continuum_regression — Continuum Regression (Stone & Brooks 1990)

Group: Nonlinear / local · Registry tolerance: 1e-06

Description

Continuum regression (interpolates PLS / OLS)

Registry note — Canonical Stone & Brooks (1990) continuum regression. Python ContinuumPyReference is a paper-faithful NumPy implementation (no widely installable Python port exists); the pls4all C++ kernel uses the same algorithm and matches bit-for-bit. The optional R JICO::continuum adapter uses a different (lambda, gamma, om) parameterization and is kept as a qualitative cross-check.

Parameters

Name	Type	Default	Notes
X	—	required	current public binding signature
y	—	required	current public binding signature
tau	float	0.5	current public binding signature
n_components	int	2	current public binding signature

Explanations

Bibliographic source

Stone, M. & Brooks, R. J. (1990). Continuum regression: Cross-validated sequentially constructed prediction embracing ordinary least squares, partial least squares and principal components regression. Journal of the Royal Statistical Society: Series B 52(2), 237–269. DOI 10.1111/j.2517-6161.1990.tb01786.x.

Mathematical principle

For the current parameter tau, each direction is proportional to \((\mathbf X^T\mathbf X)^{-\tau}\mathbf X^T\mathbf y\), equivalently emphasizing squared covariance with a \(\operatorname{Var}(\mathbf Xw)^{-2\tau}\) factor. Thus tau=0 is PLS-like and, at full rank, tau=1 is OLS-like. The implementation has no PCR endpoint.

Appropriate uses

Exploring the continuum between covariance-driven PLS and inverse-covariance/OLS-like directions.

Limits and validation

Current tau uses PLS at 0 and full-rank OLS-like weighting at 1; it has no PCR endpoint and differs from other continuum-regression parameterizations.

Implementation

The default eigensystem path implements this tau convention. The optional legacy NIPALS solver uses a different score-rescaling heuristic, so solver and tau must be reported together.

Sources and provenance

Current implementation: cpp/src/core/extra_pls.cpp.

API and bindings

C ABI (ABI 2): n4m_estimators_continuum_regression_fit. Use the linked public header for the exact signature, configuration, and result handles.

Python (verified public re-export):

from n4m.estimators.regression.latent import continuum_regression
result = continuum_regression(X, y)

Source signature: continuum_regression(X, y, *, tau: float = 0.5, n_components: int = 2) (n4m/_impl/native.py).

R (source-verified): continuum_regression(formula, data, ncomp = 2L, tau = 0.5, na.action = stats::na.omit).

The source signature has additional required inputs, so no example call is fabricated.

MATLAB / Octave (source-verified): continuum_regression(X, Y, n_components, tau).

The source signature has additional required inputs, so no example call is fabricated.

Registry parity references 📐

• 📐 ref.python_stone_brooks_1990_py (python · python) — stone-brooks-1990-py 1.0 · strict (rmse_rel ≤ 1e-06) — NumPy reference for Stone & Brooks (1990) continuum regression. First-component weight is (X’X)^{-tau} X’y, computed via the centered-X SVD; subsequent components use SIMPLS basis-v deflation of the modified cross-product matrix.

• 📐 ref.r_jico (R · r) — JICO 0.1 · strict (rmse_rel ≤ 1e-06) — R JICO::continuum (Stone & Brooks 1990). Different parameterization than pls4all — JICO uses (lambda, gamma, om) while pls4all maps a single τ. Predictions are reconstructed by regressing Y on JICO’s latent scores.

Benchmarks

Archived measurement identity. Backend labels in this table are the raw IDs recorded when the benchmark ran (including historical pls4all.* IDs). They preserve measurement provenance and do not describe a current public Python, R, or MATLAB binding; use the source-verified API and bindings section above for current entry points.

Adaptive wall-clock per cell measured against full_matrix.csv. 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 = archived binding-harness result agrees with the C++ baseline  ·  ⇄ cross-check = documented by-design selector/RNG/model, noncanonical API/facade convention, or secondary oracle  ·  ✗ 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). C++ and external rows show reference parity; archived language-harness rows show binding parity against the C++ backend. Hover the icon for role and tolerance band.

1 thread

Backend	Parity	200×50 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp	✓ ref 1e-14	2.73 ms
Python · archived pls4all benchmark
pls4all.python	✓ bind	2.63 ms
pls4all.sklearn	✓ bind	2.77 ms
R · archived pls4all benchmark
pls4all.R	✓ bind	5.55 ms
pls4all.R.formula	✓ bind	6.02 ms
pls4all.R.mdatools	✓ bind	6.62 ms
pls4all.R.pls	✓ bind	6.62 ms
Python · external
📐ref.python_stone_brooks_1990_py	source	2.27 ms🏆
R · external
📐ref.r_jico	⇄ +8e-03	31.6 ms

3 threads

Backend	Parity	200×50 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp	✓ ref 1e-14	2.70 ms
Python · archived pls4all benchmark
pls4all.python	✓ bind	2.65 ms
pls4all.sklearn	✓ bind	2.83 ms
R · archived pls4all benchmark
pls4all.R	✓ bind	5.33 ms
pls4all.R.formula	✓ bind	6.41 ms
pls4all.R.mdatools	✓ bind	6.49 ms
pls4all.R.pls	✓ bind	6.57 ms
Python · external
📐ref.python_stone_brooks_1990_py	source	2.39 ms🏆
R · external
📐ref.r_jico	⇄ +8e-03	31.6 ms

10 threads

Backend	Parity	200×50 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp	✓ ref 1e-14	2.70 ms🏆
Python · archived pls4all benchmark
pls4all.python	✓ bind	2.76 ms
pls4all.sklearn	✓ bind	2.93 ms
R · archived pls4all benchmark
pls4all.R	✓ bind	6.21 ms
pls4all.R.formula	✓ bind	6.41 ms
pls4all.R.mdatools	✓ bind	7.10 ms
pls4all.R.pls	✓ bind	6.57 ms
Python · external
📐ref.python_stone_brooks_1990_py	source	4.06 ms
R · external
📐ref.r_jico	⇄ +8e-03	49.0 ms

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See also: benchmark overview · methods index · interactive dashboard