explained variance concept

Definition	The fraction of total variance retained by a set of principal components, the identity reader's criterion for a reduction. Chapter 4. Also explained ratio.
Example	Keeping 2 of the eigenvalues 4, 3, 2, and 1 explains 7 of 10, or 70 percent.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `explained-variance`, kind concept.
Status	no ledger row names this entry. Corrections: none recorded.
Defining equation	Book equation 4.1.
Assumptions and scope	The fraction of total variance the first k components carry. It lies in the unit interval, cannot fall as k grows, and reaches one at the full dimension. With unit sensitivities it is the fraction a consumer retains, which is why it is the identity reader’s criterion. A consumer that reads other directions retains a different fraction. With variances 9 and 1 and a consumer reading only the second direction, keeping the first component explains nine tenths of the variance and none of what the consumer reads, which is the flip in its smallest form.
Prior art	none recorded
Evidence	`gtc-prototype/docs/SPECTRUM_FINDINGS.md:10-18`, `lean/DataMiningAsObservation/ExplainedVariance.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

The fraction of the spectrum the kept components carry.

Equation

Book equation 4.1.

\[d_O=\operatorname{tr}(P_C\,M_\delta)\qquad\text{against}\qquad \operatorname{tr}M_\delta=d_O\big|_{P_C=I}.\]

Conditions

The fraction of total variance the first k components carry. It lies in the unit interval, cannot fall as k grows, and reaches one at the full dimension. With unit sensitivities it is the fraction a consumer retains, which is why it is the identity reader’s criterion.
A consumer that reads other directions retains a different fraction. With variances 9 and 1 and a consumer reading only the second direction, keeping the first component explains nine tenths of the variance and none of what the consumer reads, which is the flip in its smallest form.

Conditions are curated in entries.toml rather than read from a record.

Ledger

none

First stated

TSK appendix B on principal component analysis, as chapter 4 section 4.1 of Data Mining as Observation reads it, the identity reader’s criterion for a reduction, with the program’s spectrum record in gtc-prototype/docs/SPECTRUM_FINDINGS.md:10-18.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 9 section 9.4	explained ratios, effective rank 5.19 of 8, convergence with a second method, property of the representation not the space	`gtc-prototype/docs/SPECTRUM_FINDINGS.md:10-18`

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/ExplainedVariance.lean, theorems explained_mem_unit, explained_mono, explained_full, retained_identity, retained_example, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation chapters 4, 8.

identity reader; the flip; effective rank; water-filling.

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.

← expected calibration errorF1 →