Geoscience Reference
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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.
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