localdiag {stopp}

Local diagnostics of spatio-temporal point process models

Description

This function performs local diagnostics of a model fitted for the first-order intensity of a spatio-temporal point pattern, returning the points identified as outlying following the diagnostics procedure on individual points of an observed point pattern, as introduced in Adelfio et al. (2020), and applied in D'Angelo et al. (2022) for the linear network case.

The points resulting from the local diagnostic procedure provided by this function can be inspected via the plot, print, summary, and infl functions.

Usage

localdiag(x, intensity, p = 0.95)

Arguments

x

Either a stp or a stlp object

intensity

A vector of intensity values, of the same length as the number of point in x

p

The percentile to consider as threshold for the outlying points. Default to 0.95.

Details

This function performs local diagnostics of a model fitted for the first-order intensity of a spatio-temporal point pattern, by means of the local spatio-temporal inhomogeneous K-function (Adelfio et al, 2020) documented by the function KLISTAhat of the stpp package (Gabriel et al, 2013).

The function can also perform local diagnostics of a model fitted for the first-order intensity of an spatio-temporal point pattern on a linear network, by means of the local spatio-temporal inhomogeneous K-function on linear networks (D'Angelo et al, 2021) documented by the function localSTLKinhom.

In both cases, it returns the points identified as outlying following the diagnostics procedure on individual points of an observed point pattern, as introduced in Adelfio et al. (2020), and applied in D'Angelo et al. (2022) for the linear network case.

This function computes discrepancies by means of the \chi_i^2 values, obtained following the expression

\chi_i^2=\int_L \int_T \Bigg( \frac{\big(\hat{K}^i_{I}(r,h)- \mathbb{E}[\hat{K}^i(r,h) ] \big)^2}{\mathbb{E}[\hat{K}^i(r,h) ]} \Bigg) \text{d}h \text{d}r ,

one for each point in the point pattern.

Note that the Euclidean procedure is implemented by the local K-functions of Adelfio et al. (2020), documented in KLISTAhat of the stpp package (Gabriel et al, 2013). The network case uses the local K-functions on networks (D'Angelo et al., 2021), documented in localSTLKinhom.

Value

A list object of class localdiag, containing

x

The stp object provided as input

listas

The LISTA functions, in a list object

ids

The ids of the points identified as outlying

x2

A vector with the individual contributions to the Chi-squared statistics, normalized

p

The percentile considered

Author(s)

Nicoletta D'Angelo and Giada Adelfio

References

Adelfio, G., Siino, M., Mateu, J., and Rodríguez-Cortés, F. J. (2020). Some properties of local weighted second-order statistics for spatio-temporal point processes. Stochastic Environmental Research and Risk Assessment, 34(1), 149-168.

D’Angelo, N., Adelfio, G. and Mateu, J. (2022) Local inhomogeneous second-order characteristics for spatio-temporal point processes on linear networks. Stat Papers. https://doi.org/10.1007/s00362-022-01338-4

Gabriel, E., Rowlingson, B. S., and Diggle, P. J. (2013). stpp: An R Package for Plotting, Simulating and Analyzing Spatio-Temporal Point Patterns. Journal of Statistical Software, 53(2), 1–29. https://doi.org/10.18637/jss.v053.i02

See Also

infl, plot.localdiag, print.localdiag, summary.localdiag, globaldiag

Examples

set.seed(2)
inh <- rstpp(lambda = function(x, y, t, a) {exp(a[1] + a[2]*x)}, 
             par = c(.3, 6))

mod1 <- stppm(inh, formula = ~ 1)

resmod1 <- localdiag(inh, mod1$l, p = .9)