الفهرس 
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Abstract
This dissertation’s scope is about a high-dimensional Markov chain (HD-MC) model, called high-order multivariate Markov chains (HOM-MC), and its applicability. The main concern is parsimonious modeling for HOM-MC that is vital in practice as it is intractable to use the full-parameterized model due to its excessive parameters’ number. The dissertation is of six chapters.
Chapter One reviews the literature most relevant to the study topic, elaborates the critical research problems, and outlines this dissertation’s main contributions and creative points. Chapter Two introduces the full-parameterized HOM-MC model, its prime characteristics that would be the solid ground to grasp the motivation behind the next chapters.
Chapter Three develops a novel general HOM-MC model to gather parsimony with realism. It can capture both correlation directions among categorical data sequences of possible states with a small parameters number remarkably less than the full-parameterized model. Its stationarity and parameters’ estimation procedures based on MLE and linear programming (LP) are debated besides providing simple convergence conditions to reduce complexity and speed up computations. Also, some Monte Carlo experiments are performed. It can be considered a generalization to many prior HD-MC models and other interesting ones not covered by the literature. Notably, we introduce a new facilitate HOM-MC model with a markedly reduced parameters number, and demonstrate its merits by an illustrative example and two real applications: sales demand predictions and daily exchange rates changes in Egypt.
In Chapter Four, we investigate an exciting issue of incorporating additional sequences to expand the general HOM-MC model to manipulate a large sequences number, . We propose a new gradual general HOM-MC model to reveal the relations amid the previous and extra sequences with estimating a markedly less parameters number than the general model for all sequences. Its stationarity and parameters’ estimation based on MLE and LP are discussed with detecting simple convergence conditions to diminish intricacy and expedite calculations. We derive some useful models of gradual nature from the proposed model. We amply introduce two novel gradual models for positive correlations: gradual and gradual facilitate HOM-MC models. We give two applications: daily returns in world stock market indices and daily exchange rates changes; to manifest these two models’ merits, respectively. The gradual models are shown to have better performance concern parsimony, so computations reduction and time-saving.
Chapter Five proposes a modern strategy to analyze the random phenomena evolution over time and space concurrently based on the HOM-MC framework. To combine parsimony with realism, we employ the general HOM-MC model to depict the high depth spatial-temporal dynamics. We then apply it to analyze the spatiotemporal dynamics for COVID-19 outbreak’s risk levels in WHO regions to predict the risk state of epidemiological prevalence and monitor infection control that is urgently needed nowadays. Our findings show clear evidence for the proposed model’s excellent performance for the pandemic spread realistic surveillance. We also use the application to affirm the advantages of the gradual model.
Chapter six concludes the dissertation and give some suggestions for future research.