4,437 research outputs found
Locally Adaptive Tree-based Thresholding Using the treethresh Package in R
This paper introduces the treethresh package offering accurate estimation, via thresholding, of potentially sparse heterogeneous signals and the denoising of images using wavelets. It gives considerably improved performance over other estimation methods if the underlying signal or image is not homogeneous throughout but instead has distinct regions with differing sparsity or strength characteristics. It aims to identify these different regions and perform separate estimation in each accordingly. The base algorithm offers code which can be applied directly to any one-dimensional potentially sparse sequence observed subject to noise. Also included are functions which allow two-dimensional images to be denoised following transformation to the wavelet domain. In addition to reconstructing the underlying signal or image, the package provides information on the believed partitioning of the signal or image into its differing regions
Skellam shrinkage: Wavelet-based intensity estimation for inhomogeneous Poisson data
The ubiquity of integrating detectors in imaging and other applications
implies that a variety of real-world data are well modeled as Poisson random
variables whose means are in turn proportional to an underlying vector-valued
signal of interest. In this article, we first show how the so-called Skellam
distribution arises from the fact that Haar wavelet and filterbank transform
coefficients corresponding to measurements of this type are distributed as sums
and differences of Poisson counts. We then provide two main theorems on Skellam
shrinkage, one showing the near-optimality of shrinkage in the Bayesian setting
and the other providing for unbiased risk estimation in a frequentist context.
These results serve to yield new estimators in the Haar transform domain,
including an unbiased risk estimate for shrinkage of Haar-Fisz
variance-stabilized data, along with accompanying low-complexity algorithms for
inference. We conclude with a simulation study demonstrating the efficacy of
our Skellam shrinkage estimators both for the standard univariate wavelet test
functions as well as a variety of test images taken from the image processing
literature, confirming that they offer substantial performance improvements
over existing alternatives.Comment: 27 pages, 8 figures, slight formatting changes; submitted for
publicatio
Spatially-adaptive sensing in nonparametric regression
While adaptive sensing has provided improved rates of convergence in sparse
regression and classification, results in nonparametric regression have so far
been restricted to quite specific classes of functions. In this paper, we
describe an adaptive-sensing algorithm which is applicable to general
nonparametric-regression problems. The algorithm is spatially adaptive, and
achieves improved rates of convergence over spatially inhomogeneous functions.
Over standard function classes, it likewise retains the spatial adaptivity
properties of a uniform design
Sparse modelling and estimation for nonstationary time series and high-dimensional data
Sparse modelling has attracted great attention as an efficient way of
handling statistical problems in high dimensions. This thesis considers
sparse modelling and estimation in a selection of problems such
as breakpoint detection in nonstationary time series, nonparametric
regression using piecewise constant functions and variable selection in
high-dimensional linear regression.
We first propose a method for detecting breakpoints in the secondorder
structure of piecewise stationary time series, assuming that
those structural breakpoints are sufficiently scattered over time. Our
choice of time series model is the locally stationary wavelet process
(Nason et al., 2000), under which the entire second-order structure of a
time series is described by wavelet-based local periodogram sequences.
As the initial stage of breakpoint detection, we apply a binary segmentation
procedure to wavelet periodogram sequences at each scale
separately, which is followed by within-scale and across-scales postprocessing
steps. We show that the combined methodology achieves
consistent estimation of the breakpoints in terms of their total number
and locations, and investigate its practical performance using both
simulated and real data.
Next, we study the problem of nonparametric regression by means of
piecewise constant functions, which are known to be flexible in approximating
a wide range of function spaces. Among many approaches developed
for this purpose, we focus on comparing two well-performing
techniques, the taut string (Davies & Kovac, 2001) and the Unbalanced
Haar (Fryzlewicz, 2007) methods. While the multiscale nature
of the latter is easily observed, it is not so obvious that the former
can also be interpreted as multiscale. We provide a unified, multiscale
representation for both methods, which offers an insight into the relationship
between them as well as suggesting some lessons that both
methods can learn from each other.
Lastly, one of the most widely-studied applications of sparse modelling
and estimation is considered, variable selection in high-dimensional
linear regression. High dimensionality of the data brings in many
complications including (possibly spurious) non-negligible correlations
among the variables, which may result in marginal correlation being
unreliable as a measure of association between the variables and the
response. We propose a new way of measuring the contribution of
each variable to the response, which adaptively takes into account
high correlations among the variables. A key ingredient of the proposed
tilting procedure is hard-thresholding sample correlation of the
design matrix, which enables a data-driven switch between the use of
marginal correlation and tilted correlation for each variable. We study
the conditions under which this measure can discriminate between relevant
and irrelevant variables, and thus be used as a tool for variable
selection. In order to exploit these theoretical properties of tilted correlation,
we construct an iterative variable screening algorithm and
examine its practical performance in a comparative simulation study
Finding the optimal temporal partitioning of video sequences
The existing techniques for shot partitioning either process each shot boundary independently or proceed sequentially. The sequential process assumes the last shot boundary is correctly detected and utilizes the shot length distribution to adapt the threshold for detecting the next boundary. These techniques are only locally optimal and suffer from the strong assumption about the correct detection of the last boundary. Addressing these fundamental issues, in this paper, we aim to find the global optimal shot partitioning by utilizing Bayesian principles to model the probability of a particular video partition being the shot partition. A computationally efficient algorithm based on Dynamic Programming is then formulated. The experimental results on a large movie set show that our algorithm performs consistently better than the best adaptive-thresholding technique commonly used for the task
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