Bk is the calculation of Fowlkes-Mallows index for a series of k cuts for two dendrograms.

`Bk(tree1, tree2, k, warn = dendextend_options("warn"), ...)`

- 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.

- 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.

- ...
Ignored (passed to FM_index_R).

A list (of k's length) of Fowlkes-Mallows index between two dendrogram for a scalar/vector of k values. The names of the lists' items is the k for which it was calculated.

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.

Fowlkes, E. B.; Mallows, C. L. (1 September 1983). "A Method for Comparing Two Hierarchical Clusterings". Journal of the American Statistical Association 78 (383): 553.

```
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.dendrogram(hc1)
tree2 <- as.dendrogram(hc2)
# cutree(tree1)
Bk(hc1, hc2, k = 3)
Bk(hc1, hc2, k = 2:10)
Bk(hc1, hc2)
Bk(tree1, tree2, k = 3)
Bk(tree1, tree2, k = 2:5)
system.time(Bk(hc1, hc2, k = 2:5)) # 0.01
system.time(Bk(hc1, hc2)) # 1.28
system.time(Bk(tree1, tree2, k = 2:5)) # 0.24 # after fixes.
system.time(Bk(tree1, tree2, k = 2:10)) # 0.31 # after fixes.
system.time(Bk(tree1, tree2)) # 7.85
Bk(tree1, tree2, k = 99:101)
y <- Bk(hc1, hc2, k = 2:10)
plot(unlist(y) ~ c(2:10), type = "b", ylim = c(0, 1))
# can take a few seconds
y <- Bk(hc1, hc2)
plot(unlist(y) ~ as.numeric(names(y)),
main = "Bk plot", pch = 20,
xlab = "k", ylab = "FM Index",
type = "b", ylim = c(0, 1)
)
# we are still missing some hypothesis testing here.
# for this we'll have the Bk_plot function.
}
```