Title: principal components regression
Long Title: IUPAC Gold Book - principal components regression
DOI: 10.1351/goldbook.10156
Status: current

Definition
Multivariate calibration in which a dependent variable is regressed against the scores of a chosen number of factors obtained from principal-component analysis of the predictor variable.

Notes
1) PCA decomposes the predictor variable data \(\boldsymbol{X}\) into \(k\) principal component factors \(\boldsymbol{\hat X}_{k} = \boldsymbol{T}_{k}\boldsymbol{P}_{k}^{\rm{T}}\), where \(k\) may be determined by cross validation. The dependent variable \(\boldsymbol{c}\) is then regressed against \(\boldsymbol{\hat X}_{k}\), \(\boldsymbol{c} = \boldsymbol{\hat X}_{k} \boldsymbol{\hat b}\).
2) The factorization gives orthogonal factors, but no information about the predicted variable \(\boldsymbol{c}\) is used.

Related Terms
- Multivariate calibration: https://goldbook.iupac.org//terms/view/10153
- k principal component factors: https://goldbook.iupac.org//terms/view/10107
- multilinear regression: https://goldbook.iupac.org//terms/view/10152
- partial least squares: https://goldbook.iupac.org//terms/view/10155
- principal-component analysis: https://goldbook.iupac.org//terms/view/10106
- scores: https://goldbook.iupac.org//terms/view/10112

Source
- PAC, 2016, 88, 407. 'Vocabulary of concepts and terms in chemometrics (IUPAC Recommendations 2016)' on page 435 (https://doi.org/10.1515/pac-2015-0605)

Other Outputs
- html: https://goldbook.iupac.org/terms/view/10156/html
- json: https://goldbook.iupac.org/terms/view/10156/json
- xml: https://goldbook.iupac.org/terms/view/10156/xml

Citation: Citation: 'principal components regression' in IUPAC Compendium of Chemical Terminology, 5th ed. International Union of Pure and Applied Chemistry; 2025. Online version 5.0.0, 2025. 10.1351/goldbook.10156
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Accessed: 2026-08-07T18:25:06+00:00