# Explainable Boosting Machines for Slope Failure Spatial Predictive Modeling

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Explainable Machine Learning

_{0}+ f

_{1}(x

_{1}) + f

_{2}(x

_{2}) + f

_{3}(x

_{3}) + ……. + f

_{i}(x

_{i})

_{0}) along with functions that describe the relationship between the response and each predictor variable. Essentially, the coefficients (β

_{i}) in a multiple linear regression model are replaced with learned functions (f

_{i}) that are not confined to a linear relationship. The model is additive because separate functions are learned for each predictor variable independently, which allows for an examination of the effect of each predictor variable separately [2]. In order to apply GAMs to binary classification problems, as is the case in this study, class logits are predicted as opposed to a continuous variable (Equation (2)). In this equation, p represents the probability of the sample belonging to the positive class, which is assigned a value of 1 while the negative class is assigned a value of 0 [2,10].

^{2}M) method [6,7,11]. Using GA

^{2}M, the function associated with each predictor variable is approximated using many shallow decision trees created with gradient boosting to iteratively improve model performance. More specifically, shallow decision tree generation, learning, and gradient updates are performed using a single predictor variable at a time in a round-robin fashion with a low learning rate [6,7]. Currently, the InterpretML implementation of EBM, which was used in this study, implements log loss for classification and mean square error loss (MSE) for regression as measures of error or loss [6]. Due to the low learning rate, only small updates to the model are made with the addition of each tree. This requires the model to be built by iterating through the training data over thousands of boosting iterations in which each tree only use one predictor variable. The algorithm developers argue that the low learning rate reduces the influence of the order in which features are used while iteratively cycling through the predictor variables using a round-robin method minimizes the impact of co-linearity to maintain interpretability [6,7,11]. To take into account interactions between predictor variables, two-dimensional functions (f

_{ij}(x

_{i}, x

_{j})) can be learned to relate the response variable to pairs of predictor variables. The subset of available interactions included are selected using the FAST method proposed by Lou et al. [7] that ranks all pairs of predictor variables. Adding interaction terms requires that the additive nature of GAMs be relaxed [2], and interpreting the influence of a single predictor variable will require investigating the associated one-dimensional function and any two-dimensional interaction functions that include the variable of interest [6,7].

#### 2.2. Slope Failure Mapping and Modeling

## 3. Methods

#### 3.1. Study Areas and Slope Failure Data

#### 3.2. Training Data and Predictor Variables

#### 3.3. Model Traning

#### 3.4. Model Assessment

## 4. Results

#### 4.1. Algorithm Comparisons

#### 4.2. Sample Size and Model Generalization

#### 4.3. Exploration of EBM Results

## 5. Discussion

#### 5.1. Algorithm Performance Comparison and EBM Interpretability

^{2}M method, on which EBM is based, equivalent performance or only marginal improvements in accuracy was noted between GA

^{2}M and traditional GAMs, and the major innovation of the method was highlighted as the development of the FAST method for selection of pair-wise interactions to include in the model [7]. More work on relating the predictive performance of EBMs to other GAM-based methods and comparing the spatial outputs, variable importance estimates, feature-specific functions, and two-dimensional interaction heat maps is merited. Further, since EBM relies on shallow decision trees, further investigation is necessary to explore how heterogenous the spatial outputs may be and whether or not artifacts are evident.

#### 5.2. Future Research Needs

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Conceptualization of outputs from EBM that provide model explanations. (

**a**) One-dimensional feature function for a single predictor variable; (

**b**) two-dimensional function for interaction between two predictor variables; (

**c**) global estimate of variable importance as mean absolute score; (

**d**) contribution scores for variables for predicting a new sample. Arrows indicate direction of increasing values. Scores relate to the effect of each included predictor variable or interaction on the predicted logits for the positive class.

**Figure 2.**(

**a**) Major Land Resource Areas (MLRAs) investigated in this study; (

**b**) shows the extent of (

**a**) in the contiguous United States. MLRA data are provided by the United States Department of Agriculture (USDA) [62]. (

**c**) through (

**f**) provide examples of terrain conditions, represented using LiDAR-derived hillshades, in the four MLRAs studied. Red rectangles in (

**a**) represent the areas depicted in (

**c**–

**f**). CAP = Central Allegheny Plateau, CPM = Cumberland Plateau and Mountains, EAPM = Eastern Allegheny Plateau and Mountains, and NARV = Northern Appalachian Ridges and Valleys.

**Figure 3.**Hexagon tessellation used to define training and validation regions within each MLRA with associated slope failure and pseudo absence samples.

**Figure 4.**Model comparison and assessment using the withheld validation data for each algorithm in each MLRA study area. Bars for OA and AUC PR represent an estimated 95% confidence interval. (

**a**) AUC ROC and AUC PR; (

**b**) overall accuracy and F1 score.

**Figure 5.**Receiver operating characteristic (ROC) curves for all five models in each of the four MLRA study areas. Associated AUC ROC values are provided in Table 4. (

**a**–

**d**) provide results for the CAP, CPM, EAPM, and NARV MLRAs, respectively.

**Figure 6.**Precision-recall (PR) curves for all five models in each of the four MLRA study areas. Associated AUC PR values are provided in Table 4. (

**a**–

**d**) provide results for the CAP, CPM, EAPM, and NARV MLRAs, respectively.

**Figure 7.**Impact of training sample size on model performance measured using OA, F1 score for the slope failure class, and AUC ROC within all MLRA study areas using all algorithms. Sample size represents the number of samples per class. (

**a**) Overall accuracy; (

**b**) F1 score; (

**c**) AUC ROC.

**Figure 8.**Assessment of model generalization to different MLRAs using OA and F1 score. Bars for OA represent an estimated 95% confidence interval.

**Figure 10.**EBM functions for a subset of variables and two-dimensional interaction plots for a subset of the predictor variables for the NARV model. These plots offer explanations for the global model. (

**a**) The slope; (

**b**) the surface area ratio; (

**c**) cross-sectional curvature; (

**d**) the heat load index; (

**e**) interaction between the topographic roughness index calculated using a 7 cell radius and the surface area ratio; (

**f**) interaction between topographic slope and the topographic roughness index calculated using an 11 cell radius.

**Figure 11.**Variable contribution estimates or scores for two sites as an examples of local model explanations provided by EBM. Only the top 15 contributing variables are included in the charts. (

**a**) Non-slope failure site incorrectly predicted as a failure; (

**b**) slope failure site correctly predicted, but with low probability.

**Table 1.**MLRA land areas, abbreviations used in this study, and number of mapped slope failure incidence points. Note that a statewide dataset of incidence points is not yet available since LiDAR data collection is not yet complete.

MLRA | Abbreviation | Land Area in WV | Number of Slope Failures Mapped |
---|---|---|---|

Central Allegheny Plateau | CAP | 22,281 km^{2} | 29,637 |

Cumberland Plateau and Mountains | CPM | 11,644 km^{2} | 20,712 |

Eastern Allegheny Plateau and Mountains | EAPM | 18,071 km^{2} | 12,518 |

Northern Appalachian Ridges and Valleys | NARV | 10,320 km^{2} | 1997 |

**Table 2.**Description of terrain variables used in this study. Abbreviations defined in this table are used throughout this paper.

Variable | Abbreviation | Description | Window Radius (Cells) |
---|---|---|---|

Slope Gradient | Slp | Gradient or rate of maximum change in Z as degrees of rise | 1 |

Mean Slope Gradient | SlpMn | Slope averaged over a local window | 7, 11, 21 |

Linear Aspect | LnAsp | Transform of topographic aspect to linear variable | 1 |

Profile Curvature | PrC | Curvature parallel to direction of maximum slope | 7, 11, 21 |

Plan Curvature | PlC | Curvature perpendicular to direction of maximum slope | 7, 11, 21 |

Longitudinal Curvature | LnC | Profile curvature intersecting with the plane defined by the surface normal and maximum gradient direction | 7, 11, 21 |

Cross-Sectional Curvature | CSC | Tangential curvature intersecting with the plane defined by the surface normal and a tangent to the contour—perpendicular to maximum gradient direction | 7, 11, 21 |

Slope Position | TPI | Z–Mean Z | 7, 11, 21 |

Topographic Roughness | TRI | Square root of standard deviation of slope in local window | 7, 11, 21 |

Topographic Dissection Index | TDI | $\frac{\mathrm{Z}-MinZ}{MaxZ-MinZ}$ | 7, 11, 21 |

Surface Area Ratio | SAR | $\frac{CellArea}{\mathrm{cosine}\left(slope\ast \pi \ast 180\right)}$ | 1 |

Surface Relief Ratio | SRR | $\frac{Mean\mathrm{Z}-MinZ}{MaxZ-MinZ}$ | 7, 11, 21 |

Site Exposure Index | SEI | Measure of exposure based on slope and aspect | 1 |

Heat Load Index | HLI | Measure of solar insolation based on slope, aspect, and latitude | 1 |

**Table 3.**Example binary confusion matrix and associated terminology. TP = true positive, FP = false positive, TN = true negative, and FN = false negative.

Reference Data | ||||
---|---|---|---|---|

True | False | 1—Commission Error | ||

Classification Result | True | TP | FP | Precision |

False | FN | TN | NPV | |

1—Omission Errors | Recall | Specificity |

**Table 4.**Assessment metrics calculated using the withheld validation samples in each MLRA for each algorithm. OA = overall accuracy, NPV = negative predictive value, AUC ROC = area under the receiver operating characteristics curve, and AUC PR = area under the precision-recall curve.

Study Area | Algorithm | OA | Precision | F1 Score | Recall | Specificity | NPV | AUC ROC | AUC PR |
---|---|---|---|---|---|---|---|---|---|

CAP | EBM | 0.823 | 0.857 | 0.814 | 0.776 | 0.870 | 0.795 | 0.903 | 0.909 |

CAP | kNN | 0.806 | 0.834 | 0.797 | 0.764 | 0.848 | 0.782 | 0.884 | 0.888 |

CAP | LR | 0.789 | 0.819 | 0.779 | 0.742 | 0.836 | 0.764 | 0.843 | 0.844 |

CAP | RF | 0.839 | 0.854 | 0.836 | 0.818 | 0.860 | 0.825 | 0.903 | 0.905 |

CAP | SVM | 0.854 | 0.886 | 0.848 | 0.812 | 0.896 | 0.827 | 0.911 | 0.906 |

CPM | EBM | 0.849 | 0.847 | 0.849 | 0.852 | 0.846 | 0.851 | 0.917 | 0.909 |

CPM | kNN | 0.815 | 0.801 | 0.819 | 0.838 | 0.792 | 0.830 | 0.888 | 0.880 |

CPM | LR | 0.797 | 0.799 | 0.796 | 0.794 | 0.800 | 0.795 | 0.870 | 0.839 |

CPM | RF | 0.835 | 0.829 | 0.836 | 0.844 | 0.826 | 0.841 | 0.910 | 0.899 |

CPM | SVM | 0.857 | 0.844 | 0.860 | 0.876 | 0.838 | 0.871 | 0.924 | 0.910 |

EAPM | EBM | 0.875 | 0.854 | 0.879 | 0.904 | 0.846 | 0.898 | 0.945 | 0.930 |

EAPM | kNN | 0.853 | 0.831 | 0.858 | 0.886 | 0.820 | 0.878 | 0.936 | 0.936 |

EAPM | LR | 0.850 | 0.830 | 0.854 | 0.880 | 0.820 | 0.872 | 0.931 | 0.923 |

EAPM | RF | 0.877 | 0.848 | 0.882 | 0.918 | 0.836 | 0.911 | 0.955 | 0.944 |

EAPM | SVM | 0.890 | 0.878 | 0.892 | 0.906 | 0.874 | 0.903 | 0.949 | 0.942 |

NARV | EBM | 0.870 | 0.857 | 0.872 | 0.888 | 0.852 | 0.884 | 0.947 | 0.941 |

NARV | kNN | 0.859 | 0.845 | 0.862 | 0.880 | 0.838 | 0.875 | 0.924 | 0.912 |

NARV | LR | 0.831 | 0.814 | 0.835 | 0.858 | 0.804 | 0.850 | 0.925 | 0.915 |

NARV | RF | 0.884 | 0.861 | 0.888 | 0.916 | 0.852 | 0.910 | 0.948 | 0.944 |

NARV | SVM | 0.881 | 0.879 | 0.881 | 0.884 | 0.878 | 0.883 | 0.944 | 0.940 |

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## Share and Cite

**MDPI and ACS Style**

Maxwell, A.E.; Sharma, M.; Donaldson, K.A. Explainable Boosting Machines for Slope Failure Spatial Predictive Modeling. *Remote Sens.* **2021**, *13*, 4991.
https://doi.org/10.3390/rs13244991

**AMA Style**

Maxwell AE, Sharma M, Donaldson KA. Explainable Boosting Machines for Slope Failure Spatial Predictive Modeling. *Remote Sensing*. 2021; 13(24):4991.
https://doi.org/10.3390/rs13244991

**Chicago/Turabian Style**

Maxwell, Aaron E., Maneesh Sharma, and Kurt A. Donaldson. 2021. "Explainable Boosting Machines for Slope Failure Spatial Predictive Modeling" *Remote Sensing* 13, no. 24: 4991.
https://doi.org/10.3390/rs13244991