Non-stationary Hawkes with non-exponential kernels (R port) — morie_tps_hawkes

R port of morie.tps_hawkes_advanced. Implements the Kwan, Chen and Dunsmuir (2024, arXiv:2408.09710v1) methodology for Hawkes process likelihood inference when the baseline intensity is time-varying and the excitation kernel is non-exponential (so the intensity process is non-Markovian).

Details

The complete intensity is $$\lambda(t) = \u(t) + \int_0^{t-} g(t - s) \, dN_s,$$ with kernel decomposition $g(u) = \eta \cdot \tilde g(u)$ where $\eta \in (0, 1)$ is the branching ratio (mean offspring per event) and $\tilde g$ is a probability density on $[0, \infty)$. Stationarity requires $\eta < 1$.

Supported kernels: exponential, gamma, Weibull, Lomax (Pareto-II). Supported baselines: constant and sinusoidal-with-trend $$\u(t) = \exp\bigl(a_0 + a_1 (t/T) + a_2 \sin(2\pi t / 365.25) + a_3 \cos(2\pi t / 365.25)\bigr).$$

Companion to morie_tps_hawkes_temporal_fit (exponential / constant Markovian special case) in morie.tps_stochastic.

Goodness-of-fit uses time-rescaling residuals (Brown et al. 2002 Neural Comput. 14: 325-346) and a Kolmogorov-Smirnov test against Uniform(0,1).

If the optional R package hawkes or emhawkes is installed it is consulted for the exponential-kernel constant- baseline fast path; otherwise the negative log-likelihood is computed in base R via direct O(n^2) summation. The non-Markovian kernels (gamma, Weibull, Lomax) always use the base-R path – those kernels lack the memorylessness required for O(n) recursion.

Functions

morie_tps_hawkes_advanced_fit – fit one (kernel, baseline) combination and produce a rich result with time-rescaling KS diagnostics.
morie_tps_compare_hawkes_kernels – 8-way AIC comparison across (kernel, baseline) combinations.
morie_tps_hawkes_markovian_vs_nonmarkovian – focused 2x2 comparison: classical exp/const vs gamma/ sinusoidal.

References

Kwan TKJ, Chen F, Dunsmuir WTM (2024). Likelihood inference for non-stationary Hawkes processes. arXiv:2408.09710v1.

Brown EN, Barbieri R, Ventura V, Kass RE, Frank LM (2002). The time-rescaling theorem and its application to neural spike train data analysis. Neural Computation 14: 325-346.

Mohler GO, Short MB, Brantingham PJ, Schoenberg FP, Tita GE (2011). Self-exciting point process modeling of crime. Journal of the American Statistical Association 106: 100-108.