> 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-pls.md).

# Partial least squares regression (PLS1)

NIPALS PLS1 regression for collinear and wide (p > n) feature matrices, sklearn-matched.

`pls` fits partial least squares regression for a single response vector with the NIPALS algorithm — replacing `sklearn.cross_decomposition.PLSRegression` for one `y`, with zero Python imports. PLS projects X onto latent components of maximal covariance with y, staying stable on collinear or wide feature blocks where the normal equations behind closed-form OLS/ridge degrade. It complements `regression.mojo` (closed-form OLS/ridge) and `linear_models.mojo` (iterative logistic / elastic-net).

## The modules

| Module | What it does                                                                                                                                                          |
| ------ | --------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| `pls`  | NIPALS PLS1 regression for a single response vector — fit, predict, and R², stable on collinear or wide (p > n) feature matrices where closed-form OLS/ridge degrade. |

## Functions

| Function                                                     | What it does                                                                                                                                                                                |
| ------------------------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| `pls_fit(x, n_rows, n_cols, y, n_components, max_iter, tol)` | NIPALS PLS1 fit on a flat row-major X. Per component: weights `w = E'f/‖E'f‖`, scores `t = Ew`, loadings, y-loading, then deflation; coefficients `B = W (P'W)⁻¹ q`. Returns a `PLSResult`. |
| `pls_predict(model, x_row)`                                  | Predict y for one raw feature row; the model's centering/scaling state maps back to original units. Wrong-length rows and empty models return `y_mean`.                                     |
| `pls_predict_rows(model, x, n_rows)`                         | Predict every row of a flat row-major raw feature matrix; empty list on shape mismatch.                                                                                                     |
| `pls_r2(model, x, n_rows, y)`                                | R² = 1 − SS\_res / SS\_tot of model predictions on a raw (X, y) set.                                                                                                                        |

`PLSResult` carries weights W, loadings P, scores T (component-major), y-loadings q, coefficients B (standardized space), the centering/scaling state, per-component explained-variance fractions for X and y, and per-component NIPALS iteration counts.

## Guarantees

* **Fully deterministic** — no RNG anywhere in the fit.
* X columns and y are centered and scaled internally (sample std, ddof=1; zero-variance columns get scale 1.0); predictions are in original units.
* The component count is clamped to `min(n_cols, n_rows − 1)` and extraction stops early once no X/y covariance remains — you may get fewer components than requested, never garbage ones.
* Graceful degradation: invalid shapes return an empty model; zero components (constant y, n = 1, `n_components = 0`) leave coefficients at zero so prediction falls back to `y_mean`.

## Verification

Known-value tests pin the algebra, not just behavior:

* A hand-derived 2×2 fixture matches NIPALS exactly to 1e-9: weights and loadings `(1/√2, 1/√2)`, scores `(−1, 1)`, y-loading `1/√2`, coefs `(0.5, 0.5)`, explained variance 1 for both X and y.
* A rank-2 collinear fixture (`x3 = x1 − 2·x2`, y noiseless in the column space) is recovered with exactly 2 components: R² > 0.999 and maximum prediction error below 1e-6 — the regime where plain normal equations degrade.
* Structural invariants: score vectors mutually orthogonal (normalized dot < 1e-8), weight vectors unit-norm to 1e-9, explained-variance-y sums to 1 on fully explained data.
* Edge cases proven: constant y and constant (zero-variance) columns, n = 1, shape mismatches, and over-requested component counts all degrade to the documented fallbacks.

**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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