mdendro
Extended Agglomerative Hierarchical Clustering

Package index
Search the mdendro package
Vignettes
Functions
20
Source code
3
Man pages
2
Home
/
CRAN
/
mdendro
/
A quick introduction to mdendro"

A quick introduction to mdendro"
In mdendro: Extended Agglomerative Hierarchical Clustering

Description

R package mdendro enables the calculation of agglomerative hierarchical clustering (AHC), extending the standard functionalities in several ways:

All this functionality is obtained with functions linkage, descval and descplot. Function linkage may be considered as a replacement for functions hclust (in package stats) and agnes (in package cluster). To enhance usability and interoperability, the linkage class includes several methods for plotting, summarizing information, and class conversion.

Installation

There exist two main ways to install mdendro:

Tutorial

Basics

Let us start by using the linkage function to calculate the complete linkage AHC of the UScitiesD dataset, a matrix of distances between a few US cities:

library(mdendro)
lnk <- linkage(UScitiesD, method = "complete")

Now we can plot the resulting dendrogram:

plot(lnk)

The summary of this dendrogram is:

summary(lnk)

In particular, you can recognize the calculated descriptors:

It is possible to work with similarity data without having to convert them to distances, provided they are in range [0.0, 1.0]. A typical example would be a matrix of non-negative correlations:

sim <- as.dist(Harman23.cor$cov)
lnk <- linkage(sim, type.prox = "sim")
plot(lnk, main = "Harman23")

There is also the option to choose between unweighted (default) and weighted methods:

par(mfrow = c(1, 2))
cars <- round(dist(scale(mtcars)), digits = 3)
nodePar <- list(cex = 0, lab.cex = 0.4)

lnk1 <- linkage(cars, method = "arithmetic")
plot(lnk1, main = "unweighted", nodePar = nodePar)

lnk2 <- linkage(cars, method = "arithmetic", weighted = TRUE)
plot(lnk2, main = "weighted", nodePar = nodePar)

When there are tied minimum distances in the agglomeration process, you may ignore them and proceed choosing a random pair (pair-group methods) or agglomerate them all at once (variable-group multidendrograms). With linkage you can use both approaches, being multidendrograms the default one:

par(mfrow = c(1, 2))
cars <- round(dist(scale(mtcars)), digits = 1)
nodePar <- list(cex = 0, lab.cex = 0.4)

lnk1 <- linkage(cars, method = "complete")
plot(lnk1, main = "multidendrogram", nodePar = nodePar)

lnk2 <- linkage(cars, method = "complete", group = "pair")
plot(lnk2, main = "pair-group", nodePar = nodePar)

While multidendrograms are unique, you may obtain structurally different pair-group dendrograms by just reordering the data. As a consequence, the descriptors are invariant to permutations for multidendrograms, but not for pair-group dendrograms:

cars <- round(dist(scale(mtcars)), digits = 1)
lnk1 <- linkage(cars, method = "complete")
lnk2 <- linkage(cars, method = "complete", group = "pair")

# apply random permutation to data
set.seed(666)
ord <- sample(attr(cars, "Size"))
cars_p <- as.dist(as.matrix(cars)[ord, ord])
lnk1p <- linkage(cars_p, method = "complete")
lnk2p <- linkage(cars_p, method = "complete", group = "pair")

# compare original and permuted cophenetic correlation
c(lnk1$cor, lnk1p$cor)
c(lnk2$cor, lnk2p$cor)

# compare original and permuted tree balance
c(lnk1$tb, lnk1p$tb)
c(lnk2$tb, lnk2p$tb)

In multidendrograms, the ranges (rectangles) show the heterogeneity between distances within the group, but they are optional in the plots:

par(mfrow = c(1, 2))
cars <- round(dist(scale(mtcars)), digits = 1)
nodePar <- list(cex = 0, lab.cex = 0.4)
lnk <- linkage(cars, method = "complete")
plot(lnk, col.rng = "bisque", main = "with ranges", nodePar = nodePar)
plot(lnk, col.rng = NULL, main = "without ranges", nodePar = nodePar)

Plots including ranges are only available if you directly use the plot.linkage function from mdendro. Anyway, you may still take advantage of other dendrogram plotting packages, such as dendextend and ape:

par(mfrow = c(1, 2))
cars <- round(dist(scale(mtcars)), digits = 1)
lnk <- linkage(cars, method = "complete")

lnk.dend <- as.dendrogram(lnk)
plot(dendextend::set(lnk.dend, "branches_k_color", k = 4),
     main = "dendextend package",
     nodePar = list(cex = 0.4, lab.cex = 0.5))

lnk.hcl <- as.hclust(lnk)
pal4 <- c("red", "forestgreen", "purple", "orange")
clu4 <- cutree(lnk.hcl, 4)
plot(ape::as.phylo(lnk.hcl),
     type = "fan",
     main = "ape package",
     tip.color = pal4[clu4],
     cex = 0.5)

In addition, you can also use the linkage function to plot heatmaps containing multidendrograms:

heatmap(scale(mtcars), hclustfun = linkage)

Linkage methods

The list of available AHC linkage methods is the following: single, complete, arithmetic, geometric, harmonic, versatile, ward, centroid and flexible. Their equivalences with the methods in other packages can be found below. The default method is arithmetic, which is also commonly known as average linkage or UPGMA.

par(mfrow = c(3, 3))
methods <- c("single", "complete", "arithmetic",
             "geometric", "harmonic", "versatile",
             "ward", "centroid", "flexible")

for (m in methods) {
  lnk <- linkage(UScitiesD, method = m)
  plot(lnk, cex = 0.6, main = m)
}

Two of the methods, versatile and flexible, depend on a parameter that takes values in (-Inf, +Inf) for versatile, and in [-1.0, +1.0] for flexible. Here come some examples:

par(mfrow = c(2, 3))

vals <- c(-10.0, 0.0, 10.0)
for (v in vals) {
  lnk <- linkage(UScitiesD, method = "versatile", par.method = v)
  plot(lnk, cex = 0.6, main = sprintf("versatile (%.1f)", v))
}

vals <- c(-0.8, 0.0, 0.8)
for (v in vals) {
  lnk <- linkage(UScitiesD, method = "flexible", par.method = v)
  plot(lnk, cex = 0.6, main = sprintf("flexible (%.1f)", v))
}

It is interesting to know how the descriptors depend on those parameters. Package mdendro provides two specific functions for this task, namely descval and descplot, which return just the numerical values or also the corresponding plot, respectively. For example, using versatile linkage:

par(mfrow = c(2, 3))
measures <- c("cor", "sdr", "ac", "cc", "tb")
vals <- c(-Inf, (-20:+20), +Inf)
for (m in measures)
  descplot(UScitiesD, method = "versatile",
           measure = m, par.method = vals,
           type = "o",  main = m, col = "blue")

Similarly for the flexible method:

par(mfrow = c(2, 3))
measures <- c("cor", "sdr", "ac", "cc", "tb")
vals <- seq(from = -1, to = +1, by = 0.1)
for (m in measures)
  descplot(UScitiesD, method = "flexible",
           measure = m, par.method = vals,
           type = "o",  main = m, col = "blue")

Comparison with other packages

For comparison, the same AHC can be found using functions hclust and agnes, where the default plots just show some differences in aesthetics:

library(mdendro)
lnk <- linkage(UScitiesD, method = "complete")

library(cluster)
agn <- agnes(UScitiesD, method = "complete")

# library(stats)   # unneeded, stats included by default
hcl <- hclust(UScitiesD, method = "complete")

par(mfrow = c(1, 3))
plot(lnk)
plot(agn, which.plots = 2)
plot(hcl)

Converting to class dendrogram, we can see all three are structurally equivalent:

lnk.dend <- as.dendrogram(lnk)
agn.dend <- as.dendrogram(agn)
hcl.dend <- as.dendrogram(hcl)
identical(lnk.dend, agn.dend)

par(mfrow = c(1, 2))
plot(lnk.dend, main = "lnk.dend = agn.dend", cex = 0.7)
plot(hcl.dend, main = "hcl.dend", cex = 0.7)

The cophenetic (ultrametric) matrix is readily available as component coph of the returned linkage object, and coincides with those obtained using the other functions:

hcl.coph <- cophenetic(hcl)
agn.coph <- cophenetic(agn)

all(lnk$coph == hcl.coph)
all(lnk$coph == agn.coph)

The coincidence also applies to the cophenetic correlation coefficient and agglomerative coefficient, with the advantage that linkage has them all already calculated:

hcl.cor <- cor(UScitiesD, hcl.coph)
all.equal(lnk$cor, hcl.cor)

all.equal(lnk$ac, agn$ac)

Rationale

Package mdendro was initially designed to make publicly available implementations of our work on agglomerative hierarchical clustering (AHC) [@Fernandez2008; @Fernandez2020], trying to reach a wide community through the use of the R language. We already have two implementations, a Java application called MultiDendrograms, and a Hierarchical_Clustering tool within our Radatools set of programs for the analysis of complex networks, whose Ada source code can be found in Radalib.

Apart from providing access to our algorithms, we also wanted to simplify how to use them, and improve the performance of their implementations. All this was accomplished by using state-of-the-art methods based on neighbor chains, implementing the base code in C++, and adding functionality to make the new package very similar and compatible with the main ones currently in use, namely the functions hclust in package stats, and agnes in package cluster. The result is a package mdendro that includes and largely extends the functionality of these reference functions, thus with the option of being a replacement for them.

The main novelties in mdendro, which are not available in other packages, are the following:

Multidendrograms

Description

Multidendrograms are the hierarchical structures that are obtained after applying the variable-group AHC algorithms introduced in [@Fernandez2008]. They solve the non-uniqueness problem found in the standard pair-group algorithms and implementations [@Hart1983]. This problem arises when two or more minimum distances between different clusters are equal during the amalgamation process. The standard approach consists in choosing a pair, breaking the ties between distances, and proceeding in the same way until the final hierarchical classification is obtained. However, different clusterings are possible depending on the criterion used to break the ties (usually a pair is just chosen at random), and the user is unaware of this problem.

The variable-group algorithms group more than two clusters at the same time when ties occur, given rise to a graphical representation called multidendrogram. Their main properties are:

Both hclust and agnes ignore this non-uniqueness problem, thus the need for mdendro.

Example

data <- "Name   VVMD5.1 VVMD5.2 VVMD7.1 VVMD7.2 VVMD27.1    VVMD27.2    VrZag62.1   VrZag62.2   VrZag79.1   VrZag79.2   VVS2-A1 VVS2.2
Alfrocheiro 226 238 249 253 179 189 188 200 251 251 145 153
Alvarinho   222 232 235 235 189 189 186 204 247 251 137 153
Antao Vaz   234 236 245 259 181 183 204 204 247 247 147 153
Aragonez    236 236 235 249 183 183 196 200 247 251 145 147
Arinto  226 238 239 247 181 185 186 188 247 251 145 153
Avesso  222 240 235 235 181 189 186 186 243 247 137 153
Azal    226 232 235 243 181 185 194 204 247 251 153 159
Baga    232 240 235 235 179 189 188 204 247 251 145 157
Bastardo    238 238 235 253 175 189 188 188 245 247 145 153
Bical   226 240 235 259 179 185 188 194 251 251 135 147
Camarate    234 236 239 249 181 189 188 200 247 251 147 153
Castelao    236 238 239 253 179 181 188 188 247 251 145 147
Cerceal Branco  226 236 245 253 179 181 188 204 247 251 145 159
Encruzado   226 232 235 253 183 189 194 194 247 251 151 153
Espadeiro   222 226 243 259 183 189 196 204 251 251 135 153
Fernao Pires    226 240 235 235 183 194 188 194 247 247 147 153
Folgasao    232 240 239 243 185 189 194 204 245 251 135 153
Galego Dourado  228 240 235 239 185 189 188 194 245 251 135 135
Gouveio 226 238 235 239 185 189 186 188 251 251 153 159
Jaen    226 236 245 253 181 189 188 194 247 251 147 153
Loureiro    232 232 247 259 181 185 186 196 251 251 145 153
Malvasia Fina   226 240 235 253 179 194 188 188 247 251 145 147
Malvasia Rei    228 240 235 245 185 194 188 194 251 257 135 147
Moreto  226 236 239 253 181 189 188 188 247 251 147 153
Moscatel Graudo 228 232 245 247 179 194 186 204 247 255 135 151
Moscatel Galego 226 228 235 245 179 189 186 188 251 255 135 153
Negra Mole  222 240 235 235 181 181 188 196 247 259 145 159
Perrum  236 240 235 239 181 185 188 188 243 247 135 147
Rabo de Ovelha  222 236 235 239 181 181 188 194 247 247 139 153
Rabigato    222 232 235 235 185 189 186 196 243 251 135 135
Rufete  226 236 235 253 181 189 188 194 245 247 135 159
Ramisco 226 238 235 259 181 185 188 196 247 251 135 159
Sercial 226 238 235 249 181 185 188 194 247 259 135 153
Siria   222 234 235 245 181 181 186 204 247 247 139 153
Talia   226 232 245 249 179 183 194 200 245 251 135 145
Terrantez   226 238 243 259 185 189 194 196 251 251 145 159
Tinta Barroca   228 236 235 239 181 183 188 192 245 247 145 153
Tinta Caiada    222 238 235 235 179 189 186 188 251 261 135 135
Tinto Cao   232 234 235 259 181 185 186 194 247 251 135 135
Tinta Miuda 226 238 235 235 179 183 186 188 251 259 141 153
Touriga Franca  226 228 235 239 181 183 192 194 245 247 145 153
Touriga Nacional    226 236 235 235 181 189 188 194 245 245 145 153
Trajadura   226 236 235 247 181 185 186 186 247 247 145 153
Trincadeira 234 238 235 245 181 185 188 204 247 251 135 153
Trincadeira das Pratas  238 240 235 253 189 194 188 188 251 257 143 145
Verdelho    222 232 235 253 181 189 194 196 247 251 135 153
Viosinho    232 232 235 239 185 189 186 188 243 245 135 153
Vinhao  222 226 235 259 189 189 188 196 245 251 135 137
Vital   222 240 235 235 181 194 188 188 247 253 147 153
Borracal    232 238 235 235 181 185 194 194 247 247 135 137
Fonte Cal   226 234 235 235 183 185 186 186 247 251 135 153"

dt <- read.table(textConnection(data), header = TRUE, sep = "\t", dec = ".")
# knitr::kable(dt)

n <- nrow(dt)
cols <- 2:ncol(dt)
nc <- length(cols)

m <- matrix(0.0, nrow = n, ncol = n,
            dimnames = list(dt$Name, dt$Name))

for (i in 1:(n-1)) {
    for (j in (i+1):n) {
        m[i, j] <- 1 - sum(dt[i, cols] == dt[j, cols]) / nc
        m[j, i] <- m[i, j]
    }
}

d <- as.dist(m)

Let us consider the genetic profiles of 51 grapevine cultivars at 6 microsatellite loci [@Almadanim2007]. The distance between two cultivars is defined as one minus the fraction of shared alleles, that we use to calculate a distance matrix d. The main characteristic of this kind of data is that the number of different distances is very small:

length(unique(d))

As a consequence, it becomes very easy to find tied distances during the agglomeration process. The corresponding unique multidendrogram is the following:

lnk <- linkage(d, method = "arithmetic", digits = 3)
nodePar <- list(cex = 0, lab.cex = 0.6)
plot(lnk, col.rng = NULL, nodePar = nodePar, main = "multidendrogram")

The reach of the non-uniqueness problem for this example is the existence of

This number corresponds to arithmetic linkage and a resolution of 3 decimal digits, and has been computed using the Hierarchical_Clustering tool in Radatools. We can check this fact by just calculating the binary dendrogram for random permutations of the data, and plotting the broad range of values of their cophenetic correlations, which clearly indicate the existence of many structurally different dendrograms:

nreps <- 1000
cors <- vector(length = nreps)
lnks <- list()
for (r in 1:nreps) {
    ord <- sample(attr(d, "Size"))
    d2 <- as.dist(as.matrix(d)[ord, ord])
    lnks[[r]] <- linkage(d2, group = "pair", digits = 3)
    cors[r] <- lnks[[r]]$cor
}
plot(sort(cors), main = "pair-group cophenetic correlations (same data)")

For example, the first two pair-group dendrograms calculated above are structurally different:

dend1 <- as.dendrogram(lnks[[1]])
dend2 <- as.dendrogram(lnks[[2]])
diff_12 <- dendextend::highlight_distinct_edges(dend1, dend2)
diff_21 <- dendextend::highlight_distinct_edges(dend2, dend1)

par(mfrow = c(2, 1))
nodePar <- list(cex = 0, lab.cex = 0.6)
plot(diff_12, nodePar = nodePar, main = "pair-group (original)")
plot(diff_21, nodePar = nodePar, main = "pair-group (permuted)")

The identification of ties requires the selection of the number of significant digits in our data. For example, if the original distances are experimentally obtained with a resolution of three decimal digits, two distances that differ in the sixth decimal digit should be considered as equal. If this is not taken into account, ties may be broken just by the numerical imprecision inherent to computer representations of real numbers. In the linkage function, you can control this level of resolution by adjusting its digits argument.

lnk3 <- linkage(d, method = "arithmetic", digits = 3)
lnk1 <- linkage(d, method = "arithmetic", digits = 1)

par(mfrow = c(2, 1))
nodePar <- list(cex = 0, lab.cex = 0.6)
plot(lnk3, col.rng = NULL, nodePar = nodePar,
     main = "multidendrogram (digits = 3)")
plot(lnk1, col.rng = NULL, nodePar = nodePar,
     main = "multidendrogram (digits = 1)")

Versatile linkage

Package mdendro also introduces a new family of space-conserving hierarchical clustering methods called versatile linkage [@Fernandez2020]. It is based on the use of generalized means, and includes single linkage, complete linkage and arithmetic linkage (a.k.a. average linkage or UPGMA/WPGMA) as particular cases. Additionally, versatile linkage naturally leads to the definition of two new methods, geometric linkage and harmonic linkage (hence the convenience to rename average linkage as arithmetic linkage, to emphasize the existence of different types of averages).

In function linkage, the parameter of versatile linkage is introduced using the par.method argument. It can take any real value, and the correspondence between versatile linkage and the above mentioned methods is the following:

vl_tab <- data.frame(
  linkage = c("complete", "arithmetic", "geometric", "harmonic", "single"),
  par.method = c(+Inf, +1, 0, -1, -Inf)
)
knitr::kable(vl_tab, col.names = c("linkage", "versatile linkage (`par.method`)"))

Let us show a small example in which we plot the different (multi)dendrograms as we increase the versatile linkage parameter, indicating the corresponding named methods:

d = as.dist(matrix(c( 0,  7, 16, 12,
                      7,  0,  9, 19,
                     16,  9,  0, 12,
                     12, 19, 12, 0), nrow = 4))

par(mfrow = c(2, 3))

vals <- c(-Inf, -1, 0, 1, Inf)
names <- c("single", "harmonic", "geometric", "arithmetic", "complete")
titles <- sprintf("versatile (%.1f) = %s", vals, names)

for (i in 1:length(vals)) {
  lnk <- linkage(d, method = "versatile", par.method = vals[i], digits = 2)
  plot(lnk, ylim = c(0, 20), cex = 0.6, main = titles[i])
}

Reference manual

Usage

linkage(prox, type.prox = "distance", digits = NULL,
        method = "arithmetic", par.method = 0, weighted = FALSE,
        group = "variable")

descval(prox, type.prox = "distance", digits = NULL,
        method = "versatile", par.method = c(-1,0,+1), weighted = FALSE,
        group = "variable", measure = "cor")

descplot(prox, ..., type.prox = "distance", digits = NULL,
         method = "versatile", par.method = c(-1, 0, +1), weighted = FALSE,
         group = "variable", measure = "cor", slope = 10)

Arguments:

arguments <- data.frame(
  arg = c("`prox`",
          "`type.prox`",
          "`digits`",
          "`method`",
          "`par.method`",
          "`weighted`",
          "`group`",
          "`measure`",
          "`slope`",
          "`...`"),
  desc = c("A structure of class `dist` containing non-negative proximity data (distances or similarities). All the linkage methods are used with non-squared proximity data as input, except for method `\"centroid\"`, which is meant to be used with squared Euclidean distances."
           ,
            "A character string to indicate whether the proximity data represent `\"distance\"` (default) or `\"similarity\"` between objects. Methods `\"ward\"` and `\"centroid\"` cannot be used with similarity data as input, while the rest of the linkage methods can be used with both distances and similarities."
           ,
            "An integer value specifying the precision, i.e., the number of significant decimal digits to be used for the comparisons between proximity data. This is an important parameter, since equal proximity data at a certain precision may become different by increasing its value. Thus, it may be responsible of the existence of tied proximity data. If the value of this parameter is negative or `NULL` (default), then the precision is automatically set to that of the input proximity value with the largest number of significant decimal digits."
           ,
            "A character string specifying the linkage method to be used. For `linkage()`, this should be one of: `\"single\"`, `\"complete\"`, `\"arithmetic\"`, `\"geometric\"`, `\"harmonic\"`, `\"versatile\"`, `\"ward\"`, `\"centroid\"` or `\"flexible\"`. Methods `\"versatile\"` and `\"flexible\"` are the only two methods that can be used in `descval()` and `descplot()`."
           ,
            "A real value, in the case of `linkage()`, or a vector of real values, in the case of `descval()` and `descplot()`, required as parameter for the methods `\"versatile\"` and `\"flexible\"`. The range of possible values is [-Inf, +Inf] for `\"versatile\"`, and [-1, +1] for `\"flexible\"`."
           ,
            "A logical value to choose between the weighted and the unweighted (default) versions of some linkage methods. Weighted linkage gives merging branches in a dendrogram equal weight regardless of the number of objects carried on each branch. Such a procedure weights objects unequally, contrasting with unweighted linkage that gives equal weight to each object in the clusters. This parameter has no effect on the `\"single\"` and `\"complete\"` linkages."
           ,
            "A character string to choose a grouping criterion between the `\"variable\"`-group approach (default) that returns a multidendrogram, i.e., a multifurcated dendrogram (m-ary tree), and the `\"pair\"`-group approach that returns a bifurcated dendrogram (binary tree)."
           ,
            "A character string specifying the descriptive measure to be plotted. This should be one of: `\"cor\"`, for cophenetic correlation coefficient; `\"sdr\"`, for space distortion ratio; `\"ac\"`, for agglomerative coefficient; `\"cc\"`, for chaining coefficient; or `\"tb\"`, for tree balance."
           ,
            "A real value representing the slope of a sigmoid function to map the `\"versatile\"` linkage unbounded interval (-Inf, +Inf) onto the bounded interval (-1, +1). It can be used to improve the distribution of points along the x axis."
           ,
            "Graphical parameters may also be supplied (see `par`) and are passed to `plot.default`."
            )
)
knitr::kable(arguments, col.names = c("", ""))

## S3 method for class 'linkage'
plot(x, type = c("rectangle", "triangle"),
     center = FALSE, edge.root = FALSE,
     nodePar = NULL, edgePar = list(),
     leaflab = c("perpendicular", "textlike", "none"),
     dLeaf = NULL, xlab = "", ylab = "", xaxt = "n", yaxt = "s",
     horiz = FALSE, frame.plot = FALSE, xlim, ylim,
     col.rng = "lightgray", ...)

Arguments:

arguments <- data.frame(
  arg = c("`x`",
          "`type`",
          "`center`",
          "`edge.root`",
          "`nodePar`",
          "`edgePar`",
          "`leaflab`",
          "`dLeaf`",
          "`xlab`,`ylab`",
          "`xaxt`,`yaxt`",
          "`horiz`",
          "`frame.plot`",
          "`xlim`,`ylim`",
          "`col.rng`",
          "`...`"),
  desc = c("An object of class `linkage`, typically created by `linkage()`."
           ,
            "Type of plot"
           ,
            "Logical; if `TRUE`, nodes are plotted centered with respect to the leaves in the branch. Otherwise (default), plot them in the middle of all direct child nodes."
           ,
            "Logical; if `TRUE`, draw an edge to the root node."
           ,
            "A `list` of plotting parameters to use for the nodes (see `points`), or `NULL` by default which does not draw symbols at the nodes. The list may contain components named `pch`, `cex`, `col`, `xpd`, and/or `bg` each of which can have length two for specifying separate attributes for inner nodes and leaves. Note that the default of `pch` is `1:2`, so you may want to use `pch = NA` if you specify `nodePar`."
           ,
            "A `list` of plotting parameters to use for the edge `segments`. The list may contain components named `co`l, `lty` and `lwd`. As with `nodePar`, each can have length two for differentiating leaves and inner nodes."
           ,
            "A string specifying how leaves are labeled. The default `\"perpendicular\"` writes text vertically (by default), `\"textlike\"` writes text horizontally (in a rectangle), and `\"none\"` suppresses leaf labels."
           ,
            "A number specifying the distance in user coordinates between the tip of a leaf and its label. If `NULL` as per default, 3/4 of a letter width or height is used."
           ,
            "A label for the axis."
           ,
            "A character which specifies the axis type. Specifying `\"n\"` suppresses plotting, while any value other than `\"n\"` implies plotting."
           ,
            "Logical indicating if the dendrogram should be drawn horizontally or not."
           ,
            "Logical indicating if a box around the plot should be drawn, see `plot.default`."
           ,
            "Optional x- and y-limits of the plot, passed to `plot.default`. The defaults for these show the full dendrogram."
           ,
            "Color (`\"lightgray\"` by default) to be used for plotting range rectangles in case of tied heights. If `NULL`, range rectangles are not plotted."
           ,
            "Graphical parameters (see `par`) may also be supplied and are passed to `plot.default`."
            )
)
knitr::kable(arguments, col.names = c("", ""))

Based on the plot function for objects of class dendrogram (see plot.dendrogram), the plot function for objects of class linkage adds the possibility of visualizing the existence of tied heights in a dendrogram thanks to the col.rng parameter.

Results

An object of class linkage that describes the multifurcated dendrogram obtained. The object is a list with the following components:

components <- data.frame(
  val = c("`call`",
          "`digits`",
          "`merger`",
          "`height`",
          "`range`",
          "`order`",
          "`coph`",
          "`binary`",
          "`cor`",
          "`sdr`",
          "`ac`",
          "`cc`",
          "`tb`"),
  desc = c("The call that produced the result."
           ,
            "Number of significant decimal digits used as precision. It is given by the user or automatically set to that of the input proximity value with the largest number of significant decimal digits."
           ,
            "A list of vectors of integer that describes the merging of clusters at each step of the clustering. If a number _j_ in a vector is negative, then singleton cluster _-j_ was merged at this stage. If _j_ is positive, then the merge was with the cluster formed at stage _j_ of the algorithm."
           ,
            "A vector with the proximity values between merging clusters (for the particular agglomeration) at the successive stages."
           ,
            "A vector with the range (the maximum minus the minimum) of proximity values between merging clusters. It is equal to 0 for binary clusters."
           ,
            "A vector giving a permutation of the original observations to allow for plotting, in the sense that the branches of a clustering tree will not cross."
           ,
            "Object of class `dist` containing the cophenetic (or ultrametric) proximity data in the output dendrogram, sorted in the same order as the input proximity data in `prox`."
           ,
            "A logical value indicating whether the output dendrogram is a binary tree or, on the contrary, it contains an agglomeration of more than two clusters due to the existence of tied proximity data. Its value is always `TRUE` when the `pair` grouping criterion is used."
           ,
            "Cophenetic correlation coefficient [[@Sokal1962]](#bibliography), defined as the Pearson correlation coefficient between the output cophenetic proximity data and the input proximity data. It is a measure of how faithfully the dendrogram preserves the pairwise proximity between objects."
           ,
            "Space distortion ratio [[@Fernandez2020]](#bibliography), calculated as the difference between the maximum and minimum cophenetic proximity data, divided by the difference between the maximum and minimum initial proximity data. Space dilation occurs when the space distortion ratio is greater than 1."
           ,
            "Agglomerative coefficient [[@Rousseeuw1986]](#bibliography), a number between 0 and 1 measuring the strength of the clustering structure obtained."
           ,
            "Chaining coefficient [[@Williams1966]](#bibliography), a number between 0 and 1 measuring the tendency for clusters to grow by the addition of clusters much smaller rather than by fusion with other clusters of comparable size."
           ,
            "Tree balance [[@Fernandez2020]](#bibliography), a number between 0 and 1 measuring the equality in the number of leaves in the branches concerned at each fusion in the hierarchical tree."
            )
)
knitr::kable(components, col.names = c("", ""))

Class linkage has methods for the following generic functions: print, summary, plot (see plot.linkage), as.dendrogram, as.hclust and cophenetic.

Supported linkage methods

The difference between the available hierarchical clustering methods rests in the way the proximity between two clusters is defined from the proximity between their constituent objects:

Equivalences with other packages

Except for the cases containing tied distances, the following equivalences hold between function linkage in package mdendro, function hclust in package stats, and function agnes in package cluster. When relevant, weighted (W) or unweighted (U) versions of the linkage methods and the value for par.method are indicated:

methods <- data.frame(
  linkage   = c("single", "complete", "arithmetic, U", "arithmetic, W",
                "geometric, U/W", "harmonic, U/W", "versatile, U/W, $p$",
                "---", "ward", "centroid, U", "centroid, W",
                "flexible, U, $\\beta$", "---",
                "flexible, W, $\\beta$", "---"),
  hclust    = c("single", "complete", "average", "mcquitty",
                "---", "---", "---", "ward", "ward.D2",
                "centroid", "median", "---", "---", "---", "---"),
  agnes     = c("single", "complete", "average", "weighted",
                "---", "---", "---", "---", "ward",
                "---", "---",
                "gaverage, $\\beta$",
                "gaverage, $\\alpha_1$, $\\alpha_2$, $\\beta$, $\\gamma$",
                "flexible, $(1-\\beta)/2$",
                "flexible, $\\alpha_1$, $\\alpha_2$, $\\beta$, $\\gamma$")
)
knitr::kable(methods, col.names = c("`linkage`", "`hclust`", "`agnes`"))

History

mdendro 2.1.0

New logical value added to objects of class linkage, indicating whether the output dendrogram is a binary tree.

mdendro 2.0

Fully new implementation from scratch, programmed in C++, using the efficient technique of neighbor chains, including plotting capabilities, and compatible with the main hierarchical clustering and dendrogram packages in R.

mdendro 1.0

Initial R version based on the MultiDendrograms Java core for the calculations.

Authors

Bibliography



Try the mdendro package in your browser

Any scripts or data that you put into this service are public.

mdendro documentation built on Oct. 18, 2023, 5:08 p.m.

R Package Documentation

rdrr.io home R language documentation Run R code online

Browse R Packages

CRAN packages Bioconductor packages R-Forge packages GitHub packages

We want your feedback!

Note that we can't provide technical support on individual packages. You should contact the package authors for that.
GitHub issue tracker
ian@mutexlabs.com
Personal blog

 
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.

Close