Optimal designs for parameter estimation of the Ornstein–Uhlenbeck process

Zagoraiou, Maroussa; Antognini, Alessandro Baldi

doi:10.1002/asmb.749

Cited by 21 publications

(23 citation statements)

References 31 publications

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“…This was also independently observed by Zagoraiou and Baldi-Antognini (2009). For a proof first notice that an Ornstein-Uhlenbeck process possesses a Markovian property and thus the structure of the inverse covariance matrix is tridiagonal (see for details Kiseľák and Stehlík, 2008).…”

Section: One-dimensional Examplementioning

confidence: 70%

Compound optimal spatial designs

Müller

Stehlík

2009

Environmetrics

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A single purpose design may be quite inefficient for handling a real-life problem. Therefore, we often need to incorporate more than one design criterion and a common approach is simply to construct a weighted average, which may depend upon different information matrices. Designs based upon this method have been termed compound designs. The need to satisfy more than one design criterion is particularly relevant in the context of random fields. It is evident that for precise universal kriging it is important not only to efficiently estimate the spatial trend parameters, but also the parameters of the variogram or covariance function. Both tasks could for instance be comprised by applying corresponding design criteria and constructing a compound design from there. Modern techniques for such first and second order characteristics will be suggested and reviewed in the presentation. A new hybrid stochastic exchange type optimization algorithm is proposed and an illustrating example of the design of a water-quality monitoring network is provided. COMPOUND OPTIMAL SPATIAL DESIGNS 355 Case 1. We are interested only in the trend parameters β and consider θ as known or a nuisance. Case 2. We are interested only in the covariance parameters θ. Case 3. We are interested in both sets of parameters. periment. In the following, we will define optimality of a design always strictly in the tradition of Kiefer (see e.g. Kiefer, 1959), where the inputs are selected such, that a prespecified design criterion (e.g. the determinant of the trend parameters variance-covariance matrix, so-called D-optimality) is optimized. The classic Fisher information M(β) = E[(∂ ln f (x, β))/∂β) 2 ], which is the basis for We also use now the notation [C(ξ)] ii = c(x i , x i ; θ) to emphasize the dependence of C on the design, and we assume knowledge of θ.Again all the entries can be condensed to a design criterion . If we now want to determine a Doptimal design for the whole parameter set, we can obviously-due to the orthogonality of the first and second order parameters-simply use the product of the respective determinants as an optimum design

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Section: One-dimensional Examplementioning

confidence: 70%

Compound optimal spatial designs

Müller

Stehlík

2009

Environmetrics

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“…Observe that none of the entries of the FIM I λ,ω (n) depends on the frequency parameter. Further, I λ (n)/2 coincides with the Fisher information on the covariance parameter of a real valued OU process given by Zagoraiou and Baldi Antognini (2009). Note that here we consider two-dimensional OU processes, which justifies the halving of the information, however, due to this connection the first statement of the following theorem is a direct consequence of Zagoraiou and Baldi Antognini (2009, Theorem 4.2).…”

Section: Estimation Of the Covariance Parametersmentioning

confidence: 71%

D-optimal designs for complex Ornstein–Uhlenbeck processes

Baran

Szak-Kocsis

Stehlík

2018

Journal of Statistical Planning and Inference

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Complex Ornstein-Uhlenbeck (OU) processes have various applications in statistical modelling. They play role e.g. in the description of the motion of a charged test particle in a constant magnetic field or in the study of rotating waves in time-dependent reaction diffusion systems, whereas Kolmogorov used such a process to model the so-called Chandler wobble, small deviation in the Earth's axis of rotation. In these applications parameter estimation and model fitting is based on discrete observations of the underlying stochastic process, however, the accuracy of the results strongly depend on the observation points.This paper studies the properties of D-optimal designs for estimating the parameters of a complex OU process with a trend. We show that in contrast with the case of the classical real OU process, a D-optimal design exists not only for the trend parameter, but also for joint estimation of the covariance parameters, moreover, these optimal designs are equidistant.

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“…An A-optimal design minimizes the trace of the inverse of the Fisher information matrix (FIM) on the unknown parameters, whereas E-, T-and D-optimal designs maximize the smallest eigenvalue, the trace and the determinant of the FIM, respectively (see, e.g., Pukelsheim, 1993;Abt and Welch, 1998;Pázman, 2007). The latter design criterion for regression experiments has been studied by several authors both in uncorrelated (see, e.g., Silvey, 1980) and in correlated setups (Müller and Stehlík, 2004;Kiseľák and Stehlík, 2008;Zagoraiou and Baldi Antognini, 2009;Dette et al, 2015). However, there are several situations when D-optimal designs do not exist, for instance, if one has to estimate the covariance parameter(s) of an Ornstein-Uhlenbeck (OU) process (Zagoraiou and Baldi Antognini, 2009) or sheet (Baran et al, 2015).…”

Section: Introductionmentioning

confidence: 99%

“…The latter design criterion for regression experiments has been studied by several authors both in uncorrelated (see, e.g., Silvey, 1980) and in correlated setups (Müller and Stehlík, 2004;Kiseľák and Stehlík, 2008;Zagoraiou and Baldi Antognini, 2009;Dette et al, 2015). However, there are several situations when D-optimal designs do not exist, for instance, if one has to estimate the covariance parameter(s) of an Ornstein-Uhlenbeck (OU) process (Zagoraiou and Baldi Antognini, 2009) or sheet (Baran et al, 2015). This deficiency can obviously be corrected by choosing a more appropriate design criterion.…”

Section: Introductionmentioning

confidence: 99%

K-optimal designs for parameters of shifted Ornstein–Uhlenbeck processes and sheets

Baran

2017

Journal of Statistical Planning and Inference

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Continuous random processes and fields are regularly applied to model temporal or spatial phenomena in many different fields of science, and model fitting is usually done with the help of data obtained by observing the given process at various time points or spatial locations. In these practical applications sampling designs which are optimal in some sense are of great importance. We investigate the properties of the recently introduced K-optimal design for temporal and spatial linear regression models driven by Ornstein-Uhlenbeck processes and sheets, respectively, and highlight the differences compared with the classical D-optimal sampling. A simulation study displays the superiority of the K-optimal design for large parameter values of the driving random process.

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Optimal designs for parameter estimation of the Ornstein–Uhlenbeck process

Cited by 21 publications

References 31 publications

Compound optimal spatial designs

Compound optimal spatial designs

D-optimal designs for complex Ornstein–Uhlenbeck processes

K-optimal designs for parameters of shifted Ornstein–Uhlenbeck processes and sheets

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