Gauss-Markov Estimator Research Articles

Development of a comprehensive software application for realization and dissemination of the mass scale

The main features of the RealMass Calibration software, a tool for mass standards calibration developed at the INRiM, are described. It was developed under the Matlab environment and will be distributed as an executable file on the INRiM website for free download. The software tool has a user-friendly graphical interface to be easily applied by final users. The software is capable of perform advanced mass calibrations processing at the state of the art. Although it can be use at the high-test levels for the mass scale realisation, the user-friendly features also make it suitable for routine calibrations for accredited laboratories. The elaboration of the measurements, defined by the weighing designs, can be performed by the weighted least-squares with Lagrange multipliers and the Gauss Markov estimator. It can take into account uncertainties and correlations of the input quantities, as: between the reference standards, air density measurements and between values of mass and volume of the standards. Tolls to check the consistency of the results are also provided. Validation and the applications to a real case studies are also shown.

Optimal Array Design and Directive Sensors for Guided Waves DoA Estimation.

The estimation of Direction of Arrival (DoA) of guided ultrasonic waves is an important task in many Structural Health Monitoring (SHM) applications. The aim is to locate sources of elastic waves which can be generated by impacts or defects in the inspected structures. In this paper, the array geometry and the shape of the piezo-sensors are designed to optimize the DoA estimation on a pre-defined angular sector, from acquisitions affected by noise and interference. In the proposed approach, the DoA of a wave generated by a single source is considered as a random variable that is uniformly distributed in a given range. The wave velocity is assumed to be unknown and the DoA estimation is performed by measuring the Differences in Time of Arrival (DToAs) of wavefronts impinging on the sensors. The optimization procedure of sensors positioning is based on the computation of the DoA and wave velocity parameters Cramér-Rao Matrix Bound (CRMB) with a Bayesian approach. An efficient DoA estimator is found based on the DToAs Gauss-Markov estimator for a three sensors array. Moreover, a novel directive sensor for guided waves is introduced to cancel out undesired Acoustic Sources impinging from DoAs out of the given angles range. Numerical results show the capability to filter directional interference of the novel sensor and a considerably improved DoA estimation performance provided by the optimized sensor cluster in the pre-defined angular sector, as compared to conventional approaches.

Joint estimation of gravity anomalies using second and third order potential derivatives

Abstract Satellite gradiometry data provide the framework for estimating and validating Earth's gravity field from second and third order derivatives of the Earth's gravitational potential. Such procedures are especially useful when applied locally, as they relate to local and regional characteristics of the real gravity field. In the present study a joint inversion procedure is proposed for the estimation of gravity anomalies at sea surface level from second and third order potential derivatives, based on a standard Gauss-Markov estimation model. The estimation procedure is applied for a test area stretching over Iran involving simulated grids from GOCE-only model GGM_TIM_R05 at GOCE altitude and gravity anomalies recovered at sea level. In order to validate the proposed estimation three different reductions have been considered independently, namely the removal of the long-wavelength part of the observed field through a global gravity model, the removal of the high-frequency part of the field through the incorporation of a topographic/isostatic gravity model and the application of variance component estimation. The application of a global gravity model leads to an improvement in the individual component estimation of the order of magnitude 3 per cent to 73 per cent, with a significant reduction in bias to 4 mGal. Smoother gradient components can come out according to removing the topography and taking into account for isostasy that improved up results of recovery to 25 per cent for the radial second order derivative. Finally, the implementation of variance component estimation leads to no significant improvement in results of recovered gravity anomalies.

Dark Count Resilient Time Estimators for Time-of-Flight PET

A better timing resolution for positron emission tomography (PET) detectors adds time-of-flight information to increase the image contrast. The timing resolution can be improved by timestamping multiple photons from a scintillation event and combining the information using the mean or the Gauss–Markov estimator. To circumvent the performance degradation incurred by these estimators at higher dark count rates (DCRs), two dark count resilient algorithms were studied independently in the literature: the filtered Gauss–Markov and the artificial neural networks. This paper presents parametric simulations that compare them under different DCR conditions. We studied two different types of scintillator-based digital silicon photomultipliers: a typical $1\times 1\times 10$ mm3 LYSO:Ce crystal paired with a 180 ps single photon timing resolution (SPTR) photodetector and a fast $1\times 1\times 3$ mm3 LYSO:Ce:rCa crystal paired with a 35 ps SPTR photodetector. For a single photon avalanche diode with DCR over 1 Hz/ $\mu {\mathrm{ m}}^{2}$ , simulations show a significant coincidence timing resolution (CTR) gain when using the filtered Gauss–Markov estimator, improving the CTR of the typical detector by 10% and the fast detector by 33% over the unfiltered Gauss–Markov estimator. In all evaluated cases, the filtered Gauss–Markov estimator yields the lowest CTR.

Evaluating direction of arrival uncertainty in undersea canyons with internal tides

The slopes of undersea canyon regions impart complexity to underwater acoustic fields for two reasons: intricate patterns of reflection from the seabed, and sound-speed anomalies from internal tides generated at the slopes. Both may spread the horizontal directional spectrum of sound from “line of sight” to a source. The directional spectrum is tied to the covariance matrix of the field, a fundamental quantity that can be measured or modeled. Here, sound-field horizontal-lag spatial covariance matrices and other derived quantities are generated from time-stepped 3D parabolic equation acoustic simulations made using sound-speed fields from ocean models. The covariance matrices are then inserted into the direction-of-arrival (DOA) estimation problem. Analysis of DOA estimates and error bounds is done for a conventional beamformer and for a Gauss-Markov inverse-based beamformer for a variety of signal-to-noise ratios. The method treats non-line of sight acoustic energy as a form of noise. At low signal-to-noise ratio, the Gauss-Markov estimator can perform better that the other. This analysis of field variability allows performance degradation caused by evolving ocean structures to be directly compared to other detrimental influences such as excess noise and array deformation. Detection, localization, and tracking are affected by the processes examined here.

On estimating regression coefficients in seemingly unrelated regression system

In the system of m (m ≥ 2) seemingly unrelated regressions, we show that the Gauss-Markov estimator (GME) of any regression coefficients has unique simplified form, which exactly equals to the one-step covariance-adjusted estimator of the regression coefficients, and hence we conclude that for any finite k ≥ 2 the k-step covariance-adjusted estimator degenerates to the one-step covariance-adjusted estimator and the corresponding two-stage Aitken estimator has exactly one simplified form. Also, the unique simplified expression of the GME is just the estimator presented in the Theorem 1 of Wang’ work [1988]. A new estimate of regression coefficients in seemingly unrelated regression system, Science in China, Series A 10, 1033-1040].

Enhanced design efficiency through least upper bounds

Lower and upper spectral bounds are known for positive-definite matrices in under Loewner (Uber monotone Matrixfunktionen. Math Z. 1934;38:177–216) ordering. Lower and upper singular bounds for matrices of order in derive under an induced ordering. These orderings are combined here to the following effects. Given two first-order experimental designs in their upper singular bound enhances both and in that its Fisher Information matrix dominates those for both and thus ordering essentials in Gauss–Markov estimation. Moreover, if and are dispersion matrices for linear estimators under and respectively, then is the spectral lower bound for in . In essence this algorithm identifies elements in complementary to those of and combines these into . Case studies illustrate gains to be made thereby in first and second-order designs. Specifically, two examples demonstrate that designs optimal under separate criteria may be combined into a single design dominating both. In addition, selected examples demonstrate that classical second-order designs may be improved inter se.

Mean driven balance and uniformly best linear unbiased estimators

The equivalence of ordinary least squares estimators (OLSE) and Gauss–Markov estimators for models with variance–covariance matrix \(\sigma ^2{\mathbf M}\) is extended to derive a necessary and sufficient balance condition for mixed models with mean vector \({\varvec{\mu }} = {{\mathbf X} {\varvec{\beta }}}\), with \({\mathbf {X}}\) an incidence matrix, having OLSE for \(\varvec{\beta }\) that are best linear unbiased estimator whatever the variance components. This approach leads to least squares like estimators for variance components. To illustrate the range of applications for the balance condition, interesting special models are considered.

Passive acoustic localisation using blind Gauss–Markov estimate with spectral estimation at each sensor

Time-delay estimation has essential applications in the field of radar, sonar and robotics. For a very distant source, time-delay vector estimation across an M-sensor array is realised using the generalised cross-correlation (GCC) function and the estimates combined with their covariance matrix, expressed in terms of a priori known signal and noise spectra, to yield a linear minimum variance unbiased estimator, known as the Gauss–Markov Estimate. In the absence of a priori information, spectral estimation has to be done at one of the sensors. For close range sources, the use of amplitude attenuation information across the array will improve this estimate further. This study presents a new amplitude information related Gauss–Markov estimate, which calculates the power spectral density (PSD) at all the sensors of the array and gives more accurate time delay estimates in terms of the mean-square error when compared to the earlier constant amplitude-based technique using the PSD at the closest sensor. The performance has been evaluated against signal-to-noise ratio for varying distance of a source from the receiving array. The results have been verified by simulations and experiments for a two-dimensional source localisation problem.

Construction of confidence regions for motion trajectories of objects in computer vision problems

The problem of finding an object lost for several number of frames is considered within the framework of motion analysis problem. A linear model of objects motion is designed on a finite time interval. The hypotheses on independence and normality of observation errors are tested. The Gauss-Markov estimate is found for motion parameters. Based on available measurements, hyperbolic confidence tubes are constructed for each of the coordinates of the center of mass of the moving object. The technique for derivation of the equation of an ellipse of dispersion for the two-dimensional Gauss vector of a general form is demonstrated. The form of the ellipse of dispersion for the addressed problem is found. The results of testing the algorithm for open bases of video sequences PETS and ETISEO are presented.

Let us do the twist again

Krämer (Sankhy $$\bar{\mathrm{a }}$$ 42:130–131, 1980) posed the following problem: “Which are the $$\mathbf{y}$$ , given $$\mathbf{X}$$ and $$\mathbf{V}$$ , such that OLS and Gauss–Markov are equal?”. In other words, the problem aimed at identifying those vectors $$\mathbf{y}$$ for which the ordinary least squares (OLS) and Gauss–Markov estimates of the parameter vector $$\varvec{\beta }$$ coincide under the general Gauss–Markov model $$\mathbf{y} = \mathbf{X} \varvec{\beta } + \mathbf{u}$$ . The problem was later called a “twist” to Kruskal’s Theorem, which provides conditions necessary and sufficient for the OLS and Gauss–Markov estimates of $$\varvec{\beta }$$ to be equal. The present paper focuses on a similar problem to the one posed by Krämer in the aforementioned paper. However, instead of the estimation of $$\varvec{\beta }$$ , we consider the estimation of the systematic part $$\mathbf{X} \varvec{\beta }$$ , which is a natural consequence of relaxing the assumption that $$\mathbf{X}$$ and $$\mathbf{V}$$ are of full (column) rank made by Krämer. Further results, dealing with the Euclidean distance between the best linear unbiased estimator (BLUE) and the ordinary least squares estimator (OLSE) of $$\mathbf{X} \varvec{\beta }$$ , as well as with an equality between BLUE and OLSE are also provided. The calculations are mostly based on a joint partitioned representation of a pair of orthogonal projectors.

On efficient estimators of two seemingly unrelated regressions

In this paper, following the results presented in Liu’s work [Liu, A.Y., 2002. Efficient estimation of two seemingly unrelated regression equations. Journal of Multivariate Analysis 82, 445–456], we first represent the Gauss–Markov estimator of the regression parameter as a matrix series, and hence we conclude that the observation vectors should appear in any efficient estimator in pairs. Second, we prove that the simpler form of the two-stage Aitken estimator is unique. Finally we generalize our results to the system of two seemingly unrelated regressions with unequal numbers of observations and briefly summarize our conclusions.

On the existence of the Gauss-Markov estimators in linear mixed models

The purpose of this article is to present a simpler way to verify the existence of the Gauss-Markov estimators (GME) of the expected mean which are both the best linear unbiased estimators (BLUE) of the mean in a multivariate mixed linear model and the best quadratic unbiased estimators (BQUE) of the covariance components in a new linear model built from the initial one. The results are expressed in a coordinate-free approach in order to accord them an intuitive representation in finite dimensional Hilbert spaces.

A Kinship-Based Modification of the Armitage Trend Test to Address Hidden Population Structure and Small Differential Genotyping Errors

Background/AimsWe propose a modification of the well-known Armitage trend test to address the problems associated with hidden population structure and hidden relatedness in genome-wide case-control association studies.MethodsThe new test adopts beneficial traits from three existing testing strategies: the principal components, mixed model, and genomic control while avoiding some of their disadvantageous characteristics, such as the tendency of the principal components method to over-correct in certain situations or the failure of the genomic control approach to reorder the adjusted tests based on their degree of alignment with the underlying hidden structure. The new procedure is based on Gauss-Markov estimators derived from a straightforward linear model with an imposed variance structure proportional to an empirical relatedness matrix. Lastly, conceptual and analytical similarities to and distinctions from other approaches are emphasized throughout.ResultsOur simulations show that the power performance of the proposed test is quite promising compared to the considered competing strategies. The power gains are especially large when small differential differences between cases and controls are present; a likely scenario when public controls are used in multiple studies.ConclusionThe proposed modified approach attains high power more consistently than that of the existing commonly implemented tests. Its performance improvement is most apparent when small but detectable systematic differences between cases and controls exist.

Statistical analysis of diabetes mellitus

This paper deals with an application of regression analysis to the regulation of the blood-sugar under diabetes mellitus. Section 2 gives a description of Gram-Schmidt orthogonalization, while Section 3 discusses the dierence between Gauss-Markov estimation and Least Squares Estimation. Section 4 is devoted to the statistical analysis of the blood-sugar during the night. The response change of bloodsugar is explained by three variables: time, food and physical activity (Bewegung). At the beginning of the section it is shown that the proposed method was very successful in 2007.

On Training Optimization for Estimation of Correlated MIMO Channels in the Presence of Multiuser Interference

In this paper, the problem of estimating multiple-input multiple-output (MIMO) channels in a realistic environment involving correlated channel fading and multiuser interference is considered. Four estimation schemes are studied, including the linear minimum mean squared error (LMMSE), least squares (LS), and Gauss-Markov (GM) estimators, as well as a novel scheme which is derived here as an alternative to LMMSE estimation. The MSE-optimal training sequences for each of them are provided and their requirements for side information feedback are assessed. The new scheme is shown to exhibit a performance comparable to or even better than LMMSE, at a significantly lower feedback and computational cost. The analytical comparison of the estimation schemes is supported by numerous simulation results that cover a wide range of antenna configurations, relative interference power, and channel correlation strengths. The results of this paper provide a complete picture for a palette of estimation schemes, with their relative performance and costs of training.

MV-PURE Estimator: Minimum-Variance Pseudo-Unbiased Reduced-Rank Estimator for Linearly Constrained Ill-Conditioned Inverse Problems

This paper proposes a novel estimator named minimum-variance pseudo-unbiased reduced-rank estimator (MV- PURE) for the linear regression model, designed specially for the case where the model matrix is ill-conditioned and the unknown deterministic parameter vector to be estimated is subjected to known linear constraints. As a natural generalization of the Gauss-Markov (BLUE) estimator, the MV-PURE estimator is a solution of the following hierarchical nonconvex constrained optimization problem directly related to the mean square error expression. In the first-stage optimization, under a rank constraint, we minimize simultaneously all unitarily invariant norms of an operator applied to the unknown parameter vector in view of suppressing bias of the proposed estimator. Then, in the second-stage optimization, among all pseudo-unbiased reduced-rank estimators defined as the solutions of the first-stage optimization, we find the one achieving minimum variance. We derive a closed algebraic form of the MV-PURE estimator and show that well-known estimators-the Gauss-Markov (BLUE) estimator, the generalized Marquardt's reduced-rank estimator, and the minimum-variance conditionally unbiased affine estimator subject to linear restrictions-are all special cases of the MV-PURE estimator. We demonstrate the effectiveness of the proposed estimator in a numerical example, where we employ the MV-PURE estimator to the ill-conditioned problem of reconstructing a 2-D image subjected to linear constraints from blurred, noisy observation. This example demonstrates that the MV-PURE estimator outperforms all aforementioned estimators, as it achieves smaller mean square error for all values of signal-to-noise ratio.

Uncertainty on differential measurements and its reduction using the calibration by comparison method

The paper deals with the uncertainty of differential measurements, obtained from the subtraction of a pair of absolute measurements. It is shown that if the same sensor is used to perform both measurements, a model of the sensor will reveal a correlation component between the uncertainty of each absolute measurement, reducing the uncertainty on its subtraction. The procedure followed is based on the Gauss–Markov estimation method, showing that differential measurement uncertainty vanishes when the gradient to be measured is zero. If the two absolute measurements are to be performed using different sensors, a calibration by comparison between them will result in a similar uncertainty reduction. Finally, a simulated example based on commercially available thermistor data is included.

Subspace fitting approaches for frequency estimation using real-valued data

A novel data covariance model has recently been proposed for the subspace-based estimation of multiple real-valued sine wave frequencies. In this paper, we develop weighted subspace fitting approaches using this new data model. A new parameterization of the noise subspace is proposed. This parameterization is used to solve the subspace fitting problem analytically. An expression for the residual covariance matrix is derived. This covariance matrix is further used to obtain an optimally weighted Gauss-Markov estimator. A computationally efficient suboptimal weighting is also proposed, and the associated estimator is close to the Gauss-Markov estimator in performance. The suboptimal weighting strategy is quite general and can be used in other related applications. The performance of the algorithms are illustrated using numerical simulations. The proposed subspace fitting approach shows improved resolution performance. It is also robust to additive noise.

Border detection on Common Carotid Artery using Gauss-Markov Estimation

The arteriosclerosis is on the increase with an aging or change of our living environment. For that reason, diagnosis of the common carotid artery using echocardiogram is doing to take precautions carebropathy. The arteriosclerosis of the common carotid artery is diagnosed using Intima-Media Thickness (IMT) which is obtained from echocardiogram. In order to measure IMT from echocardiogram, it is required to detect a border which is a boundary between vessel tissue layers. The method of border detection requires reproducibility, high accuracy and high-speed.In this paper, we propose the high-accuracy and high-speed detection method by Gauss-Markov estimation. About high-accuracy, it realized by attaching importance to high reliable candidate point of border. And, about high-speed, it realized by calculating matrix only one time.