In z267xu/ggmulti: High Dimensional Data Visualization
knitr::opts_chunk$set(echo = TRUE,
warning = FALSE,
message = FALSE,
fig.align = "center",
fig.width = 7,
fig.height = 5,
out.width = "60%",
collapse = TRUE,
comment = "#>",
tidy.opts = list(width.cutoff = 65),
tidy = FALSE)
library(knitr)
set.seed(12314159)
imageDirectory <- "./images/highDim"
dataDirectory <- "./data/highDim"
path_concat <- function(path1, ..., sep="/") {
# The "/" is standard unix directory separator and so will
# work on Macs and Linux.
# In windows the separator might have to be sep = "\" or
# even sep = "\\" or possibly something else.
paste(path1, ..., sep = sep)
}
library(ggplot2, quietly = TRUE)
library(dplyr, quietly = TRUE)
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 $R^2$ [@146402].
Before we start, a couple of things should be noticed:
-
In the serial axes coordinate system, no x or y (even group) are required; but other aesthetics, such as colour, fill, size, etc, are accommodated.
-
Layer geom_path is used to draw the serial lines; layer geom_histogram, geom_quantiles, and geom_density are used to draw the histograms, quantiles (not quantile regression) and densities. Users can also customize their own layer (i.e. geom_boxplot, geom_violin, etc) by editing function add_serialaxes_layers.
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
@andrews1972plots 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) = <\ve{y}_i, \ve{a}_t>]
where $\ve{y}_i$ is the $i$th responses and $\ve{a}_t$ is the orthonormal functions under certain interval. Andrew suggests to use the Fourier transformation
[\ve{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
[\ve{a}_t = {\cos(t), \cos(\sqrt{2}t), \cos(\sqrt{3}t), \cos(\sqrt{5}t), ...}]
where $t \in [0, k\pi]$ [@gnanadesikan2011methods].
```r
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 $\ve{a}_t$ 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'" [@inselberg1999don].
Reference
z267xu/ggmulti documentation built on April 12, 2024, 1:31 p.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
knitr::opts_chunk$set(echo = TRUE, warning = FALSE, message = FALSE, fig.align = "center", fig.width = 7, fig.height = 5, out.width = "60%", collapse = TRUE, comment = "#>", tidy.opts = list(width.cutoff = 65), tidy = FALSE) library(knitr) set.seed(12314159) imageDirectory <- "./images/highDim" dataDirectory <- "./data/highDim" path_concat <- function(path1, ..., sep="/") { # The "/" is standard unix directory separator and so will # work on Macs and Linux. # In windows the separator might have to be sep = "\" or # even sep = "\\" or possibly something else. paste(path1, ..., sep = sep) } library(ggplot2, quietly = TRUE) library(dplyr, quietly = TRUE)
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 $R^2$ [@146402].
Before we start, a couple of things should be noticed:
-
In the serial axes coordinate system, no
xory(evengroup) are required; but other aesthetics, such ascolour,fill,size, etc, are accommodated. -
Layer
geom_pathis used to draw the serial lines; layergeom_histogram,geom_quantiles, andgeom_densityare used to draw the histograms, quantiles (notquantileregression) and densities. Users can also customize their own layer (i.e.geom_boxplot,geom_violin, etc) by editing functionadd_serialaxes_layers.
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
@andrews1972plots 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) = <\ve{y}_i, \ve{a}_t>]
where $\ve{y}_i$ is the $i$th responses and $\ve{a}_t$ is the orthonormal functions under certain interval. Andrew suggests to use the Fourier transformation
[\ve{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.sequencein each layer or incoord_serialaxes. -
To construct a dot product serial axes plot, say Fourier transformation, "Andrew's plot", we need to set the parameter
statingeom_pathto "dotProduct". The default transformation function is the Andrew's (functionandrews). Users can customize their own, for example, Tukey suggests the following projected space
[\ve{a}_t = {\cos(t), \cos(\sqrt{2}t), \cos(\sqrt{3}t), \cos(\sqrt{5}t), ...}]
where $t \in [0, k\pi]$ [@gnanadesikan2011methods].
```r
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 $\ve{a}_t$ 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_serialaxesrather thangeom_serialaxes_...is thatcoord_serialaxescan accommodate duplicated axes in mapping aesthetics (e.g. Eulerian path, Hamiltonian path, etc). However, ingeom_serialaxes_..., duplicated axes will be omitted. -
Meaningful axes labels in
coord_serialaxescan be created automatically, while ingeom_serialaxes_..., users have to set axes labels byggplot2::scale_x_continuousorggplot2::scale_y_continuousmanually. -
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 ingeom_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'" [@inselberg1999don].
Reference
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
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