Forecasting, especially high-dimensional forecasting, is becoming more and more sought after, particularly as computing resources increase in both size and speed. Flow field forecasting is a general purpose regression-based forecasting method that has recently been expanded to high-dimensional settings. In this article, we provide an overview of the flow field forecasting methodology, with a particular emphasis on environments where the number of candidate predictor variables is large, potentially larger than the number of observations.

Flow field (FF) forecasting is a statistical framework for generalpurpose time series forecasting that can be readily adapted to various applications. Given a historical time series space, FF forecasting can actively search the space and determine which variables are most useful in prediction. FF forecasting was first developed as a univariate forecasting technique, but was extended to bivariate time series. In this paper we show that FF forecasting can further be extended to higher dimensional time series involving potentially hundreds of predictor variables. We call this implementation Tree based-flow field (TB-FF) forecasting. Using a tree-based algorithm, we sift through the predictor space in order to find the best predictor variables in a large candidate pool. We show that TB-FF forecasting technique can outperform many of the traditional techniques, especially when the time series data is non-stationary.

Regression is an often used, general means of time series forecasting. Flow field (FF) forecasting formalizes this approach, explicitly emphasizing both the chosen form of regression and the choice of predictors. FF forecasting has been previously implemented on univariate time series using Gaussian process regression. This paper addresses the practically important bivariate case, proposing a form of FF forecasting in which the flow field interpolation is accomplished by an application of nearest-neighbors regression. We call this implementation Closest history-flow field (CH-FF). CH-FF forecasting involves distance comparisons among spaces with differing dimensions. A bias adjustment is proposed to compensate for these differences, and CH-FF forecasting's resulting performance is compared in computer experiments with commonly practiced bivariate forecasting methodologies. Estimators of forecast standard error are also proposed.

The theory of supercharacters, which generalizes classical character theory, was recently introduced by P. Diaconis and I.M. Isaacs, building upon earlier work of C. André. We study supercharacter theories on (Z/nZ)d induced by the actions of certain matrix groups, demonstrating that a variety of exponential sums of interest in number theory (e.g., Gauss, Ramanujan, Heilbronn, and Kloosterman sums) arise in this manner. We develop a generalization of the discrete Fourier transform, in which supercharacters play the role of the Fourier exponential basis. We provide a corresponding uncertainty principle and compute the associated constants in several cases.

We study the representation theory of a certain finite group for which Kloosterman sums appear as character values. This leads us to consider a concrete family of commuting hermitian matrices which have Kloosterman sums as eigenvalues. These matrices satisfy a number of magical combinatorial properties and they encode various arithmetic properties of Kloosterman sums. These matrices can also be regarded as adjacency matrices for multigraphs which display Ramanujan-like behavior.