844 research outputs found
Stochastic evolution equations for nonlinear filtering of random fields in the presence of fractional Brownian sheet observation noise
The problem of nonlinear filtering of a random field observed in the presence
of a noise, modeled by a persistent fractional Brownian sheet of Hurst index
with , is studied and a suitable version of the
Bayes' formula for the optimal filter is obtained. Two types of spatial
"fractional" analogues of the Duncan-Mortensen-Zakai equation are also derived:
one tracks evolution of the unnormalized optimal filter along an arbitrary
"monotone increasing" (in the sense of partial ordering in )
one-dimensional curve in the plane, while the other describes dynamics of the
filter along the paths that are truly two-dimensional. Although the paper deals
with the two-dimensional parameter space, the presented approach and results
extend to -parameter random fields with arbitrary .Comment: 24 page
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Mini-Workshop: Levy Processes and Related Topics in Modelling
The focus of the meeting was on recent ongoing research and new ideas in the area of infinite divisibility and LeÌvy processes, with particular view to realistic modelling. As regards more applied aspects, work in mathematical finance, especially concerning modelling and measurement of volatility, figured prominently
The History of the Quantitative Methods in Finance Conference Series. 1992-2007
This report charts the history of the Quantitative Methods in Finance (QMF) conference from its beginning in 1993 to the 15th conference in 2007. It lists alphabetically the 1037 speakers who presented at all 15 conferences and the titles of their papers.
Fractional Generalizations of Zakai Equation and Some Solution Methods
The paper discusses fractional generalizations of Zakai equations arising in ïŹltering problems. The derivation of the fractional Zakai equation, existence and uniqueness of its solution, as well as some methods of solution to the fractional ïŹltering problem, including fractional version of the particle ïŹow method, are presented
Recovering edges in ill-posed inverse problems: optimality of curvelet frames
We consider a model problem of recovering a function from noisy Radon data. The function to be recovered is assumed smooth apart from a discontinuity along a curve, that is, an edge. We use the continuum white-noise model, with noise level .
Traditional linear methods for solving such inverse problems behave poorly in the presence of edges. Qualitatively, the reconstructions are blurred near the edges; quantitatively, they give in our model mean squared errors (MSEs) that tend to zero with noise level only as as . A recent innovation--nonlinear shrinkage in the wavelet domain--visually improves edge sharpness and improves MSE convergence to . However, as we show here, this rate is not optimal.
In fact, essentially optimal performance is obtained by deploying the recently-introduced tight frames of curvelets in this setting. Curvelets are smooth, highly anisotropic elements ideally suited for detecting and synthesizing curved edges. To deploy them in the Radon setting, we construct a curvelet-based biorthogonal decomposition of the Radon operator and build "curvelet shrinkage" estimators based on thresholding of the noisy curvelet coefficients. In effect, the estimator detects edges at certain locations and orientations in the Radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain.
We prove that the curvelet shrinkage can be tuned so that the estimator will attain, within logarithmic factors, the MSE as noise level . This rate of convergence holds uniformly over a class of functions which are except for discontinuities along curves, and (except for log terms) is the minimax rate for that class. Our approach is an instance of a general strategy which should apply in other inverse problems; we sketch a deconvolution example
Physics, Astrophysics and Cosmology with Gravitational Waves
Gravitational wave detectors are already operating at interesting sensitivity
levels, and they have an upgrade path that should result in secure detections
by 2014. We review the physics of gravitational waves, how they interact with
detectors (bars and interferometers), and how these detectors operate. We study
the most likely sources of gravitational waves and review the data analysis
methods that are used to extract their signals from detector noise. Then we
consider the consequences of gravitational wave detections and observations for
physics, astrophysics, and cosmology.Comment: 137 pages, 16 figures, Published version
<http://www.livingreviews.org/lrr-2009-2
Chance, long tails, and inference: a non-Gaussian, Bayesian theory of vocal learning in songbirds
Traditional theories of sensorimotor learning posit that animals use sensory
error signals to find the optimal motor command in the face of Gaussian sensory
and motor noise. However, most such theories cannot explain common behavioral
observations, for example that smaller sensory errors are more readily
corrected than larger errors and that large abrupt (but not gradually
introduced) errors lead to weak learning. Here we propose a new theory of
sensorimotor learning that explains these observations. The theory posits that
the animal learns an entire probability distribution of motor commands rather
than trying to arrive at a single optimal command, and that learning arises via
Bayesian inference when new sensory information becomes available. We test this
theory using data from a songbird, the Bengalese finch, that is adapting the
pitch (fundamental frequency) of its song following perturbations of auditory
feedback using miniature headphones. We observe the distribution of the sung
pitches to have long, non-Gaussian tails, which, within our theory, explains
the observed dynamics of learning. Further, the theory makes surprising
predictions about the dynamics of the shape of the pitch distribution, which we
confirm experimentally
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