Finite-dimensional filters with nonlinear drift. III: Duncan-Mortensen-Zakai equation with arbitrary initial condition for the linear filtering system and the Benes filtering system

1997 ◽  
Vol 33 (4) ◽  
pp. 1277-1294 ◽  
Author(s):  
Shing-Tung Yau ◽  
S.S.-T. Yau
Keyword(s):  
Initial Condition ◽  
Linear Filtering ◽  
Zakai Equation ◽  
Arbitrary Initial Condition ◽  
Finite Dimensional

scholarly journals Unbiased estimation of the solution to Zakai’s equation

2020 ◽  
Vol 26 (2) ◽  
pp. 113-129
Author(s):  
Hamza M. Ruzayqat ◽  
Ajay Jasra
Keyword(s):  
Continuous Time ◽  
Unbiased Estimation ◽  
Finite Variance ◽  
Linear Filtering ◽  
Multilevel Monte Carlo ◽  
Zakai Equation ◽  
Unbiased Estimators ◽  
First Order ◽  
Multilevel Monte Carlo Method ◽  
Filtering Problem

AbstractIn the following article, we consider the non-linear filtering problem in continuous time and in particular the solution to Zakai’s equation or the normalizing constant. We develop a methodology to produce finite variance, almost surely unbiased estimators of the solution to Zakai’s equation. That is, given access to only a first-order discretization of solution to the Zakai equation, we present a method which can remove this discretization bias. The approach, under assumptions, is proved to have finite variance and is numerically compared to using a particular multilevel Monte Carlo method.

Finite-dimensional models for hidden Markov chains

1995 ◽  
Vol 27 (01) ◽  
pp. 146-160
Author(s):  
Lakhdar Aggoun ◽  
Robert J. Elliott
Keyword(s):  
Markov Chains ◽  
Expectation Maximization ◽  
Continuous Time ◽  
Hidden Markov ◽  
Expectation Maximization Algorithm ◽  
Linear Filtering ◽  
Occupation Times ◽  
Finite Dimensional ◽  
Self Tuning ◽  
Filtering Problem

A continuous-time, non-linear filtering problem is considered in which both signal and observation processes are Markov chains. New finite-dimensional filters and smoothers are obtained for the state of the signal, for the number of jumps from one state to another, for the occupation time in any state of the signal, and for joint occupation times of the two processes. These estimates are then used in the expectation maximization algorithm to improve the parameters in the model. Consequently, our filters and model are adaptive, or self-tuning.

scholarly journals Evolving Network Model That Almost Regenerates Epileptic Data

2017 ◽  
Vol 29 (4) ◽  
pp. 937-967 ◽  
Author(s):  
G. Manjunath
Keyword(s):  
Network Model ◽  
Unique Feature ◽  
Epileptic Seizures ◽  
Time Varying ◽  
Initial Condition ◽  
Potential Utility ◽  
Arbitrary Initial Condition ◽  
Seizure Focus ◽  
The Brain ◽  
Discrete Time Dynamical Systems

In many realistic networks, the edges representing the interactions between nodes are time varying. Evidence is growing that the complex network that models the dynamics of the human brain has time-varying interconnections, that is, the network is evolving. Based on this evidence, we construct a patient- and data-specific evolving network model (comprising discrete-time dynamical systems) in which epileptic seizures or their terminations in the brain are also determined by the nature of the time-varying interconnections between the nodes. A novel and unique feature of our methodology is that the evolving network model remembers the data from which it was conceived from, in the sense that it evolves to almost regenerate the patient data even on presenting an arbitrary initial condition to it. We illustrate a potential utility of our methodology by constructing an evolving network from clinical data that aids in identifying an approximate seizure focus; nodes in such a theoretically determined seizure focus are outgoing hubs that apparently act as spreaders of seizures. We also point out the efficacy of removal of such spreaders in limiting seizures.

scholarly journals Diffusion in Heterogenous Media and Sorption—Desorption Processes

2021 ◽  
Vol 5 (4) ◽  
pp. 183
Author(s):  
Ana Paula S. Koltun ◽  
Ervin Kaminski Lenzi ◽  
Marcelo Kaminski Lenzi ◽  
Rafael Soares Zola
Keyword(s):  
Diffusion Coefficient ◽  
Kinetic Equation ◽  
Spatial Dependence ◽  
Heterogeneous Medium ◽  
Biological Fluids ◽  
Particle Diffusion ◽  
Initial Condition ◽  
Arbitrary Initial Condition ◽  
Electrolytic Cells ◽  
Heterogenous Media

We investigate particle diffusion in a heterogeneous medium limited by a surface where sorption–desorption processes are governed by a kinetic equation. We consider that the dynamics of the particles present in the medium are governed by a diffusion equation with a spatial dependence on the diffusion coefficient, i.e., K(x) = D|x|−η, with −1 < η and D = const, respectively. This system is analyzed in a semi-infinity region, i.e., the system is defined in the interval [0,∞) for an arbitrary initial condition. The solutions are obtained and display anomalous spreading, that is, the dynamics may be viewed as anomalous diffusion, which in turn is related, and hence, the model can be directly applied to several complex systems ranging from biological fluids to electrolytic cells.

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