The SPP Procedure

Distance Functions for Multitype Point Patterns

Distance functions (such as G, J, K, L, and g) can also be defined for point patterns that are "marked" with a categorical mark variable, called a type. Usually you consider mark variables that have more than one type to define distance functions. When distance functions are defined between two types, they are called cross-type distance functions. For any pair of types i and j, the cross-distance functions $G_{ij}$, $J_{ij}$, $K_{ij}$, $L_{ij}$, and $g_{ij}$ can be defined analogously to the single-type distance functions. The interpretation of cross-type distance functions is slightly different from the interpretation of single type functions. Suppose that X is the point pattern, $X_ j$ refers to the subpattern of points of type j; $X_ i$ refers to the subpattern of points of type i, and $\lambda _ j$ represents the intensity of the subpattern $X_ j$. Then the interpretation is to treat $X_ j$ as a homogeneous Poisson process and independent of $X_ i$. If the computed empirical cross-type function is identical to the function that corresponds to a homogeneous Poisson process, then $X_ i$ and $X_ j$ can be treated as independent of each other.

The empirical cross-G-function, $G_{ij}$, is defined as the distribution of the distance from a point of type i in $X_ i$ to the nearest point of type j in $X_ j$. Formally, $G_{ij}$ can be written as

\[ G_{ij}(r) = \sum _ ie(x_ i,r)\Strong{1}\{ d_{ij}\leq r\} \]

where $e(x_ i,r)$ is an edge correction and $d_{ij}$ is the distance from a point of type i to the nearest point of type j. If the two subpatterns $X_ i$ and $X_ j$ are independent of each other, then the theoretical cross-G-function is

\[ G_{ij}^{*}(r)= 1-\exp {(\lambda _ j \pi r^{2})} \]

The empirical cross-type J-function, $J_{ij}$, can be defined again in terms of the $G_{ij}$ function and the empty-space F function for subpattern $X_ j$ as

\[ J_{ij}(r) = J_{ij}(r) = \frac{1-G_{ij}(r)}{1-F_ j(r)} \]

where $F_ j(r)$ is the empty-space function for the subpattern $X_ j$. If the two subpatterns $X_ i$ and $X_ j$ are independent of each other, then the theoretical cross-J-function is $J_{ij}^{*}(r) = 1$.

The empirical cross-type K function, $K_{ij}$, is $1/\lambda _ j$ times the expected number of points of type j within a distance r of a typical point of type i. Formally, $K_{ij}$ can be written as

\[ K_{ij}(r) = K_{ij}(r) = \frac{1}{\lambda _ j \lambda _ i |W|}\sum _ i \sum _ j \Strong{1}\{ ||x_ i-x_ j||\leq r\} e(x_ i,x_ j;r) \]

where $e(x_ i,x_ j;r)$ is an edge correction. If the two subpatterns $X_ i$ and $X_ j$ are independent of each other, then the theoretical cross-K-function is $K_{ij}^{*}(r) = \pi r^{2}$.

The empirical cross-type L function, $L_{ij}$, is a transformation of $K_{ij}$. Formally, $L_{ij}$ can be written as

\[ L_{ij}(r) = L_{ij}(r) = \sqrt {\frac{K_{ij}(r)}{2\pi r}} \]

If the two subpatterns $X_ i$ and $X_ j$ are independent of each other, then the theoretical cross-type L-function is $L_{ij}^{*}(r) = r$.

The empirical cross-type pair correlation function, $g_{ij}$, is a kernel estimate of the form

\[ g_{ij}(r) = \frac{\rho (r)}{\hat{\lambda }^{2}} = \frac{1}{2\pi r \hat{\lambda _ i} \hat{\lambda _ j} |W|}\sum _ i \sum _ j \frac{k_ h(||x_ i-x_ j||-r)}{|W \cap W_{i-j}|)} \]

A border-edge-corrected version of $g_{ij}(r)$ can be written as

\[ g_{ij}(r) = \frac{1}{2 \pi r \hat{\lambda _ j}} \frac{\sum _ i \sum _ jk_ h(||x_ i-x_ j||-r)}{\sum _ i \Strong{1}\{ b_ i \geq r\} } \]

where $b_ i$ is the distance of $x_ i$ to the boundary of $W$, which is denoted as $\partial W$. If the two subpatterns $X_ i$ and $X_ j$ are independent of each other, then the theoretical cross-type pair correlation function is $g_{ij}^{*}(r) = 1$.