4,339 research outputs found
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
Signal tracking beyond the time resolution of an atomic sensor by Kalman filtering
We study causal waveform estimation (tracking) of time-varying signals in a
paradigmatic atomic sensor, an alkali vapor monitored by Faraday rotation
probing. We use Kalman filtering, which optimally tracks known linear Gaussian
stochastic processes, to estimate stochastic input signals that we generate by
optical pumping. Comparing the known input to the estimates, we confirm the
accuracy of the atomic statistical model and the reliability of the Kalman
filter, allowing recovery of waveform details far briefer than the sensor's
intrinsic time resolution. With proper filter choice, we obtain similar
benefits when tracking partially-known and non-Gaussian signal processes, as
are found in most practical sensing applications. The method evades the
trade-off between sensitivity and time resolution in coherent sensing.Comment: 15 pages, 4 figure
Signal tracking beyond the time resolution of an atomic sensor by Kalman filtering
We study causal waveform estimation (tracking) of time-varying signals in a
paradigmatic atomic sensor, an alkali vapor monitored by Faraday rotation
probing. We use Kalman filtering, which optimally tracks known linear Gaussian
stochastic processes, to estimate stochastic input signals that we generate by
optical pumping. Comparing the known input to the estimates, we confirm the
accuracy of the atomic statistical model and the reliability of the Kalman
filter, allowing recovery of waveform details far briefer than the sensor's
intrinsic time resolution. With proper filter choice, we obtain similar
benefits when tracking partially-known and non-Gaussian signal processes, as
are found in most practical sensing applications. The method evades the
trade-off between sensitivity and time resolution in coherent sensing.Comment: 15 pages, 4 figure
Forecasting trends with asset prices
In this paper, we consider a stochastic asset price model where the trend is
an unobservable Ornstein Uhlenbeck process. We first review some classical
results from Kalman filtering. Expectedly, the choice of the parameters is
crucial to put it into practice. For this purpose, we obtain the likelihood in
closed form, and provide two on-line computations of this function. Then, we
investigate the asymptotic behaviour of statistical estimators. Finally, we
quantify the effect of a bad calibration with the continuous time mis-specified
Kalman filter. Numerical examples illustrate the difficulty of trend
forecasting in financial time series.Comment: 26 pages, 11 figure
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