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Cox Process

A Poisson process whose intensity is itself random rather than fixed, generated in two steps, first a random intensity function is drawn, then points are scattered as an ordinary Poisson process given that realized intensity, also called a doubly stochastic Poisson process.

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In Plain English

An ordinary Poisson process has a fixed, known rate, events happen at, say, exactly 5 per hour on average, no matter what. A Cox process adds an extra layer of uncertainty: the rate itself is random, drawn from some distribution or stochastic process, and only once that random rate is realized does the actual Poisson-style scattering of points happen. Picture disease cases scattered across a city: the underlying risk level varies unpredictably by neighborhood (that's the random intensity), and given that risk landscape, cases occur Poisson-randomly within it. This two-layer randomness, random environment, then random points given that environment, is exactly what 'doubly stochastic' means.

Definition

A Cox process (or doubly stochastic Poisson process) is constructed in two stages: first, a random intensity function $Λ (x)$ (itself a stochastic process, e.g. a log-Gaussian process, or a mixture over a few discrete levels) is realized; then, conditional on that realization, points are generated as an inhomogeneous Poisson process with that specific intensity function $Λ (x)$ . Unconditionally (averaging over the randomness in $Λ$ ), a Cox process exhibits over-dispersion relative to an ordinary Poisson process with the same mean intensity, its variance exceeds its mean, because uncertainty about the intensity itself adds an extra source of variability on top of the Poisson counting variability. The log-Gaussian Cox process, where $lo g Λ (x)$ is a Gaussian process, is the most widely used specific form in spatial statistics, since it flexibly captures smooth, spatially correlated clustering while keeping the underlying intensity strictly positive.

Properties

Over-dispersion is the Cox process's statistical fingerprint: because the intensity itself is uncertain, the observed point counts vary more than an ordinary Poisson process's counts would, exactly as a Poisson-Gamma mixture (negative binomial) over-disperses relative to a plain Poisson count.
The log-Gaussian Cox process is popular specifically because taking the logarithm of a Gaussian process guarantees the resulting intensity $Λ (x) = e^{Z (x)}$ stays strictly positive everywhere (a requirement for any valid intensity function) while still letting the Gaussian process's covariance structure control how smoothly and strongly nearby regions' risk levels are correlated.
Cox processes are the natural modeling choice whenever clustering arises from a genuinely unobserved, spatially- or temporally-varying environmental factor (disease risk maps, wildfire ignition risk, crime hotspots), as opposed to clustering caused by events directly triggering further events, which instead calls for a self-exciting process like the Hawkes process.

At a Glance

Constructionrandom intensity drawn first, then an ordinary Poisson process given that intensity

Alternate namedoubly stochastic Poisson process

Signature effectover-dispersion, variance exceeds mean, relative to a fixed-rate Poisson process

Common specific formthe log-Gaussian Cox process, widely used in spatial statistics

Last updated September 8, 2026← Back to the Dictionary