Variance attribution (OLS + Shapley/LMG) — v1.0.0

What question it answers

Descriptive coverage says "la consommation varie de X % entre régions". This method

answers the harder question: **"de cette variation, quelle part chaque facteur

explique-t-il ?"* — e.g. "35 % s'explique par le climat, 8 % par la part résidentielle,

57 % reste inexpliqué (bâti, chauffage, revenus non mesurés)."*

Fit the full OLS model y = β₀ + Σ βⱼ·Xⱼ + ε over the cross-section.

Joint R² = share of Var(y) the factors explain together.

Split that R² into a per-factor share using the Shapley / LMG value: average the

incremental R² of each factor over all orderings of the regressors. This is the

order-independent, fair attribution when factors are correlated — a plain "add factors

one by one" split would depend on the (arbitrary) order. The shares plus

residual_share = 1 − R² sum to 1.

Report each coefficient's t-statistic (β/se) so a reader knows which factors are

statistically distinguishable from zero (|t| > 1.96 ≈ 95 %, large-n normal).

Input: y = conso_moyenne_kwh per commune; factors dju (climate), part_residentielle.

Output: `{ r_squared: 0.41, residual_share: 0.59, components: [

{ factor: "dju", share: 0.35, coefficient: …, t_stat: 6.2, significant: true },

{ factor: "part_residentielle", share: 0.06, coefficient: …, t_stat: 1.1, significant: false } ] }`.

Reading: climate explains ~35 % of the inter-commune spread and is highly significant; the

residential share adds little and isn't distinguishable from zero; 59 % is unexplained —

the missing confounders (building age, heating type, income) are named, not hidden.

This is an ESTIMATE, never a computed fact. OLS shares measure association

controlling for the supplied factors only. An omitted confounder inflates the share of

whatever it correlates with — so the share is an upper bound on a factor's true import when

confounders are missing.

The residual share + the missing-confounder list are the uncertainty. Always surface

both. A high residual means "we mostly don't know" — say so.

No causal claim from a regulated relationship. The TRV is set by the CRE; regressing it

on wholesale prices is meaningless (no exogenous variation) — such pairs are not whitelisted.

No per-share confidence interval. A rigorous CI needs a bootstrap (a PRNG), which would

break determinism/replay. We give deterministic t-based significance instead.

Closed-form OLS (Gauss–Jordan inverse) + exhaustive subset enumeration for the Shapley shares

(k ≤ 6 → ≤ 64 subsets). No Math.random, no Date, no I/O. Same inputs → byte-identical

output → replayable from (method_id, version, input_hash).