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

这个项目的重点是用非参数方法建模和预测非线性时间序列。由于其数据驱动的性质，非参数方法是灵活的，因此非常适合近似非线性特征，其函数形式通常是未知的先验。在他早期的研究中，研究者使用局部多项式回归来模拟时间序列之间的非线性关系，并假设噪声遵循自回归移动平均（ARMA）过程（Liu, 2005）。该模型称为非参数传递函数模型。非参数传递函数模型假设噪声是严格平稳的，传递函数是低维的。在这个项目中，研究者打算通过放宽这些假设来推广非参数传递函数模型。然而，直接推广基于局部多项式的模型是困难的，部分原因是当放弃假设时，与局部多项式估计量相关的估计复杂性增加了。多项式样条提供了另一种选择，因为它具有显式的函数形式，这大大简化了估计。同时保留了模型的灵活性。为了放宽平稳噪声的假设，本课题研究了一种新的传递函数用多项式样条近似的估计器。这个估计器允许噪声遵循ARIMA过程。将非参数传递函数模型扩展到高维，主要困难是“维数诅咒”。研究者计划通过使用加性模型来近似多元传递函数来克服这个问题。加性分量用多项式样条建模。基于多项式样条的估计量也可用于非参数传递函数模型中时变条件方差的建模。研究者计划用多项式样条近似传递函数和条件方差函数。该研究包括所提出的估计量的渐近性和相关问题，如可加性/单位根检验、模型选择、估计和预测。预测和过程控制一直是人类社会关注的焦点。如果没有对潜在过程的正确理解，它们就无法实现。该研究扩展了非参数传递函数模型族，为探索现实世界中的复杂关系提供了新的工具，从而提高了我们的预测和控制能力。本研究为非参数传递函数模型增加了新的功能，使它们可以用于高维传递函数、非平稳时间序列和时变条件方差的建模。这大大提高了非参数传递函数模型的适用性。所提出的方法对于实际中遇到的许多非线性特征具有足够的灵活性，计算效率高。所提出的方法可以应用于许多领域，例如，它被成功地用于预测短期用电量（Liu, Chen, Liu and Harris, 2006），这有助于我们更有效地利用能源和保护环境。该研究有助于非线性/非参数时间序列分析。它进一步扩展了非参数传递函数模型的应用领域，并为生态学、环境科学、经济学、金融学、工程学和生物学等许多领域的统计学家和研究人员提供了有用的工具。例如，该方法可用于预测波动性，这是金融时间序列分析中的一个重要问题。

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.