In this dissertation we discuss several topics in statistics concerning regression models, experimental design and optimization. When it is... Show moreIn this dissertation we discuss several topics in statistics concerning regression models, experimental design and optimization. When it is expensive to compute a function value via numerical simulation, obtaining gradient values simultaneously can improve model e ciency. In the rst and second parts, polynomial regression models with gradient information are considered. We propose an orthogonal polynomial basis with respect to an inner product involving gradients of functions, to eliminate the illconditioning of the design matrix caused by Hermite polynomial basis. Through a simpli ed nuclear reactor model, we show that compared with Hermite polynomial basis, the orthogonal polynomial basis results in a better-conditioned design matrix, and a signi cant improvement when basis polynomials are chosen adaptively, using a stepwise tting procedure. In the second part, the design problem for polynomial regression models with gradient information is addressed. A theoretical upper bound is derived on the scaled integrated mean squared error in terms of the discrepancy of the design, and this bound can be used to choose designs that are both e cient and robust under model uncertainty. Numerical experiments show that low discrepancy designs, whose empirical distribution functions match a xed target distribution, outperform random and Latin hypercube designs. Considering a speci ed regression model, we propose a relaxed optimization problem, which is a semide nite programming problem, to nd the optimal design that minimizes scaled integrated mean squared error. Numerical examples demonstrate the eligibility of the method by showing that the optimal designs we achieve coincide with the already known optimal designs for regression model without gradient information. In the third part of the dissertation, the optimal layout of wind farm is considered. To maximize the expected annual pro t gained by the wind farm, we seek for both optimal number of wind turbines and optimal positions of wind turbines based on Jensen's model. Wind speed and direction are considered as random variables with distribution approximated by empirical distribution from real data. We propose using particle swarm optimization to solve for the optimal layout of wind farm. At last, the situation that more wind turbines are added to an existing wind farm is discussed. Ph.D. in Applied Mathematics, July 2014 Show less
