Point Processes: Theory and Simulation in Python

A comprehensive guide to stochastic point processes — from Poisson foundations to self-exciting Hawkes processes — with rigorous theory and hands-on Python simulation.

About This Book

Point processes are the mathematical framework for modeling sequences of events in continuous time. Whether you are analyzing earthquake aftershocks, neural spike trains, high-frequency financial trades, or server request logs, point processes give you the tools to describe, simulate, estimate, and forecast event arrivals.

This book bridges rigorous probability theory and practical computation. Each concept is developed mathematically and then immediately grounded in Python code. By the end, you will be able to choose the right process for your data, simulate it efficiently, fit it via maximum likelihood, and validate your model using principled goodness-of-fit diagnostics.

Who This Book Is For

Data scientists and engineers working with event-stream or time-series data
Quantitative analysts modeling order-book dynamics or credit events
Neuroscientists analyzing spike train recordings
Seismologists working with earthquake catalogs
Applied mathematicians and graduate students learning stochastic processes
Anyone who has encountered Poisson processes and wants to go further

What You’ll Learn

The mathematical foundations of point processes: conditional intensity, compensators, and martingales
How to simulate homogeneous and inhomogeneous Poisson processes efficiently
The renewal process family and how inter-arrival distributions shape event regularity
Hawkes processes: self-exciting dynamics, the branching ratio, and Ogata’s thinning algorithm
Multivariate Hawkes processes and Granger causality between event streams
Cox (doubly stochastic) processes and the Log-Gaussian Cox Process
Marked point processes and the ETAS model for seismology
Maximum likelihood estimation with analytic recursive log-likelihoods
The time-rescaling theorem for rigorous goodness-of-fit testing
Advanced simulation techniques: branching representation, vectorized O(n) algorithms
Applied case studies in seismology, finance, neuroscience, and system monitoring

Book Structure

The book is organized into four parts.

Part 1 — Foundations builds the mathematical vocabulary (chapters 0–4): what a point process is, how the conditional intensity function encodes all probabilistic information, and a thorough treatment of Poisson processes and their key properties.

Part 2 — Core Process Families covers the main model classes practitioners encounter (chapters 5–8): renewal processes, univariate Hawkes processes with estimation, and multivariate Hawkes processes.

Part 3 — Advanced Models extends the toolkit (chapters 9–10): Cox processes for overdispersed data and marked point processes where events carry additional attributes.

Part 4 — Statistical Practice closes the loop (chapters 11–15): likelihood-based inference, the time-rescaling goodness-of-fit framework, advanced simulation recipes, applied case studies, and an action plan for choosing the right model.

Author Notes

All code examples use Python 3.10+. The core dependencies are numpy, scipy, matplotlib, and statsmodels — all installable via pip. The optional tick library provides a reference implementation of Hawkes processes for validation. Install the basics with:

pip install numpy scipy matplotlib statsmodels

Full code scripts live in the code/ subdirectory of this book. Each script can be run standalone and produces self-contained output or figures. Short inline snippets (≤ 20 lines) appear within the chapter text to illustrate key ideas.