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A brief introduction to the concepts and terminology of spectral analysis and a review of the standard methods for cross-spectral estimation, based on discrete time history data, are incorporated in Chapter 1. Co-spectral and quadrature-spectral estimators which are characterized by non-negative spectral windows are developed in Chapter 2. While the spectral windows for the co-spectral estimators are non-negative for all relevant values of the assignable constants, certain restrictions on these constants are necessary to assure the non-negativity of the quadrature-spectral window. The properties of these estimators are considered in detail. In Chapter 3 randomized co-spectral and quadrature spectral estimators are presented. These estimators depend on the random selection of sets of time differences, as opposed to the systematic evaluation of all possible time differences for the standard estimators. By suitable choices of probability distributions for the time differences and of weight functions, the expectations of the randomized estimators can be made equivalent to the expectations of the standard estimators or the estimators of Chapter 2. Since the randomized estimator is much simpler to use than the standard estimator, these estimators are compared in terms of their variances, given that they have equal expectations. The choice of probability distributions to yield minimum variance, given that the expectation is specified, is considered. Extremely simple co-spectral and quadrature-spectral estimators, for the case where the coefficients of the Fourier series expansions of realizations of the processes over a finite time interval can be obtained by means of suitable analog equipment, are developed in Chapter 4. The expectations, variances and covariances of these estimators are derived. / Ph. D.

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The dynamical origin of complexity is an object of intense debate and, up to moment of writing this manuscript, no unified approach exists as to how it should be properly addressed. This research work adopts the perspective of complexity as characterized by the emergence of non-Poisson renewal processes. In particular I introduce two new complex system models, namely the two-state stochastic clocks and the integrate-and-fire stochastic neurons, and investigate its coupled dynamics in different network topologies. Based on the foundations of renewal theory, I show how complexity, as manifested by the occurrence of non-exponential distribution of events, emerges from the interaction of the units of the system. Conclusion is made on the work's applicability to explaining the dynamics of blinking nanocrystals, neuron interaction in the human brain, and synchronization processes in complex networks.

Model checking linear-time properties expressed in first-order logic has non-elementary complexity, and thus various restricted logical languages are employed. In the first part of this dissertation we consider two such restricted specification logics on words: linear temporal logic (LTL) and two-variable first-order logic (FO2). LTL is more expressive but FO2 can be more succinct, and hence it is not clear which should be easier to verify. We take a comprehensive look at the issue, giving a comparison of verification problems for FO2, LTL, and various sublogics thereof across a wide range of models. In particular, we look at unary temporal logic (UTL), a subset of LTL that is expressively equivalent to FO2. We give three logic-to-automata translations which can be used to give upper bounds for FO2 and UTL and various sublogics. We apply these to get new bounds for model checking both non-deterministic systems (hierarchical and recursive state machines, games) and for probabilistic systems (Markov chains, recursive Markov chains, and Markov decision processes). Our results give a unified approach to understanding the behaviour of FO2, LTL, and their sublogics. We further consider the problem of computing maximal probabilities for interval Markov chains (and recursive interval Markov chains, stochastic context-free grammars) to satisfy LTL specifications. Using again our automata constructions we describe an expectation-maximisation algorithm to solve this problem in practice. Our algorithm can be seen as a variant of the classical Baum-Welch algorithm on hidden Markov models. We also introduce a publicly available on-line tool Tulip to perform such analysis. Finally, we investigate the extension of our techniques from words to trees. We show that the parallel between the complexity of FO2 satisfiability on general and on restricted structures breaks down as we move from words to trees, since trees allow one to encode alternating exponential time computation.

Many planning decisions related to the land phase of the hydrologic cycle involve uncertainty due to stochasticity of rainfall inputs and uncertainty in state and knowledge of hydrologic processes. Consideration of this uncertainty in planning requires quantification in the form of probability distributions. Needed probability distributions, for many cases, must be obtained by transforming distributions of rainfall input and hydrologic state through deterministic models of hydrologic processes. Probability generating functions are used to derive a recursive technique that provides the necessary probability transformation for situations where the hydrologic output of interest is the cumulative effect of a random number of stochastic inputs. The derived recursive technique is observed to be quite accurate from a comparison of probability distributions obtained independently by the recursive technique and an exact analytic method for a simple problem that can be solved with the analytic method. The assumption of Poisson occurrence of rainfall events, which is inherent in derivation of the recursive technique, is examined and found reasonable for practical application. Application of the derived technique is demonstrated with two important hydrology- related problems. It is first demonstrated for computing probability distributions of annual direct runoff from a watershed, using the USDA Soil Conservation Service (SCS direct runoff model and stochastic models for rainfall event depth and watershed state. The technique is also demonstrated for obtaining probability distributions of annual sediment yield. For this demonstration, the-deterministic transform model consists of a parametric event -based sediment yield model and the SCS models for direct runoff volume and peak flow rate. The stochastic rainfall model consists of a marginal Weibull distribution for rainfall event duration and a conditional log -normal distribution for rainfall event depth, given duration. The stochastic state model is the same as used for the direct runoff application. Probability distributions obtained with the recursive technique for both the direct runoff and sediment yield demonstration examples appear to be reasonable when compared to available data. It is, therefore, concluded that the recursive technique, derived from probability generating functions, is a feasible transform method that can be useful for coupling stochastic models of rainfall input and state to deterministic models of hydrologic processes to obtain probability distributions of outputs where these outputs are cumulative effects of random numbers of stochastic inputs.

We first review recent work on stable and multistable random processes and their localisability. Then most of the thesis concerns a new approach to these topics based on characteristic functions. Our aim is to construct processes on R, which are α(x)-multistable, where the stability index α(x) varies with x. To do this we first use characteristic functions to define α(x)-multistable random integrals and measures and examine their properties. We show that an α(x)-multistable random measure may be obtained as the limit of a sequence of measures made up of α-stable random measures restricted to small intervals with α constant on each interval. We then use the multistable random integrals to define multistable random processes on R and study the localisability of these processes. Thus we find conditions that ensure that a process locally ‘looks like’ a given stochastic process under enlargement and appropriate scaling. We give many examples of multistable random processes and examine their local forms. Finally, we examine the dimensions of graphs of α-stable random functions defined by series with α-stable random variables as coefficients.

published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

This dissertation develops an analytical-numerical approach to deterministically predict the space-time evolution of concentrations in heterogeneous geologic media conditioned on measurements of hydraulic conductivities (transmissivities) and/or hydraulic heads. Based on the new conditional Eulerian-Lagrangian transport theory by Neuman, we solve the conditional transport problem analytically at early time, and express it in pseudo-Fickian form at late time. The stochastically derived deterministic pseudo-Fickian mean concentration equation involves a conditional, space-time dependent dispersion tensor. The latter not only depends on properties of the medium and the velocity but also on the available information, and can be evaluated numerically along mean "particle" trajectories. The transport equation lends itself to accurate solution by standard Galerkin finite elements on a relatively coarse grid. This approach allows computing without using Monte Carlo simulation and explicitly the following: Concentration variance/covariance (uncertainty), origin of detected contaminant and associated uncertainty, mass flow rate across a "compliance surface", cumulative mass release and travel time probability distribution across this surface, uncertainty associated with the latter, second spatial moment of conditional mean plume about its center of mass, conditional mean second spatial moment of actual plume about its center of mass, conditional co-variance of plume center of mass, and effect of non-Gaussian velocity distribution. This approach can also account for uncertainty in initial mass and/or concentration when predicting the future evolution of a plume, whereas almost all existing stochastic models of solute transport assume the initial state to be known with certainty. We illustrate this approach by considering deterministic and uncertain instantaneous point and nonpoint sources in a two-dimensional domain with a mildly fluctuating, statistically homogeneous, lognormal transmissivity field. We take the unconditional mean velocity to be uniform, but allow conditioning on log transmissivity and hydraulic head data. Conditioning renders the velocity field statistically nonhomogeneous with reduced variances and correlation scales, renders the predicted plume irregular and non-Gaussian, and generally reduces both predictive dispersion and uncertainty.