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Log-Gaussian Cox process

Definition

A log-Gaussian Cox process is a model for randomly located events in which the logarithm of an underlying event intensity follows a Gaussian process. Conditional on that intensity, events follow a Poisson process.

What it means

A map of deliveries, tree locations or equipment faults may contain busy patches and quiet patches. Some variation can be explained by known conditions, while some remains uncertain. This model represents that uncertainty through a random surface of event intensity.

Intensity measures expected events per unit area, or per another specified unit of exposure. It is not the probability of an event at an exact coordinate. Exponentiating the Gaussian surface keeps intensity positive while allowing it to vary substantially.

The two layers matter: uncertainty about the surface differs from randomness in events generated from a particular surface. Seeing a cluster does not establish that events caused neighboring events; shared underlying conditions can also produce clustering.

How it works

Specify a mean function and covariance function for a Gaussian field Z(s), where s identifies a location. Known features can enter the mean. Set the random intensity to exp(Z(s)). For a region, integrate this intensity over its area to obtain the conditional expected count, then use a Poisson distribution for that count.

Counts in disjoint regions are independent when the entire intensity surface is given. After averaging over the uncertain surface, those counts can become dependent. Positive covariance in the latent field creates a useful model of shared spatial variation.

Inference uses observed locations and the observation window to learn about the field and its parameters. Practical calculations often approximate spatial integrals on a grid or use other numerical methods. Grid resolution, observation coverage and covariance assumptions therefore need checking alongside predictive performance.

Three practical examples

• A delivery planner models pickup locations while allowing unmeasured neighborhood activity to vary smoothly across a city.
• A forestry analyst describes tree locations using environmental features and residual spatial variation, without assuming each nearby tree attracted another.
• A maintenance team models fault locations across an inspected surface, distinguishing unobserved intensity variation from ordinary count randomness.

Think of it this way

Think of rain falling through a changing cloud cover. Conditional on a particular cover, some areas have a higher expected rate of drops; uncertainty about the cover adds another layer to the prediction.

This is only an analogy for the two layers of randomness. Real rainfall need not satisfy Poisson assumptions, and a Gaussian field does not literally describe clouds in every application.

Worked example

Why a count can have variance much larger than its mean

Consider a deliberately simple invented model in which the same random intensity Lambda applies across the whole observation region. Its logarithm is Gaussian with variance log(2) and mean log(2) minus half that variance. Standard lognormal calculations give E[Lambda] = 2 and Var(Lambda) = 4.

For a region of area 3, the conditional count N has mean and variance 3 Lambda. Averaging over Lambda gives E[N] = 6. The law of total variance gives Var(N) = E[3 Lambda] + Var(3 Lambda) = 6 + 36 = 42.

A fixed-intensity Poisson model with the same mean would have variance 6. The extra 36 comes from uncertainty in the shared intensity, not a change to the conditional Poisson rule. Two disjoint unit-area counts have covariance 4 after the shared intensity is averaged out.

This constant-field special case makes the arithmetic transparent. A realistic spatial covariance model can allow dependence to weaken with distance rather than forcing the whole map to share one rate.

Illustrative case study

A fictional parcel cooperative receives pickup requests from irregularly covered neighborhoods. Its initial map treats every blank area as low demand. An analyst notices that some areas were not served during parts of the reporting period.

The team records observation coverage, compares models with known exposure information and checks predictions on later periods. It uses an LGCP candidate to represent remaining spatial variation, but investigates whether apparent hotspots follow service availability rather than customer preference.

The result is a more careful modeling specification, not a claimed commercial improvement. A flexible intensity surface cannot repair missing coverage information by itself.

Common misconceptions and mistakes

• Gaussian event counts: the Gaussian assumption concerns the logarithm of intensity, not the observed counts.
• Independent counts everywhere: conditional independence does not imply independence after uncertainty is integrated out.
• A hotspot proves contagion: shared environmental variation can create clustering without direct event interaction.
• Exponentiating the mean gives mean intensity: for a Gaussian value with variance v, mean intensity includes the factor exp(v/2).

Business use

The model is useful when planning depends on where events occur and how uncertain their concentration is. Teams can compare predicted regional counts, uncertainty intervals and held-out spatial patterns with simpler Poisson alternatives.

Specify units, exposure, boundaries and observation completeness before choosing a sophisticated covariance function. Demand intensity should not automatically be interpreted as underlying need when recording practices differ across locations. Computational cost and sensitivity to spatial resolution also affect whether the model fits an operational workflow.

People also ask

Is this the same as Cox proportional hazards?

No. They are distinct model families that share the Cox name.

Must intensity be constant?

No. The worked example deliberately uses a constant random field; usual spatial applications allow intensity to vary by location.

Can it predict exact event coordinates?

It gives a probability model for possible point patterns and regional counts, not certainty about where the next event occurs.

Check your understanding

A conditional Poisson count has random mean M, with E[M] = 8 and Var(M) = 12. What is the unconditional count variance?

Answer: It is 8 + 12 = 20, using total variance. Replacing the random mean by its average would remove the second contribution.

Related terms

• Gaussian process
• Posterior predictive distribution
• Cox process
• Poisson point process
• lognormal distribution
• spatial covariance
• intensity function
• overdispersion

Learn next

Start with Poisson counts and the difference between intensity and probability. Then study Gaussian covariance functions, lognormal moments and the law of total variance. Finally compare simulations from a fixed-intensity Poisson model and an LGCP with the same average intensity.

More in Probabilistic machine learning

• Likelihood tempering - Likelihood tempering changes the influence of observed data by raising the likelihood to a power before combining it with the prior.
• Local level model - A local level model represents a time series as noisy observations of an underlying level that changes through a random walk.
• Local linear trend model - A local linear trend model is a state-space time-series model with both a changing level and a changing slope.
• Log-sum-exp trick - The log-sum-exp trick is a numerically stable way to calculate the logarithm of a sum of exponentials.
• Manifestation shift - Manifestation shift is a change in how a target condition produces the observable physical characteristics used to recognize it.
• Masked autoregressive flow - A masked autoregressive flow, or MAF, is a normalizing-flow model that uses masked neural networks to transform observed data into a simpler base distribution.

Sources

Møller, Syversveen and Waagepetersen, Log Gaussian Cox Processes, primary paper: https://math.arizona.edu/~jwatkins/log_Gaussian_Cox_Processes.pdf

spatstat.random documentation, rLGCP, model definition and simulation interface: https://search.r-project.org/CRAN/refmans/spatstat.random/html/rLGCP.html

The numerical example and fictional case are original teaching illustrations.

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