# `pls_diagnostic_t2` — Hotelling T² score
_Group_: **Diagnostic** · _Registry tolerance_: `10.0`
## Description
PLS Hotelling T² (§9)
> **Registry note** — R `mdatools::pls` is the only widely installable external reference. Both use SIMPLS-style but differ on score normalization conventions — tolerance is wide enough to flag the R ref's presence.
### Parameters
| Name | Type | Default | Notes |
|------|------|---------|-------|
| `n_components` | `int` | `4` | registry benchmark cell value |
## Explanations
### Bibliographic source
Hotelling, H. (1931). *The generalization of Student's ratio*. Annals of Mathematical Statistics 2(3), 360–378. — applied to PLS scores by MacGregor & Kourti 1995.
### Mathematical principle
Hotelling T² measures how unusual a sample is **within the latent score space**: $T_i^2 = \mathbf{t}_i^{\top} \boldsymbol{\Lambda}^{-1} \mathbf{t}_i$ where $\boldsymbol{\Lambda}$ is the diagonal matrix of score variances. Under multivariate normality of the scores, $\frac{n(n-k)}{k(n^2-1)} T^2 \sim F_{k, n-k}$, giving an exact upper control limit at any $\alpha$.
T² complements the Q residual (next entry): Q measures the **distance to the model** (variation in $\mathbf{X}$ that the latent space does not explain), while T² measures the **distance within the model** (unusual score combination on otherwise well-explained samples). Joint T²/Q monitoring catches both kinds of out-of-control points.
Reported per-sample as a 1-D vector aligned with the rows of the input.
### Implementation
`n4m_pls_diagnostics_compute` with stat='t2'. Reference: R `mdatools 0.15.0` (Kucheryavskiy).
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 t2_score
result = t2_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_t2", 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_t2", 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 ≤ 1e+01) — 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 ≤ 1e+01` 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 | ≈ +7e-15 | 2.55 ms🏆 | 2.31 ms | 14.0 ms | 70.8 ms | 1.24 ms🏆 | 2.66 ms | 5.74 ms | 69.1 ms | 336.0 ms🏆 | 33.1 ms🏆 | 333.6 ms | 1.7 s🏆 | 130.6 ms | 1.5 s |
pls4all.cpp.blas+omp | ≈ +7e-15 | 2.60 ms | 1.28 ms🏆 | 11.9 ms🏆 | 71.6 ms | 1.28 ms | 2.62 ms | 5.57 ms🏆 | 70.4 ms | 345.4 ms | 33.3 ms | 320.7 ms🏆 | 1.7 s | 128.5 ms | 1.5 s |
pls4all.cpp.omp | ≈ +2e-14 | 2.78 ms | 2.51 ms | 14.0 ms | 67.1 ms🏆 | 1.26 ms | 2.63 ms | 7.11 ms | 73.7 ms | 349.5 ms | 34.5 ms | 337.9 ms | 1.8 s | 128.4 ms🏆 | 1.5 s🏆 |
pls4all.cpp.ref | ≈ +2e-14 | 2.72 ms | 1.37 ms | 13.9 ms | 73.9 ms | 1.27 ms | 2.77 ms | 7.35 ms | 67.5 ms🏆 | 357.7 ms | 35.1 ms | 332.9 ms | 1.8 s | 134.8 ms | 1.5 s |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 2.57 ms | — | — | — | 1.43 ms | 3.33 ms | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | 2.74 ms | — | — | — | 1.68 ms | 2.56 ms🏆 | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ 4e-13 | 12.0 ms | — | — | — | 4.52 ms | 10.6 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ 4e-13 | 19.4 ms | — | — | — | 7.45 ms | 9.82 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ 4e-13 | 21.2 ms | — | — | — | 5.56 ms | 12.3 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ 4e-13 | 20.1 ms | — | — | — | 5.18 ms | 12.4 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +1e+01 | 4.28 ms | — | — | — | 2.25 ms | 4.81 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +1e+01 | 5.23 ms | — | — | — | 2.72 ms | 4.95 ms | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_mdatools | source | 35.9 ms | — | — | — | 18.6 ms | 20.5 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 3e-15 | — | — | — | — | 1.37 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 3e-15 | — | — | — | — | 1.91 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 5e-15 | — | — | — | — | 1.37 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 5e-15 | — | — | — | — | 1.59 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ 1e-13 | — | — | — | — | 1.25 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 1e-13 | — | — | — | — | 1.72 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | — | — | — | — | 4.26 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | — | — | — | — | 5.80 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | — | — | — | — | 5.11 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | — | — | — | — | 5.42 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +1e+01 | — | — | — | — | 2.38 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +1e+01 | — | — | — | — | 3.05 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_mdatools | source | — | — | — | — | 18.6 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 3e-15 | — | — | — | — | 1.17 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 3e-15 | — | — | — | — | 1.19 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 5e-15 | — | — | — | — | 1.19 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 5e-15 | — | — | — | — | 1.18 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ 1e-13 | — | — | — | — | 1.19 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 1e-13 | — | — | — | — | 1.31 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | — | — | — | — | 3.07 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | — | — | — | — | 3.75 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | — | — | — | — | 3.70 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | — | — | — | — | 3.71 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +1e+01 | — | — | — | — | 1.96 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +1e+01 | — | — | — | — | 2.18 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_mdatools | source | — | — | — | — | 13.6 ms | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)