# `pds` — Piecewise Direct Standardisation
_Group_: **Calibration transfer** · _Registry tolerance_: `0.5`
## Description
PDS — Piecewise Direct Standardization (§13)
From the `pls4all.sklearn.PDSTransformer` docstring:
> Piecewise Direct Standardization (Wang 1991).
> **Registry note** — Base R per-band `lm.fit` over a window of source bands — the canonical Wang 1991 PDS algorithm with no extra deps. Same algorithm as pls4all's pds_fit modulo CSV-roundtrip precision.
### Parameters
| Name | Type | Default | Notes |
|------|------|---------|-------|
| `window_half_width` | `int` | `5` | Half-width (in channels) of the PDS local regression window. |
## Explanations
### Bibliographic source
Wang, Y., Veltkamp, D. J. & Kowalski, B. R. (1991). *Multivariate instrument standardization*. Analytical Chemistry 63(23), 2750–2756. https://doi.org/10.1021/ac00023a016 — same paper as `ds`; PDS is introduced in §3 (piecewise local regression with a sliding window of width 2w+1).
### Mathematical principle
PDS generalises DS by allowing the transfer to vary across wavelength bands. For each wavelength $j$ on the primary instrument, a local regression maps a window of $\pm w$ wavelengths from the secondary instrument: $\hat{x}_{\mathrm{primary}, j} = \mathbf{x}_{\mathrm{secondary}, j-w:j+w} \cdot \mathbf{f}_j$. The full transfer matrix is then **banded**: only $\pm w$ off-diagonal columns per row are non-zero.
PDS handles wavelength-dependent inter-instrument behaviour — wavelength-axis drift, resolution differences, detector non-linearities — that DS cannot. The window half-width $w$ controls the locality: $w=0$ recovers a diagonal-only transfer, $w \to p/2$ recovers DS.
PDS is the de-facto standard in NIR / FT-IR calibration transfer; the `prospectr` R package's implementation is considered canonical.
### Implementation
`n4m_pds_fit` (TransformerMixin in tier 2). Reference: R `prospectr::pds`. Note: pls4all applies the transpose convention so that `transform(X_secondary)` returns the standardised primary-instrument estimate.
MATLAB header (`bindings/matlab/+pls4all/pds.m`):
```text
pls4all.pds Piecewise Direct Standardization (calibration transfer).
```
### 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_pds_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 pds_fit
with pls4all.Context() as ctx, pls4all.Config() as cfg:
res = pds_fit(ctx, cfg, X, y, X_target=X_target)
# 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 PDSTransformer
mdl = PDSTransformer(window_half_width=5)
mdl.fit(X, y, X_target=X_target)
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("pds", X, y,
n_components = 2L, params = list(window_half_width = 2L))
# 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.pds(X, y, 2);
% see header of bindings/matlab/+pls4all/pds.m for full
% parameter surface:
% res = pds(X_source, X_target, window_half_width)
yhat = predict(res, Xtest);
```
:::
:::{tab-item} MATLAB · pls4all (classdef)
:sync: matlab-classdef
:class-label: lang-matlab
_No idiomatic classdef wrapper — invoke `pls4all.fit("pds", X, y, …)` directly from the unified MEX factory._
:::
::::
**Registry parity references** 📐
:::{card}
:class-card: external-refs
- 📐 **`ref.r_base`** (R · r) — `base` R 4.3.3 · qualitative (rmse_rel ≤ 5e-01) — Base R `lm` per spectral band — closest installable analog to Wang 1991 Piecewise Direct Standardization. With window_half_width=0 this reduces to Direct Standardization.
:::
### 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-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 | ≈ +2e-08 | 4.50 ms | 1.25 ms | 27.3 ms | 1.2 s | 1.32 ms | 3.14 ms | 6.95 ms | 118.0 ms🏆 | 5.9 s | 39.4 ms | 675.3 ms🏆 | 22.3 s | 164.6 ms | 2.6 s |
pls4all.cpp.blas+omp | ≈ +2e-08 | 5.06 ms | 1.21 ms | 26.0 ms🏆 | 1.2 s | 1.43 ms | 3.89 ms | 5.64 ms🏆 | 124.4 ms | 4.3 s🏆 | 32.9 ms🏆 | 762.5 ms | 22.1 s | 150.4 ms | 2.5 s |
pls4all.cpp.omp | ≈ +2e-08 | 4.41 ms🏆 | 1.15 ms🏆 | 27.0 ms | 1.1 s🏆 | 1.30 ms🏆 | 3.02 ms | 6.03 ms | 122.2 ms | 4.3 s | 34.1 ms | 758.4 ms | 22.1 s🏆 | 147.9 ms🏆 | 2.5 s |
pls4all.cpp.ref | ≈ +2e-08 | 4.96 ms | 1.17 ms | 26.9 ms | 1.2 s | 1.39 ms | 3.10 ms | 8.54 ms | 125.4 ms | 5.9 s | 42.2 ms | 679.5 ms | 22.3 s | 153.3 ms | 2.5 s🏆 |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 4.99 ms | — | — | — | 1.40 ms | 2.77 ms🏆 | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 9e-16 | 4.80 ms | — | — | — | 2.87 ms | 5.79 ms | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | 37.1 ms | — | — | — | 9.68 ms | 24.9 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | 37.0 ms | — | — | — | 11.3 ms | 22.2 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | 39.8 ms | — | — | — | 11.2 ms | 18.6 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | 42.8 ms | — | — | — | 12.6 ms | 24.7 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +6e+00 | 11.1 ms | — | — | — | 4.70 ms | 8.06 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +6e+00 | 9.80 ms | — | — | — | 4.65 ms | 9.42 ms | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_base | source | 56.8 ms | — | — | — | 31.2 ms | 42.9 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 5e-09 | — | — | — | — | 1.91 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 5e-09 | — | — | — | — | 1.28 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 5e-09 | — | — | — | — | 2.00 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 5e-09 | — | — | — | — | 1.45 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 2.10 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 4e-16 | — | — | — | — | 1.54 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | — | — | — | — | 8.53 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | — | — | — | — | 10.0 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | — | — | — | — | 9.59 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | — | — | — | — | 9.49 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +6e+00 | — | — | — | — | 4.45 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +6e+00 | — | — | — | — | 4.46 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_base | source | — | — | — | — | 29.2 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 5e-09 | — | — | — | — | 1.23 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 5e-09 | — | — | — | — | 2.23 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 5e-09 | — | — | — | — | 1.22 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 5e-09 | — | — | — | — | 1.25 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 2.23 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 4e-16 | — | — | — | — | 1.39 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | — | — | — | — | 6.79 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | — | — | — | — | 7.41 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | — | — | — | — | 7.50 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | — | — | — | — | 7.30 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +6e+00 | — | — | — | — | 3.66 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +6e+00 | — | — | — | — | 4.05 ms | — | — | — | — | — | — | — | — | — |
| R · external |
|---|
📐ref.r_base | source | — | — | — | — | 21.7 ms | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)