Dr. Tharindu De Alwis

While univariate time series analysis is well established, capturing nonlinear dynamics and high-dimensional lag structure remains a significant challenge. Sufficient dimension reduction (SDR) provides a useful and powerful framework to address the curse of dimensionality in such contexts. In this study, we introduce a convolution transformation method (CM) for estimating SDR subspaces in time series, with particular focus on the time series central mean subspace (TS-CMS) and the time series central subspace (TS-CS). We develop explicit matrix-based estimators for these two subspaces and extend a Fourier transform-based method (FM), originally developed for TS-CMS estimation, to the broader TS-CS setting. The asymptotic properties of the proposed estimators are established. Through simulation studies and empirical applications, we compare the performance of the CM and FM estimators. The numerical results suggest that the CM estimator performs well for estimating the TS-CMS, while the Fourier transform-based procedure tends to perform better for estimating the TS-CS in the settings considered.

Dimension reduction has always been one of the most significant and challenging problems in the analysis of high-dimensional data. In the context of time series analysis, our focus is on the estimation and inference of conditional mean and variance functions. By using central mean and variance dimension reduction subspaces that preserve sufficient information about the response, one can estimate the unknown mean and variance functions. While several approaches exist to estimate the time series central mean and variance subspaces (TS-CMS and TS-CVS), they are often computationally intensive and impractical. By employing the Fourier transform, we derive explicit estimators for TS-CMS and TS-CVS. These estimators are consistent, asymptotically normal, and efficient. Simulation studies evaluate the method's performance, showing it is significantly more accurate and computationally efficient than existing ones. Furthermore, the method is applied to the Canadian lynx dataset.

Matrix-valued data is commonly collected over time in many scientific fields. However, existing methods for handling such data are limited and often suffer from overparametrization. In response, Chen et al. (2021) introduced the matrix autoregressive (MAR) model as an alternative to traditional time series analysis, which relies on vectorization and vector autoregression frameworks. By preserving the original structure of matrices, the MAR model avoids the loss of valuable column and row information. This approach offers a significant reduction in dimensions and enables explicit interpretations of the data. However, when applied to high-dimensional matrix time series, the MAR model faces challenges due to the large size of the coefficient matrices involved. It struggles to differentiate between relevant and irrelevant information, making it inefficient in extracting relevant information from complex data. To address these limitations, we propose envelope-based MAR (EMAR) models that effectively identify and eliminate irrelevant information. Our proposed EMAR approach achieves substantial efficiency gains in estimation and forecasting by reducing parameters and constructing a link between the mean function and covariance structure. This is achieved by utilizing the minimal reducing subspaces of covariance matrices. We establish the asymptotic properties of our proposed estimators and compare their efficiency and accuracy to existing methods through simulation studies under both normality and non-normality conditions. Furthermore, we provide two real-world applications in economics and business to demonstrate the effectiveness of our approach.