Elements of Data Science
SDS 322E

H. Sherry Zhang
Department of Statistics and Data Sciences
The University of Texas at Austin

Fall 2025

Learning objectives

• Understand the hierarchical clustering algorithm
• Understand different linkage methods
• Apply hierarchical clustering to the data and compare the result with kmeans clustering
  • calculate distance with dist()
  • compute hierarchical clustering with hclust()
  • plot the dendrogram: ggdendrogram() (or base R plot())
  • cut the dendrogram with cutree()

Hierarchical clustering with the penguins data

Follow along the class example:

usethis::create_from_github(""SDS322E-2025Fall/0903-hclust")

as_tibble(penguins)

# A tibble: 344 × 8
   species island    bill_len bill_dep flipper_len body_mass sex     year
   <fct>   <fct>        <dbl>    <dbl>       <int>     <int> <fct>  <int>
 1 Adelie  Torgersen     39.1     18.7         181      3750 male    2007
 2 Adelie  Torgersen     39.5     17.4         186      3800 female  2007
 3 Adelie  Torgersen     40.3     18           195      3250 female  2007
 4 Adelie  Torgersen     NA       NA            NA        NA <NA>    2007
 5 Adelie  Torgersen     36.7     19.3         193      3450 female  2007
 6 Adelie  Torgersen     39.3     20.6         190      3650 male    2007
 7 Adelie  Torgersen     38.9     17.8         181      3625 female  2007
 8 Adelie  Torgersen     39.2     19.6         195      4675 male    2007
 9 Adelie  Torgersen     34.1     18.1         193      3475 <NA>    2007
10 Adelie  Torgersen     42       20.2         190      4250 <NA>    2007
# ℹ 334 more rows

Similar question as in kmeans: do NA values matter?

Yes - without removing NA values, we can’t calculate distance

penguins_clean <- as_tibble(penguins) |> na.omit()

Hierarchical clustering with the penguins data

To make the illustration simpler, we will take 10 observations for each species:

penguins_small <- penguins_clean |> 
  group_by(species) |>
  slice_head(n = 10) |>
  ungroup()
penguins_small

# A tibble: 30 × 8
   species island    bill_len bill_dep flipper_len body_mass sex     year
   <fct>   <fct>        <dbl>    <dbl>       <int>     <int> <fct>  <int>
 1 Adelie  Torgersen     39.1     18.7         181      3750 male    2007
 2 Adelie  Torgersen     39.5     17.4         186      3800 female  2007
 3 Adelie  Torgersen     40.3     18           195      3250 female  2007
 4 Adelie  Torgersen     36.7     19.3         193      3450 female  2007
 5 Adelie  Torgersen     39.3     20.6         190      3650 male    2007
 6 Adelie  Torgersen     38.9     17.8         181      3625 female  2007
 7 Adelie  Torgersen     39.2     19.6         195      4675 male    2007
 8 Adelie  Torgersen     41.1     17.6         182      3200 female  2007
 9 Adelie  Torgersen     38.6     21.2         191      3800 male    2007
10 Adelie  Torgersen     34.6     21.1         198      4400 male    2007
# ℹ 20 more rows

Hierarchical clustering

Again, does scaling matter?

# scale the variables
hclust_df <- penguins_small[,3:6] |> scale() 

# compute the distance
hclust_dist <- hclust_df |> dist()

# compute the hierarchical clustering
hclust_results1 <- hclust(hclust_dist, method = "single")
hclust_results1

Call:
hclust(d = hclust_dist, method = "single")

Cluster method   : single 
Distance         : euclidean 
Number of objects: 30 

Visualize hierarchical clustering results

library(ggdendro)
ggdendrogram(hclust_results1)

Visualize hierarchical clustering results

hclust_data1 <- ggdendro::dendro_data(hclust_results1)
hclust_segments1 <- as_tibble(hclust_data1$segments)
hclust_labels1 <- as_tibble(hclust_data1$labels) |> 
  mutate(species = penguins_small$species[as.integer(label)])

p1 <- ggplot() +
  geom_segment(data = hclust_segments1,
               aes(x = x, y = y, xend = xend, yend = yend)) +
  geom_text(data = hclust_labels1,
            aes(x = x, y = y - 0.02, label = label,
                color = species), vjust = 1, size = 3) +
  labs(title = "Single Linkage") +
  theme_minimal() +
  theme(axis.title = element_blank(),
        axis.text = element_blank(),
        axis.ticks = element_blank(),
        panel.grid = element_blank(),
        legend.position = "bottom") + 
  ylim(-0.1, 2.5)
p1

Different linkage methods

• Single linkage: distance between two clusters is defined as the minimum distance between any single member of one cluster and any single member of the other cluster.
• Complete linkage: distance between two clusters is defined as the maximum distance between any single member of one cluster and any single member of the other cluster.
• Average linkage: distance between two clusters is defined as the average distance between all pairs of members from the two clusters.
• Ward’s linkage: distance between two clusters is defined based on the increase in the total within-cluster variance that would result from merging the two clusters.

Selection:

• Single linkage are most popular among statisticians.
• Average and complete linkage are generally preferred over single linkage, as they tend to yield more balanced dendrograms

Different linkage for penguins data

Full penguins data

hclust_df <- penguins_clean[,3:6] |> scale() 

# compute the distance
hclust_dist <- hclust_df |> dist()

# compute the hierarchical clustering
hclust_results <- hclust(hclust_dist, method = "average")
hclust_results

Call:
hclust(d = hclust_dist, method = "average")

Cluster method   : average 
Distance         : euclidean 
Number of objects: 333 

Full penguins data

hclust_data <- ggdendro::dendro_data(hclust_results)
hclust_segments <- as_tibble(hclust_data$segments)
hclust_labels <- as_tibble(hclust_data$labels) |> 
  mutate(species = penguins_clean$species[as.integer(label)])

ggplot() +
  geom_segment(data = hclust_segments,
               aes(x = x, y = y, xend = xend, yend = yend)) +
  geom_text(data = hclust_labels,
            aes(x = x, y = y - 0.02, label = label,
                color = species), vjust = 1, size = 3) +
  labs(title = "Average Linkage") +
  theme_minimal() +
  theme(axis.title = element_blank(),
        axis.text = element_blank(),
        axis.ticks = element_blank(),
        panel.grid = element_blank(),
        legend.position = "none") +
  ylim(-0.1, 4) 

Cut the tree into two clusters

The base R function cutree() can be used to cut the dendrogram into a specified number of clusters.

# cut the dendrogram to get 2 clusters
hclust_clusters <- cutree(hclust_results, k = 2)

dt <- penguins_clean |> 
  mutate(cluster = as.factor(hclust_clusters)) 

dt |> count(cluster, species)

# A tibble: 3 × 3
  cluster species       n
  <fct>   <fct>     <int>
1 1       Adelie      146
2 1       Chinstrap    68
3 2       Gentoo      119

dt |>   
  ggplot(aes(x = bill_dep, y = flipper_len, 
             color = cluster)) + 
  geom_point()

Compare hierarchical clustering with kmeans

Hierarchical clustering: cut into 3 groups

# A tibble: 5 × 3
  cluster species       n
  <fct>   <fct>     <int>
1 1       Adelie      144
2 1       Chinstrap     5
3 2       Adelie        2
4 2       Chinstrap    63
5 3       Gentoo      119

Kmeans clustering: choose 3 centers

# A tibble: 5 × 3
  cluster species       n
  <fct>   <fct>     <int>
1 1       Adelie      124
2 1       Chinstrap     5
3 2       Adelie       22
4 2       Chinstrap    63
5 3       Gentoo      119