The basic philosophy of Functional Data Analysis (FDA) is to think of the observed data functions as elements of a possibly infinite-dimensional function space. Most of the current research topics on FDA focus on advancing theoretical tools and extending existing multivariate techniques to accommodate the infinite-dimensional nature of data. This dissertation reports contributions on both fronts, where a unifying inverse regression theory for both the multivariate setting (Li 1991) and functional data from a Reproducing Kernel Hilbert Space (RKHS) prospective is developed. We proposed a functional multiple-index model which models a real response variable as a function of a few predictor variables called indices. These indices are random elements of the Hilbert space spanned by a second order stochastic process and they constitute the so-called Effective Dimensional Reduction Space (EDRS). To conduct inference on the EDRS, we discovered a fundamental result which reveals the geometrical association between the EDRS and the RKHS of the process. Two inverse regression procedures, a “slicing” approach and a kernel approach, were introduced to estimate the counterpart of the EDRS in the RKHS. Further the estimate of the EDRS was achieved via the transformation from the RKHS to the original Hilbert space. To construct an asymptotic theory, we introduced an isometric mapping from the empirical RKHS to the theoretical RKHS, which can be used to measure the distance between the estimator and the target. Some general computational issues of FDA were discussed, which led to the smoothed versions of the functional inverse regression methods. Simulation studies were performed to evaluate the performance of the inference procedures and applications to biological and chemometrical data analysis were illustrated.

This thesis proposes new analysis tools for simulation models in the presence of data. To achieve a representation close to reality, simulation models are typically endowed with a set of inputs, termed parameters, that represent several controllable, stochastic or unknown components of the system. Because these models often utilize computationally expensive procedures, even modern supercomputers require a nontrivial amount of time, money, and energy to run for complex systems. Existing statistical frameworks avoid repeated evaluations of deterministic models through an emulator, constructed by conducting an experiment on the code. In high dimensional scenarios, the traditional framework for emulator-based analysis can fail due to the computational burden of inference. This thesis proposes a new class of experiments where inference from half a million observations is possible in seconds versus the days required for the traditional technique. In a case study presented in this thesis, the parameter of interest is a function as opposed to a scalar or a set of scalars, meaning the problem exists in the high dimensional regime. This work develops a new modeling strategy to nonparametrically study the functional parameter using Bayesian inference. Stochastic simulations are also investigated in the thesis. I describe the development of emulators through a framework termed quantile kriging, which allows for non-parametric representations of the stochastic behavior of the output whereas previous work has focused on normally distributed outputs. Furthermore, this work studied asymptotic properties of this methodology that yielded practical insights. Under certain regulatory conditions, there is the following result: By using an experiment that has the appropriate ratio of replications to sets of different inputs, we can achieve an optimal rate of convergence. Additionally, this method provided the basic tool for the study of defect patterns and a case study is explored.

We show that recent results in [3] on risk bounds for regularized least-squares on reproducing kernel Hilbert spaces can be straightforwardly extended to the vector-valued regression setting. We first briefly introduce central concepts on operator-valued kernels. Then we show how risk bounds can be expressed in terms of a generalization of effective dimension.

Both complex dynamics and the theory of reproducing kernel Hilbert spaces have found widespread application over the last few decades. Although complex dynamics started over a century ago, the gravity of it's importance was only recently realized due to B.B. Mandelbrot's work in the 1980's. B.B. Mandelbrot demonstrated to the world that fractals, which are chaotic patterns containing a high degree of self-similarity, often times serve as better models to nature than conventional smooth models. The theory of reproducing kernel Hilbert spaces also having started over a century ago, didn't pick up until N. Aronszajn's classic was written in 1950. Since then, the theory has found widespread application to fields including machine learning, quantum mechanics, and harmonic analysis. In the paper, Infinite Product Representations of Kernel Functions and Iterated Function Systems, the authors, D. Alpay, P. Jorgensen, I. Lewkowicz, and I. Martiziano, show how a kernel function can be constructed on an attracting set of an iterated function system. Furthermore, they show that when certain conditions are met, one can construct an orthonormal basis of the associated Hilbert space via certain pull-back and multiplier operators. In this thesis we take for our iterated function system, the family of iterates of a given rational map. Thus we investigate for which rational maps their kernel construction holds as well as their orthornormal basis construction. We are able to show that the kernel construction applies to any rational map conjugate to a polynomial with an attracting fixed point at 0. Within such rational maps, we are able to find a family of polynomials for which the orthonormal basis construction holds. It is then natural to ask how the orthonormal basis changes as the polynomial within a given family varies. We are able to determine for certain families of polynomials, that the dynamics of the corresponding orthonormal basis is well behaved. Finally, we conclude with some possible avenues of future investigation.

A reproducing kernel Hilbert space (RKHS) is a Hilbert space $\mathscr{H}$ of functions with the property that the values $f(x)$ for $f \in \mathscr{H}$ are reproduced from the inner product in $\mathscr{H}$. Recent applications are found in stochastic processes (Ito Calculus), harmonic analysis, complex analysis, learning theory, and machine learning algorithms. This research began with the study of RKHSs to areas such as learning theory, sampling theory, and harmonic analysis. From the Moore-Aronszajn theorem, we have an explicit correspondence between reproducing kernel Hilbert spaces (RKHS) and reproducing kernel functions—also called positive definite kernels or positive definite functions. The focus here is on the duality between positive definite functions and their boundary spaces; these boundary spaces often lead to the study of Gaussian processes or Brownian motion. It is known that every reproducing kernel Hilbert space has an associated generalized boundary probability space. The Arveson (reproducing) kernel is $K(z,w) = \frac{1}{1-_{\C^d}}, z,w \in \B_d$, and Arveson showed, \cite{Arveson}, that the Arveson kernel does not follow the boundary analysis we were finding in other RKHS. Thus, we were led to define a new reproducing kernel on the unit ball in complex $n$-space, and naturally this lead to the study of a new reproducing kernel Hilbert space. This reproducing kernel Hilbert space stems from boundary analysis of the Arveson kernel. The construction of the new RKHS resolves the problem we faced while researching “natural” boundary spaces (for the Drury-Arveson RKHS) that yield boundary factorizations: \[K(z,w) = \int_{\mathcal{B}} K^{\mathcal{B}}_z(b)\overline{K^{\mathcal{B}}_w(b)}d\mu(b), \;\;\; z,w \in \B_d \text{ and } b \in \mathcal{B} \tag*{\it{(Factorization of} $K$).}\] Results from classical harmonic analysis on the disk (the Hardy space) are generalized and extended to the new RKHS. Particularly, our main theorem proves that, relaxing the criteria to the contractive property, we can do the generalization that Arveson's paper showed (criteria being an isometry) is not possible.

Um espaço de Hilbert de reprodução (EHR) é um espaço de Hilbert de funções construído de maneira específica e única a partir de um núcleo positivo definido. As funções do EHR tem a seguinte peculiaridade: seus valores podem ser reproduzidos através de uma operação elementar envolvendo a própria função, o núcleo gerador e o produto interno do espaço. Neste trabalho, consideramos EHR gerados por núcleos positivos definidos sobre a esfera unitária m-dimensional usual. Analisamos quais propriedades são herdadas pelos elementos do espaço, quando o núcleo gerador possui alguma hipótese de diferenciabilidade. A análise é elaborada em duas frentes: com a noção de diferenciabilidade usual sobre a esfera e com uma noção de diferenciabilidade definida por uma operação multiplicativa genérica. Esta última inclui como caso particular as derivadas fracionárias e a derivada forte de Laplace-Beltrami. Em cada um dos casos consideramos ainda propriedades específicas do mergulho do EHR em espaços de funções suaves definidos pela diferenciabilidade utilizada / A reproducing kernel Hilbert space (EHR) is a Hilbert space of functions constructed in a unique manner from a fixed positive definite generating kernel. The values of a function in a reproducing kernel Hilbert space can be reproduced through an elementary operation involving the function itself, the generating kernel and the inner product of the space. In this work, we consider reproducing kernel Hilbert spaces generated by a positive definite kernel on the usual m-dimensional sphere. The main goal is to analyze differentiability properties inherited by the functions in the space when the generating kernel carries a differentiability assumption. That is done in two different cases: using the usual notion of differentiability on the sphere and using another one defined through multiplicative operators. The second case includes the Laplace-Beltrami derivative and fractional derivatives as well. In both cases we consider specific properties of the embeddings of the reproducing kernel Hilbert space into spaces of smooth functions induced by notion of differentiability used

Um espaço de Hilbert de reprodução (EHR) é um espaço de Hilbert de funções construído de maneira específica e única a partir de um núcleo positivo definido. As funções do EHR tem a seguinte peculiaridade: seus valores podem ser reproduzidos através de uma operação elementar envolvendo a própria função, o núcleo gerador e o produto interno do espaço. Neste trabalho, consideramos EHR gerados por núcleos positivos definidos sobre a esfera unitária m-dimensional usual. Analisamos quais propriedades são herdadas pelos elementos do espaço, quando o núcleo gerador possui alguma hipótese de diferenciabilidade. A análise é elaborada em duas frentes: com a noção de diferenciabilidade usual sobre a esfera e com uma noção de diferenciabilidade definida por uma operação multiplicativa genérica. Esta última inclui como caso particular as derivadas fracionárias e a derivada forte de Laplace-Beltrami. Em cada um dos casos consideramos ainda propriedades específicas do mergulho do EHR em espaços de funções suaves definidos pela diferenciabilidade utilizada / A reproducing kernel Hilbert space (EHR) is a Hilbert space of functions constructed in a unique manner from a fixed positive definite generating kernel. The values of a function in a reproducing kernel Hilbert space can be reproduced through an elementary operation involving the function itself, the generating kernel and the inner product of the space. In this work, we consider reproducing kernel Hilbert spaces generated by a positive definite kernel on the usual m-dimensional sphere. The main goal is to analyze differentiability properties inherited by the functions in the space when the generating kernel carries a differentiability assumption. That is done in two different cases: using the usual notion of differentiability on the sphere and using another one defined through multiplicative operators. The second case includes the Laplace-Beltrami derivative and fractional derivatives as well. In both cases we consider specific properties of the embeddings of the reproducing kernel Hilbert space into spaces of smooth functions induced by notion of differentiability used

A frame is a possibly linearly dependent set of vectors in a Hilbert space that facilitates the decomposition and reconstruction of vectors. A Parseval frame is a frame that acts as its own dual frame. A Gabor frame comprises all translations and phase modulations of an appropriate window function. We show that the space of all frames on a Hilbert space indexed by a common measure space can be fibrated into orbits under the action of invertible linear deformations and that any maximal set of unitarily inequivalent Parseval frames is a complete set of representatives of the orbits. We show that all such frames are connected by transformations that are linear in the larger Hilbert space of square-integrable functions on the indexing space. We apply our results to frames on finite-dimensional Hilbert spaces and to the discretization of the Gabor frame with a band-limited window function.

Let $\Omega\subset\mathbb{R}^n$ be an open and connected subset of $\mathbb{R}^n$. We say that a function $F\colon \Omega-\Omega\to\mathbb{C}$, where $\Omega-\Omega=\{x-y\colon x,y\in\Omega\}$, is positive definite if for any $x_1,\ldots,x_m\in\Omega$ and any $c_1,\ldots,c_m\in \mathbb{C}$ we have that $\sum_{j,k=1}^m F(x_j-x_k)c_j\overline{c_k}\geq 0$. Let $F\colon\Omega-\Omega\to\mathbb{C}$ be a continuous positive definite function. We give necessary and sufficient conditions for $F$ to have an extension to a continuous and positive definite function defined on the entire Euclidean space $\mathbb{R}^n$. The conditions are formulated in terms of strong commutativity of some certain selfadjoint operators defined on a Hilbert space associated to our positive definite function.

This thesis compares and contrasts currents- and varifolds-based diffeomorphic image registration approaches for registering tree-like structures in the lung and surface of the lung. In these approaches, curve-like structures in the lung—for example, the skeletons of vessels and airways segmentation—and surface of the lung are represented by currents or varifolds in the dual space of a Reproducing Kernel Hilbert Space (RKHS). Currents and varifolds representations are discretized and are parameterized via of a collection of momenta. A momenta corresponds to a line segment via the coordinates of the center of the line segment and the tangent direction of the line segment at the center. A momentum corresponds to a mesh via the coordinates of the center of the mesh and the normal direction of the mesh at the center. The magnitude of the tangent vector for the line segment and the normal vector for the mesh are the length of the line segment and the area of the mesh respectively. A varifolds-based registration approach is similar to currents except that two varifolds representations are aligned independent of the tangent (normal) vector orientation. An advantage of varifolds over currents is that the orientation of the tangent vectors can be difficult to determine especially when the vessel and airway trees are not connected. In this thesis, we examine the image registration sensitivity and accuracy of currents- and varifolds-based registration as a function of the number and location of momenta used to represent tree like-structures in the lung and the surface of the lung. The registrations presented in this thesis were generated using the Deformetrica software package, which is publicly available at www.deformetrica.org.