landscape: Persistence landscapes
In rrrlw/ggtda: 'ggplot2' Extension to Visualize Topological Persistence

landscape

R Documentation

Persistence landscapes

Description

Visualize persistence data as a persistence landscape.

Usage

stat_landscape(
  mapping = NULL,
  data = NULL,
  geom = "landscape",
  position = "identity",
  filtration = "Rips",
  diameter_max = NULL,
  radius_max = NULL,
  dimension_max = 1L,
  field_order = 2L,
  engine = NULL,
  diagram = "landscape",
  n_levels = Inf,
  na.rm = FALSE,
  show.legend = NA,
  inherit.aes = TRUE,
  ...
)

geom_landscape(
  mapping = NULL,
  data = NULL,
  stat = "landscape",
  position = "identity",
  lineend = "butt",
  linejoin = "round",
  linemitre = 10,
  na.rm = FALSE,
  show.legend = NA,
  inherit.aes = TRUE,
  ...
)

Arguments

`mapping`	Set of aesthetic mappings created by `aes()`. If specified and `inherit.aes = TRUE` (the default), it is combined with the default mapping at the top level of the plot. You must supply `mapping` if there is no plot mapping.
`data`	The data to be displayed in this layer. There are three options: If `NULL`, the default, the data is inherited from the plot data as specified in the call to `ggplot()`. A `data.frame`, or other object, will override the plot data. All objects will be fortified to produce a data frame. See `fortify()` for which variables will be created. A `function` will be called with a single argument, the plot data. The return value must be a `data.frame`, and will be used as the layer data. A `function` can be created from a `formula` (e.g. `~ head(.x, 10)`).
`geom`	The geometric object to use to display the data, either as a `ggproto` `Geom` subclass or as a string naming the geom stripped of the `geom_` prefix (e.g. `"point"` rather than `"geom_point"`)
`position`	Position adjustment, either as a string naming the adjustment (e.g. `"jitter"` to use `position_jitter`), or the result of a call to a position adjustment function. Use the latter if you need to change the settings of the adjustment.
`filtration`	The type of filtration from which to compute persistent homology; one of `"Rips"`, `"Vietoris"` (equivalent) or `"alpha"`.
`diameter_max`, `radius_max`	Maximum diameter or radius for the simplicial filtration. Both default to `NULL`, in which case the complete filtration is constructed.
`dimension_max`	Maximum dimension of the simplicial filtration.
`field_order`	(Prime) order of the field over which to compute persistent homology.
`engine`	The computational engine to use (see 'Details'). Reasonable defaults are chosen based on `filtration`.
`diagram`	One of `"flat"`, `"diagonal"`, or `"landscape"`; the orientation for the diagram should take.
`n_levels`	The number of levels to compute and plot. If `Inf` (the default), determined to be all levels.
`na.rm`	If `FALSE`, the default, missing values are removed with a warning. If `TRUE`, missing values are silently removed.
`show.legend`	logical. Should this layer be included in the legends? `NA`, the default, includes if any aesthetics are mapped. `FALSE` never includes, and `TRUE` always includes. It can also be a named logical vector to finely select the aesthetics to display.
`inherit.aes`	If `FALSE`, overrides the default aesthetics, rather than combining with them. This is most useful for helper functions that define both data and aesthetics and shouldn't inherit behaviour from the default plot specification, e.g. `borders()`.
`...`	Other arguments passed on to `layer()`. These are often aesthetics, used to set an aesthetic to a fixed value, like `colour = "red"` or `size = 3`. They may also be parameters to the paired geom/stat.
`stat`	The statistical transformation to use on the data for this layer, either as a `ggproto` `Geom` subclass or as a string naming the stat stripped of the `stat_` prefix (e.g. `"count"` rather than `"stat_count"`)
`lineend`	Line end style (round, butt, square).
`linejoin`	Line join style (round, mitre, bevel).
`linemitre`	Line mitre limit (number greater than 1).

Details

Persistence landscapes, anticipated by some alternative coordinatizations of persistence diagrams, were proposed as Lipschitz functions that demarcate the Pareto frontiers of persistence diagrams. They can be averaged over the diagrams obtained from multiple data sets designed or hypothesized to have been generated from the same underlying topological structure.

Persistence landscapes do not currently recognize extended persistence data.

Aesthetics

stat_landscape() understands the following aesthetics (required aesthetics are in bold):

start or dataset
end or dataset
group

geom_landscape() understands the following aesthetics (required aesthetics are in bold):

x
y
alpha
colour
group
linetype
linewidth

Learn more about setting these aesthetics in vignette("ggplot2-specs", package = "ggplot2").

Computed variables

stat_landscape calculates the following variables that can be accessed with delayed evaluation.

after_stat(x), after_stat(y)
coordinates of segment endpoints of each frontier.
after_stat(dimension)
feature dimension (with 'dataset' aesthetic only).
after_stat(group)
interaction of existing 'group', dataset ID, and 'dimension'.
after_stat(level)
position of each frontier, starting from the outermost.
after_stat(slope)
slope of the landscape abscissa.

Note that start and end are dropped during the statistical transformation.

References

P Bubenik (2015) Statistical Topological Data Analysis using Persistence Landscapes. Journal of Machine Learning Research, 16 77–102. http://jmlr.org/papers/v16/bubenik15a.html

F Chazal and B Michel (2017) An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists. https://arxiv.org/abs/1710.04019

Examples

# toy example
toy.data <- data.frame(
  birth = c(0, 0, 1, 3, 4, 1.5),
  death = c(5, 3, 5, 4, 6, 3),
  dim = factor(c(0, 0, 1, 1, 2, 2))
)
# persistence diagram with landscape overlaid
ggplot(toy.data,
       aes(start = birth, end = death, colour = dim, shape = dim)) +
  theme_persist() +
  coord_equal() +
  stat_persistence() +
  stat_landscape(aes(alpha = -after_stat(level)), diagram = "diagonal") +
  lims(x = c(0, 8), y = c(0, NA)) +
  guides(alpha = "none")
# persistence landscape with diagram overlaid
ggplot(toy.data,
       aes(start = birth, end = death, colour = dim, shape = dim)) +
  theme_persist() +
  coord_equal() +
  stat_landscape(aes(linetype = after_stat(factor(level)))) +
  stat_persistence(diagram = "landscape") +
  lims(x = c(0, 8), y = c(0, NA)) +
  labs(linetype = "level")

# load library and generate dataset for comprehensive example
library("ripserr")
# noisy unit circle (Betti-1 number = 1)
n <- 100L; sd <- 0.1
set.seed(7)
t <- stats::runif(n = n, min = 0, max = 2*pi)
annulus.df <- data.frame(
  x = cos(t) + stats::rnorm(n = n, mean = 0, sd = sd),
  y = sin(t) + stats::rnorm(n = n, mean = 0, sd = sd)
)
# calculate persistence homology and format
annulus.phom <- as.data.frame(vietoris_rips(annulus.df))
annulus.phom$dimension <- as.factor(annulus.phom$dimension)
# pretty diagonal persistence diagram
ggplot(annulus.phom, aes(start = birth, end = death,
                         shape = dimension, colour = dimension)) +
  stat_persistence(diagram = "landscape") +
  theme_persist()
# pretty landscape persistence diagram
ggplot(annulus.phom, aes(start = birth, end = death,
                         shape = dimension, colour = dimension)) +
  stat_landscape(diagram = "landscape") +
  theme_persist()

# list-column of data sets to 'dataset' aesthetic
raw_data <- data.frame(obj = I(list(eurodist, 10*swiss, Nile)))
raw_data$class <- vapply(raw_data$obj, class, "")

if ("TDA" %in% rownames(utils::installed.packages())) {
  
  # barcodes
  ggplot(raw_data, aes(dataset = obj)) +
    geom_barcode(aes(color = factor(after_stat(dimension))),
                 engine = "TDA") +
    facet_wrap(facets = vars(class))
  # persistence diagram
  ggplot(raw_data, aes(dataset = obj)) +
    stat_persistence(aes(color = factor(after_stat(dimension)), shape = class),
                     engine = "GUDHI")
  # persistence landscape
  ggplot(raw_data, aes(dataset = obj)) +
    facet_wrap(facets = vars(class), scales = "free") +
    stat_landscape(aes(color = factor(after_stat(dimension))),
                   engine = "Dionysus") +
    theme(legend.position = "bottom")
  
}

if ("ripserr" %in% rownames(utils::installed.packages())) {
  
  # exclude time series data if {ripserr} v0.1.1 is installed
  if (utils::packageVersion("ripserr") == "0.1.1")
    raw_data <- raw_data[c(1L, 2L), ]
  # barcodes
  ggplot(raw_data, aes(dataset = obj)) +
    geom_barcode(aes(color = factor(after_stat(dimension))),
                 engine = "ripserr") +
    facet_wrap(facets = vars(class))
  # persistence diagram
  ggplot(raw_data, aes(dataset = obj)) +
    stat_persistence(aes(color = factor(after_stat(dimension)), shape = class),
                     engine = "ripserr")
  # persistence landscape
  ggplot(raw_data, aes(dataset = obj)) +
    facet_wrap(facets = vars(class), scales = "free") +
    stat_landscape(aes(color = factor(after_stat(dimension))),
                   engine = "ripserr") +
    theme(legend.position = "bottom")
  
}

rrrlw/ggtda documentation built on April 14, 2024, 2:24 p.m.