Space-time Hawkes process

Fits the self-exciting point process of crime in space and time of Mohler et al. (2011): $$\lambda(t, x, y) = \mu + \eta \sum_{t_j < t} \beta e^{-\beta (t - t_j)} \phi_\sigma(x - x_j, y - y_j),$$ a constant background \(\mu\) per unit area and time over the window and, after each event, eta expected offspring at exponential lags (rate beta) and normal offsets (standard deviation sigma). The log-likelihood takes the spatial kernel's mass as 1 inside the window (events near its edge lose a little offspring mass outside); it is maximised with its analytic gradient by BFGS over the logarithms of mu, beta, sigma and the logit of eta, from two starts or start. With par nothing is fitted: the log-likelihood is evaluated there. at gives the conditional intensity at query points, from the events strictly before each. hwksim() simulates from a fit.

Usage

sthwk(
  times,
  x,
  y,
  hor,
  win = NULL,
  par = NULL,
  at = NULL,
  start = NULL,
  maxit = 500
)

Arguments

times

Event times: numbers sorted in [0, hor], at least 10.

x, y

The places of the events: finite numbers in planar units.

hor

The end of the window in time: one positive number.

win

NULL (the events' bounding box) or the rectangle c(xmin, xmax, ymin, ymax) holding every event.

par

NULL to fit, or the parameters mu, eta, beta, sigma at which to evaluate.

at

NULL or query points: a data frame or list with t, x and y.

start

NULL or the starting parameters, as par.

maxit

The most BFGS steps per start: one whole number from 1.

Value

An object of class sthwk, a list: par, se and cov (mu, eta, beta, sigma); loglik, aic; conv, iter; fit (FALSE when par was given); n, hor, win, area; and lam, the intensities at at (NULL without it).

References

Mohler, G. O., Short, M. B., Brantingham, P. J., Schoenberg, F. P. and Tita, G. E. (2011). Self-exciting point process modeling of crime. Journal of the American Statistical Association, 106(493), 100-108. doi:10.1198/jasa.2011.ap09546

Reinhart, A. (2018). A review of self-exciting spatio-temporal point processes and their applications. Statistical Science, 33(3), 299-318. doi:10.1214/17-STS629

Examples

s <- hwksim(c(log(0.02), 0.5, 1), 100, win = c(0, 10, 0, 10),
            sigma = 0.3, seed = 3)
f <- sthwk(s$t, s$x, s$y, 100, win = c(0, 10, 0, 10))
f
#> sthwk: 343 events in [0, 100] over an area of 100
#>              est          se
#> mu    0.01908445 0.001555915
#> eta   0.45225241 0.042523094
#> beta  0.95205502 0.114791215
#> sigma 0.29276704 0.021369007
#> log-likelihood -1288.096
sthwk(s$t, s$x, s$y, 100, win = c(0, 10, 0, 10), par = f$par,
      at = data.frame(t = 50, x = 5, y = 5))$lam
#> [1] 0.01938062