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Fractal Fract
Volume 9
Issue 4
10.3390/fractalfract9040210
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Article

Slope Deformation Prediction Combining Particle Swarm Optimization-Based Fractional-Order Grey Model and K-Means Clustering

by
Zhenzhu Meng
1,
Yating Hu
2,*,
Shunqiang Jiang
3,
Sen Zheng
4,5,
Jinxin Zhang
1,
Zhenxia Yuan
6 and
Shaofeng Yao
1
1
School of Hydraulic Engineering, Zhejiang University of Water Resources and Electric Power, Hangzhou 310018, China
2
School of Infrastructure Construction, Nanchang University, Nanchang 330031, China
3
Hangzhou Fuyang State Owned Resources Development Group Co., Ltd., Hangzhou 311400, China
4
College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China
5
School of Civil Engineering, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
6
School of Architecture and Civil Engineering, Zhongyuan University of Technology, Zhengzhou 450007, China
*
Author to whom correspondence should be addressed.
Fractal Fract. 2025, 9(4), 210; https://doi.org/10.3390/fractalfract9040210
Submission received: 28 February 2025 / Revised: 25 March 2025 / Accepted: 27 March 2025 / Published: 28 March 2025
(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)
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Abstract

Slope deformation poses significant risks to infrastructure, ecosystems, and human safety, making early and accurate predictions essential for mitigating slope failures and landslides. In this study, we propose a novel approach that integrates a fractional-order grey model (FOGM) with particle swarm optimization (PSO) to determine the optimal fractional order, thereby enhancing the model’s accuracy, even with limited and fluctuating data. Additionally, we employ a k-means clustering technique to account for both temporal and spatial variations in multi-point monitoring data, which improves the model’s ability to capture the relationships between monitoring points and increases prediction relevance. The model was validated using displacement data collected from 12 monitoring points on a slope located in Qinghai Province near the Yellow River, China. The results demonstrate that the proposed model outperforms the traditional statistical model and artificial neural networks, achieving a significantly higher coefficient of determination R2 up to 0.9998 for some monitoring points. Our findings highlight that the model maintains robust performance even when confronted with data of varying quality—a notable advantage over conventional approaches that typically struggle under such conditions. Overall, the proposed model offers a robust and data-efficient solution for slope deformation prediction, providing substantial potential for early warning systems and risk management.
Keywords: deformation; displacement prediction; fractional-order grey model; k-means clustering; particle swarm optimization; monitoring data deformation; displacement prediction; fractional-order grey model; k-means clustering; particle swarm optimization; monitoring data

Share and Cite

MDPI and ACS Style

Meng, Z.; Hu, Y.; Jiang, S.; Zheng, S.; Zhang, J.; Yuan, Z.; Yao, S. Slope Deformation Prediction Combining Particle Swarm Optimization-Based Fractional-Order Grey Model and K-Means Clustering. Fractal Fract. 2025, 9, 210. https://doi.org/10.3390/fractalfract9040210

AMA Style

Meng Z, Hu Y, Jiang S, Zheng S, Zhang J, Yuan Z, Yao S. Slope Deformation Prediction Combining Particle Swarm Optimization-Based Fractional-Order Grey Model and K-Means Clustering. Fractal and Fractional. 2025; 9(4):210. https://doi.org/10.3390/fractalfract9040210

Chicago/Turabian Style

Meng, Zhenzhu, Yating Hu, Shunqiang Jiang, Sen Zheng, Jinxin Zhang, Zhenxia Yuan, and Shaofeng Yao. 2025. "Slope Deformation Prediction Combining Particle Swarm Optimization-Based Fractional-Order Grey Model and K-Means Clustering" Fractal and Fractional 9, no. 4: 210. https://doi.org/10.3390/fractalfract9040210

APA Style

Meng, Z., Hu, Y., Jiang, S., Zheng, S., Zhang, J., Yuan, Z., & Yao, S. (2025). Slope Deformation Prediction Combining Particle Swarm Optimization-Based Fractional-Order Grey Model and K-Means Clustering. Fractal and Fractional, 9(4), 210. https://doi.org/10.3390/fractalfract9040210

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