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 kNNs of i and j is among kNNs 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 kNNs of i and j is among kNNs 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)