High Dimensional Data Visualization

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.sequence in each layer or in coord_serialaxes.

• To construct a dot product serial axes plot, say Fourier transformation, “Andrew’s plot”, we need to set the parameter stat in geom_path to “dotProduct”. The default transformation function is the Andrew’s (function andrews). 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_serialaxes rather than geom_serialaxes_... is that coord_serialaxes can accommodate duplicated axes in mapping aesthetics (e.g. Eulerian path, Hamiltonian path, etc). However, in geom_serialaxes_..., duplicated axes will be omitted.

• Meaningful axes labels in coord_serialaxes can be created automatically, while in geom_serialaxes_..., users have to set axes labels by ggplot2::scale_x_continuous or ggplot2::scale_y_continuous manually.

• 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 in geom_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).