Calculating Fowlkes-Mallows index.

The `FM_index_R`

function also calculates the expectancy and variance of the FM Index
under the null hypothesis of no relation.

```
FM_index_R(
A1_clusters,
A2_clusters,
assume_sorted_vectors = FALSE,
warn = dendextend_options("warn"),
...
)
```

- A1_clusters
a numeric vector of cluster grouping (numeric) of items, with a name attribute of item name for each element from group A1. These are often obtained by using some k cut on a dendrogram.

- A2_clusters
a numeric vector of cluster grouping (numeric) of items, with a name attribute of item name for each element from group A2. These are often obtained by using some k cut on a dendrogram.

- assume_sorted_vectors
logical (FALSE). Can we assume to two group vectors are sorter so that they have the same order of items? IF FALSE (default), then the vectors will be sorted based on their name attribute.

- 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.

The Fowlkes-Mallows index between two vectors of clustering groups.

Includes the attributes E_FM and V_FM for the relevant expectancy and variance under the null hypothesis of no-relation.

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")
# dend1 <- as.dendrogram(hc1)
# dend2 <- as.dendrogram(hc2)
# cutree(dend1)
FM_index_R(cutree(hc1, k = 3), cutree(hc1, k = 3)) # 1
set.seed(1341)
FM_index_R(cutree(hc1, k = 3),
sample(cutree(hc1, k = 3)),
assume_sorted_vectors = TRUE) # 0.38037
FM_index_R(cutree(hc1, k = 3),
sample(cutree(hc1, k = 3)),
assume_sorted_vectors = FALSE) # 1 again :)
FM_index_R(cutree(hc1, k = 3),
cutree(hc2, k = 3)) # 0.8059
FM_index_R(cutree(hc1, k = 30),
cutree(hc2, k = 30)) # 0.4529
fo <- function(k) FM_index_R(cutree(hc1, k), cutree(hc2, k))
lapply(1:4, fo)
ks <- 1:150
plot(sapply(ks, fo) ~ ks, type = "b", main = "Bk plot for the iris dataset")
clu_1 <- cutree(hc2, k = 100) # this is a lie - since this one is NOT well defined!
clu_2 <- cutree(as.dendrogram(hc2), k = 100) # We see that we get a vector of NAs for this...
FM_index_R(clu_1, clu_2) # NA
}
```