Modeling and Forecasting High-dimensional Functional Data

This thesis summarizes the research developed along this Ph.D. trajectory. The aim of this thesis is to develop new techniques for modeling and forecasting high-dimensional functional data. The first contribution of this thesis is to propose a functional error correction model (VECM) for the forecast of multivariate functional time series data. The model utilizes functional principal component analysis to reduce the infinite-dimensional functions to low-order principal component scores; the VECM is then applied to produce the forecast. An algorithm to generate bootstrap prediction intervals is also provided. The advantage of this model is that it not only takes into account the covariance between different groups but also can cope with data for which the assumption of stationarity does not hold. The usefulness of this model is demonstrated through a series of simulation studies and applications to the age-and sex-specific mortality rates in Switzerland and the Czech Republic. Extending from the multivariate functional time series, the second contribution of this thesis is to address the problem of forecasting high-dimensional functional time series. We propose a twofold dimension reduction model, where dynamic functional principal component analysis is first applied to reduce each functional time series to a vector; we then use the factor model as a further dimension reduction technique so that only a small number of latent factors is preserved. Classic time series models can be used to forecast the factors, and conditional forecasts of the functions can be constructed. Asymptotic properties of the approximated functions are established, including both estimation error and forecast error. The proposed method avoids the curse of dimensionality problem and is easy to implement. We show the superiority of our approach via both simulation studies and an application to Japanese age- and sex- specific mortality rates. Finally, we develop a factor-augmented smoothing model for the raw functional data contaminated by high-dimensional measurement errors. The high dimensionality here concerns the dimension of the measurement error, which is in a different sense from that in the second contribution. The proposed model reduces the dimension of the measurement error with a factor model, while smoothing the functional component. We provide strong motivations from three aspects with examples. Asymptotic theorems are also established to demonstrate the effects of including factor structures on the smoothing results. As a byproduct of independent interest, an estimator for the population covariance matrix of the raw data is presented based on the proposed model. Extensive simulation studies as well as an application to Australian weather data demonstrate that these factor adjustments are extremely important in improving estimation accuracy and avoiding the curse of dimensionality