Despite their simplicity, linear models perform well at time series forecasting, even when pitted against deeper and more expensive models. A number of variations to the linear model have been proposed, often including some form of feature normalisation that improves model generalisation. In this paper we analyse the sets of functions expressible using these linear model architectures. In so doing we show that several popular variants of linear models for time series forecasting are equivalent and functionally indistinguishable from standard, unconstrained linear regression. We characterise the model classes for each linear variant. We demonstrate that each model can be reinterpreted as unconstrained linear regression over a suitably augmented feature set, and therefore admit closed-form solutions when using a mean-squared loss function. We provide experimental evidence that the models under inspection learn nearly identical solutions, and finally demonstrate that the simpler closed form solutions are superior forecasters across 72% of test settings.

翻译：尽管线性模型结构简单，但在时间序列预测任务中，即使面对更复杂、计算成本更高的深度模型，其表现依然出色。研究界已提出多种线性模型的变体，通常包含某些形式的特征归一化以提升模型泛化能力。本文分析了这些线性模型架构可表达的函数集合，从而证明时间序列预测中几种流行的线性模型变体在功能上与标准无约束线性回归等价且无法区分。我们刻画了每种线性变体的模型类别，论证了每个模型均可被重新解释为在适当增广特征集上的无约束线性回归，因此在采用均方损失函数时存在闭式解。通过实验证据表明，被检验的模型学习到的解几乎相同，并最终证明更简洁的闭式解在72%的测试场景中具有更优的预测性能。