Find a better method to represent a centroid

[axiomcura]

Maybe I am understanding your goals wrong or the pycytominer aggregation wrong, so please let me know if so. You said here in documentation that the data was aggregated by component-wise median, which I assume just involves median-ing every feature separately (or is this a typo and you meant geometric median?). I have read about component-wise median operations generalizing poorly from 1d to higher dimensional data and can produce unrealistic centroid profiles.

In the extreme case if we have very anti-correlated or correlated features the median profile would be far from anything that is typical or realistic, unless this type of correlation structure has been previously dealt with:

| Sample | Feature X | Feature Y |
|--------|-----------|-----------|
| S1     |    1      |    10     |
| S2     |    2      |     9     |
| S3     |    9      |     2     |
| S4     |   10      |     1     |
| Median    |   5.5     |   5.5     |

Originally posted by @wli51 in #60 (comment)

[axiomcura]

At the moment, it is unclear whether using cluster centroids is the right way to demonstrate improved cluster resolution. The underlying goal is to show that both consensus and replicate-level aggregate profiles fail to fully capture the underlying cellular heterogeneity present in the data.

This limitation is particularly pronounced in the CFReT cells. Failing samples inherently contain a mixture of activated and deactivated cardiac fibroblasts, resulting in a wide range of cellular states within the same population. As a result, aggregate representations tend to blur these distinctions.

One open question is whether this heterogeneity can be more effectively resolved by restricting the analysis to on-morphological features only, as described in #70. If so, we could re-generate the Sankey diagram to test whether a subset of cells within the failing population, specifically those with healthier phenotypes, map to clusters that are predominantly healthy.

Demonstrating this would further support the argument that single-cell–level information provides biologically meaningful resolution that aggregate approaches usually miss.