Stochastic Precipitation Generation for the Xilingol League Using Hidden Markov Models with Variational Bayes Parameter Estimation
Abstract
:1. Introduction
2. Materials and Methods
2.1. Data Sources
2.2. Methodology Workflow
2.3. The HMM for Precipitation
2.4. VB Parameter Estimation for the HMM
Algorithm 1 VBEM algorithm for HMMs 

Algorithm 2 Stochastic Variational Bayes for HMMs 

3. Results
3.1. Simulation Study
3.1.1. BIC Scores for Daily Precipitation
3.1.2. RMSE for Heavy Precipitation Weather
3.1.3. Scatter Plot Regression of Daily Precipitation
3.2. Analysis of Daily Precipitation in the Xilingol League
3.2.1. Annual Precipitation and NDVI Correlation Analysis
3.2.2. Seasonal Precipitation Sequences
3.2.3. Annual Precipitation Sequence
3.2.4. Analysis of Annual Precipitation Trends
4. Conclusions
5. Discussion
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. VBEM Algorithm
Appendix B. Hyperparameter Update for SVB Algorithm
Appendix C. Variational Forward–Backward Algorithm
Appendix D. Analytic Formula for KL Divergence
Appendix E. Bayesian Information Criterion
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Precipitation Level  Daily Precipitation Amounts Ranges 

Light Precipitation  <10 mm 
Moderate Precipitation  [10 mm, 25 mm] 
Heavy Precipitation  [25 mm, 50 mm] 
Torrential Precipitation  [50 mm, 100 mm] 
Heavy Storms  [100 mm, 200 mm] 
Torrential Storms  >200 mm 
Number of Hidden States  Exponential Distribution  Gamma Distribution  Pareto Distribution  

C = 3  C = 4  C = 3  C = 4  C = 3  C = 4  
3  6124.92  6118.54  6490.61  6503.58  5679.57  5680.50 
4  6131.62  6120.89  6465.26  6519.02  5711.61  5700.50 
5  6624.81  6493.07  6617.57  6626.71  5690.45  5691.82 
6  6595.85  6472.56  6669.12  6619.91  5695.55  5692.39 
Number of Hidden States  Exponential Distribution  Gamma Distribution  Pareto Distribution  

C = 3  C = 4  C = 3  C = 4  C = 3  C = 4  
3  44.94  44.79  67.01  68.61  21.70  21.74 
4  45.05  44.36  64.09  68.79  25.30  23.94 
5  82.85  73.09  72.80  76.05  3.15  4.84 
6  80.58  71.87  78.19  80.80  4.45  3.11 
Location  Longitude  Latitude  Correlation Coefficient  Partial Correlation Coefficient 

1  ${114}^{\circ}{21}^{\prime}$ E  ${44}^{\circ}{51}^{\prime}$ N  0.90  0.90 
2  ${115}^{\circ}{52}^{\prime}$ E  ${44}^{\circ}{59}^{\prime}$ N  0.91  0.93 
3  ${115}^{\circ}{54}^{\prime}$ E  ${44}^{\circ}{59}^{\prime}$ N  0.93  0.90 
Location  Distribution  Daily Precipitation  Monthly Precipitation  Annual Precipitation 

1  Exponential  2.76  35.66  154.59 
1  Gamma  0.83  26.82  63.63 
1  Pareto  0.55  35.21  81.64 
2  Exponential  2.38  46.38  167.72 
2  Gamma  1.25  36.54  123.8 
2  Pareto  0.68  36.14  110.97 
3  Exponential  2.34  39.82  68.9 
3  Gamma  1.52  34.32  50.95 
3  Pareto  0.85  32.72  6.64 
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Zhang, S.; Tuerde, M.; Hu, X. Stochastic Precipitation Generation for the Xilingol League Using Hidden Markov Models with Variational Bayes Parameter Estimation. Water 2023, 15, 3600. https://doi.org/10.3390/w15203600
Zhang S, Tuerde M, Hu X. Stochastic Precipitation Generation for the Xilingol League Using Hidden Markov Models with Variational Bayes Parameter Estimation. Water. 2023; 15(20):3600. https://doi.org/10.3390/w15203600
Chicago/Turabian StyleZhang, Shenyi, Mulati Tuerde, and Xijian Hu. 2023. "Stochastic Precipitation Generation for the Xilingol League Using Hidden Markov Models with Variational Bayes Parameter Estimation" Water 15, no. 20: 3600. https://doi.org/10.3390/w15203600