Figure 2
From: A Kriging-Based Approach to Autonomous Experimentation with Applications to X-Ray Scattering
Computed variograms for two different synthetic test functions and randomly chosen measurement locations. The dots represent the difference of data \({\rm{\Delta }}{\rho }_{ij}={(\rho ({{\bf{p}}}_{i})-\rho ({{\bf{p}}}_{j}))}^{2}\). The graph represents the fitted variogram. The variogram encompasses information about correlation between function evaluations at points at different distances and is directly used by Kriging to compute adequate measuring steps. The two test function were chosen to highlight the different behavior of the variogram in different situations; one function, that changes smoothly and the step function with a sudden drop. It is important that the variogram translates the correlation of data correctly, since the true model is never previously known. The variogram is built by fitting a predefined function to the squared difference of existing data points in a least-squares manner and is updated after each performed measurement to account for the most recent data. The squared differences in the data are normalized to unity before the fitting. C and D in equations (4) and (5) are functions of the variogram only. The variogram is a function of the euclidean distance. A low-lying variogram means high correlation of data. A steeply rising variogram means statistical low correlation of data. (a) Test function portraying a mathematical model with correlations at many different length scales. (b) Variogram, Îł, for the synthetic test function in (a). Model function values are strongly correlated locally, and also over large distances. The average behavior (red line) suggests that local information can be extrapolated to relatively far-away parts of the parameter space. (c) Test function portraying a mathematical model with small correlations over large distances. (d) Variogram, Îł, for the synthetic test function in (c). While some model function values are highly correlated, there is a large set of pairwise comparisons where the correlation is very poor. The average behavior (red line) suggests that one needs to collect information over relatively short distances in order to confidently reconstruct the model function.
