  Persistent homology is a way of determining the topological properties of a
data set. It is well known that each persistence module admits the structure of
a representation of a finite totally ordered set. In previous work, the authors
proved an analogue of the isometry theorem of Bauer and Lesnick for
representations of a certain class of finite posets. The isometry was between
the interleaving metric of Bubenik, de Silva and Scott and a bottleneck metric
which incorporated algebraic information. The key step in both isometry
theorems was proving a matching theorem, that an interleaving gives rise to a
matching of the same height. In this paper we continue this work, restricting
to those posets which arise from data while making more general the choice of
metrics. We first show that while an interleaving always produces a matching,
for an arbitrary choice of weights it will not produce one of the same height.
We then show that although the matching theorem fails in this sense, one
obtains a "shifted" matching (of the correct height) from an interleaving by
enlarging the category. We then prove an isometry theorem on this extended
category. As an application, we make precise the way in which representations
of finite partially ordered sets approximate persistence modules. Specifically,
given two finite point clouds of data, we associate a generalized sequence
(net) of algebras over which the persistence modules for both data sets can be
compared. We recover the classical interleaving distance uniformly by taking
limits.
