This thesis presents a new approach to streamflow forecasting. The approach is based on specifying the probabilities that the next flow of a stream will occur within different ranges of values. Hence, this method is different from the time series models where point estimates are given as forecasts. With this approach flood forecasting is possible by focusing on a preselected range of streamflows. A double criteria objective function is developed to assess the model performance in flood prediction. Three case studies are examined based on data from the Salt River in Phoenix, Arizona and Bird Creek near Sperry, Oklahoma. The models presented are: a first order Markov chain (FOMC), a second order Markov chain (SOMC), and a first order Markov chain with rainfall as an exogenous input (FOMCX). Three forecasts methodologies are compared among each other and against time series models. It is shown that the SOMC is better than the FOMC while the FOMCX is better than the time series models.

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This dissertation studied complex social systems that have large number of individuals and complicated functional relations among individuals. Prisoner's Dilemma (PD) including Social Dilemmas (SDs) is a type of problem arising from collective actions in social systems. Previous PD studies have limitations and are not suitable for the study of collective actions in complex social systems. The large number of individuals and the complexity of the models made the development of theoretical, analytical studies impossible. An agent-based computer simulation is used in this dissertation for investigating N-person Prisoner's Dilemma (NPD), and its new extensions. My research can be divided into three chapters (three appendixes in this dissertation). In the first problem, the classical NPD model is considered, a much faster algorithm was developed, and the long term behavior of Pavlovian agents is examined. In this study, the main feature of the classical PD model was kept by restricting the state space into two possibilities: cooperation and defection. In most social situations the state space is much more complicated. In the second study, NPD was introduced with continuous state space. A continuous variable described the cooperation level of the participating individuals. A stochastic differential equation models state change of individuals. Public media and personal influence were first introduced in the study of NPD. In the third model, we analyzed the dynamic process of fund raising for a public radio station. This model is a combination of the other two models; discrete in the sense that donating or not in a time period is discrete variable; however the amount the individuals can pledge to the station is a continuous variable. In all three models, individual personalities are considered and quantified. Major personality types that might affect the possible cooperation or defection of the agents were captured in the continuous NPD simulation; major motivations that might affect the probability of pledging at a certain time period and the pledged amount were captured in the fund raising case. During the computer simulation, the behavior of each agent and the behavior of the entire society can be monitored.

The purpose of this research is (1) to develop a multi-objective fuzzy regression (MOFR) tool to overcome the shortcomings of the existing fuzzy regression approaches while keeping the good characteristics, and (2) to study systems with uncertain elements, using the example of rainfall-runoff process to illustrate the approach. Previous research has shown that fuzzy regression performs superior compared to statistical regression in some cases. On the other hand, fuzzy regression has also been criticized because it does not allow all data points to influence the estimated parameters, it is sensitive to data outliers, and the prediction intervals become wider as more data are collected. Here, several MOFR techniques are developed to overcome these problems by enabling the decision maker select a non-dominated solution based on the tradeoff between data outliers and prediction vagueness. It is shown that MOFR provides superior results to existing fuzzy regression techniques, and the existing fuzzy regression approaches and classical least squares regression are specific cases of the MOFR framework. The methodology is illustrated with examples from rainfall-runoff modeling, more specifically, conceptual rainfall-runoff (CRR) models are analyzed here. One of the main problems in CRR modeling is dealing with the uncertainty associated with the model parameters which is related to data and/or model structure. A fuzzy CRR (FCRR) framework is proposed herein where every element of the CRR is assumed to be uncertain, taken here as fuzzy. Parameter calibration of FCRR models using newly developed fuzzy regression techniques is also investigated. Applications are provided for a linear CRR model, the experimental two-parameter (TWOPAR) and the six-parameter (SIXPAR) models. The major findings can be summarized as follows: (1) FCRR enables the decision maker to gain insight about the CRR model sensitivity to uncertainty of the model elements, (2) using MOFR for the calibration of FCRR leads to non-convex, constrained, non-linear optimization problems, (3) fuzzy least squares regression model yields to more stable parameter estimates than the non-fuzzy regression model, (4) the methodology is applicable to any dynamic system with discrete modes.

This thesis describes the design and implementation of a structured knowledge representation scheme within an object-oriented environment. The knowledge representation scheme is based on the system entity structure (SES), a labeled tree-like graph which can be used to represent families of systems. The object-oriented environment is the Knowledge Representation for Object-oriented Simulation (KROS). A brief introduction to these two systems is made and a design of an automated model structuring system based on them is discussed. A detailed example based on a visual flight simulation graphical display system (GDS) is developed to demonstrate features of the model structuring system and show its utility. The SES has been implemented within KROS and does contribute formal ideas for decomposing systems. Implementing the SES within KROS allowed the use of the object-oriented methodology that KROS provides for algorithm implementation. The standardization provided by the Common LISP language allows the resulting system to be used on a variety of machines.

Wavelet-domain hidden Markov models have proven to be useful tools for statistical signal and image processing. The hidden Markov tree (HMT) model captures the key features of the joint density of the wavelet coefficients of real-world data. One potential drawback to the HMT framework is the need for computationally expensive iterative training. We propose two reduced-parameter HMT models that capture the general structure of a broad class of real-world images. In the image HMT model, we use the fact that for real-world images the structure of the HMT is self-similar across scale, allowing us to reduce the complexity of the model to just nine parameters. In the universal HMT we fix these nine parameters, eliminating training while retaining nearly all of the key structure modeled by the full HMT. Finally, we propose a fast shift-invariant HMT estimation algorithm that outperforms all other wavelet-based estimators in the current literature.

Since their introduction a little more than 10 years ago, wavelets have revolutionized image processing. Wavelet based algorithms define the state-of-the-art for applications including image coding (JPEG-2000), restoration, and segmentation. Despite their success, wavelets have significant shortcomings in their treatment of edges. Wavelets do not parsimoniously capture even the simplest geometrical structure in images, and wavelet based processing algorithms often produce images with ringing around the edges. As a first step towards accounting for this structure, we will show how to explicitly capture the geometric regularity of contours in cartoon images using the wedgelet representation and a multiscale geometry model. The wedgelet representation builds up an image out of simple piecewise constant functions with linear discontinuities. We will show how the geometry model, by putting a joint distribution on the orientations of the linear discontinuities, allows us to weigh several factors when choosing the wedgelet representation: the error between the representation and the original image, the parsimony of the representation, and whether the wedgelets in the representation form "natural" geometrical structures. We will analyze a simple wedgelet coder based on these principles, and show that it has optimal asymptotic performance for simple cartoon images. Next, we turn our attention to piecewise smooth images; images that are smooth away from a smooth contour. Using a representation composed of wavelets and wedgeprints (wedgelets projected into the wavelet domain), we develop a quadtree based prototype coder whose rate-distortion performance is asymptotically near-optimal. We use these ideas to implement a full-scale image coder that outperforms JPEG-2000 both in peak signal to noise ratio (by 1--1.5dB at low bitrates) and visually. Finally, we shift our focus to building a statistical image model directly in the wavelet domain. For applications other than compression, the approximate shift-invariance and directional selectivity of the slightly redundant complex wavelet transform make it particularly well-suited for modeling singularity structure. Around edges in images, complex wavelet coefficients behave very predictably, exhibiting dependencies that we will exploit using a hidden Markov tree model. We demonstrate the effectiveness of the complex wavelet model with several applications: image denoising, multiscale segmentation, and feature extraction.

Due to the inherent uncertainties present in nature and given the imperfect state of our knowledge, it is impossible to guarantee the satisfactory performance of any system in an absolute sense. Therefore, an approach such as reliability based design, which offers a rational basis for taking into account in the design process the various sources of uncertainty and checking the computed probability of failure, is desirable. Structural reliability analysis also provides as a by-product various reliability sensitivity measures, which are very useful for rational decision making in structural design. In addition, the performance of large and complex structural systems can be predicted only through complicated numerical algorithms, such as the powerful finite element method. Hence, in order to evaluate the probability of failure of such systems for given limit-states or failure criteria, finite element analysis and reliability analysis must be linked together to produce the finite element reliability method. In this study, the link between a general purpose, research oriented, finite element program (FEAP) and a reliability analysis program (CALREL) is established. In order to realistically model the inelastic behavior of structural systems, several inelastic element routines are developed and implemented in FEAP. The algorithms required for computing accurately and efficiently the structural response gradient (needed in reliability analysis) with respect to basic material properties are also formulated and implemented in FEAP. Finally, finite element sensitivity and reliability analyses of several realistic structural examples are performed.

This thesis considers duality properties and their application to the sequential gradient-restoration algorithms (SGRA) for optimal control problems. Two problems are studied: (P1) the basic problem and (P2) the general problem. In Problem (P1), the minimization of a functional is considered, subject to differential constraints and final constraints, the initial state being given; in Problem (P2), the minimization of a functional is considered, subject to differential constraints, nondifferential constraints, initial constraints, and final constraints. Depending on whether the primal formulation is used or the dual formulation is used, one obtains a primal sequential gradient-restoration algorithm (PSGRA) and a dual sequential gradient-restoration algorithm (DSGRA). With particular reference to Problem (P2), it is found convenient to split the control vector into an independent control vector and a dependent control vector, the latter having the same dimension as the nondifferential constraint vector. This modification enhances the computational efficiency of both the primal formulation and the dual formulation. The basic property of the dual formulation is that the Lagrange multipliers associated with the gradient phase and the restoration phase of SGRA minimize a special functional, quadratic in multipliers, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter. This duality property yields considerable computational benefits in that the auxiliary optimal control problems associated with the gradient phase and the restoration phase of SGRA can be reduced to mathematical programming problems involving a finite number of parameters as unknowns. Several numerical examples are solved using both the primal formulation and the dual formulation.

The deficiency of the widely used LQG method is that it depends heavily on the precision of plant parameters and the noise spectrum. The robustness problem can be formalized in singular value analysis. With the help of operator theory a new method for the synthesis of linear multivariable feedback systems in H(,(INFIN)) norm is developed from the singular value analysis. The positive feature of H(,(INFIN)) norm synthesis is the transparency for robustness conditions, the weighting functions are directly related to the specifications of design requirements. In this dissertation the LQG problem is restated as an interpolation problem in H(,2) space. The interpolation problem in simplest case can be solved by an explicit formula. The H(,(INFIN)) optimal norm can be obtained from the consideration of the ratio of two H(,2) norms. The close relations and similarities between H(,(INFIN)) and H(,2) are brought out. The total H(,(INFIN)) optimal solutions can be constructed by the unitary dilation from the interpolation space. The explicit formulas in s domain for these purposes are given, including the repeated zeros case and the degenerate case. The optimal solutions must belong to the degenerate case, in this case the problem can be solved by separating the singular part of Pick matrix from the regular part by a Cholesky decomposition. These results are also developed in a recursive version for repeated zeros. The zeros and interpolation condition vectors of a system can be determined numerically by an algorithm to solve eigenvalues and eigenvectors of a pencil. To convert the two-sided problem to a one-sided problem and to convert the nonsquare problem to a square problem are related to the spectral factorization which is discussed in detail. The optimal solutions of the nonsquare problem need not be all-pass, which is related to the existence of critical point. The theory applied to the sensitivity design problem can be considered as an extension of the classical lead-lag design method from SISO to MIMO with more profound mathematical background. The robust stability problem can also be formalized and solved in the framework. The robust sensitivity design introduces a new type of mathematical problem, which can be approximated in our framework in certain situations. The regulation, tracking, filtering and optimal controller design problem under the inexactly known noise spectrum can be solved in the general model in H(,(INFIN)) space by introducing proper weighting functions.