Increasing habitat fragmentation and rapid global warming is changing the conditions for species populations and ecological communities around the world. This presents challenges for the maintenance of biodiversity and a dominant paradigm for conservation in fragmented habitats is given by island biogeography and metapopulation (or metacommunity) ecology. In this thesis I approach key concepts (area, connectivity and community assembly) in island biogeography and metacommunity ecology within the context of a dynamic land uplift archipelago. The presented work consists of two interwoven themes: (i) A methodological theme in which statistical approaches are developed to deal with the complexities of multispecies dynamic systems, and (ii) an applied theme dealing with community assembly and island biogeography of bryophytes on young land uplift islands. To describe island connectivity for entire species assemblages, an approach using functional principal component analysis (fPCA) on patch connectivity functions (the connectivity of an island as a continuous function of a variable representing the spatial scale of species dispersal capacities) was developed. In addition, a new statistical method, functional co-inertia analysis (fCoIA), for analyzing co-variation between multivariate species data and continuous functions was developed and applied to the relation between bryophyte species incidences and the island age/area-dynamics. Primarily asexual bryophyte species are dispersal limited and presence probabilities are related to island connectivity. No such patterns were found for species, at least occasionally, producing spores. Our results suggest that bryophyte dispersal is regulated by the contribution of spores to a regional spore rain and that bryophyte species with low spore output at the landscape level may be extra vulnerable under habitat fragmentation and loss. Having specialized asexual propagules increases the presence probabilities on islands, partly compensating for the dispersal limitation in asexual species. This suggests a trade-off between dispersal and establishment capacity, but also points to the importance of local dispersal for maintaining populations under the succession driven spatial turnover of microsites on the islands. Bryophyte colonization is strongly limited by habitat availability when a given habitats is rare, but there seems to exist a threshold over which other processes (e.g. dispersal limitation) become more important. Species with more vagile life history strategies appear to be stronger affected by the area of available habitats than many perennial species

As adults age, they lose the ability to produce maximal power and speed of movement. Success in daily living is often dependent upon power and speed. Thus these age-related decrements in performance can reduce physical independence and quality of life. An active lifestyle in older adulthood is associated with more successful aging. The purpose of this research program was to define the link between habitual activity and performance, specifically in regard to activities requiring power and speed. The hypothesis was that active older adults, compared to sedentary older adults, would be characterized by greater power production in maximal- and submaximal-effort tasks. Grouping older adults by activity level, coactivation was associated with activity level. Functional tasks are performed with a range of power requirements. Coactivation was used to distinguish groups in a maximal power task (Study 1) and submaximal power tasks (Study 2). In Study 1, the young adults demonstrated a greater maximal power than the older adults. While maximal power was not different between the older active and sedentary groups, the groups did differ on how they created maximal power. The active older adults produced a greater coactivation in the lower leg muscles compared to the older sedentary adults. In Study 2, the active older adults responded to different speeds during a submaximal power task with greater coactivation in the muscles of the lower leg at slow speeds compared with the sedentary older adults. Both older adults groups increased coactivation in the thigh muscles at high speeds. The sedentary older adults responded to speed with increased coactivation in the lower leg at fast speeds. The active older adults increased proximal thigh coactivation, EMG index, at the fastest speed compared with the sedentary older adults. Both older adult groups showed muscle activation adaptation to the change in task demands. The results of this dissertation increase our understanding about the link between physical activity and performance. Age-related differences in coactivation were observed during both maximal and submaximal tasks. Activity-related differences were observed suggesting the active older adults have a greater capability to adjust muscle activity to meet the challenges of community living. / text

The depth function (functional) is a modern nonparametric statistical analysis tool for (finite-dimensional) data with lots of practical applications. In the present work we focus on the possibilities of the extension of the depth concept onto a functional data case. In the case of finite-dimensional functional data the isomorphism between the functional space and the finite-dimensional Euclidean space will be utilized in order to introduce the induced functional data depths. A theorem about induced depths' properties will be proven and on several examples the possibilities and restraints of it's practical applications will be shown. Moreover, we describe and demonstrate the advantages and disadvantages of the established depth functionals used in the literature (Fraiman-Muniz depths and band depths). In order to facilitate the outcoming drawbacks of known depths, we propose new, K-band depth based on the inference extension from continuous to smooth functions. Several important properties of the K-band depth will be derived. On a final supervised classification simulation study the reasonability of practical use of the new approach will be shown. As a conclusion, the computational complexity of all presented depth functionals will be compared.

This thesis presents a contribution to the active research area of functional data analysis (FDA) and is concerned with the analysis of data from complex experimental designs in which the responses are curves. High resolution, closely correlated data sets are encountered in many research fields, but current statistical methodologies often analyse simplistic summary measures and therefore limit the completeness and accuracy of conclusions drawn. Specifically the nature of the curves and experimental design are not taken into account. Mathematically, such curves can be modelled either as sample paths of a stochastic process or as random elements in a Hilbert space. Despite this more complex type of response, the structure of experiments which yield functional data is often the same as in classical experimentation. Thus, classical experimental design principles and results can be adapted to the FDA setting. More specifically, we are interested in the functional analysis of variance (ANOVA) of experiments which use orthogonal designs. Most of the existing functional ANOVA approaches consider only completely randomised designs. However, we are interested in more complex experimental arrangements such as, for example, split-plot and row-column designs. Similar to univariate responses, such complex designs imply that the response curves for different observational units are correlated. We use the design to derive a functional mixed-effects model and adapt the classical projection approach in order to derive the functional ANOVA. As a main result, we derive new functional F tests for hypotheses about treatment effects in the appropriate strata of the design. The approximate null distribution of these tests is derived by applying the Karhunen- Lo`eve expansion to the covariance functions in the relevant strata. These results extend existing work on functional F tests for completely randomised designs. The methodology developed in the thesis has wide applicability. In particular, we consider novel applications of functional F tests to gait analysis. Results are presented for two empirical studies. In the first study, gait data of patients with cerebral palsy were collected during barefoot walking and walking with ankle-foot orthoses. The effects of ankle-foot orthoses are assessed by functional F tests and compared with pointwise F tests and the traditional univariate repeated-measurements ANOVA. The second study is a designed experiment in which a split-plot design was used to collect gait data from healthy subjects. This is commonly done in gait research in order to better understand, for example, the effects of orthoses while avoiding confounded analysis from the high variability observed in abnormal gait. Moreover, from a technical point of view the study may be regarded as a real-world alternative to simulation studies. By using healthy individuals it is possible to collect data which are in better agreement with the underlying model assumptions. The penultimate chapter of the thesis presents a qualitative study with clinical experts to investigate the utility of gait analysis for the management of cerebral palsy. We explore potential pathways by which the statistical analyses in the thesis might influence patient outcomes. The thesis has six chapters. After describing motivation and introduction in Chapter 1, mathematical representations of functional data are presented in Chapter 2. Chapter 3 considers orthogonal designs in the context of functional data analysis. New functional F tests for complex designs are derived in Chapter 4 and applied in two gait studies. Chapter 5 is devoted to a qualitative study. The thesis concludes with a discussion which details the extent to which the research question has been addressed, the limitations of the work and the degree to which it has been answered.

The present study is divided into two parts: the first is on developing the statistical analysis and modeling of mortality (or incidence) trends using Bayesian joinpoint regression and the second is on fitting differential equations from time series data to derive the rate of change of carbon dioxide in the atmosphere. Joinpoint regression model identifies significant changes in the trends of the incidence, mortality, and survival of a specific disease in a given population. Bayesian approach of joinpoint regression is widely used in modeling statistical data to identify the points in the trend where the significant changes occur. The purpose of the present study is to develop an age-stratified Bayesian joinpoint regression model to describe mortality trends assuming that the observed counts are probabilistically characterized by the Poisson distribution. The proposed model is based on Bayesian model selection criteria with the smallest number of joinpoints that are sufficient to explain the Annual Percentage Change (APC). The prior probability distributions are chosen in such a way that they are automatically derived from the model index contained in the model space. The proposed model and methodology estimates the age-adjusted mortality rates in different epidemiological studies to compare the trends by accounting the confounding effects of age. The future mortality rates are predicted using the Bayesian Model Averaging (BMA) approach. As an application of the Bayesian joinpoint regression, first we study the childhood brain cancer mortality rates (non age-adjusted rates) and their Annual Percentage Change (APC) per year using the existing Bayesian joinpoint regression models in the literature. We use annual observed mortality counts of children ages 0-19 from 1969-2009 obtained from Surveillance Epidemiology and End Results (SEER) database of the National Cancer Institute (NCI). The predictive distributions are used to predict the future mortality rates. We also compare this result with the mortality trend obtained using joinpoint software of NCI, and to fit the age-stratified model, we use the cancer mortality counts of adult lung and bronchus cancer (25-85+ years), and brain and other Central Nervous System (CNS) cancer (25-85+ years) patients obtained from the Surveillance Epidemiology and End Results (SEER) data base of the National Cancer Institute (NCI). The second part of this study is the statistical analysis and modeling of noisy data using functional data analysis approach. Carbon dioxide is one of the major contributors to Global Warming. In this study, we develop a system of differential equations using time series data of the major sources of the significant contributable variables of carbon dioxide in the atmosphere. We define the differential operator as data smoother and use the penalized least square fitting criteria to smooth the data. Finally, we optimize the profile error sum of squares to estimate the necessary differential operator. The proposed models will give us an estimate of the rate of change of carbon dioxide in the atmosphere at a particular time. We apply the model to fit emission of carbon dioxide data in the continental United States. The data set is obtained from the Carbon Dioxide Information Analysis Center (CDIAC), the primary climate-change data and information analysis center of the United States Department of Energy. The first four chapters of this dissertation contribute to the development and application of joinpiont and the last chapter discusses the statistical modeling and application of differential equations through data using functional data analysis approach.

High levels of automation in manufacturing industries are leading to data sets of increasing size and dimension. The challenge facing statisticians and field professionals is to develop methodology to help meet this demand. Functional data is one example of high-dimensional data characterized by observations recorded as a function of some continuous measure, such as time. An application considered in this thesis comes from the automotive industry. It involves a production process in which valve seats are force-fitted by a ram into cylinder heads of automobile engines. For each insertion, the force exerted by the ram is automatically recorded every fraction of a second for about two and a half seconds, generating a force profile. We can think of these profiles as individual functions of time summarized into collections of curves. The focus of this thesis is the analysis of functional process data such as the valve seat insertion example. A number of techniques are set forth. In the first part, two ways to model a single curve are considered: a b-spline fit via linear regression, and a nonlinear model based on differential equations. Each of these approaches is incorporated into a mixed effects model for multiple curves, and multivariate process monitoring techniques are applied to the predicted random effects in order to identify anomalous curves. In the second part, a Bayesian hierarchical model is used to cluster low-dimensional summaries of the curves into meaningful groups. The belief is that the clusters correspond to distinct types of processes (e.g. various types of “good” or “faulty” assembly). New observations can be assigned to one of these by calculating the probabilities of belonging to each cluster. Mahalanobis distances are used to identify new observations not belonging to any of the existing clusters. Synthetic and real data are used to validate the results.

High levels of automation in manufacturing industries are leading to data sets of increasing size and dimension. The challenge facing statisticians and field professionals is to develop methodology to help meet this demand. Functional data is one example of high-dimensional data characterized by observations recorded as a function of some continuous measure, such as time. An application considered in this thesis comes from the automotive industry. It involves a production process in which valve seats are force-fitted by a ram into cylinder heads of automobile engines. For each insertion, the force exerted by the ram is automatically recorded every fraction of a second for about two and a half seconds, generating a force profile. We can think of these profiles as individual functions of time summarized into collections of curves. The focus of this thesis is the analysis of functional process data such as the valve seat insertion example. A number of techniques are set forth. In the first part, two ways to model a single curve are considered: a b-spline fit via linear regression, and a nonlinear model based on differential equations. Each of these approaches is incorporated into a mixed effects model for multiple curves, and multivariate process monitoring techniques are applied to the predicted random effects in order to identify anomalous curves. In the second part, a Bayesian hierarchical model is used to cluster low-dimensional summaries of the curves into meaningful groups. The belief is that the clusters correspond to distinct types of processes (e.g. various types of “good” or “faulty” assembly). New observations can be assigned to one of these by calculating the probabilities of belonging to each cluster. Mahalanobis distances are used to identify new observations not belonging to any of the existing clusters. Synthetic and real data are used to validate the results.

The work presented in this dissertation centers on the theme of regression and computation methodology. Functional data is an important class of longitudinal data, and principal component analysis is an important approach to regression with this type of data. Here we present an additive hierarchical bivariate functional data model employing principal components to identify random e ects. This additive model extends the univariate functional principal component model. These models are implemented in the pfda package for R. To t the curves from this class of models orthogonalized spline basis are used to reduce the dimensionality of the t, but retain exibility. Methods for handing spline basis functions in a purely analytical manner, including the orthogonalizing process and computing of penalty matrices used to t the principal component models are presented. The methods are implemented in the R package orthogonalsplinebasis. The projects discussed involve complicated coding for the implementations in R. To facilitate this I created the NppToR utility to add R functionality to the popular windows code editor Notepad . A brief overview of the use of the utility is also included.

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.

Modern data collection methods are now frequently returning observations that should be viewed as the result of digitized recording or sampling from stochastic processes rather than vectors of finite length. In spite of great demands, only a few classification methodologies for such data have been suggested and supporting theory is quite limited. The focus of this dissertation is on discrimination and classification in this infinite dimensional setting. The methodology and theory we develop are based on the abstract canonical correlation concept of Eubank and Hsing (2005), and motivated by the fact that Fisher's discriminant analysis method is intimately tied to canonical correlation analysis. Specifically, we have developed a theoretical framework for discrimination and classification of sample paths from stochastic processes through use of the Loeve-Parzen isomorphism that connects a second order process to the reproducing kernel Hilbert space generated by its covariance kernel. This approach provides a seamless transition between the finite and infinite dimensional settings and lends itself well to computation via smoothing and regularization. In addition, we have developed a new computational procedure and illustrated it with simulated data and Canadian weather data.