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HawkesIntensity

Incremental, causal technical analysis documentation

Summary

HawkesIntensity is RTTA's streaming exponential Hawkes process intensity for event times: a constant baseline plus a self-exciting residual that decays exponentially between events and jumps when new events arrive.

Update API

result = rtta.HawkesIntensity(
    mu=1.0, alpha=0.5, beta=1.0, fillna=True
).update(time, jump=1.0)
# result.intensity, result.excitation, result.baseline

time is a real-valued event clock (seconds, exchange timestamps, or any monotonic scale consistent with beta). jump is the mark size of the event (default 1.0). batch(time, jump) or batch(time) (unit jumps) return multi-output arrays.

Theory Of Operation

A univariate exponential Hawkes process has conditional intensity

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

with baseline \(\mu\), excitation amplitude \(\alpha\), and decay \(\beta\). RTTA maintains the residual excitation \(A_t\) in a recursive form so each update is \(O(1)\): decay the previous excitation by \(e^{-\beta\Delta t}\), then add \(\alpha\cdot\operatorname{jump}\). Clustering of events raises intensity above baseline; quiet periods pull it back toward \(\mu\).

Recurrence

State is residual excitation \(A\) and last event time \(t_{\mathrm{last}}\). On the first event at time \(t\) with jump \(j\):

\[A \leftarrow \alpha j, \qquad \lambda = \mu + A\]

On a later event at time \(t\) with jump \(j\) and \(\Delta t = t - t_{\mathrm{last}}\):

\[A \leftarrow A\, e^{-\beta\,\lvert\Delta t\rvert} + \alpha j\]
\[\lambda_t = \mu + A, \qquad \operatorname{excitation}_t = A, \qquad \operatorname{baseline}_t = \mu\]

(If time goes backward, RTTA still decays with \(\lvert\Delta t\rvert\) and applies the jump so the stream remains defined.)

Implementation Notes

The recurrence is implemented in src/rtta/indicator.cpp in class HawkesIntensity. Parameters are floored so \(\mu,\alpha\ge 0\) and \(\beta > 0\).

Reference