University of Macau / Faculty of Science and Technology / Department of Computer and Information Science

Some of the more significant effects of global warming are manifested in the rise of temperatures and the increased intensity of hurricanes. This study analyzed data on Annual, January and July temperatures in Miami in the period spanning from 1949 to 2011; as well as data on central pressure and radii of maximum winds of hurricanes from 1944 to present. Annual Average, Maximum and Minimum Temperatures were found to be increasing with time. Also July Average, Maximum and Minimum Temperatures were found to be increasing with time. On the other hand, no significant trend could be detected for January Average, Maximum and Minimum Temperatures. No significant trend was detected in the central pressures and radii of maximum winds of hurricanes, while the radii of maximum winds for the largest hurricane of the year showed an increasing trend.

The State Space Model (SSM) encompasses the class of multivariate linear models, in particular, regression models with fixed, time-varying and random parameters, time series models, unobserved components models and combinations thereof. The well-known Kalman Filter (KF) provides a unifying tool for conducting statistical inferences with the SSM. A major practical problem with the KF concerns its initialization when either the initial state or the regression parameter (or both) in the SSM are diffuse. In these situations, it is common practice to either apply the KF to a transformation of the data which is functionally independent of the diffuse parameters or else initialize the KF with an arbitrarily large error covariance matrix. However neither approach is entirely satisfactory. The data transformation required in the first approach can be computationally tedious and furthermore it may not preserve the state space structure. The second approach is theoretically and numerically unsound. Recently however, De Jong (1991) has developed an extension of the KF, called the Diffuse Kalman Filter (DKF) to handle these diffuse situations. The DKF does not require any data transformation. The thesis contributes further to the theoretical and computational aspects of con ducting statistical inferences using the DKF. First, we demonstrate the appropriate initialization of the DKF for the important class of time-invariant SSM’s. This result is useful for maximum likelihood statistical inference with the SSM. Second, we derive and compare alternative pseudo-likelihoods for the diffuse SSM. We uncover some interesting characteristics of the DKF and the diffuse likelihood with the class of ARMA models. Third, we propose an efficient implementation of the DKF, labelled the collapsed DKF (CDKF). The latter is derived upon sweeping out some columns of the pertinent matrices in the DKF after an initial number of iterations. The CDKF coincides with the KF in the absence of regression effects in the SSM. We demonstrate that in general the CDKF is superior in practicality and performance to alternative algorithms proposed in the literature. Fourth, we consider maximum likelihood estimation in the SSM using an EM (Expectation-Maximization) approach. Through a judicious choice of the complete data, we develop an CDKF-EM algorithm which does not require the evaluation of lag one state error covariance matrices for the most common estimation exercise required for the SSM, namely the estimation of the covariance matrices of the disturbances in the SSM. Last we explore the topic of diagnostic testing in the SSM. We discuss and illustrate the recursive generation of residuals and the usefulness of the latters in pinpointing likely outliers and points of structural change. / Business, Sauder School of / Graduate

Thesis (Sc. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering, 1956. / "June, 1956." / Includes bibliographical references (leaf 113). / Introduction: A physically realizable nonlinear system, like a linear one, is a system whose present output is a function of the past of its input. We may regard the system as a computer that operates on the past of one time function to yield the present value of another time function. Mathematically we say that the system performs a transformation on the past of its input to yield its present output. When this transformation is linear (the case of linear systems) we can take advantage of the familiar convolution integral to obtain the present output from the past of the input and the system is said to be characterized by its response to an impulse. That is, the response of a linear system to an impulse is sufficient to determine its response to any input. When the transformation is nonlinear we no longer have a simple relation like the convolution integral relating the output to the past of the input and the system can no longer be characterized by its response to an impulse since superposition does not apply. Wiener has shown, however, that we can characterize a nonlinear system by a set of coefficients and that these coefficients can be determined from a knowledge of the response of the system to shot noise excitation. Thus, shot noise occupies the same position as a probe for investigating nonlinear systems that the impulse occupies as a probe for investigating linear systems. The first section of this thesis is devoted to the Wiener theory of nonlinear system characterization. Emphasis is placed on important concepts of this theory that are used in succeeding chapters to develop a theory for determining optimum nonlinear systems. / by Amar Gopal Bose. / Sc.D.

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Modern array sensing systems, such as distributed fiber optic sensing, are used in many applications which may record a mixture of responses to multiple physical quantities. In these applications, it may be helpful to be able to separate this mixture of responses into the signals resulting from the individual sources. This is similar to the cocktail party problem posed with Independent Component Analysis (ICA), in which we use gradient ascent and fixed point iteration optimization algorithms to achieve this separation. We then seek to apply the problem setup from ICA to mixed signals resulting from a sensor array with the goal of maintaining coherence throughout resulting spatial arrays. We propose a new post-processing technique after separation to pair up the signals from different types of physical quantities based on the Symmetric Reverse Cuthill-McKee (SRCM) and Symmetric Approximate Minimum Degree (SAMD) permutations of the coherence matrix. / Master of Science / Some modern sensing systems are able to collect data resulting from different types of sources, such as vibrations and electromagnetic waves, at the same time. This means we have signals resulting from a mixture of sources. An example of one such modern sensing system is distributed fiber optic sensors used in geoscience applications, such as seismology and subsurface imaging, which measures strain along the fiber optic cable. In many applications, it may be helpful to obtain the signals from each of these sources separately, instead of having a mixture of these sources. We propose the use of optimization algorithms, in particular two algorithms arising from Independent Component Analysis (ICA), which seek to maximize a function in order to separate these signals. We then explore changes required to the algorithms for scenarios in which we have multiple sensors spaced some distance away from each other which record signals from two different sources. We also present a method of determining which separated signals correspond to which sensors after performing signal separation.

In this thesis, we consider the identification of continuous-time multivariable systems. Direct methods of identification, i.e. identifying a continuous-time model directly from samples of input-output data, are considered briefly. Of primary consideration is the indirect method of identification, which can be considered as a two stage method. First, a discrete system model is identified from samples of input-output data. The next step is to transform this discrete time model to an equivalent continuous-time representation. The classical ZeroOrder hold (ZOH) transformation is presented primarily for comparison with the derived First-Order hold (FOH) technique. Involved in both of these methods is the transformation of the discrete-time state transition matrix to the continuous time system matrix. A new method for this transformation is presented also. This method along with the presented FOH transformation method have been published in Electronics Letters and another paper on this FOH method has been submitted as an invited paper at the 1991 IFAC Symposium on Identification. / Master of Science

The dependence structure of a stationary time series can be described by its autocorrelation function ρ^k. Consider the simple autoregressive model of order 1: y_t = αy_t-1 + u_t where α ε (-1, 1) is a fixed constant and the u_t's are i.i.d. N(O,σ²). Here ρ^k = α^|k|, k = 0, ± 1, ± 2, . . . . It can be argued that as α ranges from 1 to -1, the behavior of the corresponding AR(1) model changes from that of a slowly changing, smooth time series to that of a rapidly changing time series. This motivates a generalized AR(1) model where the coefficient itself changes stochastically with time: y_t = α(t)y_t-1 + u_t where α(t) is a random function of time. This dissertation gives necessary and sufficient conditions for the existence of a mean zero stochastic process with finite second-order moments which is a solution to the generalized AR(1) model and gives sufficient conditions for the existence of a weakly stationary solution. The theory is illustrated with a specific model structure imposed on the random coefficient α(t); α(t) is modeled as a strictly stationary, two-state Markov chain with states taking on values between 0 and 1. The resulting generalized AR(1) process is shown to be weakly stationary. Techniques are provided for estimating the parameters of this specific model and for obtaining the optimal predictor from the estimated model. / Ph. D.