Agriculture Reference
In-Depth Information
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Equation 4.11 refers to the transition from SMD to SMD, whereas Equa-
tion 4.13 shows transition from SMS to SMD. Similarly, probabilities can
be determined for sample size
i
. The numerical solutions of these equations
for different sample sizes can be obtained using computer porgrams and are
presented in figure 4.2, on the basis of a given SMS probability (
p
0.7).
Using this graph, it is possible to read probability of critical agricultural
drought of a given duration for given number of samples.
=
M
arkov Model
A
lthough dependence between successive SMS or SMD events is accounted
fo
r simply by the dependent Bernoulli model, in nature dependences are
m
ore persistent. To model critical agricultural droughts more realistically,
th
e second-order Markov process is presented. This process requires three-
in
terval basic transitional probabilities in addition to two-interval proba-
bi
lities. The SMS and SMD probabilities remain as they were in the previ-
ou
s models. The complete description of the second-order Markov model
fo
r critical drought probability predictions is presented by ¸ en (1990). The
new set of transitional probabilities can be defined as 8 (
i
[47],
Line
——
-1.1
——
Norm
PgEn
2
3
8), which
are mutually exclusive and collectively exhaustive alternatives. For exam-
ple, one of the the eight alternatives will be:
=
=
P
(
+
/
+−
)
=
P
(
X
i
>
X
0
,
X
i
−
1
>
X
0
,
X
i
−
2
>
X
0
)
[4.15]
Here,
P
(
) refers to the probability of an SMS at current interval,
given that two successive intervals included an SMD and SMS, respectively.
In contrast, mutual exclusiveness implies that
P
(
+
/
+−
[47],
+
++
−
++
=
/
)+
P
(
/
)
1;
−
−+
+
−
+−
=
+
−+
+
−
−+
=
+
−−
+
P
(
/
)
P(
/
)
1;
P
(
/
)
P
(
/
)
1 and P(
/
)
Figure 4.2
Cumulative probabilities of critical agricultural drought of given duration (length
in
years) and return period
n
(in years).
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