The metrology of unstable measurands

Abstract

Type A statistical uncertainty in measurements is usually derived from the standard deviation of the measured data. This is correct as long as the measurand is stable over time and has a meaningful constant value. In such a case the average measurement and the standard deviations are meaningful. However, as measurement methods are refined and become more precise, we can observe that a given measurand may be unstable and change with time and the uncertainty in measurement must be redefined. This is specifically true in the metrology of time which can be measured today more precisly than any other measurand. We argue that in such a case the uncertainty in the prediction of the next measurement should be used instead of the uncertainty in measurement. Both uncertainties coincide for a stable measurand. The prediction of the next measurement is achieved by means of predictors. In this paper we describe the application of linear predictors and especially optimum linear predictors to predict in the presence of various types of instability. To illustrate the issues we use clock instabilities and clock metrology as this field is most developed. A measurand can be unstable but still predictable and thus useful. This is well known in the case of white noise about a linear drift for which the optimum predictor is a linear regression. Since the deviations from prediction of optimum prediction are of white noise, we can now use simple statistics to estimate the uncertainty of the optimum or close to optimum prediction. In this paper we present the various optimum or close to optimum linear predictors optimized for different types of instability and estimate the associated prediction uncertainties.