locstppm: Fit a local Poisson process model to a spatio-temporal point pattern

Description

This function fits a Poisson process model to an observed spatio-temporal point pattern stored in a stp object, that is, a Poisson model with a set of parameters \(\theta_i\) for each point \(i\).

Usage

locstppm(
  X,
  formula,
  verbose = TRUE,
  mult = 4,
  seed = NULL,
  hs = c("global", "local"),
  npx0 = 10,
  npt0 = 10
)

Value

An object of class locstppm. A list of

IntCoefs: The fitted global coefficients
IntCoefs_local: The fitted local coefficients
X: The stp object provided as input
nX: The number of points in X
I: Vector indicating which points are dummy or data
y_resp: The response variable of the model fitted to the quadrature scheme
formula: The formula provided as input
l: Fitted intensity through the global parameters
l_local: Fitted intensity through the local parameters
mod_global: The glm object of the model fitted to the quadrature scheme
newdata: The data used to fit the model, without the dummy points
time: Time elapsed to fit the model, in minutes

Arguments

X: A stp object
formula: An object of class "formula": a symbolic description of the model to be fitted. The current version only supports formulas depending on the spatial and temporal coordinates: x, y, t.
verbose: Default to TRUE
mult: The multiplicand of the number of data points, for setting the number of dummy points to generate for the quadrature scheme
seed: The seed used for the simulation of the dummy points. Default to NULL.
hs: Character string indicating whether to select fixed or variable bandwidths for the kernel weights to be used in the log-likelihood. In any of those cases, the well-supported rule-of-thumb for choosing the bandwidth of a Gaussian kernel density estimator is employed. If hs = "global" (default), a fixed bandwidth is selected. If hs = "local", an individual bandwidth is selected for each point in the pattern X.
npx0: Number of lags for the space grid period for variable bandwidths kernel
npt0: Number of lags for the time period for variable bandwidths kernel

Author

Nicoletta D'Angelo

Details

We assume that the template model is a Poisson process, with a parametric intensity or rate function \(\lambda(\textbf{u}, t; \theta_i)\) with space and time locations \(\textbf{u} \in W, t \in T\) and parameters \(\theta_i \in \Theta.\)

Estimation is performed through the fitting of a glm using a localized version of the quadrature scheme by Berman and Turner (1992), firstly introduced in the purely spatial context by Baddeley (2017), and in the spatio-temporal framework by D'Angelo et al. (2023).

References

Baddeley, A. (2017). Local composite likelihood for spatial point processes. Spatial Statistics, 22, 261-295.

D'Angelo, N., Adelfio, G., and Mateu, J. (2023). Locally weighted minimum contrast estimation for spatio-temporal log-Gaussian Cox processes. Computational Statistics & Data Analysis, 180, 107679.

Examples

Run this code


set.seed(2)
inh <- rstpp(lambda = function(x, y, t, a) {exp(a[1] + a[2]*x)}, 
             par = c(0.005, 5))
inh_local <- locstppm(inh, formula = ~ x)

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