The aim of this thesis is to investigate of bootstrap methods (Efron, 1979), in the the performance estimation of parameter estimates in non-linear time series models, in particular SETAR models (Tong, 1993). First and higher order SETAR models in known and unknown thresholds cases are considered. To assess the performance of bootstrap methods, we first give an extensive simulation study (by using simulated normal errors), in chapters 3 and 4, to investigate large and small sample behaviours of the true sampling distributions of parameter estimates of SETAR models and how they are affected by sample size. First and higher order SETAR models in the known and unknown threshold cases are considered. An introduction to the bootstrap methods (Efron, 1979 ) is given in chapter 5. The effect of sample size on the bootstrap distributions of parameter estimates of first and higher order SETAR models in the known and unknown threshold cases ( for given order, delay and number of thresholds ) are also investigated in this chapter, via simulation and by using the same models used in the simulated normal errors 'true distribution' case ( chapters 3 & 4). The results are compared with simulated normal case in order to assess the bootstrap results. Tong and Lim (1980) method is used for fitting SETAR models to bootstrap samples, which is also used in the initial fit. Moreover, applications of bootstrap to celebrated data sets, namely, the logarithmically transformed lynx data covering the period (182-1934); and the sunspot numbers covering the period (1700- 1920), are attempted. The cyclical behaviours of bootstrap models are also examined. Finally, in chapter 5, an attempt is also made to study the problem of non-linear properties of the skeleton of a non-linear autoregressive process (Jones, 1976) via simulation and we study in particular a limit cycle behaviour.

Time series models have been widely used in simulating financial data sets. Finding a nice way to estimate the parameters is really important. One of the traditional ways is to use maximum likelihood estimation to make an approach. However, when the error terms don’t have normality, MLE would be less efficient. Quasi maximum likelihood estimation, also regarded as Gaussian MLE, would be more efficient. Considering the heavy-tailed financial data sets, we can use non-Gaussian quasi maximum likelihood, which needs less assumptions and conditions. We use real financial data sets to compare these estimators.

ARMA model selection with criterion such as AIC and BIC tends not to select a consistent ARMA model with high probability. Hence data splitting is not reliable. One technique was fairly reliable with large sample sizes, and a modification also worked.The DD plot for visualizing prediction regions can also be used to visualize three bootstrap confidence regions.

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Time-series forecasting has long been studied and remains an important research task. In scenarios where multiple time series need to be forecast, approaches that exploit the mutual impact between time series results in more accurate forecasts. This has been demonstrated in various applications, including demand forecasting and traffic forecasting, among others. Hence, this dissertation focuses on graph-based models, which leverage the internode relations to forecast more efficiently and effectively by associating time series with nodes. This dissertation begins by introducing the notion of graph time-series models in a comprehensive survey of related models. The main contributions of this survey are: (1) A novel categorization is proposed to thoroughly analyze over 20 representative graph time-series models from various perspectives, including temporal components, propagation procedures, and graph construction methods, among others. (2) Similarities and differences among models are discussed to provide a fundamental understanding of decisive factors in graph time-series models. Model challenges and future directions are also discussed. Following the survey, this dissertation develops graph time-series models that utilize complex time-series interactions to yield context-aware, real-time, and probabilistic forecasting. The first method, Context Integrated Graph Neural Network (CIGNN), targets resource forecasting with contextual data. Previous solutions either neglect contextual data or only leverage static features, which fail to exploit contextual information. Its main contributions include: (1) Integrating multiple contextual graphs; and (2) Introducing and incorporating temporal, spatial, relational, and contextual dependencies; The second method, Evolving Super Graph Neural Network (ESGNN), targets large-scale time-series datasets through training on super graphs. Most graph time-series models let each node associate with a time series, potentially resulting in a high time cost. Its main contributions include: (1) Generating multiple super graphs to reflect node dynamics at different periods; and (2) Proposing an efficient super graph construction method based on K-Means and LSH; The third method, Probabilistic Hypergraph Recurrent Neural Network (PHRNN), targets datasets under the assumption that nodes interact in a simultaneous broadcasting manner. Previous hypergraph approaches leverage a static weight hypergraph, which fails to capture the interaction dynamics among nodes. Its main contributions include: (1) Learning a probabilistic hypergraph structure from the time series; and (2) Proposing the use of a KNN hypergraph for hypergraph initialization and regularization. The last method, Graph Deep Factors (GraphDF), aims at efficient and effective probabilistic forecasting. Previous probabilistic approaches neglect the interrelations between time series. Its main contributions include: (1) Proposing a framework that consists of a relational global component and a relational local component; (2) Conducting analysis in terms of accuracy, efficiency, scalability, and simulation with opportunistic scheduling. (3) Designing an algorithm for incremental online learning. / Doctor of Philosophy / Time-series forecasting has long been studied due to its usefulness in numerous applications, including demand forecasting, traffic forecasting, and workload forecasting, among others. In scenarios where multiple time series need to be forecast, approaches that exploit the mutual impact between time series results in more accurate forecasts. Hence, this dissertation focuses on a specific area of deep learning: graph time-series models. These models associate time series with a graph structure for more efficient and effective forecasting. This dissertation introduces the notion of graph time series through a comprehensive survey and analyzes representative graph time-series models to help readers gain a fundamental understanding of graph time series. Following the survey, this dissertation develops graph time-series models that utilize complex time-series interactions to yield context-aware, real-time, and probabilistic forecasting. The first method, Context Integrated Graph Neural Network (CIGNN), incorporates multiple contextual graph time series for resource time-series forecasting. The second method, Evolving Super Graph Neural Network (ESGNN), constructs dynamic super graphs for large-scale time-series forecasting. The third method, Probabilistic Hypergraph Recurrent Neural Network (PHRNN), designs a probabilistic hypergraph model that learns the interactions between nodes as distributions in a hypergraph structure. The last method, Graph Deep Factors (GraphDF), targets probabilistic time-series forecasting with a relational global component and a relational local model. These methods collectively covers various data characteristics and model structures, including graphs, super graph, and hypergraphs; a single graph, dual graphs, and multiple graphs; point forecasting and probabilistic forecasting; offline learning and online learning; and both small and large-scale datasets. This dissertation also highlights the similarities and differences between these methods. In the end, future directions in the area of graph time series are also provided.

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

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