Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion

Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A^(-α/2)b, where A ∈ ℝ^(n×n) is a large, sparse symmetric positive definite matrix and b ∈ ℝ^n is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LL^T is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L^(-T)z, with x = A^(-1/2)z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form ϕn = A^(-α/2)b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t^(-α/2) and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A^(-α/2)b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods for sampling conditioned on linear constraints and approximating the likelihood. Finally, we consider the problem of sampling from a generalised Matern random field, which combines our techniques for solving fractional-in-space partial differential equations with our method for sampling from GMRFs.

Face recognition using fractal codes

In this paper we propose a new method for face recognition using fractal codes. Fractal codes represent local contractive, affine transformations which when iteratively applied to range-domain pairs in an arbitrary initial image result in a fixed point close to a given image. The transformation parameters such as brightness offset, contrast factor, orientation and the address of the corresponding domain for each range are used directly as features in our method. Features of an unknown face image are compared with those pre-computed for images in a database. There is no need to iterate, use fractal neighbor distances or fractal dimensions for comparison in the proposed method. This method is robust to scale change, frame size change and rotations as well as to some noise, facial expressions and blur distortion in the image

Exploration and reflection in interactive art : glass pond

Glass Pond is an interactive artwork designed to engender exploration and reflection through an intuitive, tangible interface and a simulation agent. It is being developed using iterative methods. A study has been conducted with the aim of illuminating user experience, interface, design, and performance issues.The paper describes the study methodology and process of data analysis including coding schemes for cognitive states and movements. Analysis reveals that exploration and reflection occurred as well as composing behaviours (unexpected). Results also show that participants interacted to varying degrees. Design discussion includes the artwork's (novel) interface and configuration.

Mode matching pursuit for estimating dominant modes in bulk power grid

This study presents a general approach to identify dominant oscillation modes in bulk power system by using wide-area measurement system. To automatically identify the dominant modes without artificial participation, spectral characteristic of power system oscillation mode is applied to distinguish electromechanical oscillation modes which are calculated by stochastic subspace method, and a proposed mode matching pursuit is adopted to discriminate the dominant modes from the trivial modes, then stepwise-refinement scheme is developed to remove outliers of the dominant modes and the highly accurate dominant modes of identification are obtained. The method is implemented on the dominant modes of China Southern Power Grid which is one of the largest AC/DC paralleling grids in the world. Simulation data and field-measurement data are used to demonstrate high accuracy and better robustness of the dominant modes identification approach.

Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach.

Iterative development of MobileMums : a physical activity intervention for women with young children.

Background To describe the iterative development process and final version of ‘MobileMums’: a physical activity intervention for women with young children (<5 years) delivered primarily via mobile telephone (mHealth) short messaging service (SMS). Methods MobileMums development followed the five steps outlined in the mHealth development and evaluation framework: 1) conceptualization (critique of literature and theory); 2) formative research (focus groups, n= 48); 3) pre-testing (qualitative pilot of intervention components, n= 12); 4) pilot testing (pilot RCT, n= 88); and, 5) qualitative evaluation of the refined intervention (n= 6). Results Key findings identified throughout the development process that shaped the MobileMums program were the need for: behaviour change techniques to be grounded in Social Cognitive Theory; tailored SMS content; two-way SMS interaction; rapport between SMS sender and recipient; an automated software platform to generate and send SMS; and, flexibility in location of a face-to-face delivered component. Conclusions The final version of MobileMums is flexible and adaptive to individual participant’s physical activity goals, expectations and environment. MobileMums is being evaluated in a community-based randomised controlled efficacy trial (ACTRN12611000481976).

Human-centred methods for design of mobile social technologies : a case study of agile ridesharing

Social contexts are possible information sources that can foster connections between mobile application users, but they are also minefields of privacy concerns and have great potential for misinterpretation. This research establishes a framework for guiding the design of context-aware mobile social applications from a socio-technical perspective. Agile ridesharing was chosen as the test domain for the research because its success relies upon effectively connecting people through mobile technologies.

Using qualitative mixed methods to study small health care organizations while maximising trustworthiness and authenticity

Background The primary health care sector delivers the majority of health care in western countries through small, community-based organizations. However, research into these healthcare organizations is limited by the time constraints and pressure facing them, and the concern by staff that research is peripheral to their work. We developed Q-RARA—Qualitative Rapid Appraisal, Rigorous Analysis—to study small, primary health care organizations in a way that is efficient, acceptable to participants and methodologically rigorous. Methods Q-RARA comprises a site visit, semi-structured interviews, structured and unstructured observations, photographs, floor plans, and social scanning data. Data were collected over the course of one day per site and the qualitative analysis was integrated and iterative. Results We found Q-RARA to be acceptable to participants and effective in collecting data on organizational function in multiple sites without disrupting the practice, while maintaining a balance between speed and trustworthiness. Conclusions The Q-RARA approach is capable of providing a richly textured, rigorous understanding of the processes of the primary care practice while also allowing researchers to develop an organizational perspective. For these reasons the approach is recommended for use in small-scale organizations both within and outside the primary health care sector.

Iterative estimating equations: Linear convergence and asymptotic properties

We propose an iterative estimating equations procedure for analysis of longitudinal data. We show that, under very mild conditions, the probability that the procedure converges at an exponential rate tends to one as the sample size increases to infinity. Furthermore, we show that the limiting estimator is consistent and asymptotically efficient, as expected. The method applies to semiparametric regression models with unspecified covariances among the observations. In the special case of linear models, the procedure reduces to iterative reweighted least squares. Finite sample performance of the procedure is studied by simulations, and compared with other methods. A numerical example from a medical study is considered to illustrate the application of the method.