DBSCAN, agglomerative clustering and cluster validation

1. `DBSCAN` contrast and agglomerative example

   Hands-On Machine Learning with Scikit-Learn and PyTorch - dimensionality reduction and unsupervised learning

   Book Chapter 8: DBSCAN and Other Clustering Algorithms (printed pp. 265-269). Notebook 08 cells: DBSCAN: make_moons(1000, noise=0.05) at eps=0.05 (7 clusters, many points labeled -1) versus eps=0.2 (two clusters, no noise), core_sample_indices_, components_, and KNN on the cores with a 0.2 distance cutoff to label new points; Other Clustering Algorithms: AgglomerativeClustering with complete linkage on \(\left[\begin{smallmatrix}0&2&5&8.5\end{smallmatrix}\right]\)

   Depth · Skim the Spectral Clustering bonus code; keep the eps contrast and the linkage example.

   Done
2. Density, linkage and metric limits

   scikit-learn - decomposition, clustering and outlier detection

   2.3.7 DBSCAN: dense definition from eps and min_samples, core, border and outlier rules, the eps-must-be-tuned warning with all-noise and merged-cluster failures; 2.3.6 hierarchical clustering: Ward, complete, average and single linkage with the uneven-size guidance; 2.3.11 metrics: silhouette formula with its convex-cluster bias and the adjusted Rand index correcting for chance

   Done

Focus on

• eps sets the neighborhood radius and min_samples the density floor. On the notebook moons, eps=0.05 breaks the two moons into 7 clusters and labels 7.7 percent as noise, while 0.2 finds the two moons with no noise; always sweep eps.
• Linkage defines merge distance: single follows chains, complete compacts, Ward prefers equal-variance merges. On \(\left[\begin{smallmatrix}0&2&5&8.5\end{smallmatrix}\right]\) single splits off 8.5 while complete pairs neighbors.
• DBSCAN labels new points only by extra machinery: the notebook trains KNN on cores. The clusterer itself has no predict.
• Silhouette rewards dense convex separation and can punish correct density shapes. Pair it with stability and downstream checks.

Reproduce closed-book

Traps

• Leaving eps at 0.5 on unscaled features. eps lives in feature units; scale first or the radius is meaningless.
• Calling -1 a detected anomaly set. It means not dense enough under this eps; nearby eps values relabel it.
• Reporting silhouette alone on moons. On the notebook moons a k-means split scores 0.49 and the true moon labels 0.33; the docs warn that silhouette favors convex clusters.

Practice

Self-test

1. On the notebook moons, eps=0.05 gives 7 clusters and 7.7 percent noise, while eps=0.2 gives two clean moons. What went wrong at 0.05, and what does the -1 label mean?

   Show answer Got it Missed

2. Single linkage gives \(\left[\begin{smallmatrix}0&0&0&1\end{smallmatrix}\right]\) and complete gives \(\left[\begin{smallmatrix}1&1&0&0\end{smallmatrix}\right]\) on \(\left[\begin{smallmatrix}0&2&5&8.5\end{smallmatrix}\right]\). Which is right?

   Show answer Got it Missed

3. On the notebook moons, KMeans(2) scores silhouette 0.49 and the true moon labels score 0.33. Which clustering is better?

   Show answer Got it Missed

4. Two DBSCAN runs with close eps values disagree on 5 percent of labels. What do you report?

   Show answer Got it Missed

Leads to

• 4.7 Outlier and novelty detection, thresholds and representation Treat DBSCAN noise labels as candidates, not verdicts, with a scored threshold.
• 4.8 t-SNE and UMAP as guarded exploratory views Overlay cluster labels on an honest embedding to inspect shape.

Comes after

• 4.4 k-means: objective, initialization, empty clusters and instabilityContrast partition geometry with density and merge geometry.
• 4.5 Gaussian mixtures and EMCompare soft parametric groups with density and merge structure.
• 4.8 t-SNE and UMAP as guarded exploratory viewsCheck density labels against the same guarded view.

Resources for this unit

• Hands-On Machine Learning with Scikit-Learn and PyTorch - dimensionality reduction and unsupervised learning
  Aurelien Geron (O'Reilly, 2025)primarycourse
  Read Chapters 7 and 8 and run notebooks 07 and 08 for PCA, k-means, DBSCAN, agglomerative clustering, Gaussian mixtures and anomaly detection. Stop before extended semi-supervised and face-data projects.
• scikit-learn - decomposition, clustering and outlier detection
  scikit-learnreferencereference
  Check decomposition orientation and solver behavior, cluster API semantics, and the distinction between novelty and outlier detection after the independent tests pass.