# [AniMov] Fwd: kernel homerange

Clément Calenge calenge at biomserv.univ-lyon1.fr
Wed Jul 19 14:15:16 CEST 2006

```Hi Leonardo and Carlotta,

The kernel method estimates the UD. The UD is a mathematical model which
describes the space use by an animal with the help of a bivariate
*continuous* probability density function. Thus, under this model, the
probability that the animal has been present at a given *exact* location
X,Y is equal to zero. The probability is different from zero over a
small area around the point X,Y. The probability can be computed for an
area, not for a point. Therefore, the probability to find two
relocations at exactly the same place under this model is equal to zero.
This hypothesis can be overlooked for the kernel estimation. However,
the LSCV estimation of the smoothing parameter relies more heavily on
this hypothesis of continuity than does the kernel estimation itself.

lt <- split(a[,c("x","y")], a[,1])

## consider for example the 4th animal: 101 relocations
uu <- lt[]

For example, for the 4th animal, you want to estimate the home range
using a sample of relocations. For this you need a mathematical model.
You decide that the UD is a good model, but to estimate the kernel home
range, you need a value for the smoothing parameter.
The ecological literature indicates that the LSCV is an optimal estimator:

> kernelUD(uu, h="LSCV")
...
WARNING!! No convergence in cross-validation for the following animals:
 1
Consider a new fit of UD using the ad hoc method for h.

But the LSCV does not work. As indicated previously, this maybe caused
by the fact that
several relocations are found at the same place... and indeed:

## count the number of relocations found at exactly the same point:
ii=table(paste(uu[,1], uu[,2]))
ii[ii>1]

One may find up to 5 relocations at the same point! Under such a model,
the probability of finding 5 relocations at exactly the same place is
equal to zero: the data contradict the model. So that the underlying
model of a continuous pdf does not hold. The data are therefore
inadequate for the LSCV, or, more exactly, the LSCV is inadequate for
these data. As noted in a previous mail to the list, the main question
is now: why is this model inadequate? this may be caused by several
factors, and it is difficult to determine which solution to adopt...

For example the fact that several relocations are found at the same
place may be the result of rounding errors. For example, on the field,
when relocations are taken, it may happen that the biologists round the
relocations coordinates to exact values (e.g. x = 300652.32654 rounded
to x = 300650), or round the bearings (direction of the signal relative
to the north, i.e. 30.65645843° rounded to 31°), which may also lead to
discretized relocations. In this case, the mathematical model is not
adequate, but this is due to the data collection. One may therefore
correct the data by adding a small noise to the relocations found
exactly at the same place, a noise corresponding to the imprecision in
the data. Consider the following function:

movesamepoint <- function(xy, amount)
{
fa <- paste(xy[,1], xy[,2])
ta <- table(fa)
ta <- ta[ta>1]
mov <- fa%in%names(ta)
xy[mov,1] <- jitter(xy[mov,1], amount=amount)
xy[mov,2] <- jitter(xy[mov,2], amount=amount)
return(xy)
}

Now let the error polygon on the relocations be about of 1 hectare (for
illustration purposes, this may actually be another value). That is, the
actual relocation is situated within a circle of sqrt(10000/pi) = 56,
say 50 meters around the estimated relocation. Therefore, the amount to
specifiy in the function above is 50 (the function jitter() will
therefore add a noise comprised between -50 m and 50 m).

image(kernelUD(movesamepoint(uu,50), h="LSCV"))

And the estimation is OK. Of course, in this case, it works, but if your
error is 100 m² (i.e. amount=5 meters)

> kernelUD(movesamepoint(uu,5), h="LSCV")
...
WARNING!! No convergence in cross-validation for the following animals:
 1
Consider a new fit of UD using the ad hoc method for h.
Warning message:
The algorithm did not converge
within the specified range of hlim: try to increase it in:
kernelUD(movesamepoint(uu, 5), h = "LSCV")

The problem occurs again and the question remains. In this case, the
relocations are so close that the procedure fails. But now, it is maybe
caused by biological reasons: the nature does not follow mathematical
hypotheses! The use of a given method implies a decision of the analyst:
is a given mathematical model a good one or not, given a problematic and
a dataset ? The pheasant may stay at the same relocation because the
environment at this place is favourable, or because it is the location
of its nest, or... etc.
The underlying model is inadequate, and the method should be avoided:
the LSCV is not an adequate estimator for the smoothing parameter...
Therefore:

>I don't know if there is some problems with my data.
>
The problem with your data may be that there is a rounding of the
bearings before the relocations are computed. But this problem can be
circumvented as indicated above.
If this imprecision is not the source of the failure of the procedure,
then, there is no problem with your data.
But there is a problem with the use of the method with your data: it is
inadequate. What to do? I don't know, it depends on your objectives, on
what you consider to be a home range, on what you do with the home range
after its estimation, on whether you are interested in core areas, etc.
etc.
For example, you may try to use more classical estimation "href", or to
decide on a value for the smoothing parameter (e.g. on a visual
exploration, h = 30 meters seem to be a good choice... of course this
depends on your objectives), or to decide that the kernel estimator is
not a good one and use MCP, etc. There is no automatic method...
Hope this helps,
Best wishes

Clément.

```