Time Series, Hidden Variables and Spatio-Temporal Ordinality Networks

In this work, a novel methodology for the modelling and forecasting of time series using higher-dimensional networks in R4,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{R }^{4,1}$$\end{document} space is presented. Time series data is partitioned, transformed and mapped into five-dimensional conformal space as a network which we call the ‘Spatio-Temporal Ordinality Network’ (STON). These STONs are characterised using specific Clifford Algebraic multivector functions which are found to be highly effective as featurised variables that forecast future states of the time series. The proposed STONs present as unique mathematical constructions that capture the algebra-geometric inter-dependencies in the historical behaviour of a time series system, thereby presenting governing expressions for forecasting based on historical values. Case studies on the seasonally adjusted Australian unemployment rate and the NASA Goddard Institute for Space Studies (GISS) Surface Temperature Analysis for Global Land-Ocean Temperature Index are furnished in this work and are compared against multilayer perceptrons (MLP), long short term memory (LSTM) neural networks, ARIMA and Holt-Winters methods. This formulation presents an alternative and effective modelling and forecasting paradigm for time series and multivariate systems with known or hidden variables.