# Debris Flow Prediction Based on the Fast Multiple Principal Component Extraction and Optimized Broad Learning

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## Abstract

**:**

## 1. Introduction

## 2. Overview of the Study Area

#### 2.1. Topography and Geomorphology

^{2}and 2400 km

^{2}. Its altitude range is 800~1500 m, located under the Qinling Mountains, with many mountains and gullies, belonging to the middle and low mountain terrain. The mountain’s amount of soil and stone is as high as 1,800,000 km

^{2}accounting for more than 80% of the area, forming a canyon area with significant terrain differences. At the same time, it is located in the middle of the two rivers, with many small river basins and abundant water resources. In summer and autumn, the rainfall is more, and the temperature changes significantly, which is easy to cause loose soil. The region is rich in minerals and the intensity of human activities increases the debris flow safety points.

#### 2.2. Distribution and Law of Debris Flow

## 3. Theories and Method

#### 3.1. FMPCE

**R**. The algorithm form is shown in Equation (4).

**,**$\mathit{F}(k)=\mathit{W}(k)\mathit{A}{\mathit{W}}^{\mathit{T}}(k)\mathit{RW}(k)$.

#### 3.2. Broad Learning

#### 3.2.1. Initial Structure

#### 3.2.2. Updated Structure

#### 3.3. Improved Broad Learning

## 4. Modeling Process and Result Analysis

#### 4.1. Establishment of Debris Flow Probability Prediction Model

Algorithm 1 Debris flow prediction based on the SVDBL algorithm |

Input: Training sample $(\mathit{X},\mathit{Y})$ and actual input $\mathit{Y}$ Output: Debris flow probability $\stackrel{\u2322}{\mathit{Y}}$ |

1 for $\mathrm{i}=0;\mathrm{i}\le \mathrm{n}$ do2 Initialize ${\mathit{W}}_{ei}^{}$ and ${\beta}_{ei}^{}$ 3 Calculate ${\mathit{Z}}_{{}^{i}}={\Phi}_{i}(\mathit{X}{\mathit{W}}_{ei}+{\beta}_{ei})$ 4 Calculate ${\mathit{V}}_{{\mathit{Z}}_{1}}^{\mathit{P}}$ by SVD 5 end6 for $j=0;j\le m$ do7 Initialize ${\mathit{W}}_{{h}_{j}}$ and ${\beta}_{{h}_{j}}$ 8 Calculate ${\mathit{H}}_{j}=\xi \left(\left[{\mathit{Z}}_{1}{\mathit{V}}_{{\mathit{Z}}_{1}}^{\mathit{P}},\dots ,{\mathit{Z}}_{n}{\mathit{V}}_{{\mathit{Z}}_{n}}^{\mathit{P}}\right]{\mathit{W}}_{{h}_{j}}+{\beta}_{{h}_{j}}\right)$ 9 Calculate ${\mathit{V}}_{{\mathit{H}}_{1}}^{\mathit{P}}$ by SVD 10 end11 Let ${\mathit{A}}_{}^{\left\{m,n\right\}}=\left[{\mathit{Z}}_{1}{\mathit{V}}_{{\mathit{Z}}_{1}}^{\mathit{P}},\dots ,{\mathit{Z}}_{n}{\mathit{V}}_{{\mathit{Z}}_{n}}^{\mathit{P}}\left|{H}_{1}^{}\right.{\mathit{V}}_{{\mathit{H}}_{1}}^{\mathit{P}},\dots ,{H}_{m}^{}{\mathit{V}}_{{\mathit{H}}_{m}}^{\mathit{P}}\right]$ 12 Calculate ${\left({\mathit{A}}_{}^{\left\{m,n\right\}}\right)}_{}^{+}$ by Equation (8) 13 while the training accuracy does not meet the requirements do 14 Initialize ${\mathit{W}}_{{h}_{m+1}}$ and ${\beta}_{{h}_{m+1}}$ 15 Calculate ${\mathit{H}}_{m+1}=\xi \left(\left[{\mathit{Z}}_{1}{\mathit{V}}_{{\mathit{Z}}_{1}}^{\mathit{P}},\dots ,{\mathit{Z}}_{n}{\mathit{V}}_{{\mathit{Z}}_{n}}^{\mathit{P}}\right]{\mathit{W}}_{{h}_{m+1}}+{\beta}_{{h}_{m+1}}\right)$; Update ${\mathit{A}}_{}^{m+1}$ 16 Calculate ${\mathit{V}}_{{\mathit{H}}_{m+1}}^{\mathit{P}}$ by SVD 17 Update ${\mathit{A}}_{}^{\left\{m+1,n\right\}}$ 18 Calculate ${\left({\mathit{A}}_{\mathit{F}}^{\left\{m+1,n\right\}}\right)}_{}^{+}$ and ${\mathit{W}}_{\mathit{F}}^{\left\{m+1,n\right\}}$ by Equations (22) and (23) 19 $m=m+1$ 20 end21 Calculate ${\mathit{V}}_{\mathit{F}}^{\mathit{P}}$ by SVD 22 Calculate ${\mathit{A}}_{\mathit{F}}^{}={\mathit{A}}_{\mathit{F}}^{\left\{m+1,n\right\}}{\mathit{V}}_{\mathit{F}}^{\mathit{P}}$ 23 Calculate ${\mathit{A}}_{\mathit{F}}^{+}$ and ${\mathit{W}}_{\mathit{F}}^{}={\mathit{A}}_{\mathit{F}}^{+}\mathit{Y}$ by Equation (8) 24 Let $\mathit{W}={\mathit{W}}_{\mathit{F}}^{}$ 25 Calculation of debris flow probability $\stackrel{\u2322}{\mathit{Y}}=\stackrel{\u2322}{\mathit{X}}\mathit{W}$ |

#### 4.2. Determine the Influencing Factors and Data Sources

#### 4.3. Data Preprocessing

- (1)
- Missing value processing

- (2)
- Outlier processing

- (3)
- Normalization

#### 4.4. Evaluating Indicator

^{2}, AUC value and training and test time as indicators.

- (1)
- Root mean square error

- (2)
- Mean absolute percentage error$$MAPE={\displaystyle {\sum}_{i=1}^{n}\left|\frac{y(i)-\overline{y}(i)}{y(i)}\right|}\times \frac{100}{n}$$

- (3)
- coefficient of determination R
^{2}$${R}^{2}=\frac{SSR}{SST}$$

- (4)
- AUC$$AUC=\frac{{\displaystyle {\sum}_{}^{}rank{k}_{i}-\frac{M(M+1)}{2}}}{M\times N}$$

#### 4.5. Simulation Verification and the Result Analysis

^{2}in the performance evaluation of the four models. The RMSE of SVDBL is 0.1309, 0.0818 and 0.0144, smaller than that of GD-BP, SVM and BL, respectively. MAPE of SVDBL is 0.0314, 0.0256 and 0.0190 smaller than that of GD-BP, SVM and BL, respectively. Regarding goodness of fit R2, SVDBL is 0.0777, 0.1020 and 0.0309 higher than the above three models, respectively. This shows that the proposed SVDBL model has the best performance.

^{2}. The results are shown in Figure 12 and Figure 13.

^{2}of the overall model first shows a decreasing trend and a slow and stable trend. The R

^{2}of GD-BP is basically below 0.75. The support vector machine SVM is superior to GD-BP, and the R

^{2}is slightly higher than 0.75. The R

^{2}of BL is between 0.75 and 0.8, and the performance is somewhat outstanding for the support vector machine SVM. The SVDBL is better than other models, and the R

^{2}is maintained above 0.80, indicating that the model works best.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Number | Influence Factors |
---|---|

1 | Rainfall |

2 | Soil moisture content |

3 | Pore water pressure |

4 | Watershed development degree |

5 | Watershed integrity coefficient |

6 | Vegetation coverage |

7 | Slope gradient |

8 | Lithologic |

9 | Watershed area |

10 | Relative height difference |

11 | Erosion and deposition amplitude |

12 | Gully bed gradient |

13 | Recharge section length ratio |

14 | Loose material reserves along the gully |

15 | Adverse geologic phenomena |

16 | New structure influence |

Influence Factors | Representation Significance | The Probability of Debris Flow/% | ||||
---|---|---|---|---|---|---|

<20 | 20~40 | 40~60 | 60~80 | >80 | ||

Rainfall/mm | Water source conditions required for debris flow formation, expressed by comprehensive daily rainfall | <20 | 20~50 | 50~75 | 75~100 | >100 |

Slope gradient/% | The potential energy conditions required for the collection of loose solid matter to quantify the ratio of 25° to 45° area to the basin area | <0.05 | 0.05~0.1 | 0.1~0.15 | 0.15~0.2 | >0.2 |

Gully bed gradient/‰ | Potential Energy Conditions for Debris Flow Formation | <50 | 50~100 | 100~200 | 200~400 | >400 |

Relative height difference/m | Terrain Factors of Debris Flow Formation | <20 | 20~40 | 40~60 | 60~80 | >80 |

Soil moisture content/% | The amount of water in the soil | <8 | 8~16 | 16~24 | 24~32 | >32 |

Pore water pressure/KPa | The pressure of groundwater in soil or rock, acting between particles or pores. | <10 | 10~25 | 25~40 | 40~60 | >60 |

Number | Impact Factor | ||
---|---|---|---|

Rainfall | Soil Moisture Content | Pore Water Pressure | |

1 | 18.32 | 6 | 0.20 |

2 | 23.28 | 13 | 0.13 |

3 | 17.62 | 20 | 0.48 |

4 | 38.74 | 27 | −0.26 |

… | … | … | … |

2000 | 138.62 | 33 | 0.74 |

Before Data Preprocessing | After Data Preprocessing | ||||
---|---|---|---|---|---|

Incremental nodes/Pcs | Accuracy/% | Training time/s | Incremental nodes/Pcs | Accuracy/% | Training time/s |

7000 | 80 | 5.8124 | 2000 | 93 | 1.2216 |

Model | Evaluating Indicators | ||
---|---|---|---|

RMSE | MAPE | R^{2} | |

GD-BP | 0.3706 | 0.0526 | 0.7826 |

SVM | 0.3215 | 0.0468 | 0.7583 |

BL | 0.2541 | 0.0022 | 0.8294 |

SVDBL | 0.2397 | 0.0212 | 0.8603 |

Model | GD-BP | SVM | BL | SVDBL |
---|---|---|---|---|

Average accuracy/% | 85.5 | 90 | 92.5 | 93 |

Average training time/s | 23.2906 | 18.1740 | 2.8899 | 1.2216 |

Warning Level | Possibility of Disaster | Color Identification | Explanation | Probability |
---|---|---|---|---|

I | Very low | Green | Only send information to decision-makers | <20% |

II | Lower | Blue | Push information to decision-makers and related technical personnel | 20%~40% |

III | Medium | Huang | Recommend preventive measures | 40%~60% |

IV | Higher | Orange | Take preventive measures | 60%~80% |

V | Extremely high | Red | Organizing a public emergency response | >80% |

Model | Average AUC Value/% | Average Training Time/s |
---|---|---|

GD-BP | 80.41% | 0.0817 |

SVM | 85.52% | 0.0049 |

BL | 90.37% | 0.0022 |

SVDBL | 93.16% | 0.0010 |

Model | Results of before Training Sample Expansion | Results of after Training Sample Expansion |
---|---|---|

Training Time/s | Training Time/s | |

GD-BP | 23.2906 | 31.9558 |

SVM | 18.1740 | 22.9869 |

BL | 2.8899 | 4.0573 |

SVDBL | 1.2216 | 1.2299 |

**Table 10.**Correspondence between the influence factor and the curve in Figure 7.

Artificially Selected Impact Factors | The Corresponding Curve in Figure 7 |
---|---|

1,2,3,7,12,14 | b |

1,2,3,4,5,12 | c |

1,2,7,8,10,12 | d |

1,3,6,7,10,14 | e |

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**MDPI and ACS Style**

Xu, G.; Yan, X.-E.; Cao, N.; Ma, J.; Xie, G.; Li, L. Debris Flow Prediction Based on the Fast Multiple Principal Component Extraction and Optimized Broad Learning. *Water* **2022**, *14*, 3374.
https://doi.org/10.3390/w14213374

**AMA Style**

Xu G, Yan X-E, Cao N, Ma J, Xie G, Li L. Debris Flow Prediction Based on the Fast Multiple Principal Component Extraction and Optimized Broad Learning. *Water*. 2022; 14(21):3374.
https://doi.org/10.3390/w14213374

**Chicago/Turabian Style**

Xu, Genqi, Xin-E Yan, Ning Cao, Jing Ma, Guokun Xie, and Lu Li. 2022. "Debris Flow Prediction Based on the Fast Multiple Principal Component Extraction and Optimized Broad Learning" *Water* 14, no. 21: 3374.
https://doi.org/10.3390/w14213374