Hybrid modeling of non-stationary process variations

Accurate characterization of spatial variation is essential for statistical performance analysis and modeling, post-silicon tuning, and yield analysis. Existing approaches for spatial modeling either assume that: (i) non-stationarities arise due to a smoothly varying trend component or that (ii) the process is stationary within regions associated with a predefined grid. While such assumptions may hold when profiling certain classes of variations, a number of recent modeling studies suggest that non-stationarities arise from both shifts in the process mean as well as fluctuations in the variance of the process. In order to provide a compact model for non-stationary process variations, we introduce a new hybrid spatial modeling framework that models the spatially varying random field as a union of non-overlapping rectangular regions where the process is assumed to be locally-stationary within each region. To estimate the parameters in our hybrid spatial model, we develop a host of techniques to both estimate the change-points in the random field and to find an appropriate partitioning of the chip into disjoint regions where the field is locally-stationary. We verify our models and results on measurements collected from 65nm FPGAs.