11,358 research outputs found
Recursive Bayesian inference on stochastic differential equations
This thesis is concerned with recursive Bayesian estimation of non-linear dynamical systems, which can be modeled as discretely observed stochastic differential equations. The recursive real-time estimation algorithms for these continuous-discrete filtering problems are traditionally called optimal filters and the algorithms for recursively computing the estimates based on batches of observations are called optimal smoothers. In this thesis, new practical algorithms for approximate and asymptotically optimal continuous-discrete filtering and smoothing are presented.
The mathematical approach of this thesis is probabilistic and the estimation algorithms are formulated in terms of Bayesian inference. This means that the unknown parameters, the unknown functions and the physical noise processes are treated as random processes in the same joint probability space. The Bayesian approach provides a consistent way of computing the optimal filtering and smoothing estimates, which are optimal given the model assumptions and a consistent way of analyzing their uncertainties.
The formal equations of the optimal Bayesian continuous-discrete filtering and smoothing solutions are well known, but the exact analytical solutions are available only for linear Gaussian models and for a few other restricted special cases. The main contributions of this thesis are to show how the recently developed discrete-time unscented Kalman filter, particle filter, and the corresponding smoothers can be applied in the continuous-discrete setting. The equations for the continuous-time unscented Kalman-Bucy filter are also derived.
The estimation performance of the new filters and smoothers is tested using simulated data. Continuous-discrete filtering based solutions are also presented to the problems of tracking an unknown number of targets, estimating the spread of an infectious disease and to prediction of an unknown time series.reviewe
Quantum theory of optical temporal phase and instantaneous frequency. II. Continuous time limit and state-variable approach to phase-locked loop design
We consider the continuous-time version of our recently proposed quantum
theory of optical temporal phase and instantaneous frequency [Tsang, Shapiro,
and Lloyd, Phys. Rev. A 78, 053820 (2008)]. Using a state-variable approach to
estimation, we design homodyne phase-locked loops that can measure the temporal
phase with quantum-limited accuracy. We show that post-processing can further
improve the estimation performance, if delay is allowed in the estimation. We
also investigate the fundamental uncertainties in the simultaneous estimation
of harmonic-oscillator position and momentum via continuous optical phase
measurements from the classical estimation theory perspective. In the case of
delayed estimation, we find that the inferred uncertainty product can drop
below that allowed by the Heisenberg uncertainty relation. Although this result
seems counter-intuitive, we argue that it does not violate any basic principle
of quantum mechanics.Comment: 11 pages, 6 figures, v2: accepted by PR
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
Robust Estimation of Optical Phase Varying as a Continuous Resonant Process
It is well-known that adaptive homodyne estimation of continuously varying
optical phase provides superior accuracy in the phase estimate as compared to
adaptive or non-adaptive static estimation. However, most phase estimation
schemes rely on precise knowledge of the underlying parameters of the system
under measurement, and performance deteriorates significantly with changes in
these parameters; hence it is desired to develop robust estimation techniques
immune to such uncertainties. In related works, we have already shown how
adaptive homodyne estimation can be made robust to uncertainty in an underlying
parameter of the phase varying as a simplistic Ornstein-Uhlenbeck stochastic
noise process. Here, we demonstrate robust phase estimation for a more
complicated resonant noise process using a guaranteed cost robust filter.Comment: 5 pages, 10 figures, Proceedings of the 2013 Multi-Conference on
Systems and Contro
Robust Filtering for Adaptive Homodyne Estimation of Continuously Varying Optical Phase
Recently, it has been demonstrated experimentally that adaptive estimation of
a continuously varying optical phase provides superior accuracy in the phase
estimate compared to static estimation. Here, we show that the mean-square
error in the adaptive phase estimate may be further reduced for the stochastic
noise process considered by using an optimal Kalman filter in the feedback
loop. Further, the estimation process can be made robust to fluctuations in the
underlying parameters of the noise process modulating the system phase to be
estimated. This has been done using a guaranteed cost robust filter.Comment: 5 pages, 6 figures, Proceedings of the 2012 Australian Control
Conferenc
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