A Unified Group-Theoretic Method on Improper Partial Semi-Bilateral Generating Functions

2000 Mathematics Subject Classification: 33A65, 33C20.

On the duration of negative surplus

Doutoramento em Matemática

On Some Unifications Arising from the MIMO Rician Shadowed Model

This paper shows that the proposed Rician shadowed model for multi-antenna communications allows for the unification of a wide set of models, both for multiple-input multiple-output (MIMO) and single- input single-output (SISO) communications. The MIMO Rayleigh and MIMO Rician can be deduced from the MIMO Rician shadowed, and so their SISO counterparts. Other more general SISO models, besides the Rician shadowed, are included in the model, such as the κ-μ, and its recent generalization, the κ-μ shadowed model. Moreover, the SISO η-μ and Nakagami-q models are also included in the MIMO Rician shadowed model. The literature already presents the probability density function (pdf) of the Rician shadowed Gram channel matrix in terms of the well-known gamma- Wishart distribution. We here derive its moment generating function in a tractable form. Closed- form expressions for the cumulative distribution function and the pdf of the maximum eigenvalue are also carried out.

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.

Fundamental solution and discrete random walk model for time-space fractional diffusion equation

In this paper, we consider a time-space fractional diffusion equation of distributed order (TSFDEDO). The TSFDEDO is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α∈(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of orders β 1∈(0,1) and β 2∈(1,2], respectively. We derive the fundamental solution for the TSFDEDO with an initial condition (TSFDEDO-IC). The fundamental solution can be interpreted as a spatial probability density function evolving in time. We also investigate a discrete random walk model based on an explicit finite difference approximation for the TSFDEDO-IC.

The statistics of refractive error maps : managing wavefront aberration analysis without Zernike polynomials

The refractive error of a human eye varies across the pupil and therefore may be treated as a random variable. The probability distribution of this random variable provides a means for assessing the main refractive properties of the eye without the necessity of traditional functional representation of wavefront aberrations. To demonstrate this approach, the statistical properties of refractive error maps are investigated. Closed-form expressions are derived for the probability density function (PDF) and its statistical moments for the general case of rotationally-symmetric aberrations. A closed-form expression for a PDF for a general non-rotationally symmetric wavefront aberration is difficult to derive. However, for specific cases, such as astigmatism, a closed-form expression of the PDF can be obtained. Further, interpretation of the distribution of the refractive error map as well as its moments is provided for a range of wavefront aberrations measured in real eyes. These are evaluated using a kernel density and sample moments estimators. It is concluded that the refractive error domain allows non-functional analysis of wavefront aberrations based on simple statistics in the form of its sample moments. Clinicians may find this approach to wavefront analysis easier to interpret due to the clinical familiarity and intuitive appeal of refractive error maps.

Theory and development of GPS integrity monitoring system

This thesis discusses various aspects of the integrity monitoring of GPS applied to civil aircraft navigation in different phases of flight. These flight phases include en route, terminal, non-precision approach and precision approach. The thesis includes four major topics: probability problem of GPS navigation service, risk analysis of aircraft precision approach and landing, theoretical analysis of Receiver Autonomous Integrity Monitoring (RAIM) techniques and RAIM availability, and GPS integrity monitoring at a ground reference station. Particular attention is paid to the mathematical aspects of the GPS integrity monitoring system. The research has been built upon the stringent integrity requirements defined by civil aviation community, and concentrates on the capability and performance investigation of practical integrity monitoring systems with rigorous mathematical and statistical concepts and approaches. Major contributions of this research are: • Rigorous integrity and continuity risk analysis for aircraft precision approach. Based on the joint probability density function of the affecting components, the integrity and continuity risks of aircraft precision approach with DGPS were computed. This advanced the conventional method of allocating the risk probability. • A theoretical study of RAIM test power. This is the first time a theoretical study on RAIM test power based on the probability statistical theory has been presented, resulting in a new set of RAIM criteria. • Development of a GPS integrity monitoring and DGPS quality control system based on GPS reference station. A prototype of GPS integrity monitoring and DGPS correction prediction system has been developed and tested, based on the A USN A V GPS base station on the roof of QUT ITE Building.

Statistical modeling to determine resonant frequency information present in combustion chamber pressure signals

A statistical modeling method to accurately determine combustion chamber resonance is proposed and demonstrated. This method utilises Markov-chain Monte Carlo (MCMC) through the use of the Metropolis-Hastings (MH) algorithm to yield a probability density function for the combustion chamber frequency and find the best estimate of the resonant frequency, along with uncertainty. The accurate determination of combustion chamber resonance is then used to investigate various engine phenomena, with appropriate uncertainty, for a range of engine cycles. It is shown that, when operating on various ethanol/diesel fuel combinations, a 20% substitution yields the least amount of inter-cycle variability, in relation to combustion chamber resonance.

Computed success rates of various carrier phase integer estimation solutions and their comparison with statistical success rates

The success rate of carrier phase ambiguity resolution (AR) is the probability that the ambiguities are successfully fixed to their correct integer values. In existing works, an exact success rate formula for integer bootstrapping estimator has been used as a sharp lower bound for the integer least squares (ILS) success rate. Rigorous computation of success rate for the more general ILS solutions has been considered difficult, because of complexity of the ILS ambiguity pull-in region and computational load of the integration of the multivariate probability density function. Contributions of this work are twofold. First, the pull-in region mathematically expressed as the vertices of a polyhedron is represented by a multi-dimensional grid, at which the cumulative probability can be integrated with the multivariate normal cumulative density function (mvncdf) available in Matlab. The bivariate case is studied where the pull-region is usually defined as a hexagon and the probability is easily obtained using mvncdf at all the grid points within the convex polygon. Second, the paper compares the computed integer rounding and integer bootstrapping success rates, lower and upper bounds of the ILS success rates to the actual ILS AR success rates obtained from a 24 h GPS data set for a 21 km baseline. The results demonstrate that the upper bound probability of the ILS AR probability given in the existing literatures agrees with the actual ILS success rate well, although the success rate computed with integer bootstrapping method is a quite sharp approximation to the actual ILS success rate. The results also show that variations or uncertainty of the unit–weight variance estimates from epoch to epoch will affect the computed success rates from different methods significantly, thus deserving more attentions in order to obtain useful success probability predictions.

Seeing the Wood for the Trees: A Critical Evaluation of Methods to Estimate the Parameters of Stochastic Differential Equations

Maximum-likelihood estimates of the parameters of stochastic differential equations are consistent and asymptotically efficient, but unfortunately difficult to obtain if a closed-form expression for the transitional probability density function of the process is not available. As a result, a large number of competing estimation procedures have been proposed. This article provides a critical evaluation of the various estimation techniques. Special attention is given to the ease of implementation and comparative performance of the procedures when estimating the parameters of the Cox–Ingersoll–Ross and Ornstein–Uhlenbeck equations respectively.

Stochastic modelling of T cell homeostasis for two competing clonotypes via the master equation

Stochastic models for competing clonotypes of T cells by multivariate, continuous-time, discrete state, Markov processes have been proposed in the literature by Stirk, Molina-París and van den Berg (2008). A stochastic modelling framework is important because of rare events associated with small populations of some critical cell types. Usually, computational methods for these problems employ a trajectory-based approach, based on Monte Carlo simulation. This is partly because the complementary, probability density function (PDF) approaches can be expensive but here we describe some efficient PDF approaches by directly solving the governing equations, known as the Master Equation. These computations are made very efficient through an approximation of the state space by the Finite State Projection and through the use of Krylov subspace methods when evolving the matrix exponential. These computational methods allow us to explore the evolution of the PDFs associated with these stochastic models, and bimodal distributions arise in some parameter regimes. Time-dependent propensities naturally arise in immunological processes due to, for example, age-dependent effects. Incorporating time-dependent propensities into the framework of the Master Equation significantly complicates the corresponding computational methods but here we describe an efficient approach via Magnus formulas. Although this contribution focuses on the example of competing clonotypes, the general principles are relevant to multivariate Markov processes and provide fundamental techniques for computational immunology.

A practical approach for identifying expected solution performance and robustness in operations research applications

A practical approach for identifying solution robustness is proposed for situations where parameters are uncertain. The approach is based upon the interpretation of a probability density function (pdf) and the definition of three parameters that describe how significant changes in the performance of a solution are deemed to be. The pdf is constructed by interpreting the results of simulations. A minimum number of simulations are achieved by updating the mean, variance, skewness and kurtosis of the sample using computationally efficient recursive equations. When these criterions have converged then no further simulations are needed. A case study involving several no-intermediate storage flow shop scheduling problems demonstrates the effectiveness of the approach.

Security constrained economic dispatch considering wind energy conversion systems

The available wind power is stochastic and requires appropriate tools in the OPF model for economic and reliable power system operation. This paper exhibit the OPF formulation with factors involved in the intermittency of wind power. Weibull distribution is adopted to find the stochastic wind speed and power distribution. The reserve requirement is evaluated based on the wind distribution and risk of under/over estimation of the wind power. In addition, the Wind Energy Conversion System (WECS) is represented by Doubly Fed Induction Generator (DFIG) based wind farms. The reactive power capability for DFIG based wind farm is also analyzed. The study is performed on IEEE-30 bus system with wind farm located at different buses and with different wind profiles. Also the reactive power capacity to be installed in the wind farm to maintain a satisfactory voltage profile under the various wind flow scenario is demonstrated.

Maintenance optimisation of a multi-state series-parallel system considering economic dependence and state-dependent inspection intervals

This paper presents a maintenance optimisation method for a multi-state series-parallel system considering economic dependence and state-dependent inspection intervals. The objective function considered in the paper is the average revenue per unit time calculated based on the semi-regenerative theory and the universal generating function (UGF). A new algorithm using the stochastic ordering is also developed in this paper to reduce the search space of maintenance strategies and to enhance the efficiency of optimisation algorithms. A numerical simulation is presented in the study to evaluate the efficiency of the proposed maintenance strategy and optimisation algorithms. The simulation result reveals that maintenance strategies with opportunistic maintenance and state-dependent inspection intervals are more cost-effective when the influence of economic dependence and inspection cost is significant. The study further demonstrates that the optimisation algorithm proposed in this paper has higher computational efficiency than the commonly employed heuristic algorithms.

Utilising Reliability and Condition Monitoring Data for Asset Health Prognosis

The ability to forecast machinery health is vital to reducing maintenance costs, operation downtime and safety hazards. Recent advances in condition monitoring technologies have given rise to a number of prognostic models which attempt to forecast machinery health based on condition data such as vibration measurements. This paper demonstrates how the population characteristics and condition monitoring data (both complete and suspended) of historical items can be integrated for training an intelligent agent to predict asset health multiple steps ahead. The model consists of a feed-forward neural network whose training targets are asset survival probabilities estimated using a variation of the Kaplan–Meier estimator and a degradation-based failure probability density function estimator. The trained network is capable of estimating the future survival probabilities when a series of asset condition readings are inputted. The output survival probabilities collectively form an estimated survival curve. Pump data from a pulp and paper mill were used for model validation and comparison. The results indicate that the proposed model can predict more accurately as well as further ahead than similar models which neglect population characteristics and suspended data. This work presents a compelling concept for longer-range fault prognosis utilising available information more fully and accurately.