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_yi(t) =  < y_i, a_t>

where y_i is the ith responses and 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 (−π, π). In other word, we can project a p dimensional space to an infinite (−π, π) 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 ∈ [0, kπ] (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 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).