On the theory and application of model misspecification tests in geodesy

Many geodetic testing problems concerning parametric hypotheses may be formulated within the framework of testing linear constraints imposed on a linear Gauss-Markov model. Although geodetic standard tests for such problems are computationally convenient and intuitively sound, no rigorous attempt has yet been made to derive them from a unified theoretical foundation or to establish optimality of such procedures. Another shortcoming of current geodetic testing theory is that no standard approach exists for tackling analytically more complex testing problems, concerning for instance unknown parameters within the weight matrix. To address these problems, it is proven that, under the assumption of normally distributed observation, various geodetic standard tests, such as Baarda’s or Pope’s test for outliers, multivariate significance tests, deformation tests, or tests concerning the specification of the a priori variance factor, are uniformly most powerful (UMP) within the class of invariant tests. UMP invariant tests are proven to be equivalent to likelihood ratio tests and Rao’s score tests. It is also shown that the computation of many geodetic standard tests may be simplified by transforming them into Rao’s score tests. Finally, testing problems concerning unknown parameters within the weight matrix such as autoregressive correlation parameters or overlapping variance components are addressed. It is shown that, although strictly optimal tests do not exist in such cases, corresponding tests based on Rao’s Score statistic are reasonable and computationally convenient diagnostic tools for deciding whether such parameters are significant or not. The thesis concludes with the derivation of a parametric test of normality as another application of Rao’s Score test.Zur Theorie und Anwendung von Modell-Misspezifikationstests in der Geodäsie Was das Testen von parametrischen Hypothesen betrifft, so lassen sich viele geodätische Testprobleme in Form eines Gauss-Markov-Modells mit linearen Restriktionen darstellen. Obwohl geodätische Standardtests rechnerisch einfach und intuitiv vernünftig sind, wurde bisher kein strenger Versuch unternommen, solche Tests ausgehend von einer einheitlichen theoretischen Basis herzuleiten oder die Optimalität solcher Tests zu begründen. Ein weiteres Defizit im gegenwärtigen Verständnis geodätischer Testtheorie besteht darin, dass kein Standardverfahren zum Lösen von analytisch komplexeren Testproblemen exisitiert, welche beispielsweise unbekannte Parameter in der Gewichtsmatrix betreffen. Um diesen Problemen gerecht zu werden wird bewiesen, dass unter der Annahme normalverteilter Beobachtungen verschiedene geodätische Standardtests, wie z.B. Baardas oder Popes Ausreissertest, multivariate Signifikanztests, Deformationstests, oder Tests bzgl. der Angabe des a priori Varianzfaktors, allesamt gleichmäßig beste (engl.: uniformly most powerful - UMP) invariante Tests sind. Es wird ferner bewiesen dass UMP invariante Tests äquivalent zu Likelihood-Quotienten-Tests und Raos Score-Tests sind. Ausserdem wird gezeigt, dass sich die Berechnung vieler geodätischer Standardtests vereinfachen lässt indem diese als Raos Score-Tests formuliert werden. Abschließend werden Testprobleme behandelt in Bezug auf unbekannte Parameter innerhalb der Gewichtsmatrix, beispielsweise in Bezug auf autoregressive Korrelationsparameter oder überlappende Varianzkomponenten. In solchen Fällen existieren keine im strengen Sinne besten Tests. Es wird aber gezeigt, dass entsprechende Tests, die auf Raos Score-Statistik beruhen, sinnvolle und vom Rechenaufwand her günstige Diagnose-Tools darstellen um festzustellen, ob Parameter wie die eingangs erwähnten signifikant sind oder nicht. Am Ende dieser Dissertation steht mit der Herleitung eines parametrischen Tests auf Normalverteilung eine weitere Anwendung von Raos Score-Test

On the theory and application of model misspecification tests in geodesy

Many geodetic testing problems concerning parametric hypotheses may be formulated within the framework of testing linear constraints imposed on a linear Gauss-Markov model. Although geodetic standard tests for such problems are computationally convenient and intuitively sound, no rigorous attempt has yet been made to derive them from a unified theoretical foundation or to establish optimality of such procedures. Another shortcoming of current geodetic testing theory is that no standard approach exists for tackling analytically more complex testing problems, concerning for instance unknown parameters within the weight matrix. To address these problems, it is proven that, under the assumption of normally distributed observation, various geodetic standard tests, such as Baarda's or Pope's test for outliers, multivariate significance tests, deformation tests, or tests concerning the specification of the a priori variance factor, are uniformly most powerful (UMP) within the class of invariant tests. UMP invariant tests are proven to be equivalent to likelihood ratio tests and Rao's score tests. It is also shown that the computation of many geodetic standard tests may be simplified by transforming them into Rao's score tests. Finally, testing problems concerning unknown parameters within the weight matrix such as autoregressive correlation parameters or overlapping variance components are addressed. It is shown that, although strictly optimal tests do not exist in such cases, corresponding tests based on Rao's Score statistic are reasonable and computationally convenient diagnostic tools for deciding whether such parameters are significant or not. The thesis concludes with the derivation of a parametric test of normality as another application of Rao's Score test.Zur Theorie und Anwendung von Modell-Misspezifikationstests in der Geodäsie Was das Testen von parametrischen Hypothesen betrifft, so lassen sich viele geodätische Testprobleme in Form eines Gauss-Markov-Modells mit linearen Restriktionen darstellen. Obwohl geodätische Standardtests rechnerisch einfach und intuitiv vernünftig sind, wurde bisher kein strenger Versuch unternommen, solche Tests ausgehend von einer einheitlichen theoretischen Basis herzuleiten oder die Optimalität solcher Tests zu begründen. Ein weiteres Defizit im gegenwärtigen Verständnis geodätischer Testtheorie besteht darin, dass kein Standardverfahren zum Lösen von analytisch komplexeren Testproblemen exisitiert, welche beispielsweise unbekannte Parameter in der Gewichtsmatrix betreffen. Um diesen Problemen gerecht zu werden wird bewiesen, dass unter der Annahme normalverteilter Beobachtungen verschiedene geodätische Standardtests, wie z.B. Baardas oder Popes Ausreissertest, multivariate Signifikanztests, Deformationstests, oder Tests bzgl. der Angabe des a priori Varianzfaktors, allesamt gleichmäßig beste (engl.: uniformly most powerful - UMP) invariante Tests sind. Es wird ferner bewiesen dass UMP invariante Tests äquivalent zu Likelihood-Quotienten-Tests und Raos Score-Tests sind. Ausserdem wird gezeigt, dass sich die Berechnung vieler geodätischer Standardtests vereinfachen lässt indem diese als Raos Score-Tests formuliert werden. Abschließend werden Testprobleme behandelt in Bezug auf unbekannte Parameter innerhalb der Gewichtsmatrix, beispielsweise in Bezug auf autoregressive Korrelationsparameter oder überlappende Varianzkomponenten. In solchen Fällen existieren keine im strengen Sinne besten Tests. Es wird aber gezeigt, dass entsprechende Tests, die auf Raos Score-Statistik beruhen, sinnvolle und vom Rechenaufwand her günstige Diagnose-Tools darstellen um festzustellen, ob Parameter wie die eingangs erwähnten signifikant sind oder nicht. Am Ende dieser Dissertation steht mit der Herleitung eines parametrischen Tests auf Normalverteilung eine weitere Anwendung von Raos Score-Test

On the impact of correlations on the congruence test: a bootstrap approach: Case study: B-spline surface fitting from TLS observations

The detection of deformation is one of the major tasks in surveying engineering. It is meaningful only if the statistical significance of the distortions is correctly investigated, which often underlies a parametric modelization of the object under consideration. So-called regression B-spline approximation can be performed for point clouds of terrestrial laser scanners, allowing the setting of a specific congruence test based on the B-spline surfaces. Such tests are known to be strongly influenced by the underlying stochastic model chosen for the observation errors. The latter has to be correctly specified, which includes accounting for heteroscedasticity and correlations. In this contribution, we justify and make use of a parametric correlation model called the Matérn model to approximate the variance covariance matrix (VCM) of the residuals by performing their empirical mode decomposition. The VCM obtained is integrated into the computation of the congruence test statistics for a more trustworthy test decision. Using a real case study, we estimate the distribution of the test statistics with a bootstrap approach, where no parametric assumptions are made about the underlying population that generated the random sample. This procedure allows us to assess the impact of neglecting correlations on the critical value of the congruence test, highlighting their importance

Deformation analysis using B-spline surface with correlated terrestrial laser scanner observations-a bridge under load

The choice of an appropriate metric is mandatory to perform deformation analysis between two point clouds (PC)-the distance has to be trustworthy and, simultaneously, robust against measurement noise, which may be correlated and heteroscedastic. The Hausdorff distance (HD) or its averaged derivation (AHD) are widely used to compute local distances between two PC and are implemented in nearly all commercial software. Unfortunately, they are affected by measurement noise, particularly when correlations are present. In this contribution, we focus on terrestrial laser scanner (TLS) observations and assess the impact of neglecting correlations on the distance computation when a mathematical approximation is performed. The results of the simulations are extended to real observations from a bridge under load. Highly accurate laser tracker (LT) measurements were available for this experiment: they allow the comparison of the HD and AHD between two raw PC or between their mathematical approximations regarding reference values. Based on these results, we determine which distance is better suited in the case of heteroscedastic and correlated TLS observations for local deformation analysis. Finally, we set up a novel bootstrap testing procedure for this distance when the PC are approximated with B-spline surfaces

Robust and automatic modeling of tunnel structures based on terrestrial laser scanning measurement

The terrestrial laser scanning technology is increasingly applied in the deformation monitoring of tunnel structures. However, outliers and data gaps in the terrestrial laser scanning point cloud data have a deteriorating effect on the model reconstruction. A traditional remedy is to delete the outliers in advance of the approximation, which could be time- and labor-consuming for large-scale structures. This research focuses on an outlier-resistant and intelligent method for B-spline approximation with a rank (R)-based estimator, and applies to tunnel measurements. The control points of the B-spline model are estimated specifically by means of the R-estimator based on Wilcoxon scores. A comparative study is carried out on rank-based and ordinary least squares methods, where the Hausdorff distance is adopted to analyze quantitatively for the different settings of control point number of B-spline approximation. It is concluded that the proposed method for tunnel profile modeling is robust against outliers and data gaps, computationally convenient, and it does not need to determine extra tuning constants. © The Author(s) 2019

Self-tuning robust adjustment within multivariate regression time series models with vector-autoregressive random errors

The iteratively reweighted least-squares approach to self-tuning robust adjustment of parameters in linear regression models with autoregressive (AR) and t-distributed random errors, previously established in Kargoll et al. (in J Geod 92(3):271–297, 2018. https://doi.org/10.1007/s00190-017-1062-6), is extended to multivariate approaches. Multivariate models are used to describe the behavior of multiple observables measured contemporaneously. The proposed approaches allow for the modeling of both auto- and cross-correlations through a vector-autoregressive (VAR) process, where the components of the white-noise input vector are modeled at every time instance either as stochastically independent t-distributed (herein called “stochastic model A”) or as multivariate t-distributed random variables (herein called “stochastic model B”). Both stochastic models are complementary in the sense that the former allows for group-specific degrees of freedom (df) of the t-distributions (thus, sensor-component-specific tail or outlier characteristics) but not for correlations within each white-noise vector, whereas the latter allows for such correlations but not for different dfs. Within the observation equations, nonlinear (differentiable) regression models are generally allowed for. Two different generalized expectation maximization (GEM) algorithms are derived to estimate the regression model parameters jointly with the VAR coefficients, the variance components (in case of stochastic model A) or the cofactor matrix (for stochastic model B), and the df(s). To enable the validation of the fitted VAR model and the selection of the best model order, the multivariate portmanteau test and Akaike’s information criterion are applied. The performance of the algorithms and of the white noise test is evaluated by means of Monte Carlo simulations. Furthermore, the suitability of one of the proposed models and the corresponding GEM algorithm is investigated within a case study involving the multivariate modeling and adjustment of time-series data at four GPS stations in the EUREF Permanent Network (EPN). © 2020, The Author(s)

Adjustment models for multivariate geodetic time series with vector-autoregressive errors

In this contribution, a vector-autoregressive (VAR) process with multivariate t-distributed random deviations is incorporated into the Gauss-Helmert model (GHM), resulting in an innovative adjustment model. This model is versatile since it allows for a wide range of functional models, unknown forms of auto- and cross-correlations, and outlier patterns. Subsequently, a computationally convenient iteratively reweighted least squares method based on an expectation maximization algorithm is derived in order to estimate the parameters of the functional model, the unknown coefficients of the VAR process, the cofactor matrix, and the degree of freedom of the t-distribution. The proposed method is validated in terms of its estimation bias and convergence behavior by means of a Monte Carlo simulation based on a GHM of a circle in two dimensions. The methodology is applied in two different fields of application within engineering geodesy: In the first scenario, the offset and linear drift of a noisy accelerometer are estimated based on a Gauss-Markov model with VAR and multivariate t-distributed errors, as a special case of the proposed GHM. In the second scenario real laser tracker measurements with outliers are adjusted to estimate the parameters of a sphere employing the proposed GHM with VAR and multivariate t-distributed errors. For both scenarios the estimated parameters of the fitted VAR model and multivariate t-distribution are analyzed for evidence of auto- or cross-correlations and deviation from a normal distribution regarding the measurement noise

Mems based bridge monitoring supported by image-assisted total station

In this study, the feasibility of Micro-Electro-Mechanical System (MEMS) accelerometers and an image-assisted total station (IATS) for short-and long-term deformation monitoring of bridge structures is investigated. The MEMS sensors of type BNO055 from Bosch as part of a geo-sensor network are mounted at different positions of the bridge structure. In order to degrade the impact of systematic errors on the acceleration measurements, the deterministic calibration parameters are determined for fixed positions using a KUKA youBot in a climate chamber over certain temperature ranges. The measured acceleration data, with a sampling frequency of 100 Hz, yields accurate estimates of the modal parameters over short time intervals but suffer from accuracy degradation for absolute position estimates with time. To overcome this problem, video frames of a passive target, attached in the vicinity of one of the MEMS sensors, are captured from an embedded on-axis telescope camera of the IATS of type Leica Nova MS50 MultiStation with a practical sampling frequency of 10 Hz. To identify the modal parameters such as eigenfrequencies and modal damping for both acceleration and displacement time series, a damped harmonic oscillation model is employed together with an autoregressive (AR) model of coloured measurement noise. The AR model is solved by means of a generalized expectation maximization (GEM) algorithm. Subsequently, the estimated model parameters from the IATS are used for coordinate updates of the MEMS sensor within a Kalman filter approach. The experiment was performed for a synthetic bridge and the analysis shows an accuracy level of sub-millimetre for amplitudes and much better than 0.1 Hz for the frequencies. © 2019 M. Omidalizarandi et al

Measurements of branching fraction ratios and CP-asymmetries in suppressed B^- -> D(-> K^+ pi^-)K^- and B^- -> D(-> K^+ pi^-)pi^- decays

We report the first reconstruction in hadron collisions of the suppressed decays B^- -> D(-> K^+ pi^-)K^- and B^- -> D(-> K^+ pi^-)pi^-, sensitive to the CKM phase gamma, using data from 7 fb^-1 of integrated luminosity collected by the CDF II detector at the Tevatron collider. We reconstruct a signal for the B^- -> D(-> K^+ pi^-)K^- suppressed mode with a significance of 3.2 standard deviations, and measure the ratios of the suppressed to favored branching fractions R(K) = [22.0 \pm 8.6(stat)\pm 2.6(syst)]\times 10^-3, R^+(K) = [42.6\pm 13.7(stat)\pm 2.8(syst)]\times 10^-3, R^-(K)= [3.8\pm 10.3(stat)\pm 2.7(syst]\times 10^-3, as well as the direct CP-violating asymmetry A(K) = -0.82\pm 0.44(stat)\pm 0.09(syst) of this mode. Corresponding quantities for B^- -> D(-> K^+ pi^-)pi^- decay are also reported.Comment: 8 pages, 1 figure, accepted by Phys.Rev.D Rapid Communications for Publicatio