Regression models with a scalar response and a functional predictor have been extensively studied. One approach is to approximate the functional predictor using basis function or eigenfunction expansions. In the expansion, the coefficient vector can either be fixed or random. The random coefficient vector is also known as random effects and thus the regression models are in a mixed effects framework. The random effects provide a model for the within individual covariance of the observations. But it also introduces an additional parameter into the model, the covariance matrix of the random effects. This additional parameter complicates the covariance matrix of the observations. Possibly, the covariance parameters of the model are not identifiable. We study identifiability in normal linear mixed effects models. We derive necessary and sufficient conditions of identifiability, particularly, conditions of identifiability for the regression models with a scalar response and a functional predictor using random effects. We study the regression model using the eigenfunction expansion approach with random effects. We assume the random effects have a general covariance matrix and the observed values of the predictor are contaminated with measurement error. We propose methods of inference for the regression model's functional coefficient. As an application of the model, we analyze a biological data set to investigate the dependence of a mouse's wheel running distance on its body mass trajectory.

Regression models with a scalar response and a functional predictor have been extensively studied. One approach is to approximate the functional predictor using basis function or eigenfunction expansions. In the expansion, the coefficient vector can either be fixed or random. The random coefficient vector is also known as random effects and thus the regression models are in a mixed effects framework. The random effects provide a model for the within individual covariance of the observations. But it also introduces an additional parameter into the model, the covariance matrix of the random effects. This additional parameter complicates the covariance matrix of the observations. Possibly, the covariance parameters of the model are not identifiable. We study identifiability in normal linear mixed effects models. We derive necessary and sufficient conditions of identifiability, particularly, conditions of identifiability for the regression models with a scalar response and a functional predictor using random effects. We study the regression model using the eigenfunction expansion approach with random effects. We assume the random effects have a general covariance matrix and the observed values of the predictor are contaminated with measurement error. We propose methods of inference for the regression model's functional coefficient. As an application of the model, we analyze a biological data set to investigate the dependence of a mouse's wheel running distance on its body mass trajectory.

Regression models with a scalar response and a functional predictor have been extensively studied. One approach is to approximate the functional predictor using basis function or eigenfunction expansions. In the expansion, the coefficient vector can either be fixed or random. The random coefficient vector is also known as random effects and thus the regression models are in a mixed effects framework. The random effects provide a model for the within individual covariance of the observations. But it also introduces an additional parameter into the model, the covariance matrix of the random effects. This additional parameter complicates the covariance matrix of the observations. Possibly, the covariance parameters of the model are not identifiable. We study identifiability in normal linear mixed effects models. We derive necessary and sufficient conditions of identifiability, particularly, conditions of identifiability for the regression models with a scalar response and a functional predictor using random effects. We study the regression model using the eigenfunction expansion approach with random effects. We assume the random effects have a general covariance matrix and the observed values of the predictor are contaminated with measurement error. We propose methods of inference for the regression model's functional coefficient. As an application of the model, we analyze a biological data set to investigate the dependence of a mouse's wheel running distance on its body mass trajectory. / Science, Faculty of / Statistics, Department of / Graduate

Indiana University-Purdue University Indianapolis (IUPUI) / Various methods exist to measure physical activity (PA). Subjective methods, such as diaries and surveys are relatively inexpensive ways of measuring one’s PA; how ever, they are riddled with measurement error and bias due to self-report. Wearable accelerometers offer a noninvasive and objective measure of subjects’ PA and are now widely used in observational and clinical studies. Accelerometers record high frequency data and produce an unlabeled time series at the sub-second level. An important activity to identify from such data is walking, since it is often the only form of exercise for certain populations. While much work has been done to advance the use of accelerometers in public health research, methodology is needed for quan tifying the physical characteristics of different types of PA from the raw signal. In my dissertation, I advance the accelerometry research methodology in a three-paper sequence. The first paper is a novel application of functional linear models to model the physical characteristics of walking. We emphasize the signal processing used to prepare the data for analyses, and we apply the methods to a motivating dataset collected in an elder population. The second paper addresses the classification of PA. We designed an experiment and collected the data with the purpose of extracting useful and interpretable features for differentiating among walking, descending stairs, and ascending stairs. We build subject-specific classification models utilizing a tree based classifier. We evaluate the effects of sensor location and tuning parameters on the classification rate of these models. The third paper addresses the classification of walking types at the population level. We propose a robust normalization of features extracted for each subject and compare the model classification results to evaluate the effect of feature normalization. In summary, this work provides a framework for better use of accelerometers in the study of physical activity. / 2 years

The increased concentration of greenhouse gases is associated with the global warming in the lower troposphere. For over twenty years, the space physics community has studied a hypothesis of global cooling in the thermosphere, attributable to greenhouse gases. While the global temperature increase in the lower troposphere has been relatively well established, the existence of global changes in the thermosphere is still under investigation. A central difficulty in reaching definite conclusions is the absence of data with sufficiently long temporal and sufficiently broad spatial coverage. Time series of data that cover several decades exist only in a few separated regions. The space physics community has struggled to combine the information contained in these data, and often contradictory conclusions have been reported based on the analyses relying on one or a few locations. To detect global changes in the ionosphere, we present a novel statistical methodology that uses all data, even those with incomplete temporal coverage. It is based on a new functional regression approach that can handle unevenly spaced, partially observed curves. While this research makes a solid contribution to the space physics community, our statistical methodology is very flexible and can be useful in other applied problems.

The increased concentration of greenhouse gases is associated with the global warming in the lower troposphere. For over twenty years, the space physics community has studied a hypothesis of global cooling in the thermosphere, attributable to greenhouse gases. While the global temperature increase in the lower troposphere has been relatively well established, the existence of global changes in the thermosphere is still under investigation. A central difficulty in reaching definite conclusions is the absence of data with sufficiently long temporal and sufficiently broad spatial coverage. Time series of data that cover several decades exist only in a few separated regions. The space physics community has struggled to combine the information contained in these data, and often contradictory conclusions have been reported based on the analyses relying on one or a few locations. To detect global changes in the ionosphere, we present a novel statistical methodology that uses all data, even those with incomplete temporal coverage. It is based on a new functional regression approach that can handle unevenly spaced, partially observed curves. While this research makes a solid contribution to the space physics community, our statistical methodology is very flexible and can be useful in other applied problems.

Functional data refer to data which consist of observed functions or curves evaluated at a finite subset of some interval. In this dissertation, we discuss statistical analysis, especially classification and regression when data are available in function forms. Due to the nature of functional data, one considers function spaces in presenting such type of data, and each functional observation is viewed as a realization generated by a random mechanism in the spaces. The classification procedure in this dissertation is based on dimension reduction techniques of the spaces. One commonly used method is Functional Principal Component Analysis (Functional PCA) in which eigen decomposition of the covariance function is employed to find the highest variability along which the data have in the function space. The reduced space of functions spanned by a few eigenfunctions are thought of as a space where most of the features of the functional data are contained. We also propose a functional regression model for scalar responses. Infinite dimensionality of the spaces for a predictor causes many problems, and one such problem is that there are infinitely many solutions. The space of the parameter function is restricted to Sobolev-Hilbert spaces and the loss function, so called, e-insensitive loss function is utilized. As a robust technique of function estimation, we present a way to find a function that has at most e deviation from the observed values and at the same time is as smooth as possible.

Les systèmes de contrôle aérien donneront, dans un avenir assez proche, une plus grande autonomie et liberté d’action aux pilotes (en particulier dans le cadre de la “planification 4D des trajectoires”), ce qui nécessite une prévision de trajectoire de qualité, afin d’une part d’éviter les « conflits » entre avions (avions trop proches, d’où un risque de collision), d’autre part de réguler efficacement les temps d’arrivée aux aéroports.Les trajectoires dépendent de facteurs extérieurs souvent pas, ou mal connus ; en particulier les vents ne sont pas connus avec la précision nécessaire pour une prévision de trajectoire de qualité. De sorte que la prévision de trajectoire ne peut être faite de façon utilisable qu’à court ou moyen terme, disons dans un horizon de moins de 10 minutes ou de l’ordre de 10 à 30 minutes.On appelle “trajectoire 4D” la trajectoire d’un avion, dans l’espace 4D constitué des trois dimensions d’espace, et de la dimension du temps. L’objet de cette thèse est d’établir des méthodes de prévision de trajectoires 4D à court et moyen terme (jusqu’à 10 à 30 minutes). Une telle prévision prend en compte (éventuellement implicitement) des facteurs importants tels que le type de l’appareil, les conditions atmosphériques (vent, température), de façon à pouvoir en déduire les actions précises pour résoudre les conflits potentiels, et/ou arriver à l’instant t voulu à l’aéroport.Dans ce travail, nous présentons une nouvelle approche pour la prédiction des trajectoires d’avion. Cette méthode est basée sur une régression fonctionnelle linéaire, et comprend en particulier un prétraitement des données (lissage, mais surtout synchronisation et cadencement régulier en temps), résolution de la régression par l’utilisation d’une décomposition en ondelettes. On commence par collecter un nombre important de données concernant les vols ayant existé entre deux aéroports ; ces données comportent en particulier les coordonnées, vitesses et projection de l’avion à différents temps. L’étape suivante, que nous appelons localisation, consiste à déterminer un certain nombre de trajectoires “logiquement proches”, c’est à dire, en particulier, concernant le même type d’appareil, et concernant les mêmes aéroports d’origine et de destination. Cet ensemble de trajectoires est ensuite utilisé pour construire un modèle type, qui se rapproche le plus possible de la partie connue de la trajectoire en cours, à prolonger ; ceci est réalisé grâce à une régression fonctionnelle linéaire. Le “modèle type” est alors utilisé comme prédicteur de la trajectoire en cours. Remarquons que cet algorithme n’utilise que des mesures radar, et ne prend pas en compte explicitement des données importantes physiques ou aéronautiques. Cependant les trajectoires ayant servi pour construire le modèle type dépendant elles aussi de ces données, ces dernières sont implicitement prises en compte par cette démarche.Nous avons appliqué cette méthode sur de nombreuses trajectoires entre plusieurs aéroports français, la base de données s’étendant sur plus d’un an. Près de trois millions de vols ont été pris en compte. Les résultats sont présentés en fin du manuscrit ; ils présentent en particulier l’erreur de prédiction, qui est la différence entre la prédiction de la trajectoire et la trajectoire effective (connue puisqu’il s’agit de trajectoires ayant existé, mais bien sûr non utilisée à partir de l’instant où démarre la prévision de trajectoire). Ces résultats montrent que l’erreur de prédiction relative (différence relative entre l’erreur de prédiction et la déviation standard) est de l’ordre de 2% à 16 %. Les résultats obtenus sont meilleurs que ceux obtenus avec la régression linéaire multivariée standard, comme présenté en fin du manuscrit. Rappelons que la méthode est intrinsèque, ne dépend en particulier pas de la structure de l’espace aérien / Air Traffic Management (ATM) heavily rely on the ability to predict where an aircraft will be located in a 10-minute time window. So it is clear that a controller’s workload can be reduced by an accurate and automated forecast of aircraft trajectories, since knowing whether conflicts are to be expected helps in prioritizing the actions to be taken.Despite the increasing autonomy in future air traffic management systems, especially 4D trajectory planning, the ability of trajectory prediction is still actual. As known, 4D trajectory planning implies aircraft will be properly located in a certain place at a certain time. However, such an approach is not realistic. Because, in particular, of the wind, Present Flight Management Systems are not able to predict precisely the position of an aircraft in a window larger than, say 15 minutes. For this reason, trajectory prediction problem can be stated as an actual issues at least for the near future. We consider the problem of short to mid-term aircraft trajectory prediction, that is, the estimation of where an aircraft will be located over a 10 to 30 minutes time horizon. Such a problem is central in the decision support tools, especially in conflict detection and resolution algorithms. It also appears when an air traffic controller observes traffic on the radar screen and tries to identify convergent aircraft, which may be in conflict in the near future, in order to apply maneuvers that will separate them. The problem is then to estimate where the aircraft will be located in the time horizon of 10 - 30 minutes. A 4-dimensional (4D) trajectory prediction contains data specifying the predicted horizontal and vertical position of an aircraft. The ability to accurately predict trajectories for different types of aircraft under different flight conditions, that involving external actions (pilot, ATC) and atmospheric influences (wind, temperature), is an important factor in determining the accuracy and effectiveness of ATM.In this work, we present an innovative approach for aircraft trajectory prediction in this work. It is based on local linear functional regression that considers data preprocessing, localizing and solving linear regression using wavelet decomposition. This approach starts from collecting the data set, consisting of a large enough amounts of aircraft trajectories between several airports, in order to make statistical procedures useful. It is necessary to note that ATC radar observations for a given aircraft are a discrete collection of aircraft coordinates, speed, projections, and other factors depending on the radar system. The next step, called localization, is to select a subset of trajectories of the same type of aircraft and connecting the same Origin-Destination as an aircraft trajectory to be predicted. Let us denote an aircraft trajectory to be predicted as a real trajectory. Then, the selected subset is taken as a learning data set to construct a model which is a linear functional regression model. The first part of real aircraft is also taken as a learning set to the model, and the second part is taken to compare it with the predicted part, which is a linear functional regression model. This algorithm takes into account only past radar tracks, and does not make use of any physical or aeronautical parameters.This approach has been successfully applied to aircraft trajectories between several airports on the data set (one year of air traffic over France). The data set consists of more than 2.9*10^6 flights. Several examples at the end of the manuscript show that the relative prediction error that is the difference between prediction error and standard deviation is about 2 to 16 per cents. The proposed algorithm shows better results compares to the standard multiple linear regressions that is shown from the figures at the end of the manuscript. The method is intrinsic and independent from airspace structure

abstract: Functional regression models are widely considered in practice. To precisely understand an underlying functional mechanism, a good sampling schedule for collecting informative functional data is necessary, especially when data collection is limited. However, scarce research has been conducted on the optimal sampling schedule design for the functional regression model so far. To address this design issue, efficient approaches are proposed for generating the best sampling plan in the functional regression setting. First, three optimal experimental designs are considered under a function-on-function linear model: the schedule that maximizes the relative efficiency for recovering the predictor function, the schedule that maximizes the relative efficiency for predicting the response function, and the schedule that maximizes the mixture of the relative efficiencies of both the predictor and response functions. The obtained sampling plan allows a precise recovery of the predictor function and a precise prediction of the response function. The proposed approach can also be reduced to identify the optimal sampling plan for the problem with a scalar-on-function linear regression model. In addition, the optimality criterion on predicting a scalar response using a functional predictor is derived when the quadratic relationship between these two variables is present, and proofs of important properties of the derived optimality criterion are also provided. To find such designs, an algorithm that is comparably fast, and can generate nearly optimal designs is proposed. As the optimality criterion includes quantities that must be estimated from prior knowledge (e.g., a pilot study), the effectiveness of the suggested optimal design highly depends on the quality of the estimates. However, in many situations, the estimates are unreliable; thus, a bootstrap aggregating (bagging) approach is employed for enhancing the quality of estimates and for finding sampling schedules stable to the misspecification of estimates. Through case studies, it is demonstrated that the proposed designs outperform other designs in terms of accurately predicting the response and recovering the predictor. It is also proposed that bagging-enhanced design generates a more robust sampling design under the misspecification of estimated quantities. / Dissertation/Thesis / Doctoral Dissertation Statistics 2020

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.