# Flood Prediction and Uncertainty Estimation Using Deep Learning

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. LSTM Network

^{w}x*

_{1},….x

_{n}} and output vector, Y = {y

_{1},…..,y

_{n}}. The gates consist of a sigmoid neural network layer and a point wise multiplication operator. A value of one indicates letting through all data while a value of zero does not allow any of the data to be used. The first gate layer (the “forget” gate layer, represented in “yellow” in Figure 2) takes output from the previous step (y

_{t − 1}) and current input (x

_{t}) and outputs a value between 0 and 1, indicating how much information is to be passed on. The output from the “forget” gate is represented by f

_{t}in Equation (2), where matrices U and W contain weights and recurrent connections respectively.

_{t}= σ(x

_{t}U

^{f}+ y

_{t − 1}W

^{f})

_{t}= σ(x

_{t}U

^{i}+ y

_{t − 1}W

^{i})

_{t}= tanh (x

_{t}U

^{g}+ y

_{t − 1}W

^{g})

_{t}) which has minor interactions with rest of the components. The old state (C

_{t − 1}) is multiplied by f

_{t}to allow for the possible “forgetting” of the corresponding information. In the next step, the product of i

_{t}and Ĉ

_{t}is added to provide new information to the cell state as shown in the Equation (5).

_{t}= f

_{t}C

_{t − 1}+ i

_{t}Ĉ

_{t }

_{t}) as shown in Equations (6) and (7).

_{t}= σ(x

_{t}U

^{o}+ y

_{t − 1}W

^{o})

_{t}= tanh(C

_{t}) × o

_{t}

#### 2.2. Performamnce Metrics

_{i}) values and RMSE measures square root of the mean of the squared errors. Lower values indicate a better model fit for the data for both the metrics. With RMSE, the errors are squared before the average, therefore, prioritizes larger errors. In situations where larger differences can affect the model, RMSE can be a better evaluation measure, otherwise MAE is more appropriate.

#### 2.3. Uncertainty Modeling

_{1},….x

_{n}} and outputs Y = {y

_{1},…..,y

_{n}}, the resulting function developed by the forecasting algorithm is given by Y = f

^{ω}(X), where “ω” represents the parameters of the algorithm. In this case, “ω” represents the weights and bias of the LSTM network. With Bayesian modeling, a prior distribution of the model parameters p(ω) is assumed. The corresponding likelihood distribution is defined by p(y|x, ω ). A posterior distribution is then evaluated after observing the data using Bayes’ theorem as given in Equation (10).

^{*}) on all the values of “ω” [36].

^{ω}(x*), ω) p(ω|X,Y) dω

^{i}(x) represents the network in each iteration and “p” represents the prior distribution of the network parameters.

_{m, …..,}x

_{n}) of the original training data (X = x

_{1, …..,}x

_{n}) is selected. The value of “m” or the starting point of the subset is randomly generated from a set of values, a larger range of these values results in a larger variation allowing for the control of the uncertainty estimates. Finally, the three different techniques adding input noise, dropout and data sub-selection are compared to identify the better model for this problem.

#### 2.4. Quality of Uncertainty

#### 2.5. Data

## 3. Results

#### 3.1. Comparison of Statistical and Deep Learning for Gauge Height Prediction

#### 3.2. Effect of Dropout on Model Performance

#### 3.3. Uncertainty Estimation

#### 3.4. Model Validation

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Gauge location in Meramec River [40].

**Figure 12.**Available flood mapping locations in Flood Inundation Mapper (FIM) [42].

Parameters | AIC | BIC |
---|---|---|

(0,1,1) | −258,152.156 | −258,123.893 |

(0,1,0) | −248,137.561 | −248,118.719 |

(1,1,0) | −266,952.850 | −266,924.587 |

(2,1,2) | −303,596.995 | −303,540.467 |

(1,1,2) | −303,594.450 | −303,547.344 |

(2,1,1) | −303,212.512 | −303,165.405 |

(3,1,2) | −303,479.822 | −303,413.873 |

(2,1,3) | −303,605.067 | −303,539.118 |

(1,1,3) | −303,607.796 | −303,551.268 |

(0,1,3) | −278,432.067 | −278,384.961 |

(1,1,4) | −303,601.785 | −303,535.836 |

(0,1,2) | −270,281.324 | −270,243.639 |

(0,1,4) | −283,568.922 | −283,512.394 |

(2,1,4) | −303,603.607 | −303,528.236 |

Layers | Output |
---|---|

Input Layer | (None,1,90) |

LSTM Layer | (None,20) |

Dense Layer | (None,1) |

Forecasts | 1 |

Model | RMSE | MAE |
---|---|---|

ARIMA | 2.0813 | 1.9442 |

LSTM | 1.9558 | 1.7010 |

**Table 4.**Root mean square error and mean absolute error results of LSTM and ARIMA on “Out of sample” predictions.

Model | RMSE | MAE |
---|---|---|

ARIMA | 0.8722 | 0.6732 |

LSTM | 0.3316 | 0.2595 |

Model | RMSE | MAE |
---|---|---|

Without dropout | 0.1965 | 0.1216 |

Dropout—0.2 | 0.5585 | 0.2817 |

Dropout—0.4 | 0.8825 | 0.4082 |

Dropout—0.6 | 1.6377 | 0.8777 |

Dropout—0.8 | 2.6855 | 1.6808 |

Model | RMSE | MAE | Uncertainty Area |
---|---|---|---|

Dropout | 3.4672 | 2.3763 | 2703.8913 |

Noise | 3.6540 | 2.5444 | 3039.4859 |

Data sub-selection | 3.5430 | 2.7603 | 1376.0500 |

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**MDPI and ACS Style**

Gude, V.; Corns, S.; Long, S.
Flood Prediction and Uncertainty Estimation Using Deep Learning. *Water* **2020**, *12*, 884.
https://doi.org/10.3390/w12030884

**AMA Style**

Gude V, Corns S, Long S.
Flood Prediction and Uncertainty Estimation Using Deep Learning. *Water*. 2020; 12(3):884.
https://doi.org/10.3390/w12030884

**Chicago/Turabian Style**

Gude, Vinayaka, Steven Corns, and Suzanna Long.
2020. "Flood Prediction and Uncertainty Estimation Using Deep Learning" *Water* 12, no. 3: 884.
https://doi.org/10.3390/w12030884