imputation instrument

Definition	Writing a value into a missing cell that the consumer will read. Mean imputation leaves the column mean unchanged and shrinks its variance. Chapter 2. Also imput.
Example	Filling three missing cells of a column with mean 5 leaves the mean at 5 and shrinks the variance.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `imputation`, kind instrument.
Status	no ledger row names this entry. Corrections: none recorded.
Defining equation	Book equation 2.1.
Assumptions and scope	Writing a value into a missing cell that the consumer will read. Mean imputation leaves the column mean unchanged, the filled cells add nothing to the sum of squared deviations, and the variance shrinks because the same sum of squares is spread over more rows. An imputation is a claim about the mechanism that produced the gap, and it is fit inside the fold and stated in the preregistration, because an imputation fit on all rows leaks the test fold into the training fold.
Prior art	none recorded
Evidence	`lean/DataMiningAsObservation/Imputation.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

A value written into a missing cell that the consumer will read.

Equation

Book equation 2.1.

\[P_{C_2\circ C_1}(x)=J_1(x)^{\top}\,P_{C_2}\big(C_1(x)\big)\,J_1(x),\qquad \operatorname{rank}P_{C_2\circ C_1}(x)\le\operatorname{rank}P_{C_2}\big(C_1(x)\big)\quad\text{at each row } x.\]

Conditions

Writing a value into a missing cell that the consumer will read. Mean imputation leaves the column mean unchanged, the filled cells add nothing to the sum of squared deviations, and the variance shrinks because the same sum of squares is spread over more rows.
An imputation is a claim about the mechanism that produced the gap, and it is fit inside the fold and stated in the preregistration, because an imputation fit on all rows leaks the test fold into the training fold.

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

Ledger

none

First stated

Chapter 2 section 2.3 of Data Mining as Observation, with the missing-data rule as a required preregistration field in observation-theory-campaigns/experiments/PREREG-TEMPLATE.md:47-52.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 2 section 2.3	MCAR, MAR, MNAR and the remedies, imputation before splitting	TSK 2e section 2.2; instructor working documents, not public [@bond2026course], `ECE_514-01_FA26_session-outlines.md:64-72`

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/Imputation.lean, theorems mean_imputed, sum_sq_imputed, variance_imputed_le, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation chapters 1, 2, 8.

standardization; read operator; leakage; preregistration.

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.

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