125 NTU Management Review Vol. 31 No. 1 Apr. 2021 2.3 Estimations 2.3.1 Maximum Likelihood Estimation Since the parameters in occurrence model (p00 and p10) and magnitude model (α, β1, and β2) are all approximated by the truncated Fourier series, we express the parameter set of the precipitation model as (2.11): θ(t) = { p00(t), p10(t), α(t), β1(t), β2(t)}. (2.11) Each parameter is specified by truncated Fourier series as (2.12): ( ) = 0 + sin 2 365 + cos =1 ( ) ( ) 2 365 , (2.12) where θj (t) = {p00, p10, α, β1, β2} with j = 0, 1, 2, …, 4 to distinguish different parameters. The constraints for these parameters are 0 ≤ px0(t) ≤ 1, 0 ≤ α ≤ 1 and 0 < β1 < β2. We then determine the coefficients of all parameters with MLE and provide the log-likelihood functions of both models as follows: (1) The log-likelihood functions U0 and U1 for the occurrence model: We assign Φ1 to be a vector whose elements are the coefficients of the Fourier series describing p00(t) and p10(t). The maximum likelihood estimate Φ1 of Φ1 is found by maximizing the likelihood function L(Xt│Φ1 ), or logarithm: U = logL(Xt│Φ1 ). We assume there are N years of daily precipitation magnitudes in the data. Then we calculate the historical transition probabilities using the following steps: Step 1: The transition states for each day are marked as in Table 1. For example, if January 1 is rainy, then January 2 is not rainy. In this case, we mark January 2 as “10”. Step 2: The marks of transition states are divided into 365 groups. Each group represents one calendar day which comprises N observations. Since February 29 is also included, we need to delete related transition states after having marked them. Step 3: Starting from January 1, the historical counts are calculated as shown in Table 1. For example, there are n01(t) days changing from state 0 to state 1 in N years on calendar day t. Finally, there are four series of historical counts and each series contains 365 values.
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