Measuring episodic memory: A novel approach with an indefinite number of alternative forms

Both clinical practice and clinical research settings can require successive administrations of a memory test, particularly when following the trajectory of suspected memory decline in older adults. However, relatively few verbal episodic memory tests have alternative forms. We set out to create a broad based memory test to allow for the use of an essentially unlimited number of alternative forms. Four tasks for inclusion in such a test were developed. These tasks varied the requirement for recall as opposed to recognition, the need to form an association between unrelated words, and the need to discriminate the most recent list from earlier lists, all of which proved useful. A total of 115 participants completed the battery of tests and were used to show that the test could differentiate between older and younger adults; a sub-sample of 73 participants completed alternative forms of the tests to determine test-retest reliability and the amount of learning to learn.

Uncertainty budget in the measurement of typical airborne number, surface area and mass particle distributions

The effects of particulate matter on environment and public health have been widely studied in recent years. A number of studies in the medical field have tried to identify the specific effect on human health of particulate exposure, but agreement amongst these studies on the relative importance of the particles’ size and its origin with respect to health effects is still lacking. Nevertheless, air quality standards are moving, as the epidemiological attention, towards greater focus on the smaller particles. Current air quality standards only regulate the mass of particulate matter less than 10 μm in aerodynamic diameter (PM10) and less than 2.5 μm (PM2.5). The most reliable method used in measuring Total Suspended Particles (TSP), PM10, PM2.5 and PM1 is the gravimetric method since it directly measures PM concentration, guaranteeing an effective traceability to international standards. This technique however, neglects the possibility to correlate short term intra-day variations of atmospheric parameters that can influence ambient particle concentration and size distribution (emission strengths of particle sources, temperature, relative humidity, wind direction and speed and mixing height) as well as human activity patterns that may also vary over time periods considerably shorter than 24 hours. A continuous method to measure the number size distribution and total number concentration in the range 0.014 – 20 μm is the tandem system constituted by a Scanning Mobility Particle Sizer (SMPS) and an Aerodynamic Particle Sizer (APS). In this paper, an uncertainty budget model of the measurement of airborne particle number, surface area and mass size distributions is proposed and applied for several typical aerosol size distributions. The estimation of such an uncertainty budget presents several difficulties due to i) the complexity of the measurement chain, ii) the fact that SMPS and APS can properly guarantee the traceability to the International System of Measurements only in terms of number concentration. In fact, the surface area and mass concentration must be estimated on the basis of separately determined average density and particle morphology. Keywords: SMPS-APS tandem system, gravimetric reference method, uncertainty budget, ultrafine particles.

A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation

This study considers the solution of a class of linear systems related with the fractional Poisson equation (FPE) (−∇2)α/2φ=g(x,y) with nonhomogeneous boundary conditions on a bounded domain. A numerical approximation to FPE is derived using a matrix representation of the Laplacian to generate a linear system of equations with its matrix A raised to the fractional power α/2. The solution of the linear system then requires the action of the matrix function f(A)=A−α/2 on a vector b. For large, sparse, and symmetric positive definite matrices, the Lanczos approximation generates f(A)b≈β0Vmf(Tm)e1. This method works well when both the analytic grade of A with respect to b and the residual for the linear system are sufficiently small. Memory constraints often require restarting the Lanczos decomposition; however this is not straightforward in the context of matrix function approximation. In this paper, we use the idea of thick-restart and adaptive preconditioning for solving linear systems to improve convergence of the Lanczos approximation. We give an error bound for the new method and illustrate its role in solving FPE. Numerical results are provided to gauge the performance of the proposed method relative to exact analytic solutions.

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.

Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term

In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moreover, we also present a fractional method of lines, a matrix transfer technique, and an extrapolation method for the equation. Some numerical examples are given, and the results demonstrate the effectiveness of theoretical analysis.

Numerical simulation for the 3D seepage flow with fractional derivatives in porous media

In this paper, the numerical simulation of the 3D seepage flow with fractional derivatives in porous media is considered under two special cases: non-continued seepage flow in uniform media (NCSFUM) and continued seepage flow in non-uniform media (CSF-NUM). A fractional alternating direction implicit scheme (FADIS) for the NCSF-UM and a modified Douglas scheme (MDS) for the CSF-NUM are proposed. The stability, consistency and convergence of both FADIS and MDS in a bounded domain are discussed. A method for improving the speed of convergence by Richardson extrapolation for the MDS is also presented. Finally, numerical results are presented to support our theoretical analysis.

Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation

In this paper, we consider the variable-order nonlinear fractional diffusion equation View the MathML source where xRα(x,t) is a generalized Riesz fractional derivative of variable order View the MathML source and the nonlinear reaction term f(u,x,t) satisfies the Lipschitz condition |f(u1,x,t)-f(u2,x,t)|less-than-or-equals, slantL|u1-u2|. A new explicit finite-difference approximation is introduced. The convergence and stability of this approximation are proved. Finally, some numerical examples are provided to show that this method is computationally efficient. The proposed method and techniques are applicable to other variable-order nonlinear fractional differential equations.

Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process

In this paper, we consider the following non-linear fractional reaction–subdiffusion process (NFR-SubDP): Formula where f(u, x, t) is a linear function of u, the function g(u, x, t) satisfies the Lipschitz condition and 0Dt1–{gamma} is the Riemann–Liouville time fractional partial derivative of order 1 – {gamma}. We propose a new computationally efficient numerical technique to simulate the process. Firstly, the NFR-SubDP is decoupled, which is equivalent to solving a non-linear fractional reaction–subdiffusion equation (NFR-SubDE). Secondly, we propose an implicit numerical method to approximate the NFR-SubDE. Thirdly, the stability and convergence of the method are discussed using a new energy method. Finally, some numerical examples are presented to show the application of the present technique. This method and supporting theoretical results can also be applied to fractional integrodifferential equations.

Modified alternating direction methods for solving a two-dimensional non-continuous seepage flow with fractional derivatives

In this paper, a two-dimensional non-continuous seepage flow with fractional derivatives (2D-NCSF-FD) in uniform media is considered, which has modified the well known Darcy law. Using the relationship between Riemann-Liouville and Grunwald-Letnikov fractional derivatives, two modified alternating direction methods: a modified alternating direction implicit Euler method and a modified Peaceman-Rachford method, are proposed for solving the 2D-NCSF-FD in uniform media. The stability and consistency, thus convergence of the two methods in a bounded domain are discussed. Finally, numerical results are given.

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order αset membership, variant(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order βset membership, variant(0,1) and of order αset membership, variant(1,2], respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.

Stochastic modelling of financial time series with memory and multifractal scaling

Financial processes may possess long memory and their probability densities may display heavy tails. Many models have been developed to deal with this tail behaviour, which reflects the jumps in the sample paths. On the other hand, the presence of long memory, which contradicts the efficient market hypothesis, is still an issue for further debates. These difficulties present challenges with the problems of memory detection and modelling the co-presence of long memory and heavy tails. This PhD project aims to respond to these challenges. The first part aims to detect memory in a large number of financial time series on stock prices and exchange rates using their scaling properties. Since financial time series often exhibit stochastic trends, a common form of nonstationarity, strong trends in the data can lead to false detection of memory. We will take advantage of a technique known as multifractal detrended fluctuation analysis (MF-DFA) that can systematically eliminate trends of different orders. This method is based on the identification of scaling of the q-th-order moments and is a generalisation of the standard detrended fluctuation analysis (DFA) which uses only the second moment; that is, q = 2. We also consider the rescaled range R/S analysis and the periodogram method to detect memory in financial time series and compare their results with the MF-DFA. An interesting finding is that short memory is detected for stock prices of the American Stock Exchange (AMEX) and long memory is found present in the time series of two exchange rates, namely the French franc and the Deutsche mark. Electricity price series of the five states of Australia are also found to possess long memory. For these electricity price series, heavy tails are also pronounced in their probability densities. The second part of the thesis develops models to represent short-memory and longmemory financial processes as detected in Part I. These models take the form of continuous-time AR(∞) -type equations whose kernel is the Laplace transform of a finite Borel measure. By imposing appropriate conditions on this measure, short memory or long memory in the dynamics of the solution will result. A specific form of the models, which has a good MA(∞) -type representation, is presented for the short memory case. Parameter estimation of this type of models is performed via least squares, and the models are applied to the stock prices in the AMEX, which have been established in Part I to possess short memory. By selecting the kernel in the continuous-time AR(∞) -type equations to have the form of Riemann-Liouville fractional derivative, we obtain a fractional stochastic differential equation driven by Brownian motion. This type of equations is used to represent financial processes with long memory, whose dynamics is described by the fractional derivative in the equation. These models are estimated via quasi-likelihood, namely via a continuoustime version of the Gauss-Whittle method. The models are applied to the exchange rates and the electricity prices of Part I with the aim of confirming their possible long-range dependence established by MF-DFA. The third part of the thesis provides an application of the results established in Parts I and II to characterise and classify financial markets. We will pay attention to the New York Stock Exchange (NYSE), the American Stock Exchange (AMEX), the NASDAQ Stock Exchange (NASDAQ) and the Toronto Stock Exchange (TSX). The parameters from MF-DFA and those of the short-memory AR(∞) -type models will be employed in this classification. We propose the Fisher discriminant algorithm to find a classifier in the two and three-dimensional spaces of data sets and then provide cross-validation to verify discriminant accuracies. This classification is useful for understanding and predicting the behaviour of different processes within the same market. The fourth part of the thesis investigates the heavy-tailed behaviour of financial processes which may also possess long memory. We consider fractional stochastic differential equations driven by stable noise to model financial processes such as electricity prices. The long memory of electricity prices is represented by a fractional derivative, while the stable noise input models their non-Gaussianity via the tails of their probability density. A method using the empirical densities and MF-DFA will be provided to estimate all the parameters of the model and simulate sample paths of the equation. The method is then applied to analyse daily spot prices for five states of Australia. Comparison with the results obtained from the R/S analysis, periodogram method and MF-DFA are provided. The results from fractional SDEs agree with those from MF-DFA, which are based on multifractal scaling, while those from the periodograms, which are based on the second order, seem to underestimate the long memory dynamics of the process. This highlights the need and usefulness of fractal methods in modelling non-Gaussian financial processes with long memory.

Development of a particle number and particle mass emissions inventory for an urban fleet : a study in South-East Queensland

Motor vehicles are a major source of gaseous and particulate matter pollution in urban areas, particularly of ultrafine sized particles (diameters < 0.1 µm). Exposure to particulate matter has been found to be associated with serious health effects, including respiratory and cardiovascular disease, and mortality. Particle emissions generated by motor vehicles span a very broad size range (from around 0.003-10 µm) and are measured as different subsets of particle mass concentrations or particle number count. However, there exist scientific challenges in analysing and interpreting the large data sets on motor vehicle emission factors, and no understanding is available of the application of different particle metrics as a basis for air quality regulation. To date a comprehensive inventory covering the broad size range of particles emitted by motor vehicles, and which includes particle number, does not exist anywhere in the world. This thesis covers research related to four important and interrelated aspects pertaining to particulate matter generated by motor vehicle fleets. These include the derivation of suitable particle emission factors for use in transport modelling and health impact assessments; quantification of motor vehicle particle emission inventories; investigation of the particle characteristic modality within particle size distributions as a potential for developing air quality regulation; and review and synthesis of current knowledge on ultrafine particles as it relates to motor vehicles; and the application of these aspects to the quantification, control and management of motor vehicle particle emissions. In order to quantify emissions in terms of a comprehensive inventory, which covers the full size range of particles emitted by motor vehicle fleets, it was necessary to derive a suitable set of particle emission factors for different vehicle and road type combinations for particle number, particle volume, PM1, PM2.5 and PM1 (mass concentration of particles with aerodynamic diameters < 1 µm, < 2.5 µm and < 10 µm respectively). The very large data set of emission factors analysed in this study were sourced from measurement studies conducted in developed countries, and hence the derived set of emission factors are suitable for preparing inventories in other urban regions of the developed world. These emission factors are particularly useful for regions with a lack of measurement data to derive emission factors, or where experimental data are available but are of insufficient scope. The comprehensive particle emissions inventory presented in this thesis is the first published inventory of tailpipe particle emissions prepared for a motor vehicle fleet, and included the quantification of particle emissions covering the full size range of particles emitted by vehicles, based on measurement data. The inventory quantified particle emissions measured in terms of particle number and different particle mass size fractions. It was developed for the urban South-East Queensland fleet in Australia, and included testing the particle emission implications of future scenarios for different passenger and freight travel demand. The thesis also presents evidence of the usefulness of examining modality within particle size distributions as a basis for developing air quality regulations; and finds evidence to support the relevance of introducing a new PM1 mass ambient air quality standard for the majority of environments worldwide. The study found that a combination of PM1 and PM10 standards are likely to be a more discerning and suitable set of ambient air quality standards for controlling particles emitted from combustion and mechanically-generated sources, such as motor vehicles, than the current mass standards of PM2.5 and PM10. The study also reviewed and synthesized existing knowledge on ultrafine particles, with a specific focus on those originating from motor vehicles. It found that motor vehicles are significant contributors to both air pollution and ultrafine particles in urban areas, and that a standardized measurement procedure is not currently available for ultrafine particles. The review found discrepancies exist between outcomes of instrumentation used to measure ultrafine particles; that few data is available on ultrafine particle chemistry and composition, long term monitoring; characterization of their spatial and temporal distribution in urban areas; and that no inventories for particle number are available for motor vehicle fleets. This knowledge is critical for epidemiological studies and exposure-response assessment. Conclusions from this review included the recommendation that ultrafine particles in populated urban areas be considered a likely target for future air quality regulation based on particle number, due to their potential impacts on the environment. The research in this PhD thesis successfully integrated the elements needed to quantify and manage motor vehicle fleet emissions, and its novelty relates to the combining of expertise from two distinctly separate disciplines - from aerosol science and transport modelling. The new knowledge and concepts developed in this PhD research provide never before available data and methods which can be used to develop comprehensive, size-resolved inventories of motor vehicle particle emissions, and air quality regulations to control particle emissions to protect the health and well-being of current and future generations.