# Probabilistic Forecasting of Drought Events Using Markov Chain- and Bayesian Network-Based Models: A Case Study of an Andean Regulated River Basin

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Case Study

^{3}. The monthly seasonality is accounted by the division of the ten time series for each month. All data was standardized to jointly considered variables with different units.

#### 2.2. Drought Index

_{i,k}is the DI value for the kth month in the ith year, Z

_{i,1,k}the PC1 for the kth month in the ith year, and σ

_{k}the sample standard deviation of Z

_{i,1,k}of all years. Once the DI values for each month and year were calculated, they were chronological rearranged into a single time series.

#### 2.3. Markov Chain Models

^{th}order Markov chain is (for more details see Wilks [52]):

_{ij}represents the transition probability that Y

_{tn}is equal to category j given Y

_{tn−1}is equal to category i. The estimate of the transition probabilities (${\widehat{\text{p}}}_{\text{ij}}$) can be calculated to account for the conditional relative frequencies of transitions (f

_{ij}):

_{ij}is the frequency that Y is equal to the category i at time t

_{n−1}and equal to category j at the time t

_{n}. The value of s is the number of states of the system. The numerator represents the number of transitions of category i to category j, and the denominator stands for the sum of the number of transitions of category i to any other category.

_{hij}represent the transition probability that Y

_{tn}is equal to category j, given that Y

_{tn−1}is equal to the category i and Y

_{tn−2}equal to the category h. The transition probability estimates (${\widehat{\text{p}}}_{\text{hij}}$) are obtained from conditional relative frequencies of transition counts (f

_{hij}):

_{n−2}, category i at time t

_{n−1}, and category j at the time t

_{n}, and the denominator is the sum of the number of transitions categories h,i to any other category. For this condition, the probability of the drought status of the next time period depends on the status of the two previous time periods.

#### 2.4. Bayesian Network Models

_{t1}, ..., X

_{tn}}, where the dependency ordering follows the temporal sequence and the parents of X

_{ti}is the set of all prior variables (X

_{t1}, …, X

_{ti−1}), then Equation (8) can be expressed as:

_{tn}) are calculated given the predictor variables (X

_{t1}, …, X

_{tn−1}). The calculation of the joint probability distributions in the right-hand side of Equation (11) is a relatively cumbersome task, which can be considerably simplified using copula functions.

#### 2.5. Copulas

_{1}, …, U

_{n})

^{T}, where each margin U

_{i}, i = 1, …, n, is a uniform random variable over the unit interval. Suppose the joint cumulative distribution function (CDF) of (U

_{1}, …, U

_{n})

^{T}, is as follows:

_{1}, …, X

_{n}), for all x in the domain of F, can be expressed by a n-dimensional copula, as follows [13,44]:

_{i}(X

_{i}) represents the ith univariate marginal distribution on the unit interval [0,1] and C is the cumulative copula distribution function that represents the multivariate dependence structure [55]. If U

_{1}, ..., U

_{n}are all continuous and C is unique [45], Equation (11) can be expressed as:

_{t2}) are calculated given the predictor variable (X

_{t1}), i.e., the next drought status is conditional to the current status. Similarly, if n = 3 (dependence of second order), the conditional probabilities of the forecast variable (X

_{t3}) are calculated given the predictor variables (X

_{t1},X

_{t2}), i.e., the next drought status is conditional to the current status and the status of a previous step. These cases could be called Bayesian networks of the first order (BNFO) and second order (BNSO), respectively. Thus, replacing n = 2 and n = 3 in Equation (14) leads to the following expressions:

_{s}is the drought index that causes a drought status according to the threshold defined in Section 2.2 (xd

_{0}= 0 and xd

_{1}= −1). Applying Equations (17) and (18), the probabilities of having a category 0 (no drought), i.e., DI > 0, will be equal to:

#### 2.6. Copulas Fitting

_{i}) of step 1 can be derived when the DI values are transformed to a cumulative normal distribution function, in accordance with Section 2.2. Since droughts are slowly evolving phenomena, strong temporal autocorrelation among DI is expected [14]. Steps 2 and 3 consist in modeling the temporal dependence structure by fitting the copula functions given the marginal distribution.

_{E}) and the estimated parametric copula (C

_{θ}) function under the null hypothesis that C

_{E}∈ C

_{θ}. The latter is evaluated by the p value; if the p value is greater than the significance level α the null hypothesis is accepted, conversely, the null hypothesis is rejected. P values can be obtained via the Monte Carlo method embedded in a parametric bootstrapping procedure.

#### 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 |

**Figure 4.**Scatter plots of DI respectively between the months July–August, August–September and July–September for the period 1971–2010.

#### 3.2. Markov Chain Models

**Table 3.**Transition probability matrix of the MCFO model leaving the third year out in cross-validation for the month August.

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 |

**Table 4.**Transition probability matrix of the MCSO model leaving the third year out in cross-validation for the period July–August.

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 |

**Figure 5.**Probabilities of drought forecasts leaving the third year out in cross-validation using the BNFO model for the month of September given the DI of August corresponding respectively to the drought threshold values 0 and −1.

#### 3.4. Forecast Verification

_{Clim}was calculated on the basis of reference forecasts equal to climatological relative frequencies of the time series of categorical values of the DI. Considering the average of the RPS and RPS

_{Clim}values for the entire time series, i.e., taking into account all states of drought (drought, mild drought and no-drought) and all months, and, using Equation (27) to calculate the RPSS values, the MCFO model performed better compared to the other models with the greatest value of improvement over the reference forecast equal to 0.29, followed by the MCSO and BNFO models with values of RPSS equal to 0.21 and −0.05 (negative RPSS values mean that the reference forecasts are better that the tested model). Considering only the observed drought states (drought and mild drought), the BNFO model performed better with the greatest values of RPSS equal to 0.40, followed by the BNSO and MCFO models with RPSS values equal to 0.33 and 0.19, respectively. Even considering only the most severe drought status, the BNFO model yielded better results, with the greatest RPSS value equal to 0.44, followed by the MCSO and BNSO models with RPSS values equal to 0.37 and 0.35. These results indicate that, for the given case study, the MCFO model performed better for the probabilistic forecast of dry and wet periods, while, for the probabilistic forecast of dry periods, the BN-based models are a better option.

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

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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

**AMA Style**

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 Style**

Avilés, 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