Biplot for Principal Components using ggplot2

A biplot simultaneously displays information on the observations (as points) and the variables (as vectors) in a multidimensional dataset. The 2D biplot is typically based on the first two principal components of a dataset, giving a rank 2 approximation to the data. The “bi” in biplot refers to the fact that two sets of points (i.e., the rows and columns of the data matrix) are visualized by scalar products, not the fact that the display is usually two-dimensional.

The biplot method for principal component analysis was originally defined by Gabriel (1971, 1981). Gower & Hand (1996) give a more complete treatment. Greenacre (2010) is a practical user-oriented guide to biplots. Gower et al. (2011) is the most up to date exposition of biplot methodology.

This implementation handles the results of a principal components analysis using prcomp, princomp, PCA and dudi.pca; also handles a discriminant analysis using lda.

Usage

ggbiplot(
  pcobj,
  choices = 1:2,
  scale = 1,
  pc.biplot = TRUE,
  obs.scale = 1 - scale,
  var.scale = scale,
  var.factor = 1,
  groups = NULL,
  point.size = 1.5,
  ellipse = FALSE,
  ellipse.prob = 0.68,
  ellipse.linewidth = 1.3,
  ellipse.fill = TRUE,
  ellipse.alpha = 0.25,
  labels = NULL,
  labels.size = 3,
  alpha = 1,
  var.axes = TRUE,
  circle = FALSE,
  circle.prob = 0.68,
  varname.size = 3,
  varname.adjust = 1.25,
  varname.color = "black",
  varname.abbrev = FALSE,
  axis.title = "PC",
  ...
)

Arguments

pcobj: an object returned by prcomp, princomp, PCA, dudi.pca, or lda
choices: Which components to plot? An integer vector of length 2.
scale: Covariance biplot (scale = 1), form biplot (scale = 0). When scale = 1 (the default), the inner product between the variables approximates the covariance and the distance between the points approximates the Mahalanobis distance.
pc.biplot: Logical, for compatibility with biplot.princomp(). If TRUE, use what Gabriel (1971) refers to as a "principal component biplot", with $\alpha = 1$ and observations scaled up by $sqrt(n)$ and variables scaled down by $sqrt(n)$. Then inner products between variables approximate covariances and distances between observations approximate Mahalanobis distance.
obs.scale: Scale factor to apply to observations
var.scale: Scale factor to apply to variables
var.factor: Factor to be applied to variable vectors after scaling. This allows the variable vectors to be reflected (var.factor = -1) or expanded in length (var.factor > 1) for greater visibility. reflect provides a simpler way to reflect the variables.
groups: Optional factor variable indicating the groups that the observations belong to. If provided the points will be colored according to groups and this allows data ellipses also to be drawn when ellipse = TRUE.
point.size: Size of observation points.
ellipse: Logical; draw a normal data ellipse for each group?
ellipse.prob: Coverage size of the data ellipse in Normal probability
ellipse.linewidth: Thickness of the line outlining the ellipses
ellipse.fill: Logical; should the ellipses be filled?
ellipse.alpha: Transparency value (0 - 1) for filled ellipses
labels: Optional vector of labels for the observations. Often, this will be specified as the row.names() of the dataset.
labels.size: Size of the text used for the point labels
alpha: Alpha transparency value for the points (0 = transparent, 1 = opaque)
var.axes: logical; draw arrows for the variables?
circle: draw a correlation circle? (only applies when prcomp was called with scale = TRUE and when var.scale = 1)
circle.prob: Size of the correlation circle
varname.size: Size of the text for variable names
varname.adjust: Adjustment factor the placement of the variable names, >= 1 means farther from the arrow
varname.color: Color for the variable vectors and names
varname.abbrev: logical; whether or not to abbreviate the variable names, using abbreviate.
axis.title: character; the prefix used as the axis labels. Default: "PC".
...: other arguments passed down

Value

a ggplot2 plot object of class c("gg", "ggplot")

Details

The biplot is constructed by using the singular value decomposition (SVD) to obtain a low-rank approximation to the data matrix $\mathbf{X}_{n \times p}$ (centered, and optionally scaled to unit variances) whose $n$ rows are the observations and whose $p$ columns are the variables.

Using the SVD, the matrix $\mathbf{X}$, of rank $r \le p$ can be expressed exactly as $$\mathbf{X} = \mathbf{U} \mathbf{\Lambda} \mathbf{V}' = \Sigma_i^r \lambda_i \mathbf{u}_i \mathbf{v}_i' \; ,$$

where

$\mathbf{U}$ is an $n \times r$ orthonormal matrix of observation scores; these are also the eigenvectors of $\mathbf{X} \mathbf{X}'$,
$\mathbf{\Lambda}$ is an $r \times r$ diagonal matrix of singular values, $\lambda_1 \ge \lambda_2 \ge \cdots \lambda_r$
$\mathbf{V}$ is an $r \times p$ orthonormal matrix of variable weights and also the eigenvectors of $\mathbf{X}' \mathbf{X}$.

Then, a rank 2 (or 3) PCA approximation $\widehat{\mathbf{X}}$ to the data matrix used in the biplot can be obtained from the first 2 (or 3) singular values $\lambda_i$ and the corresponding $\mathbf{u}_i, \mathbf{v}_i$ as

$$\mathbf{X} \approx \widehat{\mathbf{X}} = \lambda_1 \mathbf{u}_1 \mathbf{v}_1' + \lambda_2 \mathbf{u}_2 \mathbf{v}_2' \; .$$

The variance of $\mathbf{X}$ accounted for by each term is $\lambda_i^2$.

The biplot is then obtained by overlaying two scatterplots that share a common set of axes and have a between-set scalar product interpretation. Typically, the observations (rows of $\mathbf{X}$) are represented as points and the variables (columns of $\mathbf{X}$) are represented as vectors from the origin.

The scale factor, $\alpha$ allows the variances of the components to be apportioned between the row points and column vectors, with different interpretations, by representing the approximation $\widehat{\mathbf{X}}$ as the product of two matrices,

$$\widehat{\mathbf{X}} = (\mathbf{U} \mathbf{\Lambda}^\alpha) (\mathbf{\Lambda}^{1-\alpha} \mathbf{V}')$$

The choice $\alpha = 1$, assigning the singular values totally to the left factor, gives a distance interpretation to the row display and $\alpha = 0$ gives a distance interpretation to the column display. $\alpha = 1/2$ gives a symmetrically scaled biplot.

When the singular values are assigned totally to the left or to the right factor, the resultant coordinates are called principal coordinates and the sum of squared coordinates on each dimension equal the corresponding singular value. The other matrix, to which no part of the singular values is assigned, contains the so-called standard coordinates and have sum of squared values equal to 1.0.

References

Gabriel, K. R. (1971). The biplot graphical display of matrices with application to principal component analysis. Biometrika, 58, 453–467. doi:10.2307/2334381 .

Gabriel, K. R. (1981). Biplot display of multivariate matrices for inspection of data and diagnosis. In V. Barnett (Ed.), Interpreting Multivariate Data. London: Wiley.

Greenacre, M. (2010). Biplots in Practice. BBVA Foundation, Bilbao, Spain. Available for free at https://www.fbbva.es/microsite/multivariate-statistics/.

J.C. Gower and D. J. Hand (1996). Biplots. Chapman & Hall.

Gower, J. C., Lubbe, S. G., & Roux, N. J. L. (2011). Understanding Biplots. Wiley.

Author

Vincent Q. Vu.

Examples

data(wine)
library(ggplot2)
wine.pca <- prcomp(wine, scale. = TRUE)
ggbiplot(wine.pca, 
         obs.scale = 1, var.scale = 1, 
         varname.size = 4,
         groups = wine.class, 
         ellipse = TRUE, circle = TRUE)


data(iris)
iris.pca <- prcomp (~ Sepal.Length + Sepal.Width + Petal.Length + Petal.Width,
                    data=iris,
                    scale. = TRUE)
ggbiplot(iris.pca, obs.scale = 1, var.scale = 1,
         groups = iris$Species, point.size=2,
         varname.size = 5, 
         varname.color = "black",
         varname.adjust = 1.2,
         ellipse = TRUE, 
         circle = TRUE) +
  labs(fill = "Species", color = "Species") +
  theme_minimal(base_size = 14) +
  theme(legend.direction = 'horizontal', legend.position = 'top')