Fraction of variance unexplained¶
A regression-fit statistic equal to the proportion of dependent-variable variance left unexplained by the model's predictions.
Core Idea¶
Fraction of variance unexplained is a regression-fit statistic equal to the proportion of dependent-variable variance left unexplained by the model's predictions.
The fraction of variance unexplained is the residual variation divided by total variation for the same response and evaluation sample. Under ordinary least squares with an intercept and standard sum-of-squares decomposition, FVU=SSE/SST=1−R². Outside those conditions, the equality and even the baseline denominator require explicit definition.
Its operative boundary is not supplied by the name alone. Preserve this identity: A regression-fit statistic equal to the proportion of dependent-variable variance left unexplained by the model's predictions. Validity boundary: The numerator and denominator must use compatible residual and total variance definitions for the same regressand; generic prediction error is insufficient.
Scope of Application¶
The abstraction recurs literally within regression and prediction settings where residual error is compared with a same-sample mean baseline. The following habitats preserve the same recognition machinery; they are not invitations to extend the name metaphorically.
- Ordinary least squares. SSE/SST complements the standard R² with an intercept.
- Out-of-sample evaluation. FVU compares predictions to the held-out mean baseline and can exceed one.
- Weighted regression. both residual and total sums must use the same weights.
- Signal modeling. unmodeled signal energy is normalized by total centered energy.
- Model comparison. lower FVU indicates better squared-error fit on a common response sample.
Clarity¶
State centering, weights, degrees-of-freedom convention, evaluation sample, and treatment of missing values. Never combine training residual variance with test total variance. Interpret 'unexplained' predictively; it includes noise, misspecification, and sampling error and is not proof of absent causes.
A practical identification audit begins with the typed roles rather than the title: establish the response observations, verify the model predictions, then test the remaining conditions and exclusions.
Manages Complexity¶
The ratio makes squared prediction error scale-free relative to a simple mean benchmark. Sum-of-squares decomposition localizes what a fitted linear model captures and what remains, while boundary cases warn when the model underperforms baseline.
The compression remains accountable because each simplification has a named failure condition. Disagreement can be localized to a missing role, an invalid assumption, an ambiguous measurement, or a neighboring abstraction instead of being hidden inside an unanalyzed label.
Abstract Reasoning¶
R1. Fix one response vector, prediction vector, weights, and evaluation sample. R2. Compute residuals and their compatible sum or variance. R3. Compute total variation about the explicitly chosen baseline with the same weights. R4. Take the ratio and establish whether the OLS decomposition licenses 1−R². R5. Report uncertainty and avoid causal language unless supported by a separate design.
Knowledge Transfer¶
The statistic transfers literally across squared-error regression evaluations with compatible numerator and denominator. Measurement and decomposition are parents; generic model error or unexplained narrative variation is not FVU.
The transfer boundary is explicit: DOMAIN-SPECIFIC PASS / PRIME FAIL: The statistic is computed across regression models and datasets by comparing residual variation with total variation. Literal recognition retains the specialist vocabulary and validity conditions of regression analysis; outside that setting only broader parent operations transfer.
Relationships to Other Abstractions¶
Current abstraction Fraction of variance unexplained Domain-specific
Parents (1) — more general patterns this builds on
-
Fraction of variance unexplained is a kind of Measurement Prime
Measurement (
prime:measurement).
Hierarchy path (1) — routes to 1 parentless root
- Fraction of variance unexplained → Measurement
Neighborhood in Abstraction Space¶
Fraction of variance unexplained sits in a sparse region of the domain-specific corpus (71st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Statistical Adjustment & Estimation Effects (14 abstractions)
Nearest neighbors
- Variance function — 0.86
- Regression — 0.85
- Nonlinear Least Squares — 0.85
- Kriging — 0.84
- Dependent and independent variables — 0.84
Computed from structural-signature embeddings · 2026-09-08