Development of a General Framework for Nonlinear Prediction Using Auto-Cumulants: Theory, Methodology, and Computation

表现出非线性特征的数据经常出现在许多应用领域中，例如天气预报、信号处理等。这些特征也存在于各个国家机构收集的许多经济和人口时间序列中，用于制定对公众和社会具有重要影响的政策。然而，当前的方法严重依赖于线性方法，并且经常使用一些临时方法来处理非线性数据，使得最终的分析结果难以解释。因此，迫切需要系统地开发新的理论和方法框架，以改进预测，并考虑到时间序列数据的非线性特征。拟议的研究旨在通过开发新功能来直接满足这一需求，这些新功能将建立在现有高斯线性理论的基础上，并提供显着改进的预测。除了推进统计科学和相关科学应用之外，它还将对美国和其他国家的季节调整实践产生潜在影响，以更好地制定公共政策。该项目旨在利用高阶自累积函数和多谱工具，开发新的理论和方法来预测非高斯非线性过程。具体来说，该项目的目标包括：(i) 开发二次和高阶非线性预测器，并进行明显的改进，(ii) 扩展预测方法以用于新型所谓的二次可预测过程； (iii) 通过对 Whittle 似然的适当推广来开发非线性模型拟合，该似然是从一步超前二次预测滤波器的均方误差导出的，(iv) 开发多线性形式的自累积量的理论基础，这对于导出三阶和更高阶多项式预测变量至关重要，(v) 在 R 中开发算法和支持软件以实现该方法。该项目的结果预计将为显着改进单变量和多变量时间序列数据的预测和信号提取提供工具，这些数据表现出在许多科学领域（例如天文学、大气科学、金融、信号处理）和现实生活应用中普遍存在的非线性特征。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Data exhibiting nonlinear characteristics appear routinely in many areas of applications, such as weather forecasting, signal processing, etc. These features are also present in many economic and demographic time series collected by various national agencies for policy formulations that have important implications for the public and the society. However, the current methodology is heavily reliant upon linear approaches and some ad hoc methods are often used to handle nonlinear data, rendering the final results of analysis difficult to interpret. As a result, there is acute need for systematic development of new theoretical and methodological framework for improved prediction that takes into account the nonlinear features of the time series data. The proposed research seeks to address this need directly by developing new capabilities that will build on the existing linear theory for Gaussian and provide substantially improved prediction. In addition to advancing the statistical science and related scientific applications, it will also have potential impact on the practice of seasonal adjustments for better public policy formulation in the US and other nations.This project seeks to develop new theory and methodology for prediction for non-Gaussian, nonlinear processes, utilizing the tools of higher-order auto-cumulant functions and polyspectra. Specifically, the goals of the project include : (i) developing quadratic and higher order nonlinear predictors, with demonstrable improvements, (ii) extending forecasting approaches for a new class of so-called quadratically predictable processes; (iii) developing nonlinear models-fitting via an appropriate generalization of the Whittle likelihood, derived from the mean squared error of the one-step ahead quadratic forecasting filter, (iv) developing theoretical foundations of auto-cumulants for multi-linear forms that are paramount to derive third and higher order polynomial predictors,(v) developing algorithms and supporting software in R for implementation of the methodology. The results from the project are expected to provide tools for substantially improved forecasting and signal extraction for univariate and multivariate time series data exhibiting nonlinear characteristics that are prevalent in many areas of sciences (e.g., Astronomy, Atmospheric sciences, Finance, Signal Processing) and real life applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.