locstppm {stopp}

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
)

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

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).

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

Author(s)

Nicoletta D'Angelo

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.

See Also

stppm

Examples

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)