Gabriel (2014) proposes the following unbiased estimator for the STIK-function, based on data giving the locations of events \(x_i: i=1,\ldots,n\) on a spatio-temporal region \(S\times T\), where \(S\) is an arbitrary polygon and \(T\) is a time interval:
$$\widehat{K}(u,v)=\sum_{i=1}^{n}\sum_{j\neq i}\frac{1}{w_{ij}}\frac{1}{\lambda(x_i)\lambda(x_j)}\mathbf{1}_{\lbrace \|s_i - s_j\| \leq u \ ; \ |t_i - t_j| \leq v \rbrace},$$
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 \(w_{ij}=|S \times T|\).
If parameter infectious = TRUE, ony future events are considered and the estimator is, using an isotropic edge correction factor (Gabriel and Diggle, 2009):
$$\widehat{K}(u,v)=\frac{1}{|S\times T|}\frac{n}{n_v}\sum_{i=1}^{n_v}\sum_{j=1; j > i}^{n_v} \frac{1}{w_{ij}} \frac{1}{\lambda(x_i) \lambda(x_j)}\mathbf{1}_{\left\lbrace u_{ij} \leq u\right\rbrace}\mathbf{1}_{\left\lbrace t_j - t_i \leq v \right\rbrace}.$$
In this equation, the points \(x_i=(s_i, t_i)\) are ordered so that \(t_i < t_{i+1}\), with ties due to round-off error broken by randomly unrounding if necessary. To deal with temporal edge-effects, for each \(v\), \(n_v\) denotes the number of events for which \(t_i \leq T_1 -v\), with \(T=[T_0,T_1]\). To deal with spatial edge-effects, we use Ripley's method.
If lambda is missing in argument, STIKhat computes an estimate of the space-time (homogeneous)
K-function: $$\widehat{K}(u,v)=\frac{|S\times T|}{n_v(n-1)} \sum_{i=1}^{n_v}\sum_{j=1;j>i}^{n_v}\frac{1}{w_{ij}}\mathbf{1}_{\lbrace u_{ij}\leq u \rbrace}\mathbf{1}_{\lbrace t_j - t_i \leq v \rbrace}$$