133 NTU Management Review Vol. 31 No. 1 Apr. 2021 4.2 Estimations of Precipitation Models 4.2.1 Estimation of Occurrence Model We fit the transition probabilities of the occurrence model with the truncated Fourier series from first-order (M = 1) to fifth-order (M = 5), and estimate the coefficients by MLE. Since MLE is a numerical optimization problem, we should notice that the coefficients need to satisfy the constraint (4.1) for all transition probabilities. 0 ( ) = 0 + 3 2 65 + 3 2 65 1. (4.1) When we apply MLE in numerical computation, the initial values of coefficients are obtained by the least-squares approach. However, according to Woolhiser and Pegram (1979), the data points (transition probabilities) are estimates and have unequal variances because of varying properties of the distribution. Due to this major deficiency, we do not directly adopt the least-squares approach to estimate the coefficients. Table 3 shows the results estimated by MLE and the values of the AIC and BIC, which are used to decide the order of each transition probability. We find that the criteria of p00 are close between the 1st order and 2nd order in the AIC but in the BIC, these two orders are obviously different. According to the lowest criterion, we choose the 1st order for p00 and the 2nd order for p10. Figure 3 shows the estimated truncated Fourier series of p00 and p10, and all orders are shown in Figure B.2 of Appendix B. 4.2.2 Estimation of Magnitude Model We fit the daily rainfall data with the mixed exponential distribution, where the parameters α, β1, and β2 are described with the truncated Fourier series from first-order (M = 1) to fifth-order (M = 5). The coefficients of each parameter are estimated by MLE with constraints 0 ≤ α ≤ 1, 0 < β1 < β2. According to the estimated results and the values of the AIC and BIC in Table 4, we choose the 3rd order for α, β1, and β2. Figure 4 shows the estimated truncated Fourier series, and all orders are shown in Figure B.1 of Appendix B.
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