# `continuum_regression` — Continuum Regression (Stone & Brooks 1990) _Group_: **Nonlinear / local** · _Registry tolerance_: `1e-06` ## Description Continuum regression (interpolates PLS / OLS) From the `pls4all.sklearn.ContinuumRegression` docstring: > Continuum regression τ ∈ [0, 1] interpolates PLS (1) / OLS (0). > **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 | |------|------|---------|-------| | `n_components` | `int` | `2` | Number of latent components extracted (k). | | `tau` | `float` | `0.5` | Continuum mixing parameter in [0, 1]; 0 ≈ PLS, 1 ≈ whitened OLS / PCR-like. | ## 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*. JRSS B 52(2), 237–269. ### Mathematical principle Continuum regression introduces a single shape parameter $\tau \in [0, 1]$ that selects the loading-weight criterion: $\mathbf{w} \propto \operatorname{Cov}(\mathbf{X}\mathbf{w}, \mathbf{y})^{\tau} \cdot \operatorname{Var}(\mathbf{X}\mathbf{w})^{1-\tau}$. Special cases: $\tau = 0$ gives PCR (variance-maximising), $\tau = 1/2$ gives PLS (covariance-maximising), $\tau = 1$ gives OLS (correlation-maximising, in the appropriate limit). Cross-validating $\tau$ on a fine grid often improves RMSE over the discrete PLS / PCR choices — the optimum is rarely exactly at $\tau = 0.5$. Stone & Brooks' original treatment also cross-validates the number of components $k$ jointly with $\tau$, producing a 2-D grid. Implementation note: numerically stable continuum regression uses the parameterised power method on the matrix $\mathbf{X}^{\top}\mathbf{y}\mathbf{y}^{\top}\mathbf{X} / (\mathbf{X}^{\top}\mathbf{X})^{1-\tau}$, which avoids forming the rank-1 outer product explicitly and is what pls4all uses. ### Implementation `n4m_estimators_continuum_regression_fit` (in-sample only). R roxygen note (`sklearn_extra.R::continuum_regression`): > Continuum regression — formula entry point. MATLAB header (`bindings/matlab/+pls4all/ContinuumRegression.m`): ```text pls4all.ContinuumRegression Continuum regression (tau ∈ [0, 1]). ``` ### 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 */ n4m_context_t* ctx = n4m_context_create(); n4m_config_t* cfg = n4m_config_create(); n4m_method_result_t* res = NULL; n4m_estimators_continuum_regression_fit(ctx, cfg, &x_view, &y_view, /* hyperparams */, &res); /* … read coefficients / mask / scores via */ /* n4m_method_result_get_double_matrix / vector / scalar … */ 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._methods import continuum_regression_fit with pls4all.Context() as ctx, pls4all.Config() as cfg: res = continuum_regression_fit(ctx, cfg, X, y, n_components=4) # then: res.matrix("predictions"), res.matrix("coefficients"), # res.vector("mask"), res.scalar("intercept"), … ``` ::: :::{tab-item} Python · pls4all.sklearn :sync: python-sklearn :class-label: lang-python ```python from pls4all.sklearn import ContinuumRegression mdl = ContinuumRegression(n_components=2, tau=0.5) 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("continuum_regression", X, y, n_components = 4L, params = list(tau = 0.5)) # res is a named list with MethodResult arrays/scalars. # selected_indices / top_k_intervals are 1-based. ``` ::: :::{tab-item} R · pls4all (raw fn) :sync: r-raw :class-label: lang-r ```r library(pls4all) res <- continuum_regression_fit(X, Y, n_components, tau = 0.5) yhat <- pls4all_predict(res, X_test) ``` ::: :::{tab-item} R · pls4all (formula+S3) :sync: r-formula :class-label: lang-r ```r library(pls4all) fit <- continuum_regression(y ~ ., data = train, ncomp = 4L) yhat <- predict(fit, newdata = test) summary(fit) ``` ::: :::{tab-item} MATLAB · pls4all (MEX) :sync: matlab-mex :class-label: lang-matlab ```matlab res = pls4all.continuum_regression(X, y, 4); % see header of bindings/matlab/+pls4all/continuum_regression.m for full % parameter surface: % res = continuum_regression(X, Y, n_components, tau) yhat = predict(res, Xtest); ``` ::: :::{tab-item} MATLAB · pls4all (classdef) :sync: matlab-classdef :class-label: lang-matlab ```matlab mdl = pls4all.fit("continuum_regression", X, y, "NumComponents", 4); yhat = predict(mdl, Xtest); ``` ::: :::: **Registry parity references** 📐 :::{card} :class-card: external-refs - 📐 **`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 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  ·  ⇄ 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`](../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
BackendParity200×50 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 1e-142.73 ms
Python · pls4all
pls4all.python✓ bind2.63 ms
pls4all.sklearn✓ bind2.77 ms
R · pls4all
pls4all.R✓ bind5.55 ms
pls4all.R.formula✓ bind6.02 ms
pls4all.R.mdatools✓ bind6.62 ms
pls4all.R.pls✓ bind6.62 ms
Python · external
📐ref.python_stone_brooks_1990_pysource2.27 ms🏆
R · external
📐ref.r_jico⇄ +8e-0331.6 ms
::: :::{tab-item} 3 threads :sync: threads-3
BackendParity200×50 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 1e-142.70 ms
Python · pls4all
pls4all.python✓ bind2.65 ms
pls4all.sklearn✓ bind2.83 ms
R · pls4all
pls4all.R✓ bind5.33 ms
pls4all.R.formula✓ bind6.41 ms
pls4all.R.mdatools✓ bind6.49 ms
pls4all.R.pls✓ bind6.57 ms
Python · external
📐ref.python_stone_brooks_1990_pysource2.39 ms🏆
R · external
📐ref.r_jico⇄ +8e-0331.6 ms
::: :::{tab-item} 10 threads :sync: threads-10
BackendParity200×50 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 1e-142.70 ms🏆
Python · pls4all
pls4all.python✓ bind2.76 ms
pls4all.sklearn✓ bind2.93 ms
R · pls4all
pls4all.R✓ bind6.21 ms
pls4all.R.formula✓ bind6.41 ms
pls4all.R.mdatools✓ bind7.10 ms
pls4all.R.pls✓ bind6.57 ms
Python · external
📐ref.python_stone_brooks_1990_pysource4.06 ms
R · external
📐ref.r_jico⇄ +8e-0349.0 ms
::: :::: --- _See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)