Page 380 - Special Topic Session (STS) - Volume 3

STS551 Yousif Alyousifi et al.
For statistical modelling, the hourly API values are analysed in order to
determine the air pollution conditions for seven main cities in the central
region of Peninsular Malaysia for a three-year period from 2012 to 2014. In
this study, the data of air pollution index is classified into a three-state Markov
chain model, namely, (0 < API ≤ 100), (101 < API ≤ 200), (API > 200), which
correspond to moderate, unhealthy and very unhealthy states respectively,
representing the three different levels of air quality.

3.2 Empirical Bayesian Estimation of the Transition Probability Matrix
In the analysis of the multinomial data, as in this study, we frequently seek
to provide a count matrix with smoothed cell frequencies. In the studies by
(Seal &Hossain 2015; Rodrigues & Achcar 2012; Meshkani & Billard 1992),
Dirichlet prior has been proposed as a conjugate prior for the parameters of
the multinomial distribution. In this section, the determination of the empirical
Bayes estimator will be carried out for the multinomial distribution with
parameters = ( , ,· · · , ), for all = 1, 2,· · · , under the assumption
1

2
of conjugate Dirichlet prior. Let = ( , , … , ) denotes the row vector

1

2
of the transition count matrix = [ ], which is an observed random sample

that is assumed to follow the multinomial distribution with parameter
vector ∀ . The parameter vector is assumed to follow the Dirichlet

conjugate prior with parameter = ( , … , ) for all i = 1, 2,· · · , k, which is

1

a natural conjugate prior for the multinomial observations. Accordingly, we
have
= ( , , … , )~ Multinomial ( ; , … , ),

1
2
1
.

2

where = ( , , … , )~ Dirichlet ( , … , ) for all i = 1, 2, ... , k. It

1

1

2
can be shown that the posterior pdf can be written as

Γ(∑ =1 ( + ) + −1

( | , ) = ∏ [ ] (1)

∏ Γ ( + )
=1 =1
3.3 Characteristics of Air pollution
3.3.1 The Mean Residence Time of Air Pollution
Let represent the transition probability of a discrete-time Markov

chain { , = 0, 1, 2, … , } from state i to state j and is the residence time

for any state j and h is the number of hours. The probability of observing h
hour of residence time is given by

ℎ−1
Pr( = ℎ) = ( ) (1 − ) (2)

Then the mean residence time for any air pollution state j is given by
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