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.

An experimental and finite element investigation of the biomechanics of vertebral compression fractures

Osteoporosis is a disease characterized by low bone mass and micro-architectural deterioration of bone tissue, with a consequent increase in bone fragility and susceptibility to fracture. Osteoporosis affects over 200 million people worldwide, with an estimated 1.5 million fractures annually in the United States alone, and with attendant costs exceeding $10 billion dollars per annum. Osteoporosis reduces bone density through a series of structural changes to the honeycomb-like trabecular bone structure (micro-structure). The reduced bone density, coupled with the microstructural changes, results in significant loss of bone strength and increased fracture risk. Vertebral compression fractures are the most common type of osteoporotic fracture and are associated with pain, increased thoracic curvature, reduced mobility, and difficulty with self care. Surgical interventions, such as kyphoplasty or vertebroplasty, are used to treat osteoporotic vertebral fractures by restoring vertebral stability and alleviating pain. These minimally invasive procedures involve injecting bone cement into the fractured vertebrae. The techniques are still relatively new and while initial results are promising, with the procedures relieving pain in 70-95% of cases, medium-term investigations are now indicating an increased risk of adjacent level fracture following the procedure. With the aging population, understanding and treatment of osteoporosis is an increasingly important public health issue in developed Western countries. The aim of this study was to investigate the biomechanics of spinal osteoporosis and osteoporotic vertebral compression fractures by developing multi-scale computational, Finite Element (FE) models of both healthy and osteoporotic vertebral bodies. The multi-scale approach included the overall vertebral body anatomy, as well as a detailed representation of the internal trabecular microstructure. This novel, multi-scale approach overcame limitations of previous investigations by allowing simultaneous investigation of the mechanics of the trabecular micro-structure as well as overall vertebral body mechanics. The models were used to simulate the progression of osteoporosis, the effect of different loading conditions on vertebral strength and stiffness, and the effects of vertebroplasty on vertebral and trabecular mechanics. The model development process began with the development of an individual trabecular strut model using 3D beam elements, which was used as the building block for lattice-type, structural trabecular bone models, which were in turn incorporated into the vertebral body models. At each stage of model development, model predictions were compared to analytical solutions and in-vitro data from existing literature. The incremental process provided confidence in the predictions of each model before incorporation into the overall vertebral body model. The trabecular bone model, vertebral body model and vertebroplasty models were validated against in-vitro data from a series of compression tests performed using human cadaveric vertebral bodies. Firstly, trabecular bone samples were acquired and morphological parameters for each sample were measured using high resolution micro-computed tomography (CT). Apparent mechanical properties for each sample were then determined using uni-axial compression tests. Bone tissue properties were inversely determined using voxel-based FE models based on the micro-CT data. Specimen specific trabecular bone models were developed and the predicted apparent stiffness and strength were compared to the experimentally measured apparent stiffness and strength of the corresponding specimen. Following the trabecular specimen tests, a series of 12 whole cadaveric vertebrae were then divided into treated and non-treated groups and vertebroplasty performed on the specimens of the treated group. The vertebrae in both groups underwent clinical-CT scanning and destructive uniaxial compression testing. Specimen specific FE vertebral body models were developed and the predicted mechanical response compared to the experimentally measured responses. The validation process demonstrated that the multi-scale FE models comprising a lattice network of beam elements were able to accurately capture the failure mechanics of trabecular bone; and a trabecular core represented with beam elements enclosed in a layer of shell elements to represent the cortical shell was able to adequately represent the failure mechanics of intact vertebral bodies with varying degrees of osteoporosis. Following model development and validation, the models were used to investigate the effects of progressive osteoporosis on vertebral body mechanics and trabecular bone mechanics. These simulations showed that overall failure of the osteoporotic vertebral body is initiated by failure of the trabecular core, and the failure mechanism of the trabeculae varies with the progression of osteoporosis; from tissue yield in healthy trabecular bone, to failure due to instability (buckling) in osteoporotic bone with its thinner trabecular struts. The mechanical response of the vertebral body under load is highly dependent on the ability of the endplates to deform to transmit the load to the underlying trabecular bone. The ability of the endplate to evenly transfer the load through the core diminishes with osteoporosis. Investigation into the effect of different loading conditions on the vertebral body found that, because the trabecular bone structural changes which occur in osteoporosis result in a structure that is highly aligned with the loading direction, the vertebral body is consequently less able to withstand non-uniform loading states such as occurs in forward flexion. Changes in vertebral body loading due to disc degeneration were simulated, but proved to have little effect on osteoporotic vertebra mechanics. Conversely, differences in vertebral body loading between simulated invivo (uniform endplate pressure) and in-vitro conditions (where the vertebral endplates are rigidly cemented) had a dramatic effect on the predicted vertebral mechanics. This investigation suggested that in-vitro loading using bone cement potting of both endplates has major limitations in its ability to represent vertebral body mechanics in-vivo. And lastly, FE investigation into the biomechanical effect of vertebroplasty was performed. The results of this investigation demonstrated that the effect of vertebroplasty on overall vertebra mechanics is strongly governed by the cement distribution achieved within the trabecular core. In agreement with a recent study, the models predicted that vertebroplasty cement distributions which do not form one continuous mass which contacts both endplates have little effect on vertebral body stiffness or strength. In summary, this work presents the development of a novel, multi-scale Finite Element model of the osteoporotic vertebral body, which provides a powerful new tool for investigating the mechanics of osteoporotic vertebral compression fractures at the trabecular bone micro-structural level, and at the vertebral body level.

Numerical and experimental investigation of wedge tip radius effect on wedge plasmons

We report numerical analysis and experimental observation of strongly localized plasmons guided by triangular metal wedges and pay special attention to the effect of smooth (nonzero radius) tips. Dispersion, dissipation, and field structure of such wedge plasmons are analyzed using the compact two-dimensional finite-difference time-domain algorithm. Experimental observation is conducted by the end-fire excitation and near-field scanning optical microscope detection of the predicted plasmons on 40°silver nanowedges with the wedge tip radii of 20, 85, and 125 nm that were fabricated by the focused-ion beam method. The effect of smoothing wedge tips is shown to be similar to that of increasing wedge angle. Increasing wedge angle or wedge tip radius results in increasing propagation distance at the same time as decreasing field localization (decreasing wave number). Quantitative differences between the theoretical and experimental propagation distances are suggested to be due to a contribution of scattered bulk and surface waves near the excitation region as well as the addition of losses due to surface roughness. The theoretical and measured propagation distances are several plasmon wavelengths and are useful for a range of nano-optical applications

Three-dimensional hydrogen-bonded structures in the 1:1 proton-transfer compounds of L-tartaric acid with the associative-group monosubstituted pyridines 3-minopyridine, 3-carboxypyridine (nicotinic acid) and 2-carboxypyridine (picolinic acid).

The 1:1 proton-transfer compounds of L-tartaric acid with 3-aminopyridine [3-aminopyridinium hydrogen (2R,3R)-tartrate dihydrate, C5H7N2+·C4H5O6-·2H2O, (I)], pyridine-3-carboxylic acid (nicotinic acid) [anhydrous 3-carboxypyridinium hydrogen (2R,3R)-tartrate, C6H6NO2+·C4H5O6-, (II)] and pyridine-2-carboxylic acid [2-carboxypyridinium hydrogen (2R,3R)-tartrate monohydrate, C6H6NO2+·C4H5O6-·H2O, (III)] have been determined. In (I) and (II), there is a direct pyridinium-carboxyl N+-HO hydrogen-bonding interaction, four-centred in (II), giving conjoint cyclic R12(5) associations. In contrast, the N-HO association in (III) is with a water O-atom acceptor, which provides links to separate tartrate anions through Ohydroxy acceptors. All three compounds have the head-to-tail C(7) hydrogen-bonded chain substructures commonly associated with 1:1 proton-transfer hydrogen tartrate salts. These chains are extended into two-dimensional sheets which, in hydrates (I) and (III) additionally involve the solvent water molecules. Three-dimensional hydrogen-bonded structures are generated via crosslinking through the associative functional groups of the substituted pyridinium cations. In the sheet struture of (I), both water molecules act as donors and acceptors in interactions with separate carboxyl and hydroxy O-atom acceptors of the primary tartrate chains, closing conjoint cyclic R44(8), R34(11) and R33(12) associations. Also, in (II) and (III) there are strong cation carboxyl-carboxyl O-HO hydrogen bonds [OO = 2.5387 (17) Å in (II) and 2.441 (3) Å in (III)], which in (II) form part of a cyclic R22(6) inter-sheet association. This series of heteroaromatic Lewis base-hydrogen L-tartrate salts provides further examples of molecular assembly facilitated by the presence of the classical two-dimensional hydrogen-bonded hydrogen tartrate or hydrogen tartrate-water sheet substructures which are expanded into three-dimensional frameworks via peripheral cation bifunctional substituent-group crosslinking interactions.

Combining electrospun scaffolds with electrosprayed hydrogels leads to three-dimensional cellularization of hybrid constructs

A common problem in the design of tissue engineered scaffolds using electrospun scaffolds is the poor cellular infiltration into the structure. To tackle this issue, three approaches to scaffold design using electrospinning were investigated: selective leaching of a water-soluble fiber phase (poly ethylene oxide (PEO) or gelatin), the use of micron-sized fibers as the scaffold, and a combination of micron-sized fibers with codeposition of a hyaluronic acid-derivative hydrogel, Heprasil. These designs were achieved by modifying a conventional electrospinning system with two charged capillaries and a rotating mandrel collector. Three types of scaffolds were fabricated: medical grade poly(epsilon-caprolactone)/collagen (mPCL/Col) cospun with PEO or gelatin, mPCL/Col meshes with micron-sized fibers, and mPCL/Col microfibers cosprayed with Heprasil. All three scaffold types supported attachment and proliferation of human fetal osteoblasts. However, selective leaching only marginally improved cellular infiltration when compared to meshes obtained by conventional electrospinning. Better cell penetration was seen in mPCL/Col microfibers, and this effect was more pronounced when Heprasil regions were present in the structure. Thus, such techniques could be further exploited for the design of cell permeable fibrous meshes for tissue engineering applications.

Modeling corneal surfaces with three-dimensional basis functions

Purpose: To ascertain the effectiveness of object-centered three-dimensional representations for the modeling of corneal surfaces. Methods: Three-dimensional (3D) surface decomposition into series of basis functions including: (i) spherical harmonics, (ii) hemispherical harmonics, and (iii) 3D Zernike polynomials were considered and compared to the traditional viewer-centered representation of two-dimensional (2D) Zernike polynomial expansion for a range of retrospective videokeratoscopic height data from three clinical groups. The data were collected using the Medmont E300 videokeratoscope. The groups included 10 normal corneas with corneal astigmatism less than −0.75 D, 10 astigmatic corneas with corneal astigmatism between −1.07 D and 3.34 D (Mean = −1.83 D, SD = ±0.75 D), and 10 keratoconic corneas. Only data from the right eyes of the subjects were considered. Results: All object-centered decompositions led to significantly better fits to corneal surfaces (in terms of the RMS error values) than the corresponding 2D Zernike polynomial expansions with the same number of coefficients, for all considered corneal surfaces, corneal diameters (2, 4, 6, and 8 mm), and model orders (4th to 10th radial orders) The best results (smallest RMS fit error) were obtained with spherical harmonics decomposition which lead to about 22% reduction in the RMS fit error, as compared to the traditional 2D Zernike polynomials. Hemispherical harmonics and the 3D Zernike polynomials reduced the RMS fit error by about 15% and 12%, respectively. Larger reduction in RMS fit error was achieved for smaller corneral diameters and lower order fits. Conclusions: Object-centered 3D decompositions provide viable alternatives to traditional viewer-centered 2D Zernike polynomial expansion of a corneal surface. They achieve better fits to videokeratoscopic height data and could be particularly suited to the analysis of multiple corneal measurements, where there can be slight variations in the position of the cornea from one map acquisition to the next.

One-dimensional hydrogen-bonded structures in the 1:1 proton-transfer salts of 4,5-dichlorophthalic acid with the aliphatic Lewis bases diisopropylamine and hexamethylenetetramine

The crystal structures of the 1:1 proton-transfer compounds of 4,5-dichlorophthalic acid with the aliphatic Lewis bases diisopropylamine and hexamethylenetetramine, viz. diisopropylaminium 2-carboxy-4,5-dichlorobenzoate (1) and hexamethylenetetraminium 2-carboxy-4,5-dichlorobenzoate hemihydrate (2), have been determined. Crystals of both 1 and 2 are triclinic, space group P-1, with Z = 2 in cells with a = 7.0299(5), b = 9.4712(7), c = 12.790(1)Å, α = 99.476(6), β = 100.843(6), γ = 97.578(6)o (1) and a = 7.5624(8), b = 9.8918(8), c = 11.5881(16)Å, α = 65.660(6), β = 86.583(4), γ = 86.987(8)o (2). In each, one-dimensional hydrogen-bonded chain structures are found: in 1 formed through aminium N+-H...Ocarboxyl cation-anion interactions. In 2, the chains are formed through anion carboxyl O...H-Obridging water interactions with the cations peripherally bound. In both structures, the hydrogen phthalate anions are essentially planar with short intra-species carboxylic acid O-H...Ocarboxyl hydrogen bonds [O…O, 2.381(3) Å (1) and 2.381(8) Å (2)].

Hydrate stabilization in the three-dimensional hydrogen-bonded structure of the brucinium compound, bis(2,3-dimethoxy-10-oxostrychnidinium) biphenyl-4,4'-disulfonate hexahydrate

The crystal structure of the 2:1 proton-transfer compound of brucine with biphenyl-4,4’-disulfonate, bis(2,3-dimethoxy-10-oxostrychnidinium) biphenyl-4,4'-disulfonate hexahydrate (1) has been determined at 173 K. Crystals are monoclinic, space group P21 with Z = 2 in a cell with a = 8.0314(2), b = 29.3062(9), c = 12.2625(3) Å, β = 101.331(2)o. The crystallographic asymmetric unit comprises two brucinium cations, a biphenyl-4,4'-disulfonate dianion and six water molecules of solvation. The brucinium cations form a variant of the common undulating and overlapping head-to-tail sheet sub-structure. The sulfonate dianions are also linked head-to-tail by hydrogen bonds into parallel zig-zag chains through clusters of six water molecules of which five are inter-associated, featuring conjoint cyclic eight-membered hydrogen-bonded rings [graph sets R33(8) and R34(8)], comprising four of the water molecules and closed by sulfonate O-acceptors. These chain structures occupy the cavities between the brucinium cation sheets and are linked to them peripherally through both brucine N+-H...Osulfonate and Ocarbonyl…H-Owater to sulfonate O bridging hydrogen bonds, forming an overall three-dimensional framework structure. This structure determination confirms the importance of water in the stabilization of certain brucine compounds which have inherent crystal instability.

Improvements to the fire performance of light gauge steel floor systems

Light gauge steel frame (LSF) structures are increasingly used in commercial and residential buildings because of their non-combustibility, dimensional stability and ease of installation. A common application is in floor-ceiling systems. The LSF floor-ceiling systems must be designed to serve as fire compartment boundaries and provide adequate fire resistance. Fire-rated floor-ceiling assemblies have been increasingly used in buildings. However, limited research has been undertaken in the past and hence a thorough understanding of their fire resistance behaviour is not available. Recently a new composite floor-ceiling system has been developed to provide higher fire rating. But its increased fire rating could not be determined using the currently available design methods. Therefore a research project was conducted to investigate its structural and fire resistance behaviour under standard fire conditions. This paper presents the results of full scale experimental investigations into the structural and fire behaviour of the new LSF floor system protected by the composite ceiling unit. Both the conventional and the new floor systems were tested under structural and fire loads. It demonstrates the improvements provided by the new composite panel system in comparison to conventional floor systems. Numerical studies were also undertaken using the finite element program ABAQUS. Measured temperature profiles of floors were used in the numerical analyses and their results were compared with fire test results. Tests and numerical studies provided a good understanding of the fire behaviour of the LSF floor-ceiling systems and confirmed the superior performance of the new composite system.

Phenyl-ring disorder in the two-dimensional hydrogen-bonded structure of the 1:1 proton-transfer salt of the diazo-dye precursor 4-(phenyldiazenyl)aniline (aniline yellow) with L-tartaric acid

The structure of the 1:1 proton-transfer compound from the reaction of L-tartaric acid with the azo-dye precursor aniline yellow [4-(phenylazo)aniline], 4-(phenyldiazenyl)anilinium hydrogen 2R,3R-tartrate C12H12N3+ . C4H6O6- has been determined at 200 K. The asymmetric unit of the compound contains two independent phenylazoanilinium cations and two hydrogen L-tartrate anions. The structure is unusual in that all four phenyl rings of both cations have identical 50% rotational disorder. The two hydrogen L-tartrate anions form independent but similar chains through head-to-tail carboxylic O--H...O~carboxyl~ hydrogen bonds [graph set C7] which are then extended into a two-dimensional hydrogen-bonded sheet structure through hydroxyl O--H...O hydrogen-bonding links. The anilinium groups of the phenyldiazenyl cations are incorporated into the sheets and also provide internal hydrogen-bonding extensions while their aromatic tails layer in the structure without significant interaction except for weak \p--\p interactions [minimum ring centroid separation, 3.844(3) \%A]. The hydrogen L-tartrate residues of both anions have the common short intramolecular hydroxyl O--H...O~carboxyl~ hydogen bonds. This work has provided a solution to the unusual disorder problem inherent in the structure of this salt as well as giving another example of the utility of the hydrogen tartrate in the generation of sheet substructures in molecular assembly processes.

Robust outdoor visual localization using a three-dimensional-edge map

Visual localization systems that are practical for autonomous vehicles in outdoor industrial applications must perform reliably in a wide range of conditions. Changing outdoor conditions cause difficulty by drastically altering the information available in the camera images. To confront the problem, we have developed a visual localization system that uses a surveyed three-dimensional (3D)-edge map of permanent structures in the environment. The map has the invariant properties necessary to achieve long-term robust operation. Previous 3D-edge map localization systems usually maintain a single pose hypothesis, making it difficult to initialize without an accurate prior pose estimate and also making them susceptible to misalignment with unmapped edges detected in the camera image. A multihypothesis particle filter is employed here to perform the initialization procedure with significant uncertainty in the vehicle's initial pose. A novel observation function for the particle filter is developed and evaluated against two existing functions. The new function is shown to further improve the abilities of the particle filter to converge given a very coarse estimate of the vehicle's initial pose. An intelligent exposure control algorithm is also developed that improves the quality of the pertinent information in the image. Results gathered over an entire sunny day and also during rainy weather illustrate that the localization system can operate in a wide range of outdoor conditions. The conclusion is that an invariant map, a robust multihypothesis localization algorithm, and an intelligent exposure control algorithm all combine to enable reliable visual localization through challenging outdoor conditions.