High Dimensional Data Visualization
Wayne Oldford and Zehao Xu
2022-10-22
highDim.RmdSerialaxes coordinate
Serial axes coordinate is a methodology for visualizing the \(p\)-dimensional geometry and multivariate data. As the name suggested, all axes are shown in serial. The axes can be a finite \(p\) space or transformed to an infinite space (e.g. Fourier transformation).
In the finite \(p\) space, all axes
can be displayed in parallel which is known as the parallel coordinate;
also, all axes can be displayed under a polar coordinate that is often
known as the radial coordinate or radar plot. In the infinite space, a
mathematical transformation is often applied. More details will be
explained in the sub-section Infinite axes
A point in Euclidean \(p\)-space \(R^p\) is represented as a polyline in serial axes coordinate, it is found that a point <–> line duality is induced in the Euclidean plane \(R^2\) (A. Inselberg and Dimsdale 1990).
Before we start, a couple of things should be noticed:
In the serial axes coordinate system, no
xory(evengroup) are required; but other aesthetics, such ascolour,fill,size, etc, are accommodated.Layer
geom_pathis used to draw the serial lines; layergeom_histogram,geom_quantiles, andgeom_densityare used to draw the histograms, quantiles (notquantileregression) and densities. Users can also customize their own layer (i.e.geom_boxplot,geom_violin, etc) by editing functionadd_serialaxes_layers.
Finite axes
Suppose we are interested in the data set iris. A
parallel coordinate chart can be created as followings:
library(ggmulti)
# parallel axes plot
ggplot(iris,
mapping = aes(
Sepal.Length = Sepal.Length,
Sepal.Width = Sepal.Width,
Petal.Length = Petal.Length,
Petal.Width = Petal.Width,
colour = factor(Species))) +
geom_path(alpha = 0.2) +
coord_serialaxes() -> p
p
A histogram layer can be displayed by adding layer
geom_histogram
p +
geom_histogram(alpha = 0.3,
mapping = aes(fill = factor(Species))) +
theme(axis.text.x = element_text(angle = 30, hjust = 0.7))
A density layer can be drawn by adding layer
geom_density
p +
geom_density(alpha = 0.3,
mapping = aes(fill = factor(Species)))
A parallel coordinate can be converted to radial coordinate by
setting axes.layout = "radial" in function
coord_serialaxes.
p$coordinates$axes.layout <- "radial"
p
Note that: layers, such as
geom_histogram, geom_density, etc, are not
implemented in the radial coordinate yet.
Infinite axes
Andrews (1972) plot is a way to project multi-response observations into a function \(f(t)\), by defining \(f(t)\) as an inner product of the observed values of responses and orthonormal functions in \(t\)
\[f_{y_i}(t) = <\mathbf{y}_i, \mathbf{a}_t>\]
where \(\mathbf{y}_i\) is the \(i\)th responses and \(\mathbf{a}_t\) is the orthonormal functions under certain interval. Andrew suggests to use the Fourier transformation
\[\mathbf{a}_t = \{\frac{1}{\sqrt{2}}, \sin(t), \cos(t), \sin(2t), \cos(2t), ...\}\]
which are orthonormal on interval \((-\pi, \pi)\). In other word, we can project a \(p\) dimensional space to an infinite \((-\pi, \pi)\) space. The following figure illustrates how to construct an “Andrew’s plot”.
p <- ggplot(iris,
mapping = aes(Sepal.Length = Sepal.Length,
Sepal.Width = Sepal.Width,
Petal.Length = Petal.Length,
Petal.Width = Petal.Width,
colour = Species)) +
geom_path(alpha = 0.2,
stat = "dotProduct") +
coord_serialaxes()
p
A quantile layer can be displayed on top
p +
geom_quantiles(stat = "dotProduct",
quantiles = c(0.25, 0.5, 0.75),
linewidth = 2,
linetype = 2) 
A couple of things should be noticed:
mapping aesthetics is used to define the \(p\) dimensional space, if not provided, all columns in the dataset ‘iris’ will be transformed. An alternative way to determine the \(p\) dimensional space to set parameter
axes.sequencein each layer or incoord_serialaxes.-
To construct a dot product serial axes plot, say Fourier transformation, “Andrew’s plot”, we need to set the parameter
statingeom_pathto “dotProduct”. The default transformation function is the Andrew’s (functionandrews). Users can customize their own, for example, Tukey suggests the following projected space\[\mathbf{a}_t = \{\cos(t), \cos(\sqrt{2}t), \cos(\sqrt{3}t), \cos(\sqrt{5}t), ...\}\]
where \(t \in [0, k\pi]\) (Gnanadesikan 2011).
tukey <- function(p = 4, k = 50 * (p - 1), ...) { t <- seq(0, p* base::pi, length.out = k) seq_k <- seq(p) values <- sapply(seq_k, function(i) { if(i == 1) return(cos(t)) if(i == 2) return(cos(sqrt(2) * t)) Fibonacci <- seq_k[i - 1] + seq_k[i - 2] cos(sqrt(Fibonacci) * t) }) list( vector = t, matrix = matrix(values, nrow = p, byrow = TRUE) ) } ggplot(iris, mapping = aes(Sepal.Length = Sepal.Length, Sepal.Width = Sepal.Width, Petal.Length = Petal.Length, Petal.Width = Petal.Width, colour = Species)) + geom_path(alpha = 0.2, stat = "dotProduct", transform = tukey) + coord_serialaxes()
Note that: Tukey’s suggestion, element \(\mathbf{a}_t\) can “cover” more spheres in \(p\) dimensional space, but it is not orthonormal.
An alternative way to create a serial axes plot
Rather than calling function coord_serialaxes, an
alternative way to create a serial axes object is to add a
geom_serialaxes_... object in our model.
For example, Figure 1 to 4 can be created by calling
g <- ggplot(iris,
mapping = aes(Sepal.Length = Sepal.Length,
Sepal.Width = Sepal.Width,
Petal.Length = Petal.Length,
Petal.Width = Petal.Width,
colour = Species))
g + geom_serialaxes(alpha = 0.2)
g +
geom_serialaxes(alpha = 0.2) +
geom_serialaxes_hist(mapping = aes(fill = Species), alpha = 0.2)
g +
geom_serialaxes(alpha = 0.2) +
geom_serialaxes_density(mapping = aes(fill = Species), alpha = 0.2)
# radial axes can be created by
# calling `coord_radial()`
# this is slightly different, check it out!
g +
geom_serialaxes(alpha = 0.2) +
geom_serialaxes(alpha = 0.2) +
coord_radial()Figure 5 and 7 can be created by setting “stat” and “transform” in
geom_serialaxes; to Figure 6,
geom_serialaxes_quantile can be added to create a serial
axes quantile layer.
Some slight difference should be noticed here:
One benefit of calling
coord_serialaxesrather thangeom_serialaxes_...is thatcoord_serialaxescan accommodate duplicated axes in mapping aesthetics (e.g. Eulerian path, Hamiltonian path, etc). However, ingeom_serialaxes_..., duplicated axes will be omitted.Meaningful axes labels in
coord_serialaxescan be created automatically, while ingeom_serialaxes_..., users have to set axes labels byggplot2::scale_x_continuousorggplot2::scale_y_continuousmanually.As we turn the serial axes into interactive graphics (via package loon.ggplot), serial axes lines in
coord_serialaxes()could be turned as interactive but ingeom_serialaxes_...all objects are static.
# The serial axes is `Sepal.Length`, `Sepal.Width`, `Sepal.Length`
# With meaningful labels
ggplot(iris,
mapping = aes(Sepal.Length = Sepal.Length,
Sepal.Width = Sepal.Width,
Sepal.Length = Sepal.Length)) +
geom_path() +
coord_serialaxes()
# The serial axes is `Sepal.Length`, `Sepal.Length`
# No meaningful labels
ggplot(iris,
mapping = aes(Sepal.Length = Sepal.Length,
Sepal.Width = Sepal.Width,
Sepal.Length = Sepal.Length)) +
geom_serialaxes()Also, if the dimension of data is large, typing each variate in
mapping aesthetics is such a headache. Parameter
axes.sequence is provided to determine the axes. For
example, a serialaxes object can be created as
ggplot(iris) +
geom_path() +
coord_serialaxes(axes.sequence = colnames(iris)[-5])At very end, please report bugs here. Enjoy the high dimensional visualization! “Don’t panic… Just do it in ‘serial’” (Alfred Inselberg 1999).