The paper presents a unified approach to local likelihood estimation
for a broad class of nonparametric models, including e.g. the regression,
density, Poisson and binary response model. The method extends
the adaptive weights smoothing (AWS) procedure introduced in Polzehl
and Spokoiny (2000) in context of image denoising. Performance of the
proposed procedure is illustrated by a number of numerical examples
and applications to density or volatility estimation, classification and
estimation of the tail index parameter. We also establish a number of
important theoretical results on properties of the proposed procedure.
The adaptive weights smoothing (AWS) procedure was introduced in
Polzehl and Spokoiny (2000) in the context of image denoising. The
procedure has some remarkable properties like preservation of edges and
contrast, and (in some sense) optimal reduction of noise. The procedure
is fully adaptive and dimension free. Simulations with artificial images
show that AWS is superior to classical smoothing techniques especially
when the underlying image function is discontinuous and can be well
approximated by a piecewise constant function. However, the latter as-
sumption can be rather restrictive for a number of potential applications.
Here the AWS method is generalized to the case of an arbitrary local lin-
ear parametric structure. We also establish some important results about
properties of the AWS procedure including the so called "propagation
condition" and spatial adaptivity. The performance of the procedure is
illustrated by examples for local polynomial regression in univariate and
bivariate situations.
Data from functional magnetic resonance imaging (fMRI) consists of time series of brain images which are characterized
by a high noise level and a low signal-to-noise ratio. We provide a complete procedure for fMRI analysis. In order to reduce noise and to improve signal detection the fMRI data is spatially smoothed. However, the common application of a Gaussian filter does this at the cost of loss of information on spatial extend and shape of the activation area. We suggest to use the propagation-separation procedures introduced by Polzehl and Spokoiny (2005) instead. We show that this significantly improves the information on the spatial extend and shape of the activation region with similar results for the noise reduction. Signal detection is based on locally varying thresholds defined by random field theory. Effects of adaptive and non adaptive smoothing are illustrated by artificial examples and an analysis of real data.
Polzehl and Spokoiny (2000) introduced the adaptive weights smoothing
(AWS) procedure in the context of image denoising. The procedure
has some remarkable properties like preservation of edges and contrast,
and (in some sense) optimal reduction of noise. The procedure is fully
adaptive and dimension free. Simulations with artificial images show
that AWS is superior to classical smoothing techniques especially when
the underlying image function is discontinuous and can be well approximated
by a piecewise constant function. However, the latter assumption
can be rather restrictive for a number of potential applications. Here we
present a new method based on the ideas of propagation and separation
which extends the AWS procedure to the case of an arbitrary local linear
parametric structure. We also establish some important results about
properties of the new ‘propagation-separation’ procedure including rate
optimality in the pointwise and global sense. The performance of the
procedure is illustrated by examples for local polynomial regression and
by applications to artificial and real images.
In this paper we carry over the concept of reverse probabilistic representa-
tions developed in Milstein, Schoenmakers, Spokoiny (2004) for diffusion pro-
cesses, to discrete time Markov chains. We outline the construction of reverse
chains in several situations and apply this to processes which are connected
with jump-diffusion models and finite state Markov chains. By combining
forward and reverse representations we then construct transition density esti-
mators for chains which have root-N accuracy in any dimension and consider
some applications.
Diffusion Tensor Imaging (DTI) data is characterized by a high noise level. Thus,
estimation errors of quantities like anisotropy indices or the main diffusion direction
used for fiber tracking are relatively large and may significantly confound the accuracy
of DTI in clinical or neuroscience applications. Besides pulse sequence optimization,
noise reduction by smoothing the data can be pursued as a complementary approach
to increase the accuracy of DTI. Here, we suggest an anisotropic structural adaptive
smoothing procedure, which is based on the Propagation-Separation method and preserves
the structures seen in DTI and their different sizes and shapes. It is applied
to artificial phantom data and a brain scan. We show that this method significantly
improves the quality of the estimate of the diffusion tensor and hence enables one
either to reduce the number of scans or to enhance the input for subsequent analysis
such as fiber tracking.