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A tour is a motion graphic designed to study the joint distribution of multivariate data [Asimov (1985)](Andreas Buja and Asimov 1986). A sequence of low-dimensional projections is created by a high dimensional data set. Tours are thus used to find interesting projections. Between each plane, interpolation along a geodesic path is provided so that the points in a sub space (e.g. 2D) can be rotated smoothly. In mathematics, \(\boldsymbol{X}_{n \times p}\) represents the original data set; \(\boldsymbol{P}_{p \times d}\) is the matrix of projection vectors where \(d < p\). \[\boldsymbol{Y} = \boldsymbol{X} \boldsymbol{P}\] where \(\boldsymbol{Y}\) is the lower dimensional sub-space.
The tour was first implemented in the software
Dataviewer
[A. Buja, Asimov, and
Hurley (1986)](Hurley
1987)(Andreas Buja, Hurley, and McDonald
1987) in Symbolics Lisp machine. A smoothly moving
scatterplot could be produced to visualize the tour paths. Then, D. F. Swayne, Cook, and Buja (1991) implemented software
XGobi
in the X Window System, providing
portability across a wide variety of workstations
i.e. X terminals, personal computers, even across a network. Software
GGobi
[D. Swayne et al. (2001)](Cook, Swayne, and Buja 2007),
redesigned and extended its ancestor
XGobi
. It can be embedded in other software, like
environment R
. Package rggobi
(Lang et al. 2018) is
an R
interface of GGobi
, however, it has been
removed from the CRAN and can only be accessed from the archive.
Package tourr
(Wickham et al. 2011) implements geodesic
interpolation and variety tour generation functions (i.e. grand tour,
guided tour, etc) in R
language. The function
animate
provides tour animation as a kinematic sequence of
static displays – plots are generated and then displayed quickly in
order. In an RStudio display, for example, the user can cycle back and
forth through the sequence … but that is all. Unlike earlier tour
implementations (rggobi
) no interactive manipulation of the
plot elements is possible.
The loon package (Waddell
and Oldford 2020) is a toolkit that enables highly
interactive data visualization. The package loon.tourr
adds the full functionality of loon’s interactive graphics to
tourr
. For example, this allows interactive selection,
colouring, and deactivating of points in the tour display and linking
that display with any other loon plot. Interesting projections
discovered during the tour can be accessed at any point in the tour. In
addition, random tours displaying more than 2 dimensions are also
provided using parallel or radial coordinates and scatterplot
matrices.
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 0.62
The 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.
Cook, Swayne, and Buja (2007) introduced several
methods for choosing projections. In loon.tourr
, by
modifying the argument tour_path
, all can be realized.
Grand tour: a sequence of projections is chosen randomly
Projection pursuit guided tour: a sequence of projections is guided by an algorithm in search of “interesting” projections by optimizing a criterion function
\[\mathop{\mathrm{arg\,max}}g(\boldsymbol{Y}), \forall \boldsymbol{P}\] where \(\boldsymbol{Y} = {[{\mathbf{y}}^{\mkern-1.5mu\mathsf{T}}_1, ..., {\mathbf{y}}^{\mkern-1.5mu\mathsf{T}}_n]}^{\mkern-1.5mu\mathsf{T}} = \boldsymbol{X} \boldsymbol{P}\), \(\mathbf{y}_j\) is a \(p \times 1\) vector.
Holes:
\[g(\boldsymbol{Y}) = \frac{1 - \frac{1}{n}\sum_{i=1}^n \exp(-\frac{1}{2}{\mathbf{y}}^{\mkern-1.5mu\mathsf{T}}_i\mathbf{y}_i)}{1 - \exp{(-\frac{p}{2})}}\]
Central Mass:
\[g(\boldsymbol{Y}) = \frac{\frac{1}{n}\sum_{i=1}^n \exp(-\frac{1}{2}{\mathbf{y}}^{\mkern-1.5mu\mathsf{T}}_i \mathbf{y}_i) - \exp{(-\frac{p}{2})}}{1 - \exp{(-\frac{p}{2})}}\]
LDA
\[g(\boldsymbol{Y}) = 1 - \frac{|{\boldsymbol{P}}^{\mkern-1.5mu\mathsf{T}}\boldsymbol{W}\boldsymbol{P}|}{|{\boldsymbol{P}}^{\mkern-1.5mu\mathsf{T}}(\boldsymbol{W} + \boldsymbol{B})\boldsymbol{P}|}\]
where \(\boldsymbol{B} = \sum_{i=1}^k n_i (\bar{y}_{i.} - \bar{y}_{..}) {(\bar{y}_{i.} - \bar{y}_{..})}^{\mkern-1.5mu\mathsf{T}}\), \(\boldsymbol{W} = \sum_{i=1}^k \sum_{j=1}^{n_i} n_i (\bar{y}_{ij} - \bar{y}_{i.}) {(\bar{y}_{ij} - \bar{y}_{i.})}^{\mkern-1.5mu\mathsf{T}}\) and \(k\) is the number of groups.
Expect these, tourr
also provides some other tour
methods, for example:
frozen_tour
, one variable is designated as the
“manipulation variable”, and the projection coefficient for this
variable is controlled;little_tour
, a planned tour that travels between all
axis parallel projections;local_tour
, alternates between the starting position
and a nearby random projection and etc.GIF 3 shows the tour plot separated by variable sex into two panels.
GIF 3: facets
GIF 4 shows the tour pairs plot (the upper triangle). If we set
showSerialAxes = TRUE
, a serial axes plot (parallel or
radial axes) is displayed at the lower triangle.
GIF 4: pairs plot
Tour compound plot is a l_tour_compound
object
The loon pairs plot can be accessed by calling
l_getPlots
. It will return a l_compound
widget
that a list contains seven loon
widgets.
wp <- l_getPlots(pp)
wp
# >
# $x2y1
# [1] ".l0.pairs.plot"
# attr(,"class")
# [1] "l_plot" "loon"
#
# $x3y1
# [1] ".l0.pairs.plot1"
# attr(,"class")
# [1] "l_plot" "loon"
#
# $x4y1
# [1] ".l0.pairs.plot2"
# attr(,"class")
# [1] "l_plot" "loon"
#
# $x3y2
# [1] ".l0.pairs.plot3"
# attr(,"class")
# [1] "l_plot" "loon"
#
# $x4y2
# [1] ".l0.pairs.plot4"
# attr(,"class")
# [1] "l_plot" "loon"
#
# $x4y3
# [1] ".l0.pairs.plot5"
# attr(,"class")
# [1] "l_plot" "loon"
#
# $serialAxes
# [1] ".l0.pairs.serialaxes"
# attr(,"class")
# [1] "l_serialaxes" "loon"
#
# attr(,"class")
# [1] "l_pairs" "l_compound" "loon"
Note that all the plots in a single compound widget share the same
projection (e.g pp['projection']
).
Sometimes, layering visuals in a tour is helpful to find an interesting “pattern”.
l_layer_hull
A convex hull is the smallest convex set that contains it. We can add
such layer by l_layer_hull
The argument group
is used to clarify the group of the
set in each hull. Suppose we want to find a projection that splits the
crab species well, layering the hull could be extremely useful. If two
hulls have no intersections, this projection could be the one we are
looking for.
GIF 5: layer hull
l_layer_density2d
Two dimensional kernel density estimation can tell the dense of a distribution.
GIF 5: layer density
l_layer_trails
Display trails in tours
GIF 5: layer density
The trails can show the direction of the projection. Meanwhile, the lengths of tails represent the steps of each transformation.
The interactive layers are realized by modifying function
l_layer_callback
. It is a generic function and the
class is determined by label
. For example, we want
to create a static point layer and make it as a background of the object
pf
.
allx <- unlist(pf['x'])
ally <- unlist(pf['y'])
layers <- lapply(l_getPlots(pf),
function(p) {
l <- loon::l_layer_points(p,
x = allx,
y = ally,
color = "grey80",
label = "background")
# set the layer as the background
loon::l_layer_lower(p, l)
})
Figure 2: non-interactive layer
When the tour changes, this static layer does not change
correspondingly (in GIF 6). To make it interactive, we have to create
the following S3
function, by setting its class as
background. The parameters of this function are
target
: a l_tour
object.
layer
: a layer to be turned into
interactive.
...
includes that:
start: the start matrix of projection vectors \(\boldsymbol{P}_s\)
initialTour: the activated panel’s initial position \(\boldsymbol{Y}_s\) where \(\boldsymbol{Y}_s = \boldsymbol{X} \boldsymbol{P}_s\).
projections: A list of the activated panel’s projection matrices.
tours: A list of the activated panel’s tour path (\(\boldsymbol{Y}\)).
group: a factor defines the grouping of the activated panel.
var: scroll bar current variable.
varOld: scroll bar previous variable.
color: the activated panel’s objects’ (points’ or lines’) color.
axes: only for scatterplot, the guided axes.
labels: only for scatterplot, the guided labels.
axesLength: only for scatterplot, the guided axes length.
If the layer is added for a l_tour_compound
object
l_compound: a l_compound
widget
allTours: returns the tour path for all panels.
allInitialTour: returns the initial position for all panels.
allProjections: returns the projection matrices for all panels.
allColor: returns the color for all panels
Notice that, there is no need to use all these to build an
interactive statistical layer. For example, in function
l_layer_callback.background
, we only extract two parameters
allTours
(the tours for all panels) and var
(the current scroll bar variable).
l_layer_callback.background <- function(target, layer, ...) {
widget <- l_getPlots(target)
layer <- loon::l_create_handle(c(widget, layer))
args <- list(...)
# the overall tour paths
allTours <- args$allTours
# the scale bar variable
var <- args$var
# the current projection (bind both facets)
proj <- do.call(rbind, allTours[[var]])
loon::l_configure(layer,
x = proj[, 1],
y = proj[, 2])
}
GIF 7: interactive layer
After function l_layer_callback.background
is created
and executed, the layer is interactive.
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.