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Evaluating effective degrees-of-freedom

A classical method for uncertainty analysis with multidimensional

R Willink and B D Hall

Metrologia, 2002, 39, 361-369


A method of uncertainty analysis based on classical statistical principles is presented for a measurand that is a linear combination of multidimensional input quantities. The method assigns the measurand a combined standard uncertainty matrix and an effective degrees of freedom, which allows the measurand to be estimated by an ellipsoidal confidence region in the multidimensional space. Simulations for a 95 % nominal confidence level show the ellipsoids to contain the measurand with probability approximately 0.95, as required. The derivation of the method assumes all input uncertainties to be evaluated by the Type A method. So the method is analogous to the Welch-Satterthwaite formula for one-dimensional data, a derivation of which is given in an appendix.

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An extension to GUM methodology: degrees-of-freedom calculations for correlated multidimensional estimates

R Willink and B D Hall

ArXiv:1311.0343, Nov, 2013 (30 pages)


The Guide to the Expression of Uncertainty in Measurement advocates the use of an 'effective number of degrees of freedom' for the calculation of an interval of measurement uncertainty. However, it does not describe how this number is to be calculated when (i) the measurand is a vector quantity or (ii) when the errors in the estimates of the quantities defining the measurand (the 'input quantities') are not incurred independently. An appropriate analysis for a vector-valued measurand has been described (Metrologia 39 (2002) 361-9), and a method for a one-dimensional measurand with dependent errors has also been given (Metrologia 44 (2007) 340-9). This paper builds on those analyses to present a method for the situation where the problem is multidimensional and involves correlated errors. The result is an explicit general procedure that reduces to simpler procedures where appropriate. The example studied is from the field of radio-frequency metrology, where measured quantities are often complex-valued and can be regarded as vectors of two elements.

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