  We present a new technique to identify associations of HI emission in the
Magellanic Stream (MS) and ultraviolet (UV) absorbers from 92 QSO sight lines
near the MS. We quantify the level of associations of individual HI elements to
the main HI body of the Stream using Wasserstein distance-based models, and
derive characteristic spatial and kinematic distances of the HI emission in the
MS. With the emission-based model, we further develop a comparison metric,
which identifies the dominant associations of individual UV absorbers with
respective to the MS and nearby galaxies. For ionized gas associated with the
MS probed by CII, CIV, SiII, SiIII, SiIV, we find that the ion column densities
are generally $\sim$0.5 dex higher than those that are not associated, and that
the gas is more ionized toward the tail of the MS as indicated by the spatial
trend of the CII/CIV ratios. For nearby galaxies, we identify potential new
absorbers associated with the CGM of M33 and NGC300, and affirm the
associations of absorbers with IC1613 and WLM. For M31, we find the previously
identified gradient in column densities as a function of impact parameter, and
that absorbers with higher column densities beyond M31's virial radius are more
likely to be associated with the MS. Our analysis of absorbers associated with
the Magellanic Clouds reveals the presence of continuous and blended diffuse
ionized gas between the Stream and the Clouds. Our technique can be applied to
future applications of identifying associations within physically complex
gaseous structures.
