Contributions to statistical learning and statistical quantification in nanomaterials
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This research focuses to develop some new techniques on statistical learning including methodology, computation and application. We also developed statistical quantification in nanomaterials. For a large number of random variables with temporal or spatial structures, we proposed shrink estimates of covariance matrix to account their Markov structures. The proposed method exploits the sparsity in the inverse covariance matrix in a systematic fashion. To deal with high dimensional data, we proposed a robust kernel principal component analysis for dimension reduction, which can extract the nonlinear structure of high dimension data more robustly. To build a prediction model more efficiently, we developed an active learning via sequential design to actively select the data points into the training set. By combining the stochastic approximation and D-optimal designs, the proposed method can build model with minimal time and effort. We also proposed factor logit-models with a large number of categories for classification. We show that the convergence rate of the classifier functions estimated from the proposed factor model does not rely on the number of categories, but only on the number of factors. It therefore can achieve better classification accuracy. For the statistical nano-quantification, a statistical approach is presented to quantify the elastic deformation of nanomaterials. We proposed a new statistical modeling technique, called sequential profile adjustment by regression (SPAR), to account for and eliminate the various experimental errors and artifacts. SPAR can automatically detect and remove the systematic errors and therefore gives more precise estimation of the elastic modulus.