A maximum likelihood approach to recursive polynomial chaos parameter estimation

Pence, Benjamin L.; Fathy, Hosam K.; Stein, Jeffrey L.

doi:10.1109/acc.2010.5531345

Cited by 17 publications

(13 citation statements)

References 9 publications

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“…Hence this can be used wherever samples are needed, and this device has actually been employed, i.e. in the MCMC method [22,16,32,17] as described in Subsection 3.1, or in the EnKF method [18] described in Subsection 3.2.…”

Section: Polynomial Chaos Projectionmentioning

confidence: 99%

“…This can be then sampled by letting the Markov chain run for a sufficiently long time, although the samples are not independent in this case. With the intention of accelerating the MCMC method some authors [22,16,32,17] have introduced stochastic spectral methods into the computation. Expanding the prior random process into a polynomial chaos (PCE) or a Karhunen-Loève expansion (KLE) (e.g.…”

Section: Introductionmentioning

confidence: 99%

“…[24]), the inverse problem becomes an inference on the weights of the Karhunen-Loève modes. Pence et al [32] combine polynomial chaos theory with maximum likelihood estimation, where the parameter estimates are calculated in a recursive or iterative manner. Christen and Fox [6] have applied a local linearisation of the forward model to improve the acceptance probability of proposed moves, while in [2,20,18,39] collocation methods are employed as a more efficient sampling technique.…”

Section: Introductionmentioning

confidence: 99%

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Parameter identification in a probabilistic setting

Rosić

Kučerová

Sýkora

et al. 2013

Engineering Structures

112

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Parameter identification problems are formulated in a probabilistic language, where the randomness reflects the uncertainty about the knowledge of the true values. This setting allows conceptually easily to incorporate new information, e.g. through a measurement, by connecting it to Bayes's theorem. The unknown quantity is modelled as a (may be high-dimensional) random variable. Such a description has two constituents, the measurable function and the measure. One group of methods is identified as updating the measure, the other group changes the measurable function. We connect both groups with the relatively recent methods of functional approximation of stochastic problems, and introduce especially in combination with the second group of methods a new procedure which does not need any sampling, hence works completely deterministically. It also seems to be the fastest and more reliable when compared with other methods. We show by example that it also works for highly nonlinear non-smooth problems with non-Gaussian measures.

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Section: Polynomial Chaos Projectionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Parameter identification in a probabilistic setting

Rosić

Kučerová

Sýkora

et al. 2013

Engineering Structures

112

View full text Add to dashboard Cite

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“…We can assume the vector field models as a nonlinear system, where we only have access to the output measurements defined as the trajectory positions. Previous works related with the identification of nonlinear systems already encompass a variety of approaches [8], [9], [10], [11], [12], [13]. The algorithms used to estimate parameters include expectation-maximization [14] and particle filters [15], [16].…”

Section: Introductionmentioning

confidence: 99%

Recursive Bayesian identification of nonlinear autonomous systems

Simao

Barão

Marques

2012

2012 20th Mediterranean Conference on Control &Amp; Automation (MED)

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Abstract-This paper concerns the recursive identification of nonlinear discrete-time systems for which the original equations of motion are not known. Since the true model structure is not available, we replace it with a generic nonlinear model. This generic model discretizes the state space into a finite grid and associates a set of velocity vectors to the nodes of the grid. The velocity vectors are then interpolated to define a vector field on the complete state space. The proposed method follows a Bayesian framework where the identified velocity vectors are selected by the maximum a posteriori (MAP) criterion. The resulting algorithms allow a recursive update of the velocity vectors as new data is obtained. Simulation examples using the recursive algorithm are presented.

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“…Some of these techniques are: simulated maximum likelihood methods [such as Markov Chain Monte Carlo (MCMC) techniques], [11][12][13][14][15][16] expansion of the likelihood function using Hermite polynomial basis functions, [17,18] solving the FokkerPlanck equation numerically [19][20][21] and recursive maximum likelihood parameter estimation using polynomial chaos theory. [22] Benefits and drawbacks of these techniques are summarised by Lindstrom. [9] An approximate ML method was proposed by Kristensen et al [23] In Kristensen's method, a Gaussian distribution is assumed for the likelihood function and the mean and variance of the likelihood function are estimated using an extended Kalman filter (EKF).…”

mentioning

confidence: 99%

An approximate expectation maximisation algorithm for estimating parameters in nonlinear dynamic models with process disturbances

Karimi

McAuley

2013

Can J Chem Eng

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Stochastic terms are included in fundamental dynamic models of chemical processes to account for disturbances, input uncertainties and model mismatch. The resulting equations are called stochastic differential equations (SDEs). An approximate expectation maximisation (AEM) algorithm using B-splines is developed for estimating parameters in SDE models when the magnitude of the disturbances and model mismatch is unknown. The AEM method is evaluated using a two-state nonlinear continuous stirred tank reactor (CSTR) model. [Varziri et al., Can. J. Chem. Eng. 2008; 86: 828]). For the CSTR examples studied, the AEM algorithm provides more accurate estimates of model parameters, unknown initial conditions and disturbance intensities. SDE models and associated parameter estimates obtained using AEM will be helpful to engineers who subsequently implement on-line state estimation and process monitoring schemes because the two types of uncertainties that are considered (i.e. measurement noise and stochastic process disturbances) are consistent with the error structure used in extended Kalman filters.

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A maximum likelihood approach to recursive polynomial chaos parameter estimation

Cited by 17 publications

References 9 publications

Parameter identification in a probabilistic setting

Parameter identification in a probabilistic setting

Recursive Bayesian identification of nonlinear autonomous systems

An approximate expectation maximisation algorithm for estimating parameters in nonlinear dynamic models with process disturbances

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