5.2 Compare bioregionalizations
Pierre Denelle, Boris Leroy and Maxime Lenormand
2024-12-16
Source:vignettes/a5_2_compare_bioregionalizations.Rmd
a5_2_compare_bioregionalizations.RmdIn this vignette, we aim at comparing the assignment of sites into
different bioregions across multiple bioregionalizations, using the
function compare_bioregionalizations().
Data
We use the vegetation dataset that comes with
bioregion.
data("vegedf")
data("vegemat")
# Calculation of (dis)similarity matrices
vegedissim <- dissimilarity(vegemat, metric = c("Simpson"))
vegesim <- dissimilarity_to_similarity(vegedissim)Bioregionalization
We use the same three bioregionalization algorithms as in the visualization
vignette, i.e. a non-hierarchical, hierarchical and network
bioregionalizations.
We chose 3 bioregions for the non-hierarchical and hierarchical
bioregionalizations.
# Non hierarchical bioregionalization
vege_nhclu_kmeans <- nhclu_kmeans(vegedissim, n_clust = 3, index = "Simpson")
vege_nhclu_kmeans$cluster_info # 3## partition_name n_clust
## K_3 K_3 3
# Hierarchical bioregionalization
set.seed(1)
vege_hclu_hierarclust <- hclu_hierarclust(dissimilarity = vegedissim,
method = "mcquitty", n_clust = 3,
optimal_tree_method = "best")
vege_hclu_hierarclust$cluster_info # 3## partition_name n_clust requested_n_clust output_cut_height
## 1 K_3 3 3 0.625
# Network bioregionalization
set.seed(1)
vege_netclu_walktrap <- netclu_walktrap(vegesim,
index = names(vegesim)[3])
vege_netclu_walktrap$cluster_info # 3## partition_name n_clust
## K_3 K_3 3Compare the bioregionalizations
Before comparing the bioregionalizations, we build a common
data.frame containing the three distinct
bioregionalizations.
comp <- dplyr::left_join(vege_hclu_hierarclust$clusters,
vege_netclu_walktrap$clusters,
by = "ID")
colnames(comp) <- c("ID", "K_3_hclu", "K_3_netclu")
comp <- dplyr::left_join(comp,
vege_nhclu_kmeans$clusters,
by = "ID")
colnames(comp) <- c("ID", "K_3_hclu", "K_3_netclu", "K_3_nhclu")
head(comp)## ID K_3_hclu K_3_netclu K_3_nhclu
## 1 512 1 1 1
## 2 799 1 3 1
## 3 375 2 1 2
## 4 476 2 1 2
## 5 971 1 3 1
## 6 1282 1 3 1We can now run the function
compare_bioregionalizations().
hclu_vs_netclu <- compare_bioregionalizations(
cluster_object = comp[, c("K_3_hclu", "K_3_netclu", "K_3_nhclu")],
store_pairwise_membership = TRUE,
cor_frequency = TRUE,
store_confusion_matrix = TRUE)
str(hclu_vs_netclu)## List of 7
## $ args :List of 4
## ..$ indices : chr [1:2] "rand" "jaccard"
## ..$ cor_frequency : logi TRUE
## ..$ store_pairwise_membership: logi TRUE
## ..$ store_confusion_matrix : logi TRUE
## $ inputs : Named int [1:2] 715 3
## ..- attr(*, "names")= chr [1:2] "number_items" "number_bioregionalizations"
## $ pairwise_membership : logi [1:255255, 1:3] TRUE FALSE FALSE TRUE TRUE TRUE ...
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr [1:255255] "1_2" "1_3" "1_4" "1_5" ...
## .. ..$ : chr [1:3] "K_3_hclu" "K_3_netclu" "K_3_nhclu"
## $ freq_item_pw_membership : Named num [1:255255] 2 1 1 2 2 2 2 2 2 2 ...
## ..- attr(*, "names")= chr [1:255255] "1_2" "1_3" "1_4" "1_5" ...
## $ bioregionalization_freq_cor : Named num [1:3] 0.851 0.737 0.893
## ..- attr(*, "names")= chr [1:3] "K_3_hclu" "K_3_netclu" "K_3_nhclu"
## $ confusion_matrix :List of 3
## ..$ K_3_hclu%K_3_netclu : Named int [1:4] 63316 41892 36800 113247
## .. ..- attr(*, "names")= chr [1:4] "a" "b" "c" "d"
## ..$ K_3_hclu%K_3_nhclu : Named int [1:4] 85543 19665 11248 138799
## .. ..- attr(*, "names")= chr [1:4] "a" "b" "c" "d"
## ..$ K_3_netclu%K_3_nhclu: Named int [1:4] 66314 33802 30477 124662
## .. ..- attr(*, "names")= chr [1:4] "a" "b" "c" "d"
## $ bioregionalization_comparison:'data.frame': 3 obs. of 3 variables:
## ..$ bioregionalization_comparison: chr [1:3] "K_3_hclu%K_3_netclu" "K_3_hclu%K_3_nhclu" "K_3_netclu%K_3_nhclu"
## ..$ rand : num [1:3] 0.692 0.879 0.748
## ..$ jaccard : num [1:3] 0.446 0.735 0.508
## - attr(*, "class")= chr [1:2] "bioregion.bioregionalization.comparison" "list"compare_bioregionalizations() produces several outputs
which: - look within each bioregionalization how sites are assigned to
bioregions - compare different bioregionalizations by analysing whether
they produce similar pairwise memberships
Let’s first look at pairwise membership within bioregionalization.
Pairwise membership
The number of pairwise combinations for \(n\) sites equals \(n(n-1)/2\). So in our case, where we have 715 sites, we do end up with 2.55255^{5} pairwise combinations.
## [1] TRUEPairwise memberships look for each pairs of site whether they are assigned to the same or to a different bioregion. Let’s look at the sites 1 and 9 across the different bioregionalization:
comp[c(1, 9), ]## ID K_3_hclu K_3_netclu K_3_nhclu
## 1 512 1 1 1
## 9 948 1 3 1We can see that the sites 1 and 9 are classified in the same
bioregion in the first two bioregionalizations, but not in the third
one.
The $pairwise_membership output of
compare_bioregionalizations() shows this as a
TRUE/FALSE statement.
hclu_vs_netclu$pairwise_membership[8:10, ]## K_3_hclu K_3_netclu K_3_nhclu
## 1_9 TRUE FALSE TRUE
## 1_10 TRUE FALSE TRUE
## 1_11 TRUE FALSE TRUEThe number of times each pair of sites are clustered together
(i.e. the sum of rows of the table in $pairwise_membership)
is available in the $freq_item_pw_membership output:
hclu_vs_netclu$freq_item_pw_membership[c(1, 8)]## 1_2 1_9
## 2 2The sites 1 and 2 were never classified in the same bioregion across the three bioregionalizations. Sites 1 and 9 were classified in the same bioregion in two bioregionalizations. If we look at the total frequencies:
table(hclu_vs_netclu$freq_item_pw_membership)##
## 0 1 2 3
## 111723 41539 45403 56590we see that the most dominant situation is when sites are never assigned to the same bioregion.
Confusion matrix
The confusion matrix allows to compare different bioregionalizations by looking at the similarity of their pairwise memberships. To do so, the function computes a confusion matrix with four elements:
. \(a\) number of pairs of sites grouped in bioregionalization 1 and in bioregionalization 2 . \(b\) number of pairs of sites grouped in bioregionalization 1 but not in bioregionalization 2 . \(c\) number of pairs of sites not grouped in bioregionalization 1 but grouped in bioregionalization 2 . \(d\) number of pairs of sites not grouped in both bioregionalization 1 & 2
hclu_vs_netclu$confusion_matrix## $`K_3_hclu%K_3_netclu`
## a b c d
## 63316 41892 36800 113247
##
## $`K_3_hclu%K_3_nhclu`
## a b c d
## 85543 19665 11248 138799
##
## $`K_3_netclu%K_3_nhclu`
## a b c d
## 66314 33802 30477 124662Based on the confusion matrices, we can compute a range of indices to
indicate the agreement among bioregionalizations. As of now, we have
implemented:
Rand index \((a+d)/(a+b+c+d)\) The Rand index measures
agreement among bioregionalizations by accounting for both the pairs of
sites that are grouped, but also the pairs of sites that are not
grouped.
Jaccard index \(a/(a+b+c)\) The Jaccard index measures
agreement among bioregionalizations by only accounting for pairs of
sites that are grouped.
These two metrics are complementary, because the Jaccard index will tell if bioregionalizations are similar in their clustering structure, whereas the Rand index will tell if bioregionalizations are similar not only in the pairs of items clustered together, but also in terms of the pairs of sites that are not clustered together. For example, take two bioregionalizations which never group together the same pairs of sites. Their Jaccard index will be 0, whereas the Rand index can be > 0 due to the sites that are not grouped together.
Additional indices can be manually computed by the users on the basis of the list of confusion matrices.
In some cases, users may be interested in finding which of the
bioregionalizations is most representative of all bioregionalizations To
find it out, we can compare the pairwise membership of each
bioregionalization with the total frequency of pairwise membership
across all bioregionalizations. This correlation can be requested with
cor_frequency = TRUE.
hclu_vs_netclu$partition_freq_cor## NULLHere the third bioregionalization is the most representative of all bioregionalizations.