Probabilistic local reconstruction for k-NN regression and its application to virtual metrology in semiconductor manufacturing

Abstract

The “locally linear reconstruction” (LLR) provides a principled and k-insensitive way to determine the weights of k-nearest neighbor (k-NN) learning. LLR, however, does not provide a confidence interval for the k neighbors-based reconstruction of a query point, which is required in many real application domains. Moreover, its fixed linear structure makes the local reconstruction model unstable, resulting in performance fluctuation for regressions under different k values. Therefore, we propose a probabilistic local reconstruction (PLR) as an extended version of LLR in the k-NN regression. First, we probabilistically capture the reconstruction uncertainty by incorporating Gaussian regularization prior into the reconstruction model. This prevents over-fitting when there are no informative neighbors in the local reconstruction. We then project data into a higher dimensional feature space to capture the non-linear relationship between neighbors and a query point when a value of k is large. Preliminary experimental results demonstrated that the proposed Bayesian kernel treatment improves accuracy and k-invariance. Moreover, from the experiment on a real virtual metrology data set in the semiconductor manufacturing, it was found that the uncertainty information on the prediction outcomes provided by PLR supports more appropriate decision making.