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

Abstract

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.

Expanded uncertainty regions for complex quantities in polar coordinates

“…The covariance matrix of a measurement result may also be obtained by propagation of uncertainty through a measurement equation [2,7,8]. In that case, methods of calculating an effective number of degrees of freedom are available [9,10].…”

Evaluating the measurement uncertainty of complex quantities: a selective review

“…While elliptical and rectangular uncertainty regions can be constructed anywhere in the complex plane without regard to the measured value, expressions of uncertainty in polar coordinates when measured values are close to the origin must be treated with care. Current practice is often to report results as independent uncertainty intervals for real-valued magnitude and phase 7 , rather than an annular sector (simultaneous uncertainty intervals for magnitude and phase, forming a region of uncertainty). The coverage probability is, of course, overstated if independent intervals are misinterpreted as describing a region for the complex quantity [23].…”

“…A covariance matrix expressed in IQ coordinates can be obtained by linear transformation of a covariance matrix in when ν = 2. 7 It may not be appropriate to report phase at all when results are very close to the origin. rectangular coordinates, so no information about uncertainty is lost.…”

“…In the case of a real measurand influenced by complex quantities, a simple extension of the Welch-Satterthwaite formula is available that allows correlation between real and imaginary component estimates to be taken into account [5]. In the case of a complex measurand, an effective number of degrees of freedom associated with the uncertainty can be obtained using a different procedure [6,7].…”