sum of squared errors concept
| Definition | The sum over rows of the squared distance to the row's cluster centre, the validity index k-means minimizes. Chapter 0 section 0.15. |
|---|---|
| Example | Points 1, 2, and 6 with centres 1.5 and 6 have sum of squared errors 0.25 + 0.25 + 0 = 0.5. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id sum-of-squared-errors, kind concept. |
| 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/KMeans.lean, lean/DataMiningAsObservation/ReconstructionError.lean |
| Reviewed | not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation 9.1.
\[\mathrm{SSE}=\sum_{k=1}^{K}\sum_{i\in\mathcal C_k}\|x_i-c_k\|^{2},\qquad\text{the }P_C=I\text{ distortion summed within clusters}.\]
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
- The sum over rows of the squared distance to the row’s cluster centre, the validity index k-means minimizes. The error about any centre is the error about the mean plus the count times the squared distance between the two, so the mean minimizes it, and it is the identity reader’s distortion summed within clusters.
- Two clusterings of the same rows with the same sum of squared errors can be read oppositely by a consumer that weighs directions unequally.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Chapter 0 section 0.15 and chapter 9 section 9.1 of Data Mining as Observation.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/KMeans.lean,
theorems sse_decomposition, mean_minimizes,
assign_nearest, at observation-data-mining f3914f0; what
the check covers is stated in the book’s appendix
C.
lean/DataMiningAsObservation/ReconstructionError.lean,
theorems recon_nonneg, recon_eq_zero_iff,
recon_sum, recon_proj, 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.
Related
k-means; validity index; reconstruction error; identity reader; silhouette.
See also
Book equations stated beside the entry’s terms, not defining it: 4.2.
Ledger rows that cite the entry’s records without naming it: GO-3.
Sources-table rows that share a record with the entry without naming it: chapter 9 section 9.4, 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.