In this dissertation, we investigate various theoretical properties of optimal regression designs and develop several numerical algorithms for computing them. The results can be applied to linear, nonlinear and generalized linear models. Our work starts from how to solve the design problems for A-, As, c-, I- and L-optimality criteria on one-response model. Theoretical results are hard to derive for many regression models and criteria, and existing numerical algorithms can not compute the results efficiently when the number of support points is large. Therefore we consider to solve the design problems based on SeDuMi program in MATLAB. SeDuMi is developed to solve semidefinite programming (SDP) problems in optimization. To apply it, we derive a general transformation to connect the design problems with SDP problems, and propose a numerical algorithm based on SeDuMi to solve these SDP problems. The algorithm is quite general under the least squares estimator (LSE) and weighted least squares estimator (WLSE) and can be applied to both linear and nonlinear regression models. We continue to study the optimal designs based on one-response model when the error distribution is asymmetric. Since the second-order least squares estimator (SLSE) is more efficient than the LSE when the error distribution is not symmetric, we study optimal designs under the SLSE. We derive expressions to characterize A- and D-optimality criteria and develop a numerical algorithm for finding optimal designs under the SLSE based on SeDuMi and CVX programs in MATLAB. Several theoretical properties are also derived for optimal designs under SLSE. To check the optimality of the numerical results, we establish the Kiefer-Wolfowitz equivalence theorem and apply it to various applications. Finally, we discuss the optimal design problems for multi-response models. Our algorithm studied here is based on SeDuMi and CVX, and it can be used for linear, nonlinear and generalized linear models. The transformation invariance property and dependence on the covariance matrix of the correlated errors are derived. We also correct the errors in the literature caused by formulation issues. The results are very useful to construct optimal regression designs on discrete design space. They can be applied to any one-response and multi-response models, various optimality criteria, and several estimators including LSE, maximum likelihood estimator, best linear unbiased estimator, SLSE and WLSE. / Graduate

In some applications, we require a monotone estimate of a regression function. In others, we want to test whether the regression function is monotone. For solving the first problem, Ramsay's, Kelly and Rice's, as well as point-wise monotone regression functions in a spline space are discussed and their properties developed. Three monotone estimates are defined: least-square regression splines, smoothing splines and binomial regression splines. The three estimates depend upon a "smoothing parameter": the number and location of knots in regression splines and the usual [formula omitted] in smoothing splines. Two standard techniques for choosing the smoothing parameter, GCV and AIC, are modified for monotone estimation, for the normal errors case. For answering the second question, a test statistic is proposed and its null distribution conjectured. Simulations are carried out to check the conjecture. These techniques are applied to two data sets. / Science, Faculty of / Statistics, Department of / Graduate

Energy Consumption is a topic of great interest, especially since a surge in prices in late 2021 has caused a considerable increase in discussion around the topic. Data from the Swedish Central Bureau of statistics (SCB) and the Swedish Meteorological Institute (SMHI) were provided for macroscopic regressors. These regressors are temperature, population, GDP, day length, electricity price, electricity production, production of variable renewable energy and average income in order to predict electricity consumption. Four models were created, a full multiple linear regression model using all regressors. A reduced multiple linear regression model using a subset of the regressors determined by cross validation. A ridge model and a LASSO model. These were then used to attempt to predict the power consumption of 20% of the data set that were left out when creating the models. The LASSO model was most successful in this as it had the smallest cumulative residual and the ridge model was the worst. Since the reduced and the full model both had very high multicollinearity the conclusion was that the LASSO model is the best model out of the four.

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Source: Dissertation Abstracts International, Volume: 70-03, Section: B, page: 1743. Advisers: Wenyaw Chan; Glasser H. Jay. Includes bibliographical references.

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