Biplot: Simultaneous Display of Rows and Columns in Multivariate Data
Also known as: Gabriel biplot, PCA biplot, JK biplot, Çift grafik
A biplot is a low-dimensional graphical representation of a multivariate data matrix that simultaneously displays both the observations (rows) and the variables (columns) as points or vectors in the same plot. Introduced by Ruben Gabriel in 1971, the technique decomposes the data matrix into a rank-2 approximation using singular value decomposition, allowing the approximate value of any data entry to be read as the inner product of the corresponding row and column markers.
Key highlights
- Provides a simultaneous view of both observations and variables in a single two-dimensional plot, enabling holistic interpretation.
- The inner-product property gives a mathematically exact geometric meaning to the distances and projections visible in the plot.
- Generalizes naturally to many decomposition contexts including principal component analysis, canonical correspondence analysis, and generalized linear model biplots.
- Computationally efficient: relies on standard singular value decomposition, which is numerically stable and available in all major statistical software.
Intuition
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How it works
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When to use it
Use a biplot when you have a continuous multivariate dataset with at least three variables and want to explore structure, detect clusters, and understand variable relationships simultaneously in a single display. The method assumes numeric (ideally interval-scaled) data and works best when the first two principal components collectively explain a meaningful proportion of total variance (typically above 50%). It is less suitable for highly non-linear data, binary or nominal variables without transformation, or when more than two components are needed for an adequate summary. Alternatives include correspondence analysis for contingency tables and t-SNE or UMAP for strongly non-linear structures.
Strengths & limitations
- Provides a simultaneous view of both observations and variables in a single two-dimensional plot, enabling holistic interpretation.
- The inner-product property gives a mathematically exact geometric meaning to the distances and projections visible in the plot.
- Generalizes naturally to many decomposition contexts including principal component analysis, canonical correspondence analysis, and generalized linear model biplots.
- Computationally efficient: relies on standard singular value decomposition, which is numerically stable and available in all major statistical software.
- Quality of the display depends on the proportion of variance explained by the first two components; a low cumulative percentage makes the plot misleading.
- Interpretation requires statistical training; practitioners unfamiliar with inner-product geometry can misread distances or arrow lengths.
- Not directly applicable to binary, ordinal, or count data without appropriate pre-processing or model-based extensions.
- Overplotting can occur with large datasets, obscuring individual row markers and making the display difficult to read.
Common pitfalls
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Applications
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Frequently asked
What does the angle between two column arrows represent in a biplot?
The cosine of the angle between two column arrows approximates the correlation between those two variables. Arrows pointing in the same direction indicate high positive correlation, opposite directions indicate high negative correlation, and perpendicular arrows indicate near-zero correlation. This interpretation is exact only in the row-metric-preserving form (alpha = 1) and is approximate in other forms.
How do I choose the value of alpha?
The choice of alpha determines which distances are preserved. Use alpha = 1 (row-metric-preserving) when your primary goal is to interpret distances between observations. Use alpha = 0 (column-metric-preserving) when your focus is on variable relationships and you want the column arrows to span the same space as in principal component analysis. Alpha = 0.5 gives a symmetric, compromise display and is common in exploratory work.
Is the biplot the same as a PCA score plot?
No. A PCA score plot displays only the row (observation) scores on the first two principal components, without any representation of the variables. A biplot adds column markers (variable loadings) to the same display, enabling joint interpretation of observations and variables. The score plot is a special case of the biplot with the column markers omitted.
Sources
- 1.Gabriel, K. R. (1971). The biplot graphic display of matrices with application to principal component analysis. Biometrika, 58(3), 453–467.
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Cite this page
ScholarGate. (2026, June 2). Biplot. ScholarGate. https://scholargate.app/statistics/biplot