HDBSCAN strengths

This page explains strengths, the per-row number that HDBSCAN reports for one fit, and why it is not a probability.

The question it answers

How close to the core of its group does one fit place a row?

HDBSCAN gives every row of a single fit a number from 0 to 1. A row in the dense core of a group has a strength near 1. A row at the edge of a group has a lower one. The number describes that one fit, so it is a within-fit strength. It is not a posterior, the probability of membership given the data and a stated model.

A picture

Picture a group as a dense cloud of points on the embedding.

core of the cloud   : strength near 1.0
middle of the cloud : strength between the two
edge of the cloud   : strength near 0

The recorded runs quoted on this page ship in the repository under notebooks/runs/digits and notebooks/runs/apogee. Notebooks 01 and 02 write them, and starfold.load reads them. In the recorded run on the scikit-learn digits dataset, most grouped rows sit in cores, so the median strength is 1.00. A table whose groups are diffuse has lower strengths.

Range and reference values

  • Range: 0 to 1.

  • Perfect result: 1, the row sits at the core of its group in the fit.

  • Pure chance: not applicable. Nothing in the number compares the fit with a structureless table.

The hdbscan package sets the strength of a noise row to 0 (see the upstream docs linked below).

How to read a value

Starfold sets no cut on strengths. The bands are a rule of thumb from this project, not from a published source.

Strength

Reading

Decision

Near 1

Core row of its group in this fit

Use it to pick typical members of the group

Between 0 and 1

Edge row of its group in this fit

Expect its membership to move with small changes

0

Noise in this fit

Treat it as unassigned

For a claim that a row is reproducibly in a group, use Support and own support instead. Support is measured over refits. A strength is measured once.

What misleads

  • A strength of 1 is not a probability of 1. It is a position in one fit, which depends on the embedding and on the settings.

  • A high strength does not imply a recurrent group. In the recorded digits run the median strength over all grouped rows is 1.00, yet five of the ten groups recur at 0.73 or less (see Group recurrence).

  • Strengths are not comparable across fits. Two fits with different settings can give the same row different strengths.

  • A strength is not a measurement uncertainty. Nothing in it reflects the errors of your columns.

Where Starfold prints it

The attribute is result.strengths, one float32 value per kept row, taken from the probabilities_ attribute of the HDBSCAN fit of the selected setting on all rows. The name in the code follows HDBSCAN, and the values are within-fit strengths, not posteriors. The notebook notebooks/01_digits.ipynb printed the recorded line below.

grouped = labels >= 0
print(f"strengths of grouped rows: median {np.median(result.strengths[grouped]):.2f}")

strengths of grouped rows: median 1.00
Formula

Starfold does not compute the strengths. HDBSCAN builds a hierarchy of clusters over the mutual-reachability graph. In the hdbscan package, used on the CPU backend, the strength of a row in a selected cluster is the row’s \(\lambda\) value divided by the largest \(\lambda\) in that cluster. That gives a number between 0 and 1. The \(\lambda\) value is the inverse of the distance scale at which the row leaves the cluster. See Campello, Moulavi and Sander (2013) for the hierarchy and the hdbscan documentation for the exact membership rule.

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