  Computational topologists recently developed a method, called persistent
homology to analyze data presented in terms of similarity or dissimilarity.
Indeed, persistent homology studies the evolution of topological features in
terms of a single index, and is able to capture higher order features beyond
the usual clustering techniques. There are three descriptive statistics of
persistent homology, namely barcode, persistence diagram and more recently,
persistence landscape. Persistence landscape is useful for statistical
inference as it belongs to a space of $p-$integrable functions, a separable
Banach space. We apply tools in both computational topology and statistics to
DNA sequences taken from Clostridioides difficile infected patients treated
with an experimental fecal microbiota transplantation. Our statistical and
topological data analysis are able to detect interesting patterns among
patients and donors. It also provides visualization of DNA sequences in the
form of clusters and loops.
