Nt.def {nnspat}

$N_t$ Value (found with the definition formula)

Description

This function computes the $N_t$ value which is required in the computation of the asymptotic variance of Cuzick and Edwards $T_k$ test. Nt is defined on page 78 of (Cuzick and Edwards (1990)) as follows. $N_t= \sum \sum_{i \ne l}\sum a_{ij} a_{lj}$ (i.e, number of triplets $(i,j,l)$ $i,j$, and $l$ distinct so that $j$ is among $k$NNs of $i$ and $j$ is among $k$NNs of $l$).

This function yields the same result as the asyvarTk and varTk functions with $Nt inserted at the end.

See (Cuzick and Edwards (1990)) for more details.

Usage

Nt.def(a)

Arguments

a

The $A=(a_{ij})$ matrix. The argument a is the $A$ matrix, obtained as output fromm aij.mat.

Value

Returns the $N_t$ value standing for the number of triplets $(i,j,l)$ $i,j$, and $l$ distinct so that $j$ is among $k$NNs of $i$ and $j$ is among $k$NNs of $l$. See the description.

Author(s)

Elvan Ceyhan

References

Cuzick J, Edwards R (1990). “Spatial clustering for inhomogeneous populations (with discussion).” Journal of the Royal Statistical Society, Series B, 52, 73-104.

See Also

asyvarTk, varTk, and varTkaij

Examples

n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
k<-2 #try also 2,3
a<-aij.mat(Y,k)
Nt.def(a)