Calculator Inputs
Formula Used
This tool uses an exponential‑kernel Hawkes process, a self‑exciting point process often used to model clustered events in time.
- λ(t) = μ + Σtᵢ<t α · exp(−β (t − tᵢ))
- Λ(t) = μ t + Σtᵢ<t (α/β) · (1 − exp(−β (t − tᵢ)))
- n = α/β is the branching ratio for the exponential kernel.
Interpretation: μ sets background activity, α sets event impact, and β controls how fast that impact fades.
How to Use This Calculator
- Enter event times using a consistent time unit.
- Choose μ, α, and β based on your scenario.
- Set the evaluation time t for λ(t) and Λ(t).
- Optionally enable the grid for a time‑series table.
- Click Calculate to view results above the form.
- Use CSV or PDF buttons to export your output.
Example Data Table
Sample inputs and representative outputs for reference.
| μ | α | β | t | Event times | λ(t) | Λ(t) | n |
|---|---|---|---|---|---|---|---|
| 0.20 | 0.60 | 1.50 | 5.00 | 0.4, 1.2, 2.0, 3.7, 4.1 | ≈ 0.6656 | ≈ 1.5330 | 0.4000 |
| 0.10 | 0.80 | 2.00 | 3.00 | 0.5, 1.1, 2.4 | ≈ 0.3467 | ≈ 0.7176 | 0.4000 |
Hawkes Intensity Guide
1) What the intensity represents
A Hawkes process models events that can trigger more events. The instantaneous intensity λ(t) is the event rate at time t, conditional on the history. If λ(t)=0.8 events per second, the model predicts about 0.8 events on average in the next second, assuming the recent history remains the same. This calculator reports both λ(t) and the cumulative intensity Λ(t), which summarizes expected counts up to time t.
2) Exponential kernel and parameter meaning
The tool uses an exponential triggering kernel g(s)=α·exp(−βs) for s>0. The baseline μ sets background activity, α sets how much each event increases the rate, and β controls decay speed. Larger β means the excitement fades quickly. Smaller β spreads influence over a longer window, which often creates smoother, longer clusters.
3) Branching ratio and stability data
For the exponential kernel, the branching ratio is n=α/β. It estimates the expected number of direct offspring events caused by one event. When n<1, the process has a stable long‑run mean intensity E[λ]=μ/(1−n). When n≥1, the model can become explosive in theory. This calculator flags the stability regime and displays n to help you select realistic parameters.
4) Reading the contribution breakdown
The contribution table shows how each past event time tᵢ adds α·exp(−β(t−tᵢ)) to the current intensity. The newest events typically dominate because their lag Δt=t−tᵢ is small. If you see one or two rows contributing most of the total, your dynamics are strongly driven by recent shocks rather than background activity.
5) Time grid table for curves
The optional time grid generates λ(t) and Λ(t) across a window. As a practical rule, choose a step smaller than the characteristic decay time 1/β. For example, if β=2, then 1/β=0.5; using Δt=0.05 to 0.1 captures the curvature well while keeping the table manageable.
6) Example numbers you can reproduce
Using sample inputs μ=0.20, α=0.60, β=1.50, events [0.4,1.2,2.0,3.7,4.1], and t=5, the calculator reports an intensity around λ(t)≈0.6656 and cumulative intensity Λ(t)≈1.5330. The branching ratio is n=0.4, which indicates a stable regime and a long‑run mean intensity of about μ/(1−n)=0.3333 events per unit time.
7) Data hygiene for event histories
Hawkes models assume event times are measured precisely and consistently. Remove duplicates created by logging artifacts, and ensure all times share one unit. If you are merging streams from multiple sources, align clocks before modeling. For repeated bursts, consider whether a single burst should be decomposed into smaller events or treated as one aggregated event.
8) Exporting results for analysis
CSV export includes inputs, key results, the contribution table, and the full grid table. PDF export summarizes the same information and caps the grid to 120 rows for readability. These exports are useful for reports, parameter sweeps, or plotting λ(t) curves in your preferred analysis environment.
FAQs
1) What does λ(t) mean in practice?
It is the model’s instantaneous event rate at time t, given past events. Multiply λ(t) by a short time interval to estimate expected event count in that interval.
2) Why does the calculator compute Λ(t) too?
Λ(t) is the cumulative intensity, which approximates the expected number of events from time 0 to t under the model. It is helpful for likelihood work and sanity checks.
3) How should I choose the decay β?
β sets how fast influence fades. A practical guide is to estimate a decay time scale from data and set β≈1/(typical influence duration). Larger β means shorter memory.
4) What is the branching ratio n=α/β?
It measures expected direct offspring events per event for the exponential kernel. Values below 1 indicate a stable regime; values at or above 1 suggest potentially unstable dynamics.
5) Do events after t affect λ(t)?
No. By definition, the intensity at time t depends only on events strictly before t. The calculator filters events using tᵢ<t when computing λ(t) and Λ(t).
6) Why does the grid have a row limit warning?
Very small steps over long ranges can create thousands of rows, which may slow browsers. The tool limits the grid size to keep calculations responsive and exports manageable.
7) Can I use this for parameter fitting?
This page computes intensity and summaries for chosen parameters. For fitting, you would typically optimize μ, α, β using a likelihood objective, then validate using residuals and forecasts.