Geoscience Reference
In-Depth Information
the sample size equal to 2
n
. At a particular level α of the test, values of
h
T
below the critical value shown in the table indicate a regular distribution of
items, and values above the critical value indicate an aggregate distribution
of items. Estimates of density are liable to be biased if a significant departure
from a random distribution is indicated.
Intuitive estimates of density if items are randomly distributed are as
follows:
ˆ
D
=
(Number of items)/(Area searched)
1
=
n
/(Area of circles searched)
(6.1)
2
=Σπ
nx
/( )
i
and
ˆ
D
=
(Number of items)/(Area of half-circles searched)
2
(6.2)
2
=Σπ
n
/[ ( 2)]
i
z
An estimator of density that is more robust to the lack of random pattern
of the items is credited to Byth (1982) and takes the form
{
}
ˆ
1
2
Dn x
2
/[2][2
z
]
=
Σ
Σ
,
(6.3)
T
i
i
1/
ˆ
T
is easier to handle from a mathematical
point of view because it follows approximately a
t
distribution with
n
− 1
degrees of freedom. The standard error is given by
The reciprocal of density
D
SE(1/
ˆ
22
22
D
)
=
{8(
z s zs xs n
+
2
+
)/}
(6.4)
T
x
xz
z
where
x
is the mean of the point to the nearest-item distances,
z
is the mean
of the item to the nearest
T
T-square neighbor distances,
s
x
is the standard devi-
ation of the point to the nearest-item distances,
s
z
is the standard deviation of
the item to the nearest
T
T-square neighbor distances, and
s
xz
is the covariance
of the
x
and
z
distances.
An approximate 95% confidence interval on the reciprocal of density is
1/
ˆ
SE(1/
ˆ
Dt
[
±
⋅
D
)]
,
(6.5)
T
0.025, 1
n
−
T
where
t
0.025,
n
-1
is the value of the
t
distribution with
n
− 1 degrees of freedom
that is exceeded with probability 0.025. After obtaining a confidence interval
for the reciprocal of the density, this can be inverted to obtain confidence
limits for the density itself.