# Credal-Decision-Tree-Based Ensembles for Spatial Prediction of Landslides

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study Area

^{2}. The area is between the longitudes 108°50′ E and 109°36′ E, and the latitudes 33°25′ N and 33°56′ N (Figure 1). The lowest elevation in the study area is 516 m above sea level (a.s.l) and the highest elevation is 2763 m a.s.l. The slope angle varies from 0° to 74°. The averages of elevation and slope are 1306.96 m and 27.28°, respectively. The climate is a transition zone between subtropical and warm climates, with the characteristics of a monsoon climate due to the barrier function of the Qinling Mountains and their mountain topography. Precipitation is seasonal, with 80.5% of the annual rainfall falling in summer and early autumn, concentrated in July, August, and September. The average annual rainfall is 750 mm. Landslides frequently occur in the southeastern part of Zhashui County as illustrated in Figure 1.

## 3. Methodology

#### 3.1. Landslide Inventory Map

#### 3.2. Landslide Conditioning Factors

_{s}(m

^{2}/m) is the unit contributing area, β is the slope angle (degrees), and tan β is slope gradient (m/m). In this study, STI values were computed and then divided into five groups with an interval of 20, including <20, 20–40, 40–60, 60–80, and >80 (Figure 3f).

#### 3.3. Modeling Approaches

#### 3.3.1. Credal Decision Tree

_{φ}is defined as an assignment of masses on φ. For a general credal set on the frame X, the formula of the non-specificity state can be written as [71],

^{D}can be expressed as follows:

^{D}, the procedure to build the CDT algorithm utilizes the maximum entropy function. This function is a total uncertainty measure in the imprecise Dirichlet Model (IDM). Figure 4 shows the basic learning process. A more detailed procedure of the CDT algorithm can be found in the literature [71].

#### 3.3.2. AdaBoost

_{m}and the corresponding classification error rate is ε

_{m}. α

_{m}represents the weight of the weak classifier y

_{m}. Y

_{M}is the final strong classifier integrated from all the weak classifiers. The detailed mathematical steps are presented as follows [76].

_{m}(x

_{n}) is the prediction outcome of the weak classifier, t

_{n}is the true label, I represents the weight coefficient optimization function, and ${w}_{n}^{\left(m\right)}$ is the weight of the current weak classifier.

_{M}can be given based on α

_{m}(Figure 5),

_{m}(x) is the prediction result of each weak classifier. Figure 5 gives a basic depiction of AdaBoost.

#### 3.3.3. Random Subspace (RS)

_{i}is assigned a q-dimensional feature vector,

_{i}. It is relevant to note that q* should be smaller than q.

^{r}, written as,

^{r}is a q*-dimensional feature vector,

^{r}, namely, C

^{n}(x) (n = 1, 2, …, N).

^{n}is the ensemble size of classifier.

## 4. Results

#### 4.1. Correlation Analysis between Landslide and Conditioning Factors Using Frequency Ratio Method

#### 4.2. Application of Landslide Susceptibility Models

#### 4.3. Model Performance and Validation

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Thematic maps with landslide dataset of the study area: (

**a**) Elevation; (

**b**) slope angle; (

**c**) slope aspect; (

**d**) plan curvature; (

**e**) profile curvature; (

**f**) SPI; (

**g**) STI; (

**h**) TWI; (

**i**) lithology; (

**j**) distance to faults; (

**k**) distance to rivers; (

**l**) rainfall; (

**m**) distance to roads; (

**n**) NDVI; (

**o**) land use.

**Figure 10.**ROC curves and AUC analysis for LSM using the three models: (

**a**) goodness-of-fit from the training data; (

**b**) prediction rates from the validating data.

Group | Code | Lithology | Geological Age |
---|---|---|---|

1 | J_{2} | Monzonitic granite, quartz monzonite, granodiorite, quartz diorite | Middle Jurassic |

2 | T_{2}, T_{3} | Quartz monzonite, monzonitic granite, granodiorite | Middle and late Triassic |

3 | C_{1}, C_{2} | Lower: carbonaceous phyllite; middle: siltstone, gray-green phyllite; upper: medium-thin bedded limestone; carbonaceous slate with quartz sandstone, carbonaceous slate, slate-sandwiched sandstone, quartz conglomerate, and limestone, breccia limestone | Early and middle Carboniferous |

4 | D_{1}, D_{2}, D_{3} | Lower: sandstone sandwiches slate, sandy argillaceous limestone, and local siderite sandwiches; upper: slate and phyllite-sandwiched sandstone, dolomite, limestone, sandstone, siltstone with a small amount of slate, locally intercalated argillaceous limestone, slate mixed with fine sandstone | Devonian |

5 | S | Granite | Silurian |

6 | O | Quartz diorite, diorite, gabbro, gabbro-norite, alaskite | Ordovician |

7 | Є_{1} | Lower: black carbonaceous slate and siliceous rock; upper: variegated (dark gray, gray-purple, light gray, gray-white) limestone, dolomitic limestone; dolomite with flint | Cambrian |

8 | Z_{1}, Z_{2} | Lower: conglomerate, sandstone, shale with limestone; upper: dolomite, marl with sandstone, shale | Early and middle Sinian |

9 | Pz_{2} | Lower: mainly metamorphic quartz sandstone, meta granulite with mica-quartz schist; upper: sandy conglomerate, meta-sandstone, mica-quartz schist with a few marble layers from bottom to top | Upper Paleozoic |

10 | Pt_{1}, Qn | Biotite schist, graphite marble, clastic rock interbedded with basic lava, volcanic rock with marble, clastic rock with basic lava, volcanic rock with carbonaceous phyllite, marble, and siliceous rock | Lower Proterozoic, Qingbaikouan |

Factors | Data Source | Format Resolution/Scale |
---|---|---|

Elevation, slope angle, slope aspect, plan curvature, profile curvature, SPI, STI, TWI, distance to faults, distance to roads, distance to rivers | ASTER GDEM | Raster, 30 m |

NDVI | Landsat 8 operational land imager | Raster, 30 m |

Lithology | Geological maps | Polygon, 1:200,000 |

Rainfall | National Earth System Science Data Center | Raster, 30 m |

Land use/cover | Land use/cover maps | Polygon, 1:100,000 |

**Table 3.**Spatial relationship between conditioning factors and historical landslides using FR method.

Factor | Subclass | No. of Class Pixels | No. of Landslide Pixel | FR Value |
---|---|---|---|---|

Elevation (m) | <1000 | 434680 | 66 | 3.39 |

1000–1200 | 582467 | 35 | 1.34 | |

1200–1400 | 684338 | 13 | 0.42 | |

1400–1600 | 497394 | 4 | 0.18 | |

1600–1800 | 252707 | 0 | 0.00 | |

1800–2000 | 114189 | 0 | 0.00 | |

2000–2200 | 47075 | 0 | 0.00 | |

>2200 | 23672 | 0 | 0.00 | |

Slope (°) | 0–10 | 134147 | 21 | 3.50 |

10–20 | 488228 | 36 | 1.65 | |

20–30 | 927360 | 32 | 0.77 | |

30–40 | 794942 | 23 | 0.65 | |

40–50 | 269518 | 3 | 0.25 | |

50–60 | 21531 | 3 | 3.11 | |

60–70 | 777 | 0 | 0.00 | |

7–74 | 19 | 0 | 0.00 | |

Aspect | Flat | 162829 | 8 | 1.10 |

North | 295253 | 19 | 1.44 | |

Northeast | 355388 | 15 | 0.94 | |

East | 376859 | 21 | 1.25 | |

Southeast | 306409 | 23 | 1.68 | |

South | 321254 | 10 | 0.70 | |

Southwest | 338043 | 17 | 1.12 | |

West | 318849 | 5 | 0.35 | |

Northwest | 161638 | 5 | 0.69 | |

Plan curvature | (−11.48)–(−0.55) | 517544 | 20 | 0.86 |

(−0.55)–0.51 | 1520401 | 82 | 1.21 | |

0.51–15.57 | 598577 | 16 | 0.60 | |

Profile curvature | (−18.32)–(−0.98) | 400907 | 12 | 0.67 |

(−0.98)–0.65 | 1632087 | 74 | 1.01 | |

0.65–19.48 | 603528 | 32 | 1.18 | |

SPI | 0–20 | 29795 | 12 | 9.00 |

20–40 | 6605 | 2 | 6.77 | |

40–60 | 9282 | 2 | 4.81 | |

60–80 | 8520 | 3 | 7.87 | |

>80 | 2582320 | 99 | 0.86 | |

STI | 0–10 | 1467316 | 82 | 1.25 |

10–20 | 1148396 | 36 | 0.70 | |

20–30 | 19775 | 0 | 0.00 | |

30–40 | 841 | 0 | 0.00 | |

>40 | 194 | 0 | 0.00 | |

TWI | 0–5 | 1294286 | 27 | 0.47 |

5–6 | 714821 | 28 | 0.88 | |

6–7 | 284237 | 25 | 1.97 | |

7–8 | 126771 | 10 | 1.76 | |

>8 | 216407 | 28 | 2.89 | |

Lithology | Group 1 | 123884 | 1 | 0.18 |

Group 2 | 639471 | 3 | 0.10 | |

Group 3 | 31452 | 2 | 1.42 | |

Group 4 | 1311573 | 110 | 1.87 | |

Group 5 | 4482 | 0 | 0.00 | |

Group 6 | 89099 | 0 | 0.00 | |

Group 7 | 21375 | 1 | 1.05 | |

Group 8 | 2655 | 0 | 0.00 | |

Group 9 | 10151 | 1 | 2.20 | |

Group 10 | 402380 | 0 | 0.00 | |

Distance to faults (m) | 0–1000 | 684637 | 42 | 1.37 |

1000–2000 | 518613 | 16 | 0.69 | |

2000–3000 | 416778 | 19 | 1.02 | |

3000–4000 | 310547 | 17 | 1.22 | |

>4000 | 705947 | 24 | 0.76 | |

Distance to rivers (m) | 0–200 | 803551 | 67 | 1.86 |

200–400 | 656896 | 17 | 0.58 | |

400–600 | 454343 | 10 | 0.49 | |

600–800 | 279554 | 6 | 0.48 | |

>800 | 442178 | 18 | 0.91 | |

Rainfall (mm/yr) | 653–673 | 93654 | 0 | 0.00 |

673–693 | 466030 | 5 | 0.24 | |

693–713 | 573189 | 32 | 1.25 | |

713–733 | 724650 | 47 | 1.45 | |

733–764 | 778999 | 34 | 0.98 | |

Distance to roads (m) | 0–400 | 803551 | 47 | 1.31 |

400–800 | 656896 | 8 | 0.27 | |

800–1200 | 454343 | 6 | 0.30 | |

1200–1600 | 279554 | 5 | 0.40 | |

>1600 | 442178 | 52 | 2.63 | |

NDVI | (−0.13)–0.28 | 57074 | 17 | 6.66 |

0.28–0.41 | 169814 | 43 | 5.66 | |

0.41–0.48 | 513496 | 22 | 0.96 | |

0.48–0.54 | 989002 | 21 | 0.47 | |

0.54–0.65 | 907136 | 15 | 0.37 | |

Land use/cover | Farmland | 531645 | 46 | 1.93 |

Garden land | 1231021 | 34 | 0.62 | |

Forestland | 861886 | 34 | 0.88 | |

Commercial land | 9889 | 4 | 9.04 | |

Industrial and mining storage land | 2081 | 0 | 0.00 |

**Table 4.**Percentages of different landslide susceptibility classes for CDT, AdaCDT, and RSCDT models.

Class | CDT | AdaCDT | RSCDT |
---|---|---|---|

Very low | 44.92 | 15.84 | 35.45 |

Low | 4.26 | 15.94 | 24.43 |

Moderate | 15.70 | 7.80 | 19.65 |

High | 20.08 | 11.10 | 14.88 |

Very high | 15.05 | 49.33 | 5.59 |

Models | AUC | Standard Error | 95% Confidence Interval |
---|---|---|---|

CDT | 0.788 | 0.0304 | 0.728–0.847 |

AdaCDT | 0.821 | 0.0274 | 0.767–0.875 |

RSCDT | 0.847 | 0.0245 | 0.799–0.895 |

Models | AUC | Standard Error | 95% Confidence Interval |
---|---|---|---|

CDT | 0.771 | 0.0467 | 0.680–0.863 |

AdaCDT | 0.802 | 0.0426 | 0.719–0.886 |

RSCDT | 0.861 | 0.0375 | 0.788–0.935 |

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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gui, J.; Pérez-Rey, I.; Yao, M.; Zhao, F.; Chen, W. Credal-Decision-Tree-Based Ensembles for Spatial Prediction of Landslides. *Water* **2023**, *15*, 605.
https://doi.org/10.3390/w15030605

**AMA Style**

Gui J, Pérez-Rey I, Yao M, Zhao F, Chen W. Credal-Decision-Tree-Based Ensembles for Spatial Prediction of Landslides. *Water*. 2023; 15(3):605.
https://doi.org/10.3390/w15030605

**Chicago/Turabian Style**

Gui, Jingyun, Ignacio Pérez-Rey, Miao Yao, Fasuo Zhao, and Wei Chen. 2023. "Credal-Decision-Tree-Based Ensembles for Spatial Prediction of Landslides" *Water* 15, no. 3: 605.
https://doi.org/10.3390/w15030605