Setting-up kriging-based adaptive sampling in metrology

Statistical sampling is a fundamental tool in science, and metrology is no exception. The merit of a sample is its efficiency, i.e. a good trade-off between the information collected and the sample size. Although the sample sites are ordinarily decided prior to the measurements, a different option would be to select them one at a time. This strategy is potentially more informative as the next site can be decided based also on the measurements taken up to that time. The core of the method is to drive the next-site selection by a non-parametric model known as kriging, namely a stationary Gaussian stochastic process with a given autocorrelation structure [1,2]. The main feature of this model is the ability to promptly reconfigure itself, changing the pattern of the predictions and their uncertainty each time a new measurement comes in. Since the model is re-estimated after each added point the sampling procedure is an adaptive one. The next sampling site can be selected via a number of model-based criteria, inspired by the principles of reducing prediction uncertainty or optimizing an objective function, or a combination of the two. The methodology has been applied by the authors [3,4] to design inspection plans for measuring geometric errors using touch-probe Coordinate Measuring Machines (CMM). Results showed that both the non adaptive statistical plans (Random, Latin Hypercube sampling, uniform sampling) and two adaptive deterministic plans from the literature were largely outperformed by the proposed plans both in terms of accuracy and cost. Here we further investigate on a number of important questions related to adaptive kriging: which is the best trade-off between the number of adaptive and non-adaptive points (the latter chosen according to uniform coverage), which next-site selection criteria are more suitable to capturing extreme values of the signal in order to provide a good estimate of the geometric errors.