3,388 research outputs found
Smoothing and filtering with a class of outer measures
Filtering and smoothing with a generalised representation of uncertainty is
considered. Here, uncertainty is represented using a class of outer measures.
It is shown how this representation of uncertainty can be propagated using
outer-measure-type versions of Markov kernels and generalised Bayesian-like
update equations. This leads to a system of generalised smoothing and filtering
equations where integrals are replaced by supremums and probability density
functions are replaced by positive functions with supremum equal to one.
Interestingly, these equations retain most of the structure found in the
classical Bayesian filtering framework. It is additionally shown that the
Kalman filter recursion can be recovered from weaker assumptions on the
available information on the corresponding hidden Markov model
Inverse Problems and Data Assimilation
These notes are designed with the aim of providing a clear and concise
introduction to the subjects of Inverse Problems and Data Assimilation, and
their inter-relations, together with citations to some relevant literature in
this area. The first half of the notes is dedicated to studying the Bayesian
framework for inverse problems. Techniques such as importance sampling and
Markov Chain Monte Carlo (MCMC) methods are introduced; these methods have the
desirable property that in the limit of an infinite number of samples they
reproduce the full posterior distribution. Since it is often computationally
intensive to implement these methods, especially in high dimensional problems,
approximate techniques such as approximating the posterior by a Dirac or a
Gaussian distribution are discussed. The second half of the notes cover data
assimilation. This refers to a particular class of inverse problems in which
the unknown parameter is the initial condition of a dynamical system, and in
the stochastic dynamics case the subsequent states of the system, and the data
comprises partial and noisy observations of that (possibly stochastic)
dynamical system. We will also demonstrate that methods developed in data
assimilation may be employed to study generic inverse problems, by introducing
an artificial time to generate a sequence of probability measures interpolating
from the prior to the posterior
Imaging of a fluid injection process using geophysical data - A didactic example
In many subsurface industrial applications, fluids are injected into or withdrawn from a geologic formation. It is of practical interest to quantify precisely where, when, and by how much the injected fluid alters the state of the subsurface. Routine geophysical monitoring of such processes attempts to image the way that geophysical properties, such as seismic velocities or electrical conductivity, change through time and space and to then make qualitative inferences as to where the injected fluid has migrated. The more rigorous formulation of the time-lapse geophysical inverse problem forecasts how the subsurface evolves during the course of a fluid-injection application. Using time-lapse geophysical signals as the data to be matched, the model unknowns to be estimated are the multiphysics forward-modeling parameters controlling the fluid-injection process. Properly reproducing the geophysical signature of the flow process, subsequent simulations can predict the fluid migration and alteration in the subsurface. The dynamic nature of fluid-injection processes renders imaging problems more complex than conventional geophysical imaging for static targets. This work intents to clarify the related hydrogeophysical parameter estimation concepts
Fertility Assimilation of Immigrants: A Varying Coefficient Count Data Model
This study presents the first econometric application of the Poisson varying coefficient (PVC) model. This count data model is applied to investigate immigrant fertility adjustment after migration. Data on completed fertility are taken from the 1996 wave of the German Socioeconomic Panel (GSOEP). We find evidence in favor of the assimilation model according to which immigrant fertility converges to native levels over time. Other determinants of completed fertility are marital history and female human capital, well in accordance with theoretical predictions
The Challenge of Machine Learning in Space Weather Nowcasting and Forecasting
The numerous recent breakthroughs in machine learning (ML) make imperative to
carefully ponder how the scientific community can benefit from a technology
that, although not necessarily new, is today living its golden age. This Grand
Challenge review paper is focused on the present and future role of machine
learning in space weather. The purpose is twofold. On one hand, we will discuss
previous works that use ML for space weather forecasting, focusing in
particular on the few areas that have seen most activity: the forecasting of
geomagnetic indices, of relativistic electrons at geosynchronous orbits, of
solar flares occurrence, of coronal mass ejection propagation time, and of
solar wind speed. On the other hand, this paper serves as a gentle introduction
to the field of machine learning tailored to the space weather community and as
a pointer to a number of open challenges that we believe the community should
undertake in the next decade. The recurring themes throughout the review are
the need to shift our forecasting paradigm to a probabilistic approach focused
on the reliable assessment of uncertainties, and the combination of
physics-based and machine learning approaches, known as gray-box.Comment: under revie
The structure and formation of natural categories
Categorization and concept formation are critical activities of intelligence. These processes and the conceptual structures that support them raise important issues at the interface of cognitive psychology and artificial intelligence. The work presumes that advances in these and other areas are best facilitated by research methodologies that reward interdisciplinary interaction. In particular, a computational model is described of concept formation and categorization that exploits a rational analysis of basic level effects by Gluck and Corter. Their work provides a clean prescription of human category preferences that is adapted to the task of concept learning. Also, their analysis was extended to account for typicality and fan effects, and speculate on how the concept formation strategies might be extended to other facets of intelligence, such as problem solving
Estimating parameters in stochastic systems:a variational Bayesian approach
This work is concerned with approximate inference in dynamical systems, from a variational Bayesian perspective. When modelling real world dynamical systems, stochastic differential equations appear as a natural choice, mainly because of their ability to model the noise of the system by adding a variation of some stochastic process to the deterministic dynamics. Hence, inference in such processes has drawn much attention. Here a new extended framework is derived that is based on a local polynomial approximation of a recently proposed variational Bayesian algorithm. The paper begins by showing that the new extension of this variational algorithm can be used for state estimation (smoothing) and converges to the original algorithm. However, the main focus is on estimating the (hyper-) parameters of these systems (i.e. drift parameters and diffusion coefficients). The new approach is validated on a range of different systems which vary in dimensionality and non-linearity. These are the OrnsteinâUhlenbeck process, the exact likelihood of which can be computed analytically, the univariate and highly non-linear, stochastic double well and the multivariate chaotic stochastic Lorenz â63 (3D model). As a special case the algorithm is also applied to the 40 dimensional stochastic Lorenz â96 system. In our investigation we compare this new approach with a variety of other well known methods, such as the hybrid Monte Carlo, dual unscented Kalman filter, full weak-constraint 4D-Var algorithm and analyse empirically their asymptotic behaviour as a function of observation density or length of time window increases. In particular we show that we are able to estimate parameters in both the drift (deterministic) and the diffusion (stochastic) part of the model evolution equations using our new methods
A unified Bayesian inversion approach for a class of tumor growth models with different pressure laws
In this paper, we use the Bayesian inversion approach to study the data
assimilation problem for a family of tumor growth models described by
porous-medium type equations. The models contain uncertain parameters and are
indexed by a physical parameter , which characterizes the constitutive
relation between density and pressure. Based on these models, we employ the
Bayesian inversion framework to infer parametric and nonparametric unknowns
that affect tumor growth from noisy observations of tumor cell density. We
establish the well-posedness and the stability theories for the Bayesian
inversion problem and further prove the convergence of the posterior
distribution in the so-called incompressible limit, .
Since the posterior distribution across the index regime can
thus be treated in a unified manner, such theoretical results also guide the
design of the numerical inference for the unknown. We propose a generic
computational framework for such inverse problems, which consists of a typical
sampling algorithm and an asymptotic preserving solver for the forward problem.
With extensive numerical tests, we demonstrate that the proposed method
achieves satisfactory accuracy in the Bayesian inference of the tumor growth
models, which is uniform with respect to the constitutive relation.Comment: 29 pages, 14 figure
Recommended from our members
Uncertainty Quantification
Uncertainty quantification (UQ) is concerned with including and characterising uncertainties in mathematical models.
Major steps comprise proper description of system uncertainties, analysis and efficient quantification of uncertainties in predictions and design problems, and statistical inference on uncertain parameters starting from available measurements.
Research in UQ addresses fundamental mathematical and statistical challenges, but has also wide applicability in areas such as engineering, environmental, physical and biological applications.
This workshop focussed on mathematical challenges at the interface of applied mathematics, probability and statistics, numerical analysis, scientific computing and application domains.
The workshop served to bring together experts from those disciplines in order to enhance their interaction, to exchange ideas and to develop new, powerful methods for UQ
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