Doctor of Philosophy / Department of Electrical and Computer Engineering / Balasubramaniam Natarajan / Stochastic hybrid system (SHS) is a class of dynamical systems that experience interaction of both discrete mode and continuous dynamics with uncertainty. State estimation for SHS has attracted research interests for decades with Kalman filter based solutions dominating the area. Mode-based Kalman filter is an extended version of the traditional Kalman filter for SHS. In general, as Kalman filter is unbiased for non-hybrid system estimation, prior research efforts primarily focus on the behavior of error covariance. In SHS state estimate, mode mismatch errors could result in a bias in the mode-based Kalman filter and have impacts on the continuous state estimation quality. The relationship between mode mismatch errors and estimation stability is an open problem that this dissertation attempts to address. Specifically, the probabilistic model of mode mismatch errors can be independent and identically distributed (i.i.d.), correlated across different modes and correlated across time. The proposed approach builds on the idea of modeling the bias evolution as a transformed system. The statistical convergence of the bias dynamics is then mapped to the stability of the transformed system. For each specific model of the mode mismatch error, the system matrix of the transformed system varies which results in challenges for the stability analysis. For the first time, the dissertation derives convergence conditions that provide tolerance regions for the mode mismatch error for three mode mismatch situations. The convergence conditions are derived based on generalized spectral radius theorem, Lyapunov theorem, Schur stability of a matrix polytope and interval matrix method. This research is fundamental in nature and its application is widespread. For example, the spatially and timely correlated mode mismatch errors can effectively capture cyber-attacks and communication link impairments in a cyber-physical system. Therefore, the theory and techniques developed in this dissertation can be used to analyze topology errors in any networked system such as smart grid, smart home, transportation, flight management system etc. The main results provide new insights on the fidelity in discrete state knowledge needed to maintain the performance of a mode-based Kalman filter and provide guidance on design of estimation strategies for SHS.

In the survival and reliability data analysis, parametric and nonparametric methods are used to estimate the hazard/risk rate and survival functions. A parametric approach is based on the assumption that the underlying survival distribution belongs to some specific family of closed form distributions (normal, Weibull, exponential, etc.). On the other hand, a nonparametric approach is centered around the best-fitting member of a class of survival distribution functions. Moreover, the Kaplan-Meier and Nelson-Aalen type nonparametric approach do not assume either distribution class or closed-form distributions. Historically, well-known time-to-event processes are death of living specie in populations and failure of component in engineering systems. Recently, the human mobility, electronic communications, technological changes, advancements in engineering, medical, and social sciences have further diversified the role and scope of time-to-event processes in cultural, epidemiological, financial, military, and social sciences. To incorporate extensions, generalizations and minimize scope of existing methods, we initiate an innovative alternative modeling approach for time-to-event dynamic processes. The innovative approach is composed of the following basic components: (1) development of continuous-time state of dynamic process, (2) introduction of discrete-time dynamic intervention process, (3) formulation of continuous and discrete-time interconnected dynamic system, (4) utilizing Euler-type discretized schemes, developing theoretical dynamic algorithms, and (5) introduction of conceptual and computational state and parameter estimation procedures. The presented approach is motivated by state and parameter estimation of time-to-event processes in biological, chemical, engineering, epidemiological, medical, military, multiple-markets and social dynamic processes under the influence of discrete-time intervention processes. We initiate (1) a time-to-event process to be a probabilistic dynamic process with unitary state. Action, normal, operational, radical, survival, susceptible, etc. and its complementary states, reaction, abnormal, nonoperational, non-radical, failure, infective and so on (quantitative and qualitative variables), are considered to be illustrations of a unitary state of time-to-event dynamic processes. A unitary state is measured by a probability distribution function. Employing Newtonian dynamic modeling approach and observing the definition of hazard rate as a specific rate, survival or failure probabilistic state dynamic model is developed. This dynamic model is further extended to incorporate internal or external discrete-time dynamic intervention processes acting on unitary state time-to-event processes (2). This further demanded a formulation and development of an interconnected continuous-discrete-time hybrid, and totally discrete-time dynamic models for time-to-event processes (3). Employing the developed hybrid model, Euler-type discretized schemes, a very general fundamental conceptual analytic algorithm is outlined (4). Using the developed theoretical computational procedure in (4), a general conceptual computational data organizational and simulation schemes are presented (5) for state and parameter estimation problems in unitary state time-to-event dynamic processes. The well-known theoretical existing results in the literature are exhibited as special cases in a systematic and unified manner (6). In fact, the Kaplan-Meier and Nelson-Aalen type nonparametric estimation approaches are systematically analyzed by the developed totally discrete-time hybrid dynamic modeling process. The developed approach is applied to two data sets. Moreover, this approach does not require a knowledge of either a closed-form solution distribution or a class of distributions functions. A hazard rate need not be constant. The procedure is dynamic. In the existing literature, the failure and survival distribution functions are treated to be evolving/progressing mutually exclusively with respect to corresponding to two mutually exclusive time varying events. We refer to these two functions (failure and survival) as cumulative distributions of two mutually disjoint state output processes with respect to two mutually exclusive time-varying complementary unitary states of a time-to-event processes in any discipline of interest (7). This kind of time-to-event process can be thought of as a Bernoulli-type of deterministic/stochastic process. Corresponding to these two complementary output processes of the Bernoulli-type of stochastic process, we associate two unitary dynamic states corresponding to a binary choice options/actions (8), namely, ({action, reaction}, {normal, abnormal}, {survival, failure}, {susceptible, infective}, {operational, nonoperational}, {radical, non-radical}, and so on.) Under this consideration, we extend unitary state time-to-event dynamic model to binary state time-to-event dynamic model. Using basic tools in mathematical sciences, we initiate a Newtonian-type dynamic approach for binary state time-to-event processes in the sciences, technologies, and engineering (9). Introducing an innovative concept of “survival state dynamic principle”, an innovative interconnected nonlinear non-stationary large-scale hybrid dynamic model for number of units/species and its unitary survival state corresponding to binary state time-to-event process is formulated (10). The developed model in (10) includes dynamic model (3) as a special case. The developed approach is directly applicable to binary state time-to-event dynamic processes in biological, chemical, engineering, financial, medical, physical, military, and social sciences in a coherent manner. A by-product of this is a transformed interconnected nonlinear hybrid dynamic model with a theoretical discrete-time conceptual computational dynamic process (11). Employing the transformed discrete-time conceptual computational dynamic process, we introduce notions of data coordination, state data decomposition and aggregation, theoretical conceptual iterative processes, conceptual and computational parameter estimation and simulation schemes, conceptual and computational state simulation schemes in a systematic way (12). The usefulness of the developed interconnected algorithm is validated by using three real world data sets (13). We note that the presented algorithm does not need a closed-form representation of distribution/likelihood function. In fact, it is free from any required assumptions of the “Classical Maximum Likelihood Function Approach” in the “Survival and Reliability Analysis.” The rapid electronic communication and human mobility processes have facilitated to transform information, knowledge, and ideas almost instantly around the globe. This indeed generates heterogeneity, and it causes to form nonlinear and non-stationary dynamic processes. Moreover, the heterogeneity, non-linearity, non-stationarity, further generates two types of uncertainties, namely, deterministic, and stochastic. In view of this, it is obvious that nothing is deterministic. In short, the 21st century problems are highly nonlinear, non-stationary and under the influence of internal and external random perturbations. Using tools in stochastic analysis, interconnected deterministic models in (3) and (10) are extended to interconnected stochastic hybrid dynamic model for binary state time-to-event processes (14). The developed model is described by a large-scale nonlinear and non-stationary stochastic differential equations. Moreover, a stochastic version of a survival function is also introduced (15). Analytical, computational, statistical, and simulation algorithms/procedures are also extended and analyzed in a systematic and unified way (16). The presented interconnected stochastic model is motivated to initiate conceptual computational parameter and state estimation schemes for time-to-event statistical data (17). Again, stochastic version of computational algorithms are validated in the context of three real world data sets. The obtained parameter and state estimates show that the algorithm is independent of the choice of nonlinear transformation (18). Utilizing the developed alternative innovative procedure and the recently modified deterministic version of Local Lagged Adapted Generalized Method of Moments (LLGMM) is also extended to stochastic version in a natural way (19). This approach provides a degree of measure of confidence, prediction, and planning assessments (20). In addition, it initiates a conceptual computational parameter and state estimation and simulation schemes that is suitable for the usage of mean square sub-optimal procedure (21). The usefulness and the significance of the approach is illustrated by applying to three data sets (22). The approach provides insight for investigating various type of invariant sets, namely, sustainable/unsustainable, survival/failure, reliable/unreliable (23), and qualitative properties such as sustainability versus unsustainability, reliability versus unreliability, etc. (24) Once again, the presented algorithm is independent of any form of survival distribution functions or data sets. Moreover, it does not require a closed form survival function distribution. We also note that the introduction of intervention processes provides a measure of influence and confidence for the usage of new tools/procedures/approaches in continuous-time binary state time-to-event dynamic process (25). Moreover, the presented dynamic modeling is more feasible for its usage of investigating a more complex time-to-event dynamic process (26). The developed procedure is dynamic and indeed non-parametric (27). The dynamic approach adapts with current changes and updates statistic process (28). The dynamic nature is natural rather than the existing static and single-shot techniques (29).

Supply and demand in the World oil market are balanced through responses to price movement with considerable complexity in the evolution of underlying supply-demand expectation process. In order to be able to understand the price balancing process, it is important to know the economic forces and the behavior of energy commodity spot price processes. The relationship between the different energy sources and its utility together with uncertainty also play a role in many important energy issues. The qualitative and quantitative behavior of energy commodities in which the trend in price of one commodity coincides with the trend in price of other commodities, have always raised the questions regarding their interactions. Moreover, if there is any interaction, then one would like to know the extent of influence on each other. In this work, we undertake the study to shed a light on the above highlighted processes and issues. The presented study systematically deals with the development of stochastic dynamic models and mathematical, statistical and computational analysis of energy commodity spot price and interaction processes. Below we list the main components of the research carried out in this dissertation. (1) Employing basic economic principles, interconnected deterministic and stochastic models of linear log-spot and expected log-spot price processes coupled with non-linear volatility process are initiated. (2) Closed form solutions of the models are analyzed. (3) Introducing a change of probability measure, a risk-neutral interconnected stochastic model is derived. (4) Furthermore, under the risk-neutral measure, expectation of the square of volatility is reduced to a continuous-time deterministic delay differential equation. (5) The by-product of this exhibits the hereditary effects on the mean-square volatility process. (6) Using a numerical scheme, a time-series model is developed and utilized to estimate the state and parameters of the dynamic model. In fact, the developed time-series model includes the extended GARCH model as special case. (7) Using the Henry Hub natural gas data set, the usefulness of the linear interconnected stochastic models is outlined. (8) Using natural and basic economic ideas, interconnected deterministic and stochastic models in (1) are extended to non-linear log-spot price, expected log-spot price and volatility processes. (9) The presented extended models are validated. (10) Closed form solution and risk-neutral models of (8) are outlined. (11) To exhibit the usefulness of the non-linear interconnected stochastic model, to increase the efficiency and to reduce the magnitude of error, it was essential to develop a modified version of extended Kalman filtering approach. The modified approach exhibits the reduction of magnitude of error. Furthermore, Henry Hub natural gas data set is used to show the advantages of the non-linear interconnected stochastic model. (12) Parameter and state estimation problems of continuous time non-linear stochastic dynamic process is motivated to initiate an alternative innovative approach. This led to introduce the concept of statistic processes, namely, local sample mean and sample variance. (13) Then it led to the development of an interconnected discrete-time dynamic system of local statistic processes and (14) its mathematical model. (15) This paved the way for developing an innovative approach referred as Local Lagged adapted Generalized Method of Moments (LLGMM). This approach exhibits the balance between model specification and model prescription of continuous time dynamic processes. (16) In addition, it motivated to initiate conceptual computational state and parameter estimation and simulation schemes that generates a mean square sub-optimal procedure. (17) The usefulness of this approach is illustrated by applying this technique to four energy commodity data sets, the U. S. Treasury Bill Yield Interest Rate and the U.S. Eurocurrency Exchange Rate data sets for state and parameter estimation problems. (18) Moreover, the forecasting and confidence-interval problems are also investigated. (19) The non-linear interconnected stochastic model (8) was further extended to multivariate interconnected energy commodities and sources with and without external random intervention processes. (20) Moreover, it was essential to extend the interconnected discrete-time dynamic system of local sample mean and variance processes to multivariate discrete-time dynamic system. (21) Extending the LLGMM approach in (15) to a multivariate interconnected stochastic dynamic model under intervention process, the parameters in the multivariate interconnected stochastic model are estimated. These estimated parameters help in analyzing the short term and long term relationship between the energy commodities. These developed results are applied to the Henry Hub natural gas, crude oil and coal data sets.

Stochastic hybrid dynamic systems that incorporate both continuous and discrete dynamics have been an area of great interest over the recent years. In view of applications, stochastic hybrid dynamic systems have been employed to diverse fields of studies, such as communication networks, air traffic management, and insurance risk models. The aim of the present study is to investigate properties of some classes of stochastic hybrid dynamic systems. The class of stochastic hybrid dynamic systems investigated has random jumps driven by a non-homogeneous Poisson process and deterministic jumps triggered by hitting the boundary. Its real-valued continuous dynamic between jumps is described by stochastic differential equations of the It\^o-Doob type. Existing results of piecewise deterministic models are extended to obtain the infinitesimal generator of the stochastic hybrid dynamic systems through a martingale approach. Based on results of the infinitesimal generator, some stochastic stability results are derived. The infinitesimal generator and stochastic stability results can be used to compute the higher moments of the solution process and find a bound of the solution. Next, the study focuses on a class of multidimensional stochastic hybrid dynamic systems. The continuous dynamic of the systems under investigation is described by a linear non-homogeneous systems of It\^o-Doob type of stochastic differential equations with switching coefficients. The switching takes place at random jump times which are governed by a non-homogeneous Poisson process. Closed form solutions of the stochastic hybrid dynamic systems are obtained. Two important special cases for the above systems are the geometric Brownian motion process with jumps and the Ornstein-Uhlenbeck process with jumps. Based on the closed form solutions, the probability distributions of the solution processes for these two special cases are derived. The derivation employs the use of the modal matrix and transformations. In addition, the parameter estimation problem for the one-dimensional cases of the geometric Brownian motion and Ornstein-Uhlenbeck processes with jumps are investigated. Through some existing and modified methods, the estimation procedure is presented by first estimating the parameters of the discrete dynamic and subsequently examining the continuous dynamic piecewisely. Finally, some simulated stochastic hybrid dynamic processes are presented to illustrate the aforementioned parameter-estimation methods. One simulated insurance example is given to demonstrate the use of the estimation and simulation techniques to obtain some desired quantities.

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