Geoscience Reference
In-Depth Information
detected in the study
n
is then interpreted as the estimated expected number
of clusters, and the estimated expected cluster size for the population
Es
ˆ
()
is
a multiplier in Equation (4.3), producing
ˆ
(0)
ˆ
()
D
nf Es
L
⋅
⋅
ˆ
=
.
(4.4)
2
When objects do not occur in clusters (i.e., all objects are detected singly),
then ==
ˆ
() () 1
, and Equation (4.4) becomes Equation (4.3).
The choice of the detection function might be important for the estimates
obtained. Some nonparametric methods are
Es Es
a. fitting a curve subjectively by eye,
b. using a Fourier series approximation,
c. using an exponential power series approximation,
d. using an exponential polynomial approximation, and
e. using what is called a key function, with a series adjustment like
g
(
y
) = key(
y
){1 + series(
y
)}.
These are described as nonparametric because they do not assume a stan-
dard statistical distribution for the distance
x
of objects from the transect
line. Two parametric methods assume that
x
has
f. a negative exponential distribution and
g. a half-normal distribution.
As noted, sometimes the observed data contain extreme observations at
the right of the scale of distances. If this situation occurs, a truncation pro-
cess is suggested, that is, the removal of observations found a long way from
the track line. Buckland et al
.
(2001) recommend two rules of thumb: The
simplest one is to truncate at least 5% of the most extreme distances, and the
other alternative involves the estimation of a preliminary detection function
gx
ˆ
()
with all the distances included and then removing the large distances
x
such that
gx
ˆ
()
≥ 0.15 for the remaining distances.
It has been shown mathematically and by computer simulation that the
detection function can be made up of a mixture of more simple functions
that depend on factors such as weather, observer training, and so on, as long
as all such functions satisfy the condition that probability of detection is 1
when
x
= 0. For a full discussion on the choice of the detection function and
truncation, see the work of Buckland et al. (2001).
