pls_diagnostic_q — Q residual (squared prediction error)

Group: Diagnostic · Registry tolerance: 5.0

Description

PLS Q residuals / SPE (§9)

Registry note — R mdatools::pls$xdecomp$Q. SIMPLS-vs-NIPALS deflation ordering differences inflate the RMS divergence; both are valid Q computations on different latent bases.

Parameters

Name

Type

Default

Notes

n_components

int

4

registry benchmark cell value

Explanations

Bibliographic source

Jackson, J. E. & Mudholkar, G. S. (1979). Control procedures for residuals associated with principal component analysis. Technometrics 21(3), 341–349.

Mathematical principle

Q (also called SPE — Squared Prediction Error) is the sum of squared residuals between \(\mathbf{x}\) and its PLS reconstruction \(\hat{\mathbf{x}} = \mathbf{T}\mathbf{P}^{\top}\): \(Q_i = \|\mathbf{x}_i - \mathbf{t}_i \mathbf{P}^{\top}\|_2^2 = \sum_j (x_{ij} - \hat{x}_{ij})^2\). It measures the part of \(\mathbf{x}\) that lies orthogonal to the latent space — variation in the predictor that the model could not capture.

Under the assumption of Gaussian residuals, Jackson & Mudholkar (1979) derived a parametric upper control limit. High Q with low T² typically signals a sample with a fundamentally different spectral fingerprint from the calibration set (e.g. contamination, instrument failure); low Q with high T² signals an extreme combination of otherwise normal features.

Reported per-sample as a 1-D vector aligned with the rows of the input.

Implementation

n4m_pls_diagnostics_compute with stat=’q’. Reference: R mdatools 0.15.0.

MATLAB header (bindings/matlab/+pls4all/pls_diagnostics.m):

pls4all.pls_diagnostics  Hotelling T2, Q residuals, DModX from a SIMPLS fit.

   res = pls4all.pls_diagnostics(X, Y, n_components)
   res = pls4all.pls_diagnostics(X, Y, n_components, X_reference)

 Fits an internal SIMPLS model (store_scores=1) and evaluates row-wise

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

/* C ABI — libn4m */
n4m_context_t* ctx = n4m_context_create();
n4m_config_t*  cfg = n4m_config_create();
n4m_method_result_t* res = NULL;
n4m_pls_diagnostics_compute(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);

import pls4all
from pls4all._methods import pls_diagnostics_compute
with pls4all.Context() as ctx, pls4all.Config() as cfg:
    res = pls_diagnostics_compute(ctx, cfg, X, y, n_components=4)
# then: res.matrix("predictions"), res.matrix("coefficients"),
# res.vector("mask"), res.scalar("intercept"), …

from pls4all.sklearn import q_score
result = q_score(X, y, n_components=4)

library(pls4all)
# Unified low-level dispatcher (May 2026 R cleanup):
res <- pls4all_method("pls_diagnostic_q", X, y,
                      n_components = 4L)
# res is a named list with MethodResult arrays/scalars.
# selected_indices / top_k_intervals are 1-based.

res = pls4all.pls_diagnostics(X, y, 4);
% see header of bindings/matlab/+pls4all/pls_diagnostics.m for full
% parameter surface:
%   res = pls_diagnostics(X, Y, n_components, X_reference)
yhat = predict(res, Xtest);

No idiomatic classdef wrapper — invoke pls4all.fit("pls_diagnostic_q", X, y, …) directly from the unified MEX factory.

Registry parity references 📐

  • 📐 ref.r_mdatools (R · r) — mdatools 0.15.0 · qualitative (rmse_rel ≤ 5e+00) — R mdatools::pls with predict()$xdecomp$T2 / $Q. DModX is derived locally from $Q + DOF. mdatools uses different SIMPLS deflation / normalization conventions than pls4all, so cross-implementation parity is qualitative.

Benchmarks

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 = 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-08).

Rows tagged with 📐 are the canonical parity references for this method (declared in parity_timing.registry). 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.

BackendParity200×30 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 1e-151.22 ms
Python · pls4all
pls4all.python✓ bind1.19 ms🏆
pls4all.sklearn✓ bind1.30 ms
R · pls4all
pls4all.R✓ 3e-143.10 ms
pls4all.R.formula✓ 3e-143.57 ms
pls4all.R.mdatools✓ 3e-143.96 ms
pls4all.R.pls✓ 3e-143.47 ms
R · external
📐ref.r_mdatoolssource13.8 ms
BackendParity200×30 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 1e-151.21 ms
Python · pls4all
pls4all.python✓ bind1.21 ms🏆
pls4all.sklearn✓ bind1.27 ms
R · pls4all
pls4all.R✓ 3e-143.11 ms
pls4all.R.formula✓ 3e-143.59 ms
pls4all.R.mdatools✓ 3e-143.52 ms
pls4all.R.pls✓ 3e-143.64 ms
R · external
📐ref.r_mdatoolssource14.9 ms
BackendParity200×30 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 1e-151.23 ms
Python · pls4all
pls4all.python✓ bind1.22 ms🏆
pls4all.sklearn✓ bind1.36 ms
R · pls4all
pls4all.R✓ 3e-143.28 ms
pls4all.R.formula✓ 3e-143.83 ms
pls4all.R.mdatools✓ 3e-143.89 ms
pls4all.R.pls✓ 3e-143.74 ms
R · external
📐ref.r_mdatoolssource15.5 ms

See also: benchmark overview · methods index · interactive dashboard