Commonly used special cases

2. the locations of realized events are obtained as a random sample, with the inclusion probability in \(A_i\) proportional to the intensity function \(\lambda(x)\).

The general case, with spatially varying \(\lambda(x)\), is called a heterogeneous Poisson process.

With spatially constant \(\lambda(x) = \lambda\), we have a homogeneous Poisson process.

\[\mu_{A_i} = E[N(A_i)] = \int_{A_i}{\lambda(x) dx} = \lambda \cdot |A_i|\]

This is also called CSR = complete spatial randomness, which is tied to homogeneous spatial point processes.