Bk is the calculation of Fowlkes-Mallows index for a series of k cuts for two dendrograms. A Bk plot is simply a scatter plot of Bk versus k. This plot helps in identifiying the similarity between two dendrograms in different levels of k (number of clusters).

Bk_plot(
tree1,
tree2,
k,
rejection_line_asymptotic = TRUE,
rejection_line_permutation = FALSE,
R = 1000,
k_permutation,
conf.level = 0.95,
col_line_Bk = 1,
col_line_asymptotic = 2,
col_line_permutation = 4,
warn = dendextend_options("warn"),
main = "Bk plot",
xlab = "k (number of clusters)",
ylab = "Bk (Fowlkes-Mallows Index)",
xlim,
ylim = c(0, 1),
try_cutree_hclust = TRUE,
...
)

## Arguments

tree1

a dendrogram/hclust/phylo object.

tree2

a dendrogram/hclust/phylo object.

k

an integer scalar or vector with the desired number of cluster groups. If missing - the Bk will be calculated for a default k range of 2:(nleaves-1). No point in checking k=1/k=n, since both will give Bk=1.

logical (TRUE). Should we add a line of the Expected Bk value for each k, under the null hypothesis of no relation between the clusterings?

rejection_line_asymptotic

logical (TRUE). Should we add a line of the one sided rejection region based on the asymptotic distribution of Bk values, for each k, under the null hypothesis of no relation between the clusterings?

rejection_line_permutation

logical (FALSE). Should we add a line of the one sided rejection region based on the asymptotic distribution of Bk values, for each k, under the null hypothesis of no relation between the clusterings?

R

integer (Default is 1000). The number of Bk permutation to perform for each k. Applicable only if rejection_line_permutation is TRUE.

k_permutation

the k's to be used for permutation (sometimes we might be only interested in some k's and it is not important to run the simulation for all possible ks). If missing - k itself will be used.

conf.level

the level of one sided confidence interval used for creation of the rejection lines.

a character scalar of either "none" (default), or "bonferroni". This controls the multiple correction method to use for the critical rejection values. Currently only the Bonferroni method is implemented (based on the number of different k values).

col_line_Bk

the color of the Bk line.

col_line_asymptotic

the color of the rejection asymptotic Bk line.

col_line_permutation

the color of the rejection asymptotic Bk line.

warn

logical (default from dendextend_options("warn") is FALSE). Set if warning are to be issued, it is safer to keep this at TRUE, but for keeping the noise down, the default is FALSE. If set to TRUE, extra checks are made to varify that the two clusters have the same size and the same labels.

main

passed to plot.

xlab

passed to plot.

ylab

passed to plot.

xlim

passed to plot. If missign, xlim is from 2 to nleaves-1

ylim

passed to plot.

try_cutree_hclust

logical (TRUE). Since cutree for hclust is MUCH faster than for dendrogram - Bk_plot will first try to change the dendrogram into an hclust object. If it will fail (for example, with unbranched trees), it will continue using the cutree.dendrogram functions. If try_cutree_hclust=FALSE, it will force to use cutree.dendrogram and not cutree.hclust.

...

Ignored.

## Value

After plotting the Bk plot. Returns (invisible) the output of the elements used for constructing the plot: The Bk values, Bk permutations (if used), Bk theoratical values, etc.

## Details

From Wikipedia:

Fowlkes-Mallows index (see references) is an external evaluation method that is used to determine the similarity between two clusterings (clusters obtained after a clustering algorithm). This measure of similarity could be either between two hierarchical clusterings or a clustering and a benchmark classification. A higher the value for the Fowlkes-Mallows index indicates a greater similarity between the clusters and the benchmark classifications.

The default Bk plot comes with a line with dots (type "b") of the Bk values. Also with a fragmented (lty=2) line (of the same color) of the expected Bk line under H0, And a solid red line of the upper critical Bk values for rejection

FM_index, Bk, Bk_permutations

## Examples


if (FALSE) {

set.seed(23235)
ss <- TRUE # sample(1:150, 10 )
hc1 <- hclust(dist(iris[ss, -5]), "com")
hc2 <- hclust(dist(iris[ss, -5]), "single")
# tree1 <- as.treerogram(hc1)
# tree2 <- as.treerogram(hc2)
#    cutree(tree1)

Bk_plot(hc1, hc2, k = 2:20, xlim = c(2, 149))
Bk_plot(hc1, hc2)

Bk_plot(hc1, hc2, k = 3)
Bk_plot(hc1, hc2, k = 3:10)
Bk_plot(hc1, hc2)
Bk_plot(hc1, hc2, p.adjust.methods = "bonferroni") # higher rejection lines

# this one can take a bit of time:
Bk_plot(hc1, hc2,
rejection_line_permutation = TRUE,
k_permutation = c(2, 4, 6, 8, 10, 20, 30, 40, 50), R = 100
)
# we can see that the permutation line is VERY close to the asymptotic line.
# This is great since it means one can often use the asymptotic results
# Without having to do many simulations.

# works just as well for dendrograms:
dend1 <- as.dendrogram(hc1)
dend2 <- as.dendrogram(hc2)
Bk_plot(dend1, dend2, k = 2:3, try_cutree_hclust = FALSE) # slower than hclust, but works...
Bk_plot(hc1, dend2, k = 2:3, try_cutree_hclust = FALSE) # slower than hclust, but works...
Bk_plot(dend1, dend1, k = 2:3, try_cutree_hclust = TRUE) # slower than hclust, but works...
Bk_plot(hc1, hc1, k = 2:3) # slower than hclust, but works...
# for some reason it can't turn dend2 back to hclust :(
a <- Bk_plot(hc1, hc2, k = 2:3, try_cutree_hclust = TRUE) # slower than hclust, but works...

hc1_mixed <- as.hclust(sample(as.dendrogram(hc1)))
Bk_plot(
tree1 = hc1, tree2 = hc1_mixed,