# Dynamic Displacement Forecasting of Dashuitian Landslide in China Using Variational Mode Decomposition and Stack Long Short-Term Memory Network

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## Abstract

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## 1. Introduction

## 2. Stack Long Short-Term Memory Network

#### 2.1. LSTM Network

#### 2.2. Training Method of the SLSTM Network

## 3. Dynamic Forecasting Model Based on VMD-SLSTM Network

#### 3.1. Variational Mode Decomposition

#### 3.2. VMD-SLSTM Forecasting Model

#### 3.3. Dynamic Forecasting Process

**Step 1**The displacement is decomposed into $K$ components by the VMD: $[{p}_{1}^{i},{p}_{2}^{i},\cdots ,{p}_{t-1}^{i}]$ with $i=1,2,\cdots ,K$.

**Step 2**Each component ${\widehat{p}}_{t}^{i}$ is forecasted using the SLSTM network, respectively.

**Step 3**The sum of $K$ forecast components denotes the final forecast displacement ${\widehat{y}}_{t}$.

**Step 4**Each forecast component ${\widehat{p}}_{t}^{i}$ is added to the input, and the SLSTM network is updated to forecast the next value.

**Step 5**Steps 2–4 are repeated $q$ times, and the sum of each forecast component ${\widehat{p}}_{t+q}^{i}$ denotes the final forecast displacement ${\widehat{y}}_{t+q}$.

## 4. A Real Application Case

#### 4.1. Dashuitian Landslide

#### 4.2. Forecast Results Using the VMD-SLSTM Network

#### 4.3. Comparison with Other Forecasting Models

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Acronyms | Notations | ||

ANN | artificial neural network | $a$ | penalty parameter |

BPTT | back propagation through time | ${A}_{k}$ | instantaneous amplitude |

EEMD | ensemble empirical mode decomposition | ${b}_{f}$, ${b}_{h}$, ${b}_{u}$, ${b}_{o}$ | bias vectors |

EMD | empirical mode decomposition | ${g}_{1}$, ${g}_{2}$ | non-linear activation functions |

GPS | global positioning system | ${h}_{t}$ | state of cell at $t$ time |

IMF | intrinsic mode function | ${\tilde{h}}_{t}$ | candidate state of cell |

LSTM | long short-term memory | $K$ | mode number |

MAE | mean absolute error | ${R}_{f}$, ${R}_{h}$, ${R}_{u}$, ${R}_{o}$ | weight matrices of hidden layer connections |

RMSE | root mean square error | $U$ | weight matrix of input |

RNN | recurrent neural network | ${u}_{k}$ | amplitude modulated-frequency modulated signal |

SLSTM | stack long short-term memory network | $V$ | weight matrix of output |

VMD | variational mode decomposition | ${W}_{f}$, ${W}_{h}$, ${W}_{u}$, ${W}_{o}$ | weight matrices associated with the input unit |

$X$ | input matrix of network | ||

$Y$ | output matrix of network | ||

$\tau $ | rising step | ||

$\lambda $ | Lagrange multiplier | ||

${\omega}_{k}$ | actual center frequency | ||

${\varphi}_{k}$ | instantaneous phase |

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**Figure 3.**Landslide displacement forecasting flow chart based on the variational mode decomposition (VMD)-SLSTM network.

**Figure 10.**Forecast results of landslide displacement using three forecasting models: VMD-SLSTM, empirical mode decomposition (EMD)-SLSTM, and LSTM.

Model | MAE/mm | RMSE/mm |
---|---|---|

LSTM network | 3.99 | 4.15 |

EMD-LSTM network | 2.25 | 2.76 |

VMD-SLSTM network | 1.99 | 2.50 |

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**MDPI and ACS Style**

Xing, Y.; Yue, J.; Chen, C.; Cong, K.; Zhu, S.; Bian, Y.
Dynamic Displacement Forecasting of Dashuitian Landslide in China Using Variational Mode Decomposition and Stack Long Short-Term Memory Network. *Appl. Sci.* **2019**, *9*, 2951.
https://doi.org/10.3390/app9152951

**AMA Style**

Xing Y, Yue J, Chen C, Cong K, Zhu S, Bian Y.
Dynamic Displacement Forecasting of Dashuitian Landslide in China Using Variational Mode Decomposition and Stack Long Short-Term Memory Network. *Applied Sciences*. 2019; 9(15):2951.
https://doi.org/10.3390/app9152951

**Chicago/Turabian Style**

Xing, Yin, Jianping Yue, Chuang Chen, Kanglin Cong, Shaolin Zhu, and Yankai Bian.
2019. "Dynamic Displacement Forecasting of Dashuitian Landslide in China Using Variational Mode Decomposition and Stack Long Short-Term Memory Network" *Applied Sciences* 9, no. 15: 2951.
https://doi.org/10.3390/app9152951