This paper introduces a new framework of algebraic equivalence relations between time series and new distance metrics between them, then applies these to investigate the Australian ``Black Summer'' bushfire season of 2019-2020. First, we introduce a general framework for defining equivalence between time series, heuristically intended to be equivalent if they differ only up to noise. Our first specific implementation is based on using change point algorithms and comparing statistical quantities such as mean or variance in stationary segments. We thus derive the existence of such equivalence relations on the space of time series, such that the quotient spaces can be equipped with a metrizable topology. Next, we illustrate specifically how to define and compute such distances among a collection of time series and perform clustering and additional analysis thereon. Then, we apply these insights to analyze air quality data across New South Wales, Australia, during the 2019-2020 bushfires. There, we investigate structural similarity with respect to this data and identify locations that were impacted anonymously by the fires relative to their location. This may have implications regarding the appropriate management of resources to avoid gaps in the defense against future fires.

翻译：本文提出了一种新的时间序列间代数等价关系框架及距离度量方法，并将其应用于研究2019-2020年澳大利亚"黑色夏季"丛林大火季。首先，我们建立了一个定义时间序列等价的通用框架，其启发式意图是：若两个序列仅存在噪声差异，则视为等价。具体实现的第一种方案基于变点检测算法，比较平稳分段内的统计量（如均值或方差）。由此，我们在时间序列空间上推导出此类等价关系的存在性，使得商空间可赋予可度量化拓扑。其次，我们具体阐述了如何定义和计算一组时间序列间的距离，并据此进行聚类及附加分析。最后，我们将这些方法应用于分析2019-2020年丛林大火期间澳大利亚新南威尔士州的空气质量数据。基于该数据，我们研究了结构相似性，并识别出受火灾影响程度与其地理位置不匹配的区域。该结果对合理调配资源以避免未来火灾防御中的部署漏洞具有启示意义。