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stelfi

Introduction

stelfi fits Hawkes and Log-Gaussian Cox Point Process models using Template Model Builder. See the package’s website for more details.

Introduced in Hawkes (1971) a Hawkes process is a self-exciting temporal point process where the occurrence of an event immediately increases the chance of another. We extend this to consider self-inhibiting process and a non-homogeneous background rate. A log-Gaussian Cox process is a Poisson point process where the log-intensity is given by a Gaussian random field. We extend this to a joint likelihood formulation fitting a marked log-Gaussian Cox model.

In addition, the package offers functionality to fit self-exciting spatiotemporal point processes. Models are fitted via maximum likelihood using TMB (Template Model Builder) (Kristensen, Nielsen, Berg, Skaug, and Bell, 2016). Where included 1) random fields are assumed to be Gaussian and are integrated over using the Laplace approximation and 2) a stochastic partial differential equation model, introduced by Lindgren, Rue, and Lindström. (2011), is defined for the field(s).

Installation

From CRAN install.packages("stelfi") or development version from GitHub:

require(devtools)
devtools::install_github("cmjt/stelfi")

Model fitting functions and examples

Key arguments for each each model fitting function

Function Key arguments
fit_hawkes()
  • times - a vector of numeric occurrence times.

  • parameters - a vector of named starting values for \(\mu\) (mu), \(\alpha\) (alpha), and \(\beta\) (beta).

  • marks - optional, a vector of marks (\(m(t)\)).

fit_mhawkes()
  • times - a vector of numeric occurrence times.

  • stream - character vector specifying the stream ID of each observation in times.

  • parameters - a vector of named starting values for \(\mu\) (mu), \(\alpha\) (alpha), and \(\beta\) (beta).

fit_hawkes_cbf()

As fit_hawkes() plus

  • background - some assumed time dependent background function \(\mu(t)\).

  • background_integral - the integral of background.

  • background_parameters - parameter starting values for \(\mu(t)\).

( \(^*\)Note, \(\texttt{mu}\) in parameters will be ignored)

fit_lgcp()
  • locs - a named data frame of event locations, x, y, and t (optional).

  • sf - a polygon of the spatial domain.

  • smesh - a Delaunay triangulation of the spatial domain returned by INLA::inla.mesh.2d().

  • tmesh - optional, a temporal mesh returned by INLA::inla.mesh.1d()).

  • parameters - a vector of named starting values for \(\boldsymbol{\beta}\) (beta), \(\text{log}(\tau)\) (log_tau), \(\text{log}(\kappa)\) (log_kappa), and \(\textrm{arctan}(\rho)\) (atanh_rho, optional).

fit_mlgcp()
  • locs, sf, and smesh - as fit_lgcp().

  • marks - a matrix of marks for each observation of the point pattern.

  • parameters - a list of named parameters, as fit_lgcp() plus (betamarks), (betapp), (marks_coefs_pp ).

  • methods - integer(s) specifying mark distribution: 0, Gaussian; 1, Poisson; 2, binomial; 3, gamma.

  • strfixed - fixed structural parameters, depends on mark distribution.

  • fields - a binary vector indicating whether there is a new random field for each mark.

fit_stelfi()
  • times - as fit_hawkes().

  • locs, sf, and smesh - as fit_lgcp().

  • parameters - a list of named parameter starting values for \(\mu\) (mu), \(\alpha\) (alpha), \(\beta\) (beta), \(\sigma_x\) (xsigma) \(\sigma_y\) (ysigma), and \(\rho\) (rho).

  • GMRF - logical, should a GMRF be included as a latent spatial effect if so \(\tau\) (tau) and \(\kappa\)(kappa) supplied to parameters.

Other useful funcions

Function Key arguments Purpose
get_coefs()
  • obj - a fitted model object returned by any one of the functions in the Table above
Extract estimated parameter values from a fitted model.
get_fields()

As fit_lgcp() and

  • sd - logical, return standard deviation.
Extract estimated mean, or standard deviation, of GMRF(s).
get_weights()
  • mesh - a Delaunay triangulation of the spatial domain returned by INLA::inla.mesh. 2d().

  • sf - a polygon of the spatial domain.

Calculate mesh weights.
mesh_2_sf()
  • mesh - a Delaunay triangulation of the spatial domain returned by INLA::inla.mesh. 2d().
Transforms mesh into a sf object.
show_field()
  • x - a vector of values, one per each smesh node.

  • smesh - as fit_lgcp() .

  • sf - as fit_lgcp().

  • clip - logical, clip to domain

Plots spatial random field values.
show_hawkes()
  • obj - a fitted model object returned by fit_hawkes() or fit_hawkes_cbf().
Plot fitted Hawkes model.
show_hawkes_GOF()
  • obj - as show_hawkes().

  • plot - logical

  • return_values - logical, return compensator values

Plot goodness-of-fit metrics for a Hawkes model.
show_lambda()

As fit_lgcp() and

  • clip - logical, clip to domain
Plot estimated spatial intensity from a fitted log-Gaussian Cox process model.
sim_hawkes() As fit_hawkes() Simulate a Hawkes process.
sim_lgcp() As fit_lgcp() Simulate a realisation of a log-Gaussian Cox process.

References

Hawkes, AG. (1971) Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58: 83–90.

Lindgren, F., Rue, H., and Lindström, J. (2011) An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73: 423–498.

Kristensen, K., Nielsen, A., Berg, C. W., Skaug, H., and Bell B. M. (2016). TMB: Automatic Differentiation and Laplace Approximation. Journal of Statistical Software, 70: 1–21.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.