| Author: | Chung, Kam Hin |
| Title: | On the modelling of high dimensional time series |
| Advisors: | Yiu, Ka-fai Cedric (AMA) Wong, Heung (AMA) |
| Degree: | Ph.D. |
| Year: | 2023 |
| Subject: | Time-series analysis Mathematical statistics Statistic -- Graphic methods Hong Kong Polytechnic University -- Dissertations |
| Department: | Department of Applied Mathematics |
| Pages: | xx, 164 pages : color illustrations |
| Language: | English |
| Abstract: | Graphical models for high dimensional time series data visualize dynamics relationships among variables which however need a huge number of parameters. Some of them, indeed, fail to be estimated in some cases. To tackle these problems, this thesis introduces a new sparse graphical vector, new matrix and new sparse graphical matrix time series models, and studies an estimation problem encountered for modelling the inverse of the high dimensional covariance matrix. Our first model, the sparse graphical vector autoregressive (VAR) model, combines the sparse VAR model and the sparse Gaussian graphical model to visualize causal dynamics and conditional independence among variables. Its autoregressive (AR) coefficient matrix indicates causal dynamics of variables, while a sparse entry of the inverse of the covariance matrix (precision matrix) characterizes conditional independence between variables. Three penalized likelihood estimation methods are used and a final model is selected from all possible sparse graphical VAR models based on the Bayesian Information Criterion (BIC). Under this setting, the model has the optimal sparsity combination of the AR coefficient and precision matrices and its sparsity pattern is more robust than that in existing sparse models, which require the sparsity pre-determined. We develop an algorithm and prove that it is convergent. We also prove that the penalized maximum likelihood estimators of the model are consistent and asymptotically normally distributed. A simulation study shows that our minimax concave penalized (MCP) graphical VAR models always have smaller BIC than the popular LASSO penalized models and their estimates are unbiased. We apply our models to the Pearl River Delta air pollution data for illustration and compare the results with the existing models. Our MCP model has the minimum BIC. We, then, extend the vector AR model to a matrix AR model with a precision matrix. This model has the AR coefficient matrices in the bilinear form. The left and right coefficient matrices explain the row-wise interaction and the column-wise dependence respectively. This analyzes the structured information under two categorical variables and reduces dramatically the number of parameters. In connection with the sparse Gaussian graphical model, we consider a sparse graphical matrix AR model with the optimal sparsity obtained in the same way as our first model. Two algorithms are proposed and we proved that they are convergent. A simulation study shows satisfactory results. Economic indicator data are fitted for illustration. We compare our model results with their corresponding existing models in the literature. The prediction sum of squared errors of our models are smaller. Finally, the thesis resolves the precision matrix problem encountered in the model estimation. The positive definiteness of the matrix is not easily transformed as constraints in the maximization likelihood estimation. During the estimation, a non-positive definite matrix iterate might be obtained after some iterations and causes numerical errors in the log-likelihood function value calculation. It leads to an estimation failure. We construct two algorithms to keep the precision matrix iterates falling into the positive definite cone in the estimation process. We test our algorithms on the failure cases of the constrained VAR model and our second model. Both are found successful in estimation. |
| Rights: | All rights reserved |
| Access: | open access |
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