Functional data analysis, commonly known as FDA", refers to the analysis of information on curves of functions. Key aspects of FDA include the choice of smoothing techniques, data reduction, model evaluation, functional linear modeling and forecasting methods. FDA is applicable in numerous applications such as Bioscience, Geology, Psychology, Sports Science, Econometrics, Meteorology, etc. This dissertation main objective is to focus more specifically on Functional Linear Regression Modelling (FLRM), which is an extension of Multivariate Linear Regression Modeling. The problem of constructing a Functional Linear Regression modelling with functional predictors and functional response variable is considered in great details. Discretely observed data for each variable involved in the modelling are expressed as smooth functions using: Fourier Basis, B-Splines Basis and Gaussian Basis. The Functional Linear Regression Model is estimated by the Least Square method, Maximum Likelihood method and more thoroughly by Penalized Maximum Likelihood method. A central issue when modelling Functional Regression models is the choice of a suitable model criterion as well as the number of basis functions and an appropriate smoothing parameter. Four different types of model criteria are reviewed: the Generalized Cross-Validation, the Generalized Information Criterion, the modified Akaike Information Criterion and Generalized Bayesian Information Criterion. Each of these aforementioned methods are applied to a dataset and contrasted based on their respective results.

While multivariate data analysis is concerned with data in the form of random vectors, functional data analysis goes one big step farther, focusing on data that are infinite-dimensional, such as curves, shapes and images. We focus on functional data that are measured over time across multiple subjects. The first part of the thesis focuses on spatially correlated functional data. This correlation is modeled by correlating functional principal component scores. We propose a Spatial Principal Analysis by Conditional Expectation framework to explicitly estimate spatial correlations and reconstruct individual curves. This approach works even when the observed data per curve are extremely sparse. Assuming spatial stationarity, empirical between-curve correlations are calculated as the ratio of eigenvalues of the smoothed covariance surface Cov(Xi(s),Xi(t)) and cross-covariance surface Cov(Xi(s),Xj(t)). Then a parametric spatial correlation model is employed to fit empirical correlations. Finally, principal component scores are estimated to reconstruct the sparsely observed curves. This framework could naturally accommodate arbitrary covariance structures, but there is an enormous reduction in computation if one can assume the separability of temporal and spatial components. We propose hypothesis tests to examine the separability and isotropy effect of spatial correlation. Simulation studies and applications of empirical data show improvements in the curve reconstruction using our framework over the method where curves are assumed to be independent. In addition, asymptotic properties of estimates are discussed in details. In the second part of this work, we present a new approach to factor rotation for functional data. This is achieved by rotating the functional principal components toward a predefined space of periodic functions designed to decompose the total variation into components that are nearly-periodic and nearly-aperiodic with a predefined period. We show that the factor rotation can be obtained by the calculation of canonical correlations between appropriate spaces. Moreover, we demonstrate that our proposed rotations provide stable and interpretable results in the presence of highly complex covariance. This work is motivated by the goal of finding interpretable sources of variability in a gridded time series of vegetation index measurements obtained from remote sensing, and we demonstrate our methodology through the application of factor rotation of this data.

The functional linear model, $Y_n = Psi X_n + varepsilon_n$, with functional response and explanatory variables is considered. A simple test of the nullity of $Psi$ based on the principal component decomposition is proposed. The test statistic has asymptotic chi-squared distribution, which is also an excellent approximation in finite samples. The methodology is applied to data from terrestrial magnetic observatories. In recent years, the interaction of the auroral substorms with the equatorial and mid-latitude currents has been the subject of extensive research. We introduce a new statistical technique that allows us to test at a specified significance level whether such a dependence exists, and how long it persists. This quantitative statistical technique, relying on the concepts and tools of functional data analysis, uses directly magnetometer records in one minute resolution, and it can be applied to similar geophysical data which can be represented as daily curves. It is conceptually similar to testing the nullity of the slope in the straight line regression, but both the regressors and the responses are curves rather than points. When the regressors are daily high latitude $H$--component curves during substorm days and the responses are daily mid-- or low latitude $H$--component curves, our test shows significant dependence (the nullity hypothesis is rejected), which exists not only on the same UT day, but also extends into the next day for strong substorms. We propose a novel approach based on wavelet and functional principal component analysis to produce a cleaner index of the intensity of the symmetric ring current. We use functional canonical correlations to show that the new approach more effectively extracts symmetric global features. The main result of our work is the construction of a new index, which is an improved version of the existing wavelet-based index (WISA) and the old Dst index, in which a constant daily variation is removed. Here, we address the fact that the daily component varies from day to day and construct a ``cleaner'' index by removing non-constant daily variations. A wavelet-based method of deconvoluting the solar quiet variation from the low and mid-latitude H-component records is proposed. The resulting daily variation is non--constant, and its day--to--day variability is quantified by functional principal component scores. The procedure removes the signature of an enhanced ring current by comparing the scores at different stations. The method is fully algorithmic and is implemented in the statistical software R. R package for space physics applications is developed. It consists of several functions that compute indices of the storm activity and estimate the daily variation. Storm indices are computed automatically without any human intervention using the most recent magnetometer data available. Functional principal component analysis techniques are used to extract day-to-day variations. This package will be publicly available at Comprehensive R Archive Network (CRAN).

Functional data analysis (FDA) has grown into a substantial field of statistical research, with new methodology, numerous useful applications and interesting novel theoretical developments. My dissertation focuses on the empirical properties of functional regression models and their application to financial data. We start from testing the empirical properties of forecasts with the functional autoregressive models based on simulated and real data. We define intraday returns and consider their prediction from such returns on a market index. This is an extension to intraday data of the Capital Asset Pricing model. Finally we investigate multifactor functional models and assess their suitability for the prediction of intraday returns for various financial assets, including stock and commodity futures.

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Finite mixture models have had a profound impact on the history of statistics, contributing to modelling heterogeneous populations, generalizing distributional assumptions, and lately, presenting a convenient framework for classification and clustering. A novel approach, via Gaussian mixture distribution, is introduced for modelling receiver operating characteristic curves. The absence of a closed-form for a functional form leads to employing the Monte Carlo method. This approach performs excellently compared to the existing methods when applied to real data. In practice, the data are often non-normal, atypical, or skewed. It is apparent that non-Gaussian distributions be introduced in order to better fit these data. Two non-Gaussian mixtures, i.e., t distribution and skew t distribution, are proposed and applied to real data. A novel mixture is presented to cluster spatial and temporal data. The proposed model defines each mixture component as a mixture of autoregressive polynomial with logistic links. The new model performs significantly better compared to the most well known model-based clustering techniques when applied to real data. / Thesis / Doctor of Philosophy (PhD)

Functional data analysis (FDA) studies data in the form of measurements over a domain as whole entities. Our first focus is on the post-hoc analysis with pairwise and contrast comparisons of the popular functional ANOVA model comparing groups of functional data. Existing contrast tests assume independent functional observations within group. In reality, this assumption may not be satisfactory since functional data are often collected continually overtime on a subject. In this work, we introduce a new linear contrast test that accounts for time dependency among functional group members. For a significant contrast test, it can be beneficial to identify the region of significant difference. In the second part, we propose a non-parametric regression procedure to obtain a locally sparse estimate of functional contrast. Our work is motivated by a biomedical study using Raman spectroscopy to monitor hemodialysis treatment near real-time. With contrast test and sparse estimation, practitioners can monitor the progress of the hemodialysis within session and identify important chemicals for dialysis adequacy monitoring. In the third part, we propose a functional data model for degradation analysis of functional data. Motivated by degradation analysis application of rechargeable Li-ion batteries, we combine state-of-the-art functional linear models to produce fully functional prediction for curves on heterogenous domains. Simulation studies and data analysis demonstrate the advantage of the proposed method in predicting degradation measure than existing method using aggregation method. / Doctor of Philosophy / Functional data analysis (FDA) studies complex data structure in the form of curves and shapes. Our work is motivated by two applications concerning data from Raman spectroscopy and battery degradation study. Raman spectra of a liquid sample are curves with measurements over a domain of wavelengths that can identify chemical composition and whose values signify the constituent concentrations in the sample. We first propose a statistical procedure to test the significance of a functional contrast formed by spectra collected at beginning and at later time points during a dialysis session. Then a follow-up procedure is developed to produce a sparse representation of the contrast functional contrast with clearly identified zero and nonzero regions. The use of this method on contrast formed by Raman spectra of used dialysate collected at different time points during hemodialysis sessions can be adapted for evaluating the treatment efficacy in real time. In a third project, we apply state-of-the-art methodologies from FDA to a degradation study of rechargeable Li-ion batteries. Our proposed methods produce fully functional prediction of voltage discharge curves allowing flexibility in monitoring battery health.

The Phase I analysis of data when the quality of a process or product is characterized by a linear function is studied in this dissertation. It is assumed that each sample collected over time in the historical data set consists of several bivariate observations for which a simple linear regression model is appropriate, a situation common in calibration applications. Using a simulation study, the researcher compares the performance of some of the recommended approaches used to assess the stability of the process. Also in this dissertation, a method based on using indicator variables in a multiple regression model is proposed. This dissertation also proposes a change point approach based on the segmented regression technique for testing the constancy of the regression parameters in a linear profile data set. The performance of the proposed change point method is compared to that of the most effective Phase I linear profile control chart approaches using a simulation study. The advantage of the proposed change point method over the existing methods is greatly improved detection of sustained step changes in the process parameters. Any control chart that combines sample information over time, e.g., the cumulative sum (CUSUM) chart and the exponentially weighted moving average (EWMA) chart, has an ability to detect process changes that varies over time depending on the past data observed. The chart statistics can take values such that some shifts in the parameters of the underlying probability distribution of the quality characteristic are more difficult to detect. This is referred to as the "inertia problem" in the literature. This dissertation shows under realistic assumptions that the worst-case run length performance of control charts becomes as informative as the steady-state performance. Also this study proposes a simple new measure of the inertial properties of control charts, namely the signal resistance. The conclusions of this study support the recommendation that Shewhart limits should be used with EWMA charts, especially when the smoothing parameter is small. This study also shows that some charts proposed by Pignatiello and Runger (1990) and Domangue and Patch (1991) have serious disadvantages with respect to inertial properties. / Ph. D.

Water temperature is a critical factor for the quality and biological condition of streams. Among various factors affecting stream water temperature, air temperature is one of the most important factors related to water temperature. To appropriately quantify the relationship between water and air temperatures over a large geographic region, it is important to accommodate the spatial and temporal information of the steam temperature. In this dissertation, I devote effort to several statistical modeling techniques for analyzing bivariate spatial-temporal data in a stream temperature study. In the first part, I focus our analysis on the individual stream. A time varying coefficient model (VCM) is used to study the relationship between air temperature and water temperature for each stream. The time varying coefficient model enables dynamic modeling of the relationship, and therefore can be used to enhance the understanding of water and air temperature relationships. The proposed model is applied to 10 streams in Maryland, West Virginia, Virginia, North Carolina and Georgia using daily maximum temperatures. The VCM approach increases the prediction accuracy by more than 50% compared to the simple linear regression model and the nonlinear logistic model. The VCM that describes the relationship between water and air temperatures for each stream is represented by slope and intercept curves from the fitted model. In the second part, I consider water and air temperatures for different streams that are spatial correlated. I focus on clustering multiple streams by using intercept and slope curves estimated from the VCM. Spatial information is incorporated to make clustering results geographically meaningful. I further propose a weighted distance as a dissimilarity measure for streams, which provides a flexible framework to interpret the clustering results under different weights. Real data analysis shows that streams in same cluster share similar geographic features such as solar radiation, percent forest and elevation. In the third part, I develop a spatial-temporal VCM (STVCM) to deal with missing data. The STVCM takes both spatial and temporal variation of water temperature into account. I develop a novel estimation method that emphasizes the time effect and treats the space effect as a varying coefficient for the time effect. A simulation study shows that the performance of the STVCM on missing data imputation is better than several existing methods such as the neural network and the Gaussian process. The STVCM is also applied to all 156 streams in this study to obtain a complete data record. / Ph. D.

Modern manufacturing needs to optimize the entire product lifecycle to satisfy the customer needs. The advancement of sensing technologies has brought a data rich environment for manufacturing and provide a great opportunity for real-time, proactive quality assurance. However, due to the lack of methods for analyzing heterogeneous types of data, the transformation of data to information and knowledge for effective decision making in manufacturing is still a challenging problem. In particular, functional variables can represent the in situ process conditions and rich product performance information, and are widely encountered in various manufacturing processes. In this dissertation, I will focus on modeling of manufacturing processes with in situ process (functional) variables, and integrating these functional variables and other measured variables for the manufacturing modeling. The modeling is explored by extracting informative features through the integration of multiple functional variables, functional variables and offline setting variables, and quantitative and qualitative quality variables. After an introduction in Chapter 1, three research tasks are investigated. First, a functional variable selection problem is studied in Chapter 2 to identify the significant functional variables as well as their features in a logistic regression model. A hierarchical non-negative garrote constrained estimation method is proposed. Second, the quality-process relationships for scalar offline setting variables, functional in situ process variables, and manufacturing quality responses are studied in Chapter 3. A functional graphical model that can integrate functional variables in a graphical model is proposed and investigated. Third, the quantitative and qualitative quality responses are jointly modeled with scalar offline setting variables and functional in situ process variables in Chapter 4. A functional quantitative and qualitative model is proposed and investigated. Finally, I summarize the research contribution and discuss future research directions in Chapter 5. The proposed methodologies have broad applications in manufacturing processes with functional variables, and are demonstrated in a crystal growth process with multiple functional variables (Chapter 2), a plasma spray process with multiple scalar and functional variables (Chapter 3), and an additive manufacturing process called fused deposition modeling with quantitative and qualitative quality responses (Chapter 4). / Ph. D.