Modeling multivariable time series using regular and singular autoregressions

The primary aim of this thesis is to study the modeling of high-dimensional time series with periodic missing observations. This study is very important in different branches of science and technology such as: econometric modeling, signal processing and systems and control. For instance, in the field of econometric modeling, it is crucial to provide proper models for national economies to help policy makers with decision making and policy adjustments. These models are built upon available high-dimensional data sets, which are not usually collected at the same rate. For example, some data such as, the employment rate are available on a monthly basis while some others like the gross domestic product (GDP) are collected quarterly. Motivated by applications in econometric modeling, we mainly consider systems, which have two sets of measurement streams, one stream being available at all times and the other one is observed every N-th time. There are two major issues involved with modeling of high-dimensional time series with periodic missing observations, namely, the curse of dimensionality and missing observations. Generalized dynamic factor models (GDFMs), which have been recently introduced in the field of econometric modeling, are exploited to handle the curse of dimensionality phenomenon. Furthermore, the blocking technique from systems and control is used to tackle issues associated with the missing observations. In this thesis, we consider a class of GDFMs and assume that there exists an underlying linear time-invariant system operating at the highest sample rate and our task is to identify this model from the available mixed frequency measurements. To this end, we first provide a very detailed study about zeros of linear systems with alternate missing measurements. Zeros of this class of linear systems are examined when the parameter matrices of a minimal state space representation of a transfer function matrix corresponding to the underlying high frequency system assume generic values. Under this setting, we then illustrate situations under which linear systems with missing observations are completely zero-free. It is worthwhile noting that the obtained condition is very common in an econometric modeling context. Then we apply this result and assume that the underlying high frequency system has an autoregressive (AR) structure. Next, we study identifiability of AR systems from those population second order moments, which can be observed in principle. We propose the method of modified extended Yule-Walker equations to show that the set of identifiable AR systems is an open and dense subset of the associated parameter space i.e. AR systems are generically identifiable

Reduced order feedback control equations for linear time and frequency domain analysis

An algorithm was developed which can be used to obtain the equations. In a more general context, the algorithm computes a real nonsingular similarity transformation matrix which reduces a real nonsymmetric matrix to block diagonal form, each block of which is a real quasi upper triangular matrix. The algorithm works with both defective and derogatory matrices and when and if it fails, the resultant output can be used as a guide for the reformulation of the mathematical equations that lead up to the ill conditioned matrix which could not be block diagonalized

High Performance Reconfigurable Computing for Linear Algebra: Design and Performance Analysis

Field Programmable Gate Arrays (FPGAs) enable powerful performance acceleration for scientific computations because of their intrinsic parallelism, pipeline ability, and flexible architecture. This dissertation explores the computational power of FPGAs for an important scientific application: linear algebra. First of all, optimized linear algebra subroutines are presented based on enhancements to both algorithms and hardware architectures. Compared to microprocessors, these routines achieve significant speedup. Second, computing with mixed-precision data on FPGAs is proposed for higher performance. Experimental analysis shows that mixed-precision algorithms on FPGAs can achieve the high performance of using lower-precision data while keeping higher-precision accuracy for finding solutions of linear equations. Third, an execution time model is built for reconfigurable computers (RC), which plays an important role in performance analysis and optimal resource utilization of FPGAs. The accuracy and efficiency of parallel computing performance models often depend on mean maximum computations. Despite significant prior work, there have been no sufficient mathematical tools for this important calculation. This work presents an Effective Mean Maximum Approximation method, which is more general, accurate, and efficient than previous methods. Together, these research results help address how to make linear algebra applications perform better on high performance reconfigurable computing architectures