Agriculture Reference
In-Depth Information
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the positive run in statistics, a SMS period in the agricultural sense, and,
similarly, a negative run or SMD period. Feller (1967) also gave a definition
of runs based on recurrence theory and Bernoulli trials as follows.
A sequence of
n
events,
S
(success, SMS) and
F
(failure, SMD), contains
as many
S
runs of length
r
as there are non-overlapping, uninterrupted
blocks containing exactly
r
events
S
each. This definition is not convenient
practically because it does not say anything about the start and end of the
run (i.e., drought). In contrast, a definition of runs given by Feller (1967)
seems to be most revealing for the analysis of various drought features
because a run is defined as a succession of similar events preceded and
succeeded by different events with the number of similar events in the run
referred to as its length (figure 4.1).
[45],
Independent Bernoulli Model
Truncation of a soil moisture series
X
i
(
i
=
1,2, . . . ,
n
) at a constant level yields two complementary and mutual
distinct events—namely, SMS and SMD—with respective probabilities
p
and
q
(i.e., 1
Line
——
2.0
——
Norm
PgEn
p
). If the probability of the longest run-length,
L
(i.e.,
critical agricultural drought duration) in a sample size of
i
is equal to 1
and is denoted by
P
i
{
−
L
=
1
}
, then for sample size
i
=
1, one can simply
deduce,
P
1
{
L
=
0
} =
q
[4.2]
P
1
{
L
=
1
} =
p
[4.3]
Si
nce the occurrences of the elementary events are assumed independent
fr
om each other, the combined probabilities for
i
[45],
=
2 can be written as
P
2
{
L
=
0
} =
P
1
{
L
=
0
}
q
[4.4]
P
2
{
L
=
1
} =
P
1
{
L
=
1
}
q
+
P
1
{
L
=
0
}
p
[4.5]
P
2
{
L
=
2
} =
P
1
{
L
=
1
}
p
[4.6]
Si
mply,
P
2
{
indicates an SMD followed by another SMD. The first
te
rm on the right-hand side in
P
2
L
=
0
}
represents the SMS followed by
S
MD, and the second term represents an SMD followed by an SMS. Finally,
P
2
{
L
=
1
}
is the combination of SMS followed by another SMS event. It is
po
ssible to develop the same probability concepts for a soil moisture time
se
ries of length
n
( ¸ en, 1980a).
{
L
=
2
}
Dependent Bernoulli Model
In the derivation of drought probabilities
above, the occurrence of successive SMD and SMS is considered as inde-
pendent from each other. However, in nature, there is a tendency of SMD
to follow SMD, which implies dependence between successive occurrences.
The simplest representation of dependence can be achieved by considering
the relative situation of two successive time intervals. This leads to four
possible outcomes as transitional probabilities, which are referred to also as
conditional probability statements in the probability theory. For instance,
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