Spatial signs, symmetrized signs, ranks and signed ranks

Description

Functions to compute spatial signs, symmetrized signs, ranks and signed ranks.

Usage

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Arguments

X

a matrix or a data frame

center

a vector or a logical, see details

shape

a matrix or a logical, see details

...

arguments that can be passed on to function used for the estimation of shape.

na.action

a function which indicates what should happen when the data contain 'NA's. Default is to fail.

Details

The spatial signs of an observed vector is simply the vector, possibly affinely transformed first, multiplied by its Euclidian length. See spatial.sign for a precise definition. Symmetrized spatial signs are the spatial signs of the pairwise differences of the data

||x_i-x_j||^{-1}(x_i-x_j)

(there are n over 2 of these). Spatial rank of an observation is the average of the signs of the differences of that observation and the others:

R(x_i)=ave_j{||x_i-x_j||^{-1}(x_i-x_j)}

Spatial signed rank of an observation is defined as

Q(x_i)=(R(x_i)+ave_j{||x_i+x_j||^{-1}(x_i+x_j)})/2

If a numerical value is given for shape and/or center these are used to transform the data before the computation of signs or ranks. A logical TRUE indicates that the shape or center should be estimated. In this case an affine transformation that makes the resulting signs or ranks have a covariance matrix equal or proportional to the identity matrix and centerd on the origin is found. A logical FALSE indicates that the null value, that is, the identity matrix or the origin, should be used. Note that only signed ranks depend on a center.

The value of shape and/or location used are returned as attributes.

Author(s)

Seija Sirkia, seija.sirkia@iki.fi

References

Visuri, S., Koivunen, V. and Oja, H. (2000). Sign and rank covariance matrices. J. Statistical Planning and Inference, 91, 557-575.

See Also

spatial.sign for the signs, spatial sign and rank covariance matrices and spatial.shape for the standardizing transformations

Examples

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A<-matrix(c(1,2,-3,4),ncol=2)
X<-matrix(rnorm(100),ncol=2)%*%t(A)
def.par<-par(no.readonly=TRUE) # for resetting
layout(matrix(1:4,ncol=2,nrow=2,byrow=TRUE))
plot(X,col=c(2,rep(1,19)))
plot(spatial.symmsign(X),col=c(2,rep(1,19)),xlim=c(-1,1),ylim=c(-1,1))
theta<-seq(0,2*pi,length=1000)
lines(sin(theta),cos(theta))
plot(spatial.rank(X),col=c(2,rep(1,19)),xlim=c(-1,1),ylim=c(-1,1))
lines(sin(theta),cos(theta))
plot(spatial.signrank(X),col=c(2,rep(1,19)),xlim=c(-1,1),ylim=c(-1,1))
lines(sin(theta),cos(theta))
par(def.par)