Non-linear variation patterns and kernel preimages

Identifying important variation patterns is a key step to identifying root causes of process variability. This gives rise to a number of challenges. First, the variation patterns might be non-linear in the measured variables, while the existing research literature has focused on linear relationships. Second, it is important to remove noise from the dataset in order to visualize the true nature of the underlying patterns. Third, in addition to visualizing the pattern (preimage), it is also essential to understand the relevant features that define the process variation pattern. This dissertation considers these variation challenges. A base kernel principal component analysis (KPCA) algorithm transforms the measurements to a high-dimensional feature space where non-linear patterns in the original measurement can be handled through linear methods. However, the principal component subspace in feature space might not be well estimated (especially from noisy training data). An ensemble procedure is constructed where the final preimage is estimated as the average from bagged samples drawn from the original dataset to attenuate noise in kernel subspace estimation. This improves the robustness of any base KPCA algorithm. In a second method, successive iterations of denoising a convex combination of the training data and the corresponding denoised preimage are used to produce a more accurate estimate of the actual denoised preimage for noisy training data. The number of primary eigenvectors chosen in each iteration is also decreased at a constant rate. An efficient stopping rule criterion is used to reduce the number of iterations. A feature selection procedure for KPCA is constructed to find the set of relevant features from noisy training data. Data points are projected onto sparse random vectors. Pairs of such projections are then matched, and the differences in variation patterns within pairs are used to identify the relevant features. This approach provides robustness to irrelevant features by calculating the final variation pattern from an ensemble of feature subsets. Experiments are conducted using several simulated as well as real-life data sets. The proposed methods show significant improvement over the competitive methods.