Gejza Wimmer

Ellipse fitting by nonlinear constraints to demodulate quadrature homodyne interferometer signals and to determine the statistical uncertainty of the interferometric phase

Optical interferometers are widely used in dimensional metrology applications. Among them are many quadrature homodyne interferometers. These exhibit two sinusoidal interference signals shifted, in the ideal case, by 90° to allow a direction sensitive detection of the motion responsible for the actual phase change. But practically encountered signals exhibit additional offsets, unequal amplitudes and a phase shift that differs from 90°. In order to demodulate the interference signals the so called Heydemann correction is used in almost all cases, i.e. an ellipse is fitted to both signals simultaneously to obtain the offsets, amplitude and the phase lag. Such methods are typically based on a simplified least squares fit that leads to a system of linear equations, which can be solved directly in one step. Although many papers related to this subject have been published already only a few of them consider the uncertainties related to this demodulation scheme. In this paper we propose a new method for fitting the ellipse, based on minimization of the geometric distance between the measured and fitted signal values, which provides locally best linear unbiased estimators (BLUEs) of the unknown model parameters, and simultaneously also estimates of the related statistical uncertainties, including the uncertainties of estimated phases and/or displacements.

Univariate linear calibration via replicated errors-in-variables model

In this paper, we deal with the comparative calibration problem, i.e. with the situation when one instrument or measurement technique is calibrated against another, each of which is subject to the measurement error. We propose an approximate, small sample, calibration confidence interval of the unknown true value of the measured substance in units of the more precise instrument, given measurement in units of the less precise instrument. Here we deal with the simplest case—single-use linear univariate calibration, i.e. the case in which we assume linear relationship between the two measurement techniques (instruments), and, further, that the calibration procedure is conducted in order to obtain one value for an unknown, reported together with an interval estimate. The method for deriving the approximate confidence interval is based on estimation of the calibration line via the replicated errors-in-variables model. The model is locally linearized and the Wald-type F-statistic is constructed. An essential point in this approach is the use of the F-approximation of the distribution of the F-statistic suggested by Kenward and Roger [Kenward, M.G. and Roger, J.H., 1997, Small sample inference for fixed effects from restricted maximum likelihood. Biometrics, 53, 983–997]. The statistical properties of the proposed interval estimator are verified by simulations for wide spectrum of experimental designs and compared with two standard univariate interval estimates—the classical approach for deriving the calibration interval was proposed by Eisenhart [Eisenhart, C., 1939, The interpretation of certain methods and their use in biological and industrial research. Annals of Mathematical Statistics, 10, 162–186] and the inverse method was proposed by Krutchkoff [Krutchkoff, R.G., 1967, Classical and inverse methods of calibration. Technometrics, 9, 425–439].

Interval estimation of the mean of a normal distribution based on quantized observations

We consider the problem of making statistical inference about the mean of a normal distribution based on a random sample of quantized (digitized) observations. This problem arises, for example, in a measurement process with errors drawn from a normal distribution and with a measurement device or process with a known resolution, such as the resolution of an analog-to-digital converter or another digital instrument. In this paper we investigate the effect of quantization on subsequent statistical inference about the true mean. If the standard deviation of the measurement error is large with respect to the resolution of the indicating measurement device, the effect of quantization (digitization) diminishes and standard statistical inference is still valid. Hence, in this paper we consider situations where the standard deviation of the measurement error is relatively small. By Monte Carlo simulations we compare small sample properties of the interval estimators of the mean based on standard approach (i.e. by ignoring the fact that the measurements have been quantized) with some recently suggested methods, including the interval estimators based on maximum likelihood approach and the fiducial approach. The paper extends the original study by Hannig et al. (2007).

Estimating the standard uncertainty contribution of the straight-line fit algorithm used to determine the position and the width of a graduation line

We propose an implementation of the straight-line fit algorithm to evaluate data obtained as part of measurements of line structures, e.g. a grating line, which allows us to determine, at a conventional threshold value, the coordinates of the edges, the position and the width of its photometric profile as well as the related standard uncertainty contribution. These are derived for the (1 ? ?)100% confidence intervals with ? [0, 1]. The implementation is demonstrated using data obtained by the PTB Nanometer Comparator (Fluegge and K?ning 2001 Proc. SPIE 4401 275?83). The investigation revealed that large parts of the uncertainty of the position and the width of the line are due to the choice of the fit interval and the uncertainty of the determination of the minimum and maximum level of the photoelectric signal of the line.