Testing Independence in Regular Trajectory Parameters

Description

The function indmove tests for the independence between successive components c(dx, dy) for each burst in a regular object of class ltraj.

The function indmove.detail tests for the independence between successive dx or dy for each burst in a regular object of class ltraj.

The function testang.ltraj tests for the independence between successive angles (relative or absolute) for each burst in a regular object of class ltraj.

The function testdist.ltraj tests for the independence between successive distances between successive relocations for each burst in a regular object of class ltraj.

Usage

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indmove(ltr, nrep = 200, conflim = seq(0.95, 0.5, length=5),
        sep = ltr[[1]]$dt[1], units = c("seconds", "minutes",
                                        "hours", "days"),
        plotit = TRUE)


testang.ltraj(x, which = c("absolute", "relative"),
              nrep = 999, alter = c("two-sided","less","greater"))


testdist.ltraj(x, nrep = 999, alter = c("two-sided","less","greater"))


indmove.detail(x, detail=c("dx","dy"), nrep=999,
               alter = c("two-sided","less","greater"))

Arguments

ltr,x

an object of class ltraj

conflim

a vector giving the limits of the confidence intervals to be plotted

nrep

number of simulations

units

a character string indicating the time units for the result

alter

a character string specifying the alternative hypothesis, must be one of "greater", "less" or "two-sided" (default)

which

a character string indicating whether the absolute or relative angles are under focus

detail

a character string indicating whether "dx" or "dy" should be tested for independence

plotit

logical. Whether the results should be plotted on a graph

sep

used in the case of variable time lag between relocations. Indicates the theoretical time lag between two relocations

Details

The function indmove randomises the order of the increments c(dx, dy) in a trajectory. The criteria of the test is the Mean Squared Displacement (R^2_n) (Root & Kareiva 1984).

The function testang.ltraj randomises the order of the angles in a trajectory. The criteria of the test is f^2 = sum_(i=1)^(n-1) 2*(1 - cos(angle[i+1] - angle[i])). This measure corresponds to the mean squared length of the segment joining two successive angles on the trigonometric circle (see examples for an illustration)

The function testdist.ltraj randomises the order of the distances between successive relocations in a trajectory. The criteria of the test is sum_(i=1)^(n-1) (dist[i+1] - dist[i])^2 (Neuman 1941, Neuman et al. 1941). The same criteria is used in indmove.detail().
.

Note that these functions require "regular" trajectories, i.e. trajectories for which the relocations are separated by a constant time lag.

Finally, note that the functions testang.ltraj and testdist.ltraj are not affected by the presence of missing values in the bursts of relocations. The function indmove may be greatly affected by these missing values (they are removed prior to the test).

Value

indmove() returns a list with one component per burst. Each component is a list of two data frames. The data frame Time contains the time points at which R2n is computed for the observation (first column) and the simulations (other ones). The data frame R2n contains the values for the R2n (same dimensions).

testang.ltraj(), testdist.ltraj and indmove.detail return lists of objects of class randtest.

Author(s)

Clement Calenge clement.calenge@oncfs.gouv.fr
Stephane Dray dray@biomserv.univ-lyon1.fr

References

Root, R.B. & Kareiva, P.M. (1984) The search for resources by cabbage butterflies (Pieris Rapae): Ecological consequences and adaptive significance of markovian movements in a patchy environment. Ecology, 65: 147–165.

Neumann, J.V., Kent, R.H., Bellinson, H.R. & Hart, B.I. (1941) The mean square successive difference. Annals of Mathematical Statistics, 12: 153–162

Neumann, J.V. (1941) Distribution of the ration of the mean square successive difference to the variance. The Annals of Mathematical Statistics, 12: 367–395

See Also

ltraj

Examples

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## theoretical independence between
br <- simm.brown(1:1000)
testang.ltraj(br)
testdist.ltraj(br)

## Not run: 
indmove(br)

## End(Not run)

## Illustration of the statistic used for the test of the independence
## of the angles
opar <- par(mar = c(0,0,4,0))
plot(0,0, asp=1, xlim=c(-1, 1), ylim=c(-1, 1), ty="n", axes=FALSE,
main="Criteria f for the measure of independence between successive
angles at time i-1 and i") 
box()
symbols(0,0,circle=1, inches=FALSE, lwd=2, add=TRUE)
abline(h=0, v=0)
x <- c( cos(pi/3), cos(pi/2 + pi/4))
y <- c( sin(pi/3), sin(pi/2 + pi/4))
arrows(c(0,0), c(0,0), x, y)
lines(x,y, lwd=2, col="red")
text(0, 0.9, expression(f^2 == 2*sum((1 - cos(alpha[i]-alpha[i-1])),
i==1, n-1)), col="red")
foo <- function(t, alpha)
{
  xa <- sapply(seq(0, alpha, length=20), function(x) t*cos(x))
  ya <- sapply(seq(0, alpha, length=20), function(x) t*sin(x))
  lines(xa, ya)
}
foo(0.3, pi/3)
foo(0.1, pi/2 + pi/4)
foo(0.11, pi/2 + pi/4)
text(0.34,0.18,expression(alpha[i]), cex=1.5)
text(0.15,0.11,expression(alpha[i-1]), cex=1.5)
par(opar)

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