# `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`):
```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_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} 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 · 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`](../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**: qualitative — shape/smoke comparison only. The external library and pls4all do not produce numerically equivalent output for this method (see the MethodSpec notes); the `rmse_rel_tol ≤ 5e+00` budget is set wide on purpose. Treat ~ shape as *“we ran both, both finished”*, not as numerical agreement.
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) | 200×30 (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 | ≈ +1e-15 | 2.63 ms🏆 | 2.15 ms | 14.0 ms | 70.0 ms | 1.26 ms🏆 | 2.56 ms🏆 | 7.42 ms | 64.4 ms🏆 | 326.8 ms🏆 | 31.9 ms🏆 | 336.1 ms | 1.9 s | 124.9 ms🏆 | 1.4 s |
pls4all.cpp.blas+omp | ≈ +1e-15 | 2.90 ms | 2.21 ms | 13.1 ms | 65.7 ms🏆 | 1.45 ms | 2.76 ms | 5.61 ms | 66.4 ms | 341.0 ms | 33.4 ms | 327.7 ms🏆 | 1.9 s | 133.6 ms | 1.4 s🏆 |
pls4all.cpp.omp | ≈ +1e-15 | 2.76 ms | 1.25 ms🏆 | 12.0 ms🏆 | 69.9 ms | 1.36 ms | 2.60 ms | 5.55 ms🏆 | 68.4 ms | 335.8 ms | 35.9 ms | 341.3 ms | 1.9 s | 130.9 ms | 1.5 s |
pls4all.cpp.ref | ≈ +1e-15 | 3.03 ms | 1.96 ms | 13.8 ms | 70.1 ms | 1.37 ms | 3.16 ms | 8.04 ms | 68.5 ms | 342.6 ms | 33.5 ms | 329.5 ms | 1.9 s🏆 | 134.9 ms | 1.5 s |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 2.76 ms | — | — | — | 1.36 ms | 2.65 ms | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | 2.97 ms | — | — | — | 1.67 ms | 2.70 ms | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ 4e-13 | 11.9 ms | — | — | — | 4.17 ms | 11.0 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ 4e-13 | 20.6 ms | — | — | — | 6.95 ms | 9.41 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ 4e-13 | 20.1 ms | — | — | — | 5.09 ms | 10.6 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ 4e-13 | 20.6 ms | — | — | — | 6.14 ms | 11.4 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +3e+01 | 4.57 ms | — | — | — | 2.38 ms | 4.64 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +3e+01 | 4.53 ms | — | — | — | 2.78 ms | 5.13 ms | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_mdatools | source | 33.2 ms | — | — | — | 19.6 ms | 22.7 ms | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 3 threads
:sync: threads-3
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×30 (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 | ~ shape 1e-15 | — | — | — | — | 1.39 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 1e-15 | — | — | — | — | 1.25 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 1e-15 | — | — | — | — | 1.27 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 1e-15 | — | — | — | — | 1.28 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ 3e-14 | — | — | — | — | 1.63 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 3e-14 | — | — | — | — | 2.01 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | — | — | — | — | 4.34 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | — | — | — | — | 5.14 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | — | — | — | — | 5.19 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | — | — | — | — | 5.36 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +3e+01 | — | — | — | — | 2.11 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +3e+01 | — | — | — | — | 2.52 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_mdatools | source | — | — | — | — | 19.8 ms | — | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 10 threads
:sync: threads-10
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×30 (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 | ~ shape 1e-15 | — | — | — | — | 1.19 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 1e-15 | — | — | — | — | 1.28 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 1e-15 | — | — | — | — | 1.20 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 1e-15 | — | — | — | — | 1.19 ms🏆 | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ 3e-14 | — | — | — | — | 1.49 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 3e-14 | — | — | — | — | 1.31 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | — | — | — | — | 3.01 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | — | — | — | — | 3.85 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | — | — | — | — | 3.61 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | — | — | — | — | 3.83 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +3e+01 | — | — | — | — | 1.94 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +3e+01 | — | — | — | — | 2.26 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_mdatools | source | — | — | — | — | 15.9 ms | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)