Spatial Clustering

Last updated on 2026-10-25 | Edit this page

Overview

Questions

  • What is spatial clustering and when is it useful?
  • Why must coordinates be in meters before clustering?
  • How do K-Means, hierarchical clustering, DBSCAN and HDBSCAN differ?
  • When is a cluster of points really a hotspot?

Objectives

  • Prepare point data for clustering by reprojecting to a meter based CRS
  • Find dense hotspots with DBSCAN and HDBSCAN and recognize noise points
  • Divide points into zones with K-Means and choose k with the elbow method
  • Choose a clustering method that fits the research question
  • Explain how each method works, what it assumes, and how to choose its parameters
  • Recognize the population at risk, data collection effects and other pitfalls in hotspot maps

Why spatial clustering?

Clustering finds groups of points that sit close together: crime hotspots, disease outbreaks, clusters of archaeological finds, or concentrations of businesses. It turns thousands of points into a few areas you can describe and compare.

Method You choose Noise points? Best for
K-Means Number of clusters (k) No, every point is assigned Dividing an area into zones
Hierarchical Number of clusters or a cut height No Seeing structure at several scales (small datasets)
DBSCAN Search radius (eps) and min_samples Yes Hotspots of any shape at a known distance
HDBSCAN min_cluster_size Yes Hotspots with varying density

The notebooks use about 200,000 crime reports from the Chicago Data Portal and focus on robbery.

Three panels of the same synthetic points: three dense groups of different sizes and shapes, one of them long and curved, plus scattered points across the whole area. Left, K-Means with k equal to 3: every point is colored, the area is split into three regions, and the scattered points are assigned to the nearest group. Middle, DBSCAN: the three dense groups are colored, the curved group is kept whole, and the scattered points are gray noise. Right, a close-up explaining DBSCAN with a circle of radius eps around points: core points with at least five points inside a circle of radius eps, a border point inside a core point's circle but with few neighbors of its own, and an isolated noise point.
K-Means and DBSCAN on the same points.

Points, patterns and the population at risk

Clustering looks for places where events (robberies, disease cases, archaeological finds) are concentrated. It is related to, but different from, the previous episode: spatial autocorrelation asks whether values attached to areas (such as median age by county) are similar among neighbors.

Before calling a cluster a “hotspot”, ask what you are comparing it with. Events happen where people are: a busy downtown has more robberies than a quiet suburb partly because far more people pass through it. A dense cluster of points may reflect the population at risk rather than an unusually high risk. Common ways to handle this are to compare with a background (rates per resident or per visitor, or a control set of points such as all reported crimes), or to aggregate the points to areas and analyze rates with the methods from the PySAL episode.

How the data were collected shapes them too. Crime reports reflect reporting and policing as well as crime itself, and the Chicago portal shows locations at the block level only to protect victims’ privacy, so many reports share exactly the same coordinates.

1. Load points and reproject to meters

PYTHON 

import pandas as pd
import geopandas as gpd
import numpy as np

# read the CSV straight from GitHub into memory (nothing to download)
crimes = pd.read_csv("https://raw.githubusercontent.com/SpatialTurn/DataAnalysis/main/data/CrimesChicago20220225.csv")
crimes.columns = crimes.columns.str.strip()
robbery = crimes[crimes["PRIMARY DESCRIPTION"] == "ROBBERY"].dropna(subset=["LATITUDE", "LONGITUDE"])

robbery = gpd.GeoDataFrame(
    robbery,
    geometry=gpd.points_from_xy(robbery["LONGITUDE"], robbery["LATITUDE"]),
    crs="EPSG:4326",
).to_crs(epsg=32616)          # UTM zone 16N: x and y in meters

coords = np.column_stack([robbery.geometry.x, robbery.geometry.y])

Latitude and longitude are angles. One degree of longitude in Chicago is about 85 km while one degree of latitude is about 111 km, so distances in degrees are distorted. Always reproject first.

2. DBSCAN: density based hotspots

PYTHON 

from sklearn.cluster import DBSCAN

db = DBSCAN(eps=1000, min_samples=50).fit(coords)   # eps in meters
robbery["dbscan"] = db.labels_                       # -1 means noise

A point with at least min_samples neighbors within eps meters is a core point; nearby core points form a cluster. Choose eps with a k-distance plot (see notebook 3b).

3. HDBSCAN: no radius needed

PYTHON 

from sklearn.cluster import HDBSCAN

robbery["hdbscan"] = HDBSCAN(min_cluster_size=50, min_samples=10).fit_predict(coords)

4. K-Means: zones

PYTHON 

from sklearn.cluster import KMeans

robbery["kmeans"] = KMeans(n_clusters=6, n_init=10, random_state=42).fit_predict(coords)

5. Hierarchical clustering and the dendrogram

PYTHON 

from scipy.cluster.hierarchy import linkage, dendrogram, fcluster

sample = robbery.sample(2000, random_state=1)        # hierarchical is slow on large data
tree = linkage(np.column_stack([sample.geometry.x, sample.geometry.y]), method="ward")
dendrogram(tree, truncate_mode="lastp", p=20, no_labels=True)
sample["hier"] = fcluster(tree, t=6, criterion="maxclust")

Plot any result with robbery.plot(column="dbscan", categorical=True, markersize=1).

How the methods work

DBSCAN

DBSCAN (Density-Based Spatial Clustering of Applications with Noise) sorts every point into one of three kinds:

  • a core point has at least min_samples points (counting itself) within eps;
  • a border point lies within eps of a core point but does not have enough neighbors of its own;
  • a noise point is neither, and gets the label -1.

Core points that are within eps of each other join the same cluster, so clusters can take any shape, following a river or a commercial street.

Choosing eps. Make a k-distance plot: for every point, find the distance to its k-th nearest neighbor (with k equal to min_samples), sort those distances and plot them. Most points sit on a flat part of the curve; where the curve bends sharply upward (the “knee”), points start to be isolated. The distance at the knee is a good first choice for eps.

PYTHON 

import matplotlib.pyplot as plt
from sklearn.neighbors import NearestNeighbors

k = 50                                                    # same as min_samples
dist, _ = NearestNeighbors(n_neighbors=k).fit(coords).kneighbors(coords)
plt.plot(np.sort(dist[:, -1]))
plt.ylabel(f"distance to neighbor {k} (m)")
plt.xlabel("points, sorted")
plt.show()

Choosing min_samples. Larger values ask for denser clusters and label more points as noise. With thousands of points, values in the tens are common; with a few hundred, try 5 to 10.

DBSCAN’s weakness is that one eps applies to the whole map, so it cannot find both a tight downtown cluster and a looser suburban one at the same time.

HDBSCAN

HDBSCAN (Hierarchical DBSCAN) avoids choosing eps. In effect it tries every value of eps at once, builds a tree of how clusters split apart as the distance shrinks, and keeps the clusters that last over a wide range of distances (the most stable ones). You set min_cluster_size, the smallest group worth calling a cluster. It finds clusters of different densities, labels noise as -1, and reports how strongly each point belongs to its cluster (probabilities_).

K-Means

K-Means places k centers, assigns every point to its nearest center, moves each center to the mean of its points, and repeats until nothing changes. It assumes clusters that are roughly round and of similar size, and it assigns every point, so outliers pull the centers toward them. The result depends on where the centers start; n_init=10 runs ten starts and keeps the best one.

To choose k, plot the inertia (the total squared distance from points to their centers) for k = 2, 3, 4 and so on, and look for the elbow where adding clusters stops helping much. The silhouette score (sklearn.metrics.silhouette_score, from -1 to 1) measures how well each point fits its own cluster compared with the next nearest one; higher is better. On large datasets, compute it on a sample with sample_size=2000.

Hierarchical clustering

Agglomerative hierarchical clustering starts with every point as its own cluster and repeatedly merges the two closest clusters. The linkage rule defines “closest”: single linkage (the nearest pair of points, which tends to make long chains), complete linkage (the farthest pair), average linkage, or Ward (the merge that adds the least variance, which makes compact clusters). In the dendrogram, the height of each join is the distance at which two groups merge; cutting the tree at a height gives the clusters. Memory use grows with the square of the number of points, which is why the example uses a sample.

Challenge

Challenge 1: K-Means or DBSCAN?

Map the K-Means and DBSCAN results side by side. Which one would you use to report “robbery hotspots” to a city council, and why?

DBSCAN (or HDBSCAN). K-Means assigns every robbery to a zone, even isolated ones, so its “clusters” cover the whole city. DBSCAN keeps only dense areas and labels scattered points as noise, which is what a hotspot means.

Challenge

Challenge 2: Adjusting DBSCAN

Run DBSCAN with eps=500 and eps=1500 (meters). How do the number of clusters and the share of noise points change?

PYTHON 

for eps in [500, 1500]:
    labels = DBSCAN(eps=eps, min_samples=50).fit(coords).labels_
    print(eps, labels.max() + 1, "clusters", f"{(labels == -1).mean():.0%} noise")

A smaller eps gives more, smaller clusters and more noise. A larger eps merges clusters into a few large ones.

Pitfalls

  • Degrees instead of meters. Always reproject first (Section 1).
  • Duplicate locations. Block-level coordinates put many reports on the same spot. Every duplicate counts as a neighbor, which can make an area look denser than it is.
  • Parameters shape the result. Report eps, min_samples, min_cluster_size or k, and check how the map changes with other values (Challenge 2).
  • No significance test. Unlike LISA, these methods do not test whether a cluster could arise by chance. Comparing with random points, or using a scan statistic, is needed for that.
  • Edges. Points near the edge of the study area, such as the city limits or the Lake Michigan shore, have fewer neighbors.
  • Consequences. Hotspot maps can stigmatize neighborhoods and influence where police patrol, which in turn produces more reports in the same places.
Challenge

Challenge 3: Reading a k-distance plot

Your k-distance plot (k = 50) stays fairly flat up to about 800 m and then rises steeply. Which eps would you try first? What would happen with eps=200 or eps=3000?

Start with an eps of about 800 m, at the knee of the curve. With eps=200, few points have 50 neighbors that close, so most points become noise and only the very densest spots remain as small clusters. With eps=3000, almost every point is a core point, so clusters merge into one or a few very large ones that cover much of the city and say little about hotspots.

Discussion

Hotspot or crowd?

DBSCAN finds a large robbery cluster in the Loop, Chicago’s downtown. Is the Loop a dangerous place, or just a crowded one? What extra data would help you decide?

Math

DBSCAN looks at the neighborhood of each point \(p\): all points within distance \(\varepsilon\) (eps). If that neighborhood holds at least min_samples points, \(p\) is a core point. Core points within \(\varepsilon\) of each other join the same cluster, along with the border points near them. Every other point is noise.

K-Means instead chooses \(k\) centers \(\mu_1, \dots, \mu_k\) that minimize the total squared distance \(\sum_i \lVert x_i - \mu_{c(i)} \rVert^2\), where \(c(i)\) is the center closest to point \(x_i\). This total is the inertia used in the elbow plot.

Key Points
  • Reproject to meters before clustering so distances are real
  • DBSCAN and HDBSCAN find dense hotspots of any shape and label scattered points as noise
  • K-Means splits all points into k round zones; choose k with the elbow method
  • Hierarchical clustering shows how groups merge, but is slow on large datasets
  • Pick the method that matches your question: hotspots or zones
  • Choose eps with a k-distance plot and k with the elbow method or silhouette score
  • A dense cluster may reflect the population at risk, duplicate locations or reporting, not high risk

Notebooks

Level Notebook What you will do
Beginner 3a. Exploring Crime Points Summarize Chicago crimes and turn them into points in meters
Intermediate 3b. DBSCAN and HDBSCAN Choose eps, find robbery hotspots and compare with HDBSCAN
Advanced 3c. K-Means and Comparing Methods Use the elbow method, a dendrogram and a side by side comparison