Karl-Rudolf Koch

MAXIMUM LIKELIHOOD ESTIMATE OF VARIANCE COMPONENTS

SummaryUsing the orthogonal complement likehood function, an iterative procedure for the maximum likelihood estimates of the variance and covariance components is derived. It is shown that these estimates are identical with the reproducing estimates of the locally best invariant quadratic unbiased estimation of variance and covariance components. Successive approximations of the maximum likelihood estimates are given in addition.

Evaluation of uncertainties in measurements by Monte Carlo simulations with an application for laserscanning

Abstract The trapezoidal and triangular distribution recommended by the “Guide to the Expression of Uncertainty in Measurement (GUM)” are derived to obtain methods for drawing random samples from them. It is demonstrated that variances, covariances and confidence intervals of these distributions can be estimated by the random variates. This method is also applicable for the convolution of different types of distributions like the triangular and the normal distribution. Monte Carlo simulations give in addition estimates of the expected values, variances, covariances and confidence regions of unknown parameters derived from the measurements. Furthermore, hypotheses for the parameters can be tested. This method is applied for the determination of the surface of a motorway by laserscanning. The variances of a laserscanner are specified which are sufficiently small to decide whether detected deformations of the surface are permissible or not.

Least squares adjustment and collocation

SummaryFor the estimation of parameters in linear models best linear unbiased estimates are derived in case the parameters are random variables. If their expected values are unknown, the well known formulas of least squares adjustment are obtained. If the expected values of the parameters are known, least squares collocation, prediction and filtering are derived. Hence in case of the determination of parameters, a least squares adjustment must precede a collocation because otherwise the collocation gives biased estimates. Since the collocation can be shown to be equivalent to a special case of the least squares adjustment, the variance of unit weight can be estimated for the collocation also. This estimate gives the scale factor for the covariance matrices being used in the collocation. In addition, the methods of testing hypotheses and establishing confidence intervals for the parameters of the least squares adjustment may be applied to the collocation.

Robust estimations for the nonlinear Gauss Helmert model by the expectation maximization algorithm

For deriving the robust estimation by the EM (expectation maximization) algorithm for a model, which is more general than the linear model, the nonlinear Gauss Helmert (GH) model is chosen. It contains the errors-in-variables model as a special case. The nonlinear GH model is difficult to handle because of the linearization and the Gauss Newton iterations. Approximate values for the observations have to be introduced for the linearization. Robust estimates by the EM algorithm based on the variance-inflation model and the mean-shift model have been derived for the linear model in case of homoscedasticity. To derive these two EM algorithms for the GH model, different variances are introduced for the observations and the expectations of the measurements defined by the linear model are replaced by the ones of the GH model. The two robust methods are applied to fit by the GH model a polynomial surface of second degree to the measured three-dimensional coordinates of a laser scanner. This results in detecting more outliers than by the linear model.

Determining uncertainties of correlated measurements by Monte Carlo simulations applied to laserscanning

Abstract The determination of uncertainties according to the “Guide to the Expression of Uncertainty in Measurement (GUM)” is extended to correlated measurements. The correlations are estimated by repeated observations. Uniform, trapezoidal and triangular distributions recommended by GUM are generalized to multivariate distributions by Monte Carlo simulations. To introduce the Type B component of uncertainty of GUM, which expresses uncertainty based on experience, two methods are discussed. The first one introduces individual corrections to the measurements with identical variances, the second one an identical correction with the same variance to each observation. Correlations are caused by the latter method. The correlations of the coordinates of points measured by the laserscanner Leica HDS 3000 are estimated from repeated observations. Except for the correlations of the three coordinates of the same points, the correlations turn out to be small. Nevertheless, they should be considered for the uncertainties of distances derived from the coordinates, especially if an addition constant for the measured distances of the laserscanner is introduced.

The geopotential from gravity measurements, levelling data and satellite results

The geodetic boundary value problem is formulated which uses as boundary values the differences between the geopotential of points at the surface of the continents and the potential of the geoid. These differences are computed by gravity measurements and levelling data. In addition, the shape of the geoid over the oceans is assumed to be known from satellite altimetry and the shape of the continents from satellite results together with three-dimensional triangulation. The boundary value problem thus formulated is equivalent to Dirichlets exterior problem except for the unknown potential of the geoid. This constant is determined by an integral equation for the normal derivative of the gravitational potential which results from the first derivative of Greens fundamental formula. The general solution, which exists, of the integral equation gives besides the potential of the geoid the solution of the geodetic boundary value problem. In addition approximate solutions for a spherical surface of the earth are derived.

Robust estimation by expectation maximization algorithm

A mixture of normal distributions is assumed for the observations of a linear model. The first component of the mixture represents the measurements without gross errors, while each of the remaining components gives the distribution for an outlier. Missing data are introduced to deliver the information as to which observation belongs to which component. The unknown location parameters and the unknown scale parameter of the linear model are estimated by the EM algorithm, which is iteratively applied. The E (expectation) step of the algorithm determines the expected value of the likelihood function given the observations and the current estimate of the unknown parameters, while the M (maximization) step computes new estimates by maximizing the expectation of the likelihood function. In comparison to Huber’s M-estimation, the EM algorithm does not only identify outliers by introducing small weights for large residuals but also estimates the outliers. They can be corrected by the parameters of the linear model freed from the distortions by gross errors. Monte Carlo methods with random variates from the normal distribution then give expectations, variances, covariances and confidence regions for functions of the parameters estimated by taking care of the outliers. The method is demonstrated by the analysis of measurements with gross errors of a laser scanner.

Gibbs sampler by sampling-importance-resampling

We propose an alternative to the usual time–independent Born–Oppenheimer approximation that is specifically designed to describe molecules with symmetrical Hydrogen bonds. In our approach, the masses of the Hydrogen nuclei are scaled differently from those of the heavier nuclei, and we employ a specialized form for the electron energy level surface. Consequently, anharmonic effects play a role in the leading order calculations of vibrational levels. Although we develop a general theory, our analysis is motivated by an examination of symmetric bihalide ions, such as F H F− or Cl HCl−. We describe our approach for the F H F− ion in detail.

Minimal detectable outliers as measures of reliability

The concept of reliability was introduced into geodesy by Baarda (A testing procedure for use in geodetic networks. Publications on Geodesy, vol. 2. Netherlands Geodetic Commission, Delft, 1968). It gives a measure for the ability of a parameter estimation to detect outliers and leads in case of one outlier to the MDB, the minimal detectable bias or outlier. The MDB depends on the non-centrality parameter of the