# Integration of Information Theory, K-Means Cluster Analysis and the Logistic Regression Model for Landslide Susceptibility Mapping in the Three Gorges Area, China

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Study Area

#### 2.1.1. General Characteristics and Geological Setting

^{2}in the southwest of Hubei Province. Its latitudes and longitudes lie between 30°54′59″N to 31°03′32″N and 110°18′44″E to 110°52′30″E, respectively, while its highest point reaches 2000 m above sea level, as shown in Figure 1. This study area belongs to the subtropical monsoon climate zone and is characterized by abundant rainfall and humidity. The average annual precipitation for the period from 2001–2010 in Badong County and Zigui County in Hubei Province is 1069.2 mm and 944.5 mm, respectively, while the highest annual precipitation reached 1148.7 mm in 2008. Furthermore, most of the rainfall in this area is concentrated from May–September of each year, accounting for 70% of the annual precipitation.

#### 2.1.2. Slope Failures and Causative Factors

^{2}, covering 5.24% of the study area, as shown in Figure 1c. It can also be observed from Figure 1c that the area of these landslides varies widely, e.g., the largest Fanjiaping landslide has an area of 1.51 km

^{2}, while the smallest Kuihua street landslide is only 2068.8 m

^{2}.

- √
- A Landsat-8 OLI image obtained on 14 April 2015, with the path/row number of 125/38. To perform feature extraction, we have performed a series of operations on this multispectral image. This process includes radiometric correction to avoid radiometric errors or distortions over the whole image, geometric correction to avoid geometric distortion due to Earth’s rotation and other imaging conditions from the image and atmospheric correction to remove the effects of the atmosphere on the reflectance values of the image. Meanwhile, Bands 4 and 5 of the image are used for computing the normalized difference vegetable index (NDVI), whereas Bands 3 and 6 of the image are used for computing the normalized difference water index (NDWI).
- √
- The 1:50,000-scale geological maps provided by Hubei Geological Bureau for the exaction of geological factors, including lithology and distance to fault.
- √
- ASTER GDEM Version 2 (V2) data, representing the surface in raster format, for the extraction of geomorphological and hydrological factors, including elevation, distance to rivers, the terrain roughness index (TRI), the terrain position index (TPI), slope gradient, catchment area, catchment slope, terrain curvature, the topographic wetness index (TWI), terrain surface convexity, terrain surface texture, slope aspect and slope form.

#### 2.2. The Proposed Framework

#### 2.2.1. Information Coefficient Based on Shannon’s Entropy Index

#### 2.2.2. Certainty Factor

#### 2.2.3. K-means Clustering Analysis

^{2}norm of $\left({x}_{j}^{i}-{c}_{i}\right)$.

#### 2.2.4. Multicollinearity Analysis

_{j}is regressed on all other predictor variables in the model. The VIF value is computed as follows:

_{j}is not linearly related to the others; if ${R}_{j}^{2}$ is close to 1, then $\mathrm{VIF}\to \infty $ and $\mathrm{TOL}=0$, indicating that X

_{j}is highly related to the others. If the VIF value is above the threshold value of 5 or 10 [75], the corresponding regression coefficients are collinear and should be removed from the predictive model.

#### 2.2.5. Logistic Regression

#### 2.3. Objective Evaluation Measures

## 3. Results

#### 3.1. The Construction of the Proposed Framework

#### 3.1.1. Choosing the Number of Classes for Each Causative Factor

#### 3.1.2. Selecting Causative Factors for Each Grid Cell

#### 3.1.3. Clustering Grid Cells into Different Groups

#### 3.1.4. Multicollinearity Analysis of the Selected Causative Factors

#### 3.2. Validation and Comparison

## 4. Discussion

#### 4.1. Impact of K

^{2}measure is 0.252 using the proposed framework with $\mathrm{K}=3$, which is appropriate for LSM of the study area. To validate the proposed framework, we constructed a regional LR model with $\mathrm{K}=\left\{2,\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}4\right\}$ (LR_K2, LR_K3 and LR_K4) for comparison, and the landslide susceptibility maps of the three LR-based methods are illustrated in Figure 7. As K was increased from 2–4, more accurate landslide susceptibility maps were obtained, comparing the results in Figure 7a–c. For instance, the high or very high susceptibility zones in Figure 7c produced by the LR_K4 method mainly correlate with the actual landslide areas, whereas most of the non-landslide areas in this map are categorized as medium or low susceptibility zones.

#### 4.2. The Suitability for Urban Development

- near the county of Badong, southwest south and southeast of this county.
- near the county of Xietan, west, northwest, north, northeast and east of this county.
- near the county of Shazhenxi, northwest, north, northeast and south of this county.
- near the county of Guizhou, north, northeast, east and southeast of this county.
- near the county of Guojiaba, south and southeast of this county.
- near the county of Xiangxi, north, northeast and east of this county.
- near the county of Quyuan, northwest, north, northeast, east and southeast of this county.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Location of the study area. Site maps of (

**a**) China, (

**b**) Hubei province and (

**c**) the study area (a true color composite image using Bands 4, 3 and 2 of Landsat-8 OLI data).

**Figure 3.**Illustration of the proposed framework for landslide susceptibility mapping (LSM). LR, logistic regression.

**Figure 4.**Maps showing classification using different causative factors. (

**a**) Elevation, (

**b**) distance to river, (

**c**) NDVI, (

**d**) NDWI, (

**e**) TRI, (

**f**) TPI, (

**g**) slope gradient, (

**h**) distance to fault, (

**i**) catchment area, (

**j**) catchment slope, (

**k**) terrain curvature, (

**l**) TWI, (

**m**) terrain surface convexity, (

**n**) terrain surface texture, (

**o**) lithology, (

**p**) slope aspect and (

**q**) slope form.

**Figure 5.**The landslide susceptibility maps of the study area produced by the four methods. (

**a**) Logistic regression; (

**b**) support vector machine; (

**c**) decision tree; (

**d**) the proposed framework ($\mathrm{K}=3$).

**Figure 6.**ROC curves based on the randomly-selected training-validation samples produced by the four methods. (

**a**) The success rate curve; (

**b**) the prediction rate curve.

**Figure 7.**The landslide susceptibility maps of our study area produced by the proposed framework with different values of K. (

**a**) $\mathrm{K}=2$; (

**b**) $\mathrm{K}=3$; (

**c**) $\mathrm{K}=4$.

**Figure 8.**The impact of K on the ROC curves of the proposed framework using randomly selected training-validation samples. (

**a**) The success rate curve; (

**b**) the prediction rate curve.

**Table 1.**Information coefficients of each causative factor. The highest ones in each factor are indicated in bold and underlined. TRI, terrain roughness index; TPI, terrain position index; TWI, topographic wetness index.

Causative Factor | Information Coefficient | ||||
---|---|---|---|---|---|

2 Classes | 3 Classes | 4 Classes | 5 Classes | 6 Classes | |

Elevation | 0.8655 | 0.3912 | 0.4209 | NC | NC |

Distance to river | 0.8040 | 0.4542 | 0.3068 | 0.2743 | NC |

NDVI | 0.2898 | 0.1578 | 0.1223 | 0.1172 | 0.1053 |

NDWI | 0.2776 | 0.1439 | 0.1486 | 0.1428 | 0.1175 |

TRI | 0.2343 | 0.2270 | 0.2472 | 0.2398 | NC |

TPI | 0.0011 | 0.0715 | 0.0889 | 0.1021 | 0.1089 |

Slope gradient | 0.1144 | 0.1229 | 0.1140 | 0.1123 | 0.1180 |

Distance to fault | 0.0193 | 0.0037 | 0.0048 | 0.0322 | 0.0181 |

Catchment area | 0.0531 | 0.0386 | 0.0294 | 0.0315 | 0.0360 |

Catchment slope | 0.1175 | 0.1350 | 0.1230 | 0.1279 | 0.1258 |

Terrain curvature | 0.0080 | 0.0384 | 0.0640 | 0.0808 | 0.0933 |

TWI | 0.0858 | 0.1798 | 0.1690 | 0.1424 | 0.1539 |

Terrain surface convexity | 0.0582 | 0.0847 | 0.0933 | 0.0781 | 0.0729 |

Terrain surface texture | 0.3848 | 0.3495 | 0.3139 | 0.3040 | 0.2940 |

Category | Value Range | Description | Stability |
---|---|---|---|

1 | $\mathrm{CF}<-0.6$ | Basically no landslides occurred | stable |

2 | $-0.6\le \mathrm{CF}<-0.2$ | Landslides are less likely to occur | relatively stable |

3 | $-0.2\le \mathrm{CF}<0.2$ | Uncertain whether landslides will occur | uncertain |

4 | $0.2\le \mathrm{CF}<0.6$ | Landslides are more likely to occur | unstable |

5 | $\mathrm{CF}\ge 0.6$ | The possibility of landslides is great | extremely unstable |

Causative Factor | Classes | Percentage of Landslide | Percentage of Class | CF |
---|---|---|---|---|

Elevation | 80~700 | 99.00 | 65.50 | 0.3599 |

>700~2000 | 1.00 | 34.50 | −0.9728 | |

Distance to river | <2000 | 96.80 | 48.57 | 0.5300 |

2000~6900 | 3.20 | 51.43 | −0.9412 | |

NDVI | −0.7186~0.4140 | 16.24 | 4.46 | 0.7713 |

>0.4140~0.9190 | 83.76 | 95.54 | −0.1301 | |

NDWI | −0.8879~0.4503 | 88.29 | 96.79 | 0.0929 |

>0.4503~0.8025 | 11.71 | 3.21 | 0.7718 | |

Catchment area | 900~18,590 | 96.13 | 91.52 | −0.0624 |

>18,590~412,609 | 3.87 | 8.48 | 0.4134 | |

Terrain surface texture | 4.4474~28.4901 | 82.07 | 45.10 | 0.4792 |

>28.4901~61.9701 | 17.93 | 54.90 | −0.6869 | |

Slope gradient | <20 | 48.77 | 35.17 | 0.2967 |

20~35 | 44.61 | 44.25 | 0.0086 | |

>35~75 | 6.62 | 20.58 | −0.6920 | |

Lithology | mudstone, shale and Quaternary deposits | 3.94 | 22.48 | −0.8660 |

sandstones and thinly bedded limestones | 51.82 | 26.97 | 0.5153 | |

limestones and massive sandstones | 44.24 | 50.55 | −0.1257 | |

TWI | 0.8704~3.4408 | 23.12 | 51.64 | −0.5676 |

>3.4408~6.2721 | 76.49 | 47.49 | 0.4033 | |

>6.2721~10.4071 | 0.39 | 0.87 | −0.5626 | |

Catchment slope | <0.3224 | 36.43 | 28.02 | 0.2457 |

0.3224~0.4946 | 56.68 | 47.52 | 0.1719 | |

>0.4946~1.1306 | 6.89 | 24.46 | −0.7308 | |

TRI | <8.8293 | 58.80 | 39.26 | 0.3535 |

8.8293~15.0834 | 35.76 | 41.89 | −0.1542 | |

>15.0834~26.1201 | 5.26 | 16.50 | −0.6947 | |

>26.1201~94.1794 | 0.18 | 2.35 | −0.9277 | |

Terrain surface convexity | >13.9169~42.4579 | 12.13 | 5.10 | 0.6168 |

>42.4579~48.8004 | 31.42 | 25.02 | 0.2167 | |

>48.8004~53.7838 | 41.06 | 43.07 | −0.0495 | |

>53.7838~71.9051 | 15.39 | 26.81 | −0.4412 | |

Distance to fault | <1200 | 24.54 | 28.50 | −0.1464 |

1200~2400 | 22.32 | 27.37 | −0.1940 | |

>2400~3600 | 29.61 | 22.88 | 0.2418 | |

>3600~5400 | 20.84 | 16.01 | 0.2463 | |

>5400~8700 | 2.69 | 5.24 | −0.5011 | |

TPI | −90.8220~−18.5199 | 0.52 | 2.13 | −0.7665 |

>−18.5199~−8.4604 | 6.04 | 10.30 | −0.4283 | |

>−8.4604~−2.7133 | 26.97 | 22.60 | 0.1724 | |

>−2.7133~4.1138 | 48.00 | 35.04 | 0.2873 | |

>4.1138~12.2871 | 17.17 | 23.56 | −0.2835 | |

>12.2871~70.1288 | 1.30 | 6.37 | −0.8060 | |

Terrain curvature | −2.1060~−0.4418 | 0.21 | 1.07 | −0.8173 |

>−0.4418~−0.1991 | 3.83 | 7.44 | −0.5006 | |

>−0.1991~−0.0604 | 20.45 | 20.08 | 0.0194 | |

>−0.0604~0.0782 | 56.76 | 45.41 | 0.2128 | |

>0.0782~0.2863 | 17.65 | 22.28 | −0.2178 | |

>0.2863~2.3318 | 1.10 | 3.72 | −0.7184 | |

Slope aspect | Flat | 0.26 | 0.57 | −0.5560 |

North | 23.54 | 14.83 | 0.3936 | |

North–East | 15.95 | 12.57 | 0.2249 | |

East | 9.67 | 11.86 | −0.1940 | |

South–East | 7.27 | 10.62 | −0.3292 | |

South | 13.52 | 12.47 | 0.0826 | |

South-West | 6.43 | 10.86 | −0.4234 | |

West | 9.46 | 14.52 | −0.3620 | |

North–West | 13.90 | 11.70 | 0.1690 | |

Slope form | V/V | 29.74 | 28.71 | 0.0367 |

GE/V | 2.70 | 1.63 | 0.4238 | |

X/V | 10.60 | 11.15 | −0.0522 | |

V/GR | 4.05 | 3.54 | 0.1340 | |

GE/GR | 1.30 | 0.58 | 0.5896 | |

X/GR | 3.27 | 3.05 | 0.0728 | |

V/X | 13.05 | 13.84 | −0.0605 | |

GE/X | 3.72 | 2.37 | 0.3867 | |

X/X | 31.57 | 35.13 | −0.1100 |

**Table 4.**The optimal combinations of the causative factors for each cluster and their regression coefficients with the proposed framework when K = 3. SE, selected; RC, regression coefficient.

Causative Factor/Intercept | No Clustering | $\mathbf{K}=3$ | ||||||
---|---|---|---|---|---|---|---|---|

Cluster 1 | Cluster 2 | Cluster 3 | ||||||

Intercept | −8.125 | −8.783 | −17.097 | 1.415 | ||||

SE | RC | SE | RC | SE | RC | SE | RC | |

Elevation | √ | 1.838 | ||||||

Distance to river | √ | 2.801 | √ | 4.473 | ||||

NDVI | √ | −0.597 | √ | 0.709 | ||||

NDWI | √ | 0.276 | ||||||

Catchment area | √ | −0.156 | ||||||

Terrain surface texture | √ | 1.295 | √ | 0.906 | √ | 0.908 | ||

Slope gradient | √ | 0.255 | √ | 5.476 | √ | 4.256 | ||

Lithology | √ | −1.191 | ||||||

TWI | √ | 0.671 | √ | 4.470 | √ | −13.190 | ||

Catchment slope | √ | 0.400 | √ | 3.643 | ||||

TRI | √ | 1.340 | √ | 3.859 | ||||

Terrain surface convexity | √ | 0.836 | √ | 13.433 | ||||

Distance to fault | √ | 0.286 | ||||||

TPT | √ | 0.573 | √ | 1.819 | ||||

Terrain curvature | √ | −0.648 | √ | −1.322 | √ | −0.346 | ||

Slope aspect | √ | −0.297 | √ | −0.898 | ||||

Slope form | √ | 0.012 | √ | −0.184 |

**Table 5.**The multicollinearity analysis of the causative factors for each cluster corresponding to Table 4. TOL, tolerance; VIF, variance inflation factor.

Causative Factor | Cluster 1 | Cluster 2 | Cluster 3 |
---|---|---|---|

TOL/VIF | |||

Elevation | |||

Distance to river | 0.940/1.064 | ||

NDVI | 0.944/1.059 | ||

NDWI | |||

Catchment area | |||

Terrain surface texture | 0.977/1.023 | 0.944/1.060 | |

Slope gradient | 0.648/1.543 | 0.402/2.489 | |

Lithology | |||

TWI | 0.741/1.350 | 0.408/2.451 | |

Catchment slope | 0.612/1.634 | ||

TRI | 0.401/2.494 | ||

Terrain surface convexity | 0.538/1.860 | ||

Distance to fault | |||

TPI | 0.775/1.291 | ||

Terrain curvature | 0.812/1.231 | 0.907/1.102 | |

Slope aspect | 0.961/1.041 | ||

Slope form | 0.999/1.001 |

Methods | Overall Accuracy |
---|---|

LR | 80.26% |

SVM | 83.74% |

DT | 84.13% |

LR_K3 | 85.32% |

**Table 7.**The optimal combinations of the causative factors for each cluster and their regression coefficients with the proposed framework when K = 2 and 4.

Causative Factor/Intercept | $\mathbf{K}=2$ | $\mathbf{K}=4$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Cluster 1 | Cluster 2 | Cluster 1 | Cluster 2 | Cluster 3 | Cluster 4 | |||||||

Intercept | −3.600 | −0.508 | −17.628 | −30.784 | −50.719 | −12.063 | ||||||

SE | RC | SE | RC | SE | RC | SE | RC | SE | RC | SE | RC | |

Elevation | √ | 3.321 | √ | 4.392 | √ | 20.754 | ||||||

Distance to river | √ | 4.473 | √ | 7.241 | √ | 13.396 | √ | 3.392 | ||||

NDVI | √ | 0.709 | ||||||||||

NDWI | √ | 3.447 | ||||||||||

Catchment area | ||||||||||||

Terrain surface texture | √ | 0.906 | √ | 0.908 | √ | 17.378 | √ | 2.841 | ||||

Slope gradient | √ | 5.476 | ||||||||||

Lithology | √ | 1.350 | √ | 3.513 | ||||||||

TWI | √ | 4.470 | √ | 14.754 | √ | −3.863 | √ | 0.400 | ||||

Catchment slope | √ | 3.643 | √ | 3.990 | ||||||||

TRI | ||||||||||||

Terrain surface convexity | √ | 7.693 | √ | 31.940 | ||||||||

Distance to fault | ||||||||||||

TPT | √ | 1.819 | √ | 3.560 | √ | −2.669 | √ | 0.778 | ||||

Terrain curvature | √ | −1.322 | √ | −1.286 | ||||||||

Slope aspect | √ | −0.898 | √ | −1.151 | ||||||||

Slope form | √ | −0.184 | √ | −0.519 |

**Table 8.**The multicollinearity analysis of the causative factors for each cluster corresponding to Table 7.

Causative Factor | $\mathbf{K}=2$ (TOL/VIF) | $\mathbf{K}=4$ (TOL/VIF) | ||||
---|---|---|---|---|---|---|

Cluster 1 | Cluster 2 | Cluster 1 | Cluster 2 | Cluster 3 | Cluster 4 | |

Elevation | 0.569/1.756 | 0.861/1.162 | 0.464/2.154 | |||

Distance to river | 0.934/1.070 | 0.563/1.777 | 0.406/2.462 | 0.985/1.016 | ||

NDVI | ||||||

NDWI | 0.915/1.093 | 0.969/1.032 | ||||

Catchment area | ||||||

Terrain surface texture | 0.975/1.026 | 0.670/1.493 | 0.946/1.057 | |||

Slope gradient | ||||||

Lithology | 0.977/1.023 | 0.852/1.173 | 0.864/1.157 | |||

TWI | 0.789/1.268 | 0.949/1.054 | 0.774/1.291 | 0.627/1.594 | ||

Catchment slope | 0.806/1.240 | 0.632/1.581 | ||||

TRI | ||||||

Terrain surface convexity | 0.958/1.043 | 0.780/1.282 | ||||

Distance to fault | ||||||

TPI | 0.975/1.026 | 0.931/1.074 | 0.912/1.097 | 0.638/1.567 | ||

Terrain curvature | 0.908/1.101 | 0.998/1.002 | ||||

Slope aspect | 0.997/1.003 | 0.879/1.138 | ||||

Slope form | 0.685/1.460 |

Clusters | Statistics | ||||
---|---|---|---|---|---|

−2ln Likelihood | −2ln L0 | Goodness of Fit | Pseudo R^{2} | ||

No clustering | 68,666.849 | 36,819.425 | 69,332.399 | 0.463 | |

$\mathrm{K}=2$ | Cluster 1 | 15,559.511 | 8542.172 | 215.389 | 0.451 |

Cluster 2 | 27,105.308 | 10,245.806 | 2833.666 | 0.622 | |

$\mathrm{K}=3$ | Cluster 1 | 21,731.018 | 1738.480 | 1887.318 | 0.920 |

Cluster 2 | 62,526.249 | 46,769.634 | 3962.812 | 0.252 | |

Cluster 3 | 12,830.029 | 3412.788 | 2805.024 | 0.734 | |

$\mathrm{K}=4$ | Cluster 1 | 24,014.293 | 17,722.546 | 1954.458 | 0.262 |

Cluster 2 | 16,054.908 | 5025.186 | 19,513.817 | 0.687 | |

Cluster 3 | 10,818.555 | 2704.639 | 2940.536 | 0.750 | |

Cluster 4 | 11,601.296 | 7865.679 | 34,575.656 | 0.322 |

Methods | Overall Accuracy |
---|---|

LR_K2 | 81.85% |

LR_K3 | 85.32% |

LR_K4 | 91.76% |

Factors | Potential Rating | Weights W_{i} | ||||
---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | ||

Elevation (m) | >1000 | >750–1000 | >500–750 | 250–500 | <250 | 0.080 |

Distance to river (m) | >4000 | >3000–4000 | >2000–3000 | 1000–2000 | <1000 | 0.078 |

Distance to main towns (m) | >4000 | >3000–4000 | >2000–3000 | 1000–2000 | <1000 | 0.212 |

Landslide susceptibility map | Very high | High | Medium | Low | 0.320 | |

Slope gradient (°) | >25 | >15–25 | >10–15 | 5–10 | <5 | 0.246 |

Slope aspect | N | NE, NW | E, W | SE, SW | S, Flat | 0.064 |

© 2017 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, Q.; Wang, Y.; Niu, R.; Peng, L. Integration of Information Theory, K-Means Cluster Analysis and the Logistic Regression Model for Landslide Susceptibility Mapping in the Three Gorges Area, China. *Remote Sens.* **2017**, *9*, 938.
https://doi.org/10.3390/rs9090938

**AMA Style**

Wang Q, Wang Y, Niu R, Peng L. Integration of Information Theory, K-Means Cluster Analysis and the Logistic Regression Model for Landslide Susceptibility Mapping in the Three Gorges Area, China. *Remote Sensing*. 2017; 9(9):938.
https://doi.org/10.3390/rs9090938

**Chicago/Turabian Style**

Wang, Qian, Yi Wang, Ruiqing Niu, and Ling Peng. 2017. "Integration of Information Theory, K-Means Cluster Analysis and the Logistic Regression Model for Landslide Susceptibility Mapping in the Three Gorges Area, China" *Remote Sensing* 9, no. 9: 938.
https://doi.org/10.3390/rs9090938