Spatial modeling and monitoring considering long-range dependence

Abstract

Spatial modeling and monitoring are critical in geometric characterization and quality control of material/product surfaces. With advances in metrology technology, a long-range dependence (LRD) effect has recently been detected in spatial data over different fields. The spatial LRD refers to a type of dependence that decays slowly over the distance with heavy tails and non-summable autocovariances so that the correlation is high among surface measurements across long spatial distances. Physically, the spatial LRD effect can be caused by specific spatial patterns such as certain material textures, surface profiles, or manufacturing defects. In literature, although various Markovian and non-Markovian spatial models have been proposed to study material surfaces, none of them has yet considered the LRD effect, which can lead to inefficient surface characterization and inaccurate surface quality control. To overcome the challenge, in this article, we first propose a novel spatial model that can capture the spatial LRD on material surfaces. Both isotropic and anisotropic scenarios of the model are developed based on the Lévy fractional Brownian random field and the fractional Brownian sheet, respectively. Subsequently, based on the proposed spatial model we develop an LRD-integrated quality control framework to monitor surface quality via generalized likelihood ratio test. Comprehensive simulation studies and a real case study using images of wood surfaces are conducted to validate the proposed approach. Results show that the proposed model that integrates LRD significantly outperforms multiple existing models in anomaly detection, and traditional models mis-detect out-of-control surfaces when the spatial LRD actually presents.