# `pcr` — Principal Components Regression
_Group_: **Core PLS** · _Registry tolerance_: `1e-06`
## Description
Principal Components Regression
From the `pls4all.sklearn.PCR` docstring:
> Principal Components Regression — fits a least-squares regression
on the SVD of X.
> **Registry note** — PCR via SVD on X then linear regression; references are sklearn Pipeline(PCA(svd_solver='full') + LinearRegression) and R `pls::pcr`.
### Parameters
| Name | Type | Default | Notes |
|------|------|---------|-------|
| `n_components` | `int` | `2` | Number of latent components extracted (k). |
| `center_x` | `bool` | `True` | Subtract the column mean of X before fitting. |
| `scale_x` | `bool` | `True` | Standardize X columns to unit variance before fitting. |
| `center_y` | `bool` | `True` | Subtract the column mean of y before fitting. |
| `scale_y` | `bool` | `False` | Standardize y columns to unit variance before fitting. |
| `tol` | `float` | `1e-06` | Convergence tolerance for iterative solvers (NIPALS / power-iteration). |
| `max_iter` | `int` | `500` | Maximum iterations for iterative solvers. |
| `store_scores` | `bool` | `False` | If True, keep the latent score matrix (`x_scores_`) after fit. |
## Explanations
### Bibliographic source
Massy, W. F. (1965). *Principal Components Regression in Exploratory Statistical Research*. JASA 60(309), 234–256.
### Mathematical principle
PCR sidesteps the multicollinearity of $\mathbf{X}$ by regressing on its orthogonal principal-component scores rather than on the raw columns. The factorisation $\mathbf{X} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^{\top}$ (SVD) yields scores $\mathbf{T}_k = \mathbf{U}_k\boldsymbol{\Sigma}_k$ for the top $k$ components, and the regression $\mathbf{Y} = \mathbf{T}_k\mathbf{Q}_k + \mathbf{E}$ is fit by ordinary least squares.
Unlike PLS, PCR is **unsupervised in its dimensionality reduction**: the first $k$ directions maximise the variance of $\mathbf{X}$ regardless of how relevant they are to $\mathbf{Y}$. This makes PCR a useful baseline for diagnosing whether the predictive directions in a calibration set really do coincide with the high-variance directions (in which case PCR ≈ PLS) or not (in which case PLS is strictly preferable at the same $k$).
Coefficients in the original feature scale are recovered as $\mathbf{B} = \mathbf{V}_k \boldsymbol{\Sigma}_k^{-1} \mathbf{T}_k^{\top}\mathbf{Y}$. Total cost is dominated by the partial SVD: $O(np\min(n,p))$ for a full decomposition, or $O(npk)$ with a truncated method (Lanczos, randomised SVD).
### Implementation
`Algorithm.PCR` + `Solver.SVD` in libn4m. Reference implementations are scikit-learn's `Pipeline(PCA(n_components=k), LinearRegression())` and R `pls::pcr`.
MATLAB header (`bindings/matlab/+pls4all/PcrRegression.m`):
```text
pls4all.PcrRegression — Principal Component Regression model.
Example:
mdl = pls4all.PcrRegression(X, y, 5);
yhat = predict(mdl, Xnew);
```
### Usage
Every pls4all binding tab dispatches into the same C kernel; the external libraries listed at the bottom of the page are the parity references registered in `benchmarks.parity_timing.registry`. Switch tabs to read the same fit in your language. The R package now ships drop-in-compatible facades for the CRAN `pls` package (`plsr`, `pcr`, `mvr`) and for the `mdatools::pls(x, y, ...)` matrix idiom — those tabs appear only on the methods that have a meaningful equivalence.
**pls4all bindings**
::::{tab-set}
:class: pls4all-bindings
:::{tab-item} C ABI · libn4m
:sync: c
:class-label: lang-c
```c
/* C ABI — libn4m direct MethodResult path */
n4m_context_t* ctx = n4m_context_create();
n4m_config_t* cfg = n4m_config_create();
n4m_config_set_n_components(cfg, 4);
n4m_method_result_t* res = NULL;
n4m_pcr_fit(ctx, cfg, &x_view, &y_view, &res);
/* res contains coefficients, predictions, x_mean/x_scale, y_mean/y_scale,
* weights_w, loadings_p, rotations_r, rmse and n_components. */
n4m_method_result_destroy(res);
n4m_config_destroy(cfg);
n4m_context_destroy(ctx);
```
:::
:::{tab-item} Python · pls4all (raw)
:sync: python-raw
:class-label: lang-python
```python
import pls4all
from pls4all import Algorithm, Solver
with pls4all.Context() as ctx, pls4all.Config() as cfg:
cfg.algorithm = Algorithm.PCR
cfg.solver = Solver.SVD
cfg.n_components = 4
with pls4all.Model.fit(ctx, cfg, X, y) as mdl:
y_hat = mdl.predict(X_test)
```
:::
:::{tab-item} Python · n4m direct
:sync: python-n4m
:class-label: lang-python
```python
import n4m
from n4m.sklearn import NativePCRRegressor
res = n4m.pcr(X, y, n_components=4, scale_x=True)
mdl = NativePCRRegressor(n_components=4, scale_x=True).fit(X, y)
y_hat = mdl.predict(X_test)
```
:::
:::{tab-item} Python · pls4all.sklearn
:sync: python-sklearn
:class-label: lang-python
```python
from pls4all.sklearn import PCR
mdl = PCR(n_components=2, center_x=True, scale_x=True, center_y=True, scale_y=False, tol=1e-06, max_iter=500, store_scores=False)
mdl.fit(X, y)
y_hat = mdl.predict(X_test)
```
:::
:::{tab-item} R · pls4all_method()
:sync: r-dispatcher
:class-label: lang-r
```r
library(pls4all)
# Unified low-level dispatcher (May 2026 R cleanup):
res <- pls4all_method("pcr", X, y,
n_components = 4L)
# res is a named list with MethodResult arrays/scalars.
# selected_indices / top_k_intervals are 1-based.
```
:::
:::{tab-item} R · `pls` package compat
:sync: r-pls-compat
:class-label: lang-r
```r
library(pls4all)
# Drop-in for CRAN `pls::pcr` (same signature).
fit <- pcr(y ~ ., ncomp = 4L, data = train,
validation = "CV", segments = 10L)
yhat <- predict(fit, newdata = test, ncomp = 4L)
RMSEP(fit)
```
:::
:::{tab-item} R · `mdatools` compat
:sync: r-mdatools
:class-label: lang-r
```r
library(pls4all)
# Drop-in for `mdatools::pls(x, y, ncomp, method = "pcr")`.
fit <- pls_mdatools(X, y, ncomp = 4L, method = "pcr",
center = TRUE, scale = FALSE)
yhat <- predict(fit, newdata = X_test, ncomp = 4L)
```
:::
:::{tab-item} MATLAB · pls4all (MEX)
:sync: matlab-mex
:class-label: lang-matlab
```matlab
res = pls4all.pcr(X, y, 4);
% see header of bindings/matlab/+pls4all/pcr.m for full
% parameter surface:
% [coefs, x_mean, y_mean, predictions] = pcr(X, Y, n_components)
yhat = predict(res, Xtest);
```
:::
:::{tab-item} MATLAB · pls4all (classdef)
:sync: matlab-classdef
:class-label: lang-matlab
```matlab
mdl = pls4all.fit("pcr", X, y, "NumComponents", 4);
yhat = predict(mdl, Xtest);
```
:::
::::
**Registry parity references** 📐
:::{card}
:class-card: external-refs
- 📐 **`ref.python_scikit_learn`** (python · python) — `scikit-learn` 1.8.0 · strict (rmse_rel ≤ 1e-06) — sklearn Pipeline(PCA(svd_solver='full') + LinearRegression).
- 📐 **`ref.r_pls`** (R · r) — `pls` 2.8.5 · strict (rmse_rel ≤ 1e-06) — R pls::pcr(scale=FALSE).
:::
### Benchmarks
Adaptive wall-clock per cell measured against [`full_matrix.csv`](../benchmarks/overview.md). Only backends that implement this method are listed; libraries without the method are omitted.
**Verdict** · ✓ ref / ≈ ref / ~ shape mark a reference-gate pass at strict / relaxed / qualitative tolerance · ✓ bind = pls4all binding agrees with the C++ baseline · ✗ divergent · ⚠ error · — not run. The fastest backend per column is marked 🏆.
**Reference gate**: strict — numeric equivalence (`rmse_rel_tol ≤ 1e-06`).
Rows tagged with **📐** are the canonical parity references for this method (declared in [`parity_timing.registry`](../benchmarks/methodology.md)). C++ and external rows show reference parity; pls4all language bindings show binding parity against the C++ backend. Hover the icon for role and tolerance band.
::::{tab-set}
:class: parity-tabs
:::{tab-item} 1 thread
:sync: threads-1
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×50 (ms) | 250×50 (ms) | 500×50 (ms) | 500×500 (ms) | 500×2500 (ms) | 2500×50 (ms) | 2500×500 (ms) | 2500×2500 (ms) | 10000×50 (ms) | 10000×500 (ms) |
|---|
| C++ native · libn4m |
|---|
pls4all.cpp.blas | ≈ +9e-12 | 108.6 ms | 1.86 ms | 1.2 s | 195.9 s🏆 | 2.60 ms🏆 | 3.60 ms | 10.3 ms | 2.9 s | 709.4 s🏆 | 46.6 ms | 3.0 s🏆 | — | 171.2 ms | 4.1 s |
pls4all.cpp.blas+omp | ≈ +9e-12 | 107.0 ms | 1.82 ms🏆 | 1.4 s | 209.9 s | 2.77 ms | 3.34 ms | 9.78 ms🏆 | 2.8 s🏆 | 742.2 s | 42.7 ms🏆 | 3.2 s | — | 169.0 ms🏆 | 4.0 s🏆 |
pls4all.cpp.omp | ≈ +9e-12 | 109.0 ms | 1.90 ms | 1.4 s | 205.8 s | 2.70 ms | 3.55 ms | 10.9 ms | 2.9 s | 713.8 s | 43.0 ms | 3.3 s | — | 173.2 ms | 5.2 s |
pls4all.cpp.ref | ≈ +9e-12 | 109.0 ms | 1.97 ms | 1.2 s🏆 | 212.3 s | 2.68 ms | 3.69 ms | 11.0 ms | 2.9 s | 719.9 s | 49.8 ms | 3.3 s | — | 178.3 ms | 5.4 s |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 108.1 ms | — | — | — | 2.71 ms | 3.33 ms🏆 | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | 108.7 ms | — | — | — | 3.48 ms | 4.05 ms | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ 2e-11 | 2.7 s | — | — | — | 7.98 ms | 26.2 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ 2e-11 | 2.7 s | — | — | — | 12.6 ms | 28.0 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ 2e-11 | 2.7 s | — | — | — | 10.3 ms | 28.8 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ 2e-11 | 2.7 s | — | — | — | 16.4 ms | 34.9 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +3e+00 | 117.3 ms | — | — | — | 4.23 ms | 4.85 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +3e+00 | 119.2 ms | — | — | — | 4.83 ms | 5.90 ms | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_scikit_learn | source | 4.09 ms🏆 | — | — | — | 3.28 ms | 4.69 ms | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_pls | ✗ +1e+00 | 25.9 ms | — | — | — | 12.6 ms | 15.8 ms | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 3 threads
:sync: threads-3
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×50 (ms) | 250×50 (ms) | 500×50 (ms) | 500×500 (ms) | 500×2500 (ms) | 2500×50 (ms) | 2500×500 (ms) | 2500×2500 (ms) | 10000×50 (ms) | 10000×500 (ms) |
|---|
| C++ native · libn4m |
|---|
pls4all.cpp.blas | ✓ ref 3e-12 | — | — | — | — | 3.46 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ✓ ref 3e-12 | — | — | — | — | 3.29 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ✓ ref 3e-12 | — | — | — | — | 3.39 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ✓ ref 3e-12 | — | — | — | — | 3.09 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ 2e-15 | — | — | — | — | 2.76 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 2e-15 | — | — | — | — | 3.29 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | — | — | — | — | 6.81 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | — | — | — | — | 9.69 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | — | — | — | — | 9.28 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | — | — | — | — | 15.7 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +3e+00 | — | — | — | — | 3.87 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +3e+00 | — | — | — | — | 5.14 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_scikit_learn | source | — | — | — | — | 3.45 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_pls | ✗ +1e+00 | — | — | — | — | 12.6 ms | — | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 10 threads
:sync: threads-10
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×50 (ms) | 250×50 (ms) | 500×50 (ms) | 500×500 (ms) | 500×2500 (ms) | 2500×50 (ms) | 2500×500 (ms) | 2500×2500 (ms) | 10000×50 (ms) | 10000×500 (ms) |
|---|
| C++ native · libn4m |
|---|
pls4all.cpp.blas | ✓ ref 3e-12 | — | — | — | — | 2.55 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ✓ ref 3e-12 | — | — | — | — | 2.47 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ✓ ref 3e-12 | — | — | — | — | 2.62 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ✓ ref 3e-12 | — | — | — | — | 2.54 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ 2e-15 | — | — | — | — | 2.53 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 2e-15 | — | — | — | — | 2.76 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | — | — | — | — | 5.92 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | — | — | — | — | 7.09 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | — | — | — | — | 7.52 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | — | — | — | — | 12.5 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +3e+00 | — | — | — | — | 3.72 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +3e+00 | — | — | — | — | 3.97 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_scikit_learn | source | — | — | — | — | 5.56 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_pls | ✗ +1e+00 | — | — | — | — | 13.6 ms | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)