A conditioning factor selection framework considering sample heterogeneity in debris flow susceptibility mapping

Abstract

Debris flow susceptibility mapping (DFSM) is critical for disaster prevention, while challenges still exist in addressing selecting conditioning factors. This study aims to propose a novel framework for debris flow conditioning factor selection considering the sample heterogeneity problem. Utilizing the fuzzy C-means clustering technique, the study area was segmented into multiple homogeneous subareas. The predictive capacity of the conditioning factors was assessed by applying the information gain ratio approach. This evaluation was conducted on both the total dataset prior to clustering and the homogeneous datasets derived from the clustering procedure. Then random forest modeling was implemented on all the datasets following the elimination of the two conditioning factors exhibiting the weakest predictive ability. The prediction results of models built on homogeneous datasets need to be merged for evaluation. The total dataset and homogeneous datasets with all conditioning factors retained were also involved in model training for comparison. The results showed that reasonable conditioning factor selection could significantly improve the model performance. In addition, the conditioning factor selection framework proposed in this study that considers sample heterogeneity could provide better conditioning factor combinations for DFSM.

Introduction

A debris flow is a fast-moving, destructive mixture of water, mud, and debris, causing enormous loss of life and property every year¹. Given the persistent threat of debris flows, developing effective control measures is essential^2,3. Recently, debris flow susceptibility mapping (DFSM) has proven indispensable for identifying high-risk areas and formulating strategic mitigation policies⁴.

Debris flow susceptibility assessment has developed significantly in recent years, driven by the integration of multidisciplinary approaches and evolving computational techniques^5,6. From qualitative evaluation methods based on expert experience to traditional statistical methods, the reliability of DFSM has improved a lot⁷. However, the emergence of machine learning algorithms has revolutionized this field, particularly through their superior nonlinear processing capabilities and robustness^8–11 Recently, machine learning algorithms have been increasingly employed¹², including support vector machine¹³, random forest (RF)¹⁴, logistic regression¹⁵, etc. In addition, integration frameworks have also been developed to combine multiple machine learning algorithms to further improve performance^16–18. It should be noted that the formation mechanism of debris flows is complex, involving nonlinear interactions among multiple conditioning factors. Traditional linear models may not be able to capture such complex relationships. On the other hand, debris flow samples sometimes have certain defects, which would poss greater challenges to algorithms that require high data quality. Furthermore, some ensemble algorithms impose more computational demands, making them difficult to apply. In this study, RF was chosen as the primary modeling algorithm. RF was chosen as the modeling algorithm due to its excellent nonlinear modeling capability, good tolerance for outliers, and low demand for computational resources.

In addition to the development of algorithms, the selection of conditioning factors is another effective approach for better model performance^19,20. However, there are no generalized rules for determining the optimal combination of conditioning factors. Generally speaking, conditioning factors should be determined according to the situation of specific study areas. Kavzoglu et al. proposed a genetic algorithm-based feature selection method to compose the optimal combination of landslide conditioning factors for Macka District of Trabzon²¹. Sameen et al. employed an unsupervised conditioning factor optimization approach to identify the seven most significant conditioning factors in the region of Chukha Dzongkhag, Bhutan²². Zhou et al. used GeoDetector and RFE to explore the conditioning factor combination for Wuxi County²³. Overall, conditioning factor selection methods currently available include filter, wrapper and embedded methods^24,25. Filter methods implies a lower computational cost and is suitable for high-dimensional data. It evaluate conditioning factors based on statistical properties such as pearson correlation, mutual information and maximal information coefficient²⁶. Wrapper methods could capture conditioning factor interactions and model-specific relevance, and is ideal for small-to-medium conditioning factor sets. It treats conditioning factor selection as a search problem, using predictive models to evaluate subsets. The key approaches include recursive feature elimination, forward selection and backward selection²⁷. The embedded methods could balance efficiency and efficacy by combining selection and model training and handle multicollinearity via regularization. It integrates conditioning factor selection within model training. The prominent techniques include L₁ regularization and tree based method²⁸. L₁ regularization achieves automatic feature selection by adding the sum of the absolute values of parameters as a penalty term to the model’s loss function, thereby forcing unimportant feature coefficients to approach zero. Tree-based models assess conditioning factor importance by analyzing the contribution of each feature during the construction of decision trees, thereby enabling conditioning factor selection. However, current conditioning factor selection studies are usually conducted based on an entire specific study area, which ignored the presence of sample heterogeneity.

In this study, sample heterogeneity mainly refers to the fact that debris flow samples at different locations in the same study area would also have different formation mechanisms²⁹. In other words, the same conditioning factors can play a different role during the formation process of debris flows at different locations, which would also have an effect on the model performance. Conditioning factors selected while ignoring sample heterogeneity might only represent overall data characteristics without capturing local patterns, which can lead to models performing well on training sets but experiencing a sharp decline in predictive performance on new data or specific subgroups. Furthermore, when the sample heterogeneity is not well controlled, the selected conditioning factors might be highly sensitive to minor variations in the data, causing significant fluctuations in feature importance rankings across different subsamples and resulting in model instability. Therefore, conditioning factor selection studies that consider sample heterogeneity are quite necessary. Guo et al. systematically reviewed the theory of spatial stratified heterogeneity and four types of stratification methods, providing a theoretical basis for identifying and addressing the spatial non-stationarity of geographical attributes³⁰. Zhao et al. evaluated eight spatial stratified random sampling schemes including cluster-based ones, and verified that cluster stratification based on environmental variables can significantly improve the efficiency of regional evaluation³¹. Guan et al. integrated the cellular automaton with geographically weighted regression (GWR), simulated the spatial heterogeneity of ecosystem service value loss induced by urban expansion, and highlighted the advantages of GWR in addressing spatial nonstationarity³². Smith et al. proposed that the sample heterogeneity problem can be solved by additional training for samples in different distributions, respectively³³. Yang et al. introduced the idea of defining customized parameters for individual samples during model training³⁴. In this study, the fuzzy C-means clustering approach was employed to partition the study area into multiple homogeneous subareas, which were then analyzed separately to identify conditioning factors with strong predictive capabilities.

In summary, this research aims to introduce a novel framework for selecting conditioning factors. Fuzzy C-means (FCM) clustering was utilized to address sample heterogeneity, while the information gain ratio (IGR) technique was employed to select relevant conditioning factors. Modeling was carried out using the random forest (RF) algorithm, with performance assessed via receiver operating characteristic (ROC) curves. The proposed approach offers a more efficient strategy for conditioning factor selection and supports broader adoption of machine learning in DFSM applications.

Materials

Study area and debris flow inventory

Beichuan County is located in the northwestern section of China’s Sichuan Basin, covering a total area of roughly 3084 km². Its position at the junction of the Longmenshan and Minshan ranges results in a highly complex geological environment. It is a seismically active area that experienced catastrophic impacts during the Wenchuan earthquake, reducing the stability of the geotechnical body. In terms of topography, the steep terrain and deep valleys could provide ideal conditions for debris flow initiation. The upper reaches of valleys exhibit funnel-shaped or dendritic drainage systems, facilitating rapid water and debris convergence. Vertical drops greater than 3,700 m would generate significant gravitational potential energy, which could cause slope failures during heavy rainfall. On the other hand, the fragile lithology and tectonic activity contribute to abundant loose material. The area was dominated by sandstone, shale, and limestone, which are prone to weathering and disintegration. Since the area was deeply affected by earthquake, post-quake slopes remain unstable, with frequent rockfalls and superficial landslides. As for the hydrometeorological triggers, the annual precipitation averages 1270 mm, with > 50% concentrated in July to August. In addition, historical records indicate that the 24-hour rainfall exceeds 280 mm, intense and heavy rainfall is necessary for debris flow formation. Finally, based on the historical records, a total of 148 debris flow catchments were collected as samples for model training. The spatial distribution of debris flow samples is shown in Fig. 1. The time span covers from the Wenchuan earthquake in 2008 to the present. Multiple debris flow events within a single debris flow catchment were identified as the same sample. The same number of non-debris flow samples are randomly selected from each dataset to avoid data imbalance. The prediction of susceptibility to debris flows can be transformed into a binary classification problem. Finally, 70% of the samples from each dataset were used for training, while 30% were used for validation. The division ratio is primarily determined based on sample size and previous studies⁶. Adequate samples are required to ensure both effective training and sufficient validation of the model. The use of simple partitioning is to reduce computational complexity. The partitioning of each dataset was carried out randomly to ensure a consistent distribution between the training dataset and the validation dataset.

Fig. 1.

The debris flow inventory map of Beichuan County. (Produced by ArcGIS 10.8)

Conditioning factors

In machine learning based studies, conditioning factor refers to the input features of the samples. The initial determination of debris flow conditioning factors is usually subjective, but generally needs to be guided by the following rules: (1) Conditioning factors need to cover all major elements of debris flow formation, including the dimensions of geology, topography, climate, physical sources, vegetation and human activities; (2) Conditioning factors that can be quantified through remote sensing, GIS technology or field surveys should be prioritized; (3) Prioritization should be given to conditioning factors that capture the distinctive features of the study area. Finally, combining previous studies and data availability³⁵, as shown in Figs. 2 and 14 conditioning factors were prepared.

Fig. 2.

Conditioning factors: (a) Altitude; (b) Slope; (c) Aspect; (d) Plane curvature; (e) Profile curvature; (f) TWI; (g) NDVI; (h) Rainfall; (i) Distance to roads; (j) Distance to faults; (k) Lithology; (l) Watershed area; (m) Watershed circularity; (n) Melton ratio. (Produced by ArcGIS 10.8)

Topographic factors usually play a significant role in DFSM studies. In this study, all topographic factors were derived from a 30-meter digital elevation model (DEM) sourced from the Geospatial Data Cloud. Altitude directly affects topographic energy and precipitation distribution. Higher altitude areas are characterized by large diurnal temperature differences, strong physical weathering and the formation of loose accumulations³⁶. Steeper slopes would increase shear stress, reduce stability, accelerate surface runoff, enhance scouring capacity, and promote the initiation of loose materials³⁷. Aspect is another conditioning factor. Shady slopes with low solar radiation and high soil moisture have low shear strength and are prone to destabilization. Sunny slopes are prone to accelerated rock disintegration due to large temperature differences. These conditioning factors collectively favor the build-up of material sources for debris flows³⁸. In governing the mechanisms of debris flow, plane curvature and profile curvature usually serve as critical variables³⁹. High topographic wetness index (TWI) values indicate high soil saturation, which tends to trigger soil liquefaction and reduce shear strength, thus providing loose material⁴⁰.

With lower vegetation cover, the root fixation effect is usually quite weak. The surface runoff would be accelerated, and the soil erosion would be increased⁴¹. Finally, the loose material sources is directly increased. In this study, normalized difference vegetation index (NDVI) was selected to characterize the vegetative state, which is also from the Geospatial Data Cloud.

Heavy rainfall acts as the primary trigger for debris flows, leading to soil saturation, an increase in the weight of rock and soil masses, and a decline in shear strength⁴². Consequently, rainfall was incorporated as one of the conditioning factors. The rainfall data is sourced from a website: worldclim.org.

Fault zones are characterized by rock fragmentation and fissure development, which could provide a rich source of materials for debris flows. On the other hand, active faults may also trigger earthquakes, indirectly triggering landslides and collapses. Distance to faults was therefore chosen as a conditioning factor⁴³. In addition, distance to roads was selected as another conditioning factor since road construction requires excavation of slopes, which destabilizes the terrain and increases the amount of loose accumulations⁴⁴. At the same time, it changes the path of surface runoff and intensifies erosion. The faults were obtained based on a 1:50000 geological map, and the roads were obtained based on Google imagery.

Lithology determines the resistance of rocks to weathering and the supply of loose material sources. Weak rock formations are prone to weathering and fragmentation and provide a large amount of scattered material⁴⁵. The lithology was obtained based on a 1:50000 geological map.

Indicators related to watershed morphology significantly influence the initiation of debris flows. Using the ArcGIS platform, key parameters including watershed area, circularity ratio, and Melton ratio were derived from the DEM⁴⁶.

Finally, the conditioning factors were used as independent variables, and the occurrence of debris flows (represented by 0 and 1) was used as the dependent variable to be input into the model for training.

Methodology

FCM and silhouette score

FCM clustering provides a mathematically rigorous framework for partitioning data amid ambiguity. Its probabilistic formulation reflects inherent uncertainties in pattern recognition tasks, offering superior adaptability to noisy, overlapping structures compared to hard clustering counterparts^47,48. Taking debris flow samples as an example, traditional hard clustering methods would categorize a debris flow sample exclusively into a specific class. On the contrary, FCM could calculate the membership degree of the debris flow sample to different classes. Clearly, debris flow samples cannot always be easily classified into a single category and FCM is more compatible with the complex formation mechanism of debris flows. In addition, some spatial clustering methods such as GeoDetector were also widely used recently⁴⁹. However, geographic data commonly exhibit spatial autocorrelation. Spatial clustering methods readily yield contiguous large spatial blocks, which might merely reflect geographic proximity, obscuring deeper attribute-based similarity patterns that span regions. In contrast, the FCM method is not confined to spatial distribution and holds a broader application prospect. Therefore, the FCM method was selected for this study. The relevant steps and formulas are as follows:

1. Membership constraint: Each sample x_i has a membership degree u_ij to cluster j. The sum of sample’s membership degrees across all C clusters must equal 1:

1

• (2)Cluster center update: The center c_j of cluster j is the weighted average of all samples, where the weights are their membership degrees raised to a fuzziness coefficient. In accordance with prior studies⁵⁰, the value of m was determined as 2:

2

• (3)Objective Function: Through the minimization of the objective function, new cluster centers could be derived. ||x_i-x_j|| represents the Euclidean distance between sample x_i and x_j.

3

• (4)Membership degree update: Based on the current cluster centers, the membership degree of sample x_i to cluster j is updated inversely proportional to its distance to each center:

4

The algorithm iterates between updating cluster centers (Eq. 2) and membership degrees (Eq. 4) until the objective function J converges. Final cluster assignment is determined by the highest membership degree for each sample. In this study, all the conditioning factors were considered as inputs to the FCM algorithm after normalization. Categorical variables were converted into numerical form based on one-hot encoding. The implementation of the FCM algorithm was based on the Python 3.13 using Pycharm. Except for the number of clusters, which requires repeated modification, the remaining hyperparameters were set to their default values.

To evaluate the clustering effectiveness, we also introduced the silhouette score, which is an excellent internal evaluation metric for unsupervised clustering⁵¹. Its key advantage is that it requires no ground truth labels and assesses clustering quality based solely on the data’s intrinsic structure. The formula for a single sample i is calculated as follows:

1. Compute a(i) (mean intra-cluster distance): The average distance between sample i and all other points in the same cluster. A smaller value indicates the sample is well-matched to its own cluster.

2. Compute b(i) (mean nearest-cluster distance): The average distance between sample i and all points in the nearest cluster to which it does not belong. A larger value indicates the sample is well-separated from other clusters.

3. Combine to compute the silhouette score for sample i, s(i):

5

The final silhouette score is the mean of all samples’ s(i), with a value range of [-1,1]. The closer the silhouette score is to 1, the better the clustering results.

Information gain ratio

The information gain ratio (IGR) method refines feature selection in decision trees by harmonizing discriminative ability with attribute complexity. Its mechanism mitigates biases inherent in information gain method, promoting more generalizable and interpretable results⁵². The higher the IGR value, the stronger the predictive ability of the conditioning factor. The formulas are as follows:

The information gain ratio R(D, X) is defined as the ratio of the information gain G(D, X) to the intrinsic information I(X) of the factor:

6

The information gain G(D, X) quantifies the reduction in uncertainty about the target variable (debris flow occurrence) after knowing X. It is calculated as the difference between the entropy of the dataset H(D) and the conditional entropy H(D|X):

7

The entropy of the dataset H(D) measures the overall uncertainty of the target variable. For a binary classification (debris flow occurrence: 0 or 1), it is defined as:

8

Where p_k represents the proportion of samples belonging to class k in dataset D (e.g., p₁ is the proportion of debris flow samples).

The conditional entropy H(D|X) measures the remaining uncertainty about the target variable given the conditioning factor X. It is computed by weighting the entropy within each partition of D induced by the values of X :

9

Where x_i denotes the i-th unique value or category of the conditioning factor X; P(X = x_i) is the probability of samples in D where factor X takes the value x_i ; H(D|X = x_i) is the entropy of the target variable calculated only from the subset of samples where X = x_i, using the same formula as (8) with class proportions p_k computed within that subset.

Finally, the intrinsic information I(X) penalizes factors with many values. It is the entropy of the conditioning factor X itself:

10

Where P(X = x_i) is the same probability as defined in Eq. (9).

Random forest

The random forest (RF) algorithm constitutes an ensemble learning approach that operates by aggregating the outputs of multiple decision trees, thereby enhancing its predictive performance and generalization capacity⁵³. The algorithm is mainly based on two synergistic randomization mechanisms: (1) In bootstrap aggregating, each tree is constructed from a separate data subset created through replacement sampling of the original dataset; (2) In feature subspace randomization, where node splitting utilizes an optimal criterion derived from a randomly selected subset of features for classification rather than the entire feature set. This dual randomization strategy could mitigate over-fitting by reducing model variance while maintaining low bias, and diminishes inter-tree correlation, thereby amplifying the ensemble’s robustness against noise and outliers⁵⁴. The implementation of the FCM algorithm was based on the Python scikit-learn library and Pycharm was applied. In the modeling process, the number of decision trees N and the number of conditioning factors m employed for tree splitting were two key parameters that require tuning. Based on the grid search method and previous studies⁵⁵, N and m were determined to be 200 and 3, respectively. The remaining parameters remain at their default values.

Model evaluation method

The ROC curve is a fundamental graphical tool for evaluating the performance of binary classifiers across diverse classification thresholds. This metric characterizes the relationship between true positive rate and the false positive rate by systematically varying the discrimination threshold⁵⁶. Sensitivity and specificity are two critical variables for plotting a ROC curve. Sensitivity can measure the model’s ability to “identify” all positive samples, while specificity measures the model’s ability to “eliminate” negative examples and avoid false positives. The relevant formulas are as follows:

11

12

Where TP, TN, FP, and FN denote true positive, true negative, false positive, and false negative observations. Varying the classification threshold yields multiple pairs of sensitivity and 1-specificity values, and the ROC curve could be obtained. In practice, the area under the curve (AUC) is a vital measure of predictive performance.

In addition, accuracy and F1 score were also used to evaluate the model in this study. Accuracy is the overall ability of a model to make “correct judgments”. It is a very intuitive and easy to understand comprehensive indicator. F1 score mainly focuses on the predictive quality of positive samples, it can play an important role in some scenarios where the sample is imbalanced^57,58. The relevant formula is as follows:

13

Where the meanings of TP, FP and FN are the same as those in Eqs. (11) and (12).

Results

Study area division results

Based on the FCM clustering method, the study area could be divided into a specific number of homogenized regions. It should be noted that the FCM method was applied to all the debris flow catchments to perform clustering. In general, A higher cluster count generally mitigates the detrimental effects of sample heterogeneity in the resulting model. On the other hand, each cluster should also retain a certain number of samples to build a stable model. Figure 3 exhibited the clustering results of the study area with different cluster numbers. Table 1 showed the silhouette scores of different clustering schemes. It can be seen that the clustering effect is best when the number of clusters is 4. Table 2 showed the number and the proportion of the debris flow samples in each cluster after clustering. The results showed that the number of debris flow samples in some clusters began to be insufficient when the total number of clusters reached 5. Therefore, the study are was finally divided into 4 homogenized regions.

Fig. 3.

Clustering results corresponding to different numbers of clusters: (a) 2 clusters; (b) 3 clusters; (c) 4 clusters; (d) 5 clusters. (Produced by ArcGIS 10.8)

Table 1.

Silhouette scores of different clustering schemes.

Number of clusters	Silhouette score
2	0.632
3	0.798
4	0.856
5	0.739

Table 2.

Statistical results of clustering.

Number of clusters	Cluster	Number of debris flow samples	Percentage of debris flow samples (%)
2	Cluster 1	40	27.0
	Cluster 2	108	73.0
3	Cluster 1	65	43.9
	Cluster 2	69	46.6
	Cluster 3	14	9.5
4	Cluster 1	33	22.3
	Cluster 2	9	6.1
	Cluster 3	56	37.8
	Cluster 4	50	33.8
5	Cluster 1	36	24.3
	Cluster 2	73	49.3
	Cluster 3	1	0.7
	Cluster 4	8	5.4
	Cluster 5	30	20.3

Conditioning factor analysis results

In practice, the IGR value is obtained based on all debris flow samples and non-debris flow samples in each dataset. The calculation of IGR value typically involves dividing the conditioning factors into several categories. For continuous variables, the specific division strategy was shown in Fig. 2. As presented in Table 3, the predictive ability of the conditioning factors exhibited considerable variation across the different datasets, which indicated the distinct main controlling factors for debris flow formation in different subregions. In addition, most conditioning factors are more predictive in homogenized datasets, demonstrating the necessity of considering the sample heterogeneity problem. Finally, the total dataset and each of the homogeneous datasets were involved in subsequent modeling after removing two conditioning factors with the worst predictive ability, which is a comprehensive judgment of the balance between model complexity and information retention. For comparative purposes, the total dataset and homogenized datasets containing all conditioning factors were also employed in the subsequent modeling phase.

Table 3.

The IGR values of conditioning factors in different datasets.

Conditioning factors	Total dataset	Cluster 1	Cluster 2	Cluster 3	Cluster 4
Altitude	0.0577	0.0010	0.1786	0.0035	0.0677
Slope	0.0336	0.0487	0.0671	0.0555	0.1098
Aspect	0.0056	0.0234	0.1954	0.0181	0.0385
Plane curvature	0.0036	0.0103	0.0465	0.0044	0.0913
Profile curvature	0.0106	0.0176	0.0475	0.0103	0.0609
TWI	0.0258	0.0397	0.0236	0.1138	0.1071
NDVI	0.0170	0.0098	0.0445	0.0145	0.0825
Rainfall	0.0843	0.0425	0.0437	0.0756	0.1508
Distance to roads	0.0321	0.0104	0.1648	0.0139	0.0588
Distance to faults	0.0277	0.0177	0.0697	0.0593	0.0669
Lithology	0.0630	0.0071	0.1052	0.1010	0.0985
Watershed area	0.0065	0.0096	0.0864	0.0011	0.0548
Watershed circularity	0.0071	0.0069	0.0936	0.0053	0.0550
Melton ratio	0.0192	0.0009	0.0926	0.0110	0.1115

Model performance

To mitigate the impact of data imbalance, an equal number of debris flow samples and non-debris flow samples in the training dataset were prepared for each dataset to participate in model training³⁶. Global models for the total dataset and personalized models for the homogenized datasets were established based on the RF algorithm. To facilitate comparison with global models, the prediction results of the 4 personalized models on training datasets and validation datasets were merged respectively to evaluate the model performance. Figure 4 presented the ROC curves, while Table 4 summarized the model performance metrics across different modeling scenarios. The results showed that the personalized models performed better, underscoring the importance of accounting for sample heterogeneity. Furthermore, the selection of conditioning factors could help the models to achieve better performance.

Fig. 4.

ROC curves derived from: (a) training datasets with all conditioning factors; (b) training datasets with selected conditioning factors; (c) validation datasets with all conditioning factors; (d) validation datasets with selected conditioning factors.

Table 4.

Model performance metrics across different modeling scenarios.

Conditioning factors	Model type	Datasets	AUC	Sensitivity	Specificity	Accuracy	F1 score
All	Global	Training	0.914	0.83	0.78	0.81	0.81
		Validation	0.833	0.77	0.73	0.75	0.74
	Personalized	Training	0.931	0.89	0.87	0.88	0.89
		Validation	0.912	0.84	0.83	0.83	0.84
Selected	Global	Training	0.934	0.96	0.94	0.95	0.88
		Validation	0.856	0.83	0.79	0.81	0.82
	Personalized	Training	0.955	0.97	0.95	0.96	0.90
		Validation	0.920	0.87	0.86	0.87	0.85

Debris flow susceptibility maps

As the primary output of a debris flow susceptibility modeling (DFSM) study, the accuracy and reliability of the resulting susceptibility maps are essential in assessing the overall project’s success. Figure 5 exhibited the susceptibility maps generated by various models. The susceptibility maps based on personalized models were composed of the prediction results of four personalized models for each cluster. The susceptibility of each map was categorized into four classes, namely very low (0-0.25), low (0.25–0.5), medium (0.5–0.75), and high (0.75-1). The distribution of debris flow samples on each susceptibility map were shown in Table 5. The results showed that the debris flow susceptibility maps generated by the models after conditioning factor selection can predict more debris flow samples as moderate or high susceptibility. As shown in Fig. 5 (c), 87.1% of the debris flow samples are distributed in moderate to high susceptibility, while the proportion in Fig. 5 (a) is 85.1%. On the other hand, models that consider sample heterogeneity could generate more reasonable susceptibility debris flow susceptibility maps. As shown in Fig. 5 (d), the proportion of debris flow samples with moderate or high susceptibility is 92.5%, which is consistent with the best performance of the personalized models underwent conditioning factor selection. In addition, the proportion of very low susceptibility level in the susceptibility maps based on this model is also the highest, which reflects the stronger discriminative ability and higher decision confidence of the model. By addressing sample heterogeneity and screening key conditioning factors, the model can more clearly identify large areas lacking triggering conditions (such as specific terrain, hydrology, or source combinations) as truly safe ‘very low’ susceptibility areas, and more accurately focus on high-susceptibility areas. In practice, this model has significant value in practical disaster prevention planning and helps optimize resource allocation.

Fig. 5.

Debris flow susceptibility maps derived from different models: (a) global model for all conditioning factors; (b) personalized models for all conditioning factors; (c) global model for selected conditioning factors; (d) personalized models for selected conditioning factors. (Produced by ArcGIS 10.8)

Table 5.

Statistical results of debris flow susceptibility maps.

Model	Susceptibility	Number of samples	Percentage of samples	Number of debris flow samples	Percentage of debris flow samples
Global model for all conditioning factors	Very low	657	16.9%	4	2.7%
	Low	1296	33.3%	18	12.2%
	Moderate	828	21.2%	22	14.8%
	High	1112	28.6%	104	70.3%
Personalized models for all conditioning factors	Very low	1260	32.4%	4	2.7%
	Low	1521	39.1%	10	6.8%
	Moderate	473	12.2%	21	14.2%
	High	639	16.3%	113	76.3%
Global model for selected conditioning factors	Very low	670	17.2%	5	3.4%
	Low	1283	33.0%	14	9.5%
	Moderate	499	12.8%	27	18.2%
	High	1441	37.0%	102	68.9%
Personalized models for selected conditioning factors	Very low	1953	50.2%	3	2.1%
	Low	828	21.2%	8	5.4%
	Moderate	264	6.8%	23	15.5%
	High	848	21.8%	114	77.0%

Comparing the spatial patterns of high-susceptibility areas in Fig. 5, it can be observed that the distribution of high-susceptibility areas of the global model is relatively scattered and has fewer continuous areas, indicating that its prediction is influenced by the global average effect and lacks spatial specificity. The high-risk areas of the personalized model exhibited more obvious spatial clustering, and have a higher degree of spatial coupling with known fault zones, steep river valleys, and historical debris flow points. Especially in Fig. 5 (d), the high-susceptibility areas are the most compact and coherent in space, mainly concentrated along the main fault zones and deep river valleys in the central and southern parts of the study area, which is highly consistent with the geological and topographic control mechanisms of debris flows in this region. The evolution of this spatial pattern further proves that a framework that considers heterogeneity can produce more geographically significant and more consistent susceptibility zones with formation mechanisms.

Discussion

Implications and solutions to the sample heterogeneity problem

The problem of heterogeneity in DFSM originates primarily from disparities in the geographical distribution of debris flow samples. Previous DFSM studies typically assumed that the samples within the study area share the same debris flow formation mechanism. However, for large-scale study areas, the genetic mechanisms of debris flows across distinct blocks would exhibit greater diversity, resulting in more pronounced sample heterogeneity. For example, in this study, distance to roads, distance to faults, and lithology are three condition factors reflecting different source material origins. For cluster 1, distance to faults exhibited the strongest predictive ability. For Cluster 2, distance to roads demonstrated the best predictive ability. For clusters 3 and 4, lithology played the most important role. The results showed that debris flow samples in different clusters have different main provenance sources. In this context, machine learning models might produce biased decisions due to distribution differences implicit in the data. If the samples in the training set are dominated by a particular distribution, it is difficult for the model to achieve accurate predictions for samples in other distributions. Moreover, the issue of sample heterogeneity amplifies the model’s dependence on initialization settings, creating additional difficulties for effective training.

Addressing the challenge of sample heterogeneity necessitates the formulation of tailored methodologies for samples exhibiting diverse distributions. Yang et al. specified individualized parameters for samples in different distributions to enhance model performance³⁴. Zhang et al. introduced a multitask learning framework to develop a model that simultaneously handles several related tasks, and by incorporating domain-specific knowledge, the model’s cross-task generalization capability can be enhanced⁵⁹.Wang et al. combined a global model trained on all samples with personalized models trained on samples in specific distributions to obtain a new model to balance the predictive ability⁶⁰. Shi et al. suggested improving model stability by establishing correlations between the parameter sets of collective and individualized models via regularization techniques⁶¹.

Overall, the solutions proposed by previous studies to address sample heterogeneity are highly dependent on model parameter settings and model coupling, which would be challenging sometimes. This study employed the fuzzy C-means (FCM) algorithm to partition the target area into multiple environmentally uniform subareas for individualized analysis. The results showed that considering sample heterogeneity could generate more predictive conditioning factors, better-performing models and more plausible maps of debris flow susceptibility.

In fact, the method proposed in this study is based on the fundamental assumption that samples that are close in the conditioning factor space share similar mechanisms for debris flow formation, which is grounded in both geomorphological theory and machine learning principles. From a geomorphological perspective, debris flow initiation is governed by a combination of topographic, geological, hydrological, and anthropogenic factors that collectively define the “preparatory” and “triggering” conditions. When samples exhibit similar values across a comprehensive set of conditioning factors, they likely represent comparable environmental settings in terms of slope stability, material availability, and triggering thresholds. From a machine learning standpoint, the use of unsupervised clustering (such as FCM) to partition the study area implicitly assumes that the feature space captures the underlying structure of the dataset. If the selected conditioning factors are physically meaningful and collectively representative of the debris flow formation mechanism, then proximity in this high-dimensional space should reflect similarity in the dominant controlling processes. This assumption is supported by the empirical results of this study, which showed that different clusters exhibit distinct dominant factors. Overall, the assumption relied upon in this study are quite reliable, bolstering the persuasiveness of the proposed solutions to sample heterogeneity.

Conditioning factor selection framework considering sample heterogeneity

Conditioning factor selection methods have their merits and can help to improve models performance. For RF models used in this study, an appropriate conditioning factor extraction strategy could reduce the risk of overfitting, enhance training and prediction efficiency, and improve the interpretability of the model. In addition, conditioning factor selection considering sample heterogeneity could provide a more accurate physical explanation for the formation mechanism of debris flows in different subregions. In this study, based on the IGR values, clusters 1, 2, 3, and 4 can be named as slope-dominated cluster, aspect-dominated cluster, hydrology-dominated cluster, and rainfall-triggered cluster, respectively. For the slope-dominated cluster, the predictive ability of slope is the strongest for the occurrence of debris flows, indicating that topographic steepness is a key driving factor for the formation of debris flows in this subregion. For the aspect-dominated cluster, the predictive ability of aspect is significantly higher than that of other factors, indicating that aspect-related factors such as sunlight, temperature, and humidity differences played a dominant role in the formation mechanism of debris flows in this subregion. For the hydrology-dominated cluster, TWI reflects soil moisture and confluence conditions, and its high predictive ability indicates that the occurrence of debris flows in this subregion is primarily controlled by hydrological processes and soil saturation levels. For the rainfall-triggered cluster, the predictive ability of rainfall factors is the most prominent, indicating that debris flows in this sub-region are primarily triggered directly by heavy rainfall events, with rainfall being the most sensitive external inducing factor in the area.

However, few studies have considered the sample heterogeneity problem. Within the scope of sizable DFSM investigations, a given set of conditioning factors invariably assumes distinct roles across separate blocks. In this study, we conducted independent studies for samples in different distributions, with each subset removing its two conditioning factors with the worst predictive ability.The results showed that this framework better enhanced the model performance and improved the quality of debris flow susceptibility maps compared to traditional frameworks.

Limitations and future works

This study proposed a novel conditioning factor selection framework, yet several limitations should be acknowledged. Firstly, the performance of the Fuzzy C-Means (FCM) clustering is sensitive to the pre-defined number of clusters and initialization. The determination of four clusters, while justified by sample size and silhouette score, remains somewhat subjective and may not fully capture the inherent environmental heterogeneity. Secondly, the clustering was conducted solely based on the conditioning factors, which might not reflect the true differentiation of debris flows. Thirdly, subregions with few samples would risk model overfitting, reducing the generalization ability of the models. In addition, due to the scarcity of samples, all the debris flow samples were utilized in the data processing process prior to modeling, which might result in a certain degree of data leakage, making it impossible to accurately evaluate the model performance. Finally, the framework’s real-world application requires high-resolution data and significant computational resources, posing practical challenges. In the future, some works can be carried out to improve this study: (1) develop adaptive clustering strategies; (2) incorporate broader environmental variables into clustering; (3) apply few-shot learning to improve model robustness in data-scarce subregions; (4) validate the framework’s transferability in different regions; (5) conduct effective data segmentation and utilization and (6) enhance model interpretability using explainable techniques for practical hazard management.

Conclusion

DFSM is critical for disaster prevention, while challenges still exist in addressing selecting conditioning factors. This paper explored the an innovative conditioning factor selection framework. The following conclusions could be drawn:

1. Addressing sample heterogeneity significantly improves model performance. Partitioning the study area into homogeneous subareas using Fuzzy C-Means clustering effectively mitigates the issue of sample heterogeneity. Personalized models developed on these subareas demonstrated significantly higher predictive accuracy than the global model, validating the necessity of considering spatial heterogeneity in both factor selection and modeling.

2. Local conditioning factor selection outperforms global selection. Selecting conditioning factors based on IGR within each homogeneous subarea identifies more locally representative key factor combinations. This approach enhances the model’s generalizability and interpretability more effectively than strategies based on global factor selection.

3. The integrated framework produces more reliable susceptibility maps. The integrated framework combining unsupervised clustering, local factor optimization, and Random Forest modeling not only achieves higher predictive accuracy but also generates debris flow susceptibility maps with clearer geographical significance and risk classification. This provides a more reliable decision-making basis for disaster risk management.

Author contributions

Ruiyuan Gao: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Writing–original draft; Ang Wang: Supervision, Data curation, Investigation, Project administration, Writing – review & editing. Di Wu: Data curation, Investigation, Validation.

Funding

This work was supported by the the Research Foundation for Doctors of Huanghe Science and Technology College.

Data availability

The authors declare that the data supporting the findings of this study are available within the paper. If any raw data files are needed in another format, they are available from the corresponding author upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.