On quaternionic functional analysis

In this article, we will show that the category of quaternion vector spaces, the category of (both one-sided and two sided) quaternion Hilbert spaces and the category of quaternion $B^*$-algebras are equivalent to the category of real vector spaces, the category of real Hilbert spaces and the category of real $C^*$-algebras respectively. We will also give a Riesz representation theorem for quaternion Hilbert spaces and will extend two results of Kulkarni (namely, we will give the full versions of the Gelfand-Naimark theorem and the Gelfand theorem for quaternion $B^*$-algebras). On our way to these results, we compare, clarify and unify the term "quaternion Hilbert spaces" in the literatures.Comment: to appear in the Mathematical Proceedings of the Cambridge Philosophical Societ

Warped Functional Analysis of Variance

This article presents an Analysis of Variance model for functional data that explicitly incorporates phase variability through a time-warping component, allowing for a unified approach to estimation and inference in presence of amplitude and time variability. The focus is on single-random-factor models but the approach can be easily generalized to more complex ANOVA models. The behavior of the estimators is studied by simulation, and an application to the analysis of growth curves of flour beetles is presented. Although the model assumes a smooth latent process behind the observed trajectories, smoothness of the observed data is not required; the method can be applied to the sparsely observed data that is often encountered in longitudinal studies

Static Analysis of Functional Programs

In this paper, the static analysis of programs in the functional programming language Miranda* is described based on two graph models. A new control-flow graph model of Miranda definitions is presented, and a model with four classes of callgraphs. Standard software metrics are applicable to these models. A Miranda front end for Prometrix, ¿, a tool for the automated analysis of flowgraphs and callgraphs, has been developed. This front end produces the flowgraph and callgraph representations of Miranda programs. Some features of the metric analyser are illustrated with an example program. The tool provides a promising access to standard metrics on functional programs

Longitudinal Functional Data Analysis

We consider analysis of dependent functional data that are correlated because of a longitudinal-based design: each subject is observed at repeated time visits and for each visit we record a functional variable. We propose a novel parsimonious modeling framework for the repeatedly observed functional variables that allows to extract low dimensional features. The proposed methodology accounts for the longitudinal design, is designed for the study of the dynamic behavior of the underlying process, and is computationally fast. Theoretical properties of this framework are studied and numerical investigation confirms excellent behavior in finite samples. The proposed method is motivated by and applied to a diffusion tensor imaging study of multiple sclerosis. Using Shiny (Chang et al., 2015) we implement interactive plots to help visualize longitudinal functional data as well as the various components and prediction obtained using the proposed method.Comment: 32 pages, 4 figure

Local functional principal component analysis

Covariance operators of random functions are crucial tools to study the way random elements concentrate over their support. The principal component analysis of a random function X is well-known from a theoretical viewpoint and extensively used in practical situations. In this work we focus on local covariance operators. They provide some pieces of information about the distribution of X around a fixed point of the space x₀. A description of the asymptotic behaviour of the theoretical and empirical counterparts is carried out. Asymptotic developments are given under assumptions on the location of x₀ and on the distributions of projections of the data on the eigenspaces of the (non-local) covariance operator