Special issue on Statistical and Probabilistic Methods for Metrology

Abstract

This special issue of Metrologia is the first that is not devoted to units, or constants, or measurement techniques in some specific field of metrology, but to the generic topic of statistical and probabilistic methods for metrology. The number of papers on this subject in measurement journals, and in Metrologia in particular, has continued to increase over the years, driven by the publication of the Guide to the Expression of Uncertainty in Measurement (GUM) [1] and the Mutual Recognition Arrangement (MRA) of the CIPM [2]. The former stimulated metrologists to think in greater depth about the appropriate modelling of their measurements, in order to provide uncertainty evaluations associated with measurement results. The latter obliged the metrological community to investigate reliable measures for assessing the calibration and measurement capabilities declared by the national metrology institutes (NMIs).Furthermore, statistical analysis of measurement data became even more important than hitherto, with the need, on the one hand, to treat the greater quantities of data provided by sophisticated measurement systems, and, on the other, to deal appropriately with relatively small sets of data that are difficult or expensive to obtain.The importance of supporting the GUM and extending its provisions was recognized by the formation in the year 2000 of Working Group 1, Measurement uncertainty, of the Joint Committee for Guides in Metrology. The need to provide guidance on key comparison data evaluation was recognized by the formation in the year 2001 of the BIPM Director's Advisory Group on Uncertainty. A further international initiative was the revision, in the year 2004, of the remit and title of a working group of ISO/TC 69, Application of Statistical Methods, to reflect the need to concentrate more on statistical methods to support measurement uncertainty evaluation.These international activities are supplemented by national programmes such as the Software Support for Metrology programme in the UK, which includes within its main themes generic items related to modelling, uncertainty evaluation and key comparisons. There are also teams concentrating on statistics within a metrology environment, the largest of which is the Statistical Engineering Division at NIST. There are, however, key pockets of mathematical and statistical expertise at all major and many of the smaller NMIs. Academia also makes considerable input to metrological thinking. The papers in this special issue reflect the above considerations—and more.There are several offerings relating to the GUM: (a) the manner in which the GUM is evolving, especially through Supplements to the GUM, (b) a comparison of the GUM, the GUM Supplement concerned with the propagation of distributions and Bayesian statistics, in the context of linear calibration, (c) theoretical and practical aspects of the use of a Monte Carlo method for propagating distributions, (d) the use of a generalization of the sensitivity coefficients in the GUM to correlated quantities, and (e) considerations on obtaining best estimates when the model is non-linear.At a more fundamental level, a systematic and versatile approach to developing the model of measurement, on which uncertainty evaluation is of course based, is presented, and a paper is included on principles of probability and statistics that promote sound decision-making.The evaluation of key comparison data is represented in terms of contributions relating to (a) models of key comparisons, with measures of operability and interoperability, (b) a Bayesian procedure for providing PDFs from which the measures required by the MRA can be extracted, (c) an extension of the En measure familiar to many metrologists, and (d) the use of the median and weighted median as the key comparison reference value in the presence of discrepant measurement results.The remaining contributions concern the analysis of measurement data, including spectral analysis. Covered are (a) a comparison of conventional and Bayesian approaches to the evaluation of data subject only to random effects, (b) element-wise weighted least squares and its comparison with conventionally weighted and total least squares, (c) the generalized weighted mean of correlated quantities (with an application to key comparisons), (d) the uncertainty associated with the average of autocorrelated quantities, (e) the fitting of three-dimensional geometric elements to coordinate data, (f) the fitting of calibration curves from data having general uncertainty structure, (g) the estimation of the power spectrum of clock noise, (h) Kalman filtering, also in the presence of clock noise, and (i) an improved Allan deviation-like statistic.The realization of such a rich issue involved considerable efforts from many contributors. Thanks are due to the former Editor, Peter Martin, who first conceived the idea of this issue, to Jeffrey Williams, the present Editor, for his support to its realization, to the authors for their contributions, and to the referees who gave their time to review and comment on the papers.