Equivalence relations and $L^p$ distances between time series

We introduce a general framework for defining equivalence and measuring distances between time series, and a first concrete method for doing so. We prove the existence of equivalence relations on the space of time series, such that the quotient spaces can be equipped with a metrizable topology. We illustrate algorithmically how to calculate such distances among a collection of time series, and perform clustering analysis based on these distances. We apply these insights to analyse the recent bushfires in NSW, Australia. There, we introduce a new method to analyse time series in a cross-contextual setting.