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Parameter Estimation for Partially Observed Hypoelliptic Diffusions
TL;DR: In this paper, a deterministic scan Gibbs sampler alternating between missing data in the unobserved solution components, and parameters is used to model a variety of phenomena in applications ranging from molecular dynamics to audio signal analysis.
Abstract: Hypoelliptic diffusion processes can be used to model a variety of phenomena in applications ranging from molecular dynamics to audio signal analysis. We study parameter estimation for such processes in situations where we observe some components of the solution at discrete times. Since exact likelihoods for the transition densities are typically not known, approximations are used that are expected to work well in the limit of small inter-sample times $\Delta t$ and large total observation times $N\Delta t$. Hypoellipticity together with partial observation leads to ill-conditioning requiring a judicious combination of approximate likelihoods for the various parameters to be estimated. We combine these in a deterministic scan Gibbs sampler alternating between missing data in the unobserved solution components, and parameters. Numerical experiments illustrate asymptotic consistency of the method when applied to simulated data. The paper concludes with application of the Gibbs sampler to molecular dynamics data.
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01 Jan 2009
TL;DR: This short course is devoted to a few statistical problems related to the observation of a given process on a fixed time interval, when the observations occur at regularly spaced discrete times.
Abstract: This short course is devoted to a few statistical problems related to the observation of a given process on a fixed time interval, when the observations occur at regularly spaced discrete times. This kind of observations may occur in many different contexts, but they are particularly relevant in finance: we do have now huge amounts of data on the prices of various assets, exchange rates, and so on, typically ”tick data” which are recorded at every transaction time. So we are mainly concerned with the problems which arise in this context, and the concrete applications we will give are all pertaining to finance.
128 citations
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17 May 2012
TL;DR: Pavliotis et al. as discussed by the authors proposed an estimation method for multiscale diffusion coefficient estimation based on bridge processes and unbiased Monte Carlo for diffusions using high frequency data.
Abstract: Estimating functions for diffusion-type processes, Michael Sorensen Introduction Low frequency asymptotics Martingale estimating functions The likelihood function Non-martingale estimating functions High-frequency asymptotics High-frequency asymptotics in a fixed time-interval Small-diffusion asymptotics Non-Markovian models General asymptotic results for estimating functions Optimal estimating functions: General theory The econometrics of high frequency data, Per. A. Mykland and Lan Zhang Introduction Time varying drift and volatility Behavior of estimators: Variance Asymptotic normality Microstructure Methods based on contiguity Irregularly spaced data Statistics and high frequency data, Jean Jacod Introduction What can be estimated? Wiener plus compound Poisson processes Auxiliary limit theorems A first LNN (Law of Large Numbers) Some other LNNs A first CLT CLT with discontinuous limits Estimation of the integrated volatility Testing for jumps Testing for common jumps The Blumenthal-Getoor index Importance sampling techniques for estimation of diffusion models, Omiros Papaspiliopoulos and Gareth Roberts Overview of the chapter Background IS estimators based on bridge processes IS estimators based on guided processes Unbiased Monte Carlo for diffusions Appendix: Typical problems of the projection-simulation paradigm in MC for diffusions Appendix: Gaussian change of measure Non parametric estimation of the coefficients of ergodic diffusion processes based on high frequency data, Fabienne Comte, Valentine Genon-Catalot, and Yves Rozenholc Introduction Model and assumptions Observations and asymptotic framework Estimation method Drift estimation Diffusion coefficient estimation Examples and practical implementation Bibliographical remarks Appendix. Proof of Proposition.13 Ornstein-Uhlenbeck related models driven by Levy processes, Peter J. Brockwell and Alexander Lindner Introduction Levy processes Ornstein-Uhlenbeck related models Some estimation methods Parameter estimation for multiscale diffusions: an overview, Grigorios A. Pavliotis, Yvo Pokern, and Andrew M. Stuart Introduction Illustrative examples Averaging and homogenization Subsampling Hypoelliptic diffusions Nonparametric drift estimation Conclusions and further work
79 citations
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TL;DR: In this article, the authors consider stochastic extensions in order to capture unknown influences (changing behaviors, public interventions, seasonal effects etc) and assign diffusion processes to the time-varying parameters, and their inferential procedure is based on a suitably adjusted adaptive particle MCMC algorithm.
Abstract: Epidemics are often modelled using non-linear dynamical systems observed through partial and noisy data. In this paper, we consider stochastic extensions in order to capture unknown influences (changing behaviors, public interventions, seasonal effects etc). These models assign diffusion processes to the time-varying parameters, and our inferential procedure is based on a suitably adjusted adaptive particle MCMC algorithm. The performance of the proposed computational methods is validated on simulated data and the adopted model is applied to the 2009 H1N1 pandemic in England. In addition to estimating the effective contact rate trajectories, the methodology is applied in real time to provide evidence in related public health decisions. Diffusion driven SEIR-type models with age structure are also introduced.
78 citations
TL;DR: In this paper, the authors derived effective properties for the dynamics of a coarse-grained variable (i.e., a smooth function with value in a high-dimensional space) using conditional expectations, and showed that these properties yield an effective dynamics which accurately reproduces the residence times in the potential energy wells.
Abstract: The question of coarse-graining is ubiquitous in molecular dynamics. In this article, we are interested in deriving effective properties for the dynamics of a coarse-grained variable $\xi(x)$, where $x$ describes the configuration of the system in a high-dimensional space $\R^n$, and $\xi$ is a smooth function with value in $\R$ (typically a reaction coordinate). It is well known that, given a Boltzmann-Gibbs distribution on $x \in \R^n$, the equilibrium properties on $\xi(x)$ are completely determined by the free energy. On the other hand, the question of the effective dynamics on $\xi(x)$ is much more difficult to address. Starting from an overdamped Langevin equation on $x \in \R^n$, we propose an effective dynamics for $\xi(x) \in \R$ using conditional expectations. Using entropy methods, we give sufficient conditions for the time marginals of the effective dynamics to be close to the original ones. We check numerically on some toy examples that these sufficient conditions yield an effective dynamics which accurately reproduces the residence times in the potential energy wells. We also discuss the accuracy of the effective dynamics in a pathwise sense, and the relevance of the free energy to build a coarse-grained dynamics.
76 citations
Cites background from "Parameter Estimation for Partially ..."
...[33, 20]), or to postulate a parametric form for the effective dynamics and to identify its coefficients by numerical simulation on the complete system [27, 36]....
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TL;DR: In this paper, the authors proposed a method to solve the problem of energy-efficient wireless sensor networks for the first time, which was later validated by the National Science Foundation and U.S. Department of Energy.
71 citations
References
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49,597 citations
01 Jan 2001
TL;DR: In this paper, the clssical filleting and prediclion problem is re-examined using the Bode-Shannon representation of random processes and the?stat-tran-sition? method of analysis of dynamic systems.
Abstract: The clssical filleting and prediclion problem is re-examined using the Bode-Shannon representation of random processes and the ?stat-tran-sition? method of analysis of dynamic systems. New result are: (1) The formulation and Methods of solution of the problm apply, without modification to stationary and nonstationary stalistics end to growing-memory and infinile -memory filters. (2) A nonlinear difference (or differential) equalion is dericed for the covariance matrix of the optimal estimalion error. From the solution of this equation the coefficients of the difference, (or differential) equation of the optimal linear filter are obtained without further caleulations. (3) Tke fillering problem is shoum to be the dual of the nois-free regulator problem. The new method developed here, is applied to do well-known problems, confirming and extending, earlier results. The discussion is largely, self-contatained, and proceeds from first principles; basic concepts of the theory of random processes are reviewed in the Appendix.
15,391 citations
TL;DR: The CHARMM (Chemistry at Harvard Macromolecular Mechanics) as discussed by the authors is a computer program that uses empirical energy functions to model macromolescular systems, and it can read or model build structures, energy minimize them by first- or second-derivative techniques, perform a normal mode or molecular dynamics simulation, and analyze the structural, equilibrium, and dynamic properties determined in these calculations.
Abstract: CHARMM (Chemistry at HARvard Macromolecular Mechanics) is a highly flexible computer program which uses empirical energy functions to model macromolecular systems. The program can read or model build structures, energy minimize them by first- or second-derivative techniques, perform a normal mode or molecular dynamics simulation, and analyze the structural, equilibrium, and dynamic properties determined in these calculations. The operations that CHARMM can perform are described, and some implementation details are given. A set of parameters for the empirical energy function and a sample run are included.
14,725 citations
"Parameter Estimation for Partially ..." refers background or methods in this paper
...These periodic ansatz functions are a natural choice for dihedral angle potentials, in fact, the dihedral angle potential given in Brooks (1983) is of this form....
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...The coordinate x in this equation stands for cartesian coordinates of the four extended atoms making up the butane molecule, see [9] for details of the CHARMM forcefield used here....
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...Here, the time evolution of the Cartesian coordinates of the four extended atoms of Butane (see Figure 9) is simulated using a damped-driven Hamiltonian system; details of the force field used can be found in Brooks (1983)....
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TL;DR: In this article, a particle which is caught in a potential hole and which, through the shuttling action of Brownian motion, can escape over a potential barrier yields a suitable model for elucidating the applicability of the transition state method for calculating the rate of chemical reactions.
7,289 citations
"Parameter Estimation for Partially ..." refers background in this paper
...question is relatively recent, under the name of the ‘Kramers problem’ it dates back to Kramers (1940) with a brief summary in section 5.3.6a of Gardiner (1985)....
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...While this phrasing of the question is relatively recent, under the name of the ”Kramers problem” it dates back to Kramers (1940) with a brief summary in section 5.3.6a of Gardiner (1985)....
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01 Jan 1983
7,182 citations
"Parameter Estimation for Partially ..." refers methods in this paper
...by a hypoelliptic diffusion. While this phrasing of the question is relatively recent, under the name of the ”Kramers problem” it dates back to Kramers (1940) with a brief summary in section 5.3.6a of Gardiner (1985). 2 Pokern, Stuart, Wiberg Another application, audio signalanalysis, is referredto in Giannopoulos and Godsill (2001) where a continuous time ARMA model is used, see also Godsill and Yang (2006) for ...
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