An approximately unbiased estimator for the space-time pair correlation function, based on data giving the locations of events \(x_i: i=1,...n\) on a spatio-temporal region \(S \times T\), where \(S\) is an arbitrary polygon and \(T\) a time interval:
$$\widehat{g}(u,v)=\frac{1}{4\pi u}\sum_{i=1}^{n}\sum_{j \neq i} \frac{1}{w_{ij}}\frac{k_{s}(u-\|s_i-s_j\|)k_{t}(v-|t_i-t_j|)}{\lambda(x_i) \lambda(x_j)},$$
where \(\lambda(x_i)\) is the intensity at \(x_i = (s_i,t_i)\) and \(w_{ij}\) is an edge correction factor to deal with spatial-temporal edge effects. The edge correction methods implemented are:
isotropic: \(w_{ij} = |S \times T| w_{ij}^{(t)} w_{ij}^{(s)}\), where the temporal edge correction factor \(w_{ij}^{(t)} = 1\) if both ends of the interval of length \(2 |t_i - t_j|\) centred at \(t_i\) lie within \(T\) and \(w_{ij}^{(t)}=1/2\) otherwise and \(w_{ij}^{(s)}\) is the proportion of the circumference of a circle centred at the location \(s_i\) with radius \(\|s_i -s_j\|\) lying in \(S\) (also called Ripley's edge correction factor).
border: \(w_{ij}=\frac{\sum_{j=1}^{n}\mathbf{1}\lbrace d(s_j,S)>u \ ; \ d(t_j,T) >v\rbrace/\lambda(x_j)}{\mathbf{1}_{\lbrace d(s_i,S) > u \ ; \ d(t_i,T) >v \rbrace}}\), where \(d(s_i,S)\) denotes the distance between \(s_i\) and the boundary of \(S\) and \(d(t_i,T)\) the distance between \(t_i\) and the boundary of \(T\).
modified.border: \(w_{ij} = \frac{|S_{\ominus u}|\times|T_{\ominus v}|}{\mathbf{1}_{\lbrace d(s_i,S) > u \ ; \ d(t_i,T) >v \rbrace}}\), where \(S_{\ominus u}\) and \(T_{\ominus v}\) are the eroded spatial and temporal region respectively, obtained by trimming off a margin of width \(u\) and \(v\) from the border of the original region.
translate: \(w_{ij} =|S \cap S_{s_i-s_j}| \times |T \cap T_{t_i-t_j}|\), where \(S_{s_i-s_j}\) and \(T_{t_i-t_j}\)
are the translated spatial and temporal regions.
none: No edge correction is performed and \(w_{ij}=|S \times T|\).
\(k_s()\) and \(k_t()\) denotes kernel functions with bandwidth \(h_s\) and \(h_t\). Experience with pair correlation function estimation recommends box kernels (the default), see Illian et al. (2008). Epanechnikov, Gaussian and biweight kernels are also implemented. Whatever the kernel function, if the bandwidth is missing, a value is obtain from the function dpik of the package KernSmooth. Note that the bandwidths play an important role and their choice is crucial in the quality of the estimators as they heavily influence their variance.