Kriging and Design of Experiments on Circular Domains

Abstract : This research is motivated by the problem of reconstructing a spatial profile in microelectronics. More precisely, the aim is to reconstruct a variable defined on a disk (called wafer) from few measurements, typically less than 20. Furthermore, the spatial profile sometimes contains radial or angular patterns, due to the technological processes involved in their fabrication, such as rotations or diffusions. Among spatial statistics techniques, Kriging (or Gaussian process regression) may be the preferred choice of modelizers. Firstly it provides a measure of uncertainty, due to its stochastic nature. Secondly, it is parameterized in a flexible way by a function, called kernel, that allows incorporating a priori information. We introduce so-called polar Gaussian processes, defined as Gaussian processes on the cylinder of polar coordinates. The corresponding kernel is defined as a combination of a kernel for the radius, and a kernel on the circle for the angle. This typically allows taking into account radial and angular correlations. A construction from the ANOVA decomposition also allows a complete visualization of the different effects (radial, angular and interaction). Of course, the problem of learning on a disk is closely linked to design of experiments. After reviewing the main designs classes, we introduce Latin cylinder designs (LCD), that generalize Latin hypercubes to polar coordinates, and propose two kinds of maximin LCDs. The whole methodology is applied to toy functions, as well as case studies. We observe that polar Gaussian processes significantly outperform the standard Kriging technique, when the spatial profile contains radial or angular patterns. Finally, we evoke two connected works: A relocation strategy based on the IMSE criterion, and an extension in higher dimensions. In particular, it is observed that reconstructing a radial function is done much more accurately with polar Gaussian processes.
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Submitted on : Monday, August 1, 2016 - 11:46:27 AM
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  • HAL Id : hal-01350664, version 1


Olivier Roustant, Espéran Padonou. Kriging and Design of Experiments on Circular Domains. The Fourth International Conference on the Interface between Statistics and Engineering 2016 (ICISE2016), Department of Chemical and Managerial Engineering, Manufacturing and Information Sciences, University of Palermo, Jun 2016, Palerme, Italy. ⟨hal-01350664⟩



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