  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.
