ggbiplot | R Documentation |
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
.
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",
...
)
pcobj |
an object returned by |
choices |
Which components to plot? An integer vector of length 2. |
scale |
Covariance biplot ( |
pc.biplot |
Logical, for compatibility with |
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
( |
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 |
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 |
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
|
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 |
axis.title |
character; the prefix used as the axis labels. Default: |
... |
other arguments passed down |
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.
a ggplot2 plot object of class c("gg", "ggplot")
Vincent Q. Vu.
Gabriel, K. R. (1971). The biplot graphical display of matrices with application to principal component analysis. Biometrika, 58, 453–467. \Sexpr[results=rd]{tools:::Rd_expr_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.
reflect
, ggscreeplot
;
biplot
for the original stats package version;
fviz_pca_biplot
for the factoextra package version.
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')
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