128,447 research outputs found
Fixed-point smoothing in Hilbert spaces
The fixed-point smoothing estimator and smoothing error covariance operator equations are derived for infinite-dimensional linear systems using both the Wiener-Hoph theory in Hilbert spaces developed by Falb and the abstract evolution theory. Since it is clear that the prediction problems can be solved by the same approach, the present results in conjunction with the work of Falb on filtering give a complete treatment of the infinite-dimensional linear estimation problem from the viewpoint of Wiener-Hoph theory. Finally, based on the optimal smoothing estimator in Hilbert space, the fixed-point smoothing estimator is derived for a linear distributed parameter system of parabolic type
An Optimal Control Derivation of Nonlinear Smoothing Equations
The purpose of this paper is to review and highlight some connections between
the problem of nonlinear smoothing and optimal control of the Liouville
equation. The latter has been an active area of recent research interest owing
to work in mean-field games and optimal transportation theory. The nonlinear
smoothing problem is considered here for continuous-time Markov processes. The
observation process is modeled as a nonlinear function of a hidden state with
an additive Gaussian measurement noise. A variational formulation is described
based upon the relative entropy formula introduced by Newton and Mitter. The
resulting optimal control problem is formulated on the space of probability
distributions. The Hamilton's equation of the optimal control are related to
the Zakai equation of nonlinear smoothing via the log transformation. The
overall procedure is shown to generalize the classical Mortensen's minimum
energy estimator for the linear Gaussian problem.Comment: 7 pages, 0 figures, under peer reviewin
Bounded state space
This investigation is divided functionally into three different areas: (1) study of bounded state space, (2) nonlinear smoothing theory, and (3) system identification. (1) Study of bounded state space: necessary and sufficient conditions for an optimal control are obtained for a bounded state space optimal control problem. The difficulty of determining the so-called jump conditions is eliminated; however, the problem of determining the points where the response either enters or leaves the boundary still remains unsolved. (2) Nonlinear smoothing theory: nonlinear fixed-interval, fixed-point and fixed-lag smoothing of a random signal generated by a stochastic differential equation are investigated. Results on the asymptotic stability of a linear constant-parameter fixed-interval smoothing filter are obtained. (3) System identification: a particular stochastic modelling problem is solved. An Ito stochastic integral equation is used to mathematically model a black box having multiple inputs and multiple outputs. A new method for identifying system parameters is presented
Filtering and Smoothing for Linear Discrete-Time Distributed Parameter Systems Based on Wiener-Hopf Theory with Application to Estimation of Air Pollution
Optimal filtering and smoothing algorithms for linear discrete-time distributed parameter systems are derived by a unified approach based on the Wiener-Hopf theory. The Wiener-Hopf equation for the estimation problems is derived using the least-squares estimation error criterion. Using the basic equation, three types of the optimal smoothing estimators are derived, namely, fixed-point, fixed-interval, and fixed-lag smoothers. Finally, the results obtained are applied to estimation of atmospheric sulfur dioxide concentrations in the Tokushima prefecture of Japan
Mild solutions of semilinear elliptic equations in Hilbert spaces
This paper extends the theory of regular solutions ( in a suitable
sense) for a class of semilinear elliptic equations in Hilbert spaces. The
notion of regularity is based on the concept of -derivative, which is
introduced and discussed. A result of existence and uniqueness of solutions is
stated and proved under the assumption that the transition semigroup associated
to the linear part of the equation has a smoothing property, that is, it maps
continuous functions into -differentiable ones. The validity of this
smoothing assumption is fully discussed for the case of the Ornstein-Uhlenbeck
transition semigroup and for the case of invertible diffusion coefficient
covering cases not previously addressed by the literature. It is shown that the
results apply to Hamilton-Jacobi-Bellman (HJB) equations associated to infinite
horizon optimal stochastic control problems in infinite dimension and that, in
particular, they cover examples of optimal boundary control of the heat
equation that were not treatable with the approaches developed in the
literature up to now
Optimum filters and smoothers design for carrier phase and frequency tracking
The report presents the application of fixed lag smoothing algorithms to the problem of estimation of the phase and frequency of a sinusoidal carrier received in the presence of process noise and additive observation noise. A suboptimal structure consists of a phase-locked loop (PLL) followed by post-loop correction to the phase and frequency estimates. When the PLL is operating under high signal-to-noise ratio, the phase detector is approximately linear, and the smoother equations then correspond to the optimal linear equations for an equivalent linear signal model. The performance of such a smoother can be predicted by linear filtering theory. However, if the PLL is operating near the threshold region of the signal-to-noise ratio, the phase detector cannot be assumed to be linear. Then the actual performance of the smoother can significantly differ from that predicted by linear theory. In this report we present both the theoretical and simulated performance of such smoothers derived on the basis of various models for the phase and frequency processes
Fixed lag smoothers for carrier phase and frequency tracking
The application of fixed lag smoothing algorithms are presented for the problem of estimation of the phase and frequency of a sinusoidal carrier received in the presence of process noise and additive observation noise. A suboptimal structure consists of a phase-locked loop (PLL) followed by a post loop correction to the phase and frequency estimates. When the PLL is operating under a high signal-to-noise ratio, the phase detector is approximately linear, and the smoother equations then correspond to the optimal linear equations for an equivalent linear signal model. The performance of such a smoother can be predicted by the linear filtering theory. However, if the PLL is operating near the threshold region of the signal to noise ratio, the phase detector cannot be assumed to be linear. Then the actual performance of the smoother can significantly differ from that predicted by linear filtering theory. Both the theoretical and simulated performance of such smoothers derived on the basis of various models for the phase of frequency processes are presented
Orthogonalized smoothing for rescaled spike and slab models
Rescaled spike and slab models are a new Bayesian variable selection method
for linear regression models. In high dimensional orthogonal settings such
models have been shown to possess optimal model selection properties. We review
background theory and discuss applications of rescaled spike and slab models to
prediction problems involving orthogonal polynomials. We first consider global
smoothing and discuss potential weaknesses. Some of these deficiencies are
remedied by using local regression. The local regression approach relies on an
intimate connection between local weighted regression and weighted generalized
ridge regression. An important implication is that one can trace the effective
degrees of freedom of a curve as a way to visualize and classify curvature.
Several motivating examples are presented.Comment: Published in at http://dx.doi.org/10.1214/074921708000000192 the IMS
Collections (http://www.imstat.org/publications/imscollections.htm) by the
Institute of Mathematical Statistics (http://www.imstat.org
Modeling the reconstructed BAO in Fourier space
The density field reconstruction technique, which was developed to partially
reverse the nonlinear degradation of the Baryon Acoustic Oscillation (BAO)
feature in the galaxy redshift surveys, has been successful in substantially
improving the cosmology constraints from recent galaxy surveys such as Baryon
Oscillation Spectroscopic Survey (BOSS). We estimate the efficiency of the
reconstruction method as a function of various reconstruction details. To
directly quantify the BAO information in nonlinear density fields before and
after reconstruction, we calculate the cross-correlations (i.e., propagators)
of the pre(post)-reconstructed density field with the initial linear field
using a mock galaxy sample that is designed to mimic the clustering of the BOSS
CMASS galaxies. The results directly provide the BAO damping as a function of
wavenumber that can be implemented into the Fisher matrix analysis. We focus on
investigating the dependence of the propagator on a choice of smoothing filters
and on two major different conventions of the redshift-space density field
reconstruction that have been used in literature. By estimating the BAO
signal-to-noise for each case, we predict constraints on the angular diameter
distance and Hubble parameter using the Fisher matrix analysis. We thus
determine an optimal Gaussian smoothing filter scale for the signal-to-noise
level of the BOSS CMASS. We also present appropriate BAO fitting models for
different reconstruction methods based on the first and second order Lagrangian
perturbation theory in Fourier space. Using the mock data, we show that the
modified BAO fitting model can substantially improve the accuracy of the BAO
position in the best fits as well as the goodness of the fits.Comment: 21 pages, 7 figures, 1 table. Minor revisions. Matches version
accepted by MNRA
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