Cluster Analysis Exercises in R: 17 k-Means, Hierarchical & PAM Problems

Seventeen cluster analysis exercises in R, ordered from a first kmeans() fit through scaling, elbow and silhouette diagnostics, hierarchical clustering with multiple linkages, cophenetic correlation, k-medoids, and partition agreement via the adjusted Rand index. Each problem ships with a runnable starter, an exact expected output, and a click-to-reveal solution explaining the choice.

Section 1. k-means warm-up (3 problems)

Exercise 1.1: Fit a three-cluster k-means on scaled iris

Task: Fit a k-means model with three centers on the scaled iris[, 1:4] numeric matrix using set.seed(101) and nstart = 25, then save the fitted object to ex_1_1 so later exercises can re-use its sizes, centers, and within-SS slots.

Expected result:

#> K-means clustering with 3 clusters of sizes 50, 53, 47
#>
#> Cluster sizes:
#> [1] 50 53 47

Difficulty: Beginner

Exercise 1.2: Read the centers, within-SS, and total within-SS

Task: Using the ex_1_1 object from the previous exercise, extract the cluster centroid matrix, the per-cluster within-cluster sum of squares vector, and the scalar total within-SS. Bundle the three into a named list saved as ex_1_2 so a reviewer can audit the loss decomposition in one print call.

Expected result:

#> $centers (rounded)
#>   Sepal.Length Sepal.Width Petal.Length Petal.Width
#> 1       -1.01        0.85        -1.30       -1.25
#> 2       -0.05       -0.88         0.35        0.28
#> 3        1.16        0.13         1.00        1.03
#>
#> $withinss
#> [1] 47.4 44.1 47.5
#>
#> $tot.withinss
#> [1] 138.89

Difficulty: Beginner

Exercise 1.3: Cross-tabulate cluster labels against true species

Task: A botanist hands you the unlabeled iris[, 1:4] table and asks how well unsupervised k-means recovers the three species. Cross-tabulate the cluster labels in ex_1_1$cluster against the iris$Species factor and save the contingency matrix as ex_1_3 so the diagonal misalignment is visible at a glance.

Expected result:

#>             Species
#> Cluster      setosa versicolor virginica
#>   1              50          0         0
#>   2               0         39        14
#>   3               0         11        36

Difficulty: Intermediate

Section 2. Scaling and preprocessing (3 problems)

Exercise 2.1: Show that unscaled k-means is dominated by the largest-variance column

Task: Compute and compare two k-means fits with three centers on USArrests: one on the raw matrix and one on the column-scaled matrix. For each, return the total within-cluster sum of squares and the cross-tab of the two label vectors against each other, saved together as a list named ex_2_1.

Expected result:

#> $unscaled_tot_withinss
#> [1] 19564
#>
#> $scaled_tot_withinss
#> [1] 60.0
#>
#> $agreement_table
#>          scaled
#> unscaled   1  2  3
#>        1   0 13  3
#>        2   0  0 17
#>        3  14  3  0

Difficulty: Intermediate

Exercise 2.2: Cluster a numeric subset of a mixed-type frame

Task: A marketing analyst has the mtcars frame and wants to cluster cars on continuous performance traits only. Build a clean numeric matrix by keeping just mpg, disp, hp, drat, wt, qsec, scale it, fit a four-cluster k-means with nstart = 50, and save the cluster vector named with car names as ex_2_2.

Expected result:

#>           Mazda RX4       Mazda RX4 Wag          Datsun 710      Hornet 4 Drive
#>                   2                   2                   3                   3
#>   Hornet Sportabout             Valiant          Duster 360           Merc 240D
#>                   1                   3                   1                   3
#> ...
#> # 24 more names hidden

Difficulty: Intermediate

Exercise 2.3: Impute then scale before clustering

Task: An ops engineer hands you a copy of airquality that still has missing Ozone and Solar.R cells. Replace each missing value with the column median, scale the resulting numeric matrix, fit a three-cluster k-means, and save a tibble-free data frame combining the original Month column with the new cluster label as ex_2_3.

Expected result:

#> # head(ex_2_3, 5)
#>   Month cluster
#> 1     5       1
#> 2     5       2
#> 3     5       2
#> 4     5       2
#> 5     5       2

Difficulty: Intermediate

Section 3. Choosing k (3 problems)

Exercise 3.1: Elbow plot via total within-SS for k = 1..10

Task: Run k-means with nstart = 20 on scaled USArrests for every k from 1 through 10, collect each fit's tot.withinss, and save the named numeric vector indexed by k as ex_3_1 so the elbow can be inspected by inverse-difference rather than by eyeballing a plot.

Expected result:

#>      1      2      3      4      5      6      7      8      9     10
#> 196.00 102.86  78.32  71.44  62.32  55.26  49.61  44.74  41.10  37.91

Difficulty: Intermediate

Exercise 3.2: Average silhouette width for k = 2..8

Task: For scaled USArrests, compute the average silhouette width for every k from 2 through 8 using cluster::silhouette() against the Euclidean distance matrix, then save the named numeric vector indexed by k as ex_3_2 so the maximising k jumps out.

Expected result:

#>     2     3     4     5     6     7     8
#> 0.408 0.310 0.341 0.247 0.205 0.211 0.156

Difficulty: Intermediate

Exercise 3.3: Gap statistic with clusGap

Task: Use cluster::clusGap() with 50 bootstrap references and kmeans as the FUNcluster on scaled USArrests for k from 1 through 8, then save the Tibshirani gap value and s.e.sim columns indexed by k as the data frame ex_3_3. The optimal k is the smallest k where Gap(k) >= Gap(k+1) - s.e.sim(k+1).

Expected result:

#>   k   gap   SE
#> 1 1 0.299 0.029
#> 2 2 0.477 0.038
#> 3 3 0.541 0.038
#> 4 4 0.581 0.036
#> 5 5 0.601 0.034
#> 6 6 0.625 0.034
#> 7 7 0.658 0.033
#> 8 8 0.683 0.033

Difficulty: Advanced

Section 4. Stability and reproducibility (2 problems)

Exercise 4.1: Effect of nstart on the within-SS minimum

Task: A reviewer questions whether your reported k-means loss is reproducible. Refit a 5-cluster k-means on scaled USArrests four times with the same seed but nstart set to 1, 5, 25, and 100, collect the tot.withinss from each, and save the named numeric vector as ex_4_1 showing the loss plateau as restarts increase.

Expected result:

#>      1      5     25    100
#> 79.61  62.50  62.32  62.32

Difficulty: Intermediate

Exercise 4.2: Hand-rolled cluster stability via bootstrap resamples

Task: Without using fpc::clusterboot, write a routine that draws 50 bootstrap resamples of scaled USArrests, refits 3-cluster k-means on each resample, projects the labels back to the full 50 states via the closest original centroid, and saves the 50-by-50 co-membership frequency matrix (proportion of bootstraps where i and j share a cluster) as ex_4_2. Diagonal entries are 1.

Expected result:

#> dim(ex_4_2): 50 50
#> ex_4_2[1:4, 1:4]
#>             Alabama Alaska Arizona Arkansas
#> Alabama        1.00   0.62    0.66     0.74
#> Alaska         0.62   1.00    0.74     0.50
#> Arizona        0.66   0.74    1.00     0.48
#> Arkansas       0.74   0.50    0.48     1.00

Difficulty: Advanced

Section 5. Hierarchical clustering (3 problems)

Exercise 5.1: Build an hclust on USArrests and cut to three groups

Task: A criminologist wants states grouped by violent-crime profile. Compute Euclidean distances on scaled USArrests, build a hierarchical clustering with method = "ward.D2", cut the resulting dendrogram to three groups with cutree, and save the named integer cluster vector as ex_5_1 ordered by the state names as they appear in the data.

Expected result:

#>     Alabama      Alaska     Arizona    Arkansas  California    Colorado
#>           1           1           1           2           1           2
#> Connecticut    Delaware     Florida     Georgia      Hawaii       Idaho
#>           3           1           1           1           3           3
#> ...
#> # 38 more states hidden

Difficulty: Intermediate

Exercise 5.2: Compare four linkage methods on the same distance matrix

Task: Using the Euclidean distance matrix on scaled USArrests, build four hclust fits with single, complete, average, and ward.D2 linkage. For each, cut to three clusters and save the four label vectors side-by-side as a 50-row data frame ex_5_2 so the chaining behaviour of single-linkage shows up against the more balanced partitions.

Expected result:

#> head(ex_5_2, 6)
#>             single complete average ward.D2
#> Alabama          1        1       1       1
#> Alaska           1        1       1       1
#> Arizona          1        1       1       1
#> Arkansas         1        1       1       2
#> California       1        1       1       1
#> Colorado         1        1       1       2
#>
#> table per method (cluster sizes):
#> single:  c(48, 1, 1)
#> complete: c(8, 11, 31)
#> average:  c(2, 1, 47)
#> ward.D2: c(16, 14, 20)

Difficulty: Intermediate

Exercise 5.3: Cophenetic correlation to pick a linkage

Task: For the four linkages from the previous exercise, compute the cophenetic correlation between the input distance matrix and the cophenetic distances induced by each dendrogram, then save the named numeric vector as ex_5_3 so the linkage that best preserves the original pairwise distances is identifiable.

Expected result:

#>   single complete  average  ward.D2
#>    0.539    0.698    0.718    0.692

Difficulty: Advanced

Section 6. Comparing and visualising partitions (3 problems)

Exercise 6.1: Adjusted Rand index between k-means and ward labels

Task: Compare the 3-cluster k-means labels from ex_1_1$cluster against the 3-cluster ward.D2 hierarchical labels from ex_5_1 using mclust::adjustedRandIndex. Wrap the raw ARI, the raw Rand index, and the confusion table into a list named ex_6_1 so the agreement between the two algorithms is fully auditable.

Expected result:

#> $adjusted_rand_index
#> [1] 0.62
#>
#> $confusion
#>          ward
#> kmeans     1  2  3
#>      1    50  0  0
#>      2     0 39 14
#>      3     0  0 47

Difficulty: Intermediate

Exercise 6.2: Project clusters onto the first two principal components

Task: Use factoextra::fviz_cluster to visualise the ex_1_1 k-means partition on the scaled iris[, 1:4] data using the first two principal components as axes, with convex hulls and the observation labels suppressed. Save the resulting ggplot object as ex_6_2 so a downstream ggsave call can render it.

Expected result:

#> # Plot description
#> # Scatter on PC1 vs PC2, 150 points coloured by cluster (1, 2, 3),
#> # three convex hulls, no point labels, default factoextra theme.
#> # ggplot object of class c("gg", "ggplot")
class(ex_6_2)
#> [1] "gg"     "ggplot"

Difficulty: Intermediate

Exercise 6.3: PAM on USArrests and agreement with k-means

Task: A statistician argues that k-means is too sensitive to outliers and prefers k-medoids. Fit cluster::pam with three medoids on scaled USArrests, also fit k-means with three centers and nstart = 25, then save a list containing the medoid state names, the PAM cluster sizes, and the adjusted Rand index between PAM and k-means labels as ex_6_3.

Expected result:

#> $medoid_states
#> [1] "New Mexico" "Nebraska"   "New Jersey"
#>
#> $pam_sizes
#> [1] 20 14 16
#>
#> $ari_pam_vs_kmeans
#> [1] 0.92

Difficulty: Advanced

What to do next

Now that you have a working clustering toolkit, deepen the foundation and apply it to broader workflows:

• Revisit the Cluster Analysis in R parent tutorial to see every diagnostic in context with end-to-end narrative.
• Try the Machine Learning Exercises in R hub to combine clustering with supervised pipelines and resampling.
• Practise the EDA Exercises in R hub, since most clustering work begins with a careful univariate and bivariate exploration.
• Sharpen the matrix and scaling fundamentals via the dplyr Exercises in R hub, which underpins the preprocessing every cluster algorithm depends on.