HIGH PERFORMANCE, LOW COST SUBSPACE DECOMPOSITION AND POLYNOMIAL ROOTING FOR REAL TIME DIRECTION OF ARRIVAL ESTIMATION: ANALYSIS AND IMPLEMENTATION

This thesis develops high performance real-time signal processing modules for direction of arrival (DOA) estimation for localization systems. It proposes highly parallel algorithms for performing subspace decomposition and polynomial rooting, which are otherwise traditionally implemented using sequential algorithms. The proposed algorithms address the emerging need for real-time localization for a wide range of applications. As the antenna array size increases, the complexity of signal processing algorithms increases, making it increasingly difficult to satisfy the real-time constraints. This thesis addresses real-time implementation by proposing parallel algorithms, that maintain considerable improvement over traditional algorithms, especially for systems with larger number of antenna array elements. Singular value decomposition (SVD) and polynomial rooting are two computationally complex steps and act as the bottleneck to achieving real-time performance. The proposed algorithms are suitable for implementation on field programmable gated arrays (FPGAs), single instruction multiple data (SIMD) hardware or application specific integrated chips (ASICs), which offer large number of processing elements that can be exploited for parallel processing. The designs proposed in this thesis are modular, easily expandable and easy to implement. Firstly, this thesis proposes a fast converging SVD algorithm. The proposed method reduces the number of iterations it takes to converge to correct singular values, thus achieving closer to real-time performance. A general algorithm and a modular system design are provided making it easy for designers to replicate and extend the design to larger matrix sizes. Moreover, the method is highly parallel, which can be exploited in various hardware platforms mentioned earlier. A fixed point implementation of proposed SVD algorithm is presented. The FPGA design is pipelined to the maximum extent to increase the maximum achievable frequency of operation. The system was developed with the objective of achieving high throughput. Various modern cores available in FPGAs were used to maximize the performance and details of these modules are presented in detail. Finally, a parallel polynomial rooting technique based on Newton’s method applicable exclusively to root-MUSIC polynomials is proposed. Unique characteristics of root-MUSIC polynomial’s complex dynamics were exploited to derive this polynomial rooting method. The technique exhibits parallelism and converges to the desired root within fixed number of iterations, making this suitable for polynomial rooting of large degree polynomials. We believe this is the first time that complex dynamics of root-MUSIC polynomial were analyzed to propose an algorithm. In all, the thesis addresses two major bottlenecks in a direction of arrival estimation system, by providing simple, high throughput, parallel algorithms.

Data-driven estimates of the number of clusters in multivariate time series

An important problem in unsupervised data clustering is how to determine the number of clusters. Here we investigate how this can be achieved in an automated way by using interrelation matrices of multivariate time series. Two nonparametric and purely data driven algorithms are expounded and compared. The first exploits the eigenvalue spectra of surrogate data, while the second employs the eigenvector components of the interrelation matrix. Compared to the first algorithm, the second approach is computationally faster and not limited to linear interrelation measures.

Graft coronary vasculopathy in cardiac transplantation--evaluation of risk factors by multivariate analysis

The development of coronary vasculopathy is the main determinant of long-term survival in cardiac transplantation. The identification of risk factors, therefore, seems necessary in order to identify possible treatment strategies. Ninety-five out of 397 patients, undergoing orthotopic cardiac transplantation from 10/1985 to 10/1992 were evaluated retrospectively on the basis of perioperative and postoperative variables including age, sex, diagnosis, previous operations, renal function, cholesterol levels, dosage of immunosuppressive drugs (cyclosporin A, azathioprine, steroids), incidence of rejection, treatment with calcium channel blockers at 3, 6, 12, and 18 months postoperatively. Coronary vasculopathy was assessed by annual angiography at 1 and 2 years postoperatively. After univariate analysis, data were evaluated by stepwise multiple logistic regression analysis. Coronary vasculopathy was assessed in 15 patients at 1 (16%), and in 23 patients (24%) at 2, years. On multivariate analysis, previous operations and the incidence of rejections were identified as significant risk factors (P < 0.05), whereas the underlying diagnosis had borderline significance (P = 0.058) for the development of graft coronary vasculopathy. In contrast, all other variables were not significant in our subset of patients investigated. We therefore conclude that the development of coronary vasculopathy in cardiac transplant patients mainly depends on the rejection process itself, aside from patient-dependent factors. Therapeutic measures, such as the administration of calcium channel blockers and regulation of lipid disorders, may therefore only reduce the progress of native atherosclerotic disease in the posttransplant setting.

Multivariate modelling of responses to conditional items: New possibilities for latent class analysis

Questionnaire data may contain missing values because certain questions do not apply to all respondents. For instance, questions addressing particular attributes of a symptom, such as frequency, triggers or seasonality, are only applicable to those who have experienced the symptom, while for those who have not, responses to these items will be missing. This missing information does not fall into the category 'missing by design', rather the features of interest do not exist and cannot be measured regardless of survey design. Analysis of responses to such conditional items is therefore typically restricted to the subpopulation in which they apply. This article is concerned with joint multivariate modelling of responses to both unconditional and conditional items without restricting the analysis to this subpopulation. Such an approach is of interest when the distributions of both types of responses are thought to be determined by common parameters affecting the whole population. By integrating the conditional item structure into the model, inference can be based both on unconditional data from the entire population and on conditional data from subjects for whom they exist. This approach opens new possibilities for multivariate analysis of such data. We apply this approach to latent class modelling and provide an example using data on respiratory symptoms (wheeze and cough) in children. Conditional data structures such as that considered here are common in medical research settings and, although our focus is on latent class models, the approach can be applied to other multivariate models.

Stationarity of multivariate particle systems

A particle system is a family of i.i.d. stochastic processes with values translated by Poisson points. We obtain conditions that ensure the stationarity in time of the particle system in RdRd and in some cases provide a full characterisation of the stationarity property. In particular, a full characterisation of stationary multivariate Brown–Resnick processes is given.

Time-Series Panel Analysis (TSPA): Multivariate Modeling of Temporal Associations in Psychotherapy Process.

Objective: Processes occurring in the course of psychotherapy are characterized by the simple fact that they unfold in time and that the multiple factors engaged in change processes vary highly between individuals (idiographic phenomena). Previous research, however, has neglected the temporal perspective by its traditional focus on static phenomena, which were mainly assessed at the group level (nomothetic phenomena). To support a temporal approach, the authors introduce time-series panel analysis (TSPA), a statistical methodology explicitly focusing on the quantification of temporal, session-to-session aspects of change in psychotherapy. TSPA-models are initially built at the level of individuals and are subsequently aggregated at the group level, thus allowing the exploration of prototypical models. Method: TSPA is based on vector auto-regression (VAR), an extension of univariate auto-regression models to multivariate time-series data. The application of TSPA is demonstrated in a sample of 87 outpatient psychotherapy patients who were monitored by postsession questionnaires. Prototypical mechanisms of change were derived from the aggregation of individual multivariate models of psychotherapy process. In a 2nd step, the associations between mechanisms of change (TSPA) and pre- to postsymptom change were explored. Results: TSPA allowed a prototypical process pattern to be identified, where patient's alliance and self-efficacy were linked by a temporal feedback-loop. Furthermore, therapist's stability over time in both mastery and clarification interventions was positively associated with better outcomes. Conclusions: TSPA is a statistical tool that sheds new light on temporal mechanisms of change. Through this approach, clinicians may gain insight into prototypical patterns of change in psychotherapy.

M-Functionals of Multivariate Scatter

This survey provides a self-contained account of M-estimation of multivariate scatter. In particular, we present new proofs for existence of the underlying M-functionals and discuss their weak continuity and differentiability. This is done in a rather general framework with matrix-valued random variables. By doing so we reveal a connection between Tyler's (1987) M-functional of scatter and the estimation of proportional covariance matrices. Moreover, this general framework allows us to treat a new class of scatter estimators, based on symmetrizations of arbitrary order. Finally these results are applied to M-estimation of multivariate location and scatter via multivariate t-distributions.

On The Breakdown Properties of some Multivariate M-Functionals

For probability distributions on ℝq, a detailed study of the breakdown properties of some multivariate M-functionals related to Tyler's [Ann. Statist. 15 (1987) 234] ‘distribution-free’ M-functional of scatter is given. These include a symmetrized version of Tyler's M-functional of scatter, and the multivariate t M-functionals of location and scatter. It is shown that for ‘smooth’ distributions, the (contamination) breakdown point of Tyler's M-functional of scatter and of its symmetrized version are 1/q and inline image, respectively. For the multivariate t M-functional which arises from the maximum likelihood estimate for the parameters of an elliptical t distribution on ν ≥ 1 degrees of freedom the breakdown point at smooth distributions is 1/(q + ν). Breakdown points are also obtained for general distributions, including empirical distributions. Finally, the sources of breakdown are investigated. It turns out that breakdown can only be caused by contaminating distributions that are concentrated near low-dimensional subspaces.