Structure assignment problems in linear systems: Algebraic and geometric methods

The Determinantal Assignment Problem (DAP) is a family of synthesis methods that has emerged as the abstract formulation of pole, zero assignment of linear systems. This unifies the study of frequency assignment problems of multivariable systems under constant, dynamic centralized, or decentralized control structure. The DAP approach is relying on exterior algebra and introduces new system invariants of rational vector spaces, the Grassmann vectors and Plücker matrices. The approach can handle both generic and non-generic cases, provides solvability conditions, enables the structuring of decentralisation schemes using structural indicators and leads to a novel computational framework based on the technique of Global Linearisation. DAP introduces a new approach for the computation of exact solutions, as well as approximate solutions, when exact solutions do not exist using new results for the solution of exterior equations. The paper provides a review of the tools, concepts and results of the DAP framework and a research agenda based on open problems

The approximate determinantal assignment problem

The Determinantal Assignment Problem (DAP) is one of the central problems of Algebraic Control Theory and refers to solving a system of non-linear algebraic equations to place the critical frequencies of the system to specied locations. This problem is decomposed into a linear and a multi-linear subproblem and the solvability of the problem is reduced to an intersection of a linear variety with the Grassmann variety. The linear subproblem can be solved with standard methods of linear algebra, whereas the intersection problem is a problem within the area of algebraic geometry. One of the methods to deal with this problem is to solve the linear problem and then and which element of this linear space is closer - in terms of a metric - to the Grassmann variety. If the distance is zero then a solution for the intersection problem is found, otherwise we get an approximate solution for the problem, which is referred to as the approximate DAP. In this thesis we examine the second case by introducing a number of new tools for the calculation of the minimum distance of a given parametrized multi-vector that describes the linear variety implied by the linear subproblem, from the Grassmann variety as well as the decomposable vector that realizes this least distance, using constrained optimization techniques and other alternative methods, such as the SVD properties of the so called Grassmann matrix, polar decompositions and mother tools. Furthermore, we give a number of new conditions for the appropriate nature of the approximate polynomials which are implied by the approximate solutions based on stability radius results. The approximate DAP problem is completely solved in the 2-dimensional case by examining uniqueness and non-uniqueness (degeneracy) issues of the decompositions, expansions to constrained minimization over more general varieties than the original ones (Generalized Grassmann varieties), derivation of new inequalities that provide closed-form non-algorithmic results and new stability radii criteria that test if the polynomial implied by the approximate solution lies within the stability domain of the initial polynomial. All results are compared with the ones that already exist in the respective literature, as well as with the results obtained by Algebraic Geometry Toolboxes, e.g., Macaulay 2. For numerical implementations, we examine under which conditions certain manifold constrained algorithms, such as Newton's method for optimization on manifolds, could be adopted to DAP and we present a new algorithm which is ideal for DAP approximations. For higher dimensions, the approximate solution is obtained via a new algorithm that decomposes the parametric tensor which is derived by the system of linear equations we mentioned before

Advances in point process filters and their application to sympathetic neural activity

This thesis is concerned with the development of techniques for analyzing the sequences of stereotypical electrical impulses within neurons known as spikes. Sequences of spikes, also called spike trains, transmit neural information; decoding them often provides details about the physiological processes generating the neural activity. Here, the statistical theory of event arrivals, called point processes, is applied to human muscle sympathetic spike trains, a peripheral nerve signal responsible for cardiovascular regulation. A novel technique that uses observed spike trains to dynamically derive information about the physiological processes generating them is also introduced. Despite the emerging usage of individual spikes in the analysis of human muscle sympathetic nerve activity, the majority of studies in this field remain focused on bursts of activity at or below cardiac rhythm frequencies. Point process theory applied to multi-neuron spike trains captured both fast and slow spiking rhythms. First, analysis of high-frequency spiking patterns within cardiac cycles was performed and, surprisingly, revealed fibers with no cardiac rhythmicity. Modeling spikes as a function of average firing rates showed that individual nerves contribute substantially to the differences in the sympathetic stressor response across experimental conditions. Subsequent investigation of low-frequency spiking identified two physiologically relevant frequency bands, and modeling spike trains as a function of hemodynamic variables uncovered complex associations between spiking activity and biophysical covariates at these two frequencies. For example, exercise-induced neural activation enhances the relationship of spikes to respiration but does not affect the extremely precise alignment of spikes to diastolic blood pressure. Additionally, a novel method of utilizing point process observations to estimate an internal state process with partially linear dynamics was introduced. Separation of the linear components of the process model and reduction of the sampled space dimensionality improved the computational efficiency of the estimator. The method was tested on an established biophysical model by concurrently computing the dynamic electrical currents of a simulated neuron and estimating its conductance properties. Computational load reduction, improved accuracy, and applicability outside neuroscience establish the new technique as a valuable tool for decoding large dynamical systems with linear substructure and point process observations

On the Parallel Implementation of the Lehman Factoring Algorithm

Abstract not provided

Making Sense of Direction: Proximity and Order in Asymmetric Paired Comparison Data

In a square asymmetric matrix, the relationships among objects in the lower triangular half-matrix, differ from the relationships among the same objects in the upper triangular half. Square, asymmetric matrices can arise in similarity and preference data, when the direction of comparison is important. An asymmetric matrix can be rendered symmetric by averaging corresponding entries above and below the main diagonal. The difference between the original and the symmetric matrix is purely asymmetric, or skew-symmetric. The symmetric and skew-symmetric pans are orthogonal. An eigenvector-eigenvalue decomposition analyses the asymmetries into rank 2 skew-symmetric matrices, having an optimum least squares fit to the asymmetries (Gower, 1977). In this dissertation I derive an alternating least squares, nonmetric analogue of the canonical decomposition of asymmetry, suitable for ordinal-level data. In simulation studies, the nonmetric version gives better metric and nonmetric recovery, than does the canonical decomposition, when the asymmetries have been distorted by a range-compressing monotonic transform. The nonmetric technique appears to out-perform the canonical decomposition in detecting simplexes, and possibly in recovering multiplicative bias coefficients. However, canonical decomposition gives superior recovery after range-expanding monotonic transforms, and in the presence of error. An eigenvalue ratio test is proposed for determining the number of eigenvectors to extract in the canonical decomposition. The test quantifies changes in the slope of the log eigenvalue plot. In simulation studies the test appears to maintain its anticipated Type I error rate. The test is under-powered , which may help it to extract only well-identified eigenvectors. Finally, directional similarity judgments were collected for all possible pairs of exemplars of two semantic categories. The exemplars differed in typicality. After Tversky (1977) this should produce asymmetries related to the typicality. No asymmetries were found, however. Power analysis indicated that a correlation ratio for the asymmetries of .05 could have been detected 90% of the time. An extreme groups analysis also did not indicate asymmetry. The first eigenvector underlying the symmetric data, however, was highly correlated with typicality. Hence, Tversky\u27s model was not supported

Phase-Distortion-Robust Voice-Source Analysis

This work concerns itself with the analysis of voiced speech signals, in particular the analysis of the glottal source signal. Following the source-filter theory of speech, the glottal signal is produced by the vibratory behaviour of the vocal folds and is modulated by the resonances of the vocal tract and radiation characteristic of the lips to form the speech signal. As it is thought that the glottal source signal contributes much of the non-linguistic and prosodical information to speech, it is useful to develop techniques which can estimate and parameterise this signal accurately. Because of vocal tract modulation, estimating the glottal source waveform from the speech signal is a blind deconvolution problem which necessarily makes assumptions about the characteristics of both the glottal source and vocal tract. A common assumption is that the glottal signal and/or vocal tract can be approximated by a parametric model. Other assumptions include the causality of the speech signal: the vocal tract is assumed to be a minimum phase system while the glottal source is assumed to exhibit mixed phase characteristics. However, as the literature review within this thesis will show, the error criteria utilised to determine the parameters are not robust to the conditions under which the speech signal is recorded, and are particularly degraded in the common scenario where low frequency phase distortion is introduced. Those that are robust to this type of distortion are not well suited to the analysis of real-world signals. This research proposes a voice-source estimation and parameterisation technique, called the Power-spectrum-based determination of the Rd parameter (PowRd) method. Illustrated by theory and demonstrated by experiment, the new technique is robust to the time placement of the analysis frame and phase issues that are generally encountered during recording. The method assumes that the derivative glottal flow signal is approximated by the transformed Liljencrants-Fant model and that the vocal tract can be represented by an all-pole filter. Unlike many existing glottal source estimation methods, the PowRd method employs a new error criterion to optimise the parameters which is also suitable to determine the optimal vocal-tract filter order. In addition to the issue of glottal source parameterisation, nonlinear phase recording conditions can also adversely affect the results of other speech processing tasks such as the estimation of the instant of glottal closure. In this thesis, a new glottal closing instant estimation algorithm is proposed which incorporates elements from the state-of-the-art techniques and is specifically designed for operation upon speech recorded under nonlinear phase conditions. The new method, called the Fundamental RESidual Search or FRESS algorithm, is shown to estimate the glottal closing instant of voiced speech with superior precision and comparable accuracy as other existing methods over a large database of real speech signals under real and simulated recording conditions. An application of the proposed glottal source parameterisation method and glottal closing instant detection algorithm is a system which can analyse and re-synthesise voiced speech signals. This thesis describes perceptual experiments which show that, iunder linear and nonlinear recording conditions, the system produces synthetic speech which is generally preferred to speech synthesised based upon a state-of-the-art timedomain- based parameterisation technique. In sum, this work represents a movement towards flexible and robust voice-source analysis, with potential for a wide range of applications including speech analysis, modification and synthesis

A multiple-SIMD architecture for image and tracking analysis

The computational requirements for real-time image based applications are such as to warrant the use of a parallel architecture. Commonly used parallel architectures conform to the classifications of Single Instruction Multiple Data (SIMD), or Multiple Instruction Multiple Data (MIMD). Each class of architecture has its advantages and dis-advantages. For example, SIMD architectures can be used on data-parallel problems, such as the processing of an image. Whereas MIMD architectures are more flexible and better suited to general purpose computing. Both types of processing are typically required for the analysis of the contents of an image. This thesis describes a novel massively parallel heterogeneous architecture, implemented as the Warwick Pyramid Machine. Both SIMD and MIMD processor types are combined within this architecture. Furthermore, the SIMD array is partitioned, into smaller SIMD sub-arrays, forming a Multiple-SIMD array. Thus, local data parallel, global data parallel, and control parallel processing are supported. After describing the present options available in the design of massively parallel machines and the nature of the image analysis problem, the architecture of the Warwick Pyramid Machine is described in some detail. The performance of this architecture is then analysed, both in terms of peak available computational power and in terms of representative applications in image analysis and numerical computation. Two tracking applications are also analysed to show the performance of this architecture. In addition, they illustrate the possible partitioning of applications between the SIMD and MIMD processor arrays. Load-balancing techniques are then described which have the potential to increase the utilisation of the Warwick Pyramid Machine at run-time. These include mapping techniques for image regions across the Multiple-SIMD arrays, and for the compression of sparse data. It is envisaged that these techniques may be found useful in other parallel systems