# Canonical variate analysis

Canonical variate analysis (CVA) is a multivariate statistical method that finds linear combinations of measured variables which maximally separate pre-defined groups, or, in a second sense, linear combinations of two variable sets that are maximally correlated. It is used for classification. The name covers two related traditions: one, tied to discriminant analysis, maximizes the ratio of between-group to within-group variance<sup>[1](http://www.filogenetica.org/bibliografia/morfometria/ordenacion/Campbell&Atchley%201979CVA.pdf)</sup><sup> • </sup><sup>[2](https://support.nag.com/numeric/cl/nagdoc_latest/html/g03/g03acc.html)</sup>; the other, tied to canonical correlation analysis (CCA), maximizes correlation between two sets of variates.<sup>[3](https://doi.org/10.1093/biomet/28.3-4.321)</sup> [Terminology](https://www.edgepedia.org/terminology) in the literature is genuinely tangled: CVA is often equated with linear discriminant analysis (LDA), while CCA is sometimes itself called canonical variates analysis.<sup>[4](https://bio723-class.github.io/Bio723-book/canonical-variates-analysis.html)</sup><sup> • </sup><sup>[5](https://jlahne.github.io/r-opus-v2/canonical-variate-analysis.html)</sup>

| Key fact | Detail |
|---|---|
| Quantity maximized | Ratio of between-group to within-group variance, \( \lambda = \mathbf{l}'\mathbf{B}\mathbf{l} / \mathbf{l}'\mathbf{W}\mathbf{l} \)<sup>[6](https://link.springer.com/article/10.1007/s10260-025-00831-y)</sup> |
| Eigenvalue problem | \( \mathbf{B}\mathbf{l} = \lambda \mathbf{W}\mathbf{l} \), equivalently \( (\mathbf{B} - f\mathbf{W})\mathbf{c} = 0 \)<sup>[6](https://link.springer.com/article/10.1007/s10260-025-00831-y)</sup><sup> • </sup><sup>[1](http://www.filogenetica.org/bibliografia/morfometria/ordenacion/Campbell&Atchley%201979CVA.pdf)</sup> |
| Number of variates | \( h = \min(v,\, g-1) \) with non-zero roots, for \( v \) variables and \( g \) groups<sup>[1](http://www.filogenetica.org/bibliografia/morfometria/ordenacion/Campbell&Atchley%201979CVA.pdf)</sup> |
| Key requirement | Within-group scatter matrix must be nonsingular, which requires \( v \le n-g \) and full column rank of the within-group residuals<sup>[6](https://link.springer.com/article/10.1007/s10260-025-00831-y)</sup> |
| Significance testing | Bartlett's 1938 chi-square statistic with \( (v-k) \times (g-k-1) \) degrees of freedom<sup>[7](https://genstat.kb.vsni.co.uk/knowledge-base/cva/)</sup> |
| Historical origin | Canonical variates and canonical correlations introduced by H. Hotelling, Biometrika, 1936<sup>[3](https://doi.org/10.1093/biomet/28.3-4.321)</sup> |
| Named variant | Canonical ridge, H. D. Vinod, Journal of Econometrics, 1976<sup>[8](https://doi.org/10.1016/0304-4076%2876%2990010-5)</sup> |

## How it works

In the group-separation sense, CVA takes an \( n \times p \) data matrix whose samples fall in \( g \) groups and seeks the linear combination of variables that best represents the group means, separated by Mahalanobis distances (distance measure accounting for variable correlations and scale), in a low-dimensional space.<sup>[9](https://www.casperalbers.nl/files/canonicalanalysis2014.pdf)</sup> Let \( \mathbf{B} \) denote the between-group and \( \mathbf{W} \) the within-group sums-of-squares-and-products matrices. The coefficient vector \( \mathbf{c} \) is chosen to maximize the ratio

\[ f = \frac{\mathbf{c}'\mathbf{B}\mathbf{c}}{\mathbf{c}'\mathbf{W}\mathbf{c}}, \]

which leads to the fundamental eigenequation \( (\mathbf{B} - f\mathbf{W})\mathbf{c} = 0 \)<sup>[1](http://www.filogenetica.org/bibliografia/morfometria/ordenacion/Campbell&Atchley%201979CVA.pdf)</sup>, written equivalently as the two-sided problem \( \mathbf{B}\mathbf{l} = \lambda \mathbf{W}\mathbf{l} \).<sup>[6](https://link.springer.com/article/10.1007/s10260-025-00831-y)</sup> CVA was originally defined as optimizing exactly this ratio of two quadratic forms, requiring the maximal eigenvalue of a two-sided eigenvalue problem.<sup>[9](https://www.casperalbers.nl/files/canonicalanalysis2014.pdf)</sup>

Rank limits the output: for \( g \) groups and \( v \) variables there are \( h = \min(v, g-1) \) canonical vectors with non-zero canonical roots, and when \( g-1 < v \) the group means lie in a \( g-1 \)-dimensional subspace.<sup>[1](http://www.filogenetica.org/bibliografia/morfometria/ordenacion/Campbell&Atchley%201979CVA.pdf)</sup> Equivalently, the number of linear combinations needed is \( \operatorname{rank}(\mathbf{B}) \le \min(v, g-1) \), reaching the maximum only when the group means span the full possible rank.<sup>[6](https://link.springer.com/article/10.1007/s10260-025-00831-y)</sup>

In the two-set sense, CCA identifies \( r \le \min(p, q) \) linear combinations of \( x \) and \( y \) that are maximally correlated, solving \( \max\ \mathbf{u}^\top \Sigma_{XY} \mathbf{v} \) subject to unit-variance and orthogonality constraints; the pairs \( (x\mathbf{u}_i, y\mathbf{v}_i) \) are the canonical variates, \( \mathbf{u}_i, \mathbf{v}_i \) the canonical directions, and \( \lambda_i = \mathbf{u}_i^\top \Sigma_{XY} \mathbf{v}_i \) the canonical correlations.<sup>[10](http://jmlr.org/papers/volume27/25-0196/25-0196.pdf)</sup> The solutions satisfy eigenvalue-type equations \( \Sigma_X^{-1} \Sigma_{XY} \Sigma_Y^{-1} \Sigma_{YX} U = U \Lambda^2 \) and its symmetric counterpart for \( V \).<sup>[10](http://jmlr.org/papers/volume27/25-0196/25-0196.pdf)</sup>

## How it is done

A practitioner works through the following steps<sup>[11](https://metricgate.com/docs/canonical-variate-analysis/)</sup>:

1. Assemble the \( n \times p \) data matrix with group labels for each row.
2. Compute the within-group scatter matrix \( \mathbf{W} \) and between-group scatter matrix \( \mathbf{B} \).
3. Solve the eigenvalue problem, computing the eigendecomposition of \( \mathbf{W}^{-1}\mathbf{B} \) (or via singular value decomposition, which some implementations use<sup>[2](https://support.nag.com/numeric/cl/nagdoc_latest/html/g03/g03acc.html)</sup>; the canonical vectors can also be obtained from a standard or generalized eigenvalue problem<sup>[12](https://ar5iv.labs.arxiv.org/html/1711.02391)</sup>).
4. Retain the top \( \min(g-1, v) \) eigenvectors, ordered by descending eigenvalue, and project the data to obtain canonical scores.<sup>[11](https://metricgate.com/docs/canonical-variate-analysis/)</sup>
5. Test significance of successive dimensions with Bartlett's chi-square statistic, asymptotically chi-square with \( (v-k) \times (g-k-1) \) degrees of freedom, tested for \( k = 0, 1, \ldots, \min(g-1, v)-1 \).<sup>[7](https://genstat.kb.vsni.co.uk/knowledge-base/cva/)</sup>
6. Inspect a scree diagram of the eigenvalues in decreasing order, which indicates the proportion of the between- to within-group variance ratio captured and helps decide how many variates are needed.<sup>[13](https://simfit.uk/canonical_variates.pdf)</sup>

Interpretation uses structure correlations, the correlations of the original variables with the canonical images (scores), which convey how the scores align with the variable axes<sup>[12](https://ar5iv.labs.arxiv.org/html/1711.02391)</sup>; plotting these rather than raw coefficients is the usual practice.<sup>[5](https://jlahne.github.io/r-opus-v2/canonical-variate-analysis.html)</sup> Latent vectors are commonly scaled so the average within-group variability in each canonical variate dimension is 1, so loadings for roots below 1 mark dimensions with more within- than between-group variation.<sup>[7](https://genstat.kb.vsni.co.uk/knowledge-base/cva/)</sup> Group-mean scores are centered so their size-weighted centroid sits at the origin.<sup>[7](https://genstat.kb.vsni.co.uk/knowledge-base/cva/)</sup>

## Origin

Canonical variates and canonical correlations were introduced by H. Hotelling in "Relations between two sets of variates" ([Biometrika](https://www.edgepedia.org/biometrika), 1936), which defines them as the invariant relations between two sets of variates representable by correlation coefficients.<sup>[3](https://doi.org/10.1093/biomet/28.3-4.321)</sup> A 1935 paper, "The most predictable criterion" in the Journal of Educational Psychology, immediately precedes it.<sup>[14](https://doi.org/10.1037/h0058165)</sup><sup> • </sup><sup>[15](https://journals.sagepub.com/doi/10.3102/00346543043004433)</sup>

The attribution of group-separation CVA is not settled across traditions <sup>[1](http://www.filogenetica.org/bibliografia/morfometria/ordenacion/Campbell&Atchley%201979CVA.pdf)</sup>, while other accounts credit Hotelling's 1936 paper with the introduction of canonical variates<sup>[3](https://doi.org/10.1093/biomet/28.3-4.321)</sup><sup> • </sup><sup>[10](http://jmlr.org/papers/volume27/25-0196/25-0196.pdf)</sup> and one source credits the book-length treatment "Canonical variate analysis" by Robert Gittins (Biomathematics, 1985).<sup>[16](https://doi.org/10.1007/978-3-642-69878-1_4)</sup><sup> • </sup><sup>[6](https://link.springer.com/article/10.1007/s10260-025-00831-y)</sup> These reflect two distinct lineages, discriminant analysis and canonical correlation, rather than a single priority claim that the sources resolve.

## Variants

**Regularization.** The canonical ridge, proposed by H. D. Vinod in the [Journal of Econometrics](https://www.edgepedia.org/journal-of-econometrics) in 1976, adapts the ridge regression idea to the canonical framework by adding penalty terms \( k_1 \) and \( k_2 \) to the diagonals of the sample covariance matrices, addressing the case where variables exceed the sample size.<sup>[8](https://doi.org/10.1016/0304-4076%2876%2990010-5)</sup><sup> • </sup><sup>[17](https://ar5iv.labs.arxiv.org/html/1501.01231)</sup> Ridge CCA corresponds to an \( \ell_2 \) penalty on the canonical directions, while sparse CCA methods penalize the \( \ell_1 \)-norm.<sup>[18](https://arxiv.org/html/2403.02979v1)</sup>

**Nonlinear and multi-view.** Kernel CCA maps data into a high-dimensional feature space induced by a kernel and then performs linear CCA, so nonlinear relationships can be found; two-stage kernel CCA selects sparse features and multiple nonlinear associations.<sup>[19](https://link.springer.com/article/10.1186/s12859-017-1543-x)</sup> Generalized CCA extends the method to more than two views.<sup>[20](https://arxiv.org/html/2502.18756)</sup> Sparse, deep, and Bayesian extensions address small-sample, nonlinear, and high-dimensional settings.<sup>[12](https://ar5iv.labs.arxiv.org/html/1711.02391)</sup>

## Applications

In morphometrics and systematic biology, CVA separates morphologically similar organisms, assesses intergroup affinities, and allocates individuals to pre-existing groups.<sup>[1](http://www.filogenetica.org/bibliografia/morfometria/ordenacion/Campbell&Atchley%201979CVA.pdf)</sup> In sensory science, principal component analysis of product mean scores ignores panelist variability, so CVA of the product effect in a two-way MANOVA is a natural alternative, generating successive components that maximize product discrimination.<sup>[21](https://onlinelibrary.wiley.com/doi/10.1111/joss.12160)</sup> [Canonical correlation analysis](https://www.edgepedia.org/canonical-correlation-analysis) in the two-set sense remains relevant in domains including genetics and neuroscience.<sup>[22](https://par.nsf.gov/servlets/purl/10324406)</sup>

## Limitations and alternatives

**Assumptions and failure modes.** CVA assumes homogeneity of within-group covariance; violations can distort the canonical axes and inflate apparent separation.<sup>[11](https://metricgate.com/docs/canonical-variate-analysis/)</sup> The within-class scatter matrix must be nonsingular, so the number of variables \( v \) should not exceed the number of observations \( n \); covariance-matrix regularization is one remedy when \( v > n \).<sup>[6](https://link.springer.com/article/10.1007/s10260-025-00831-y)</sup> The method is sensitive to multicollinearity, which makes \( \mathbf{W} \) near-singular, and without kernel extensions it captures only linear boundaries.<sup>[11](https://metricgate.com/docs/canonical-variate-analysis/)</sup> Significance tests should be treated with caution unless \( n - g \) is very much larger than \( v \).<sup>[7](https://genstat.kb.vsni.co.uk/knowledge-base/cva/)</sup>

**Comparison with alternatives.** PCA maximizes total variance regardless of group structure, while CVA maximizes the ratio of between-group to within-group variance.<sup>[11](https://metricgate.com/docs/canonical-variate-analysis/)</sup> CVA is closely related to canonical correlation analysis and often equated with (linear) discriminant analysis, sometimes called Fisher's linear discriminant analysis; it is a supervised method requiring group identifiers.<sup>[5](https://jlahne.github.io/r-opus-v2/canonical-variate-analysis.html)</sup> A further interpretive caveat: CVA does not in general produce orthogonal linear functions, so plotting canonical variates as Cartesian axes slightly distorts the space.<sup>[5](https://jlahne.github.io/r-opus-v2/canonical-variate-analysis.html)</sup>

**Recent methodological work.** A 2026 extension builds CVA biplots via the generalized singular value decomposition, removing the \( v \le n \) restriction for biplot display; it is implemented in the wideRhino R package.<sup>[6](https://link.springer.com/article/10.1007/s10260-025-00831-y)</sup>

## References

1. [The Geometry of Canonical Variate Analysis (Campbell & Atchley, 1979, Systematic Zoology)](http://www.filogenetica.org/bibliografia/morfometria/ordenacion/Campbell&Atchley%201979CVA.pdf)
2. [nag_mv_canon_var (g03acc), NAG C Library documentation](https://support.nag.com/numeric/cl/nagdoc_latest/html/g03/g03acc.html)
3. [H. HOTELLING (1936). RELATIONS BETWEEN TWO SETS OF VARIATES. Biometrika.](https://doi.org/10.1093/biomet/28.3-4.321)
4. [Chapter 14 Canonical Variates Analysis, Biology 723](https://bio723-class.github.io/Bio723-book/canonical-variates-analysis.html)
5. [Chapter 5 Canonical Variate Analysis, The R Opus v2](https://jlahne.github.io/r-opus-v2/canonical-variate-analysis.html)
6. [A canonical variate analysis biplot based on the generalized singular value decomposition (Statistical Methods & Applications, 2025)](https://link.springer.com/article/10.1007/s10260-025-00831-y)
7. [CVA directive, Genstat Knowledge Base 2024](https://genstat.kb.vsni.co.uk/knowledge-base/cva/)
8. [Canonical ridge and econometrics of joint production (Journal of Econometrics, 1976)](https://doi.org/10.1016/0304-4076%2876%2990010-5)
9. [Canonical Analysis: Ranks, Ratios and Fits (Albers & Gower, 2014)](https://www.casperalbers.nl/files/canonicalanalysis2014.pdf)
10. [Canonical Correlation Analysis as Reduced Rank Regression in High Dimensions (JMLR, 2025)](http://jmlr.org/papers/volume27/25-0196/25-0196.pdf)
11. [Canonical Variate Analysis Calculator, MetricGate documentation](https://metricgate.com/docs/canonical-variate-analysis/)
12. [A Tutorial on Canonical Correlation Methods](https://ar5iv.labs.arxiv.org/html/1711.02391)
13. [Canonical variates analysis, Simfit documentation](https://simfit.uk/canonical_variates.pdf)
14. [H. Hotelling (1935). The most predictable criterion.. Journal of Educational Psychology.](https://doi.org/10.1037/h0058165)
15. [Canonical Variate Analysis and Related Techniques (Review of Educational Research)](https://journals.sagepub.com/doi/10.3102/00346543043004433)
16. [Robert Gittins (1985). Canonical variate analysis. Biomathematics.](https://doi.org/10.1007/978-3-642-69878-1_4)
17. [Sparse canonical correlation analysis from a predictive point of view](https://ar5iv.labs.arxiv.org/html/1501.01231)
18. [Regularised Canonical Correlation Analysis: graphical lasso, biplots and beyond (arXiv, 2024)](https://arxiv.org/html/2403.02979v1)
19. [Sparse kernel canonical correlation analysis for discovery of nonlinear interactions in high-dimensional data (BMC Bioinformatics, 2017)](https://link.springer.com/article/10.1186/s12859-017-1543-x)
20. [Nonlinear Sparse Generalized Canonical Correlation Analysis for Multi-view High-dimensional Data (arXiv, 2025)](https://arxiv.org/html/2502.18756)
21. [Canonical Variate Analysis of Sensory Profiling Data (Journal of Sensory Studies, 2015)](https://onlinelibrary.wiley.com/doi/10.1111/joss.12160)
22. [Canonical correlation analysis in high dimensions with structured regularization](https://par.nsf.gov/servlets/purl/10324406)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Multivariate association and dimension reduction › Titles A to Me*

*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*

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