An alternative expression for the variance factors in using Iterated Almost Unbiased Estimation

Hsu, Rongshin

doi:10.1007/s001900050234

Cited by 25 publications

(5 citation statements)

References 2 publications

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“…To facilitate the calculation of the variance factor, it is approximated by Eq. (29): 33 fi≈[dei]ZtTWk˜iWZt,i=1,2,DE=I−H(HTR−1H)−1HTR−1,where [dei] represents the sum of all diagonal elements in the i’th group of matrix DE.…”

Section: Methodsmentioning

confidence: 99%

Settlement monitoring data fusion approach for high-speed railways based on GNSS and InSAR

Qiu,

Wang,

Zhang

et al. 2023

J. Appl. Rem. Sens.

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.The high-precision and high spatio-temporal resolution settlement time series data generated by integrating Global Navigation Satellite System (GNSS) and Interferometric Synthetic-Aperture Radar (InSAR) data are of great value for the safe operation of high-speed railways. The GNSS monitoring stations and InSAR monitoring area present zonal distribution along the high-speed railways, which causes the spatial matrix of the GNSS and InSAR data fusion model to be an ill-conditioned matrix. The inverse of the ill-conditioned space matrix is unstable, and it is difficult to estimate the optimal parameter value of the fusion model, which leads to low accuracy and large fluctuations in the fusion of GNSS and InSAR data. We propose a spatio-temporal filter fusion (STFF) method to solve the influence of the ill-conditioned space matrix on the fusion of GNSS and InSAR data, which realizes the high-precision fusion of GNSS and InSAR data along the high-speed railway. First, we propose the kriging model of the adaptive spectral correction method to construct the spatial model of GNSS and InSAR data, avoiding the generation of the ill-conditioned space matrix. Second, iterative almost unbiased estimation is used to obtain the weights of GNSS and InSAR data, avoiding the occurrence of negative variance components. The experimental data are fused using STFF, the spatio-temporal Kalman filter (STKF) method, and the spatio-temporal random effect (STRE) fusion method, respectively. The research results show that the comprehensive performance of STFF is significantly better than that of STKF and STRE. In the spatial domain, the root mean square error (RMSE) reduced by 47% and 46% compared with STKF and STRE, respectively, and the structural similarity index increased by 22% and 16% compared with STKF and STRE, respectively. In the time domain, the RMSE reduced by 29% and 34% compared with STKF and STRE, respectively.

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Section: Methodsmentioning

confidence: 99%

Settlement monitoring data fusion approach for high-speed railways based on GNSS and InSAR

Qiu,

Wang,

Zhang

et al. 2023

J. Appl. Rem. Sens.

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“…If the ratio of the a priori values for the variance factors is correct (when they are known up to a factor α) this formula gives unbiased estimators. An alternative formula for almost unbiased estimation of variance factors is given by Hsu (1999). One can show that the above estimator leads, at the point of convergence, to the results of the iterated estimates of the local MINQUE, BIQUE or REML estimates (see Koch, 1986Koch, , 1987Ou, 1989;Koch, 1999).…”

Section: Simplified Methodsmentioning

confidence: 99%

Least-Squares Variance Component Estimation. Theory and GPS Applications

Amiri-Simkooei¹

2007

Publications on Geodesy

135

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Data processing in geodetic applications often relies on the least-squares method, for which one needs a proper stochastic model of the observables. Such a realistic covariance matrix allows one first to obtain the best (minimum variance) linear unbiased estimator of the unknown parameters; second, to determine a realistic precision description of the unknowns; and, third, along with the distribution of the data, to correctly perform hypothesis testing and assess quality control measures such as reliability. In many practical applications the covariance matrix is only partly known. The covariance matrix is then usually written as an unknown linear combination of known cofactor matrices. The estimation of the unknown (co)variance components is generally referred to as variance component estimation (VCE).In this thesis we study the method of least-squares variance component estimation (LS-VCE) and elaborate on theoretical and practical aspects of the method. We show that LS-VCE is a simple, flexible, and attractive VCE-method. The LS-VCE method is simple because it is based on the well-known principle of least-squares. With this method the estimation of the (co)variance components is based on a linear model of observation equations. The method is flexible since it works with a user-defined weight matrix. Different weight matrix classes can be defined which all automatically lead to unbiased estimators of (co)variance components. LS-VCE is attractive since it allows one to apply the existing body of knowledge of least-squares theory to the problem of (co)variance component estimation. With this method, one can 1) obtain measures of discrepancies in the stochastic model, 2) determine the covariance matrix of the (co)variance components, 3) obtain the minimum variance estimator of (co)variance components by choosing the weight matrix as the inverse of the covariance matrix, 4) take the a-priori information on the (co)variance component into account, 5) solve for a nonlinear (co)variance component model, 6) apply the idea of robust estimation to (co)variance components, 7) evaluate the estimability of the (co)variance components, and 8) avoid the problem of obtaining negative variance components.LS-VCE is capable of unifying many of the existing VCE-methods such as MINQUE, BIQUE, and REML, which can be recovered by making appropriate choices for the weight matrix. An important feature of the LS-VCE method is the capability of applying hypothesis testing to the stochastic model, for which we rely on the w-test, v-test, and overall model test. We aim to find an appropriate structure for the stochastic model which includes the relevant noise components into the covariance matrix. The w-test statistic is introduced to see whether or not a certain noise component is likely to be present in the observations, which consequently can be included in the stochastic model. Based on the normal distribution of the original observables we determine the mean and the variance of the w-test statistic, which are zero and one, respectivel...

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“…There are other methods for estimating the variance and covariance components, such as the best invariant quadratic unbiased estimator by Koch (1999); Crocetto et al (2000) gave a simplified version. Hsu (1999) described an alternative that uses iterated almost unbiased estimation. Based on Baarda's data snooping method, Schwarz and Kok (1993) found that blunders are easily detected by testing the normalized gray-level residuals.…”

Section: Measurement Variancesmentioning

confidence: 99%

Optimum estimator for SPOT-pan and ALS-lidar intensity image registration via tie points

Chang

Hsu

2011

Journal of the Chinese Institute of Engineers

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The characteristics of a Syste`me Pour d'Observation de la Terre (SPOT) panchromatic image of a geographic area such as an airfield are different from those of an airborne laser scanner lidar intensity image of the same area. This study investigates the feasibility of locating tie points shared by the target (airborne lidar-intensity) and search (spaceborne SPOT-pan) images. At an approximate point position, image segmentation and enhancement are employed so that a pair of target and search window images tend to resemble each other. The similarity is then measured by utilizing a least-squares image matching algorithm which involves variance-component estimation. The proposed method has a 3-15% higher success rate than an equal-weight matching method. Coordinate differences of the conjugate image points serve as reference value to be used in the least-squares collocation method for pixel-by-pixel transformation. The image registration accuracy from a SPOT-pan to a lidar intensity image is estimated to be AE1.0 pixel.

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An alternative expression for the variance factors in using Iterated Almost Unbiased Estimation

Cited by 25 publications

References 2 publications

Settlement monitoring data fusion approach for high-speed railways based on GNSS and InSAR

Settlement monitoring data fusion approach for high-speed railways based on GNSS and InSAR

Least-Squares Variance Component Estimation. Theory and GPS Applications

Optimum estimator for SPOT-pan and ALS-lidar intensity image registration via tie points

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