mathematics

Hawkes process

Hawkes process is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Hawkes process rather than just read about it. In short: In probability theory and statistics, a Hawkes process is an age-dependent branching process driven by immigration from an inhomogeneous Poisson process. The process, named after Alan G.

Key takeaways

Hawkes process belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hawkes process to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hawkes process from memory before moving on to harder problems.

Reference excerpt

In probability theory and statistics, a Hawkes process is an age-dependent branching process driven by immigration from an inhomogeneous Poisson process. The process, named after Alan G. Hawkes, is also called a self-exciting point process., i.e., the occurrence of an event increases the likelihood of another event occurring. It has arrivals at times 0 < t 1 < t 2 < t 3 < ⋯ {\textstyle 0<t_{1}<t_{2}<t_{3}<\cdots } where the infinitesimal probability of an arrival during the time interval [ t , t + d t ) {\textstyle [t,t+dt)} is

λ t d t = ( μ ( t ) + ∑ t k : t k < t ϕ ( t − t k ) ) d t . {\displaystyle \lambda _{t}\,dt=\left(\mu (t)+\sum _{t_{k}\,:\,t_{k}\,<\,t}\phi (t-t_{k})\right)\,dt.}

The function μ {\textstyle \mu } is the intensity of an underlying Poisson process. The first arrival occurs at time t 1 {\textstyle t_{1}} and immediately after that, the intensity becomes μ ( t ) + ϕ ( t − t 1 ) {\textstyle \mu (t)+\phi (t-t_{1})} , and at the time t 2 {\textstyle t_{2}} of the second arrival the intensity jumps to μ ( t ) + ϕ ( t − t 1 ) + ϕ ( t − t 2 ) {\textstyle \mu (t)+\phi (t-t_{1})+\phi (t-t_{2})} and so on. During the time interval ( t k , t k + 1 ) {\textstyle (t_{k},t_{k+1})} , the process is the sum of k + 1 {\textstyle k+1} independent processes with intensities μ ( t ) , ϕ ( t − t 1 ) , … , ϕ ( t − t k ) . {\textstyle \mu (t),\phi (t-t_{1}),\ldots ,\phi (t-t_{k}).} The arrivals in the process whose intensity is ϕ ( t − t k ) {\textstyle \phi (t-t_{k})} are the "daughters" of the arrival at time t k . {\textstyle t_{k}.} The integral ∫ 0 ∞ ϕ ( t ) d t {\displaystyle \int _{0}^{\infty }\phi (t)\,dt} is the average number of daughters of each arrival and is called the branching ratio. Thus viewing some arrivals as descendants of earlier arrivals, we have a Galton–Watson branching process. The number of such descendants is finite with probability 1 if branching ratio is 1 or less. If the branching ratio is more than 1, then each arrival has positive probability of having infinitely many descendants.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Hawkes process

Start with the simplest possible case. Write down what Hawkes process claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hawkes process before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hawkes process ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hawkes process

In research: Hawkes process appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hawkes process in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hawkes process is common in secondary-school and first-year university syllabi. It links to neighbouring topics Mathematical finance, Point processes, Stochastic processes, so understanding it makes those chapters shorter.
In everyday life: Look for Hawkes process outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Hawkes process” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Hawkes process in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hawkes process means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hawkes process out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hawkes process in simple terms?

In probability theory and statistics, a Hawkes process is an age-dependent branching process driven by immigration from an inhomogeneous Poisson process. The process, named after Alan G.

Why does Hawkes process matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hawkes process?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hawkes process.