Basics
The crabs (Campbell and Mahon 1974)
data frame (stored in package MASS) contains 200
observations and 8 features, describing 5 morphological measurements on
50 crabs each of two colour forms and both sexes. GIF 1 shows the tours
and color represents the species, “B” (blue) or “O” (orange).
| sp | sex | index | FL | RW | CL | CW | BD |
|---|---|---|---|---|---|---|---|
| B | M | 1 | 8.1 | 6.7 | 16.1 | 19.0 | 7.0 |
| B | M | 2 | 8.8 | 7.7 | 18.1 | 20.8 | 7.4 |
| B | M | 3 | 9.2 | 7.8 | 19.0 | 22.4 | 7.7 |
| B | M | 4 | 9.6 | 7.9 | 20.1 | 23.1 | 8.2 |
| B | M | 5 | 9.8 | 8.0 | 20.3 | 23.0 | 8.2 |
| B | M | 6 | 10.8 | 9.0 | 23.0 | 26.5 | 9.8 |
This plot is interactive. Users can pan, zoom or select on this plot. The default number of random bases is 30 and the steps between two serial projections are 40. So, we have 1200 + 1 (start position) projections in total. To navigate the tour, scroll the rightmost bar down, the projection is transformed from one to another. If, unfortunately, none of the projections is interesting, click the “refresh” button at the left-bottom corner. New random tours are created.
Immediately below the plot, behind the “refresh” button, there are several radio buttons, “data”, “variable”, “observation” and “sphere” that represent the scaling methods \(f\) of \(\boldsymbol{X}\).
The first of which is “data”, where \(f(\boldsymbol{X}) = \boldsymbol{X}\);
If
scaling = "variable", \(f(\mathbf{x}_j) = \frac{1}{a - b}(\mathbf{x}_j - b\mathbf{1})\) where \(\boldsymbol{X} = [\mathbf{x}_{1}, ..., \mathbf{x}_{p}]\), \(\mathbf{x}_{j}\) is a \(n \times 1\) vector, \(a = \max{(\mathbf{x}_j)}\) and \(b = \min{(\mathbf{x}_j)}\);The third is “observation”, where \(f(\mathbf{x}_i) = \frac{1}{c - d}(\mathbf{x}_i - d\mathbf{1})\), \(\boldsymbol{X} = {[{\mathbf{x}_{1}}^{\mkern-1.5mu\mathsf{T}}, ..., {\mathbf{x}_{n}}^{\mkern-1.5mu\mathsf{T}}]}^{\mkern-1.5mu\mathsf{T}}\), \(\mathbf{x}_{i}\) is a \(p \times 1\) vector, \(c = \max{(\mathbf{x}_i)}\) and \(d = \min{(\mathbf{x}_i)}\);
If the scaling method is “sphere”, where \(f(\boldsymbol{X}) = \boldsymbol{X}^\star \boldsymbol{V}\) where \(\boldsymbol{X}^\star = (\boldsymbol{I} - \frac{1}{n}\mathbf{1}{\mathbf{1}}^{\mkern-1.5mu\mathsf{T}})\boldsymbol{X} = \boldsymbol{U} \boldsymbol{D} \boldsymbol{V}\).
Notice that the tour is based on the transformed data set \[\boldsymbol{Y} = f(\boldsymbol{X}) \boldsymbol{P}\]
color <- rep("skyblue", nrow(crabs))
color[crabs$sp == "O"] <- "orange"
cr <- crabs[, c("FL", "RW", "CL", "CW", "BD")]
p0 <- l_tour(cr, color = color)
GIF 1: 2D grand tour
The projection dimension (the dimension of the \(\boldsymbol{P}\)) is controlled by
tour_path = grand_tour(d) where \(d\) represents the dimensions and the
default is \(2\). Unlike the 2
dimensional Cartesian coordinate, higher dimensional space (\(>2\)) will be embedded in the parallel
coordinate or radial coordinate. In GIF 2, the \(d\) is set as 4.
GIF 2: 4D grand tour
An l_tour object is returned by p1 (or
p0).
The loon serialaxes plot for p1 (or scatterplot for
p0) can be accessed by calling l_getPlots
The matrix of projection vectors \(\boldsymbol{P}_{4 \times 4}\) can be
returned by function l_cget or a simple [
round(p1["projection"], 2)
# >
# [,1] [,2] [,3] [,4]
# [1,] -0.12 -0.93 -0.01 -0.35
# [2,] -0.57 0.21 0.63 -0.36
# [3,] -0.06 0.14 0.14 -0.38
# [4,] -0.37 0.23 -0.76 -0.47
# [5,] -0.72 -0.14 -0.12 0.62The radial coordinate can be converted to a parallel coordinate by
calling [<- in the console or trigger the “parallel”
radio button of “axes layout” on p1’s inspector.
Additionally, Andrews plot can be shown by runing the following code in console
The interactive graphics can be turned to static either by
or
Figure 1: Andrews curve
To investigate more about loon, please visit great-northern-diver-loon.
