# Forecasting Reservoir Water Levels Using Deep Neural Networks: A Case Study of Angat Dam in the Philippines

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## Abstract

**:**

## 1. Introduction

- We provide an analysis of deep neural networks for both short-and long-term forecasting horizons using multiple exogenous variables;
- We propose the use of a time series cross-validation method in order to obtain more robust estimates of prediction accuracy;
- We include simple baseline methods, namely the naive/persistence and seasonal mean methods, in order to properly contextualize the relative performance of the more sophisticated models;
- We examine prediction accuracy based on the inclusion or exclusion of several related exogenous variables.

## 2. Models and Methodology

#### 2.1. Study Area

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#### 2.2. Data Description

#### 2.3. Forecasting Methods

#### 2.3.1. Baseline Methods

#### 2.3.2. ARIMA

#### 2.3.3. Gradient Boosting Machines

#### 2.3.4. Deep Neural Networks

#### 2.4. One-Step and Multi-Step Forecasting

#### 2.5. Evaluating Model Performance

^{2}statistic as a goodness-of-fit measure. These are defined as,

## 3. Results and Discussion

#### 3.1. Time Series Description and Characteristics

#### 3.2. Model Performance Results and Analysis

^{2}statistics, which illustrates the goodness-of-fit of the models for each prediction horizon. In the discussions below, we focus our attention on the MAE, RMSE, and MAPE metrics as these statistics more appropriately measure prediction accuracy and how well the model generalizes any unseen (i.e., future) data.

#### 3.3. Effects of Exogenous Variables on Multi-Step Forecasting Performance

^{2}values were very minimal across the combinations. As such, we omit them from our results.

#### 3.4. Practical Implications

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The study area consists of the Angat Dam, located within the Angat Watershed. The Angat Dam is a multipurpose structure used in irrigation, water supply, hydroelectric power, and flood control. The dam is the main source of Metro Manila’s water supply system, serving more than 1 million households or about 6 million Filipinos in the capital of the Philippines. Image from [19].

**Figure 2.**Historical water levels (above mean sea level) of the Angat Dam, from 1 January 2001 to 30 April 2021, in meters. The time series shows a drastic drop in 2019 (corresponding to the 2019 Metro Manila Water Crisis) that has only been seen on one other occasion, in 2010. Blue observations denote the training set and green observations denote the test set.

**Figure 3.**Historical daily rainfall for the Angat Dam in millimeters. Spikes in the observations are attributed to periods of intense rain caused by the monsoon season or typhoons. Since the Philippines has a distinct rainy season, the annual seasonality of rainfall is reflected in this plot. The yearly peaks usually occur in the middle of this rainy season. Blue observations denote the training set and green observations denote the test set.

**Figure 4.**Oceanic Niño Index. The index is used to monitor the El Niño-Southern Oscillation and is calculated by averaging sea surface temperature anomalies in an area of the east-central equatorial Pacific Ocean. Monthly observations are transformed to daily frequency using a naïve interpolation. Blue observations denote the training set and green observations denote the test set. Data obtained from [22].

**Figure 5.**Historical irrigation releases from the Angat Dam in cubic meters per second (cms). As the Angat Dam is also the primary source of irrigation for nearby farmlands, a portion of the water stored is allocated regularly for irrigation use. Blue observations denote the training set and green observations denote the test set.

**Figure 6.**Autocorrelation Plot of Historical Water Levels for lags up to 730. Seasonality is illustrated by the periodic shape, with peaks occurring every 365 days.

**Figure 7.**Autocorrelation Plot of the Historical Water Levels after applying seasonal differencing with $l=365$ and first-order differencing. First-order differencing is applied when fitting the ARIMA model with $d=1$.

**Figure 8.**A representation of an LSTM cell. From left to right, the σ-blocks are called the forget, input, and output gates. The forget gate controls the amount of information allowed to flow in from the past, while the input and output gates look at the current input and previous hidden state to determine the outputs of the cell. In the diagram above, the σ refers to a sigmoid activation function.

**Figure 9.**Our DNN architecture. The encoder LSTM takes in lagged observations of the water level $y$ and exogenous variables $x$ and encodes them into a hidden state $h$ and a cell state $c$. These encoder states are then used as the initial states for the decoder LSTM, which accepts known future dates $d$ as input. The decoder outputs are then passed to a time distributed dense layer, which generates the forecasts.

**Figure 10.**TSCV for one-step-ahead forecasting. The blue dots refer to the training set while the green dots refer to the test set. The sampling distribution of an error statistic is estimated over a sequence of constructed train-test partitions.

**Figure 11.**Sample forecasts for each prediction horizon and each forecasting method. The black line depicts the actual observed water level, while the dashed lines indicate the model forecasts. We note that forecasts such as these are generated at each step of TSCV and their error and goodness-of-fit statistics are calculated at each step. As such, we note that these plots do not reflect the average performance of each model.

**Figure 12.**Sample 180-day and 90-day forecasts for the DNN models with 95% prediction intervals estimated using bootstrapped residuals.

**Table 1.**List of variables used in this study. All variables have a daily granularity, and the units for each are as indicated.

Variable | Units | Training Set | Test Set |
---|---|---|---|

Water Level | meters | 1 January 2001 to 31 December 2017 | 1 January 2018 to 30 April 2021 |

Rainfall | millimeters | ||

Oceanic Niño Index | |||

Irrigation Releases | cubic meters per second |

Model | Parameter | Value |
---|---|---|

ARIMA | $p$ | 1 |

$d$ | 1 | |

$q$ | 2 |

Model | Forecast Horizon | Window Size | Parameters | Value |
---|---|---|---|---|

GBM (Multivariate) | 1 | 7 | loss | least squares |

30 | 60 | learning rate | 0.1 | |

90 | 180 | max depth | 5 | |

180 | 365 | n estimators | 50 |

Model | Forecast Horizon | Window Size | LSTM Units | Parameters | Value |
---|---|---|---|---|---|

DNN-U (Univariate) | 1 | 365 | 64 | activation | tanh |

30 | epochs | 50 | |||

DNN-M (Multivariate) | 90 | batch size optimizer | 63 Adam | ||

180 | loss | Huber |

1-Day Forecast | ||||
---|---|---|---|---|

Model | MAE | RMSE | MAPE | R^{2} |

Naïve | 0.291 (0.424) | 0.291 (0.424) | 0.002 (0.002) | 0.999 |

Seasonal Mean | 8.088 (6.080) | 8.088 (6.080) | 0.043 (0.036) | 0.560 |

ARIMA | 0.261 (0.467) | 0.261 (0.467) | 0.001 (0.002) | 0.999 |

GBM | 0.256 (0.411) | 0.256 (0.411) | 0.001 (0.001) | 0.999 |

DNN-U | 0.198 (0.390) | 0.198 (0.390) | 0.001 (0.002) | 0.999 |

DNN-M | 0.239 (0.372) | 0.239 (0.372) | 0.001 (0.002) | 0.999 |

30-Day Forecast | ||||
---|---|---|---|---|

Model | MAE | RMSE | MAPE | R^{2} |

Naïve | 3.344 (2.631) | 3.909 (2.991) | 0.017 (0.014) | 0.890 |

Seasonal Mean | 8.136 (5.818) | 8.371 (5.795) | 0.043 (0.034) | 0.560 |

ARIMA | 3.379 (2.732) | 4.008 (3.148) | 0.017 (0.014) | 0.896 |

GBM | 3.059 (2.501) | 4.557 (2.818) | 0.016 (0.005) | 0.890 |

DNN-U | 2.938 (2.368) | 3.322 (2.576) | 0.015 (0.013) | 0.906 |

DNN-M | 2.892 (2.263) | 3.273 (2.493) | 0.015 (0.013) | 0.910 |

90-Day Forecast | ||||
---|---|---|---|---|

Model | MAE | RMSE | MAPE | R^{2} |

Naïve | 8.390 (5.469) | 9.825 (6.157) | 0.044 (0.029) | 0.358 |

Seasonal Mean | 8.210 (4.954) | 8.987 (5.069) | 0.044 (0.030) | 0.565 |

ARIMA | 7.934 (5.287) | 9.339 (6.021) | 0.041 (0.028) | 0.411 |

GBM | 6.209 (3.603) | 8.261 (3.989) | 0.033 (0.006) | 0.638 |

DNN-U | 5.371 (2.965) | 6.243 (3.312) | 0.028 (0.016) | 0.735 |

DNN-M | 5.103 (3.091) | 5.962 (3.510) | 0.027 (0.017) | 0.746 |

180-Day Forecast | ||||
---|---|---|---|---|

Model | MAE | RMSE | MAPE | R^{2} |

Naïve | 13.408 (7.079) | 15.475 (7.856) | 0.070 (0.040) | −0.702 |

Seasonal Mean | 8.469 (3.791) | 9.911 (3.845) | 0.046 (0.023) | 0.472 |

ARIMA | 11.431 (4.947) | 13.614 (5.422) | 0.060 (0.026) | −0.360 |

GBM | 7.335 (2.743) | 9.578 (3.109) | 0.039 (0.005) | 0.518 |

DNN-U | 7.113 (2.982) | 8.668 (3.445) | 0.038 (0.018) | 0.547 |

DNN-M | 6.652 (3.092) | 8.128 (3.767) | 0.036 (0.019) | 0.581 |

**Table 9.**Average estimates for MAE and RMSE, and their standard deviations calculated using TSCV for a 180-day forecast across different combinations of exogenous variables.

180-Day Forecast (DNN) | ||
---|---|---|

Exog. Variables | MAE | RMSE |

Rain, Irrigation, ONI | 6.652 (3.092) | 8.128 (3.767) |

Rain, Irrigation | 6.458 (3.432) | 7.997 (4.003) |

Rain, ONI | 6.739 (3.479) | 8.290 (4.213) |

Irrigation, ONI | 7.040 (2.708) | 8.523 (3.341) |

Rain | 6.601 (3.352) | 8.142 (3.942) |

Irrigation | 6.551 (3.248) | 7.999 (3.827) |

ONI | 6.814 (2.450) | 8.258 (3.025) |

No Exog. | 7.113 (2.982) | 8.668 (3.445) |

**Table 10.**Average estimates for MAE and RMSE, and their standard deviations calculated using TSCV for a 90-day forecast across different combinations of exogenous variables.

90-Day Forecast (DNN) | ||
---|---|---|

Exog. Variables | MAE | RMSE |

Rain, Irrigation, ONI | 5.103 (3.091) | 5.962 (3.510) |

Rain, Irrigation | 5.034 (3.317) | 5.870 (3.729) |

Rain, ONI | 5.146 (3.393) | 5.986 (3.812) |

Irrigation, ONI | 5.367 (2.869) | 6.225 (3.216) |

Rain | 5.108 (3.278) | 5.960 (3.698) |

Irrigation | 5.370 (3.022) | 6.223 (3.479) |

ONI | 5.375 (2.773) | 6.272 (3.114) |

No Exog. | 5.371 (2.965) | 6.243 (3.312) |

**Table 11.**Average estimates for MAE and RMSE, and their standard deviations calculated using TSCV for a 30-day forecast across different combinations of exogenous variables.

30-Day Forecast (DNN) | ||
---|---|---|

Exog. Variables | MAE | RMSE |

Rain, Irrigation, ONI | 2.892 (2.263) | 3.273 (2.493) |

Rain, Irrigation | 2.880 (2.512) | 3.253 (2.692) |

Rain, ONI | 2.953 (2.437) | 3.321 (2.632) |

Irrigation, ONI | 3.039 (2.375) | 3.398 (2.588) |

Rain | 2.763 (2.283) | 3.160 (2.516) |

Irrigation | 2.901 (2.249) | 3.315 (2.486) |

ONI | 2.942 (2.318) | 3.338 (2.543) |

No Exog. | 2.938 (2.368) | 3.322 (2.576) |

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**MDPI and ACS Style**

Ibañez, S.C.; Dajac, C.V.G.; Liponhay, M.P.; Legara, E.F.T.; Esteban, J.M.H.; Monterola, C.P.
Forecasting Reservoir Water Levels Using Deep Neural Networks: A Case Study of Angat Dam in the Philippines. *Water* **2022**, *14*, 34.
https://doi.org/10.3390/w14010034

**AMA Style**

Ibañez SC, Dajac CVG, Liponhay MP, Legara EFT, Esteban JMH, Monterola CP.
Forecasting Reservoir Water Levels Using Deep Neural Networks: A Case Study of Angat Dam in the Philippines. *Water*. 2022; 14(1):34.
https://doi.org/10.3390/w14010034

**Chicago/Turabian Style**

Ibañez, Sebastian C., Carlo Vincienzo G. Dajac, Marissa P. Liponhay, Erika Fille T. Legara, Jon Michael H. Esteban, and Christopher P. Monterola.
2022. "Forecasting Reservoir Water Levels Using Deep Neural Networks: A Case Study of Angat Dam in the Philippines" *Water* 14, no. 1: 34.
https://doi.org/10.3390/w14010034