The average rate speaks in a calm voice.

It says: 180 trades per minute, 40 incidents per day, 3 failures per rack-hour, 12 alerts per shift.

Then the pager starts drumming on the desk.

The problem is not only that the rate increased. The problem is that one event changed the odds of the next event. The clock acquired memory.

That is the Hawkes-process move:

an event is not only an observation
it is also a shove to the future intensity

When Averages Stay Calm

Suppose two event streams have the same long-run average rate.

One is close to Poisson: event arrivals are conditionally independent given a constant rate. The other arrives in bursts: one event kicks the short-term rate up, then that extra rate decays.

If the dashboard reports only events per minute, these streams can look interchangeable. They are not interchangeable. Capacity planning, alert fatigue, market-impact estimates, and contagion risk all care about clustering.

A point process writes this in its conditional intensity:

\[\lambda(t \mid \mathcal{H}_t),\]

the instantaneous event rate given the history so far. A homogeneous Poisson process has no history in the rate:

\[\lambda(t \mid \mathcal{H}_t) = \mu.\]

An exponential Hawkes process adds memory:

\[\lambda(t) = \mu + \sum_{t_i < t} \alpha e^{-\beta(t-t_i)}.\]

Each event adds a jump of size \(\alpha\). The jump decays at rate \(\beta\). The ratio

\[n = \frac{\alpha}{\beta}\]

is the branching ratio: the expected number of direct offspring events caused by one event in the cluster representation. In the linear Hawkes model, a stable stationary process needs \(n < 1\). As \(n\) approaches one, the process gets near critical; clusters can become large and slow to die.

Hawkes’ 1971 papers introduced self-exciting and mutually exciting point processes in exactly this spirit: events can make future events more likely, and multiple streams can excite each other.1

Two Clocks With the Same Average

The lab below simulates a univariate exponential Hawkes process using Ogata’s thinning method.2 It compares the path to a Poisson process with the same stationary mean rate:

\[\bar\lambda = \frac{\mu}{1-n}.\]

The Hawkes path is initialized at that stationary mean intensity, so the default contrast is not a cold-start trick. Both processes are deterministic for a given seed.

Hawkes events Same-mean Poisson Conditional intensity Mostly endogenous event

Toy univariate exponential Hawkes process. The comparison Poisson process uses the same stationary mean rate. The time-rescale diagnostic uses the Hawkes model that generated the path, not an estimated model.

In the default deterministic run, the counts are deliberately close:

Hawkes events:   177
Poisson events:  173

But the clustering is not close:

Hawkes Fano factor:   5.46
Poisson Fano factor:  0.63
Hawkes gap CV:        1.51
Poisson gap CV:       1.05

The Fano factor is variance divided by mean for equal-width bin counts. A Poisson process has Fano factor one in the ideal model. Finite samples wiggle, but a Fano factor around five is not a wiggle. It is the chart saying: this stream clumps.

Move Branching ratio down to 0%. The Hawkes process collapses toward a Poisson process with rate \(\mu\). Move it toward 90% or 95%. The average rate rises as \(\mu/(1-n)\), but the more important visual change is physical: event clusters take over the time line.

Move Decay beta. Higher decay means excitation burns off quickly: sharper, shorter bursts. Lower decay means memory persists. Two models can have the same branching ratio but different burst shapes because \(\alpha=n\beta\) and \(\beta\) decide how concentrated the offspring risk is in time.

Why Markets Keep Coming Back to Hawkes

High-frequency finance is made of event times: trades, quote changes, order submissions, cancellations, price moves, and cross-asset responses. The natural question is rarely “how many per minute?” alone. It is:

did this event raise the short-term odds of another event?
did events in stream A raise the odds in stream B?
how fast did that effect decay?

Bacry, Mastromatteo, and Muzy review how Hawkes processes entered market microstructure: volatility at transaction time scales, market stability, contagion, systemic risk, execution, and order-book dynamics.3 The appeal is straightforward. A multivariate Hawkes model turns a cloud of event timestamps into a matrix of excitation kernels.

That does not make the model safe by default.

Self-excitation can mean several things:

  1. genuine contagion;
  2. order splitting by one large participant;
  3. common reaction to an unobserved news shock;
  4. market-making inventory feedback;
  5. timestamping or batching artifacts;
  6. an omitted covariate that changes the baseline rate.

A Hawkes fit names dependence in event time. It does not prove the causal story.

Make the Clock Confess

For a correctly specified point-process model, the integrated intensity between successive events,

\[z_i = \int_{t_{i-1}}^{t_i} \lambda(s)\,ds,\]

should behave like independent exponential random variables with mean one. This is the time-rescaling theorem used in point-process diagnostics. Ogata’s work on earthquake occurrence models made residual analysis a central practical tool: transform the fitted process, then check whether the residual process looks Poisson.4

The lab’s ledger reports a simple Kolmogorov-Smirnov distance for those time-rescaled gaps under the true generating model. It is not an estimator benchmark. It is there to emphasize that a point-process model is testable on the clock, not only on the total count.

If you fit a Hawkes model to real data, this is where many pretty stories get less pretty.

Do Not Worship the Branching Ratio

The branching ratio is often the headline because it has a seductive interpretation: the expected fraction of events that are endogenous is roughly \(n\) in the stationary linear Hawkes model. If \(n=0.65\), one is tempted to say “65% of events are caused by previous events.”

Careful.

That statement lives inside the model. If the baseline is misspecified, if there are intraday seasonality effects, if one latent news process is omitted, or if several agents react to the same external signal, the fitted branching ratio can absorb structure that is not literal contagion.

This is why I like the lab’s plainer language:

branching ratio controls how much event risk is recycled into the future

That is a model statement. It is useful without pretending to be causal by itself.

Receipts I Would Keep

For any production event stream where bursts matter, I would not stop at rate charts. I would log and monitor:

  1. event timestamps at the finest trustworthy resolution;
  2. explicit marks such as event type, venue, host, route, side, size, or actor;
  3. calendar-time seasonality so baseline movement does not masquerade as self-excitation;
  4. bin-count Fano factors at several window sizes;
  5. fitted excitation kernels with confidence intervals;
  6. time-rescaled residual diagnostics;
  7. out-of-sample burst prediction, not only in-sample likelihood.

Hawkes processes are useful because they move the hidden state into the open. The intensity is not a constant. It is a live object that history keeps pushing.

The average rate answers:

how busy was the clock?

The Hawkes question is sharper:

what did the last event do to the next minute?

  1. Alan G. Hawkes, “Spectra of Some Self-Exciting and Mutually Exciting Point Processes”, Biometrika, 1971. See also “Point Spectra of Some Mutually Exciting Point Processes”, Journal of the Royal Statistical Society: Series B, 1971. ↩

  2. Yosihiko Ogata, “On Lewis’ Simulation Method for Point Processes”, IEEE Transactions on Information Theory, 1981. ↩

  3. Emmanuel Bacry, Iacopo Mastromatteo, and Jean-Francois Muzy, “Hawkes Processes in Finance”, Market Microstructure and Liquidity, 2015. ↩

  4. Yosihiko Ogata, “Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes”, Journal of the American Statistical Association, 1988. ↩