Hawkes
Hawkes Process
A Hawkes process is a self-exciting point process, meaning the occurrence of an event increases the probability of future events occurring in the same process. It is a stochastic process used to model the timing of events that tend to cluster together, often exhibiting bursty behavior.
The key characteristic of a Hawkes process is its conditional intensity function, which determines the rate at which events occur at a given time, conditioned on the history of previous events. The intensity function typically comprises a background rate (also known as the base rate or spontaneous rate) and a term that depends on the times and magnitudes of previous events.
Specifically, the conditional intensity, denoted by λ(t), usually takes the form:
λ(t) = μ + Σti < t g(t - ti)
where:
- μ is the background rate, representing the constant rate of events occurring independently of other events.
- ti are the times of past events (where i is an index over all events that have occurred before time t).
- g(t - ti) is the excitation function or kernel, which describes how the occurrence of an event at time ti affects the intensity at a later time t. Common choices for g(t) include exponential functions, power-law functions, and Gaussian functions. The excitation function typically decays as the time difference (t - ti) increases, reflecting the diminishing influence of past events on the current intensity.
Hawkes processes find applications in various fields, including:
- Finance: Modeling high-frequency trading and price movements.
- Social Media: Analyzing the spread of information and viral phenomena.
- Seismology: Studying earthquake aftershocks.
- Neuroscience: Modeling neural spiking activity.
- Epidemiology: Tracking the spread of infectious diseases.
- Criminology: Analyzing crime patterns and hotspots.
The self-exciting nature of Hawkes processes makes them particularly suitable for modeling phenomena where events trigger or influence subsequent events, leading to cascading or clustered patterns. Parameter estimation for Hawkes processes can be achieved using methods such as maximum likelihood estimation (MLE). Different variations of the Hawkes process exist, including multivariate Hawkes processes, which model the interactions between multiple event streams, and non-parametric Hawkes processes, where the excitation function is learned directly from data without assuming a specific functional form.