficients mainly due to the noise \epsilon_n. Removing
all data coefficients being in magnitude below a certain threshold yields a
reconstruction of the original signal. In order to properly balance the amount
of noise to be removed and the relevant signal features to be kept, a precise
understanding of the statistical properties of thresholding is important. For
that purpose we derive the asymptotic distribution of max_{\omega \in \Omega_n}
|| for a wide class of redundant frames
(\phi_\omega^n: \omega \in \Omega_n}. Based on our theoretical results we give
a rationale for universal extreme value thresholding techniques yielding
asymptotically sharp confidence regions and smoothness estimates corresponding
to prescribed significance levels. The results cover many frames used in
imaging and signal recovery applications, such as redundant wavelet systems,
curvelet frames, or unions of bases. We show that `generically' a standard
Gumbel law results as it is known from the case of orthonormal wavelet bases.
However, for specific highly redundant frames other limiting laws may occur. We
indeed verify that the translation invariant wavelet transform shows a
different asymptotic behaviour.Comment: [Content: 39 pages, 4 figures] Note that in this version 4 we have
slightely changed the title of the paper and we have rewritten parts of the
introduction. Except for corrected typos the other parts of the paper are the
same as the original versions
Constructing a quasilinear moving average using the scaling function
The scaling function from multiresolution analysis can be used to constuct a smoothing tool in the context of time series analysis. We give a time series smoothing function for which we show the properties of a quasilinear moving average. Furthermore; we discuss its features and especially derive the distributional properties of our quasilinear moving average given some simple underlying stochastic processes. Eventually we compare it to existing smoothing methods in order to motivate its application --Scaling function,Quasilinear moving average,Influence function
A Statistical Study of Wavelet Coherence for Stationary and Nonstationary Processes
The coherence function measures the correlation between a pair of random
processes in the frequency domain. It is a well studied and understood concept,
and the distributional properties of conventional coherence estimators for
stationary processes have been derived and applied in a number of physical
settings.
In recent years the wavelet coherence measure has been used to analyse
correlations between a pair of processes in the time-scale domain, typically in
hypothesis testing scenarios, but it has proven resistant to analytic study with
resort to simulations for statistical properties. As part of the null hypothesis
being tested, such simulations invariably assume joint stationarity of the
series. In this thesis two methods of calculating wavelet coherence have been
developed and distributional properties of the wavelet coherence estimators
have been fully derived.
With the first method, in an analogous framework to multitapering, wavelet
coherence is estimated using multiple orthogonal Morse wavelets. The second
coherence estimator proposed uses time-domain smoothing and a single Morlet
wavelet. Since both sets of wavelets are complex-valued, we consider the case of
wavelet coherence calculated from discrete-time complex-valued and stationary
time series. Under Gaussianity, the Goodman distribution is shown, for large
samples, to be appropriate for wavelet coherence. The true wavelet coherence
value is identified in terms of its frequency domain equivalent and degrees
of freedom can be readily derived. The theoretical results are verified via
simulations.
The notion of a spectral function is considered for the nonstationary case.
Particular focus is given to Priestley’s evolutionary process and a Wold-Cramér
nonstationary representation where time-varying spectral functions can be
clearly defined. Methods of estimating these spectra are discussed, including
the continuous wavelet transform, which when performed with a Morlet
wavelet and temporal smoothing is shown to bear close resemblance to Priestley’s
own estimation procedure.
The concept of coherence for bivariate evolutionary nonstationary processes
is discussed in detail. In such situations it can be shown that the coherence
function, as in the stationary case, is invariant of time. It is shown that
for spectra that vary slowly in time the derived statistics of the temporally
smoothed wavelet coherence estimator are appropriate. Further to this the
similarities with Priestleys spectral estimator are exploited to derive distributional
properties of the corresponding Priestley coherence estimator.
A well known class of the evolutionary and Wold-Cramér nonstationary
processes are the modulated stationary processes. Using these it is shown that
bivariate processes can be constructed that exhibit coherence variation with
time, frequency, and time-and-frequency. The temporally smoothed Morlet
wavelet coherence estimator is applied to these processes. It is shown that
accurate coherence estimates can be achieved for each type of coherence, and
that the distributional properties derived under stationarity are applicable
On a class of self-similar processes with stationary increments in higher order Wiener chaoses
We study a class of self-similar processes with stationary increments
belonging to higher order Wiener chaoses which are similar to Hermite
processes. We obtain an almost sure wavelet-like expansion of these processes.
This allows us to compute the pointwise and local H\"older regularity of sample
paths and to analyse their behaviour at infinity. We also provide some results
on the Hausdorff dimension of the range and graphs of multidimensional
anisotropic self-similar processes with stationary increments defined by
multiple Wiener integrals.Comment: 22 page
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