5  Intensity function

5.1 Motivation

In the previous chapter, we introduced point processes as random counting mechanisms.

A natural next question is:

How do we describe where points tend to occur?

At the most basic level, we want to measure the average concentration of points across space.

This leads to the concept of first-order structure.

Roughly speaking, first-order structure describes how the expected number of points varies from place to place.

  • If points are equally likely everywhere, the process is spatially uniform.

  • If some regions tend to contain more points than others, the process is spatially inhomogeneous.

5.2 From expected counts to local rate

Let \(X\) be a point process on a spatial domain \(W \subset \mathbb{R}^d\), and let \(N(B)\) denote the number of points falling in a region \(B \subset W\).

One way to summarize the average behavior is through expected count in a region:

\[ \mathbb{E}[N(B)]. \]

This tells us the expected number of points \(B\), but it does not yet describe how that expectation is distributed across space.

To get a local version of this, we ask:

What is the expected number of points in a very small region around location \(u\)?

This motivates the intensity function.

5.3 Definition of intensity function

Informally, the intensity function \(\lambda(u)\) gives the expected number of points per unit area near location \(u\).

Formally, if it exists, the intensity function is defined by

\[ \lambda(u) = \lim_{du \to 0}\frac{\mathbb{E}[N(du)]}{|du|} \]

where \(du\) denotes a small region containing \(u\), and \(|du|\) is its volume (area in 2D).

Note

Interpretation:

  • \(\mathbb{E}[N(du)]\) is the expected number of points in a tiny region near \(u\)
  • dividing by \(|du|\) converts this into an expected number of points per unit area
  • so \(\lambda(u)\) is a local average density of points

Equivalently, for a sufficiently small region \(du\) around \(u\),

\[ \mathbb{E}[N(du)] \approx \lambda(u)\,|du| \]

Note

“sufficiently small” yucky.

This approximation is often the most useful way to think about intensity.

5.4 Mean measure

Before introducing \(\lambda(u)\), it is useful to define the mean measure.

For any region \(B \subset W\), define

\[ \mu(B) = \mathbb{E}[N(B)] \]

The function \(\mu(\cdot)\) is called the mean measure of the point process.

It records the expected number of points in every region.

If this mean measure admits a density with respect to volume, then that density is precisely the intensity function:

\[ \mu(B) = \int_B \lambda(u)\ du \]

So the intensity function is just the spatial density of the mean measure.

Tip

This is exactly analogous to ordinary probability:

  • a distribution function gives total probability of sets
  • a density function gives probability per unit length/area/volume

Here:

  • the mean measure \(\mu(B)\) gives expected count in a set
  • the intensity \(\lambda(u)\) gives expected count per unit area at location \(u\)
Note

Straight booty cheeks not intuitive at all right now:

If this mean measure admits a density with respect to volume

????? Really?

And if \(\mu(B) = \mathbb{E}[N(B)]\), why even define it as its own thing?????

5.5 Interpretation of \(\lambda(u)\)

The intensity function is a first-order summary.

It describes the average abundance of points across space, but it does not describe the interaction between points.

For example, two processes can have the same intensity function but one may be clustered, while the other may be regular.

So the intensity tells us about where points are expected to occur on average, but not whether they attract or repel one another.

That is why it belongs to the first-order structure, rather than the second-order structure.

5.6 Homogeneous intensity

The simplest case is when the intensity is constant:

\[ \lambda(u) = \lambda \quad \forall u \in W. \]

In this case, the process is aid to have constant first-order structure, or to be first-order homogeneous.

Then for any region \(B\),

\[ \mathbb{E}[N(B)] = \int_B \lambda(u)\ du = \lambda \int_B 1\ du = \lambda |B| \]

So the expected number of points in a region depends only on its size, not on its location in \(W\).

Note

Interpretation:

If \(\lambda(u)\) is constant, then no location is favored over any other in terms of average point density.

This does not automatically mean the process is Poisson.

It only means that the first-order structure is spatially uniform.

5.7 Inhomogeneous intensity

More generally, the intensity may vary with location:

\[ \lambda(u) \neq \lambda(v) \quad \text{for some} \ u, v \in W \]

In this case, the process is inhomogeneous.

Then the expected count in a region is

\[ \mathbb{E}[N(B)] = \int_B \lambda(u)\ du \]

Regions where \(\lambda(u)\) is larger are expected to contain more points, on average.

This allows us to model spatial trends such as:

  • envionmental gradients
  • varying population density
  • spatial covariate effects
  • boundary effects or hotspots

5.8 Example: Poisson processes

The intensity function already appeared in the previous chapter for Poisson processes.

5.8.1 Homogeneous Poisson process

For a homogeneous Poisson process with parameter \(\lambda\),

\[ \mathbb{E}[N(B)] = \lambda |B| \]

so the intensity function is simply

\[ \lambda(u) = \lambda \]

5.8.2 Inhomogeneous Poisson process

For an inhomogeneous Poisson process with intensity function \(\lambda(u)\),

\[ \mathbb{E}[N(B)] = \int_B \lambda(u)\ du \]

So in the Poisson case, the intensity function fully determines the first-order structure.

Note

Still not clear to me what exactly first-order structure is?

Like is it LITERALLY just \(\mathbb{E}[N(B)]\)???

However, outside of a Poisson setting, many different point processes may share the same intensity function.

5.9 Why intensity matters

The intensity function is important because it is usually the first quantity we try to understand from spatial point pattern data.

Note

The intensity function is important because we try to understand it first?????

Dog water

It helps answer questions such as:

  • Are points more common in some parts of the window than others?
  • Is there a large-scale spatial trend?
  • Is the pattern approximately uniform?
  • Should we remove first-order inhomogeneity before studying clustering?

In practice, estimating \(\lambda(u)\) is often the first step in spatial point process analysis.

5.10 What the intensity does not tell us

It is important to not over-interpret the intensity function.

A high value of \(\lambda(u)\) means that points are more common on average near \(u\), but it does not mean that points are interacting there.

For example, a process may have:

  • constant intensity, but strong clustering
  • varying intensity, but no interaction
  • both inhomogeneity and clustering

So the intensity alone is not enough to describe the full spatial structure.

To study dependence between points, we need second-order summaries, which we will introduce in later chapters.

5.11 Uncertainty in intensity estimation

Estimating \(\lambda(u)\) from a single realisation introduces variability.

For simple estimators based on local counts, variability arises from the randomness in the number of observed points.

Under a Poisson assumption, we have

\[ \mathrm{Var}(N(B)) = \mathbb{E}[N(B)], \]

which suggests that

\[ \mathrm{Var}(\hat{\lambda}(u)) \propto \frac{\lambda(u)}{|B(u,r)|}. \]

For kernel estimators, the variance depends on the bandwidth \(h\), with smaller values of \(h\) leading to more variable estimates.

Note

Key idea:

There is a bias–variance trade-off:

  • small bandwidth → low bias, high variance
  • large bandwidth → high bias, low variance

In practice, uncertainty in intensity estimates is often assessed using simulation-based methods, such as parametric bootstrap.

However, in many applications, intensity estimation is primarily used as a descriptive tool rather than for formal inference.

Note

I would like to flesh this out a bit.

Especially: WHY are simulation based methods preferred in practice?

A: Simulation-based methods are preferred because analytic uncertainty is either unavailable, unreliable, or too complicated once you leave the Poisson world.

Analytic expressions for uncertainty quickly become intractable in spatial point processes due to dependence between points, edge effects, and smoothing. Simulation-based methods provide a flexible alternative that naturally accounts for these complexities.

5.12 Summary

The intensity function \(\lambda(u)\) is the fundamental descriptor of first-order structure in a point process.

It tells us the expected number of points per unit area near location \(u\), and satisfies

\[ \mathbb{E}[N(B)] = \int_B \lambda(u)\,du. \]

  • If \(\lambda(u)\) is constant, the process is first-order homogeneous.
  • If \(\lambda(u)\) varies with \(u\), the process is inhomogeneous.

The intensity function describes average spatial abundance, but not interaction between points.

That is why it is only the beginning of the structural description of a point process.