Dissertation (PhD)--University of Stellenbosch, 2002 / ENGLISH ABSTRACT: Singular spectrum analysis (SSA) originated in the field of Physics. The technique is non-parametric by nature and inter alia finds application in atmospheric sciences, signal processing and recently in financial markets. The technique can handle a very broad class of time series that can contain combinations of complex periodicities, polynomial or exponential trend. Forecasting techniques are reviewed in this study, and a new coordinate free joint-horizon k-period-ahead forecasting formulation is derived. The study also considers model selection in SSA, from which it become apparent that forward validation results in more stable model selection. The roots of SSA are outlined and distributional assumptions of signal senes are considered ab initio. Pitfalls that arise in the multivariate statistical theory are identified. Different approaches of recurrent one-period-ahead forecasting are then reviewed. The forecasting approaches are all supplied in algorithmic form to ensure effortless adaptation to computer programs. Theoretical considerations, underlying the forecasting algorithms, are also considered. A new coordinate free joint-horizon kperiod- ahead forecasting formulation is derived and also adapted for the multichannel SSA case. Different model selection techniques are then considered. The use of scree-diagrams, phase space portraits, percentage variation explained by eigenvectors, cross and forward validation are considered in detail. The non-parametric nature of SSA essentially results in the use of non-parametric model selection techniques. Finally, the study also considers a commercial software package that is available and compares it with Fortran code, which was developed as part of the study. / AFRIKAANSE OPSOMMING: Singulier spektraalanalise (SSA) het sy oorsprong in die Fisika. Die tegniek is nieparametries van aard en vind toepassing in velde soos atmosferiese wetenskappe, seinprossesering en onlangs in finansiële markte. Die tegniek kan 'n wye verskeidenheid tydreekse hanteer wat kombinasies van komplekse periodisiteite, polinomiese- en eksponensiële tendense insluit. Vooruitskattingstegnieke word ook in hierdie studie beskou, en 'n nuwe koërdinaatvrye gesamentlike horison k-periodevooruitskattingformulering word afgelei. Die studie beskou ook model seleksie in SSA, waaruit duidelik blyk dat voorwaartse validasie meer stabiele model seleksie tot gevolg het. Die agtergrond van SSA word ab initio geskets en verdelingsaannames van seinreekse beskou. Probleemgevalle wat voorkom in die meervoudige statistiese teorie word duidelik geïdentifiseer. Verskeie tegnieke van herhalende toepassing van een-periode-vooruitskatting word daarna beskou. Die benaderings tot vooruitskatting word in algororitmiese formaat verskaf wat die aanpassing na rekenaarprogrammering vergemaklik. Teoretiese vraagstukke, onderliggend aan die vooruitskattings-algortimes, word ook beskou. 'n Nuwe koërdinaatvrye gesamentlike horison k-periode-vooruitskattingsformulering word afgelei en aangepas vir die multikanaal SSA geval. Verskillende model seleksie tegnieke is ook beskou. Die gebruik van "scree"- diagramme, fase ruimte diagramme, persentasie variasie verklaar deur eievektore, kruis- en voorwaartse validasie word ook aangespreek. Die nie-parametriese aard van SSA noop die gebruik van nie-parametriese model seleksie tegnieke. Die studie vergelyk laastens 'n kommersiële sagtewarepakket met die Fortran bronkode wat as deel van hierdie studie ontwikkel is.

Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: See full text. / AFRIKAANSE OPSOMMING: Sien volteks.

This thesis is devoted to the analysis of problems that arise when long products of matrices taken in a given set are constructed. A typical application is the stability of switched linear systems. The stability of a discrete-time linear system is a classical engineering problem that has been well understood for long: the dynamics can be expressed in terms of the eigenvalues of the matrix ruling the system. A more complicated problem arises when the dynamical system can switch, that is, if the matrix changes over time. If this matrix is taken from a given set but can be chosen arbitrarily in this set at every time, the stability problem turns to the computation of a quantity, the joint spectral radius of the set of matrices, introduced in the early sixties. While this quantity appears to be hard to compute, it has acquired more and more importance during the last decades, and new applications of the joint spectral radius in engineering or mathematics are frequently discovered. It has for instance been proved useful for the analysis of regularity of fractals, for the continuity of wavelets, or for autonomous agents detection in sensor networks. In the first part of this thesis, we present a theoretical survey of the joint spectral radius, including old and new results. The joint spectral subradius, which is its stabilizability counterpart, is also considered. In a second part, we study some applications related to long products of matrices. We first analyse in detail a problem in coding theory, that has been recently shown to involve a joint spectral radius computation. We then propose a new application of the joint spectral radius (and related quantities) to a classical problem in number theory, namely the counting of overlap-free words. We then turn to problems related with autonomous agents detection: we analyse the trackability of sensor networks, and introduce and analyse a new notion, namely the observability of sensor networks.

Computed Tomography (CT) is a medical imaging modality which acquires anatomical data via the unique x-ray attenuation of materials. Yet, some clinically important materials remain difficult to distinguish with current CT technology. Spectral CT is an emerging technology which acquires multiple CT datasets for specific x-ray spectra. These spectra provide a fingerprint that allow materials to be distinguished that would otherwise look the same on conventional CT. The unique characteristics of spectral CT data motivates research into novel visualization techniques. In this thesis, we aim to provide the foundation for visualizing spectral CT data. Our initial investigation of similar multi-variate data types identified intermixing as a promising visualization technique. This promoted the development of a generic, modular and extensible intermixing framework. Therefore, the contribution of our work is a framework supporting the construction, analysis and storage of algorithms for visualizing spectral CT studies. To allow evaluation, we implemented the intermixing framework in an application called MARSCTExplorer along with a standard set of volume visualization tools. These tools provide user-interaction as well as supporting traditional visualization techniques for comparison. We evaluated our work with four spectral CT studies containing materials indistinguishable by conventional CT. Our results confirm that spectral CT can distinguish these materials, and reveal how these materials might be visualized with our intermixing framework.

The linear receptivity and stability of plane idealized detonation with one-step Arrhenius type reaction kinetics is explored in the case of three-dimensional perturbations to a Zel'dovich-von Neumann-Doering base flow. This is explored in both overdriven and explicitly Chapman-Jouguet detonation. Additionally, the use of a multi-domain spectral collocation method for solving the conventional stability problem is explored within the context of normal-mode detonation. An extension of the stability analysis to confined detonations in a slightly porous walled tube is also carried out. Finally, an asymptotic analysis of a detonation with two-step reaction kinetics in the limit of large activation energy and for general overdrive and reaction order is performed yielding a nonlinear evolution equation for perturbations that produce stable limit cycle solutions.

In this dissertation, I design a processing approach, implement and test several solutions to combining spatial and spectral processing of multisource data. The measured spectral information is assumed to come from a multispectral or hyperspectral imaging system with low spatial resolution. Thematic content from a higher spatial resolution sensor is used to spatially localize different materials by their spectral signature. This approach results in both spectralunmixing and sharpening, a spatial-spectral fusion. The main real imagery example, fusion of polarimetric synthetic aperture radar (SAR) with hyperspectral imagery, poses a unique challenge due to the phenomenological differences between the sensors.Theoretical models for electro-optical image formation and scene reflectivity are shown to lead naturally to a set of pixel mixing equations. Several solutions for the spatial unmixing form of these equations are examined, based on the method of least squares. In particular, a method for introducing thematic content into the solution for spatial unmixing is defined using weighted least squares. Finally, and most significantly, a spatial-spectral fusion algorithm based on the theory of projection onto convex sets (POCS) is presented. Theoretical aspects of POCS are briefly discussed, showing how the use of constraints in the form of closed convex sets drives the solution. Then, constraints are derived that are intimately tied to the underlying theoretical models. Simulated imagery is used to characterize the different constraintcombinations that can be used in a POCS-based fusion algorithm.The fusion algorithms are applied to real imagery from two data sets, a Landsat ETM+ scene over Tucson, AZ and an AVIRIS/AirSAR scene over Tombstone, AZ. The results of the fusion are analyzed using scattergrams and correlation statistics. The POCS-based fusion algorithm is shown to produce a reasonable fusion of the AVIRIS/AirSAR data, with some sharpening of spatial-spectral features.

A Computed Tomographic Imaging Spectrometer (CTIS) is an imaging spectrometer which can acquire a hyper-spectral data set in a single snapshot (one focal plane array integration time) with no moving parts. A specially designed dispersing element, which separates light from the three-dimensional object cube into a grid of two-dimensional prismatic diffraction orders, is the key element in the instrument. The capabilities of the CTIS instrument can be improved by employing a more optimized grating design.There were two main goals to this research: (1) to design a novel CTIS disperser that will improve CTIS capabilities over the previous 5x5 disperser and (2) to integrate the new disperser into the CTIS and evaluate its performance compared to the 5x5 disperser. Six new disperser ideas were evaluated based on their performance in a number of computer simulations to determine the most optimal dispersion pattern. A new CTIS disperser incorporating a novel radial design pattern was developed and tested. Reconstruction results of various spatial and spectral targets are presented. Capabilities of the new CTIS instrument incorporating the radial grating are compared to the previous instrument employing a 5x5 disperser. While both dispersers perform similarly for point-source objects, the radial grating performs better than the previous disperser for extended sources.

Although Robert Cogan's New Images of Musical Sound won the Society of Music Theory's Outstanding Publication Award in 1987, his musical application of spectral analysis has seen little use over the past 25 years. Spectral images are most effective at illustrating the timbre of sound, and harmony, not timbre, is the key structural component of most Western music. There are, however, some compositions in which timbre plays a critical role. Chief among these is Edgard Varèse's Ionisation, an epic percussion ensemble piece built around 40 instruments and 13 musicians. Previous analyses by Jean-Charles François and Varèse protege Chou Wen-Chung have emphasized the importance of timbre to Ionisation's construction, but are limited in their exploration of timbral qualities. Modern digital spectral analysis allows for a more accurate picture of the individual timbres that make up Ionisation and define the broader textures and structures that give the piece meaning.

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Towards having a one size fits all solution to cocaine detection at borders; this thesis proposes a systematic cocaine detection methodology that can use raw data output from a fibre optic sensor to produce a set of unique features whose decisions can be combined to lead to reliable output. This multidisciplinary research makes use of real data sourced from cocaine analyte detecting fibre optic sensor developed by one of the collaborators - City University, London. This research advocates a two-step approach: For the first step, the raw sensor data are collected and stored. Level one fusion i.e. analyses, pre-processing and feature extraction is performed at this stage. In step two, using experimentally pre-determined thresholds, each feature decides on detection of cocaine or otherwise with a corresponding posterior probability. High level sensor fusion is then performed on this output locally to combine these decisions and their probabilities at time intervals. Output from every time interval is stored in the database and used as prior data for the next time interval. The final output is a decision on detection of cocaine. The key contributions of this thesis includes investigating the use of data fusion techniques as a solution for overcoming challenges in the real time detection of cocaine using fibre optic sensor technology together with an innovative user interface design. A generalizable sensor fusion architecture is suggested and implemented using the Bayesian and Dempster-Shafer techniques. The results from implemented experiments show great promise with this architecture especially in overcoming sensor limitations. A 5-fold cross validation system using a 12 13 - 1 Neural Network was used in validating the feature selection process. This validation step yielded 89.5% and 10.5% true positive and false alarm rates with 0.8 correlation coefficient. Using the Bayesian Technique, it is possible to achieve 100% detection whilst the Dempster Shafer technique achieves a 95% detection using the same features as inputs to the DF system.