Optimality of equidistant sampling designs for the Brownian motion with a quadratic drift

Harman, Radoslav; Štulajter, František

doi:10.1016/j.jspi.2011.02.025

Cited by 8 publications

(13 citation statements)

References 6 publications

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“…Here we state an auxiliary result, which gives an explicit form for the inverse of the matrix Σ = (K(t i , t j )) N i,j=1 , with a triangular covariance kernel K. We did not find this result (as formulated below) in the literature. Versions of Lemma A.1, however, have been derived independently by different authors; see, for example, Lemma 7.3.2 in Zhigljavsky (1991) and formula (8) in Harman andŠtulajter (2011). The proof follows from straightforward checking the condition…”

Section: A1 Explicit Form Of the Inverse Of The Covariance Matrix Ofmentioning

confidence: 99%

Optimal designs in regression with correlated errors

2016

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This paper discusses the problem of determining optimal designs for regression models, when the observations are dependent and taken on an interval. A complete solution of this challenging optimal design problem is given for a broad class of regression models and covariance kernels. We propose a class of estimators which are only slightly more complicated than the ordinary least-squares estimators. We then demonstrate that we can design the experiments, such that asymptotically the new estimators achieve the same precision as the best linear unbiased estimator computed for the whole trajectory of the process. As a by-product we derive explicit expressions for the BLUE in the continuous time model and analytic expressions for the optimal designs in a wide class of regression models. We also demonstrate that for a finite number of observations the precision of the proposed procedure, which includes the estimator and design, is very close to the best achievable. The results are illustrated on a few numerical examples.

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Section: A1 Explicit Form Of the Inverse Of The Covariance Matrix Ofmentioning

confidence: 99%

Optimal designs in regression with correlated errors

2016

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“…and Brus, D. J., Kempen, B. & Heuvelink G. B. M. 19, 22 , but deferent from the finding of some other scholars 17, 18 . The equidistant sampling method also has certain shortcomings, i.e., the overall number of units may contain hidden forms or “unqualified samples”, and the investigators may neglect them during sample selection.…”

Section: Discussionmentioning

confidence: 88%

“…The advantage of this method is that the operation is simple, but it is easy to cause polarization of the system, thus affecting the accuracy of the model and test. The calibration set and the validation set of the equidistant sampling method can make the extracted samples uniformly distributed on the whole, which can eliminate the polarization caused by the simple random sampling 19 . At present, studies exploring the interaction impact capacity between sensitive wavelengths are insufficient, particularly in the areas of modeling and test samples.…”

Section: Introductionmentioning

confidence: 99%

Improving the Accuracy of the Hyperspectral Model for Apple Canopy Water Content Prediction using the Equidistant Sampling Method

Zhao

Zhu

Cheng

et al. 2017

Sci Rep

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The influence of the equidistant sampling method was explored in a hyperspectral model for the accurate prediction of the water content of apple tree canopy. The relationship between spectral reflectance and water content was explored using the sample partition methods of equidistant sampling and random sampling, and a stepwise regression model of the apple canopy water content was established. The results showed that the random sampling model was Y = 0.4797 − 721787.3883 × Z3 − 766567.1103 × Z5 − 771392.9030 × Z6; the equidistant sampling model was Y = 0.4613 − 480610.4213 × Z2 − 552189.0450 × Z5 − 1006181.8358 × Z6. After verification, the equidistant sampling method was verified to offer a superior prediction ability. The calibration set coefficient of determination of 0.6599 and validation set coefficient of determination of 0.8221 were higher than that of the random sampling model by 9.20% and 10.90%, respectively. The root mean square error (RMSE) of 0.0365 and relative error (RE) of 0.0626 were lower than that of the random sampling model by 17.23% and 17.09%, respectively. Dividing the calibration set and validation set by the equidistant sampling method can improve the prediction accuracy of the hyperspectral model of apple canopy water content.

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“…The covariance structure (7) has the so-called product form. Besides the model with the simplest product covariance structure-the Brownian motion with a time-dependent drift [4], [16]-we can find the product covariance structure also in other design problems, see, for instance, [3], [5], [10]. In particular, H a r m a n andŠ t u l a j t e r [3], [5] analysed the autonomous nonstationary Ornstein-Uhlenbeck process discussed in Introduction, which in its original form coincides with the SDE (1) with a constant volatility σ(t) ≡ σ, and assumed the corresponding linear regression model to have a more general response function of the form E[X t ] = (a 1 + b 1 e −κt )θ 1 + (a 2 + b 2 e −κt )θ 2 with a = (a 1 , a 2 ) ′ and b = (b 1 , b 2 ) ′ linearly independent.…”

Section: Planning Of Experiments For a Nonautonomous O-u Processmentioning

confidence: 99%

“…The fact that Σ(τ ) has a product structure and the results of H a r m a n anď S t u l a j t e r [4] imply that for…”

Section: Information Matrix and Optimal Designsmentioning

confidence: 99%

Planning of experiments for a nonautonomous ornstein-uhlenbeck process

Lacko¹

2012

Tatra Mountains Mathematical Publications

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We study exact optimal designs for processes governed by mean- -reversion stochastic differential equations with a time dependent volatility and known mean-reversion speed. It turns out that any mean-reversion It¯o process has a product covariance structure.We prove the existence of a nondegenerate optimal sampling design for the parameter estimation and derive the information matrix corresponding to the observation of the full path. The results are demonstrated on a process with exponential volatility.

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Optimality of equidistant sampling designs for the Brownian motion with a quadratic drift

Cited by 8 publications

References 6 publications

Optimal designs in regression with correlated errors

Optimal designs in regression with correlated errors

Improving the Accuracy of the Hyperspectral Model for Apple Canopy Water Content Prediction using the Equidistant Sampling Method

Planning of experiments for a nonautonomous ornstein-uhlenbeck process

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