Karhunen-Loeve expansions and their applications.

The Karhunen-Loeve Expansion (K-L expansion) is a bi-orthogonal stochastic process expansion. In the field of stochastic process, the Karhunen-Loeve expansion decomposes the process into a series of orthogonal functions with the random coefficients. The essential idea of the expansion is to solve the Fredholm integral equation, associated with the covariance kernel of the process, which defines a Reproducing Kernel Hilbert Space (RKHS). This either has an analytical solution or special numerical methods are needed. This thesis applies the Karhunen-Loeve expansion to some fields of statistics. The first two chapters review the theoretical background of the Karhunen-Loeve expansion and introduce the numerical methods, including the integral method and the expansion method, when the analytical solution to the expansion is unavailable. Chapter 3 applies the theory of the Karhunen-Loeve expansion to the field of the design experiment using a criteria called "maximum entropy sampling". Under such setting, a type of duality is set up between maximum entropy sampling and the D- optimal design of the classical optimal design. Chapter 4 uses the Karhunen-Loeve expansion to calculate the conditional mean and variance for a given set of observations, with application to prediction. Chapter 5 extends the theory of the Karhunen- Loeve expansion from the univariate setting to the multivariate setting: multivariate space, univariate time. Adaptations of numerical methods of Chapter 2 are also provided for the multivariate setting, with a full matrix development. Chapter 6 applies the numerical method developed in Chapter 5 to the emerging area of multivariate functional data analysis with a detailed example on a trivariate autoregressive process

Deconvolution of Quantized-Input Linear Systems: An Information-Theoretic Approach

The deconvolution problem has been drawing the attention of mathematicians, physicists and engineers since the early sixties. Ubiquitous in the applications, it consists in recovering the unknown input of a convolution system from noisy measurements of the output. It is a typical instance of inverse, ill-posed problem: the existence and uniqueness of the solution are not assured and even small perturbations in the data may cause large deviations in the solution. In the last fifty years, a large amount of estimation techniques have been proposed by different research communities to tackle deconvolution, each technique being related to a peculiar engineering application or mathematical set. In many occurrences, the unknown input presents some known features, which can be exploited to develop ad hoc algorithms. For example, prior information about regularity and smoothness of the input function are often considered, as well as the knowledge of a probabilistic distribution on the input source: the estimation techniques arising in different scenarios are strongly diverse. Less effort has been dedicated to the case where the input is known to be affected by discontinuities and switches, which is becoming an important issue in modern technologies. In fact, quantized signals, that is, piecewise constant functions that can assume only a finite number of values, are nowadays widespread in the applications, given the ongoing process of digitization concerning most of information and communication systems. Moreover, hybrid systems are often encountered, which are characterized by the introduction of quantized signals into physical, analog communication channels. Motivated by such consideration, this dissertation is devoted to the study of the deconvolution of continuous systems with quantized input; in particular, our attention will be focused on linear systems. Given the discrete nature of the input, we will show that the whole problem can be interpreted as a paradigmatic digital transmission problem and we will undertake an Information-theoretic approach to tackle it. The aim of this dissertation is to develop suitable deconvolution algorithms for quantized-input linear systems, which will be derived from known decoding procedures, and to test them in different scenarios. Much consideration will be given to the theoretical analysis of these algorithms, whose performance will be rigorously described in mathematical terms

Proceedings of the Workshop on Applications of Distributed System Theory to the Control of Large Space Structures

Two general themes in the control of large space structures are addressed: control theory for distributed parameter systems and distributed control for systems requiring spatially-distributed multipoint sensing and actuation. Topics include modeling and control, stabilization, and estimation and identification

Nonlinear Systems

The editors of this book have incorporated contributions from a diverse group of leading researchers in the field of nonlinear systems. To enrich the scope of the content, this book contains a valuable selection of works on fractional differential equations.The book aims to provide an overview of the current knowledge on nonlinear systems and some aspects of fractional calculus. The main subject areas are divided into two theoretical and applied sections. Nonlinear systems are useful for researchers in mathematics, applied mathematics, and physics, as well as graduate students who are studying these systems with reference to their theory and application. This book is also an ideal complement to the specific literature on engineering, biology, health science, and other applied science areas. The opportunity given by IntechOpen to offer this book under the open access system contributes to disseminating the field of nonlinear systems to a wide range of researchers

Identification of Random Dynamic Force Using an Improved Maximum Entropy Regularization Combined with a Novel Conjugate Gradient

We propose a novel mathematical algorithm to offer a solution for the inverse random dynamic force identification in practical engineering. Dealing with the random dynamic force identification problem using the proposed algorithm, an improved maximum entropy (IME) regularization technique is transformed into an unconstrained optimization problem, and a novel conjugate gradient (NCG) method was applied to solve the objective function, which was abbreviated as IME-NCG algorithm. The result of IME-NCG algorithm is compared with that of ME, ME-CG, ME-NCG, and IME-CG algorithm; it is found that IME-NCG algorithm is available for identifying the random dynamic force due to smaller root mean-square-error (RMSE), lower restoration time, and fewer iterative steps. Example of engineering application shows that L-curve method is introduced which is better than Generalized Cross Validation (GCV) method and is applied to select regularization parameter; thus the proposed algorithm can be helpful to alleviate the ill-conditioned problem in identification of dynamic force and to acquire an optimal solution of inverse problem in practical engineering

Modified Maximum Entropy Method and Estimating the AIF via DCE-MRI Data Analysis

Background: For the kinetic models used in contrast-based medical imaging, the assignment of the arterial input function named AIF is essential for the estimation of the physiological parameters of the tissue via solving an optimization problem. Objective: In the current study, we estimate the AIF relayed on the modified maximum entropy method. The effectiveness of several numerical methods to determine kinetic parameters and the AIF is evaluated-in situations where enough information about the AIF is not available. The purpose of this study is to identify an appropriate method for estimating this function. Materials and Methods: The modified algorithm is a mixture of the maximum entropy approach with an optimization method, named the teaching-learning method. In here, we applied this algorithm in a Bayesian framework to estimate the kinetic parameters when specifying the unique form of the AIF by the maximum entropy method. We assessed the proficiency of the proposed method for assigning the kinetic parameters in the dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI), when determining AIF with some other parameter-estimation methods and a standard fixed AIF method. A previously analyzed dataset consisting of contrast agent concentrations in tissue and plasma was used. Results and Conclusions: We compared the accuracy of the results for the estimated parameters obtained from the MMEM with those of the empirical method, maximum likelihood method, moment matching ("method of moments"), the least-square method, the modified maximum likelihood approach, and our previous work. Since the current algorithm does not have the problem of starting point in the parameter estimation phase, it could find the best and nearest model to the empirical model of data, and therefore, the results indicated the Weibull distribution as an appropriate and robust AIF and also illustrated the power and effectiveness of the proposed method to estimate the kinetic parameters