Virtual calibration environment for a-priori estimation of measurement uncertainty

During product engineering of a measuring instrument, the question is which measures are necessary to achieve the highest possible measurement accuracy. In this context, a measuring instrument´s target uncertainty is an essential part of its requirement specifications, because it is an indicator for the measurement´s overall quality. This paper introduces an algebraic framework to determine the confidence and prediction intervals of a calibration curve; the matrix based framework greatly simplifies the associated proofs and implementation details. The regression analysis for discrete orthogonal polynomials is derived, and new formulae for the confidence and prediction intervals are presented for the first time. The orthogonal basis functions are numerically more stable and yield more accurate results than the traditional polynomial Vandermonde basis; the methods are thereby directly compared. The new virtual environment for measurement and calibration of cyber-physical systems is well suited for establishing the error propagation chain through an entire measurement system, including complicated tasks such as data fusion. As an example, an adaptable virtual lens model for an optical measurement system is established via a reference measurement. If the same hardware setup is used in different systems, the uncertainty can be estimated a-priori to an individual system´s calibration, making it suitable for industrial applications. With this model it is possible to determine the number of required calibration nodes for system level calibration in order to achieve a predefined measurement uncertainty. Hence, with this approach, systematic errors can be greatly reduced and the remaining random error is described by a probabilistic model. Verification is performed via numerical experiments using a non-parametric Kolmogorov-Smirnov test and Monte Carlo simulation.