# Evaluation of Aeolian Sand Collapsibility Based on Physical Indicators in the Mu Us Desert, China
## 5 Construction of Collapsibility Evaluation Method
### Establishment of SVM Regression Model  
An SVM is a generalized linear classifier that performs a binary classification of data in a supervised learning manner; its concept was proposed by [PERSON] in 1963 [53]. Compared to logistic regression and neural networks, vector machines provide a clearer and more efficient means of learning complex nonlinear equations. It was first proposed for improvements and applications in object recognition and was later found to produce high-quality results for time predictions and regressions [54; 55; 56; 57; 58].  
For the SVM, the main idea is to maximize the interval in the data feature space and then transform it into a solution to a convex quadratic programming problem. SVM regression (SVR) is based on the SVM; introduces the deviation, \(\epsilon\), value; and constructs an isolation zone with a width of \(2\epsilon\) centred on \(f(x)=\omega^{T}x+b\). In contrast to previous machine learning techniques, there is no loss if the sample falls within the interval zone. Therefore, it can wrap as many samples as possible, and the fewest samples possible fall outside the interval zone [59; 60; 61].  
In this study, 14 sets of sample data were collected. Ten sets of sample data were used as training sets (Table 6) and four sets of sample data were used as test sets (Table 7). The water content, dry density, specific gravity, void ratio, and saturation were used as input variables, and the collapsibility coefficient was used as the output variable.  
Using the LIBSVM 3.31 software package ([[https://www.csie.ntu.edu.tw/~cjlin/libsvm/#d-ownload](https://www.csie.ntu.edu.tw/~cjlin/libsvm/#d-ownload)]([https://www.csie.ntu.edu.tw/~cjlin/libsvm/#d-ownload](https://www.csie.ntu.edu.tw/~cjlin/libsvm/#d-ownload)), accessed on 17 February 2023), the Gaussian radial basis kernel function (RBF) was selected, which has a good fit for nonlinearity ([PERSON], 2018) and has been applied for loess collapsibility ([PERSON], 2018). Specially, LIBSVM 3.31 is an open-source software library predicated on the support vector machine (SVM) paradigm, and is primarily utilized for classification tasks, encompassing binary and multi-class categorization, as well as regression problems.  
\[K\Big{(}x,x^{T}\Big{)}=exp\Big{(}-\gamma\cdot\Big{\|}x-x^{\prime}\Big{\|}^{2} \Big{)} \tag{12}\]  
where \(\gamma\) is a variable parameter. In addition to selecting the kernel function, the optimal penalty factor (\(c\)) and radial basis function parameters (\(g\)) of the model need to be determined.  
The particle swarm algorithm performed more effectively than the other two methods, the grid search and genetic algorithm, while finding the optimal parameters for the SVM classifier ([PERSON], 2018; [PERSON], 2018; [PERSON], 2018). This study used the particle swarm algorithm (PSO-SVR) to optimize these two parameters. The optimization results were as follows: penalty factor, \(c=13.3514\), and radial basis function parameter, \(g=1.0635\). Among them, 10 particles and 20 iterations were used, and the learning factors (\(c_{1},c_{2}\)) were both set to 1.5 in the parameter settings of the PSO method.