Attribute the two PCA cross-validation schemes correctly

[kgdunn]

Two claims in latent-variable-modelling/principal-component-analysis/determining-the-number-of-components-to-use-in-the-model-with-cross-validation.rst are wrong. Both are live on the site at <https://learnche.org/pid/latent-variable-modelling/principal-component-analysis/determining-the-number-of-components-to-use-in-the-model-with-cross-validation>. They surfaced while researching a question about why a PCA is cross-validated cell by cell rather than row by row, and they are unrelated to the batch case studies in #273, which is why this is a separate pull request.

The Wold attribution

The section describes building the model on the retained rows, projecting the held-out rows onto the loadings to get their scores, reconstructing and taking residuals, and calls it "Wold's cross-validation procedure". Wold's 1978 paper, the one that introduced cross-validation for component models, did not hold out rows. It held out a scattered pattern of individual elements along pseudo-diagonals and estimated them as missing values with NIPALS.

The consequence sat further down, where the text had Bro and co-authors recommending element deletion "instead". That stages a correction of Wold that never happened, because Wold was already deleting elements thirty years earlier.

Changed:

• The procedure is named for what it is, a row-wise procedure, in both the description and the caution that follows it.
• The review's contribution is stated as the independence requirement it actually is: a prediction must not use the value it is predicting.
• The element-wise remedy is credited to Wold, with Eastment and Krzanowski (1982), Technometrics, 24, 73-77 named for the two-model route to the same property, and the row-wise procedure described as the later simplification.

I could not read Wold (1978) directly, since it is paywalled with no open copy. Four mutually independent secondary sources agree on what it did: Owen and Perry in the Annals of Applied Statistics, Perry's Stanford thesis, Camacho and Ferrer, and the Bioconductor pcaMethods documentation.

The Simca-P label

The section says the bar plot is "Simca-P's output, from the row-wise scheme", and builds a "same data, both schemes" comparison on that. Simca-P cross-validates a PCA element-wise, by the Eastment and Krzanowski route: rows are held out to estimate the loadings, columns to estimate the scores. Its row-wise cross-validation is reserved for PLS and OPLS, which is the likely source of the mix-up. The Simca 15 User Guide states this in section 16.8.1.

The recorded curve agrees. In figures/pca/barplot-for-R2-and-Q2.R the Simca-P series falls at components 3, 4, 6 and 7. The row-projection scheme cannot produce that: its loadings are nested, so the reconstruction error can only shrink as components are added, and a Q² of one minus PRESS over the total sum of squares cannot dip. The ProSensus series recorded beside it never falls and tracks R² to within a few points, which is what the row-projection scheme does produce.

Changed:

• The heading and framing become "the same data, two element-wise predictors", naming what each of the two actually does.
• The closing sentence had drawn a contrast that no longer exists, since it credited the element-wise scheme with not letting a held-out value contribute to its own prediction as though the other curve did. It now draws the conclusion that follows: two schemes forming their predictions differently read these data the same way.

Verification

• make check-code-file on this file, which runs the chapter in order against the released library: 4 blocks, all passed. The element-wise curve is unchanged at 0.292, 0.404, 0.374, 0.343, 0.341, 0.240, 0.100, 0.193, 0.696, 0.850, 0.807.
• make text: zero warnings.
• No number in the section moved. No code block was touched.
• No em-dashes in the added prose.
• CITATION.cff: version 2026.09.10, date-released 2026-09-10.

Paired change

The same mislabel is in the figure's legend. kgdunn/figures#96 corrects it and regenerates pca/q2-across-packages.png. That one should merge first, since the PDF build resolves figures from the figures repository's default branch.

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