MLE for partially observed diffusions: direct maximization vs. the em algorithm

Campillo, Fabien; Gland, François Le

doi:10.1016/0304-4149(89)90041-0

Cited by 57 publications

(33 citation statements)

References 9 publications

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“…However, the analysis of this process is hampered by the fact that usually only a 1D-series of values of X(t) will be available in practice. In the mathematical community this problem is known as "partially observed diffusions" [9,10]. There are approaches to deal with such problems.…”

Section: Introductionmentioning

confidence: 99%

Analyzing a stochastic process driven by Ornstein-Uhlenbeck noise

Lehle

Peinke

2018

Phys. Rev. E

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A scalar Langevin-type process X(t) that is driven by Ornstein-Uhlenbeck noise η(t) is non-Markovian. However, the joint dynamics of X and η is described by a Markov process in two dimensions. But even though there exists a variety of techniques for the analysis of Markov processes, it is still a challenge to estimate the process parameters solely based on a given time series of X. Such a partially observed 2D-process could, e.g., be analyzed in a Bayesian framework using Markov chain Monte Carlo methods. Alternatively, an embedding strategy can be applied, where first the joint dynamic of X and its temporal derivativeẊ is analyzed. Subsequently the results can be used to determine the process parameters of X and η. In this paper, we propose a more direct approach that is purely based on the moments of the increments of X, which can be estimated for different time-increments τ from a given time series. From a stochastic Taylor-expansion of X, analytic expressions for these moments can be derived, which can be used to estimate the process parameters by a regression strategy.

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Section: Introductionmentioning

confidence: 99%

Analyzing a stochastic process driven by Ornstein-Uhlenbeck noise

Lehle

Peinke

2018

Phys. Rev. E

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“…The same problem is considered in [4] where two-sided stochastic calculus was used to compute . To motivate the robust smoothers presented below, consider computing the smoothed estimate of the last term in (4). One would have liked to have interchanged the conditional expectation and the integral.…”

Section: A Signal Model and Objectivesmentioning

confidence: 99%

“…These smoothed estimates are required in computing the maximum likelihood parameter estimate via the EM algorithm; see Section II-B. The same problem is considered in [4] where two-sided stochastic calculus was used to compute . To motivate the robust smoothers presented below, consider computing the smoothed estimate of the last term in (4).…”

Section: A Signal Model and Objectivesmentioning

confidence: 99%

“…These robust smoothers can be used in maximum likelihood parameter estimation via the expectation maximization (EM) algorithm. The EM algorithm (see Section II-B) is a widely used numerical method for computing the maximum likelihood parameter estimate for partially observed stochastic dynamical systems; see, for example, [23], [4], and [14]. Unlike this paper, in [14] and [9], two-sided stochastic calculus involving Skorohod and generalized Stratonovich integrals are used to derive smoothers for computing estimates of the functionals required in the EM algorithm for HMMs and linear Gaussian state space models, respectively.…”

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confidence: 99%

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Robust continuous-time smoothers without two-sided stochastic integrals

Krishnamurthy

Elliott

2002

IEEE Trans. Automat. Contr.

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Abstract-We consider the problem of fixed-interval smoothing of a continuous-time partially observed nonlinear stochastic dynamical system. Existing results for such smoothers require the use of two-sided stochastic calculus. The main contribution of this paper is to present a robust formulation of the smoothing equations. Under this robust formulation, the smoothing equations are nonstochastic parabolic partial differential equations (with random coefficients) and, hence, the technical machinery associated with two sided stochastic calculus is not required. Furthermore, the robust smoothed state estimates are locally Lipschitz in the observations, which is useful for numerical simulation. As examples, finite dimensional robust versions of the Benes and hidden Markov model smoothers and smoothers for piecewise linear dynamics are derived; these finite-dimensional smoothers do not involve stochastic integrals.Index Terms-Continuous time, hidden Markov models (HMMs), maximum likelihood estimation, nonlinear smoothing, piecewise linear models, stochastic differential equations.

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“…Let Y T denote the σ-algebra generated by the process Y . The hypotheses ensure that the probability measures in M ε are mutually absolutely continuous, thus, according to Campillo and LeGland [4], the likelihood function for estimating the parameter θ given Y T can be expressed as…”

Section: Statistical Model and Residual Definitionmentioning

confidence: 99%

Asymptotic Normality in Partially Observed Diffusions with Small Noise: Application to FDI

LeGland¹,

Wang²

Stochastic Theory and Control

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Abstract. The problem of residual evaluation for fault detection in partially observed diffusions is investigated, using the local asymptotic approach, under the small noise asymptotics. The score function (i.e. the gradient of the log-likelihood function) evaluated at the nominal value of the parameter, and suitably normalized, is used as residual. It is proved that this residual is asymptotically Gaussian, with mean zero under the null hypothesis, with a different mean (depending linearly on the parameter change) and the same covariance matrix under the contiguous alternative hypothesis. This result relies on the local asymptotic normality (LAN) property for the family of probability distributions of the observation process, which is also proved.

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MLE for partially observed diffusions: direct maximization vs. the em algorithm

Cited by 57 publications

References 9 publications

Analyzing a stochastic process driven by Ornstein-Uhlenbeck noise

Analyzing a stochastic process driven by Ornstein-Uhlenbeck noise

Robust continuous-time smoothers without two-sided stochastic integrals

Asymptotic Normality in Partially Observed Diffusions with Small Noise: Application to FDI

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