  This paper proposes a stable volume and a stable volume variant, referred to
as a stable sub-volume, for more reliable data analysis using persistent
homology. In prior research, an optimal cycle and similar ideas have been
proposed to identify the homological structure corresponding to each
birth-death pair in a persistence diagram. While this is helpful for data
analysis using persistent homology, the results are sensitive to noise. In this
paper, stable volumes and stable sub-volumes are proposed to solve this
problem. For a special case, we prove that a stable volume is the robust part
of an optimal volume against noise. We implemented stable volumes and
sub-volumes on HomCloud, a data analysis software package based on persistent
homology, and show examples of stable volumes and sub-volumes.
