182 research outputs found
Identifiability for Blind Source Separation of Multiple Finite Alphabet Linear Mixtures
We give under weak assumptions a complete combinatorial characterization of
identifiability for linear mixtures of finite alphabet sources, with unknown
mixing weights and unknown source signals, but known alphabet. This is based on
a detailed treatment of the case of a single linear mixture. Notably, our
identifiability analysis applies also to the case of unknown number of sources.
We provide sufficient and necessary conditions for identifiability and give a
simple sufficient criterion together with an explicit construction to determine
the weights and the source signals for deterministic data by taking advantage
of the hierarchical structure within the possible mixture values. We show that
the probability of identifiability is related to the distribution of a hitting
time and converges exponentially fast to one when the underlying sources come
from a discrete Markov process. Finally, we explore our theoretical results in
a simulation study. Our work extends and clarifies the scope of scenarios for
which blind source separation becomes meaningful
Autocovariance estimation in regression with a discontinuous signal and -dependent errors: A difference-based approach
We discuss a class of difference-based estimators for the autocovariance in
nonparametric regression when the signal is discontinuous (change-point
regression), possibly highly fluctuating, and the errors form a stationary
-dependent process. These estimators circumvent the explicit pre-estimation
of the unknown regression function, a task which is particularly challenging
for such signals. We provide explicit expressions for their mean squared errors
when the signal function is piecewise constant (segment regression) and the
errors are Gaussian. Based on this we derive biased-optimized estimates which
do not depend on the particular (unknown) autocovariance structure. Notably,
for positively correlated errors, that part of the variance of our estimators
which depends on the signal is minimal as well. Further, we provide sufficient
conditions for -consistency; this result is extended to piecewise
Holder regression with non-Gaussian errors.
We combine our biased-optimized autocovariance estimates with a
projection-based approach and derive covariance matrix estimates, a method
which is of independent interest. Several simulation studies as well as an
application to biophysical measurements complement this paper.Comment: 41 pages, 3 figures, 3 table
Lower bounds for volatility estimation in microstructure noise models
In this paper we derive lower bounds in minimax sense for estimation of the
instantaneous volatility if the diffusion type part cannot be observed directly
but under some additional Gaussian noise. Three different models are
considered. Our technique is based on a general inequality for Kullback-Leibler
divergence of multivariate normal random variables and spectral analysis of the
processes. The derived lower bounds are indeed optimal. Upper bounds can be
found in Munk and Schmidt-Hieber [18]. Our major finding is that the Gaussian
microstructure noise introduces an additional degree of ill-posedness for each
model, respectively.Comment: 16 page
Nonparametric estimation of the volatility function in a high-frequency model corrupted by noise
We consider the models Y_{i,n}=\int_0^{i/n}
\sigma(s)dW_s+\tau(i/n)\epsilon_{i,n}, and \tilde
Y_{i,n}=\sigma(i/n)W_{i/n}+\tau(i/n)\epsilon_{i,n}, i=1,...,n, where W_t
denotes a standard Brownian motion and \epsilon_{i,n} are centered i.i.d.
random variables with E(\epsilon_{i,n}^2)=1 and finite fourth moment.
Furthermore, \sigma and \tau are unknown deterministic functions and W_t and
(\epsilon_{1,n},...,\epsilon_{n,n}) are assumed to be independent processes.
Based on a spectral decomposition of the covariance structures we derive series
estimators for \sigma^2 and \tau^2 and investigate their rate of convergence of
the MISE in dependence of their smoothness. To this end specific basis
functions and their corresponding Sobolev ellipsoids are introduced and we show
that our estimators are optimal in minimax sense. Our work is motivated by
microstructure noise models. Our major finding is that the microstructure noise
\epsilon_{i,n} introduces an additionally degree of ill-posedness of 1/2;
irrespectively of the tail behavior of \epsilon_{i,n}. The method is
illustrated by a small numerical study.Comment: 5 figures, corrected references, minor change
Shape Constrained Regularisation by Statistical Multiresolution for Inverse Problems: Asymptotic Analysis
This paper is concerned with a novel regularisation technique for solving
linear ill-posed operator equations in Hilbert spaces from data that is
corrupted by white noise. We combine convex penalty functionals with
extreme-value statistics of projections of the residuals on a given set of
sub-spaces in the image-space of the operator. We prove general consistency and
convergence rate results in the framework of Bregman-divergences which allows
for a vast range of penalty functionals. Various examples that indicate the
applicability of our approach will be discussed. We will illustrate in the
context of signal and image processing that the presented method constitutes a
locally adaptive reconstruction method
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