Marked Spatio-Temporal Point Process Simulator

A set of Python tools called STPPG for simulating any form of Marked Spatio-temporal self-exciting point processes.

Usage

There are two critical files included in this repo:

• stppg.py includes basic generators for homogeneous and inhomogeneous univariate point process, as well as various types of intensity classes and kernel functions.
• utils.py includes plotting functions for visualizing the point process simulation.

For simulating any parametric point process defined in references, we first need to define the parametric form of its conditional density and event space (time, location, and marks).

• To define the form of the conditional density of a point process, a Lam object defined in stppg.py needs to be substantiated accordingly. The most important parameters of a conditional intensity Lam is defined in its kernel function. See the example below.
• For spatio-temporal point processes, time is specified by 1D two-elements list T, and location & mark space is jointly specified by 2D list S, where first two sub-lists indicate the location X and Y, and the following sub-lists indicate the mark space.

Examples

A simple example of simulating a spatio-temporal hawkes point process equipped with a standard diffusion kernel by using stppg.

from stppg import StdDiffusionKernel, HawkesLam
from utils import plot_spatio_temporal_points, plot_spatial_intensity

np.random.seed(0)
np.set_printoptions(suppress=True)

# parameters initialization
mu     = .1
kernel = StdDiffusionKernel(C=1., beta=1., sigma_x=.1, sigma_y=.1)
lam    = HawkesLam(mu, kernel, maximum=1e+3)
pp     = SpatialTemporalPointProcess(lam)

# generate points
points, sizes = pp.generate(
    T=[0., 10.], S=[[-1., 1.], [-1., 1.]], 
    batch_size=500, verbose=True)

# plot intensity of the process over the time
plot_spatial_intensity(lam, points[0], S=[[0., 10.], [-1., 1.], [-1., 1.]],
    t_slots=1000, grid_size=50, interval=50)

And see the console output below.

[2018-09-21T08:28:59.974629-04:00] generate samples (10167, 3) from homogeneous poisson point process
[2018-09-21T08:28:59.975163-04:00] 0 raw samples have been checked. 0 samples have been retained.
[2018-09-21T08:29:00.097072-04:00] 1000 raw samples have been checked. 1 samples have been retained.
[2018-09-21T08:29:00.327207-04:00] 2000 raw samples have been checked. 11 samples have been retained.
[2018-09-21T08:29:00.643629-04:00] 3000 raw samples have been checked. 21 samples have been retained.
[2018-09-21T08:29:01.061958-04:00] 4000 raw samples have been checked. 51 samples have been retained.
[2018-09-21T08:29:01.592077-04:00] 5000 raw samples have been checked. 71 samples have been retained.
[2018-09-21T08:29:02.242894-04:00] 6000 raw samples have been checked. 88 samples have been retained.
[2018-09-21T08:29:02.974756-04:00] 7000 raw samples have been checked. 100 samples have been retained.
[2018-09-21T08:29:03.779483-04:00] 8000 raw samples have been checked. 110 samples have been retained.
[2018-09-21T08:29:04.741294-04:00] 9000 raw samples have been checked. 128 samples have been retained.
[2018-09-21T08:29:05.729931-04:00] 10000 raw samples have been checked. 146 samples have been retained.
[2018-09-21T08:29:05.904117-04:00] thining samples (147, 3) based on Spatio-temporal Hawkes point process intensity with mu=0, beta=1 and Diffusion-type Kernel.
[[0.06299644 0.9183208  0.97126597]
 [0.11153312 0.280956   0.05509999]
 [0.121267   0.36991751 0.65443569]
 [0.12390124 0.49035887 0.64267112]
 [0.13272256 0.9463777  0.02484997]
     ... ...
 [0.97093684 0.13255896 0.15690197]
 [0.97663217 0.6115613  0.34787382]
 [0.97863127 0.21207182 0.19613095]
 [0.98051828 0.43338621 0.67951877]
 [0.99105454 0.21976152 0.26218951]]
[2018-09-21T08:29:05.911032-04:00] preparing the dataset 1000 × (50, 50) for plotting.
[2018-09-21T08:32:26.300910-04:00] start animation.

The animations below show the progression of conditional intensities through time with different types of kernel functions.

Standard Diffusion Kernel	Spatial-Variant Gaussian Diffusion Kernel	Spatial-Variant Gaussian Mixture Diffusion Kernel

References

• Shixiang Zhu, Yao Xie. "Reinforcement Learning of Spatio-Temporal Point Processes."
• Y. Ogata. "Space-Time Point-Process Models for Earthquake Occurrences"
• F. Musmeci, D. Vere-Jones. "A Space-Time Clustering Model for Historical Earthquakes"
• S. Zhu and Y. Xie. "Crime Linkage Detection by Spatio-Temporal-Textual Point Processes"