145,541 research outputs found
Gradient-Based Estimation of Uncertain Parameters for Elliptic Partial Differential Equations
This paper addresses the estimation of uncertain distributed diffusion
coefficients in elliptic systems based on noisy measurements of the model
output. We formulate the parameter identification problem as an infinite
dimensional constrained optimization problem for which we establish existence
of minimizers as well as first order necessary conditions. A spectral
approximation of the uncertain observations allows us to estimate the infinite
dimensional problem by a smooth, albeit high dimensional, deterministic
optimization problem, the so-called finite noise problem in the space of
functions with bounded mixed derivatives. We prove convergence of finite noise
minimizers to the appropriate infinite dimensional ones, and devise a
stochastic augmented Lagrangian method for locating these numerically. Lastly,
we illustrate our method with three numerical examples
Concurrent estimation of noise and compact-binary signal parameters in gravitational-wave data
Gravitational-wave parameter estimation for compact binary signals typically
relies on sequential estimation of the properties of the detector Gaussian
noise and of the binary parameters. This procedure assumes that the noise
variance, expressed through its power spectral density, is perfectly known in
advance. We assess the impact of this approximation on the estimated parameters
by means of an analysis that simultaneously estimates the noise and compact
binary parameters, thus allowing us to marginalize over uncertainty in the
noise properties. We compare the traditional sequential estimation method and
the new full marginalization method using events from the GWTC-3 catalog. We
find that the recovered signals and inferred parameters agree to within their
statistical measurement uncertainty. At current detector sensitivities,
uncertainty about the noise power spectral density is a subdominant effect
compared to other sources of uncertainty.Comment: 10 pages, 9 figure
Computational aspects of Bayesian spectral density estimation
Gaussian time-series models are often specified through their spectral
density. Such models present several computational challenges, in particular
because of the non-sparse nature of the covariance matrix. We derive a fast
approximation of the likelihood for such models. We propose to sample from the
approximate posterior (that is, the prior times the approximate likelihood),
and then to recover the exact posterior through importance sampling. We show
that the variance of the importance sampling weights vanishes as the sample
size goes to infinity. We explain why the approximate posterior may typically
multi-modal, and we derive a Sequential Monte Carlo sampler based on an
annealing sequence in order to sample from that target distribution.
Performance of the overall approach is evaluated on simulated and real
datasets. In addition, for one real world dataset, we provide some numerical
evidence that a Bayesian approach to semi-parametric estimation of spectral
density may provide more reasonable results than its Frequentist counter-parts
Spectral unmixing of Multispectral Lidar signals
In this paper, we present a Bayesian approach for spectral unmixing of
multispectral Lidar (MSL) data associated with surface reflection from targeted
surfaces composed of several known materials. The problem addressed is the
estimation of the positions and area distribution of each material. In the
Bayesian framework, appropriate prior distributions are assigned to the unknown
model parameters and a Markov chain Monte Carlo method is used to sample the
resulting posterior distribution. The performance of the proposed algorithm is
evaluated using synthetic MSL signals, for which single and multi-layered
models are derived. To evaluate the expected estimation performance associated
with MSL signal analysis, a Cramer-Rao lower bound associated with model
considered is also derived, and compared with the experimental data. Both the
theoretical lower bound and the experimental analysis will be of primary
assistance in future instrument design
Bayesian spectral modeling for multiple time series
We develop a novel Bayesian modeling approach to spectral density estimation for multiple time series. The log-periodogram distribution for each series is modeled as a mixture of Gaussian distributions with frequency-dependent weights and mean functions. The implied model for the log-spectral density is a mixture of linear mean functions with frequency-dependent weights. The mixture weights are built through successive differences of a logit-normal distribution function with frequency-dependent parameters. Building from the construction for a single spectral density, we develop a hierarchical extension for multiple time series. Specifically, we set the mean functions to be common to all spectral densities and make the weights specific to the time series through the parameters of the logit-normal distribution. In addition to accommodating flexible spectral density shapes, a practically important feature of the proposed formulation is that it allows for ready posterior simulation through a Gibbs sampler with closed form full conditional distributions for all model parameters. The modeling approach is illustrated with simulated datasets, and used for spectral analysis of multichannel electroencephalographic recordings (EEGs), which provides a key motivating application for the proposed methodology
Analytical maximum likelihood estimation of stellar magnetic fields
The polarised spectrum of stellar radiation encodes valuable information on
the conditions of stellar atmospheres and the magnetic fields that permeate
them. In this paper, we give explicit expressions to estimate the magnetic
field vector and its associated error from the observed Stokes parameters. We
study the solar case where specific intensities are observed and then the
stellar case, where we receive the polarised flux. In this second case, we
concentrate on the explicit expression for the case of a slow rotator with a
dipolar magnetic field geometry. Moreover, we also give explicit formulae to
retrieve the magnetic field vector from the LSD profiles without assuming mean
values for the LSD artificial spectral line. The formulae have been obtained
assuming that the spectral lines can be described in the weak field regime and
using a maximum likelihood approach. The errors are recovered by means of the
hermitian matrix. The bias of the estimators are analysed in depth.Comment: accepted for publication in MNRA
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