The well-known problem of fitting a straight line to data with uncertainties in both coordinates is revisited. An algorithm is developed which treats x- and y-data in a symmetrical way. The problem is reduced to a one-dimensional search for a minimum. Global convergence and stability are assured by determining the angle of the straight line with respect to the abscissa instead of the slope. As opposed to previous publications on the subject, the complete uncertainty matrix is calculated, i.e. variances and covariance of the fitting parameters. The algorithm is tested using Pearson's data with York's weights. Although the algorithm is implemented in MATLAB, implementation in a different programming language is straightforward using the formulae presented. An application example is given, a calibration line for dosimetry based on electron spin resonance of alanine is investigated.
Within the scope of the efforts concerning the redefinition of the SI base unit "kg" the final results of the international research project aiming at the redetermination of the Avogadro constant are ready to be announced. Among other quantities the volume of two spheres which are made of a highly enriched 28 Si single crystal had to be determined. For this purpose a special Fizeau interferometer for the measurement of the spheres' diameters has been developed at PTB. This paper reports the final results of the volumes, the uncertainties and also the latest findings regarding systematic corrections including the effects of surface layers on the pure silicon core. The results are confirmed by density comparison measurements.
The well-known problem of fitting a straight line to data with uncertainties in both coordinates is revisited. An algorithm which treats x- and y-data in a symmetrical way and which had been published previously is generalized to the case when there are correlations. Taking known correlations into account helps to reduce the uncertainties of the parameters of the fit which is of major importance in metrology. Although the algorithm is implemented in MATLAB, implementation in a different programming language is straightforward using the formulae presented. The effectiveness of the algorithm is demonstrated with simulated data as well as with experimental data. As application examples, measurements of the temperature coefficient for alanine dosimetry are used.
The Guide to the Expression of Uncertainty in Measurement (GUM) is the de facto standard for the evaluation of measurement uncertainty in metrology. Recently, evaluation of measurement uncertainty has been proposed on the basis of probability density functions (PDFs) using a Monte Carlo method. The relation between this PDF approach and the standard method described in the GUM is outlined. The Monte Carlo method required for the numerical calculation of the PDF approach is described and illustrated by its application to two examples. The results obtained by the Monte Carlo method for the two examples are compared to the corresponding results when applying the GUM.
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