Equivalence relations and $L^p$ distances between time series with application to the Black Summer Australian bushfires
This work addresses the problem of comparing and clustering time series for domain-specific applications like environmental monitoring, though it appears incremental as it builds on existing change point and statistical methods.
The paper introduces a framework of algebraic equivalence relations and distance metrics for time series to analyze structural similarity, applying it to air quality data from the 2019-2020 Australian bushfires to identify anomalously impacted locations.
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.