Graduation Year

2019

Document Type

Dissertation

Degree

Ph.D.

Degree Name

Doctor of Philosophy (Ph.D.)

Degree Granting Department

Mathematics and Statistics

Major Professor

Gangaram S. Ladde, Ph.D.

Committee Member

Leslaw Skrzypek, Ph.D.

Committee Member

Mohamed Elhamdadi, Ph.D.

Committee Member

Brian Curtin, Ph.D.

Keywords

Binary State, Invariant Sets, Modified LLGMM, Stochastic Hybrid System, Survival Principle

Abstract

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).

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