Chapter 10: Marked Point Processes

So far, every event has been just a time. In practice, events carry additional information: an earthquake has a magnitude, a trade has a size and direction, a neural spike has a waveform. A marked point process augments the event times with marks — random variables attached to each event.

10.1 Definition

A marked point process is a collection of pairs {(tᵢ, mᵢ)}, where:

• tᵢ ∈ [0, ∞) is the event time (the ground process)
• mᵢ ∈ M is the mark (a real number, a vector, a category, etc.)

The ground process N_g(t) = |{i : tᵢ ≤ t}| is the point process of times, ignoring marks.

The full conditional intensity factors as:

λ*(t, m) = λ_g*(t) · p(m | t, F_{t-})

where:

• λ_g*(t) is the conditional intensity of the ground process
• p(m | t, F_{t-}) is the conditional mark distribution given the current time and history

10.2 Independent Marks

The simplest case is independent marks: the mark distribution does not depend on the event time or past history:

p(m | t, F_{t-}) = p(m)

In this case, the log-likelihood separates:

ℓ = ℓ_ground + ℓ_marks
ℓ_ground = Σᵢ log λ_g*(tᵢ) − ∫ λ_g*(t) dt
ℓ_marks  = Σᵢ log p(mᵢ)

Ground process and mark distribution can be estimated independently. The ground process can be any of the models covered in previous chapters (HPP, NHPP, Hawkes, Cox).

10.3 History-Dependent Marks

More interesting is when marks depend on the history. In seismology, for example, larger mainshocks tend to produce larger aftershocks — the mark distribution changes as the process evolves.

If p(m | t, F_{t-}) = p(m | λ_g*(t)), i.e., marks depend on the current intensity, the log-likelihood no longer separates, and joint estimation is required.

10.4 Mark-Dependent Intensity: The ETAS Model

The most influential marked point process model in seismology is the Epidemic-Type Aftershock Sequence (ETAS) model (Ogata, 1988). Each earthquake at time tᵢ with magnitude mᵢ contributes to future seismicity according to:

λ*(t) = μ + Σ_{tᵢ < t} K · exp(α(mᵢ − Mc)) / (t − tᵢ + c)^p

where:

• μ is the background rate
• K is the productivity parameter
• α controls magnitude-dependent productivity
• Mc is the completeness magnitude threshold
• c, p are the modified Omori law parameters (typically p ≈ 1.1)
• Magnitudes are i.i.d. Gutenberg-Richter: p(m) = β_GR · exp(−β_GR(m − Mc)) for m ≥ Mc

The key feature: large events (mᵢ ≫ Mc) generate exponentially more offspring. This is why major earthquakes produce long, intense aftershock sequences.

10.5 Simulation of ETAS

Simulation combines branching structure with magnitude-dependent productivity:

def simulate_etas(mu, K, alpha, c, p, beta_gr, Mc, T):
    events = []  # list of (time, magnitude)

    def add_offspring(t_parent, m_parent, depth=0):
        if depth > 50:
            return  # safety limit
        n_offspring = np.random.poisson(K * np.exp(alpha * (m_parent - Mc)))
        for _ in range(n_offspring):
            # Sample delay from modified Omori (truncated power law)
            delay = c * ((np.random.uniform()**(-1/(p-1))) - 1)
            t_child = t_parent + delay
            if t_child < T:
                m_child = Mc + np.random.exponential(1.0 / beta_gr)
                events.append((t_child, m_child))
                add_offspring(t_child, m_child, depth + 1)

    # Background events
    n_bg = np.random.poisson(mu * T)
    bg_times = np.random.uniform(0, T, n_bg)
    for t in bg_times:
        m = Mc + np.random.exponential(1.0 / beta_gr)
        events.append((t, m))
        add_offspring(t, m)

    events.sort(key=lambda x: x[0])
    return events

See code/10_marked_point_processes.py for the full implementation.

10.6 The Gutenberg-Richter Law

In seismology, earthquake magnitudes follow the Gutenberg-Richter law:

log₁₀ N(≥m) = a − b · m

where N(≥m) is the number of earthquakes with magnitude ≥ m, and b ≈ 1 is the b-value (nearly universal). This is an exponential distribution on the magnitude scale:

p(m) = β_GR · exp(−β_GR · (m − Mc)),   m ≥ Mc
β_GR = b · ln(10)

The MLE of β_GR from observed magnitudes m₁, ..., mₙ (all ≥ Mc) is:

β̂_GR = n / Σᵢ(mᵢ − Mc) = 1 / (m̄ − Mc)

10.7 Key Takeaways

• A marked point process augments event times with marks (magnitudes, sizes, types).
• The conditional intensity factorizes as λ*(t, m) = λ_g*(t) · p(m | t, history).
• With independent marks, the log-likelihood separates — ground process and marks are estimated independently.
• The ETAS model is the canonical seismological application: magnitude-dependent Hawkes process with a power-law kernel.
• The Gutenberg-Richter law (exponential magnitudes) leads to a simple MLE for the magnitude distribution.