> For the complete documentation index, see [llms.txt](https://docs.algenta.ai/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.algenta.ai/machine-learning/ml-ensembles.md).

# Model ensembling

Stacking and weighted voting ensembles over arbitrary base-model predictions.

`ensembles` combines predictions from multiple models — zero Python imports — replacing sklearn.ensemble's `VotingClassifier`, `VotingRegressor`, and the `StackingRegressor` ridge meta-learner. It operates on flat prediction matrices so any model type can feed it, and complements `voting`, which is social choice theory and unrelated to model ensembling.

## The modules

| Module      | What it does                                                                                                                                                                                                                      |
| ----------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| `ensembles` | Combines predictions from multiple models — weighted/majority voting, weighted regression blending, and closed-form ridge stacking — replacing sklearn.ensemble's `VotingClassifier`, `VotingRegressor`, and `StackingRegressor`. |
| `voting`    | Social choice theory: plurality, Borda, and Condorcet winners, D'Hondt/Sainte-Laguë and Droop/Hare apportionment, and the Banzhaf power index — unrelated to model ensembling despite the name.                                   |

## Functions

| Function                                                                | What it does                                                                                                                            |
| ----------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------- |
| `voting_classify(pred_labels, n_models, n_samples, n_classes, weights)` | Majority / weighted-majority vote over per-model class labels; exact ties break toward the lowest class index.                          |
| `blend_regression(preds, n_models, n_samples, weights)`                 | Weight-normalized average of regression predictions per sample (`[1, 3]` behaves as `[0.25, 0.75]`).                                    |
| `stack_fit_ridge(oof_preds, n_samples, n_models, y, l2)`                | Closed-form ridge meta-weights from out-of-fold predictions: solves the normal equations `(XᵀX + l2·I) w = Xᵀy` exactly (no intercept). |
| `stack_predict(preds, n_samples, n_models, weights)`                    | Applies meta-weights raw — not normalized, since ridge weights already encode scale: `out[s] = Σ_m weights[m] · preds[s, m]`.           |

Layout: `voting_classify` / `blend_regression` take model-major input (entry \[m, s] at `m * n_samples + s`); `stack_fit_ridge` / `stack_predict` take sample-major input (entry \[s, m] at `s * n_models + m`).

## Usage guidance

* **Fully deterministic.** No RNG anywhere: identical inputs give identical outputs. Voting ties break to the lowest class index (test: two models predicting {1, 0} with uniform weights → 0), and labels outside `[0, n_classes)` contribute nothing to the vote.
* **Weight fallbacks.** A weights list of the wrong length (e.g. empty) means uniform 1.0; `blend_regression` also falls back to the uniform mean when `|sum(weights)| < 1e-12` (e.g. `[1, −1]`).
* **Singularity is signaled, not hidden.** The internal Gaussian-elimination solver (partial pivoting) returns the empty list when a pivot magnitude falls below 1e-12: duplicated models with `l2 = 0` yield empty; passing `l2 > 0` regularizes, and the duplicates then split the weight symmetrically, summing to \~1. Inconsistent dimensions also return empty across all four functions.

## Verification

Known-value tests pin every function to hand-computed results at 1e-12 tolerance: a 3-model × 4-sample majority vote returns `[0, 1, 2, 2]`, and weights `[0.2, 0.2, 0.6]` flip sample 0 to give `[1, 1, 2, 2]`; blending predictions `[1, 2]` / `[3, 6]` with weights `[0.25, 0.75]` gives `[2.5, 5.0]`, identically for unnormalized `[1, 3]`. For the ridge meta-learner, X = \[1, 2], y = \[2, 4] gives w = 10/6 at `l2 = 1` and exactly 2 at `l2 = 0` (exact least squares); when model 1 equals y and model 2 is a constant, `stack_fit_ridge` recovers weights ≈ \[1, 0] and the fit + predict roundtrip reproduces y within 0.05.

**Next:** promote what you trained with [statistical model comparison](/machine-learning/ml-model-comparison.md), or browse the [engine catalog](/engine-runtime/catalog.md).


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