  Local maxima and minima, or extremal events, in experimental time series can
be used as a coarse summary to characterize data. However, the discrete
sampling in recording experimental measurements suggests uncertainty on the
true timing of extrema during the experiment. This in turn gives uncertainty in
the timing order of extrema within the time series. Motivated by applications
in genomic time series and biological network analysis, we construct a weighted
directed acyclic graph (DAG) called an extremal event DAG using techniques from
persistent homology that is robust to measurement noise. Furthermore, we define
a distance between extremal event DAGs based on the edit distance between
strings. We prove several properties including local stability for the extremal
event DAG distance with respect to pairwise $L_{\infty}$ distances between
functions in the time series data. Lastly, we provide algorithms, publicly free
software, and implementations on extremal event DAG construction and
comparison.
