hierarchical clustering instrument
| Definition | A clustering that merges the closest pair of clusters at each step, under a linkage that reads the nearest pair, the farthest, or the average, so that merge heights never decrease. Chapter 9. Also linkage, dendrogram. |
|---|---|
| Example | Points at 0, 1, 5, and 6 merge 0 with 1 and 5 with 6 at height 1, then the two pairs at height 4 under single linkage. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id hierarchical-clustering, kind instrument. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation 9.1. |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | lean/DataMiningAsObservation/Hierarchical.lean |
| Reviewed | not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation 9.1.
\[\mathrm{SSE}=\sum_{k=1}^{K}\sum_{i\in\mathcal C_k}\|x_i-c_k\|^{2},\qquad\text{the }P_C=I\text{ distortion summed within clusters}.\]
Conditions
- A clustering that merges the closest pair of clusters at each step under a linkage that reads the nearest pair, the farthest pair, or the average. When the merged pair was the closest at some height, no updated distance falls below it, so the merge heights never decrease, single linkage is never above complete, and average linkage lies between them.
- The dendrogram inherits the distance’s quotient at every level, so the tree is a certificate about the chosen quotient and not about the data.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Chapter 9 section 9.1 of Data Mining as Observation, after TSK chapter 7.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/Hierarchical.lean,
theorems single_ge, complete_ge,
average_ge, single_le_complete,
average_between, at observation-data-mining f3914f0; what
the check covers is stated in the book’s appendix
C.
Used in
Data Mining as Observation chapters 9.
Related
k-means; DBSCAN; Euclidean distance; validity index; quotient.
See also
Book equations stated beside the entry’s terms, not defining it: 3.1.
Sources-table rows that share a record with the entry without naming it: chapter 9 section 9.1, chapter 10 section 10.1.
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.