Probabilistic Forecasting of Drought Events Using Markov Chain- and Bayesian Network-Based Models: A Case Study of an Andean Regulated River Basin
Abstract
:1. Introduction
2. Materials and Methods
2.1. Case Study
2.2. Drought Index
2.3. Markov Chain Models
2.4. Bayesian Network Models
2.5. Copulas
2.6. Copulas Fitting
2.7. Forecast Verification
3. Results and Discussion
3.1. Drought Index
PC1 | January | February | March | April | May | June | July | August | September | October | November | December |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Eigenvalues | 6.21 | 7.26 | 7.89 | 7.13 | 7.64 | 7.08 | 7.10 | 7.16 | 7.08 | 6.77 | 5.86 | 5.71 |
Explained variance | 62% | 73% | 79% | 71% | 76% | 71% | 71% | 72% | 71% | 68% | 59% | 57% |
Category | Drought State | Frequency |
---|---|---|
0 | no drought | 218 |
1 | mild drought | 185 |
2 | drought | 77 |
3.2. Markov Chain Models
State i | State j | ||
---|---|---|---|
0 | 1 | 2 | |
0 | 0.72 | 0.28 | 0.00 |
1 | 0.07 | 0.86 | 0.07 |
2 | 0.00 | 0.57 | 0.43 |
States h-i | State j | ||
---|---|---|---|
0 | 1 | 2 | |
0-0 | 0.75 | 0.25 | 0.00 |
0-1 | 0.00 | 1.00 | 0.00 |
0-2 | 0.33 | 0.33 | 0.33 |
1-0 | 0.50 | 0.50 | 0.00 |
1-1 | 0.09 | 0.82 | 0.09 |
1-2 | 0.00 | 0.33 | 0.67 |
2-0 | 0.33 | 0.33 | 0.33 |
2-1 | 0.00 | 1.00 | 0.00 |
2-2 | 0.00 | 0.75 | 0.25 |
3.3. Bayesian Network Models
p Value | ||||||||||||
Copulas | January | February | March | April | May | June | July | August | September | October | November | December |
Normal | 0.292 | 0.233 | 0.077 | 0.329 | 0.368 | 0.527 | 0.369 | 0.727 | 0.189 | 0.222 | 0.524 | 0.664 |
t | 0.057 | 0.108 | 0.066 | 0.358 | 0.338 | 0.313 | 0.464 | 0.432 | 0.127 | 0.322 | 0.326 | 0.326 |
Clayton | 0.007 | 0.022 | 0.052 | 0.022 | 0.009 | 0.034 | 0.005 | 0.020 | 0.004 | 0.002 | 0.021 | 0.017 |
Frank | 0.363 | 0.910 | 0.854 | 0.534 | 0.535 | 0.565 | 0.804 | 0.864 | 0.207 | 0.170 | 0.426 | 0.193 |
S-statistic | ||||||||||||
Copulas | January | February | March | April | May | June | July | August | September | October | November | December |
Normal | 0.028 | 0.027 | 0.033 | 0.024 | 0.023 | 0.021 | 0.022 | 0.018 | 0.029 | 0.029 | 0.022 | 0.020 |
t | 0.040 | 0.033 | 0.034 | 0.024 | 0.024 | 0.025 | 0.022 | 0.024 | 0.034 | 0.028 | 0.027 | 0.027 |
Clayton | - | - | 0.049 | - | - | - | - | - | - | - | - | - |
Frank | 0.027 | 0.018 | 0.018 | 0.023 | 0.022 | 0.023 | 0.019 | 0.019 | 0.030 | 0.034 | 0.026 | 0.031 |
p Value | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Copulas | January | February | March | April | May | June | July | August | September | October | November | December |
Normal | 0.083 | 0.164 | 0.103 | 0.582 | 0.698 | 0.651 | 0.400 | 0.276 | 0.459 | 0.542 | 0.433 | 0.437 |
t | 0.007 | 0.079 | 0.087 | 0.406 | 0.523 | 0.254 | 0.236 | 0.110 | 0.305 | 0.294 | 0.064 | 0.088 |
Clayton | 0.001 | 0.006 | 0.009 | 0.010 | 0.005 | 0.008 | 0.004 | 0.000 | 0.001 | 0.000 | 0.000 | 0.000 |
Frank | 0.314 | 0.718 | 0.648 | 0.588 | 0.661 | 0.605 | 0.865 | 0.136 | 0.108 | 0.068 | 0.074 | 0.259 |
S-statistic | ||||||||||||
Copulas | January | February | March | April | May | June | July | August | September | October | November | December |
Normal | 0.047 | 0.038 | 0.042 | 0.027 | 0.026 | 0.026 | 0.031 | 0.035 | 0.032 | 0.030 | 0.032 | 0.032 |
t | - | 0.048 | 0.046 | 0.033 | 0.031 | 0.036 | 0.039 | 0.046 | 0.038 | 0.039 | 0.051 | 0.049 |
Clayton | - | - | - | - | - | - | - | - | - | - | - | - |
Frank | 0.045 | 0.032 | 0.034 | 0.035 | 0.033 | 0.034 | 0.028 | 0.055 | 0.061 | 0.064 | 0.063 | 0.049 |
3.4. Forecast Verification
Model | January | February | March | April | May | June | July | August | September | October | November | December |
---|---|---|---|---|---|---|---|---|---|---|---|---|
(a) No Drought, Mild Drought and Drought | ||||||||||||
MCFO | −0.40 | 0.27 | 0.63 | 0.52 | 0.37 | 0.41 | 0.48 | 0.45 | 0.33 | 0.20 | 0.15 | 0.15 |
MCSO | −0.46 | −0.32 | 0.63 | 0.58 | 0.25 | 0.32 | 0.50 | 0.52 | 0.18 | 0.19 | 0.05 | 0.16 |
BNFO | −0.50 | 0.13 | 0.09 | 0.06 | −0.20 | 0.08 | 0.04 | 0.00 | −0.11 | −0.15 | 0.02 | −0.09 |
BNSO | −0.60 | −0.33 | −0.11 | −0.08 | −0.29 | −0.08 | −0.14 | −0.13 | −0.30 | −0.33 | −0.09 | −0.27 |
(b) Mild Drought and Drought | ||||||||||||
MCFO | −0.35 | 0.12 | 0.53 | 0.49 | 0.27 | 0.19 | 0.42 | 0.31 | 0.04 | 0.12 | 0.09 | 0.18 |
MCSO | −0.28 | −0.28 | 0.53 | 0.53 | 0.15 | 0.07 | 0.48 | 0.47 | −0.14 | 0.10 | −0.03 | 0.16 |
BNFO | 0.29 | 0.53 | 0.48 | 0.59 | 0.29 | 0.38 | 0.47 | 0.47 | −0.07 | 0.32 | 0.45 | 0.62 |
BNSO | 0.28 | 0.50 | 0.37 | 0.53 | 0.23 | 0.20 | 0.38 | 0.39 | −0.27 | 0.18 | 0.42 | 0.53 |
4. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Avilés, A.; Célleri, R.; Solera, A.; Paredes, J. Probabilistic Forecasting of Drought Events Using Markov Chain- and Bayesian Network-Based Models: A Case Study of an Andean Regulated River Basin. Water 2016, 8, 37. https://doi.org/10.3390/w8020037
Avilés A, Célleri R, Solera A, Paredes J. Probabilistic Forecasting of Drought Events Using Markov Chain- and Bayesian Network-Based Models: A Case Study of an Andean Regulated River Basin. Water. 2016; 8(2):37. https://doi.org/10.3390/w8020037
Chicago/Turabian StyleAvilés, Alex, Rolando Célleri, Abel Solera, and Javier Paredes. 2016. "Probabilistic Forecasting of Drought Events Using Markov Chain- and Bayesian Network-Based Models: A Case Study of an Andean Regulated River Basin" Water 8, no. 2: 37. https://doi.org/10.3390/w8020037