Simulate a spatiotemporal Hawkes process (immigrant-offspring branching)

Exact simulation via the branching construction (Hawkes & Oakes 1974): immigrants are a homogeneous Poisson process of rate \(\mu\) over the region and horizon; each event spawns \(\mathrm{Poisson}(\alpha)\) offspring at exponential-\(\beta\) time lags and Gaussian-\(\sigma\) spatial offsets, recursively. Requires \(\alpha < 1\).

Usage

morie_hawkes_st_simulate(
  params,
  end_time,
  region,
  seed = NULL,
  max_events = 100000L
)

Arguments

params: List list(mu, alpha, beta, sigma) with alpha < 1.
end_time: Horizon \(T\).
region: Numeric c(xmin, xmax, ymin, ymax).
seed: Optional integer seed.
max_events: Safety cap on total events (default 1e5).

Value

A data frame with columns t, x, y, gen (generation: 0 = immigrant), sorted by t.

Examples

morie_hawkes_st_simulate(list(mu = 0.1, alpha = 0.4, beta = 1, sigma = 0.5),
                         end_time = 10, region = c(0, 5, 0, 5), seed = 42)
#>            t           x            y gen
#> 1  0.8243756  3.74397693  1.168516993   0
#> 2  1.1748736  4.85483305  0.007852771   0
#> 3  1.3466660  4.53300704  1.202723698   0
#> 4  1.3871017  1.99242706  3.228159392   0
#> 5  2.5542882  2.15875624  3.596779188   0
#> 6  2.8613953  3.42584865  4.248448593   0
#> 7  3.0232615  3.71634690  4.157070240   1
#> 8  3.7551686  3.77592737  3.965903376   2
#> 9  3.8810828  2.83244212  4.628223743   0
#> 10 3.9020347  0.85632165  0.428060325   0
#> 11 4.1281555  2.28771696  4.292334417   1
#> 12 4.4696963  2.57206467  3.337132573   0
#> 13 4.5774178  2.17885792  1.081927075   0
#> 14 4.6229282  4.78788298  0.039423694   0
#> 15 4.7499708  3.09419104  2.908020013   0
#> 16 5.1421178  3.38638415  0.449902582   0
#> 17 5.1909595  0.03667073  4.140792426   0
#> 18 5.6033275  1.66713606  0.789526041   0
#> 19 5.6399547  0.42076010  4.607465590   1
#> 20 5.6496289  1.16232010 -0.301546675   1
#> 21 5.7239866  0.40821383  4.351140462   2
#> 22 5.9515608  3.03374277  2.477623719   1
#> 23 6.2377884  1.06997161  0.223656700   1
#> 24 6.4174552  4.16458040  1.356433074   0
#> 25 6.5699229  3.05889322  0.214943980   0
#> 26 6.9866012  4.83210310 -0.515970746   1
#> 27 7.0506478  1.89779620  0.702395471   0
#> 28 7.1911225  0.18715516  2.396992821   0
#> 29 7.3443370  0.23942207 -0.523155833   2
#> 30 7.3658831  1.03829486  3.466024102   0
#> 31 7.3759562  4.91408599  1.042849785   0
#> 32 7.5930237 -0.25573298  3.093051009   1
#> 33 7.6094189  0.41903896  2.807879376   1
#> 34 7.9091706  0.47307532  4.158824876   2
#> 35 8.1105514  3.79772134  4.665170637   0
#> 36 8.2138803  3.68915141  4.705161913   1
#> 37 8.3044763  0.01974169  0.947369677   0
#> 38 8.3600426  3.37803637  0.001194483   0
#> 39 9.0403139  1.73374124  1.795141529   0
#> 40 9.0573813  1.30543982  1.526091847   0
#> 41 9.3232019  0.81786304  0.292284810   2
#> 42 9.3467225  4.86769957  0.987051711   0
#> 43 9.4001452  4.43877453  1.877449823   0
#> 44 9.4666823  0.19468246  2.818234208   0
#> 45 9.7822643  3.19989385  2.572038542   0
#> 46 9.8889173  3.92346388  3.879116813   0
#> 47 9.9895303  4.31780912  0.748964750   1