# 5.2 Compare partitions

#### Pierre Denelle, Boris Leroy and Maxime Lenormand

#### 2024-11-03

Source:`vignettes/a5_2_compare_partitions.Rmd`

`a5_2_compare_partitions.Rmd`

In this vignette, we aim at comparing the assignment of sites into
different bioregions across multiple bioregionalizations, using the
function `compare_partitions()`

.

## 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,
index = names(vegedissim)[3],
comat = vegemat,
method = "mcquitty", n_clust = 3)
vege_hclu_hierarclust$cluster_info # 3
```

```
## partition_name n_clust requested_n_clust output_cut_height
## 1 K_3 3 3 0.4638672
```

```
# 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 3
```

## Compare the partitions

Before comparing the partitions, 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 1003 1 3 1
## 2 1004 1 3 3
## 3 1005 1 3 1
## 4 1006 1 3 3
## 5 1007 1 3 1
## 6 1008 1 3 1
```

We can now run the function `compare_partitions()`

.

```
hclu_vs_netclu <- compare_partitions(
cluster_object = comp[, c("K_3_hclu", "K_3_netclu", "K_3_nhclu")],
store_pairwise_membership = TRUE,
cor_frequency = TRUE,
store_confusion_matrix = TRUE)
hclu_vs_netclu
```

```
## Partition comparison:
## - 3 partitions compared
## - 715 items in the clustering
## - Requested indices: rand jaccard
## - Metric summary:
## rand jaccard
## Min 0.6761474 0.4271905
## Mean 0.7632459 0.5483558
## Max 0.8654130 0.7100855
## - Correlation between each partition and the total frequency of item pairwise membership computed:
## # Range: 0.731 - 0.891
## # Partition(s) most representative (i.e., highest correlation):
## K_3_nhclu
## Correlation = 0.891
## - Item pairwise membership stored in outputs
## - Confusion matrices of partition comparisons stored in outputs
```

`compare_partitions()`

produces several outputs which: -
look within each partition/bioregionalization how sites are assigned to
bioregions - compare different partitions/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] TRUE`

Pairwise 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 1003 1 3 1
## 9 1011 1 3 1
```

We 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_partitions()`

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 TRUE TRUE
## 1_10 TRUE FALSE TRUE
## 1_11 TRUE FALSE TRUE
```

The 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 3
```

The 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
## 109881 42717 47932 54725
```

we 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
## 61650 44199 38466 110940
##
## $`K_3_hclu%K_3_nhclu`
## a b c d
## 84143 21706 12648 136758
##
## $`K_3_netclu%K_3_nhclu`
## a b c d
## 66314 33802 30477 124662
```

Based on the confusion matrices, we can compute a range of indices to
indicate the agreement among partitions. As of now, we have
implemented:*Rand index*
$(a+d)/(a+b+c+d)$
The Rand index measures agreement among partitions 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 partitions by only accounting
for pairs of sites that are grouped.

These two metrics are complementary, because the Jaccard index will tell if partitions are similar in their clustering structure, whereas the Rand index will tell if partitions 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 partitions 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
partitions is most representative of all partitions. To find it out, we
can compare the pairwise membership of each partition with the total
frequency of pairwise membership across all partitions. This correlation
can be requested with `cor_frequency = TRUE`

.

`hclu_vs_netclu$partition_freq_cor`

```
## K_3_hclu K_3_netclu K_3_nhclu
## 0.8352365 0.7307649 0.8908211
```

Here the third bioregionalization is the most representative of all partitions.