Analytical and numerical solutions of the space and time fractional bloch-torrey equation

Fractional order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brown-ian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As MRI is applied with increasing temporal and spatial resolution, the spin dynamics are being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments where processes are often anisotropic. Anomalous diffusion in the human brain using fractional order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch-Torrey equation using fractional order calculus with respect to time and space (see R.L. Magin et al., J. Magnetic Resonance, 190 (2008) 255-270). However effective numerical methods and supporting error analyses for the fractional Bloch-Torrey equation are still limited. In this paper, the space and time fractional Bloch-Torrey equation (ST-FBTE) is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE, and the stability and convergence of the INM are investigated. We prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.

Probabilistic multi-tensor estimation using the tensor distribution function

Diffusion weighted magnetic resonance (MR) imaging is a powerful tool that can be employed to study white matter microstructure by examining the 3D displacement profile of water molecules in brain tissue. By applying diffusion-sensitized gradients along a minimum of 6 directions, second-order tensors can be computed to model dominant diffusion processes. However, conventional DTI is not sufficient to resolve crossing fiber tracts. Recently, a number of high-angular resolution schemes with greater than 6 gradient directions have been employed to address this issue. In this paper, we introduce the Tensor Distribution Function (TDF), a probability function defined on the space of symmetric positive definite matrices. Here, fiber crossing is modeled as an ensemble of Gaussian diffusion processes with weights specified by the TDF. Once this optimal TDF is determined, the diffusion orientation distribution function (ODF) can easily be computed by analytic integration of the resulting displacement probability function.

The tensor distribution function

Diffusion weighted magnetic resonance imaging is a powerful tool that can be employed to study white matter microstructure by examining the 3D displacement profile of water molecules in brain tissue. By applying diffusion-sensitized gradients along a minimum of six directions, second-order tensors (represented by three-by-three positive definite matrices) can be computed to model dominant diffusion processes. However, conventional DTI is not sufficient to resolve more complicated white matter configurations, e.g., crossing fiber tracts. Recently, a number of high-angular resolution schemes with more than six gradient directions have been employed to address this issue. In this article, we introduce the tensor distribution function (TDF), a probability function defined on the space of symmetric positive definite matrices. Using the calculus of variations, we solve the TDF that optimally describes the observed data. Here, fiber crossing is modeled as an ensemble of Gaussian diffusion processes with weights specified by the TDF. Once this optimal TDF is determined, the orientation distribution function (ODF) can easily be computed by analytic integration of the resulting displacement probability function. Moreover, a tensor orientation distribution function (TOD) may also be derived from the TDF, allowing for the estimation of principal fiber directions and their corresponding eigenvalues.

Prediction of highway truck conveyance configuration under over-limit prohibition and conveyance equilibrium

Aiming at the shortage of prevailing prediction methods about highway truck conveyance configuration in over-limit freight research that transferring the goods attributed to over-limit portion to another fully loaded truck of the same configuration and developing the truck traffic volume synchronously, a new way to get accumulated probability function of truck power tonnage in basal year by highway truck classified by wheel and axle type load mass spectrum investigation was presented. Logit models were used to forecast overall highway freight diversion and single cargo tonnage diversion when the weight rules and strict of enforcement intensity of overload were changed in scheme year. Assumption that the probability distribution of single truck loadage should be consistent with the probability distribution of single goods freighted, the model describes the truck conveyance configuration in the future under strict over-limit prohibition. The model was used and tested in Highway Over-limit Research Project in Anhui by World Bank.

Multiscale analysis of protein functions and stochastic modelling of gene transcriptional regulatory networks

Genomic and proteomic analyses have attracted a great deal of interests in biological research in recent years. Many methods have been applied to discover useful information contained in the enormous databases of genomic sequences and amino acid sequences. The results of these investigations inspire further research in biological fields in return. These biological sequences, which may be considered as multiscale sequences, have some specific features which need further efforts to characterise using more refined methods. This project aims to study some of these biological challenges with multiscale analysis methods and stochastic modelling approach. The first part of the thesis aims to cluster some unknown proteins, and classify their families as well as their structural classes. A development in proteomic analysis is concerned with the determination of protein functions. The first step in this development is to classify proteins and predict their families. This motives us to study some unknown proteins from specific families, and to cluster them into families and structural classes. We select a large number of proteins from the same families or superfamilies, and link them to simulate some unknown large proteins from these families. We use multifractal analysis and the wavelet method to capture the characteristics of these linked proteins. The simulation results show that the method is valid for the classification of large proteins. The second part of the thesis aims to explore the relationship of proteins based on a layered comparison with their components. Many methods are based on homology of proteins because the resemblance at the protein sequence level normally indicates the similarity of functions and structures. However, some proteins may have similar functions with low sequential identity. We consider protein sequences at detail level to investigate the problem of comparison of proteins. The comparison is based on the empirical mode decomposition (EMD), and protein sequences are detected with the intrinsic mode functions. A measure of similarity is introduced with a new cross-correlation formula. The similarity results show that the EMD is useful for detection of functional relationships of proteins. The third part of the thesis aims to investigate the transcriptional regulatory network of yeast cell cycle via stochastic differential equations. As the investigation of genome-wide gene expressions has become a focus in genomic analysis, researchers have tried to understand the mechanisms of the yeast genome for many years. How cells control gene expressions still needs further investigation. We use a stochastic differential equation to model the expression profile of a target gene. We modify the model with a Gaussian membership function. For each target gene, a transcriptional rate is obtained, and the estimated transcriptional rate is also calculated with the information from five possible transcriptional regulators. Some regulators of these target genes are verified with the related references. With these results, we construct a transcriptional regulatory network for the genes from the yeast Saccharomyces cerevisiae. The construction of transcriptional regulatory network is useful for detecting more mechanisms of the yeast cell cycle.

Probability analysis of machine angle stability with non-gaussian wind power input

The security of power transfer across a given transmission link is typically a steady state assessment. This paper develops tools to assess machine angle stability as affected by a combination of faults and uncertainty of wind power using probability analysis. The paper elaborates on the development of the theoretical assessment tool and demonstrates its efficacy using single machine infinite bus system.

On the probability density function of the product of Rayleigh distributed random variables

In this paper we investigate the distribution of the product of Rayleigh distributed random variables. Considering the Mellin-Barnes inversion formula and using the saddle point approach we obtain an upper bound for the product distribution. The accuracy of this tail-approximation increases as the number of random variables in the product increase.

Bearing-only SLAM : a vision-based navigation system for autonomous robots

To navigate successfully in a previously unexplored environment, a mobile robot must be able to estimate the spatial relationships of the objects of interest accurately. A Simultaneous Localization and Mapping (SLAM) sys- tem employs its sensors to build incrementally a map of its surroundings and to localize itself in the map simultaneously. The aim of this research project is to develop a SLAM system suitable for self propelled household lawnmowers. The proposed bearing-only SLAM system requires only an omnidirec- tional camera and some inexpensive landmarks. The main advantage of an omnidirectional camera is the panoramic view of all the landmarks in the scene. Placing landmarks in a lawn field to define the working domain is much easier and more flexible than installing the perimeter wire required by existing autonomous lawnmowers. The common approach of existing bearing-only SLAM methods relies on a motion model for predicting the robot’s pose and a sensor model for updating the pose. In the motion model, the error on the estimates of object positions is cumulated due mainly to the wheel slippage. Quantifying accu- rately the uncertainty of object positions is a fundamental requirement. In bearing-only SLAM, the Probability Density Function (PDF) of landmark position should be uniform along the observed bearing. Existing methods that approximate the PDF with a Gaussian estimation do not satisfy this uniformity requirement. This thesis introduces both geometric and proba- bilistic methods to address the above problems. The main novel contribu- tions of this thesis are: 1. A bearing-only SLAM method not requiring odometry. The proposed method relies solely on the sensor model (landmark bearings only) without relying on the motion model (odometry). The uncertainty of the estimated landmark positions depends on the vision error only, instead of the combination of both odometry and vision errors. 2. The transformation of the spatial uncertainty of objects. This thesis introduces a novel method for translating the spatial un- certainty of objects estimated from a moving frame attached to the robot into the global frame attached to the static landmarks in the environment. 3. The characterization of an improved PDF for representing landmark position in bearing-only SLAM. The proposed PDF is expressed in polar coordinates, and the marginal probability on range is constrained to be uniform. Compared to the PDF estimated from a mixture of Gaussians, the PDF developed here has far fewer parameters and can be easily adopted in a probabilistic framework, such as a particle filtering system. The main advantages of our proposed bearing-only SLAM system are its lower production cost and flexibility of use. The proposed system can be adopted in other domestic robots as well, such as vacuum cleaners or robotic toys when terrain is essentially 2D.

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.

Rademacher and Gaussian complexities: Risk bounds and structural results

We investigate the use of certain data-dependent estimates of the complexity of a function class, called Rademacher and Gaussian complexities. In a decision theoretic setting, we prove general risk bounds in terms of these complexities. We consider function classes that can be expressed as combinations of functions from basis classes and show how the Rademacher and Gaussian complexities of such a function class can be bounded in terms of the complexity of the basis classes. We give examples of the application of these techniques in finding data-dependent risk bounds for decision trees, neural networks and support vector machines.

Vector quantization based Gaussian modeling for speaker verification

Gaussian mixture models (GMMs) have become an established means of modeling feature distributions in speaker recognition systems. It is useful for experimentation and practical implementation purposes to develop and test these models in an efficient manner particularly when computational resources are limited. A method of combining vector quantization (VQ) with single multi-dimensional Gaussians is proposed to rapidly generate a robust model approximation to the Gaussian mixture model. A fast method of testing these systems is also proposed and implemented. Results on the NIST 1996 Speaker Recognition Database suggest comparable and in some cases an improved verification performance to the traditional GMM based analysis scheme. In addition, previous research for the task of speaker identification indicated a similar system perfomance between the VQ Gaussian based technique and GMMs

Gaussian process models for sensor-centric robot localisation

This paper presents an approach to building an observation likelihood function from a set of sparse, noisy training observations taken from known locations by a sensor with no obvious geometric model. The basic approach is to fit an interpolant to the training data, representing the expected observation, and to assume additive sensor noise. This paper takes a Bayesian view of the problem, maintaining a posterior over interpolants rather than simply the maximum-likelihood interpolant, giving a measure of uncertainty in the map at any point. This is done using a Gaussian process framework. To validate the approach experimentally, a model of an environment is built using observations from an omni-directional camera. After a model has been built from the training data, a particle filter is used to localise while traversing this environment

Planning most-likely paths from overhead imagery

This paper is about planning paths from overhead imagery, the novelty of which is taking explicit account of uncertainty in terrain classification and spatial variation in terrain cost. The image is first classified using a multi-class Gaussian Process Classifier which provides probabilities of class membership at each location in the image. The probability of class membership at a particular grid location is then combined with a terrain cost evaluated at that location using a spatial Gaussian process. The resulting cost function is, in turn, passed to a planner. This allows both the uncertainty in terrain classification and spatial variations in terrain costs to be incorporated into the planned path. Because the cost of traversing a grid cell is now a probability density rather than a single scalar value, we can produce not only the most-likely shortest path between points on the map, but also sample from the cost map to produce a distribution of paths between the points. Results are shown in the form of planned paths over aerial maps, these paths are shown to vary in response to local variations in terrain cost.