# Characterization and Prediction of Water Stress Using Time Series and Artificial Intelligence Models

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Description of the Data and Study Area

#### 2.2. Standardized Precipitation Index (SPI)

_{0}= 2.515517, C

_{1}= 0.802853, C

_{2}= 0.010328, d

_{1}= 1.432788, d

_{2}=0.189269, and d

_{3}= 0.001308. The SPI for time scales 3 (SPI3) and 6 (SPI6) represent short-term and mid-term meteorological droughts, respectively [37]. The SPI is categorized in to different groups as mention in Table 1 below:

#### 2.3. ARIMA Model

_{t}time series to be used as predictors. ‘q’ is the order of the ‘moving average’ (MA) term, assuming the number of lagged forecast errors that should go into the ARIMA model. ‘d’ is the number of differences required to make the time series stationary; the most common approach of differencing is, subtract the previous value from the current value.

#### 2.4. Artificial Neural Network (ANN)

_{t}of a single-layer or multi-layer feed-forward autoregressive network is expressed as follows:

#### 2.5. Support Vector Regression (SVR)

_{i}is the scalar output, and N is the size of data set. The general equation SVR can be written as follows [41]:

_{i}is the actual value, $f\left({x}_{i}\right)$ is the estimated value, and $\epsilon $ is the margin of tolerance where no penalty is given to errors.

## 3. Results

#### 3.1. Calculated SPI

#### 3.2. ARIMA Model

#### 3.3. ANN Model

#### 3.4. SVR Model

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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SPI Range | Category |
---|---|

+2 to more | Extremely wet |

1.5 to1.99 | Very wet |

1.0 to 1.49 | Moderately wet |

−0.99 to 0.99 | Near normal |

−1.0 to −1.49 | Moderately dry |

−1.5 to −1.99 | Severely dry |

−2 to less | Extremely dry |

Year | Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1982 | −1.21 | |||||||||||

1983 | −1.56 | −2.04 | −2.25 | −3.15 | −2.13 | −1.30 | −1.48 | |||||

1984 | −1.56 | −1.23 | −1.89 | |||||||||

1985 | −1.15 | |||||||||||

1988 | −1.30 | |||||||||||

1989 | −1.19 | −1.43 | −1.22 | −1.08 | −1.34 | |||||||

1990 | −1.34 | −1.64 | −1.78 | −1.17 | ||||||||

1991 | −1.51 | |||||||||||

1992 | −2.06 | −1.11 | −1.49 | |||||||||

1993 | −2.04 | −1.12 | −1.71 | |||||||||

1994 | −1.21 | |||||||||||

1997 | −1.30 | −1.48 | ||||||||||

1998 | −1.60 | −1.00 | −1.92 | |||||||||

1999 | −1.14 | |||||||||||

2002 | −1.26 | |||||||||||

2003 | −1.37 | −1.48 | −1.67 | |||||||||

2004 | −1.15 | −1.13 | ||||||||||

2007 | −2.08 | |||||||||||

2010 | −1.32 | −1.67 | ||||||||||

2015 | −2.25 | |||||||||||

2018 | −2.08 | −2.49 | −1.27 | −1.97 | −1.11 | −1.80 | ||||||

2019 | −1.12 | −1.82 | −1.66 |

Year | Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1983 | −1.18 | −1.08 | −2.66 | |||||||||

1985 | −1.37 | −1.33 | −1.36 | |||||||||

1986 | −1.86 | −1.53 | ||||||||||

1989 | −1.27 | −1.35 | −1.19 | |||||||||

1990 | −1.11 | −1.18 | ||||||||||

1991 | −2.42 | |||||||||||

1996 | −1.37 | −1.28 | −1.07 | |||||||||

2000 | −1.08 | −1.02 | −1.04 | |||||||||

2002 | −1.21 | −1.86 | −1.56 | −1.98 | ||||||||

2003 | −1.72 | −1.24 | ||||||||||

2007 | −1.53 | |||||||||||

2011 | −2.20 | −1.85 | ||||||||||

2016 | −2.66 | |||||||||||

2018 | −1.95 | −1.62 | ||||||||||

2019 | −1.66 |

**Table 4.**Parameter estimation of ARIMA by maximum likelihood method for SPI3 and SPI6 time scales of Hyderabad region.

Time Scales | Model | Parameters | Estimation | S. E | Z Value | Prob. | Model Fitting | Box–Pierce Non-Correlation Test | ||
---|---|---|---|---|---|---|---|---|---|---|

SPI3 | ARIMA (1,0,2) | AR1 | −0.018 | 0.130 | −0.143 | p < 0.0001 | Log likelihood | −544.14 | Original | Residuals |

MA1 | 0.674 | 0.121 | 5.559 | p < 0.0001 | BIC | 1112.79 | Chi-square = 148.39 (p < 0.001) | Chi-square = 0.005 (p = 0.938) | ||

MA2 | 0.393 | 0.066 | 5.956 | p < 0.0001 | AIC | 1096.27 | ||||

SPI6 | ARIMA (1,0,0) | AR1 | 0.717 | 0.032 | 22.03 | p < 0.0001 | Log likelihood | −479.53 | Chi-square = 233.19 (p < 0.001) | Chi-square = 0.078 (p = 0.779) |

AIC | 963.07 | |||||||||

BIC | 971.31 |

Sample | Dimension | SPI3 | SPI6 | ||
---|---|---|---|---|---|

Statistics | Probability | Statistics | Probability | ||

eps(1) | m = 2 | 15.63 | p < 0.0001 | 15.68 | p < 0.0001 |

m = 3 | 15.37 | p < 0.0001 | 18.09 | p < 0.0001 | |

eps(2) | m = 2 | 14.23 | p < 0.0001 | 12.72 | p < 0.0001 |

m = 3 | 13.11 | p < 0.0001 | 12.09 | p < 0.0001 | |

eps(3) | m = 2 | 13.90 | p < 0.0001 | 12.51 | p < 0.0001 |

m = 3 | 12.34 | p < 0.0001 | 11.18 | p < 0.0001 | |

eps(4) | m = 2 | 15.40 | p < 0.0001 | 13.92 | p < 0.0001 |

m = 3 | 13.76 | p < 0.0001 | 12.43 | p < 0.0001 |

Parameters | SPI3 | SPI6 |
---|---|---|

Input lag | 5 | 4 |

Output variable/dependent | 1 | 1 |

Hidden nodes | 1 | 1 |

Hidden layers | 2 | 4 |

Model | (5:2S:1L) | (4:4S:1L) |

Total number of parameters | 15 | 25 |

Network type | Feed forward | Feed forward |

Activation function (I:H) | Sigmoidal | Sigmoidal |

Activation function (H:O) | Identity | Identity |

Box–pierce non-correlation test for residuals | 0.018 | 0.124 |

p-value | 0.873 | 0.724 |

Model | SPI3 | SPI6 |
---|---|---|

Parameters | Values | Values |

Kernel function | RBF | RBF |

No. of S. V’s | 451 | 337 |

Cost | 7.9 | 7.7 |

Gamma | 0.16 | 0.16 |

Epsilon | 0.01 | 0.1 |

Cross validation error | 0.043 | 0.029 |

Box–Pierce non-correlation test for residuals | 0.19 | 0.31 |

p-value | 0.27 | 0.57 |

Category | SPI3 | SPI6 |
---|---|---|

Extremely wet | - | - |

Very wet | - | - |

Moderately wet | - | - |

Near normal | - | - |

Moderately dry | Oct-1982, Sep-1983, Oct-1983, Jul-1984, Dec-1985, Dec-1988, Jan-1989, Feb-1989, Apr-1989, Nov-1989, Dec-1989, Jan-1990, Apr-1990, Jun-1992, Jul-1992, Jun-1993, Oct-1994, Aug-1997, Sep-1997, Jun-1998, Nov-1999, Oct-2002, Mar-2003, Jun-2003, Sep-2004, Nov-2004, Apr-2010, Sep-2018, Nov-2018, Jan-2019 | Jan-1983, Feb-1983, Jun-1985, Jul-1985, Aug-1985, Jan-1989, Feb-1989, Mar-1989, Jan-1990, Feb-1990, Jul-1996, Aug-1996, Sep-1996, Fep-2000, Nov-2000, Dec-2000, Aug-2002, Feb-2003 |

Severely dry | Jan-1983, Jun-1984, Aug-1984, Feb-1990, Mar-1990, Mar-1991, May-1998, Jul-1998, Jul-2003, May-2010, Oct-2018, Dec-2018, May-2019, Jun-2019 | Oct-1986, Nov-1986, Sep-2002, Oct-2002, Dec-2002, Jan-2003, Jun-2007, Mar-2011, Nov-2018, Dec-2018, Jan-2019 |

Extremely dry | Feb-1983, Mar-1983, May-1983, Mar-1992, Mar-1993, Jul-1993, Mar-2007, Jul-2015, Jan-2018, Feb-2018 | May-1983, May-1991, Feb-2011, Mar-2016 |

Model | Parameter | ARIMA | ANN | SVR |
---|---|---|---|---|

SPI3 | MSE | 0.438 | 0.418 | 0.378 |

RMSE | 0.496 | 0.459 | 0.413 | |

SPI6 | MSE | 0.328 | 0.275 | 0.237 |

RMSE | 0.363 | 0.317 | 0.289 |

Period | Actual | Forecasted | ||||||
---|---|---|---|---|---|---|---|---|

ARIMA | ANN | SVR | ||||||

SPI3 | SPI6 | SPI3 | SPI6 | SPI3 | SPI6 | SPI3 | SPI6 | |

Jun-20 | 0.917 | 0.851 | 0.241 | 0.083 | 0.132 | 0.254 | 0.197 | 0.064 |

Jul-20 | 1.477 | 1.418 | 0.194 | 0.11 | 0.07 | 0.182 | 0.713 | 0.669 |

Aug-20 | 0.133 | 0.12 | −0.004 | 0.082 | −0.089 | 0.131 | 0.554 | 0.83 |

Sep-20 | 0.226 | 0.262 | −0.689 | 0.085 | −0.068 | 0.094 | −0.085 | 0.352 |

Oct-20 | 0.279 | 0.400 | −0.812 | 0.059 | −0.083 | 0.067 | −0.186 | −0.501 |

Nov-20 | 1.734 | 0.369 | −0.987 | 0.433 | 0.357 | 0.048 | 0.403 | 0.338 |

Dec-20 | 1.142 | 0.322 | −0.988 | 0.025 | −0.072 | 0.035 | 0.87 | 0.294 |

Jan-21 | −0.342 | 0.23 | −0.567 | 0.096 | −0.069 | 0.025 | −0.549 | 0.25 |

Feb-21 | −0.636 | 1.646 | −0.288 | −0.028 | −0.074 | 0.018 | −0.804 | 0.839 |

Mar-21 | −0.853 | 0.84 | −0.755 | −0.013 | −0.07 | 0.013 | −0.025 | 1.416 |

Apr-21 | −0.138 | −0.526 | 1.988 | −0.02 | −0.072 | 0.009 | −0.994 | 0.399 |

May-21 | 1.709 | 1.491 | 1.832 | −0.025 | −0.07 | 0.007 | −0.111 | 0.227 |

MSE | 1.317 | 0.72 | 0.867 | 0.672 | 0.686 | 0.495 | ||

RMSE | 1.734 | 0.848 | 0.931 | 0.82 | 0.828 | 0.703 |

**Table 11.**Diebold–Mariano (DM) test for comparing performance of different time series models for training and testing data set for both SPI3 and SPI6.

Model | Data Type | M1, M2 | M1, M3 | M2, M3 |
---|---|---|---|---|

SPI3 | Training set | 5.62 (p < 0.0001) | 5.05 (p < 0.0001) | −0.38 (p < 0.0001) |

Testing set | 3.67 (p < 0.0001) | 2.57 (p < 0.0001) | −0.12 (p < 0.0001) | |

SPI6 | Training set | 7.77 (p < 0.0001) | 6.24 (p < 0.0001) | 4.07 (p < 0.0001) |

Testing set | 0.75 (p < 0.0001) | 0.62 (p < 0.0001) | 0.52 (p < 0.0001) |

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Gorlapalli, A.; Kallakuri, S.; Sreekanth, P.D.; Patil, R.; Bandumula, N.; Ondrasek, G.; Admala, M.; Gireesh, C.; Anantha, M.S.; Parmar, B.;
et al. Characterization and Prediction of Water Stress Using Time Series and Artificial Intelligence Models. *Sustainability* **2022**, *14*, 6690.
https://doi.org/10.3390/su14116690

**AMA Style**

Gorlapalli A, Kallakuri S, Sreekanth PD, Patil R, Bandumula N, Ondrasek G, Admala M, Gireesh C, Anantha MS, Parmar B,
et al. Characterization and Prediction of Water Stress Using Time Series and Artificial Intelligence Models. *Sustainability*. 2022; 14(11):6690.
https://doi.org/10.3390/su14116690

**Chicago/Turabian Style**

Gorlapalli, Amuktamalyada, Supriya Kallakuri, Pagadala Damodaram Sreekanth, Rahul Patil, Nirmala Bandumula, Gabrijel Ondrasek, Meena Admala, Channappa Gireesh, Madhyavenkatapura Siddaiah Anantha, Brajendra Parmar,
and et al. 2022. "Characterization and Prediction of Water Stress Using Time Series and Artificial Intelligence Models" *Sustainability* 14, no. 11: 6690.
https://doi.org/10.3390/su14116690