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  • Primer
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Principal component analysis

Nature Reviews Methods Primers volume 2, Article number: 100 (2022) Cite this article

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A Publisher Correction to this article was published on 08 March 2023

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Abstract

Principal component analysis is a versatile statistical method for reducing a cases-by-variables data table to its essential features, called principal components. Principal components are a few linear combinations of the original variables that maximally explain the variance of all the variables. In the process, the method provides an approximation of the original data table using only these few major components. This Primer presents a comprehensive review of the method’s definition and geometry, as well as the interpretation of its numerical and graphical results. The main graphical result is often in the form of a biplot, using the major components to map the cases and adding the original variables to support the distance interpretation of the cases’ positions. Variants of the method are also treated, such as the analysis of grouped data, as well as the analysis of categorical data, known as correspondence analysis. Also described and illustrated are the latest innovative applications of principal component analysis: for estimating missing values in huge data matrices, sparse component estimation, and the analysis of images, shapes and functions. Supplementary material includes video animations and computer scripts in the R environment.

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Fig. 1: PCA of the indicators in the World Happiness Report.
Fig. 2: Schematic view of the PCA workflow.
Fig. 3: Schematic view of dimension reduction in PCA.
Fig. 4: PCA of the child cancer data.
Fig. 5: Correspondence analysis of the Barents Sea fish data, 1999–2004, explaining the between-year variance.
Fig. 6: Movie recommender system via matrix completion.
Fig. 7: PCA of visualizable objects: images, shapes and functions.

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Code availability

Several datasets and the R scripts that produce certain results in this Primer can be found on GitHub at: https://github.com/michaelgreenacre/PCA.

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Acknowledgements

This review is dedicated to the memory of Professor Cas Troskie, who was the head of the Department of Statistics at the University of Cape Town, both teacher and mentor to M.G. and T.H., and who planted the seeds of principal component analysis in them at an early age. T.H. was partially supported by grants DMS2013736 and IIS1837931 from the National Science Foundation, and grant 5R01 EB001988-21 from the National Institutes of Health. E.T. was supported by the Stanford Data Science Institute.

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Authors and Affiliations

  1. Department of Economics and Business, Universitat Pompeu Fabra and Barcelona School of Management, Barcelona, Spain

    Michael Greenacre

  2. Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam, Rotterdam, Netherlands

    Patrick J. F. Groenen

  3. Departments of Statistics and Biomedical Science, Stanford University, Stanford, CA, USA

    Trevor Hastie

  4. Department of Political Sciences, University of Naples Federico II, Naples, Italy

    Alfonso Iodice D’Enza

  5. Department of Primary Education, Democritus University of Thrace, Alexandroupolis, Greece

    Angelos Markos

  6. Department of Statistics, Stanford University, Stanford, CA, USA

    Elena Tuzhilina

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  1. Michael Greenacre
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  3. Trevor Hastie
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Contributions

Introduction (M.G. & T.H.); Experimentation (M.G., P.J.F.G. & T.H.); Results (M.G., P.J.F.G., T.H. & E.T.); Applications (M.G., P.J.F.G., T.H. & E.T.); Reproducibility and data deposition (M.G., A.I.D’E. & A.M.); Limitations and optimizations (M.G., T.H., A.I.D’E., A.M. & E.T.); Outlook (M.G., T.H., A.I.D’E., A.M. & E.T.); Overview of the Primer (all authors).

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Correspondence to Michael Greenacre.

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The authors declare no competing interests.

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Peer review information

Nature Reviews Methods Primers thanks Age Smilde, Carles Cuadras and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Related links

amap: https://CRAN.R-project.org/package=amap

elasticnet: https://CRAN.R-project.org/package=elasticnet

fdapace: https://CRAN.R-project.org/package=fdapace

irlba: https://CRAN.R-project.org/package=irlba

Musical illustration of the SVD: https://www.youtube.com/watch?v=JEYLfIVvR9I

onlinePCA: https://CRAN.R-project.org/package=onlinePCA

PCAtools: https://github.com/kevinblighe/PCAtools

pca3d: https://CRAN.R-project.org/package=pca3d

RSDA: https://CRAN.R-project.org/package=RSDA

RSpectra: https://CRAN.R-project.org/package=RSpectra

softImpute: https://CRAN.R-project.org/package=softImpute

stats: https://www.R-project.org/

symbolicDA: https://CRAN.R-project.org/package=symbolicDA

vegan: https://CRAN.R-project.org/package=vegan

Supplementary information

Supplementary Video 1 A three-dimensional animation of the centroid analysis of the four tumour groups.

43586_2022_184_MOESM2_ESM.mp4

Supplementary Video 2 A dynamic transition from the regular PCA to the PCA of the four tumour group centroids, as weight is transferred from the individual tumours to the tumour group centroids. This shows how the centroid analysis separates the groups better in the two-dimensional PCA solution, as well as how the highly contributing genes change.

43586_2022_184_MOESM3_ESM.mp4

Supplementary Video 3 A dynamic transition from the PCA of the group centroids to the corresponding sparse PCA solution. This shows how most genes are shrunk to the origin, and are thus eliminated, while the others are generally shrunk to the axes, which means they are contributing to only one PC. A few genes still contribute to both PCs.

Glossary

Active variables

Variables used to construct the principal component analysis solution.

Biplot

Joint representation in principal component analysis of the sampling units (usually the rows of the data matrix) represented as points in a scatterplot, often using the principal components as coordinates and variables (the columns) obtained from the right singular vectors shown as arrows.

Biplot axis

Axis in the direction of the variable arrow in a biplot.

Bootstrap

Process aimed at assessing the statistical variability of a solution by repeatedly creating a bootstrap dataset derived from the original dataset through sampling the cases with replacement and computing the solution each time.

Covariance matrix

Matrix containing the covariances between all pairs of variables.

Dense

In the context of a data matrix, the presence of very few or no zeros; in the context of principal component analysis, the presence of no zeros in the principal component coefficients.

Eigenvalue

In principal component analysis, a value indicating the accounted variance by a principal component.

Eigenvalue decomposition

Reconstruction of any square and symmetric matrix through a sum of rank-one matrices of the outer product of an eigenvector with itself (vvT) times the corresponding eigenvalue.

Eigenvector

In principal component analysis, this provides the linear combination for a principal component.

Euclidean distance

The measure of distance between two points defined as the length, in the physical sense, of the shortest straight line connecting these points.

Least-squares matrix approximation

Approximation of a data matrix such that the sum over all squared differences is minimized, between values in the data matrix and the corresponding approximated values.

Linear combination

For a set of variables, a sum of scalar coefficients times the variables.

Low-rank matrix approximation

Approximation of a matrix by one of lower rank.

Nonlinear multivariate analysis

General strategy that optimally assigns numerical values to the categories of a categorical variable and, in the context of principal component analysis, this strategy helps to increase the variance accounted for by the principal components.

Passive variables

Variables that are not used to determine the principal component analysis solution and are fitted into the solution afterwards, also called supplementary variables.

Permutation test

General computational method that compares a statistic of observed data with the distribution of the statistic simulated many times using data with the values randomly permuted under a certain null hypothesis.

Principal axis

The same as a dimension in principal component analysis and equivalent to the direction corresponding to maximal variance projections of the sampling units and uncorrelated to other principal axes.

Principal coordinates

The coordinates of the sampling units or variables on a dimension that have average sum of squares equal to the variance accounted for by that dimension.

Regressed

In the context of principal component analysis, using multiple regression to predict a variable from the principal components.

Scree plot

Plot of eigenvalue by dimension often used for selecting the number of principal component analysis dimensions by those above the straight line (scree) that goes approximately through the higher dimensions.

Shrinkage penalty

The addition to the objective function of an additional objective to reduce the absolute value of certain quantities being estimated; for example, the singular values in matrix completion, or the principal component coefficients in sparse principal component analysis.

Singular value

In principal component analysis, the square root of the variance accounted for by a principal component.

Singular value decomposition

Reconstruction of any matrix by the weighted sum of rank-one matrices consisting of the outer product of the left and right singular vectors (uvT) multiplied by their corresponding positive singular value.

Singular vectors

In principal component analysis (PCA), the vectors of the singular value decomposition that lead to the row and column coordinates in a PCA biplot.

Sparsity

In the context of a data matrix, the presence of many zeros; in the context of principal component analysis, the presence of many zeros in the principal component coefficients.

Standard coordinates

Coordinates in a principal component analysis that are standardized to have the average sum of squares equal to 1.

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Greenacre, M., Groenen, P.J.F., Hastie, T. et al. Principal component analysis. Nat Rev Methods Primers 2, 100 (2022). https://doi.org/10.1038/s43586-022-00184-w

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