Home Statistical Dictionary Marked Point Process

Marked Point Process

A point process in which every point carries an additional random attribute, or 'mark,' beyond its plain location, such as an earthquake's magnitude or a tree's trunk diameter attached to its position.

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In Plain English

A plain point process tells you where or when things happened, a marked point process adds how big, how severe, or what kind. Earthquake epicenters marked with magnitude, disease cases marked with severity, trees marked with species and diameter, every marked point process starts from an ordinary point pattern and attaches one more piece of information to each point. This turns questions from 'where do points cluster?' into richer ones like 'are the larger-magnitude events also the more spatially clustered ones?'

Definition

A marked point process is a point process ${x_{i}}$ on a space $E$ together with an associated sequence of marks ${m_{i}}$ , one attached to each point, where $m_{i}$ takes values in some mark space $M$ (which can be continuous, e.g. a magnitude, categorical, e.g. a species label, or itself multivariate). Formally, it can be represented as an ordinary point process on the product space $E \times M$ . Key summary tools extend naturally: the mark distribution describes the marginal behavior of marks, while mark correlation functions test whether marks near each other in $E$ tend to be similar (positive mark correlation) or dissimilar (negative mark correlation), beyond what would be expected if marks were assigned independently of location. When the marks themselves influence the underlying point locations (e.g. a large earthquake mark increasing the local rate of aftershock points, as in the Hawkes-process-based ETAS model of seismicity), the marked point process framework merges naturally with self-exciting process models.

Properties

Whether the marks are independent of the point pattern's locations or genuinely tied to it is itself a substantive scientific question, mark independence would mean, e.g., that earthquake magnitude has nothing to do with where or when earthquakes cluster, which is testable directly via the mark correlation function.
The most consequential real-world marked point process is arguably the ETAS (Epidemic-Type Aftershock Sequence) model of seismicity, which combines a marked point process (earthquake location, time, magnitude) with self-exciting (Hawkes-process) dynamics, where larger-magnitude marks trigger disproportionately more aftershocks.
Marked point processes generalize naturally to multivariate point processes (multiple, distinguishable types of points sharing the same space, treated as a special case of discrete marks) and to processes where the marks are themselves functions or curves (functional marks), extending well beyond simple scalar attributes.

At a Glance

Structurean ordinary point process, plus an attached random mark at every point

Mark spacecan be continuous (magnitude), categorical (species), or multivariate

Key questionare marks near each other spatially/temporally similar (positive mark correlation)?

Common applicationearthquake catalogs (location + magnitude), disease case data (location + severity)

Last updated September 8, 2026← Back to the Dictionary