Point processes

evergreen#foundations#point-processes#timing

Bar: state the Poisson process three ways, and explain Hawkes self-excitation + its stability condition from memory.

What a point process is

A model for random events in time (clicks, requests, purchases): the random object is the set of timestamps. Summarized by the counting process $N(t)$ = number of events up to $t$ , or by the conditional intensity $\lambda(t)$ = instantaneous event rate given the past.

Poisson process (the baseline)

A homogeneous Poisson process with rate $\lambda$ is equivalently:

Counts: events in any interval of length $\Delta$ are $\text{Poisson}(\lambda\Delta)$ ;
Interarrivals: gaps are i.i.d. $\text{Exponential}(\lambda)$ , mean $1/\lambda$ — memoryless;
Independent increments: disjoint intervals are independent.

P(N(\Delta)=k)=e^{-\lambda\Delta}\frac{(\lambda\Delta)^k}{k!}

Worked: $\lambda=2$ events/min. $P(\text{0 in 1 min})=e^{-2}\approx0.135$ ; expected gap $=0.5$ min; expected events in 5 min $=10$ .

Inhomogeneous Poisson: rate varies, $\lambda(t)$ . Counts in $[a,b]$ are $\text{Poisson}\!\big(\int_a^b\lambda(t)\,dt\big)$ . This captures time-of-day / seasonality in traffic — the deterministic daily curve.

Hawkes process (self-excitation = burstiness)

Real clickstreams are clustered: one event makes more events imminent (a session, a viral spike). Hawkes makes the past raise the intensity:

\lambda(t)=\mu+\sum_{t_i<t}\phi(t-t_i),\qquad \phi(s)=\alpha\,e^{-\beta s}\ (\beta>\alpha>0)

$\mu$ = baseline (exogenous) rate; each event adds a decaying bump $\phi$ .
Branching ratio $n=\int_0^\infty\phi(s)\,ds=\alpha/\beta$ = expected direct offspring per event.
Stability: stationary only if $n<1$ . As $n\to1$ , clusters explode; $n\ge1$ is a runaway cascade.

Interpretation (branching/Galton–Watson view): each event independently triggers children; total events per "immigrant" $=1/(1-n)$ . That single number controls how bursty the stream looks.

Why this matters here

Timing is half of a clickstream. Markov gives what page next; point processes give when. The starter generator uses a Markov chain for pages and a Hawkes process for arrival times (Generating background traffic).
Inhomogeneous $\lambda(t)$ models seasonality; Hawkes models bursty sessions and bot floods.
A change in $\mu$ or $n$ at a known time = planted drift to measure detection delay (Concept drift in production).

By-hand exercise (meets the bar)

For $\lambda=3$ /min, compute $P(N=2$ in 1 min $)$ . (Answer: $e^{-3}3^2/2!=9e^{-3}/2\approx0.224$ .)
A Hawkes process has $\alpha=0.6,\beta=1.0$ . Is it stable? What's the expected cluster size per immigrant? (Answer: $n=0.6<1$ , stable; cluster size $1/(1-0.6)=2.5$ .)