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

Equivalence relations and $L^p$ distances between time series with application to the Black Summer Australian bushfires