Branching-ratio arithmetic of a self-exciting (Hawkes) crime process

Every recorded event triggers on average n further events (the branching ratio, the integral of the triggering kernel). The offspring of one background event form a Galton-Watson tree with mean offspring n: generation k has expected size \(n^k\) (Research.P10.generation_mean); for \(n < 1\) the expected cluster size is \(1/(1-n)\) and the stationary mean rate is \(\mu/(1-n)\) for background rate \(\mu\) (cluster_size_of_lt_one, stationary_rate); the share of events that are triggered rather than background is exactly n (endogeneity_share). For \(n \ge 1\) the expected cluster size is unbounded (cluster_size_diverges_of_ge_one): a fitted branching ratio at or above one is a model that predicts unbounded crime, not a forecast, and is the first thing to check in any near-repeat fit.

Usage

morie_contagion_branching(n, mu = 1, generations = 10L)

Arguments

n: Branching ratio, non-negative.
mu: Background rate (events per unit time), positive.
generations: How many generation means to report.

Value

A list with n, subcritical, generation_means, expected_cluster_size (Inf when \(n \ge 1\)), stationary_rate, endogeneity_share and theorems.

Examples

morie_contagion_branching(n = 0.4, mu = 2)
#> $n
#> [1] 0.4
#> 
#> $subcritical
#> [1] TRUE
#> 
#> $generation_means
#>  [1] 1.000000000 0.400000000 0.160000000 0.064000000 0.025600000 0.010240000
#>  [7] 0.004096000 0.001638400 0.000655360 0.000262144
#> 
#> $expected_cluster_size
#> [1] 1.666667
#> 
#> $stationary_rate
#> [1] 3.333333
#> 
#> $endogeneity_share
#> [1] 0.4
#> 
#> $theorems
#> [1] "Research.P10.generation_mean"                
#> [2] "Research.P10.cluster_size_of_lt_one"         
#> [3] "Research.P10.stationary_rate"                
#> [4] "Research.P10.cluster_size_diverges_of_ge_one"
#> [5] "Research.P10.endogeneity_share"              
#> 
morie_contagion_branching(n = 1.1, mu = 2)$expected_cluster_size   # Inf
#> [1] Inf