123 NTU Management Review Vol. 31 No. 1 Apr. 2021 For example, p10 is the probability that it does not rain on day t but it rains on day t-1. It should be noticed that in (2.4) and (2.5) p01 = 1 - p00 and p11 = 1 - p10 since there are only two states. According to Wilks (2011), the precipitation time series usually result in p01 < p00 and p10 < p11, which means a rainy day is more likely to be followed by a rainy day rather than a sunny day. Similarly, a sunny day is more likely to be followed by a sunny day rather than a rainy day. From now on, we use px0 as the abbreviation for p00 and p10, x = 0 or 1. According to the characteristic (1), the time-varying transition probabilities px0 will not be constants during a year, so px0 is approximated by a truncated Fourier series like (2.6), which could capture the seasonal feature. 0( ) = 0 + sin 3 2 65 + cos 3 2 65 =1 ( ) ( ) , (2.6) where cx0, cx i, dx i are coefficients; M is the order of Fourier series; and according to Cao et al. (2004), the maximum order is set as 5. To simplify the model, we assume the transition probabilities stay constant between years, px0 (t + 365) = px0 (t). (2.7) which means we only discuss the behavior of precipitation during a year while the varying differences across years are not in our consideration. For a time-homogeneous Markov chain, the distribution of steady states is an important property that we usually care about. According to Alexandridis and Zapranis (2013), the stationary probability π1 in a first-order, two-state Markov chain, which describes the unconditional probability of precipitation, is given by 1 = 01 1+ 01 11 = 01 01 + 10. (2.8) And the stationary probability π0 for state 0 is 0 =1 1 = 01 10 + 10. (2.9) For the steady states, both π0 and π1 should be constants. It implies that
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