3,923 research outputs found
Gaussian Approximations of Small Noise Diffusions in Kullback-Leibler Divergence
We study Gaussian approximations to the distribution of a diffusion. The
approximations are easy to compute: they are defined by two simple ordinary
differential equations for the mean and the covariance. Time correlations can
also be computed via solution of a linear stochastic differential equation. We
show, using the Kullback-Leibler divergence, that the approximations are
accurate in the small noise regime. An analogous discrete time setting is also
studied. The results provide both theoretical support for the use of Gaussian
processes in the approximation of diffusions, and methodological guidance in
the construction of Gaussian approximations in applications
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
Complex dynamics of elementary cellular automata emerging from chaotic rules
We show techniques of analyzing complex dynamics of cellular automata (CA)
with chaotic behaviour. CA are well known computational substrates for studying
emergent collective behaviour, complexity, randomness and interaction between
order and chaotic systems. A number of attempts have been made to classify CA
functions on their space-time dynamics and to predict behaviour of any given
function. Examples include mechanical computation, \lambda{} and Z-parameters,
mean field theory, differential equations and number conserving features. We
aim to classify CA based on their behaviour when they act in a historical mode,
i.e. as CA with memory. We demonstrate that cell-state transition rules
enriched with memory quickly transform a chaotic system converging to a complex
global behaviour from almost any initial condition. Thus just in few steps we
can select chaotic rules without exhaustive computational experiments or
recurring to additional parameters. We provide analysis of well-known chaotic
functions in one-dimensional CA, and decompose dynamics of the automata using
majority memory exploring glider dynamics and reactions
Importance Sampling: Intrinsic Dimension and Computational Cost
The basic idea of importance sampling is to use independent samples from a
proposal measure in order to approximate expectations with respect to a target
measure. It is key to understand how many samples are required in order to
guarantee accurate approximations. Intuitively, some notion of distance between
the target and the proposal should determine the computational cost of the
method. A major challenge is to quantify this distance in terms of parameters
or statistics that are pertinent for the practitioner. The subject has
attracted substantial interest from within a variety of communities. The
objective of this paper is to overview and unify the resulting literature by
creating an overarching framework. A general theory is presented, with a focus
on the use of importance sampling in Bayesian inverse problems and filtering.Comment: Statistical Scienc
Long-time asymptotics of the filtering distribution for partially observed chaotic dynamical systems
The filtering distribution is a time-evolving probability distribution on the state of a dynamical system given noisy observations. We study the large-time asymptotics of this probability distribution for discrete-time, randomly initialized signals that evolve according to a deterministic map Ψ. The observations are assumed to comprise a low-dimensional projection of the signal, given by an operator P, subject to additive noise. We address the question of whether these observations contain sufficient information to accurately reconstruct the signal. In a general framework, we establish conditions on Ψ and P under which the filtering distributions concentrate around the signal in the small-noise, long-time asymptotic regime. Linear systems, the Lorenz ’63 and ’96 models, and the Navier–Stokes equation on a two-dimensional torus are within the scope of the theory. Our main findings come as a by-product of computable bounds, of independent interest, for suboptimal filters based on new variants of the 3DVAR filtering algorith
Comparing the Min–Max–Median/IQR Approach with the Min–Max Approach, Logistic Regression and XGBoost, maximising the Youden index
Although linearly combining multiple variables can provide adequate diagnostic performance, certain algorithms have the limitation of being computationally demanding when the number of variables is sufficiently high. Liu et al. proposed the min–max approach that linearly combines the minimum and maximum values of biomarkers, which is computationally tractable and has been shown to be optimal in certain scenarios. We developed the Min–Max–Median/IQR algorithm under Youden index optimisation which, although more computationally intensive, is still approachable and includes more information. The aim of this work is to compare the performance of these algorithms with well-known Machine Learning algorithms, namely logistic regression and XGBoost, which have proven to be efficient in various fields of applications, particularly in the health sector. This comparison is performed on a wide range of different scenarios of simulated symmetric or asymmetric data, as well as on real clinical diagnosis data sets. The results provide useful information for binary classification problems of better algorithms in terms of performance depending on the scenario
Controlling Unpredictability with Observations in the Partially Observed Lorenz '96 Model
In the context of filtering chaotic dynamical systems it is well-known that
partial observations, if sufficiently informative, can be used to control the
inherent uncertainty due to chaos. The purpose of this paper is to investigate,
both theoretically and numerically, conditions on the observations of chaotic
systems under which they can be accurately filtered. In particular, we
highlight the advantage of adaptive observation operators over fixed ones. The
Lorenz '96 model is used to exemplify our findings.
We consider discrete-time and continuous-time observations in our theoretical
developments. We prove that, for fixed observation operator, the 3DVAR filter
can recover the system state within a neighbourhood determined by the size of
the observational noise. It is required that a sufficiently large proportion of
the state vector is observed, and an explicit form for such sufficient fixed
observation operator is given. Numerical experiments, where the data is
incorporated by use of the 3DVAR and extended Kalman filters, suggest that less
informative fixed operators than given by our theory can still lead to accurate
signal reconstruction. Adaptive observation operators are then studied
numerically; we show that, for carefully chosen adaptive observation operators,
the proportion of the state vector that needs to be observed is drastically
smaller than with a fixed observation operator. Indeed, we show that the number
of state coordinates that need to be observed may even be significantly smaller
than the total number of positive Lyapunov exponents of the underlying system
Gaussian Approximations of Small Noise Diffusions in Kullback-Leibler Divergence
We study Gaussian approximations to the distribution of a diffusion. The approximations are easy to compute: they are defined by two simple ordinary differential equations for the mean and the covariance. Time correlations can also be computed via solution of a linear stochastic differential equation. We show, using the Kullback–Leibler divergence, that the approximations are accurate in the small noise regime. An analogous discrete time setting is also studied. The results provide both theoretical support for the use of Gaussian processes in the approximation of diffusions, and methodological guidance in the construction of Gaussian approximations in applications
ValidaciĂłn de un nombre en Biscutella (Brassicaceae) del este de la PenĂnsula IbĂ©rica
The name Biscutella marinae is applied to an endemic plant from co-astal sand-dune ecosystems of northern Alicante. It however was not published accor-ding to the Melbourne Code, and therefore it still remains nomenclaturally invalid. In the present contribution it is validated, and new data are reported that complete the available information on that endemic.El nombre Biscutella marinae se aplica a un endemismo de los ecosistemas de dunas costeras del norte de Alicante. Sin embargo, su publicaciĂłn ini-cial no se hizo conforme al Codigo de Melbourne, por lo que dicho nombre no es váli-do nomenclaturalmente. Por ello, aquĂ se valida y se aportan datos que completan la in-formaciĂłn existente sobre este endemismo.This work was partly supported by the I+D+i research project CGL2011–30140 from MICINN (MÂş de EconomĂa y Competitividad, Spanish Government). The Andrew W. Mellon Foundation, New York, supported the type di-gitization for the Global Plant Initiative (GPI)
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