Page 424 - Contributed Paper Session (CPS) - Volume 6

CPS2008 Syafrina A.H et al.
and showed bad image to Penang since it should be a city of international
standard. Therefore, this study will be performed to handle these problems.
Statistical Models
Two models for daily rainfall amount are tested with their probability
density functions (pdf) as follows. Daily rainfall amount is represented by x.i)
Gamma distribution with two parameters α and β represent the shape and
scale parameters.

1 −
() = () (−1) ; , , > 0 (2.1)

Γ()
where
∞

Γ() = ∫ −1 −
0
(2.2)
Weibull distribution with two parameters and represent the shape and
scale parameters.

() = ( ) −1 exp [− ( ) ] , > 0, > 0, > 0 (2.3)

Maximum Likelihood Estimation (MLE)
Estimating the parameters is useful in order to fit the distributions to
rainfall data. The shape and scale parameters will be treated as unknown
values and it is assumed to be independent and identically distributed (i.i.d)
for the joint density for all observations of the dataset (Nielson, 2011). MLE
starts with the mathematical expressions which is called likelihood function of
the sample data. These values of the parameter that maximize the sample
likelihood are known as the maximum likelihood estimates. It provides a
consistent approach to parameter estimation problems. Suppose that random
variables , … , have a joint density or frequency function ( , , … , |).

1
1
2
Given observed values = , where = 1, … , , the likelihood of as a

function of , , … , is defined as

1
2
() = ( , , … , |) (2.4)
2
1

The joint density can be considered as a function of rather than as a function
of the . The MLE of is the value of that maximizes the likelihood makes

the observed data most probable. If the are assumed to be i.i.d., their joint

density is the product of the marginal densities, and the likelihood is
() = ∏ ( |) (2.5)

=1
it is usually easier to maximize its natural logarithm (which is equivalent since
the logarithm is a monotonic function). For an i.i.d. sample, the log likelihood
is
(2.6)
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