Monitoring spatial uniformity of particle distributions in manufacturing processes using the K function

Data in the form of spatial point patterns are frequently encountered in some manufacturing processes such as the nanoparticle reinforced composite materials and defects on semiconductor wafers. Their spatial characteristics contain rich information about the fabrication processes and are often strongly related to the product quality. The distributional characteristics of a spatial point pattern can be summarized by functional profiles such as the popular Ripley's K function. By analyzing the K function, we can effectively monitor the distributional behaviors of the spatial point data. In this study, statistical properties of the K function are investigated, and a Gaussian process is found to be appropriate in characterizing its behavior under complete spatial randomness. A control chart is proposed based on the results to monitor the uniformity of point patterns. Our proposed approach has been compared with existing methods through numerical simulations and shown superior performances.