La théorie des graphes a longtemps été étudiée en mathématiques et en probabilité en tant qu’outil pour décrire la dépendance entre les nœuds. Cependant, ce n’est que récemment qu’elle a été mise en œuvre sur des données, donnant naissance à l’analyse statistique des réseaux réels.La topologie des réseaux économiques et financiers est remarquablement complexe: elle n’est généralement pas observée, et elle nécessite ainsi des procédures inférentielles adéquates pour son estimation, d’ailleurs non seulement les nœuds, mais la structure de la dépendance elle-même évolue dans le temps. Des outils statistiques et économétriques pour modéliser la dynamique de changement de la structure du réseau font défaut, malgré leurs besoins croissants dans plusieurs domaines de recherche. En même temps, avec le début de l’ère des “Big data”, la taille des ensembles de données disponibles devient de plus en plus élevée et leur structure interne devient de plus en plus complexe, entravant les processus inférentiels traditionnels dans plusieurs cas. Cette thèse a pour but de contribuer à ce nouveau champ littéraire qui associe probabilités, économie, physique et sociologie en proposant de nouvelles méthodologies statistiques et économétriques pour l’étude de l’évolution temporelle des structures en réseau de moyenne et haute dimension. / Graph theory has long been studied in mathematics and probability as a tool for describing dependence between nodes. However, only recently it has been implemented on data, giving birth to the statistical analysis of real networks.The topology of economic and financial networks is remarkably complex: it is generally unobserved, thus requiring adequate inferential procedures for it estimation, moreover not only the nodes, but the structure of dependence itself evolves over time. Statistical and econometric tools for modelling the dynamics of change of the network structure are lacking, despite their increasing requirement in several fields of research. At the same time, with the beginning of the era of “Big data” the size of available datasets is becoming increasingly high and their internal structure is growing in complexity, hampering traditional inferential processes in multiple cases.This thesis aims at contributing to this newborn field of literature which joins probability, economics, physics and sociology by proposing novel statistical and econometric methodologies for the study of the temporal evolution of network structures of medium-high dimension.

The thesis is dedicated to time series analysis for functional data and contains three original parts. In the first part, we derive statistical tests for the presence of a periodic component in a time series of functions. We consider both the traditional setting in which the periodic functional signal is contaminated by functional white noise, and a more general setting of a contaminating process which is weakly dependent. Several forms of the periodic component are considered. Our tests are motivated by the likelihood principle and fall into two broad categories, which we term multivariate and fully functional. Overall, for the functional series that motivate this research, the fully functional tests exhibit a superior balance of size and power. Asymptotic null distributions of all tests are derived and their consistency is established. Their finite sample performance is examined and compared by numerical studies and application to pollution data. In the second part, we consider vector autoregressive processes (VARs) with innovations having a singular covariance matrix (in short singular VARs). These objects appear naturally in the context of dynamic factor models. The Yule-Walker estimator of such a VAR is problematic, because the solution of the corresponding equation system tends to be numerically rather unstable. For example, if we overestimate the order of the VAR, then the singularity of the innovations renders the Yule-Walker equation system singular as well. Moreover, even with correctly selected order, the Yule-Walker system tends be close to singular in finite sample. We show that this has a severe impact on predictions. While the asymptotic rate of the mean square prediction error (MSPE) can be just like in the regular (non-singular) case, the finite sample behavior is suffering. This effect turns out to be particularly dramatic in context of dynamic factor models, where we do not directly observe the so-called common components which we aim to predict. Then, when the data are sampled with some additional error, the MSPE often gets severely inflated. We explain the reason for this phenomenon and show how to overcome the problem. Our numerical results underline that it is very important to adapt prediction algorithms accordingly. In the third part, we set up theoretical foundations and a practical method to forecast multiple functional time series (FTS). In order to do so, we generalize the static factor model to the case where cross-section units are FTS. We first derive a representation result. We show that if the first r eigenvalues of the covariance operator of the cross-section of n FTS are unbounded as n diverges and if the (r+1)th eigenvalue is bounded, then we can represent the each FTS as a sum of a common component driven by r factors and an idiosyncratic component. We suggest a method of estimation and prediction of such a model. We assess the performances of the method through a simulation study. Finally, we show that by applying our method to a cross-section of volatility curves of the stocks of S&P100, we have a better prediction accuracy than by limiting the analysis to individual FTS. / Doctorat en Sciences économiques et de gestion / info:eu-repo/semantics/nonPublished

Esta dissertação faz um estudo da aplicação da técnica estatística denominada Análise de Dados Funcionais (ADF) à geração de descritores usados em reconhecimento de padrões, mais especificamente, no reconhecimento de objetos de interesse em imagens. Estes objetos podem ser representados por vetores de características, também chamados de assinaturas, obtidos por uma técnica chamada de Dimensão Fractal Multiescala (DFM). Ocorre que estes vetores apresentam alta dimensionalidade (número de elementos), fazendo-se assim necessário o uso de uma abordagem que reduza este número de valores, sem que haja uma grande perda da informação transmitida pela assinatura. Neste contexto, diversas técnicas de extração de um reduzido conjunto de descritores da assinatura são apresentadas pela literatura. Entre estas, as mais populares são Fourier e \\emph, ambas relativamente simples de se apresentar e com resultados satisfatórios. A proposta aqui apresentada é de se utilizar ADF em combinação com DFM na geração de descritores de padrões. Os resultados obtidos com o uso desta abordagem na geração de descritores demostraram que a técnica possibilita bons resultados, mesmo em situações em que não é possível o uso de muitos descritores. Os experimentos demostraram que ADF apresenta um bom potencial para aplicação neste tipo de problema, permitindo que o método de classificação alcance bons resultados mesmo com poucos descritores. São sugeridos trabalhos futuros em que ADF possa ser usada, pesquisando-se por métodos ainda mais eficazes. / This work studies the application of a statistical technique named Functional Data Analysis (FDA) for the generation of descriptors. These descriptors can be used for pattern recognition, more specifically, for the recognition of relevant objects in an image. These objects can be represented by features vectors, also known as signatures, obtained by a technique named Multi-scale Fractal Dimension (MFD). These vectors present a high dimensionality (number of elements), causing to be necessary the use of an approach for the reduction of this number of values, but without a large loss of information carried by the signature. In this context, several techniques for the extraction of a reduced set of signature descriptors are studied in the literature. Among these techniques, the most classic are Fourier and wavelets, both with simple presentation and providing satisfactory results. The proposal presented here is the use of FDA combined with MFD for the generation of pattern descriptors. The results obtained by the use of this approach for the generation of descriptors showed that this technique allows the obtention of good results, even in situations in wich is not possible the use of many descriptors. FDA was also applied to the extraction of descriptors of MFD texture signatures. Also in this case, the results were interesting. The experiments showed the FDA presents a good potential for the application to this type of problem, allowing the obtention of good results even by using a few descriptors. It is suggested future works in which FDA can be used, researching for still more efficient methods.

In many instances arising prominently, but not exclusively, in imaging problems, it is important to condense the salient information so as to obtain a low-dimensional approximant of the data. This thesis is concerned with two basic situations which call for such a dimension reduction. The first of these is the statistical recovery of smooth edges in regression and density surfaces. The edges are understood to be contiguous curves, although they are allowed to meander almost arbitrarily through the plane, and may even split at a finite number of points to yield an edge graph. A novel locally-parametric nonparametric method is proposed which enjoys the benefit of being relatively easy to implement via a `tracking' approach. These topics are discussed in Chapters 2 and 3, with pertaining background material being given in the Appendix. In Chapter 4 we construct concomitant confidence bands for this estimator, which have asymptotically correct coverage probability. The construction can be likened to only a few existing approaches, and may thus be considered as our main contribution. ¶ Chapter 5 discusses numerical issues pertaining to the edge and confidence band estimators of Chapters 2-4. Connections are drawn to popular topics which originated in the fields of computer vision and signal processing, and which surround edge detection. These connections are exploited so as to obtain greater robustness of the likelihood estimator, such as with the presence of sharp corners. ¶ Chapter 6 addresses a dimension reduction problem for spatial data where the ultimate objective of the analysis is the discrimination of these data into one of a few pre-specified groups. In the dimension reduction step, an instrumental role is played by the recently developed methodology of functional data analysis. Relatively standar non-linear image processing techniques, as well as wavelet shrinkage, are used prior to this step. A case study for remotely-sensed navigation radar data exemplifies the methodology of Chapter 6.

In this thesis, we develop several novel degradation models based on techniques from functional data analysis. These models are suitable for characterizing different types of sensor-based degradation signals, whether they are censored at a certain fixed time point or truncated at the failure threshold. Our proposed models can also be easily extended to accommodate for the effects of environmental conditions on degradation processes. Unlike many existing degradation models that rely on the existence of a historical sample of complete degradation signals, our modeling framework is well-suited for modeling complete as well as incomplete (sparse and fragmented) degradation signals. We utilize these models to predict and continuously update, in real time, the residual life distributions of partially degraded components. We assess and compare the performance of our proposed models and existing benchmark models by using simulated signals and real world data sets. The results indicate that our models can provide a better characterization of the degradation signals and a more accurate prediction of a system's lifetime under different signal scenarios. Another major advantage of our models is their robustness to the model mis-specification, which is especially important for applications with incomplete degradation signals (sparse or fragmented).

With the rise autonomous and robotic systems in field applications, the need for dexterous, highly adaptable end effectors has become a major research topic. Control mechanisms of robotics hands with a high number independent actuators is recognized as a complex, high dimensional problem, with exponentially complex algorithms. However, recent studies have shown that human hand motion possesses very high joint correlation which translates into a set of predefined postures, or synergies. The hand produces a motion using a complementing contribution of multiple joints, called synergies. The similarities place variables onto a common dimensional space, effectively reducing the number of independent variables. In this thesis, we analyze the motion of the hand during a set of objects grasps using mul- tivariate Principal Component Analysis (mPCA) to extract both the principal variables and their correlation during grasping. We introduce the use of Functional PCA (fPCA) primarily on princi- pal components to study the dynamic requirements of the motion. The goal is to defined a set of synergies common and specific to all motions. We expand the analysis by classifying the objects grasps, or tasks, using their functional components, or harmonics over the entire motion. A set of groups are described based on these classification that confirms empirical findings. Lastly, we evaluate the motions generated from the analysis by applying them onto robotic hands. The results from the mPCA and fPCA procedures are used to map the principal components from each motion onto underactuated robotic designs. We produce a viable routine that indicates how the mapping is performed, and finally, we implement the motion generated onto a real hand. The resultant robotic motion was evaluated on how it mimics the human motion.

In recent years, Condition Monitoring (CM), which can be performed via several sensor channels, has been recognized as an effective paradigm for failure prevention of operational equipment or processes. However, the complexity caused by asynchronous data collection with different and/or time-varying sampling/transmission rates has long been a hindrance in the effective use of multichannel data in constructing empirical models. The problem becomes more challenging when sensor readings are incomplete. Traditional sensor data recovery techniques are often prohibited in asynchronous CM environments, not to mention sparse datasets. The proposed Functional Principal Component Analysis (FPCA) methodologies, e.g., nonparametric FPC model and semi-parametric functional regression model, provide new sensor data recovery techniques to improve the reliability and robustness of multichannel CM systems. Based on the FPCA results obtained from historical asynchronous data, the deviation from the smoothing trajectory of each sensor signal can be described by a set of unit-specific model parameters. Furthermore, the relationships among these sensor signals can be identified and used to construct regression models for the correlated signals. For real-time or online implementation, use of these models along with the parameters adjusted by real-time CM data become powerful tools for dealing with asynchronous CM data while recovering lost data when needed. To improve the robustness and predictability in dealing with asynchronous data, which may be skewed in probability distribution, robust methods were developed based on Functional Data Analysis (FDA) and Local Quantile Regression (LQR) models. Case studies examining turbofan aircraft engines and an experimental two-tank flow-control loop are used to demonstrate the effectiveness and adaptability of the proposed sensor data recovery techniques. The proposed methods may also find a variety of applications in systems of other industries, such as nuclear power plants, wind turbines, railway systems, economic fields, etc., which may face asynchronous sampling and/or missing data collection problems.

函數型資料分析的是一組曲線資料，通常定義域為一段時間範圍。常見的如某一個地區人口在成長期的身高紀錄表或是氣候統計資料。函數型資料主要特色曲線間常有共同趨勢，而且個別曲線反應共同趨勢時也有時間和強度上的差異。本文研究主要是使用Kneip 和 Ramsay提出，結合對齊程序和主成分分析的想法作為模型架構，來分析函數型資料的特性。首先在對齊過程中，使用時間轉換函數(warping function)，解決觀測資料上時間的差異；並使用主成分分析方法，幫助研究者探討資料的主要特性。基於函數型資料被預期的共同趨勢性，我們可以利用此一特色作為各種類型資料分類上的依據。此外本研究會對幾種選取主成分個數的方法，進行綜合討論與比較。 / In this thesis, a procedure combining curve alignment and functional principal component analysis is studied. The procedure is proposed by Kneip and Ramsay .In functional principal component analysis, if the data curves are roughly linear combinations of k basis curves, then the data curves are expected to be explained well by principle component curves. The goal of this study is to examine whether this property still holds when curves need to be aligned. It is found that, if the aligned data curves can be approximated well by k basis curves, then applying Kneip and Ramsay's procedure to the unaligned curves gives k principal components that can explain the aligned curves well. Several approaches for selecting the number of principal components are proposed and compared.

Esta dissertação faz um estudo da aplicação da técnica estatística denominada Análise de Dados Funcionais (ADF) à geração de descritores usados em reconhecimento de padrões, mais especificamente, no reconhecimento de objetos de interesse em imagens. Estes objetos podem ser representados por vetores de características, também chamados de assinaturas, obtidos por uma técnica chamada de Dimensão Fractal Multiescala (DFM). Ocorre que estes vetores apresentam alta dimensionalidade (número de elementos), fazendo-se assim necessário o uso de uma abordagem que reduza este número de valores, sem que haja uma grande perda da informação transmitida pela assinatura. Neste contexto, diversas técnicas de extração de um reduzido conjunto de descritores da assinatura são apresentadas pela literatura. Entre estas, as mais populares são Fourier e \\emph, ambas relativamente simples de se apresentar e com resultados satisfatórios. A proposta aqui apresentada é de se utilizar ADF em combinação com DFM na geração de descritores de padrões. Os resultados obtidos com o uso desta abordagem na geração de descritores demostraram que a técnica possibilita bons resultados, mesmo em situações em que não é possível o uso de muitos descritores. Os experimentos demostraram que ADF apresenta um bom potencial para aplicação neste tipo de problema, permitindo que o método de classificação alcance bons resultados mesmo com poucos descritores. São sugeridos trabalhos futuros em que ADF possa ser usada, pesquisando-se por métodos ainda mais eficazes. / This work studies the application of a statistical technique named Functional Data Analysis (FDA) for the generation of descriptors. These descriptors can be used for pattern recognition, more specifically, for the recognition of relevant objects in an image. These objects can be represented by features vectors, also known as signatures, obtained by a technique named Multi-scale Fractal Dimension (MFD). These vectors present a high dimensionality (number of elements), causing to be necessary the use of an approach for the reduction of this number of values, but without a large loss of information carried by the signature. In this context, several techniques for the extraction of a reduced set of signature descriptors are studied in the literature. Among these techniques, the most classic are Fourier and wavelets, both with simple presentation and providing satisfactory results. The proposal presented here is the use of FDA combined with MFD for the generation of pattern descriptors. The results obtained by the use of this approach for the generation of descriptors showed that this technique allows the obtention of good results, even in situations in wich is not possible the use of many descriptors. FDA was also applied to the extraction of descriptors of MFD texture signatures. Also in this case, the results were interesting. The experiments showed the FDA presents a good potential for the application to this type of problem, allowing the obtention of good results even by using a few descriptors. It is suggested future works in which FDA can be used, researching for still more efficient methods.

Lung cancer has been considered one of the leading causes of deaths while cancer re- mains the second most common cause of deaths in the USA. Understanding the behavior of a disease over time could yield important information to make decisions about the disease. Statistical models could provide crucial clues and help to make a decision about the dis- ease, budget allocation, evaluation, and implement prevention. Longitudinal trend analysis of the diseases helps to understand long term effects and nature. Cholesterol level is one of the most contributing risk factors for Coronary Heart Disease. Studying cholesterol statistically helps to know more about its nature and provides crucial information to mitigate its effectiveness in diagnosing its impact to public health. In our study, we have analyzed lung cancer mortality in the USA based on age at death, period at death, and birth cohort to investigate its nature in longitudinal effects. The attempt has been made to estimate mortality rate based on age for different age groups and to find the relative risk of mortality due to period effect and relative risk due to birth cohort for lung cancer in the United States. Our statistical analysis and modeling are based on the data obtained from Surveillance Epidemiology and End Results (SEER) program of the United States. We have also investigated the probabilistic behavior of average cholesterol level based on gender and ethnicity. The study reveals significant differences with respect to the distribution they follow and their basic inferences which could be beneficial to draw conclusions in various ways in addressing related issues. At the same time, the change of cholesterol level over time for an individual might be a good source to study the association of cholesterol level, coronary heart disease and their effects on age. The cholesterol data is obtained from inter-university Consortium for Political and Social Research and National Health and Nutrition Examination Survey (NHANS) of the United States. Understanding the average change in total serum cholesterol level over time as people get older could be vital to explore it. We have studied the longitudinal behavior of the association of sex and time with cholesterol level. It is observed that age, sex, and time have an individual effect and can impact differently upon collective considerations. Their adverse effect in increasing cholesterol level could promote to worsen the cholesterol re- lated issues and hence heart related diseases. We believe our study pivots knowing more about target population of cholesterol level and helps to have the useful inference about cholesterol levels for public health. Finally, we also analyzed the average cholesterol data using a functional data analysis approach to understand its nature and effect on age. Since functional data analysis approach presents more flexibility in modeling, it could provide more insight in studying cholesterol level.