| Scenario | μ | α | β | T | Event Times | Interpretation |
|---|---|---|---|---|---|---|
| Moderate clustering | 0.35 | 0.80 | 1.40 | 12 | 0.4, 1.1, 1.8, 2.2, 4.9, 5.1, 5.8, 8.3 | Bursts appear, but decay eventually dominates. |
| Low contagion | 0.50 | 0.25 | 1.80 | 10 | 0.7, 2.6, 3.9, 6.5, 8.1 | Events behave closer to a background process. |
| Near-critical clustering | 0.20 | 0.95 | 1.00 | 15 | 0.3, 0.9, 1.4, 3.2, 3.5, 7.8, 8.0, 11.1 | Strong persistence with slow stabilization. |
λ(t) = μ + Σ α e-β(t - tᵢ)Each past event
tᵢ raises the current rate, then its effect decays exponentially.
n = α / βThis summarizes the average self-excitation strength. Values below one are typically stable.
ℓ = Σ log λ(tᵢ) − μT − Σ (α/β)(1 − e−β(T − tᵢ))This is useful when comparing parameter choices for the same event history.
n / (1 − n) for n < 1This measures how many additional events one event is expected to trigger on average.
ln(2) / βThis is the time required for an event’s excitation effect to drop by half.
- Enter the background rate μ, excitation size α, and decay rate β.
- Set the observation horizon T so it covers your final observed event.
- Paste event times into the event field using commas, spaces, or new lines.
- Choose an evaluation time to inspect the current conditional intensity.
- Enter a forecast horizon if you want an expected event count over a future interval.
- Submit the form to generate summary metrics, a per-event table, and a Plotly graph.
- Use the CSV button for structured export and the PDF button for a shareable report snapshot.
1) What is a self-exciting process?
It is a point-process model where each event temporarily increases the probability of future events. That makes it useful for clustered arrivals, bursts, cascades, and contagion-like timing patterns.
2) What does the branching ratio tell me?
The branching ratio α/β measures average triggering strength. Values below one usually indicate stable behavior. Values near one imply persistent clustering. Values above one suggest explosive or nonstationary dynamics.
3) Why does the calculator need event times instead of counts?
This model is time-sensitive. Exact event timings determine how much excitation remains at each moment, which directly affects intensity, likelihood, and the shape of the resulting process curve.
4) What does current intensity mean?
Current intensity is the instantaneous event rate at the evaluation time. It combines the background rate with all surviving excitation from earlier events, so it changes as events occur and decay.
5) How should I interpret log-likelihood?
Log-likelihood is mainly a comparison metric. Larger values usually indicate a better fit for the same observed event sequence. It is most useful when comparing different parameter settings or estimation results.
6) Why are some outputs unavailable in unstable cases?
When α/β is one or greater, long-run mean quantities may not exist or may not be reliable. The calculator still shows immediate metrics, but stationary summaries are intentionally withheld.
7) What units should I use?
Use one consistent time unit throughout the model. If event times are in minutes, then μ, α, and β must also be interpreted relative to minutes. Consistency matters more than the unit itself.
8) Can I use this for alerts, trades, earthquakes, or clicks?
Yes. Any domain with clustered event arrivals can be analyzed, provided event times are meaningful and the exponential-kernel Hawkes structure is a reasonable approximation for triggering behavior.