Geoscience Reference
In-Depth Information
or
N
≈ (
z
1
+
z
2
+ . . . +
z
n
) +
N
(1 −
p
)
n
.
Solving this equation for
N
, we have a model for the relationship between
N
,
total catch, and
p
, which is
N
≈ (
z
1
+
z
2
+ . . . +
z
n
)/{1 − (1 −
p
)
n
}.
(5.1)
Continuing the numerical example and assuming the total catch is 67,
Equation (5.1) yields the result that
N
≈ (67)/{1 − (0.8)
5
)} = 99.7.
Hence, with real-life data Equation (5.1) gives an estimate of
N
.
The expected catch in the
i
ith sample is
z
i
=
N
(1 −
p
)
i
-1
p
. Taking the loga-
rithm of both sides of this equation yields the approximate linear relationship
log
e
(
z
i
) ≈ log
e
(
N
) + log
e
(
p
) + (
i
− 1) log
e
(1 −
p
),
or
log
e
(
z
i
) ≈
a
+ (
i
− 1)
b
,
so that
y
i
≈
a
+
bx
i
,
(5.2)
where
y
i
= log
e
(
z
i
),
x
i
=
i
− 1, and the slope of the line is
b
= log
e
(1 −
p
). This
equation therefore gives a model for the natural logarithm of the number
removed in the
i
ith sample.
On the basis of Equations (5.1) and (5.2), Soms (1985) proposed the fol-
lowing method for estimating
N
and
p
. First, the values
y
i
= log
e
(
z
i
) and
x
i
=
i
− 1 are calculated for each of the sample,
i
= 1, 2, . . . ,
n
. Next, the equa-
tion
y
i
=
a
+
bx
i
is fitted by ordinary linear regression methods to obtain
an estimate of the slope
b
. Then, because
b
in the regression equation is
b
= log
e
(1 −
p
) in Equation (5.2), an estimate of
p
can be found by solving the
equation
log(1
ˆ
)
e
−
pb
to give
ˆ
1exp()
.
p
=−
b
(5.3)
Finally, Equation (5.1) suggests the estimator
{
}
ˆ
ˆ
)
n
Nzz
=+++
(
z
)/1 (1
−−
p
,
2
…
1
n
