asbio (version 1.9-7)

prp: Perpendicularity

Description

Calculates a perpendicularity index, \(\eta\), for animal spatial movements. The index has a [0, 1] range with 0 indicating a perfectly parallel movement with respect to boundary or edge and 1 indicating perfectly perpendicular movement. Other summaries are also provided.

Usage

prp(Time, S.X, S.Y, N.X, N.Y, habitat = NULL, near.angle = NULL, 
F.0.NA = TRUE)

Value

Returns a list with four or five items.

lines

A matrix with n - 1 rows containing line lengths for the lines A, B, C, D, and F. See figure in examples below.

angles

A matrix with n - 1 rows containing line lengths for the angles \(\kappa\), \(\gamma\) and \(\delta\). See Figure in examples below.

moment.by.moment

This component provides a matrix with n - 1 rows. Included are the columns: End.time, Eta.Index, Delta, Habitat, and Brdr chng. The columns Habitat, and Brdr chng are excluded if habitat = NULL or near.angle = NULL.

P.summary

Contains averages and standard errors for \(\eta\).

crossing.summary

Crossing binomial summaries. Provided if habitat and near.angle are specified.

Arguments

Time

A numeric vector containing the times when spatial coordinates were recorded.

S.X

X-coordinates of animal.

S.Y

Y-coordinates of animal.

N.X

X-coordinate of nearest point on boundary. These data can be obtained from function near.bound or from ARCGIS Near output.

N.Y

Y-coordinate of nearest point on boundary. These data can be obtained from function near.bound or from ARCGIS Near output.

habitat

A character vector of habitat categories.

near.angle

A numeric vector containing the angle of azimuth to the nearest point on the boundary with respect to a four quadrant system. NE = \(0^{\circ}\) to \(90^{\circ}\), NW is > \(90^{\circ}\) and \(\le 180^{\circ}\), SE is < \(0^{\circ}\) and \(\le - 90^{\circ}\) is > \(-90^{\circ}\) and \(\le -180^{\circ}\). This output can be obtained from function bound.angle or from ARCGIS Near output.

F.0.NA

A logical argument specifying whether or not a time interval in which F = 0 should be made NA (see Figure from examples)

Author

Ken Aho

Details

This index for perpendicularity, \(\eta\) is based on the following rules:

if \(\delta \le 90^o\) then \(\eta\) = \(\delta/90^o\); if \(90^o < \delta \le 135^o\) then \(\eta\) = \([90^o - (\delta - 90^o)]/90^o\); if \(135^o < \delta \le 180^o\) then \(\eta\) = \((\delta - 90^o)/90^o\)

For notation create Figures from examples.

References

Kie, J.G., A.A. Ager, and R.T. Bowyer (2005) Landscape-level movements of North American elk (Cervus elaphus): effects of habitat patch structure and topography. Landscape Ecology 20:289-300.

McGarigal K., SA Cushman, M.C. Neel, and E. Ene (2002) FRAGSTATS: Spatial Pattern Analysis Program for Categorical Maps. Computer software program produced by the authors at the University of Massachusetts, Amherst.

See Also

near.bound, bound.angle

Examples

if (FALSE) {
###Diagram describing prp output.  
y<-rnorm(100,0,5)
plot(seq(1,100),sort(y),type="l",xaxt="n",yaxt="n",lwd=2,xlab="",ylab="")
op <- par(font=3)

segments(52,-12,46,sort(y)[46],lty=1,col=1,lwd=1)##A
segments(90,-8,85,sort(y)[85],lty=1,col=1,lwd=1)##B
segments(46,sort(y)[46],85,sort(y)[85],lty=1)##F
segments(90,-8,46,sort(y)[46],lty=2)##D
arrows(52,-12,90,-8,length=.1,lwd=3)##C
arrows(20,-12,20,8,lty=2,col="gray",length=.1)#North
arrows(20,sort(y)[46],95,sort(y)[46],length=.1,lty=2,col="gray")
arrows(20,-12,95,-12,length=.1,lty=2,col="gray")#East

text(20,9,"N",col="gray");text(97,-12, "E", col= "gray");text(97,sort(y)[46], "E",
col= "gray")
text(49.5,-12.5,"a");text(92.5,-8.5,"b")
text(45.5,-5.5,"A",font=4,col=1);text(70,-9,"C",font=4,col=1);text(91.5,-1.75,"B",
font=4,col=1)
text(44,sort(y)[46]+1,"c");text(67.5,-2.5,"D",font=4,col=1);text(65,3.9,"F",font=4,
col=1)
text(87,sort(y)[87]+1,"d");text(57,-10,expression(kappa),col=1);
text(81,sort(y)[87]-3,expression(gamma),col=1);text(57,1.3,expression(theta),col=1)
text(64,-11.5,expression(beta),col=1)

library(plotrix)
draw.arc(50,-12,6,1.35,col=1);draw.arc(50,-12,6,.3,col=1);draw.arc(50,-12,6,0.02,
col=1)
draw.arc(46,sort(y)[46],7,.01,col=1);draw.arc(46,sort(y)[46],7,.5,col="white")
draw.arc(85,sort(y)[85],6,-2.7,col=1);draw.arc(85,sort(y)[85],6,-1.4,col="white",
lwd=2)
legend("topleft",c(expression(paste(kappa, " = acos[(",C^2," + ",X^2," - ",D^2,")
/2CX]")),
expression(paste(gamma," = acos[(",Y^2," + ",F^2," - ",D^2,")/2YF]")),
expression(paste(theta," = atan[(",y[f]," - ",y[n],")/(",x[f]," - ",x[n],")]")),
expression(paste(beta, " = atan[(",y[epsilon]," - ",y[alpha],")/(",x[epsilon],
" - ",x[alpha],")]"))),
bty="n",cex=.9,inset=-.025)

###Figure for demo dataset.
bX<-seq(0,49)/46

bY<-c(4.89000,4.88200,4.87400,4.87300,4.88000,4.87900,4.87900,4.90100,4.90800,
4.91000,4.93300,4.94000,4.91100,4.90000,4.91700,4.93000,4.93500,4.93700,
4.93300,4.94500,4.95900,4.95400,4.95100,4.95800,4.95810,4.95811,4.95810,
4.96100,4.96200,4.96300,4.96500,4.96500,4.96600,4.96700,4.96540,4.96400,
4.97600,4.97900,4.98000,4.98000,4.98100,4.97900,4.98000,4.97800,4.97600,
4.97700,4.97400,4.97300,4.97100,4.97000)

X<-c(0.004166667,0.108333333,0.316666667,0.525000000,0.483333333,0.608333333,
0.662500000,0.683333333,0.900000000,1.070833333)
Y<-c(4.67,4.25,4.26,4.50,4.90,4.10,4.70,4.40,4.20,4.30)

plot(bX,bY,type="l",lwd=2,xlab="",ylab="",ylim=c(4,5.1))
lines(X,Y)

for(i in 1:9)arrows(X[i],Y[i],X[i+1],Y[i+1],length=.1,lwd=1,angle=20)
mx<-rep(1,9)
my<-rep(1,9)
for(i in 1:9)mx[i]<-mean(c(X[i],X[i+1]))
for(i in 1:9)my[i]<-mean(c(Y[i],Y[i+1]))
for(i in 1:9)text(mx[i],my[i],i,font=2,cex=1.3)

nn<-near.bound(X,Y,bX,bY)
prp(seq(1,10),X,Y,nn[,1],nn[,2])$moment.by.moment
par(op)
}