Temporal Point Process

The core mathematical object defining a temporal point process. It represents the instantaneous rate of event occurrence at time t given the entire history of past events up to that moment.

• Notation: λ*(t) = λ(t | H_t)
• Interpretation: λ*(t)dt is the probability of an event in [t, t+dt)
• Self-exciting processes: Past events increase λ*(t)
• Self-correcting processes: Past events decrease λ*(t)

This function fully characterizes the process and is the target of estimation in most modeling tasks.

The simplest and most fundamental temporal point process, where events occur independently at a constant average rate.

• Homogeneous Poisson: Constant intensity λ, events are memoryless
• Non-homogeneous Poisson: Intensity λ(t) varies with time but not with event history
• Key property: The number of events in disjoint intervals are independent
• Limitation: Cannot capture clustering or inhibitory effects

Serves as the null model against which more complex self-exciting or self-correcting processes are compared.

A self-exciting temporal point process where each event increases the probability of future events in the near term, creating clustered event patterns.

• Intensity form: λ*(t) = μ + Σ α · exp(-β(t - t_i))
• μ: Baseline intensity
• α: Jump size (excitation strength)
• β: Decay rate of excitation
• Applications: Earthquake aftershock modeling, social media cascades, high-frequency trading, neural spike trains

Captures the 'rich-get-richer' temporal dynamics observed in many real-world phenomena.

An extension where each event carries an associated mark (category, value, or vector) alongside its timestamp, enabling joint modeling of when and what.

• Mark space: Can be discrete (user action type) or continuous (purchase amount)
• Conditional intensity: λ*(t, k) for mark type k
• Decomposition: λ*(t, k) = λ*(t) · p*(k | t)
• Use case: Modeling sequences of heterogeneous user actions (clicks, adds-to-cart, purchases) as a single unified process

Essential for next-event prediction tasks where both timing and event type matter.

Modern approaches that parameterize the conditional intensity function using deep neural networks instead of hand-crafted parametric forms.

• RNN-based: Use recurrent networks to encode event history H_t into a hidden state, then decode into λ*(t)
• Transformer-based: Apply self-attention over the event sequence for parallelized history encoding
• Advantage: Learn complex, non-linear dependencies from data without strong parametric assumptions
• Training: Maximum likelihood estimation over observed event sequences

Enables modeling of intricate user behavior patterns that classical parametric processes cannot capture.

An alternative characterization of a temporal point process through the probability distribution of waiting times between consecutive events.

• Relationship to intensity: f(τ) = λ*(τ) · exp(-∫ λ*(s)ds)
• Exponential distribution: Implied by constant intensity (memoryless)
• Weibull distribution: Captures increasing or decreasing hazard rates
• Log-normal distribution: Common in human activity modeling

Useful for simulation, goodness-of-fit testing, and understanding the rhythm of user behavior patterns.