High Dimensional Data Visualization

Wayne Oldford and Zehao Xu

2021-03-10

Serialaxes 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 \(R^2\) (Inselberg and Dimsdale 1990).

Before we start, a couple of things should be noticed:

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",
            # no scaling is applied
            scaling = "none")  + 
  coord_serialaxes()
p

A quantile layer can be displayed on top

p + 
 geom_quantiles(stat = "dotProduct",
                scaling = "none",
                quantiles = c(0.25, 0.5, 0.75),
                size = 2,
                linetype = 2) 

A couple of things should be noticed:

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_radar()` or `coord_polar(is_linear = TRUE)`
# this is slightly different, check it out! 
g + 
  geom_serialaxes(alpha = 0.2) + 
  geom_serialaxes(alpha = 0.2) + 
  coord_radar()

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:

# 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’” (Inselberg 1999).

Reference

Andrews, David F. 1972. “Plots of High-Dimensional Data.” Biometrics, 125–0136.

Gnanadesikan, Ram. 2011. “Methods for Statistical Data Analysis of Multivariate Observations.” In, 321:207–0218. John Wiley & Sons.

Inselberg, A., and B. Dimsdale. 1990. “Parallel Coordinates: A Tool for Visualizing Multi-Dimensional Geometry.” In Proceedings of the First Ieee Conference on Visualization: Visualization ‘90, 361–0378.

Inselberg, Alfred. 1999. “Don’t Panic... Just Do It in Parallel!” Computational Statistics 14 (1): 53–077.