Wmat: The incidence matrix `W` for the NN digraph

Description

Returns the \(W=(w_ij)\) matrix which is used to compute \(Q\), \(R\) and \(T\) values in the NN structure. \(w_{ij}=I(\) point \(j\) is a NN of point \(i))\) i.e., \(w_{ij}=1\) if point \(j\) is a NN of point \(i\) and 0 otherwise.

The argument ties is a logical argument (default=FALSE) to take ties into account or not. If TRUE the function takes ties into account by making \(w_{ij}=1/m\) if point \(j\) is a NN of point \(i\) and there are \(m\) tied NNs and 0 otherwise. If FALSE, \(w_{ij}=1\) if point \(j\) is a NN of point \(i\) and 0 otherwise. The matrix \(W\) is equivalent to \(A=(a_{ij})\) matrix with \(k=1\), i.e., Wmat(X)=aij.mat(X,k=1).

The argument is.ipd is a logical argument (default=TRUE) to determine the structure of the argument x. If TRUE, x is taken to be the inter-point distance (IPD) matrix, and if FALSE, x is taken to be the data set with rows representing the data points.

Usage

Wmat(x, ties = FALSE, is.ipd = TRUE, ...)

Value

The incidence matrix \(W=(w_ij)\)

where \(w_{ij}=I(\) point \(j\) is a NN of point \(i))\), i.e., \(w_{ij}=1\) if point \(j\) is a NN of point \(i\) and 0 otherwise.

Arguments

x: The IPD matrix (if is.ipd=TRUE) or a data set of points in matrix or data frame form where points correspond to the rows (if is.ipd = FALSE).
ties: A logical parameter (default=FALSE) to take ties into account in computing the \(W\) matrix, so, if it is TRUE, \(w_{ij}=1/m\) if point \(j\) is a NN of point \(i\) and there are \(m\) tied NNs and 0 otherwise and if FALSE, \(w_{ij}=1\) if point \(j\) is a NN of point \(i\) and 0 otherwise.
is.ipd: A logical parameter (default=TRUE). If TRUE, x is taken as the inter-point distance matrix, otherwise, x is taken as the data set with rows representing the data points.
...: are for further arguments, such as method and p, passed to the dist function.

Author

Elvan Ceyhan

Examples

Run this code

n<-3
X<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(X)
Wmat(ipd)
Wmat(X,is.ipd = FALSE)

n<-5
Y<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(Y)
Wmat(ipd)
Wmat(Y,is.ipd = FALSE)
Wmat(Y,is.ipd = FALSE,method="max")

Wmat(Y,is.ipd = FALSE)
aij.mat(Y,k=1)

#1D data points
X<-as.matrix(runif(5)) # need to be entered as a matrix with one column
#(i.e., a column vector), hence X<-runif(5) would not work
ipd<-ipd.mat(X)
Wmat(ipd)
Wmat(X,is.ipd = FALSE)

#with ties=TRUE in the data
Y<-matrix(round(runif(15)*10),ncol=3)
ipd<-ipd.mat(Y)
Wmat(ipd,ties=TRUE)
Wmat(Y,ties=TRUE,is.ipd = FALSE)