RUI: Nonstationary and High Dimensioanl Nonparametric Transfer Function Models Using Polynomial Splines

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This project focuses on modeling and forecasting nonlinear time series using nonparametric methods. Because of their data-driven nature, nonparametric methods are flexible therefore ideal for approximating nonlinear features whose functional forms are usually unknown a priori. In his earlier study, the investigator used local polynomial regression to model the nonlinear relationship between time series and assumed that the noise follows an Autoregressive-Moving Average (ARMA) process (Liu, 2005). This model is named nonparametric transfer function model. The nonparametric transfer function model assumes the noise to be strictly stationary, and the transfer function to be low-dimensional. In this project, the investigator intends to generalize the nonparametric transfer function models by relaxing these assumptions. However, generalizing the local polynomial-based model directly is difficult partly because of the added estimation complexity related to local polynomial estimators when the assumptions are dropped. Polynomial spline provides an alternative because it has an explicit functional form, which greatly simplifies the estimation. At the same time it retains the flexibility of the model. To relax the assumption of stationary noise, in this project the investigator studies a new estimator in which the transfer function is approximated with polynomial splines. This estimator allows the noise to follow an ARIMA process. To extend the nonparametric transfer function models to higher dimensions, the main difficulty is the ``curse of dimensionality''. The investigator plans to overcome this problem by using an additive model to approximate the multivariate transfer function. The additive components are modeled with polynomial splines. The polynomial spline-based estimator can also be used to model time-varying conditional variance in nonparametric transfer function model. The investigator plans to approximate the transfer function and the conditional variance function by polynomial splines. The proposed study includes the asymptotic behavior of the proposed estimators and related issues, such as tests for additivity/unit root, model selection, estimation and forecasting.Forecasting and process control have been a constant interest in human society. They cannot be achieved without proper understanding of the underlying process. The proposed research extends the family of nonparametric transfer function models and provides new tools to explore complex relations in real-world, therefore enhances our abilities of forecasting and control. The proposed research adds new capabilities to the nonparametric transfer function models so they can be used to model high-dimensional transfer function, nonstationary time series, and time-varying conditional variance. As a result, the applicability of the nonparametric transfer function models is greatly enhanced. The proposed procedures are flexible enough for many nonlinear features encountered in practice, they are computationally efficient. The proposed methods can be applied in many areas, for example, it is used successfully in forecasting short-term electricity usage (Liu, Chen, Liu and Harris, 2006), which helps us use energy more efficiently and protect the environment. The proposed research contributes to nonlinear/nonparametric time series analysis. It further expands the application areas of the nonparametric transfer function models and provides useful tools to statisticians and researchers in many areas including ecology, environmental sciences, economics, finance, engineering and biology. For example, the proposed approach can be used to forecast volatility, which is an important issue in financial time series analysis.