Page 381 - Special Topic Session (STS) - Volume 3

STS551 Yousif Alyousifi et al.
1
( | ) = (3)

(1− )
3.3.2 The Steady-State Probability of Air Pollution
Suppose that { , = 0, 1, 2, … , } be a discrete-time Markov chain with

state space S, and let P be the transition probability matrix of the Markov
chain. A vector = [ ⋯ ] is said to be the steady-state probability of air

1
pollution state if elements of are non-negative and satisfy the conditions
= ∑ =1 and ∑ =1 = 1 , = 1, 2, … , , indicating that the proportion

of time in which the stochastic process stays in a particular state. In a matrix
form, the linear equations system of the equation above can be solved based
on the following form
−1

= [( − ) + ] , = [ ] (4)
×

where I is an identity matrix, E is a unit matrix, e is a unit vector and [( − ) +
] is a nonsingular matrix (Kulkarni 2011; Tijms 2003; Sanusi et.al 2014). If the
value of is high, then the probability of occurrences of state j is high.

4. Results and Conclusion
The mean residence time (MRST), which describes the duration for each
state of air pollution event, is determined. It can be seen that most areas
experienced MRST for the moderate state to be approximately between 288
to 2359 hours, while for unhealthy and very unhealthy states, the duration of
the MRST is shorter, approximately between 4 to 28 and between 2 to 32 hours
respectively. This implies that, on the average, the moderate air pollution
condition is expected to occur longer than the unhealthy and very unhealthy
air pollution conditions in the study areas.
The steady state probability values (SSP) of air quality status for each
station are determined to represent the long-term probability of occurrence
of a particular air pollution state. It can be seen that the probabilities of the air
pollution states (unhealthy and very unhealthy states) slightly varies, with
values of from 0.00008 to 0.03026 and for moderate state the average value
is 0.975. Therefore, most areas have a high probability of being in the
moderate state as opposed to unhealthy and very unhealthy states.
In this study, it is demonstrated that the empirical Bayes method could
successfully smoothed out the zero probability in the transition probability
matrices. In addition, it also provides a more precise probability values as
opposed to those found based on maximum likelihood method.

References
1. Alyousifi, Y., Masseran, N., & Ibrahim, K. (2017). Modeling the stochastic
dependence of air pollution index data. Stochastic Environmental
Research and Risk Assessment, 1-9.

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