Point Processes

The masspcf.point_process module provides samplers for spatial point processes, returning PointCloudTensor objects. All samplers support deterministic seeding via Generator (see Random Generation).

Poisson point process

sample_poisson() generates point clouds from a homogeneous spatial Poisson process. Each element of the output tensor is a point cloud with a random number of points:

\[N \sim \text{Poisson}(\lambda \cdot V)\]

where \(\lambda\) is the rate (intensity) and \(V\) is the volume of the sampling region. Points are placed uniformly in the region.

Show code

def plot_poisson_samples(color="steelblue"):
    from masspcf.point_process import sample_poisson
    import masspcf as mpcf

    gen = mpcf.random.Generator(seed=42)
    X = sample_poisson((3,), dim=2, rate=50.0, generator=gen)

    fig, axes = plt.subplots(1, 3, figsize=(9, 3))
    for i, ax in enumerate(axes):
        pc = np.asarray(X[i])
        ax.scatter(pc[:, 0], pc[:, 1], s=8, color=color, alpha=0.7)
        ax.set_xlim(0, 1)
        ax.set_ylim(0, 1)
        ax.set_aspect("equal")
        ax.set_title(f"Sample {i + 1} ({pc.shape[0]} points)")
        ax.set_xlabel("$x_1$")
        ax.set_ylabel("$x_2$")
    fig.tight_layout()
    return fig

Basic usage:

from masspcf.point_process import sample_poisson

# 100 point clouds in R^2, rate 50, in the unit square
X = sample_poisson((100,), dim=2, rate=50.0)

# 3-D point clouds in a custom box, seeded for reproducibility
import masspcf as mpcf
gen = mpcf.random.Generator(seed=42)

X = sample_poisson(
    (10, 20),
    dim=3,
    rate=100.0,
    lo=[0.0, 0.0, -1.0],
    hi=[1.0, 2.0, 1.0],
    generator=gen,
)

By default the sampling region is \([0, 1]^d\). Custom bounds are specified with lo and hi, each an array of length dim.