Esta dissertação apresenta um método adaptado do estudo de repetitividade e reprodutibilidade para analisar a capacidade e o desempenho de sistemas de medição, no contexto da análise de dados funcionais. Dado funcional é a variável de resposta dada por uma coleção de dados que formam um perfil ou uma curva. O método adaptado contribui para o avanço do estado da arte sobre a análise de sistemas de medição. O método proposto é uma alternativa ao uso de métodos tradicionais de análise, que usados de forma equivocada, podem deteriorar a qualidade dos produtos monitorados através de variáveis de resposta funcionais. O método proposto envolve a adaptação de testes de hipótese e da análise de variância de um e dois fatores usados em comparações de populações, na avaliação de sistemas de medições. A proposta de adaptação foi baseada na utilização de distâncias entre curvas. Foi usada a Distância de Hausdorff como uma medida de proximidade entre as curvas. A adaptação proposta à análise de variância foi composta de três abordagens. Os métodos adaptados foram aplicados a um estudo simulado de repetitividade e reprodutibilidade. O estudo foi estruturado para analisar cenários em que o sistema de medição foi aprovado e reprovado. O método proposto foi denominado de ANOVA das Distâncias. / This work presents a method to analyze a measurement system's performance in a functional data analysis context, based on repeatability and reproducibility studies. Functional data are a collection of data points organized as a profile or curve. The proposed method contributes to the state of the art on measurement system analysis. The method is an alternative to traditional methods often used mistakenly, leading to deterioration in the quality of products monitored through functional responses. In the proposed method we adapt hypothesis tests and one-way and two-way ANOVA to be used in measurement system analysis. The method is grounded on the use of distances between curves. For that matter the Hausdorff distance was chosen as a measure of proximity between curves. Three ANOVA approaches were proposed and applied in a simulated repeatability and reproducibility study. The study was structured to analyze scenarios in which the measurement system was approved or rejected. The proposed method was named ANOVA of the distances.

Esta dissertação apresenta um método adaptado do estudo de repetitividade e reprodutibilidade para analisar a capacidade e o desempenho de sistemas de medição, no contexto da análise de dados funcionais. Dado funcional é a variável de resposta dada por uma coleção de dados que formam um perfil ou uma curva. O método adaptado contribui para o avanço do estado da arte sobre a análise de sistemas de medição. O método proposto é uma alternativa ao uso de métodos tradicionais de análise, que usados de forma equivocada, podem deteriorar a qualidade dos produtos monitorados através de variáveis de resposta funcionais. O método proposto envolve a adaptação de testes de hipótese e da análise de variância de um e dois fatores usados em comparações de populações, na avaliação de sistemas de medições. A proposta de adaptação foi baseada na utilização de distâncias entre curvas. Foi usada a Distância de Hausdorff como uma medida de proximidade entre as curvas. A adaptação proposta à análise de variância foi composta de três abordagens. Os métodos adaptados foram aplicados a um estudo simulado de repetitividade e reprodutibilidade. O estudo foi estruturado para analisar cenários em que o sistema de medição foi aprovado e reprovado. O método proposto foi denominado de ANOVA das Distâncias. / This work presents a method to analyze a measurement system's performance in a functional data analysis context, based on repeatability and reproducibility studies. Functional data are a collection of data points organized as a profile or curve. The proposed method contributes to the state of the art on measurement system analysis. The method is an alternative to traditional methods often used mistakenly, leading to deterioration in the quality of products monitored through functional responses. In the proposed method we adapt hypothesis tests and one-way and two-way ANOVA to be used in measurement system analysis. The method is grounded on the use of distances between curves. For that matter the Hausdorff distance was chosen as a measure of proximity between curves. Three ANOVA approaches were proposed and applied in a simulated repeatability and reproducibility study. The study was structured to analyze scenarios in which the measurement system was approved or rejected. The proposed method was named ANOVA of the distances.

Com o recente avanço do poder computacional, a amostragem de curvas indexadas espacialmente tem crescido principalmente em dados ecológicos, atmosféricos e ambientais, o que conduziu a adaptação de métodos geoestatísticos para o contexto de Análise de Dados Funcionais. O objetivo deste trabalho é estudar métodos de krigagem para Dados Funcionais, adaptando os métodos de interpolação espacial em Geoestatística. Mais precisamente, em um conjunto de dados funcionais pontualmente fracamente estacionário e isotrópico, desejamos estimar uma curva em um ponto não monitorado no espaço buscando estimadores não viciados com erro quadrático médio mínimo. Apresentamos três abordagens para aproximar uma curva em sítio não monitorado, demonstramos resultados que simplificam o problema de otimização postulado pela busca de estimadores ótimos não viciados, implementamos os modelos em MATLAB usando ondaletas, que é mais adequada para captar comportamentos localizados, e comparamos os três modelos através de estudos de simulação. Ilustramos os métodos através de dois conjuntos de dados reais: um conjunto de dados de temperatura média diária das províncias marítimas do Canadá (New Brunswick, Nova Scotia e Prince Edward Island) coletados em 82 estações no ano 2000 e um conjunto de dados da CETESB (Companhia Ambiental do Estado de São Paulo) referentes ao índice de qualidade de ar MP10 em 22 estações meteorológicas na região metropolitana da cidade de São Paulo coletados no ano de 2014. / The advance of the computational power in last decades has been generating a considerable increase in datasets of spatially indexed curves, mainly in ecological, atmospheric and environmental data, what have leaded to adjustments of geostatistcs for the context of Functional Data Analysis. The goal of this work is to adapt the kriging methods from geostatistcs analysis to the framework of Functional Data Analysis. More precisely, we shall interpolate a curve in an unvisited spot searching for an unbiased estimator with minimum mean square error for a pointwise weakly stationary and isotropic functional dataset. We introduce three different approaches to estimate a curve in an unvisited spot, we demonstrate some results simplifying the optimization problem postulated by the optimality from these estimators, we implement the three models in MATLAB using wavelets and we compare them by simulation. We illustrate the ideas using two dataset: a real climatic dataset from Canadian maritime provinces (New Brunswick, Nova Scotia and Prince Edward Island) sampled at year 2000 in 82 weather station consisting of daily mean temperature and data from CETESB (environmental agency from the state of São Paulo, Brazil) sampled at 22 weather station in the metropolitan region of São Paulo city at year 2014 consisting of the air quality index PM10.

This thesis focuses on two topics, computational methods for large spatial datasets and functional data ranking. Both are tackling the challenges of big and high-dimensional data. The first topic is motivated by the prohibitive computational burden in fitting Gaussian process models to large and irregularly spaced spatial datasets. Various approximation methods have been introduced to reduce the computational cost, but many rely on unrealistic assumptions about the process and retaining statistical efficiency remains an issue. We propose a new scheme to approximate the maximum likelihood estimator and the kriging predictor when the exact computation is infeasible. The proposed method provides different types of hierarchical low-rank approximations that are both computationally and statistically efficient. We explore the improvement of the approximation theoretically and investigate the performance by simulations. For real applications, we analyze a soil moisture dataset with 2 million measurements with the hierarchical low-rank approximation and apply the proposed fast kriging to fill gaps for satellite images. The second topic is motivated by rank-based outlier detection methods for functional data. Compared to magnitude outliers, it is more challenging to detect shape outliers as they are often masked among samples. We develop a new notion of functional data depth by taking the integration of a univariate depth function. Having a form of the integrated depth, it shares many desirable features. Furthermore, the novel formation leads to a useful decomposition for detecting both shape and magnitude outliers. Our simulation studies show the proposed outlier detection procedure outperforms competitors in various outlier models. We also illustrate our methodology using real datasets of curves, images, and video frames. Finally, we introduce the functional data ranking technique to spatio-temporal statistics for visualizing and assessing covariance properties, such as separability and full symmetry. We formulate test functions as functions of temporal lags for each pair of spatial locations and develop a rank-based testing procedure induced by functional data depth for assessing these properties. The method is illustrated using simulated data from widely used spatio-temporal covariance models, as well as real datasets from weather stations and climate model outputs.

Indiana University-Purdue University Indianapolis (IUPUI) / The research ideas included in this dissertation are motivated by a large sexually trans mitted infections (STIs) study (IU Phone study), which is also an ecological momentary assessment (EMA) study implemented by Indiana University from 2008 to 2013. EMA, as a group of methods used to collect subjects’ up-to-date behaviors and status, can increase the accuracy of this information by allowing a participant to self-administer a survey or diary entry, in their own environment, as close to the occurrence of the behavior as possible. IU Phone study’s high reporting level shows one of the benefits gain from introducing EMA in STIs study. As a prospective study lasting for 84 days, participants in IU Phone study undergo STI testing and complete EMA forms with project-furnished cellular telephones according to the predetermined schedules. At pre-selected eight-hour intervals, participants respond to a series of questions to identify sexual and non-sexual interactions with specific partners including partner name, relationship satisfaction and sexual satisfaction with this partner, time of each coital event and condom use for each event. etc. STIs lab results of all the participants are collected weekly as well. We are interested in several variables related to the risk of infection and sexual or non-sexual behaviors, especially the relationship among the longitudinal processes of those variables. New statistical models and applications are established to deal with the data with complex dependence and sampling data structures. The methodologies covers various of statistical aspect like generalized mixed models, mul tivariate models and autoregressive and cross-lagged model in longitudinal data analysis, misclassification adjustment in imperfect diagnostic tests, and variable-domain functional regression in functional data analysis. The contribution of our work is we bridge the meth ods from different areas with EMA data in the IU Phone study and also build up a novel understanding of the association among all the variables of interest from different perspec tives based on the characteristic of the data. Besides all the statistical analyses included in this dissertation, variety of data visualization techniques also provide informative support in presenting the complex EMA data structure.

This thesis consists of two parts.The first part considers further development of a model used for marital fertility, the Coale-Trussell's fertility model, which is based on age-specific fertility rates. A new model is suggested using individual fertility data and a waiting time after pregnancies. The model is named the waiting model and can be understood as an alternating renewal process with age-specific intensities. Due to the complicated form of the waiting model and the way data is presented, as given in the United Nation Demographic Year Book 1965, a normal approximation is suggested together with a normal approximation of the mean and variance of the number of births per summarized interval. A further refinement of the model was then introduced to allow for left truncated and censored individual data, summarized as table data. The waiting model suggested gives better understanding of marital fertility and by a simulation study it is shown that the waiting model outperforms the Coale-Trussell model when it comes to estimating the fertility intensity and to predict the mean and variance of the number of births for a population. The second part of the thesis focus on developing functional clustering methods.The methods are motivated by and applied to varved (annually laminated) sediment data from lake Kassj\"on in northern Sweden. The rich but complex information (with respect to climate) in the varves, including the shapes of the seasonal patterns, the varying varve thickness, and the non-linear sediment accumulation rates makes it non-trivial to cluster the varves. Functional representations, smoothing and alignment are functional data tools used to make the seasonal patterns comparable.Functional clustering is used to group the seasonal patterns into different types, which can be associated with different weather conditions. A new non-parametric functional clustering method is suggested, the Bagging Voronoi K-mediod Alignment algorithm, (BVKMA), which simultaneously clusters and aligns spatially dependent curves. BVKMA is used on the varved lake sediment, to infer on climate, defined as frequencies of different weather types, over longer time periods. Furthermore, a functional model-based clustering method is proposed that clusters subjects for which both functional data and covariates are observed, allowing different covariance structures in the different clusters. The model extends a model-based functional clustering method proposed by James and Suger (2003). An EM algorithm is derived to estimate the parameters of the model.

L'étude de la relation entre les variations d'abondance du phytoplancton et les facteurs environnementaux (naturels ou anthropiques) dans les zones saumâtres peu profondes est essentielle à la compréhension et à la gestion de cet écosystème complexe. Les relations existant entre les variables physico-chimiques (température, salinité et les nutriments) et les assemblages de phytoplancton de l'étang de Berre ont été analysées à partir d'un suivi écologique mensuel de 16 années (1994-2010). A l'aide des données recueillies par cette étude à long terme, des questions en relation avec la gestion de ce milieu ont été abordées grâce à l'application d'analyses statistiques et à la représentation originale des données. Depuis 2004, la nouvelle politique de relargage d'eau douce a provoqué de forts changements dans la salinité globale de la lagune : une diminution de la stratification et une raréfaction des phénomènes d'anoxie dans sa partie la plus profonde. Un changement dans la structure de la communauté phytoplanctonique a également été observé en association avec l'évolution des conditions environnementales. Une augmentation de la richesse spécifique phytoplanctonique, et plus précisément, l'émergence d'espèces à affinité marine a permis de mettre en évidence la première étape d'une marinisation de la lagune. Ces résultats soulignent l'impact significatif d'un nouvelle politique de gestion de cette zone côtière particulière. Nous nous sommes ensuite intéressés à la dynamique du phytoplancton à l'échelle de la journée reflet des variations rapides de l'environnement. / The study of the relationship between variations in phytoplankton abundance and environmental forces (natural or anthropogenic) in shallow brackish areas is essential to both understanding and managing this complex ecosystem. Over a 16 year (1994-2011) monthly monitoring program the relationships between physicochemical variables (temperature, salinity and nutrients) and phytoplankton assemblages of the Berre Lagoon were analyzed. Using data collected from this long-term study, we have addressed environmental management issues through the application of advanced statistical analyses and original data displays. These analyses and data displays can readily be applied to other data sets related to the environment, with the aim of informing both researcher and practitioner. Since 2004, a new policy for freshwater discharge has induced strong changes in the global salinity of the lagoon : a weakened stratification and a rarefaction of anoxia phenomena in its deepest part. A shift in the structure of the phytoplankton community has been observed in association with changes in environmental conditions. An increase of phytoplanktonic species richness, and more precisely, the emergence of species with marine affinity highlights the first step of a marinization of the lagoon. The results underline the significant impact of a new management policy in this specific coastal zone. We then focused on the response of phytoplankton to quick environmental variations. An original approach for automated high frequency analysis of phytoplankton was adopted with the use of an autonomous flow cytometer (CytoSense).

Master of Science / Department of Statistics / Leigh Murray / Using remotely sensed reflectance data is an appealing tool for controlling invasive species of grasses by rangeland managers. Recent developments in functional data analysis include the functional boxplot (FBP) which is shown here to be a useful tool in the visualization of reflectance data. Functional boxplots are a novel method of visually inspecting functional data and determining the presence of outliers in the data. Implementation and interpretation of FBPs are both straightforward and intuitive. The goal of this study is to examine the use of FBPs for visualizing reflectance data, and to determine the efficacy of using the FBP to distinguish between native tall grasses and invasive Old World Bluestem (OWB, Bothriochloa spp.) monocultures in a Kansas prairie. Validation trials were conducted in order to determine the stability of the FBP when used to analyze spectral data. FBPs were shown to be highly stable for use with both native and OWB grasses at all times and subsets of wavelengths tested. Identification trials were conducted by introducing a single OWB observation to a test set of native tall grass observations and constructing a FBP. Results indicate that using observations recorded early in the growing season, the functional boxplot is able to successfully identify the OWB observation as an outlier in a test set of native tall grass observations with an estimated probability 100% and 95.45% when considering the visible and cellular spectrums, respectively. A 95% lower bound for the probability of successfully identifying the OWB observation using the cellular spectrum in May is found to be 89.67%.

Afin d'étudier les processus biogéochimiques de l'Océan Austral, des balises posées sur des éléphants de mer ont permis de récolter en 2009-2010 des profils de variables océanographiques (Chlorophylle a (Chl a), température, salinité, lumière) dans une zone s'étalant du sud des îles Kerguelen jusqu'au continent Antarctique. Cette thèse se penche en particulier sur les données de Chl a, car celle-ci est contenue dans les organismes photosynthétiques qui jouent un rôle essentiel de pompe à carbone. Mais les profils verticaux de Chl a, récoltés peu fréquemment, ne permettent pas d'obtenir une cartographie de cette variable dans cette zone de l'océan. Cependant, nous disposons de profils de lumière, échantillonnés plus souvent. L'objectif était alors de développer une méthodologie permettant de reconstruire de manière indirecte les profils de Chl a à partir des profils de lumière, et qui prenne en compte les caractéristiques de ce type de données qui se présentent naturellement comme des données fonctionnelles. Pour cela, nous avons abordé la décomposition des profils à reconstruire ou explicatifs sur une base de splines, ainsi que les questions d'ajustement associées. Un modèle linéaire fonctionnel a été utilisé, permettant de prédire des profils de Chl a à partir des dérivées des profils de lumière. Il est montré que l'utilisation d'un tel modèle permet d'obtenir une bonne qualité de reconstruction pour accéder aux variations hautes fréquences des profils de Chl a à fine échelle. Enfin, une interpolation par krigeage fonctionnel permet de prédire la concentration en Chl a de nuit, car les mesures de lumière acquises à ce moment-là ne peuvent pas être exploitées. / To study biogeochemical processes in the Southern Ocean, tags placed on elephant seals allowed to collect during 2009-2010 oceanographic variables profiles (Chlorophyll a (Chl a), temperature, salinity, light) in an area ranging from southern Kerguelen until the Antarctic continent. This thesis focuses on Chl a data as it is contained in photosynthetic organisms and these ones play an essential role in the oceanic carbon cycle. The infrequently collected vertical Chl a profiles don't provide a mapping of this variable in this area of the ocean. However, we have light profiles sampled more often. The aim of this thesis was then to develop a methodology for reconstructing indirectly Chl a profiles from light profiles, and that takes into account characteristics of this kind of data that naturally occur as functional data. For this, we adressed the profiles decomposition to rebuild or explanations on splines basis, as well as issues related adjustment. A functional linear model was used to predict Chl a profiles from light profiles derivatives. It was shown that the use of such a model provides a good quality of reconstruction to access high frequency variations of Chl a profiles at fine scale. Finally, a functional kriging interpolation predicted the Chl a concentration during night, as light measurements acquired at that time can't be exploited. In the future, the methodology aims to be applied to any type of functional data.

L'objet principal de cette thèse est de développer des estimateurs adaptatifs en statistique pour données fonctionnelles. Dans une première partie, nous nous intéressons au modèle linéaire fonctionnel et nous définissons un critère de sélection de la dimension pour des estimateurs par projection définis sur des bases fixe ou aléatoire. Les estimateurs obtenus vérifient une inégalité de type oracle et atteignent la vitesse de convergence minimax pour le risque lié à l'erreur de prédiction. Pour les estimateurs définis sur une collection de modèles aléatoires, des outils de théorie de la perturbation ont été utilisés pour contrôler les projecteurs aléatoires de manière non-asymptotique. D'un point de vue numérique, cette méthode de sélection de la dimension est plus rapide et plus stable que les méthodes usuelles de validation croisée. Dans une seconde partie, nous proposons un critère de sélection de fenêtre inspiré des travaux de Goldenshluger et Lepski, pour des estimateurs à noyau de la fonction de répartition conditionnelle lorsque la covariable est fonctionnelle. Le risque de l'estimateur obtenu est majoré de manière non-asymptotique. Des bornes inférieures sont prouvées ce qui nous permet d'établir que notre estimateur atteint la vitesse de convergence minimax, à une perte logarithmique près. Dans une dernière partie, nous proposons une extension au cadre fonctionnel de la méthodologie des surfaces de réponse, très utilisée dans l'industrie. Ce travail est motivé par une application à la sûreté nucléaire. / The main purpose of this thesis is to develop adaptive estimators for functional data.In the first part, we focus on the functional linear model and we propose a dimension selection device for projection estimators defined on both fixed and data-driven bases. The prediction error of the resulting estimators satisfies an oracle-type inequality and reaches the minimax rate of convergence. For the estimator defined on a data-driven approximation space, tools of perturbation theory are used to solve the problems related to the random nature of the collection of models. From a numerical point of view, this method of dimension selection is faster and more stable than the usual methods of cross validation.In a second part, we consider the problem of bandwidth selection for kernel estimators of the conditional cumulative distribution function when the covariate is functional. The method is inspired by the work of Goldenshluger and Lepski. The risk of the estimator is non-asymptotically upper-bounded. We also prove lower-bounds and establish that our estimator reaches the minimax convergence rate, up to an extra logarithmic term.In the last part, we propose an extension to a functional context of the response surface methodology, widely used in the industry. This work is motivated by an application to nuclear safety.