silhouette concept
| Definition | For a row, the difference between its mean distance to the nearest other cluster and its mean distance to its own cluster, divided by the larger of the two. Equation 0.30. Also validity ind. |
|---|---|
| Example | A point with mean distance 1 to its own cluster and 4.5 to the nearest other has silhouette 3.5 over 4.5, or 0.778. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id silhouette, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation 0.30. |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | lean/DataMiningAsObservation/Silhouette.lean |
| Reviewed | not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation 0.30.
\[\mathrm{SSE}=\sum_{k}\sum_{i\in\mathcal C_k}\|x_i-c_k\|^{2},\qquad s_i=\frac{b_i-a_i}{\max(a_i,b_i)}.\]
Conditions
- For a row, the difference between its mean distance to the nearest other cluster and its mean distance to its own cluster, over the larger of the two. It lies between minus one and one, is positive exactly when the row is closer to its own cluster, and is unchanged when every distance is scaled by the same factor.
- It is computed under the identity reader on the distances it is given, so it validates a clustering for that reader and not for a consumer that reads other directions, which is why chapter 9 asks the recognizer to name the manifold instead.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Rousseeuw, silhouettes, 1987, as TSK chapter 7 presents it and chapter 0 section 0.15 states it, and chapter 9 of Data Mining as Observation, where it is one validity index among those the recognizer replaces.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/Silhouette.lean,
theorems silhouette_mem, silhouette_pos_iff,
silhouette_scale, silhouette_self, at
observation-data-mining f3914f0; what the check covers is stated in the
book’s appendix
C.
Used in
Data Mining as Observation chapters 0, 9, 10.
Related
recognizer; vacuity threshold; identity reader; quotient.
See also
Book equations stated beside the entry’s terms, not defining it: 9.1.
Status
Generated 2026-09-10 by encyclopedia/generate.py; book
at observation-data-mining f3914f0; the commit of every record is listed
in the encyclopedia’s provenance.