GeoLinear

Boosted piecewise-linear models on cooperative geometry partitions.

---

What is GeoLinear?

GeoLinear discovers cooperative geometry regimes in your data — groups of observations where features interact in similar ways — and fits an interpretable linear model inside each regime. Predictions are accumulated across boosting rounds.

The key insight is that many real-world relationships are piecewise-linear in cooperative geometry. A global linear model averages over regimes and loses the regime-specific signal. GeoLinear finds the regimes automatically (via HVRT) and lets the coefficients vary across them.

Round 1: HVRT partitions X → {cooperative, competitive, mixed} regimes
Ridge fits within each partition on residuals
Round 2: New partitioning on updated residuals → new local Ridge models
...
Final prediction = Σ (learning_rate × stage_predictions) + intercept

Each partition's Ridge coefficients are directly interpretable as local relativities — exactly the quantities actuaries file with regulators.

---

Why it matters for actuaries

Insurance pricing models must be both accurate and interpretable. Regulators require filed relativities; black-box models are inadmissible. The usual compromise — vanilla GLM — leaves accuracy on the table when the true relationship is regime-switching.

GeoLinear bridges this gap:

1. Fit GeoLinear on the training portfolio. Each partition's Ridge coefficients are the relativities for that cooperative geometry segment.
2. File a meta-GLM: fit OLS(X → ŷ_GL) to compress the ensemble into a single interpretable GLM. The compression loss is typically small (R² drop < 2%).
3. Audit trail: every prediction can be traced to a specific partition and its local coefficient vector.

See examples/insurance_pricing_demo.py for a worked actuarial example including relativities tables and the GLM bridge.

---

Installation

pip install geolinear

Requires a C++17 compiler and CMake (automatically satisfied on most systems; the cmake PyPI package is a reliable fallback).

For the examples:

pip install geolinear[examples] # adds xgboost, optuna, matplotlib

---

Quick start

Regression

fromgeolinearimportGeoLinearfromsklearn.datasetsimportfetch_california_housingfromsklearn.model_selectionimporttrain_test_splitfromsklearn.metricsimportr2_scoreX, y=fetch_california_housing(return_X_y=True)
X_tr, X_te, y_tr, y_te=train_test_split(X, y, test_size=0.2, random_state=0)
model=GeoLinear(n_rounds=20, learning_rate=0.1, alpha=1.0)
model.fit(X_tr, y_tr)
print(r2_score(y_te, model.predict(X_te))) # ~0.72 default params

Classification

fromgeolinearimportGeoLinearClassifierfromsklearn.datasetsimportload_breast_cancerfromsklearn.model_selectionimporttrain_test_splitfromsklearn.metricsimportroc_auc_scoreX, y=load_breast_cancer(return_X_y=True)
X_tr, X_te, y_tr, y_te=train_test_split(X, y, test_size=0.2, random_state=0)
clf=GeoLinearClassifier(n_rounds=20, alpha=1.0)
clf.fit(X_tr, y_tr)
print(roc_auc_score(y_te, clf.predict_proba(X_te)[:, 1])) # ~0.993

Pipeline + GridSearchCV

fromsklearn.pipelineimportPipelinefromsklearn.preprocessingimportStandardScalerfromsklearn.model_selectionimportGridSearchCVfromgeolinearimportGeoLinearpipe=Pipeline([("scaler", StandardScaler()), ("gl", GeoLinear())])
grid=GridSearchCV(pipe, {"gl__alpha": [0.1, 1.0, 10.0], "gl__n_rounds": [10, 20]}, cv=5)
grid.fit(X_tr, y_tr)

Optuna HPO

importoptunafromgeolinearimportGeoLinearfromsklearn.model_selectionimportcross_val_scoredefobjective(trial):
model=GeoLinear(
n_rounds=trial.suggest_int("n_rounds", 10, 100),
learning_rate=trial.suggest_float("learning_rate", 0.01, 0.3, log=True),
alpha=trial.suggest_float("alpha", 0.01, 20.0, log=True),
y_weight=trial.suggest_float("y_weight", 0.1, 1.0),
base_learner=trial.suggest_categorical("base_learner", ["ridge", "lasso"]),
hvrt_n_partitions=trial.suggest_int("hvrt_n_partitions", 3, 14),
min_samples_partition=trial.suggest_int("min_samples_partition", 3, 25),
hvrt_model=trial.suggest_categorical("hvrt_model", ["hvrt", "pyramid_hart", "fast_hvrt"]),
use_t_feature=trial.suggest_categorical("use_t_feature", [False, True]),
refit_interval=trial.suggest_categorical("refit_interval", [0, 1, 2]),
)
returncross_val_score(model, X_tr, y_tr, cv=5, scoring="r2").mean()
study=optuna.create_study(direction="maximize")
study.optimize(objective, n_trials=60)

---

Ecosystem: which tool to use?

GeoLinear is part of a family of cooperative-geometry libraries. The right choice depends on what "interpretable" means in your domain and the structure of your data.

Domain	Primary need	Recommended
Insurance pricing	Filed relativities; regulator-auditable linear tariff	GeoLinear
Credit risk / scoring	Scorecard linearity; SR 11-7 model risk compliance	GeoLinear
Utility rate-setting	Filed tariff schedules with linear rate factors	GeoLinear
Healthcare / clinical	XGBoost-level performance with fully deterministic, auditable predictions	GeoXGB
Ecology / environmental	Nonlinear regime detection; richer explanation than SHAP via 100% determinism	GeoXGB
Public policy	Algorithmic accountability without linearity constraints	GeoXGB
Personalised interventions	Per-entity longitudinal data; individual treatment trajectories	AutoITE

Rule of thumb: if your regulator or ethics board requires a linear equation you can file or defend, use GeoLinear. If you need XGBoost-class accuracy with fully deterministic predictions that provide richer interpretability than SHAP, use GeoXGB. If you have repeated observations per individual and want to model how treatment effects evolve over time for each entity, use AutoITE.

---

Benchmark results

80-trial Optuna HPO, GeoLinear (v0.3.0) vs XGBoost. use_t_feature included in GeoLinear's HPO search space so the optimizer can exploit cooperative geometry when it exists.

Regression R² — v0.3.0 (80-trial HPO)

DGP	GL-HPO	use_t	XGB-HPO	Gap
T-regime	0.569	✓	0.296	+0.273 GL wins
3-regime	0.358	✓	0.329	+0.029 GL wins
Friedman1	0.984	—	0.988	−0.003 (tie)
Linear	0.965	—	0.957	+0.008 GL wins
CalHousing	0.584	✓	0.856	−0.271 XGB wins

Score: 3 wins · 1 tie · 1 loss.

use_t_feature is selected by HPO whenever the data has cooperative structure (T-regime, 3-regime). On smooth or axis-aligned DGPs (Friedman1, Linear) HPO correctly opts out. CalHousing remains XGBoost territory: spatial autocorrelation and threshold effects are better captured by axis-aligned splits than by pairwise cooperative geometry.

Classification AUC

Dataset	GL-Ridge (default)	GL-HPO	XGB-HPO
Synthetic	0.940	0.957	0.981
BreastCancer	0.993	0.972	0.993

Default-parameter GL-Ridge matches XGBoost-HPO on BreastCancer (AUC 0.993 each).

---

API reference

GeoLinear (regressor)

Parameter	Type	Default	Description
n_rounds	int	20	Boosting rounds
learning_rate	float	0.1	Shrinkage per round
y_weight	float	0.5	HVRT outcome-blend (0 = geometry-only, 1 = y-driven)
base_learner	str	"ridge"	"ridge", "ols", or "lasso"
alpha	float	1.0	L2 regularisation within each partition
min_samples_partition	int	5	Minimum samples to fit a partition model
hvrt_n_partitions	int|None	None	Target partitions (None = HVRT auto-tune)
hvrt_min_samples_leaf	int|None	None	HVRT min leaf size
hvrt_inner_rounds	int	1	HVRT T-residual inner rounds per stage
partition_inner_rounds	int	1	Base-learner rounds within each partition
refit_interval	int	0	0 = fresh HVRT each round; k>0 = refit every k rounds (faster)
use_t_feature	bool	False	Append per-sample T-statistic (S²−Q) as an extra linear feature
use_coop_weights	bool	False	Scale features by |corr(z_k, S−z_k)|² before linear fit
random_state	int	42	Seed (incremented per round for diverse partitionings)

GeoLinearClassifier accepts the same parameters.

Key methods

model.fit(X, y) # fitmodel.predict(X) # regression predictionsclf.predict_proba(X) # class probabilities, shape (n, 2)model.feature_importances(feature_names) # weighted mean |coef| across partitions/stagesmodel.stages_# list of (None, dict[partition_id, RidgeModel])

---

License

GNU Affero General Public License v3.0 or later. See LICENSE.