The proliferation of renewables, electric vehicles, and power electronic devices calls for innovative approaches to learn, optimize, and plan the power system. The uncertain and volatile nature of the integrated components necessitates using swift and probabilistic solutions. Gaussian process regression is a machine learning paradigm that provides closed-form predictions with quantified uncertainties. The key property of Gaussian processes is the natural ability to integrate the sensitivity of the labels with respect to features, yielding improved accuracy. This dissertation tailors Gaussian process regression for three applications in power systems. First, a physics-informed approach is introduced to infer the grid dynamics using synchrophasor data with minimal network information. The suggested method is useful for a wide range of applications, including prediction, extrapolation, and anomaly detection. Further, the proposed framework accommodates heterogeneous noisy measurements with missing entries. Second, a learn-to-optimize scheme is presented using Gaussian process regression that predicts the optimal power flow minimizers given grid conditions. The main contribution is leveraging sensitivities to expedite learning and achieve data efficiency without compromising computational efficiency. Third, Bayesian optimization is applied to solve a bi-level minimization used for strategic investment in electricity markets. This method relies on modeling the cost of the outer problem as a Gaussian process and is applicable to non-convex and hard-to-evaluate objective functions. The designed algorithm shows significant improvement in speed while attaining a lower cost than existing methods. / Doctor of Philosophy / The proliferation of renewables, electric vehicles, and power electronic devices calls for innovative approaches to learn, optimize, and plan the power system. The uncertain and volatile nature of the integrated components necessitates using swift and probabilistic solutions. This dissertation focuses on three practically important problems stemming from the power system modernization. First, a novel approach is proposed that improves power system monitoring, which is the first and necessary step for the stable operation of the network. The suggested method applies to a wide range of applications and is adaptable to use heterogeneous and noisy measurements with missing entries. The second problem focuses on predicting the minimizers of an optimization task. Moreover, a computationally efficient framework is put forth to expedite this process. The third part of this dissertation identifies investment portfolios for electricity markets that yield maximum revenue and minimum cost.

In typical normal theory regression, the assumption of homogeneity of variances is often not appropriate. Instead of treating the variances as a nuisance and transforming away the heterogeneity, the structure of the variances may be of interest and it is desirable to model the variances. Aitkin (1987) proposes a parametric dual model in which a log linear dependence of the variances on a set of explanatory variables is assumed. Aitkin's parametric approach is an iterative one providing estimates for the parameters in the mean and variance models through joint maximum likelihood. Estimation of the mean and variance parameters are interrelatedas the responses in the variance model are the squared residuals from the fit to the means model. When one or both of the models (the mean or variance model) are misspecified, parametric dual modeling can lead to faulty inferences. An alternative to parametric dual modeling is to let the data completely determine the form of the true underlying mean and variance functions (nonparametric dual modeling). However, nonparametric techniques often result in estimates which are characterized by high variability and they ignore important knowledge that the user may have regarding the process. Mays and Birch (1996) have demonstrated an effective semiparametric method in the one regressor, single-model regression setting which is a "hybrid" of parametric and nonparametric fits. Using their techniques, we develop a dual modeling approach which is robust to misspecification in either or both of the two models. Examples will be presented to illustrate the new technique, termed here as Dual Model Robust Regression. / Ph. D.

This dissertation is concerned with the evaluation of a robust modification of existing methodology within the classical inference framework. This results in an F-test based on the robust weights used in arriving at the M or Bounded-Influence estimates. These estimates are known to be robust to outliers and highly influential points, respectively. The first part of this evaluation involves a Monte Carlo power study, under violations of the classical assumptions, of this F-test based on robust weights and several other proposed robust tests. It is shown in simulation studies that, under certain conditions, the F-test based on robust weights is a much more powerful test than the classical F -test, and compares favorably to all other proposals studied. The second part involves the development of the influence curve (IC) for the F-test based on robust weights and one empirical approximation to the IC, the Sample Influence Curve (SIC). It is shown for two sample data sets that the SIC demonstrates the resistance to unusual points of the F-test based on robust weights. / Ph. D.

The linear calibration problem, frequently referred to as inverse regression or the discrimination problem can be stated briefly as the problem of estimating the independent variable x in a regression situation for a measured value of the dependent variable y. The literature on this problem deals primarily with the Classical method where the Classical estimator is obtained by expressing the linear model as y_i = α + βx_i + ε_i , obtaining the least squares estimator for y for a given value of x and inverting the relationship. A second estimator for calibration, the Inverse estimator, is obtained by expressing the linear model as x_i = γ + δy_i + ε’_i and using the resulting least squares estimator to estimate x. The experimental design problem for the Inverse estimator is explored first in this dissertation using the criterion of minimizing the average or integrated mean squared error, and the resulting optimal and near optimal designs are then compared with those for the Classical estimator which were recently derived by Ott and Nycrs. Optimal designs are developed for a linear approximation when the true model is linear and when it is quadratic. In both cases, the optimal designs depend on unknown model parameters and are not realistically useable. However, designs are shown to exist which are near optimal and do not depend on the unknown model parameters. For the linear approximation to the quadratic model, these near optimal designs depend on N, the number of observations used to estimate the model parameters, and specific designs are developed and set forth in tables for N = 5(1)20(2)30(5)50. The cost of misclassifying a quadratic model as linear is discussed from a design point of view as well as the cost of protecting against a possible quadratic effect, The costs are expressed in terms of the percent deviation from the average mean squared error that would be obtained if the model were classified correctly, The derived designs for the Inverse estimator are compared with the recently derived designs for the Classical estimator using as a measure of comparison the ratio of minimum average mean squared errors obtained by using the optimal design for both estimators. Further comparisons are also made between optimal designs for the Classical estimator and the derived near optimal designs for the Inverse estimator using the ratio of the corresponding average mean squared errors as a measure of comparison. Parallels are drawn between forward regression (estimating, the dependent variable for a given value of the independent variable) and inverse regression using both the Classical and Inverse methods. / Ph. D.

This study analyzes the wage and non-wage factors underlying non-metropolitan-to-metropolitan commuting decisions of households in five non-metropolitan counties in Northern Virginia. The potential fiscal and planning implications of these decisions are also discussed. Chapter one contains a description of the study area, problem statement and objectives. Chapter two reviews related literature on commuting, housing and job location, as well as rent and wage gradients. Chapter three provides a theoretical framework for analyzing household commuting decisions. Chapter four presents descriptive statistics, and introduces a switching regression system of equations to simultaneously estimate factors influencing commuting decisions and earnings in non-metropolitan and metropolitan labor markets. Chapter five reports the regression results, and simulates wage gaps and the distance of the metropolitan labor market draw for different groups of workers. Chapter six discusses potential fiscal implications of commuting and potential policies to manage growth in commuting. The empirical result shows that the major incentive for workers to commute is a large age gap between metropolitan and non-metropolitan labor market areas. Household responsibilities, housing preference and ability to find local jobs represent non-wage factors underlying commuting decisions. Two study findings suggest that the local fiscal implications of non-metropolitan-to-metropolitan commuting households may be limited. First, commuting households are found to have fewer school-aged children, and require less local expenditures on education. Second, commuting households are more likely to be homeowners, have more rooms in their homes, and provide a larger tax base. / Master of Science

Master of Science

For this study, I have constructed a logistic regression model, using existing datasets of environmental parameters to predict the probability of As concentrations above 5 parts per billion (ppb) in Virginia groundwater and to evaluate if geologic or other characteristics are linked to elevated As concentrations. Measured As concentrations in groundwater from the Virginia Tech Biological Systems Engineering (BSE) Household Water Quality dataset were used as the dependent variable to train (calibrate) the model. Geologic units, lithology, soil series and texture, land use, and physiographic province were used as regressors in the model. Initial models included all regressors, but during model refinement, attention was focused solely on geologic units. Two geologic units, Triassic-aged sedimentary rocks and Devonian-aged shales/sandstones, were identified as significant in the model; the presence of these units at a spatial location results in a higher probability for As occurrences in groundwater. Measured As concentrations in groundwater from an independent dataset collected by the Virginia Department of Health were used to test (validate) the model. Due to the structure of the As datasets, which included As concentrations mostly (95-99%) = 5 ppb, and thus few (1-5%) data in the range > 5 ppb, the regression model cannot be used reliably to predict As concentrations in other parts of the state. However, our results are useful for identifying areas of Virginia, defined by underlying geology, that are more likely to have elevated As concentrations in groundwater. Results of this work suggest that homeowners with wells installed in these geologic units have their wells tested for As and regulators closely monitor public supply wells in these areas for As. / Master of Science

When near linear relationships exist among the columns of regressor variables, the variances of the least squares estimators of the regression coefficients become very large. The least squares estimator of the vector of the regression coefficients, which can be written in terms of latent roots and latent vectors of X'X, tends to place heavy weights on the latent vectors corresponding to small latent roots of X'X. Thus, the estimates of regression coefficients corresponding to the regressors involved in multicollinearities tend to be dominated by the multicollinearities. Therefore, the least squares estimators could estimate the true parameters poorly and could be very unreliable. In order to overcome the ill-effects of multicollinearities on the least squares estimator, the procedure of principal components regression deletes those components corresponding to the small latent roots of X'X. Then we regress <u>y</u> on the retained components using ordinary least squares. When principal components regression is used as an alternative to the least squares in the presence of a near singular X'X matrix, its performance depends strongly on the orientation of the deleted components to the vector of regression coefficients. In this paper, we present a modification of the principal components procedure in which components associated with near singularities are dampened but are not completely deleted. The resulting estimator was compared in a Monte Carlo study with the least squares estimator and the principal component estimator using mean squared error as the basis of comparison. The results indicate that the modified principal components estimator will perform better than either of the other two estimators over a wide range of orientations and signal-to-noise ratios and that it provides a reasonable compromise choice when the orientation is unknown. / Ph. D.

This thesis reviews and discusses the so-called “Group Averages method" in the linear regression, the quadratic regression, and the functional relation situations. In the linear and quadratic regression situations, under the assumption of X_i equally spaced, the efficiency of the Group Averages estimator is quite satisfactory as compared with Least Squares estimators. In the functional relation situation we used the Group Averages method and the Maximum Likelihood method for estimation of parameters. To compare their efficiencies we used the variance of the Group Averages estimator which was given by Dorff and Gurland [3], and developed the variance of Maximum Likelihood estimators. Under the assumption of X_i equally spaced, we round the efficiency of the Group Averages estimator to be quite satisfactory. However, caution is needed for using the Group Averages method in functional relationships, since it requires the following condition to be satisfied: Pr {|d_i| ≥ ½ c} negligible Where c = Min. |X_i+1 - X_i|. / Master of Science

This study seeks to investigate weather-related flight delay trends at 13 United States airports. Flight delay data were collected from 2004-2019 and normalized by airport operations data. Using Support Vector Regression (SVR), visual trends were identified. Further analysis was conducted by comparing all four meteorological seasons through computing 95% bootstrap confidence intervals on their means. Finally, precipitation and snowfall data were correlated with normalized delays to investigate how they are related. This study found that the season with the highest normalized delay values is heavily dependent upon location. Most airports saw a decrease in the SVR line at some point since 2004, but have since leveled off. It was also discovered that while precipitation trends are not changing drastically, delay variability has decreased at many airports in the last 10 years, which may be indicative of more effective mitigation strategies.