Point Pattern Basics

Serge Rey

Point Pattern Basics


  • Objectives & Terminology
  • Examples

Objectives of Point Pattern Analysis

Point Pattern Analysis Objectives

Goals

  • Pattern detection
  • Assessing the presence of clustering
  • Identification of individual clusters

General Approaches

  • Estimate intensity of the process
  • Formulating an idealized model and investigating deviations from expectations
  • Formulating a stochastic model and fitting it to the data

Point Pattern Analysis Definitions

Spatial Point Pattern: A set of events, irregularly distributed within a region \(A\) and presumed to have been generated by some form of stochastic mechanism.

Spatial Point Process \(\left\{Y(A),A \subset \mathbb{R} \right\}\)

  • \(Y(A)\): For each subset \(A\) of the spatial region \(\mathbb{R}\), (\(\mathbb{R}^2\), representing a two-dimensional space), the random variable \(Y(A)\) represents the number of events occurring within area \(A\).

  • Events in Area: An “event” refers to any occurrence of interest (e.g., sightings of a specific species, crimes, accidents) within the spatial region. Thus, \(Y(A)\) counts the occurrences in the specified region \(A\).

  • Collection of Random Variables: For each possible area \(A\), there is a corresponding count \(Y(A)\). Together, these counts form a random field or point process over the space \(\mathbb{R}\).

Events, points, locations

  • Event: an occurrence of interest
  • Point: any location in study area
  • Event location: a particular point where an event occurs

Point Pattern Analysis Definitions

Region: \(A\)

  • Most often planar (two-dimensional Euclidean space)
  • One dimensional applications also possible
  • Three-dimensional increasingly popular (space + time)
  • Point processes on networks (non-planar)

Space-Time Point Patterns

Space-Time Point Patterns

Point Patterns on Networks

Point Patterns

Unmarked Point Patterns

  • Only location is recorded

  • Attribute is binary (presence, absence)

Marked Point Patterns

  • Location is recorded

  • Non-binary stochastic attribute

  • e.g., sales at a retail store, dbh of tree

Realizations

Mapped Point Patterns

  • All events are recorded and mapped

  • Complete enumeration of events

  • Full information on the realization from the process

Sampled Point Patterns

  • Sample of events are recorded and mapped

  • Complete enumeration of events impossible or intractable

  • Partial information on the realization from the process

  • Presence/“absence” data (ecology, forestry)

Research Questions

Research Questions

  • Location Only are points randomly located or patterned

  • Location and Value

    • marked point pattern
    • is combination of location and value random or patterned

Both Cases: What is the Underlying Process?

Points on a Plane (Planar Point Pattern Anaysis)

Classic Point Pattern Analysis

  • points on an isotropic plane
  • no effect of translation and rotation
  • classic examples: tree seedlings, rocks, etc

Distance

  • no directional effects
  • no translational effects
  • straight line distance only

Events: Point Map

Points in Context

Intensity

First Moment

  • number of points \(N\), area of study \(|A|\)
  • intensity: \(\lambda = N/|A|\)
  • area depends on bounds, often arbitrary

Artificial Boundaries

  • bounding box (rectangle, square)
  • other (city boundary)

Bounding Box

District Boundary

Convex Hull

  • Tightest fit various algorithms

  • Rescaled Convex Hull (Ripley-Rasson)

    • adjust to properly reflect spatial domain of point process
    • use centroid of convex hull
    • rescale by \(1/[\sqrt{(1-m/N)}]\)
    • \(m\): number of vertices of convex hull

Convex Hull

Multiple Boundaries

Intensity Estimates

Area Intensity
\(km^2\) \(cases/km^2\)
District Boundary 315.155 3.29
Bounding Box 310.951 3.33
Convex Hull 229.421 4.52


N=1036

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Geoprocessing Points