In this work, a novel methodology for the modelling and forecasting of time series using higher-dimensional networks in \(\mathbf{R }^{4,1}\) 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.

中文翻译：

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时间序列，隐藏变量和时空序数网络

在这项工作中，使用\（\ mathbf {R} ^ {4,1} \）中的高维网络对时间序列进行建模和预测的新方法提出了空间。时间序列数据作为网络被划分，转换和映射到五维共形空间，我们称其为“时空常规网络”（STON）。这些STON使用特定的Clifford代数多矢量函数进行了表征，这些函数被发现作为预测时间序列未来状态的特征化变量非常有效。拟议的STON以独特的数学结构形式呈现，它们捕获了时间序列系统的历史行为中的代数-几何相互依存关系，从而提供了基于历史值进行预测的控制表达式。这项工作提供了关于季节性调整后的澳大利亚失业率的案例研究以及美国宇航局戈达德空间研究所（GISS）全球陆地-海洋温度指数的地表温度分析，并与多层感知器（MLP），长期短期记忆进行了比较（ LSTM）神经网络，ARIMA和Holt-Winters方法。此公式为具有已知或隐藏变量的时间序列和多元系统提供了一种替代的有效建模和预测范式。