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

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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.