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.

Apparatus for monitoring load bearing rehabilitation exercises of a transfemoral amputee fitted with an osseointegrated fixation : a proof-of-concept study

The purpose of this proof-of-concept study was to determine the relevance of direct measurements to monitor the load applied on the osseointegrated fixation of transfemoral amputees during static load bearing exercises. The objectives were (A) to introduce an apparatus using a three-dimensional load transducer, (B) to present a range of derived information relevant to clinicians, (C) to report on the outcomes of a pilot study and (D) to compare the measurements from the transducer with those from the current method using a weighing scale. One transfemoral amputee fitted with an osseointegrated implant was asked to apply 10 kg, 20 kg, 40 kg and 80 kg on the fixation, using self-monitoring with the weighing scale. The loading was directly measured with a portable kinetic system including a six-channel transducer, external interface circuitry and a laptop. As the load prescribed increased from 10 kg to 80 kg, the forces and moments applied on and around the antero-posterior axis increased by 4 fold anteriorly and 14 fold medially, respectively. The forces and moments applied on and around the medio-lateral axis increased by 9 fold laterally and 16 fold from anterior to posterior, respectively. The long axis of the fixation was overloaded and underloaded in 17 % and 83 % of the trials, respectively, by up to ±10 %. This proof-of-concept study presents an apparatus that can be used by clinicians facing the challenge of improving basic knowledge on osseointegration, for the design of equipment for load bearing exercises and for rehabilitation programs.

Required hours of practice for learner drivers : a comparison between two Australian jurisdictions

Problem: This study considers whether requiring learner drivers to complete a set number of hours while on a learner licence affects the amount of hours of supervised practice that they undertake. It compares the amount of practice that learners in Queensland and New South Wales report undertaking. At the time the study was conducted, learner drivers in New South Wales were required to complete 50 hours of supervised practice while those from Queensland were not. Method: Participants were approached outside driver licensing centres after they had just completed their practical driving test to obtain their provisional (intermediate) licence. Those agreeing to participate were interviewed over the phone later and asked a range of questions to obtain information including socio-demographic details and amount of supervised practice completed. Results: There was a significant difference in the amount of practice that learners reported undertaking. Participants from New South Wales reported completing a significantly greater amount of practice (M = 73.3 hours, sd = 29.12 hours) on their learner licence than those from Queensland (M = 64.1 hours, sd = 51.05 hours). However, the distribution of hours of practice among the Queensland participants was bimodal in nature. Participants from Queensland reported either completing much less or much more practice than the New South Wales average. Summary: While it appears that the requirement that learner drivers complete a set number of hours may increase the average amount of hours of practice obtained, it may also serve to discourage drivers from obtaining additional practice, over and above the required hours. Impact on Industry: The results of this study suggest that the implications of requiring learner drivers to complete a set number of hours of supervised practice are complex. In some cases, policy makers may inadvertently limit the amount of hours learners obtain to the mandated amount rather than encouraging them to obtain as much practice as possible.

The use of high dynamic range luminance mapping in the assessment, understanding and defining of visual issues in post occupancy building assessment

The international focus on embracing daylighting for energy efficient lighting purposes and the corporate sector’s indulgence in the perception of workplace and work practice “transparency” has spurned an increase in highly glazed commercial buildings. This in turn has renewed issues of visual comfort and daylight-derived glare for occupants. In order to ascertain evidence, or predict risk, of these events; appraisals of these complex visual environments require detailed information on the luminances present in an occupant’s field of view. Conventional luminance meters are an expensive and time consuming method of achieving these results. To create a luminance map of an occupant’s visual field using such a meter requires too many individual measurements to be a practical measurement technique. The application of digital cameras as luminance measurement devices has solved this problem. With high dynamic range imaging, a single digital image can be created to provide luminances on a pixel-by-pixel level within the broad field of view afforded by a fish-eye lens: virtually replicating an occupant’s visual field and providing rapid yet detailed luminance information for the entire scene. With proper calibration, relatively inexpensive digital cameras can be successfully applied to the task of luminance measurements, placing them in the realm of tools that any lighting professional should own. This paper discusses how a digital camera can become a luminance measurement device and then presents an analysis of results obtained from post occupancy measurements from building assessments conducted by the Mobile Architecture Built Environment Laboratory (MABEL) project. This discussion leads to the important realisation that the placement of such tools in the hands of lighting professionals internationally will provide new opportunities for the lighting community in terms of research on critical issues in lighting such as daylight glare and visual quality and comfort.

Analysis of phasor measurement method in tracking the power frequency of distorted signals

This paper presents an analysis of phasor measurement method for tracking the fundamental power frequency to show if it has the performance necessary to cope with the requirements of power system protection and control. In this regard, several computer simulations presenting the conditions of a typical power system signal especially those highly distorted by harmonics, noise and offset, are provided to evaluate the response of the Phasor Measurement (PM) technique. A new method, which can shorten the delay of estimation, has also been proposed for the PM method to work for signals free of even-order harmonics.

Development of a method to measure upper eyelid pressure on the cornea

Purpose: The cornea is known to be susceptible to forces exerted by eyelids. There have been previous attempts to quantify eyelid pressure but the reliability of the results is unclear. The purpose of this study was to develop a technique using piezoresistive pressure sensors to measure upper eyelid pressure on the cornea. Methods: The technique was based on the use of thin (0.18 mm) tactile piezoresistive pressure sensors, which generate a signal related to the applied pressure. A range of factors that influence the response of this pressure sensor were investigated along with the optimal method of placing the sensor in the eye. Results: Curvature of the pressure sensor was found to impart force, so the sensor needed to remain flat during measurements. A large rigid contact lens was designed to have a flat region to which the sensor was attached. To stabilise the contact lens during measurement, an apparatus was designed to hold and position the sensor and contact lens combination on the eye. A calibration system was designed to apply even pressure to the sensor when attached to the contact lens, so the raw digital output could be converted to actual pressure units. Conclusions: Several novel procedures were developed to use tactile sensors to measure eyelid pressure. The quantification of eyelid pressure has a number of applications including eyelid reconstructive surgery and the design of soft and rigid contact lenses.