R/concept_lattice.R
In neuroimaginador/fcaR: Formal Concept Analysis
#' @title
#' R6 class for a concept lattice
#'
#' @description
#' This class implements the data structure and methods for concept lattices.
#'
#' @examples
#' # Build a formal context
#' fc_planets <- FormalContext$new(planets)
#'
#' # Find the concepts
#' fc_planets$find_concepts()
#'
#' # Find join- and meet- irreducible elements
#' fc_planets$concepts$join_irreducibles()
#' fc_planets$concepts$meet_irreducibles()
#'
#' # Get concept support
#' fc_planets$concepts$support()
#'
#' @export
#' @import R6
#'
ConceptLattice <- R6::R6Class(
classname = "ConceptLattice",
inherit = ConceptSet,
public = list(
#' @description
#' Create a new \code{ConceptLattice} object.
#'
#' @param extents (\code{dgCMatrix}) The extents of all concepts
#' @param intents (\code{dgCMatrix}) The intents of all concepts
#' @param objects (character vector) Names of the objects in the formal context
#' @param attributes (character vector) Names of the attributes in the formal context
#' @param I (\code{dgCMatrix}) The matrix of the formal context
#'
#' @return
#' A new \code{ConceptLattice} object.
#'
#' @export
initialize = function(extents, intents,
objects, attributes,
I = NULL) {
private$objects <- objects
private$attributes <- attributes
private$pr_extents <- extents
private$pr_intents <- intents
if (!capabilities()["long.double"] & getRversion() < "4.1.0") {
private$can_plot <- FALSE
}
super$initialize(extents, intents,
objects, attributes,
I)
},
#' @description
#' Plot the concept lattice
#'
#' @param object_names (logical) If \code{TRUE}, plot object names, otherwise omit them from the diagram.
#' @param to_latex (logical) If \code{TRUE}, export the plot as a \code{tikzpicture} environment that can be included in a \code{LaTeX} file.
#' @param ... Other parameters to be passed to the \code{tikzDevice} that renders the lattice in \code{LaTeX}, or for the figure caption. See \code{Details}.
#'
#' @details
#' Particular parameters that control the size of the \code{tikz} output are: \code{width}, \code{height} (both in inches), and \code{pointsize} (in points), that should be set to the font size used in the \code{documentclass} header in the \code{LaTeX} file where the code is to be inserted.
#'
#' If a \code{caption} is provided, the whole \code{tikz} picture will be wrapped by a \code{figure} environment and the caption set.
#'
#' @return If \code{to_latex} is \code{FALSE}, it returns nothing, just plots the graph of the concept lattice. Otherwise, this function returns the \code{LaTeX} code to reproduce the concept lattice.
#' @export
#'
plot = function(object_names = TRUE,
to_latex = FALSE,
...) {
if (self$size() == 0) {
warning("No concepts.", call. = FALSE)
return(invisible(NULL))
}
if (!private$can_plot) {
warning("The R system has not the needed capabilities to plot.",
call. = FALSE)
return(invisible(FALSE))
}
if (!requireNamespace("hasseDiagram", quietly = TRUE)) {
warning("You have not installed the 'hasseDiagram' package, which is needed to plot the lattice.",
call. = FALSE)
return(invisible(FALSE))
}
if ((super$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
lattice_plot(extents = private$pr_extents,
intents = private$pr_intents,
subconcept_matrix = private$subconcept_matrix,
objects = private$objects,
attributes = private$attributes,
object_names = object_names,
to_latex = to_latex,
...)
},
#' @description
#' Sublattice
#'
#' @param ... See Details.
#'
#' @details
#' As argument, one can provide both integer indices or \code{Concepts}, separated by commas. The corresponding concepts are used to generate a sublattice.
#'
#' @return
#' The generated sublattice as a new \code{ConceptLattice} object.
#'
#' @export
sublattice = function(...) {
idx <- private$to_indices(...)
if (length(idx) > 0) {
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
idx <- .get_sublattice(private$subconcept_matrix,
starting_idx = idx)
if (length(idx) > 1) {
my_intents <- private$pr_intents[, idx]
my_extents <- private$pr_extents[, idx]
} else {
my_intents <- .extract_column(private$pr_intents, idx)
my_extents <- .extract_column(private$pr_extents, idx)
}
cl <- ConceptLattice$new(extents = my_extents,
intents = my_intents,
objects = private$objects,
attributes = private$attributes,
I = private$I)
return(cl)
}
},
#' @description Top of a Lattice
#'
#' @return The top of the Concept Lattice
#' @export
#'
#' @examples
#' fc <- FormalContext$new(planets)
#' fc$find_concepts()
#' fc$concepts$top()
#'
top = function() {
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
idx <- which(Matrix::colSums(private$subconcept_matrix) == self$size())
self[idx]$to_list()[[1]]
},
#' @description Bottom of a Lattice
#'
#' @return The bottom of the Concept Lattice
#' @export
#'
#' @examples
#' fc <- FormalContext$new(planets)
#' fc$find_concepts()
#' fc$concepts$bottom()
#'
bottom = function() {
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
idx <- which(Matrix::colSums(private$subconcept_matrix) == 1)
self[idx]$to_list()[[1]]
},
#' @description
#' Join-irreducible Elements
#'
#' @return
#' The join-irreducible elements in the concept lattice.
#'
#' @export
#'
join_irreducibles = function() {
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
if (is.null(private$reduced_matrix)) {
private$reduced_matrix <- .reduce_transitivity(private$subconcept_matrix)
}
idx <- which(Matrix::colSums(private$reduced_matrix) == 1)
self[idx]
},
#' @description
#' Meet-irreducible Elements
#'
#' @return
#' The meet-irreducible elements in the concept lattice.
#'
#' @export
#'
meet_irreducibles = function() {
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
M <- .reduce_transitivity(Matrix::t(private$subconcept_matrix))
idx <- which(Matrix::colSums(M) == 1)
self[idx]
},
#' @description
#' Decompose a concept as the supremum of meet-irreducible concepts
#'
#' @param C A list of \code{Concept}s
#' @return
#' A list, each field is the set of meet-irreducible elements whose supremum is the corresponding element in \code{C}.
#'
#' @export
#'
decompose = function(C) {
irreducible <- self$meet_irreducibles()
irr_intents <- irreducible$intents()
C_intents <- C$intents()
ss <- lapply(seq(C$size()),
function(i) {
r <- Matrix::Matrix(C_intents[, i],
sparse = TRUE)
if (sum(r) == 0) return(C[i])
id <- Matrix::which(.subset(irr_intents,
r))
if (length(id) > 1) {
MM <- .subset(irr_intents[, id])
id <- id[Matrix::rowSums(MM) == 1]
}
decomposition <- irreducible[id]
return(decomposition)
})
return(ss)
},
#' @description
#' Supremum of Concepts
#'
#' @param ... See Details.
#'
#' @details
#' As argument, one can provide both integer indices or \code{Concepts}, separated by commas. The corresponding concepts are used to compute their supremum in the lattice.
#'
#' @return
#' The supremum of the list of concepts.
#'
#' @export
supremum = function(...) {
idx <- private$to_indices(...)
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
return(self[join(private$subconcept_matrix, idx)]$to_list()[[1]])
},
#' @description
#' Infimum of Concepts
#'
#' @param ... See Details.
#'
#' @details
#' As argument, one can provide both integer indices or \code{Concepts}, separated by commas. The corresponding concepts are used to compute their infimum in the lattice.
#'
#' @return
#' The infimum of the list of concepts.
#'
#' @export
infimum = function(...) {
idx <- private$to_indices(...)
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
return(self[meet(private$subconcept_matrix, idx)]$to_list()[[1]])
},
#' @description
#' Subconcepts of a Concept
#'
#' @param C (numeric or \code{SparseConcept}) The concept to which determine all its subconcepts.
#'
#' @return
#' A list with the subconcepts.
#' @export
subconcepts = function(C) {
idx <- private$to_indices(C)
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
# Get the index of all subconcepts
M <- Matrix::t(private$subconcept_matrix)[idx, ]
candidates <- Matrix::which(M > 0)
self[candidates]
},
#' @description
#' Superconcepts of a Concept
#'
#' @param C (numeric or \code{SparseConcept}) The concept to which determine all its superconcepts.
#'
#' @return
#' A list with the superconcepts.
#' @export
superconcepts = function(C) {
idx <- private$to_indices(C)
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
# Get the index of all superconcepts
M <- private$subconcept_matrix[idx, ]
candidates <- which(M > 0)
self[candidates]
},
#' @description
#' Lower Neighbours of a Concept
#'
#' @param C (\code{SparseConcept}) The concept to which find its lower neighbours
#'
#' @return
#' A list with the lower neighbours of \code{C}.
#'
#' @export
lower_neighbours = function(C) {
idx <- private$to_indices(C)
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
if (is.null(private$reduced_matrix)) {
private$reduced_matrix <- .reduce_transitivity(private$subconcept_matrix)
}
self[which(private$reduced_matrix[, idx] > 0)]
},
#' @description
#' Upper Neighbours of a Concept
#'
#' @param C (\code{SparseConcept}) The concept to which find its upper neighbours
#'
#' @return
#' A list with the upper neighbours of \code{C}.
#'
#' @export
upper_neighbours = function(C) {
idx <- private$to_indices(C)
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
if (is.null(private$reduced_matrix)) {
private$reduced_matrix <- .reduce_transitivity(private$subconcept_matrix)
}
self[which(private$reduced_matrix[idx, ] > 0)]
}
),
private = list(
subconcept_matrix = NULL,
reduced_matrix = NULL,
can_plot = TRUE,
concept_list_to_indices = function(concept_list) {
extents <- lapply(concept_list,
function(l) l$get_extent()$get_vector())
extents <- Reduce(cbind, extents)
indices <- .equal_sets(x = extents,
y = private$pr_extents)
indices <- arrayInd(Matrix::which(indices),
.dim = dim(indices))[, 2]
return(indices)
},
to_indices = function(...) {
dots <- list(...)
if (is.list(dots[[1]])) {
dots <- unlist(dots)
}
if (is.logical(dots[[1]])) {
return(which(dots[[1]]))
}
csets_idx <- sapply(dots,
function(l) inherits(l, "ConceptSet"))
if (length(csets_idx) > 0) {
csets <- sapply(dots[csets_idx],
function(l) l$to_list()) %>%
unlist()
} else {
csets <- c()
}
sets <- sapply(dots,
function(l)
inherits(l, "Concept"))
sets <- c(csets, dots[which(sets)])
indices <- sapply(dots, is.numeric)
idx <- c()
if (length(indices) > 0) {
idx <- c(idx, Reduce(c, dots[indices]))
}
if (length(sets) > 0) {
idx <- c(idx,
private$concept_list_to_indices(sets))
}
return(idx)
}
)
)
neuroimaginador/fcaR documentation built on Dec. 10, 2023, 10:23 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.
#' @title
#' R6 class for a concept lattice
#'
#' @description
#' This class implements the data structure and methods for concept lattices.
#'
#' @examples
#' # Build a formal context
#' fc_planets <- FormalContext$new(planets)
#'
#' # Find the concepts
#' fc_planets$find_concepts()
#'
#' # Find join- and meet- irreducible elements
#' fc_planets$concepts$join_irreducibles()
#' fc_planets$concepts$meet_irreducibles()
#'
#' # Get concept support
#' fc_planets$concepts$support()
#'
#' @export
#' @import R6
#'
ConceptLattice <- R6::R6Class(
classname = "ConceptLattice",
inherit = ConceptSet,
public = list(
#' @description
#' Create a new \code{ConceptLattice} object.
#'
#' @param extents (\code{dgCMatrix}) The extents of all concepts
#' @param intents (\code{dgCMatrix}) The intents of all concepts
#' @param objects (character vector) Names of the objects in the formal context
#' @param attributes (character vector) Names of the attributes in the formal context
#' @param I (\code{dgCMatrix}) The matrix of the formal context
#'
#' @return
#' A new \code{ConceptLattice} object.
#'
#' @export
initialize = function(extents, intents,
objects, attributes,
I = NULL) {
private$objects <- objects
private$attributes <- attributes
private$pr_extents <- extents
private$pr_intents <- intents
if (!capabilities()["long.double"] & getRversion() < "4.1.0") {
private$can_plot <- FALSE
}
super$initialize(extents, intents,
objects, attributes,
I)
},
#' @description
#' Plot the concept lattice
#'
#' @param object_names (logical) If \code{TRUE}, plot object names, otherwise omit them from the diagram.
#' @param to_latex (logical) If \code{TRUE}, export the plot as a \code{tikzpicture} environment that can be included in a \code{LaTeX} file.
#' @param ... Other parameters to be passed to the \code{tikzDevice} that renders the lattice in \code{LaTeX}, or for the figure caption. See \code{Details}.
#'
#' @details
#' Particular parameters that control the size of the \code{tikz} output are: \code{width}, \code{height} (both in inches), and \code{pointsize} (in points), that should be set to the font size used in the \code{documentclass} header in the \code{LaTeX} file where the code is to be inserted.
#'
#' If a \code{caption} is provided, the whole \code{tikz} picture will be wrapped by a \code{figure} environment and the caption set.
#'
#' @return If \code{to_latex} is \code{FALSE}, it returns nothing, just plots the graph of the concept lattice. Otherwise, this function returns the \code{LaTeX} code to reproduce the concept lattice.
#' @export
#'
plot = function(object_names = TRUE,
to_latex = FALSE,
...) {
if (self$size() == 0) {
warning("No concepts.", call. = FALSE)
return(invisible(NULL))
}
if (!private$can_plot) {
warning("The R system has not the needed capabilities to plot.",
call. = FALSE)
return(invisible(FALSE))
}
if (!requireNamespace("hasseDiagram", quietly = TRUE)) {
warning("You have not installed the 'hasseDiagram' package, which is needed to plot the lattice.",
call. = FALSE)
return(invisible(FALSE))
}
if ((super$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
lattice_plot(extents = private$pr_extents,
intents = private$pr_intents,
subconcept_matrix = private$subconcept_matrix,
objects = private$objects,
attributes = private$attributes,
object_names = object_names,
to_latex = to_latex,
...)
},
#' @description
#' Sublattice
#'
#' @param ... See Details.
#'
#' @details
#' As argument, one can provide both integer indices or \code{Concepts}, separated by commas. The corresponding concepts are used to generate a sublattice.
#'
#' @return
#' The generated sublattice as a new \code{ConceptLattice} object.
#'
#' @export
sublattice = function(...) {
idx <- private$to_indices(...)
if (length(idx) > 0) {
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
idx <- .get_sublattice(private$subconcept_matrix,
starting_idx = idx)
if (length(idx) > 1) {
my_intents <- private$pr_intents[, idx]
my_extents <- private$pr_extents[, idx]
} else {
my_intents <- .extract_column(private$pr_intents, idx)
my_extents <- .extract_column(private$pr_extents, idx)
}
cl <- ConceptLattice$new(extents = my_extents,
intents = my_intents,
objects = private$objects,
attributes = private$attributes,
I = private$I)
return(cl)
}
},
#' @description Top of a Lattice
#'
#' @return The top of the Concept Lattice
#' @export
#'
#' @examples
#' fc <- FormalContext$new(planets)
#' fc$find_concepts()
#' fc$concepts$top()
#'
top = function() {
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
idx <- which(Matrix::colSums(private$subconcept_matrix) == self$size())
self[idx]$to_list()[[1]]
},
#' @description Bottom of a Lattice
#'
#' @return The bottom of the Concept Lattice
#' @export
#'
#' @examples
#' fc <- FormalContext$new(planets)
#' fc$find_concepts()
#' fc$concepts$bottom()
#'
bottom = function() {
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
idx <- which(Matrix::colSums(private$subconcept_matrix) == 1)
self[idx]$to_list()[[1]]
},
#' @description
#' Join-irreducible Elements
#'
#' @return
#' The join-irreducible elements in the concept lattice.
#'
#' @export
#'
join_irreducibles = function() {
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
if (is.null(private$reduced_matrix)) {
private$reduced_matrix <- .reduce_transitivity(private$subconcept_matrix)
}
idx <- which(Matrix::colSums(private$reduced_matrix) == 1)
self[idx]
},
#' @description
#' Meet-irreducible Elements
#'
#' @return
#' The meet-irreducible elements in the concept lattice.
#'
#' @export
#'
meet_irreducibles = function() {
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
M <- .reduce_transitivity(Matrix::t(private$subconcept_matrix))
idx <- which(Matrix::colSums(M) == 1)
self[idx]
},
#' @description
#' Decompose a concept as the supremum of meet-irreducible concepts
#'
#' @param C A list of \code{Concept}s
#' @return
#' A list, each field is the set of meet-irreducible elements whose supremum is the corresponding element in \code{C}.
#'
#' @export
#'
decompose = function(C) {
irreducible <- self$meet_irreducibles()
irr_intents <- irreducible$intents()
C_intents <- C$intents()
ss <- lapply(seq(C$size()),
function(i) {
r <- Matrix::Matrix(C_intents[, i],
sparse = TRUE)
if (sum(r) == 0) return(C[i])
id <- Matrix::which(.subset(irr_intents,
r))
if (length(id) > 1) {
MM <- .subset(irr_intents[, id])
id <- id[Matrix::rowSums(MM) == 1]
}
decomposition <- irreducible[id]
return(decomposition)
})
return(ss)
},
#' @description
#' Supremum of Concepts
#'
#' @param ... See Details.
#'
#' @details
#' As argument, one can provide both integer indices or \code{Concepts}, separated by commas. The corresponding concepts are used to compute their supremum in the lattice.
#'
#' @return
#' The supremum of the list of concepts.
#'
#' @export
supremum = function(...) {
idx <- private$to_indices(...)
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
return(self[join(private$subconcept_matrix, idx)]$to_list()[[1]])
},
#' @description
#' Infimum of Concepts
#'
#' @param ... See Details.
#'
#' @details
#' As argument, one can provide both integer indices or \code{Concepts}, separated by commas. The corresponding concepts are used to compute their infimum in the lattice.
#'
#' @return
#' The infimum of the list of concepts.
#'
#' @export
infimum = function(...) {
idx <- private$to_indices(...)
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
return(self[meet(private$subconcept_matrix, idx)]$to_list()[[1]])
},
#' @description
#' Subconcepts of a Concept
#'
#' @param C (numeric or \code{SparseConcept}) The concept to which determine all its subconcepts.
#'
#' @return
#' A list with the subconcepts.
#' @export
subconcepts = function(C) {
idx <- private$to_indices(C)
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
# Get the index of all subconcepts
M <- Matrix::t(private$subconcept_matrix)[idx, ]
candidates <- Matrix::which(M > 0)
self[candidates]
},
#' @description
#' Superconcepts of a Concept
#'
#' @param C (numeric or \code{SparseConcept}) The concept to which determine all its superconcepts.
#'
#' @return
#' A list with the superconcepts.
#' @export
superconcepts = function(C) {
idx <- private$to_indices(C)
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
# Get the index of all superconcepts
M <- private$subconcept_matrix[idx, ]
candidates <- which(M > 0)
self[candidates]
},
#' @description
#' Lower Neighbours of a Concept
#'
#' @param C (\code{SparseConcept}) The concept to which find its lower neighbours
#'
#' @return
#' A list with the lower neighbours of \code{C}.
#'
#' @export
lower_neighbours = function(C) {
idx <- private$to_indices(C)
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
if (is.null(private$reduced_matrix)) {
private$reduced_matrix <- .reduce_transitivity(private$subconcept_matrix)
}
self[which(private$reduced_matrix[, idx] > 0)]
},
#' @description
#' Upper Neighbours of a Concept
#'
#' @param C (\code{SparseConcept}) The concept to which find its upper neighbours
#'
#' @return
#' A list with the upper neighbours of \code{C}.
#'
#' @export
upper_neighbours = function(C) {
idx <- private$to_indices(C)
if ((self$size() > 0) & (is.null(private$subconcept_matrix))) {
private$subconcept_matrix <- .subset(private$pr_extents)
}
if (is.null(private$reduced_matrix)) {
private$reduced_matrix <- .reduce_transitivity(private$subconcept_matrix)
}
self[which(private$reduced_matrix[idx, ] > 0)]
}
),
private = list(
subconcept_matrix = NULL,
reduced_matrix = NULL,
can_plot = TRUE,
concept_list_to_indices = function(concept_list) {
extents <- lapply(concept_list,
function(l) l$get_extent()$get_vector())
extents <- Reduce(cbind, extents)
indices <- .equal_sets(x = extents,
y = private$pr_extents)
indices <- arrayInd(Matrix::which(indices),
.dim = dim(indices))[, 2]
return(indices)
},
to_indices = function(...) {
dots <- list(...)
if (is.list(dots[[1]])) {
dots <- unlist(dots)
}
if (is.logical(dots[[1]])) {
return(which(dots[[1]]))
}
csets_idx <- sapply(dots,
function(l) inherits(l, "ConceptSet"))
if (length(csets_idx) > 0) {
csets <- sapply(dots[csets_idx],
function(l) l$to_list()) %>%
unlist()
} else {
csets <- c()
}
sets <- sapply(dots,
function(l)
inherits(l, "Concept"))
sets <- c(csets, dots[which(sets)])
indices <- sapply(dots, is.numeric)
idx <- c()
if (length(indices) > 0) {
idx <- c(idx, Reduce(c, dots[indices]))
}
if (length(sets) > 0) {
idx <- c(idx,
private$concept_list_to_indices(sets))
}
return(idx)
}
)
)
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.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.