# `pls_diagnostic_q` — Q residual (squared prediction error) _Group_: **Diagnostic** · _Registry tolerance_: `1e-08` ## 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_metrics_pls_diagnostics_compute` with stat='q'. Reference: R `mdatools 0.15.0`. R roxygen note (`diagnostics.R::pls_diagnostics`): > PLS diagnostics: T², Q, DModX from a fitted model. MATLAB header (`bindings/matlab/+pls4all/pls_diagnostics.m`): ```text 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** ::::{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_metrics_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); ``` ::: :::{tab-item} Python · pls4all (raw) :sync: python-raw :class-label: lang-python ```python 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"), … ``` ::: :::{tab-item} Python · pls4all.sklearn :sync: python-sklearn :class-label: lang-python ```python from pls4all.sklearn import q_score result = q_score(X, y, n_components=4) ``` ::: :::{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("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. ``` ::: :::{tab-item} R · pls4all (raw fn) :sync: r-raw :class-label: lang-r ```r library(pls4all) res <- pls_diagnostics(model, X) yhat <- pls4all_predict(res, X_test) ``` ::: :::{tab-item} MATLAB · pls4all (MEX) :sync: matlab-mex :class-label: lang-matlab ```matlab 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); ``` ::: :::{tab-item} MATLAB · pls4all (classdef) :sync: matlab-classdef :class-label: lang-matlab _No idiomatic classdef wrapper — invoke `pls4all.fit("pls_diagnostic_q", X, y, …)` directly from the unified MEX factory._ ::: :::: **Registry parity references** 📐 :::{card} :class-card: external-refs - 📐 **`ref.r_mdatools`** (R · r) — `mdatools` 0.15.0 · strict (rmse_rel ≤ 1e-08) — 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`](../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-08`). 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
BackendParity200×30 (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
::: :::{tab-item} 3 threads :sync: threads-3
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
::: :::{tab-item} 10 threads :sync: threads-10
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](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)