Agriculture Reference
In-Depth Information
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46
P
(
−
/
+
) implies the probability of SMD (
−
) at current time interval on the
condition (given) that there is SMS (
) in the following time interval. In
contrast, according to the joint probability definition, it is possible to state
(1) transition from an SMS to an SMS with probability
P
(
+
+
/
+
); (2) tran-
sition from an SMD to an SMS with probability
P
(
−
/
+
); (3) transition
from an SMS to an SMD with probability
P
(
) ; and, finally, (4) tran-
sition from a soil moisture to an SMD with probability
P
(
+
/
−
−
/
−
). Four soil
moisture joint probability statements are
P
(
+
,
+
)
=
P
(
+
/
+
)
P
(
+
);
P
(
−
,
+
)
=
). Similar
to the independent Bernoulli case, there are two state probabilities: SMS
P
(
P
(
−
/
+
)
P
(
+
);
P
(
+
,
−
)
=
P
(
+
/
−
)
P
(
−
) and
P
(
−
,
−
)
=
P
(
−
/
−
)
P
(
−
) probabilities. Because transition and state probabili-
ties are independent of each other, the relationships between them can be
written as:
+
) and SMD
P
(
−
[46],
P
(
+
)
=
P(
+
/
+
)P (
+
)
+
P(
+
/
−
)P (
−
)
[4.7]
P
(
−
)
=
P(
−
/
+
)P (
+
)
+
P(
−
/
−
)P (
−
)
[4.8]
Line
——
8.0
——
Norm
PgEn
w
here equation 4.7 expresses the probability of an SMS in the current time
interval with its first right-hand side term as the probability of SMS
P
(
+
), in
the previous time interval with its transition
P
(
) from SMS to SMS, and
the second term on the right-hand side representing the transition
P
(
+
/
+
)
from SMD in the previous time interval to SMS in the current time inter-
val. Equation 4.8 has similar interpretations. Furthermore, due to the mu-
tual exclusiveness of probabilities, the following sequences of probability
statements are also valid. Any time interval may have either SMS or SMD
cases with state probabilities
P
(
+
/
−
[46],
), respectively, whereas transi-
tional probabilities are valid between two successive time intervals, given
that the state is in SMS in the previous interval. The derivation mechanism
of agricultural drought probabilities are the same as independent Bernoulli
case, but a slight change of notation is necessary due to the dependent na-
ture of the successive events. Hence, the probability of the longest critical
drought duration,
L
, being equal to an integer value,
j
, in a sample size
of
i
with a surplus state at the final stage will be denoted by
P
i
+
{
+
)or
P
(
−
L
=
j
}
.
Accordingly one can write
P
1
{
L
=
0
} =
P(
−
)
=
q
[4.9]
P
1
{
L
=
1
} =
P(
+
)
=
p
[4.10]
If two successive time intervals (
i
=
2) are considered, it is possible to
ob
tain the following by enumeration:
P
2
{
P
1
L
=
0
} =
{
L
=
0
}
P(
−
/
−
)
[4.11]
P
2
{
P
1
L
=
1
} =
{
L
=
0
}
P(
+
/
−
)
[4.12]
P
2
{
P
1
L
=
1
} =
{
L
=
1
}
P(
−
/
+
)
[4.13]
P
2
P
1
{
L
=
2
} =
{
L
=
1
}
P(
+
/
+
)
[4.14]
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