Multi-kernel optimized relevance vector machine for probabilistic prediction of concrete dam displacement

  • Siyu Chen
  • Chongshi GuEmail author
  • Chaoning LinEmail author
  • Kang Zhang
  • Yantao Zhu
Original Article


The observation data of dam displacement can reflect the dam’s actual service behavior intuitively. Therefore, the establishment of a precise data-driven model to realize accurate and reliable safety monitoring of dam deformation is necessary. This study proposes a novel probabilistic prediction approach for concrete dam displacement based on optimized relevance vector machine (ORVM). A practical optimization framework for parameters estimation using the parallel Jaya algorithm (PJA) is developed, and various simple kernel/multi-kernel functions of relevance vector machine (RVM) are tested to obtain the optimal selection. The proposed model is tested on radial displacement measurements of a concrete arch dam to mine the effect of hydrostatic, seasonal and irreversible time components on dam deformation. Four algorithms, including support vector regression (SVR), radial basis function neural network (RBF-NN), extreme learning machine (ELM) and the HST-based multiple linear regression (HST-MLR), are used for comparison with the ORVM model. The simulation results demonstrate that the proposed multi-kernel ORVM model has the best performance for predicting the displacement out of range of the used measurements dataset. Meanwhile, the ORVM model has the advantages of probabilistic output and can provide reasonable confidence interval (CI) for dam safety monitoring. This study lays the foundation for the application of RVM in the field of dam health monitoring.


Optimized relevance vector machine Multi-kernel Jaya optimization algorithm Dam health monitoring Prediction model 



Relevance vector machine


Optimized relevance vector machine


Confidence interval






Multiple linear regression


Partial least squares regression


Stepwise regression


Parallel Jaya algorithm


Artificial neural network


Multilayer perceptron


Single hidden layer feedforward neural networks


Adaptive neural fuzzy inference system


Multivariate adaptive regression splines


Gaussian process regression


Radial basis function neural network


Extreme learning machine


Support vector machine


Support vector regression


HST-based multiple linear regression


Multi-kernel Gaussian kernel + polynomial kernel


Multi-kernel Gaussian kernel + Laplace kernel


Multi-kernel Laplace kernel + polynomial kernel


Gaussian kernel-based optimized relevance vector machine


SumGP kernel-based optimized relevance vector machine


Coefficient of determination


Root mean square error


Mean absolute error


Maximum absolute error


Average width of confidence interval


Average variance of confidence interval



The authors are grateful to the financial sponsorship from National Natural Science Foundation of China (Grant Nos. 51739003, 51779086), National Key R&D Program of China (2018YFC0407104, 2016YFC0401601), Special Project Funded of National Key Laboratory (20165042112) and Key R&D Program of Guangxi (AB17195074).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.


  1. 1.
    Wu ZR (2003) Safety monitoring theory and its application of hydraulic structures. Higher Education, BeijingGoogle Scholar
  2. 2.
    Zhao EF (2018) Dam Safety Monitoring Data Analysis Theory & Assessment Methods. Hohai University Press,Google Scholar
  3. 3.
    Shi YQ, Yang JJ, Wu JL, He JP (2018) A statistical model of deformation during the construction of a concrete face rockfill dam. Structural Control & Health Monitoring. CrossRefGoogle Scholar
  4. 4.
    Gu CS, Wu ZR (2006) Safety monitoring of dams and dam foundations-theories & methods and their application. Hohai University Press,Google Scholar
  5. 5.
    Salazar F, Toledo MA, Onate E, Moran R (2015) An empirical comparison of machine learning techniques for dam behaviour modelling. Struct Saf 56:9–17. CrossRefGoogle Scholar
  6. 6.
    Salazar F, Morán R, Toledo MA, Oñate E (2015) Data-Based Models for the Prediction of Dam Behaviour: A Review and Some Methodological Considerations. Archives of Computational Methods in Engineering 24(1):1–21. CrossRefzbMATHGoogle Scholar
  7. 7.
    Mata J, de Castro AT, da Costa JS (2014) Constructing statistical models for arch dam deformation. Structural Control & Health Monitoring 21(3):423–437. CrossRefGoogle Scholar
  8. 8.
    Lin CN, Li TC, Liu XQ, Zhao LH, Chen SY, Qi HJ (2019) A deformation separation method for gravity dam body and foundation based on the observed displacements. Structural Control & Health Monitoring. CrossRefGoogle Scholar
  9. 9.
    Sun PM, Bao TF, Gu CS, Jiang M, Wang T, Shi ZW (2016) Parameter sensitivity and inversion analysis of a concrete faced rock-fill dam based on HS-BPNN algorithm. Science China-Technological Sciences 59(9):1442–1451. CrossRefGoogle Scholar
  10. 10.
    Mata J (2011) Interpretation of concrete dam behaviour with artificial neural network and multiple linear regression models. Engineering Structures 33(3):903–910. CrossRefGoogle Scholar
  11. 11.
    Stojanovic B, Milivojevic M, Ivanovic M, Milivojevic N, Divac D (2013) Adaptive system for dam behavior modeling based on linear regression and genetic algorithms. Advances in Engineering Software 65:182–190. CrossRefGoogle Scholar
  12. 12.
    Gu CS, Li B, Xu GL, Yu H (2010) Back analysis of mechanical parameters of roller compacted concrete dam. Science China-Technological Sciences 53(3):848–853. CrossRefzbMATHGoogle Scholar
  13. 13.
    Xi GY, Yue JP, Zhou BX, Tang P (2011) Application of an artificial immune algorithm on a statistical model of dam displacement. Computers & Mathematics with Applications 62(10):3980–3986. MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gu CS, Wang YC, Peng Y, Xu BS (2011) Ill-conditioned problems of dam safety monitoring models and their processing methods. Science China-Technological Sciences 54(12):3275–3280. CrossRefzbMATHGoogle Scholar
  15. 15.
    Hariri-Ardebili MA, Pourkamali-Anaraki F (2018) Simplified reliability analysis of multi hazard risk in gravity dams via machine learning techniques. Arch Civ Mech Eng 18(2):592–610. CrossRefGoogle Scholar
  16. 16.
    Hariri-Ardebili MA, Pourkamali-Anaraki F (2018) Support vector machine based reliability analysis of concrete dams. Soil Dynamics and Earthquake Engineering 104:276–295. CrossRefGoogle Scholar
  17. 17.
    Hariri-Ardebili MA, Barak S (2019) A series of forecasting models for seismic evaluation of dams based on ground motion meta-features. Engineering Structures. CrossRefGoogle Scholar
  18. 18.
    Hariri-Ardebili MA, Pourkamali-Anaraki F (2019) Matrix completion for cost reduction in finite element simulations under hybrid uncertainties. Applied Mathematical Modelling 69:164–180. MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hariri-Ardebili MA, Sudret B (2019) Polynomial chaos expansion for uncertainty quantification of dam engineering problems. Engineering Structures. CrossRefGoogle Scholar
  20. 20.
    Moody J, Darken CJ (1989) Fast Learning in Networks of Locally-Tuned Processing Units. Neural Computation 1(2):281–294. CrossRefGoogle Scholar
  21. 21.
    Vapnik V, Golowich SE, Smola A (1997) Support vector method for function approximation, regression estimation, and signal processing. Adv Neur In 9:281–287Google Scholar
  22. 22.
    Chen SY, Gu CS, Lin CN, Zhao EF, Song JT (2018) Safety Monitoring Model of a Super-High Concrete Dam by Using RBF Neural Network Coupled with Kernel Principal Component Analysis. Mathematical Problems in Engineering 2018:1–13. CrossRefGoogle Scholar
  23. 23.
    Kang F, Li JJ, Zhao SZ, Wang YJ (2019) Structural health monitoring of concrete dams using long-term air temperature for thermal effect simulation. Engineering Structures 180:642–653. CrossRefGoogle Scholar
  24. 24.
    Kang F, Liu J, Li JJ, Li SJ (2017) Concrete dam deformation prediction model for health monitoring based on extreme learning machine. Structural Control & Health Monitoring. CrossRefGoogle Scholar
  25. 25.
    Huang GB, Zhu QY, Siew CK (2006) Extreme learning machine: Theory and applications. Neurocomputing 70(1–3):489–501. CrossRefGoogle Scholar
  26. 26.
    Liu CG, Gu CS, Chen B (2017) Zoned elasticity modulus inversion analysis method of a high arch dam based on unconstrained Lagrange support vector regression (support vector regression arch dam). Engineering with Computers 33(3):443–456. CrossRefGoogle Scholar
  27. 27.
    Su HZ, Chen ZX, Wen ZP (2016) Performance improvement method of support vector machine-based model monitoring dam safety. Structural Control & Health Monitoring 23(2):252–266. CrossRefGoogle Scholar
  28. 28.
    Rankovic V, Grujovic N, Divac D, Milivojevic N (2014) Development of support vector regression identification model for prediction of dam structural behaviour. Struct Saf 48:33–39. CrossRefGoogle Scholar
  29. 29.
    Bui K-TT, Tien Bui D, Zou J, Van Doan C, Revhaug I (2016) A novel hybrid artificial intelligent approach based on neural fuzzy inference model and particle swarm optimization for horizontal displacement modeling of hydropower dam. Neural Computing and Applications 29(12):1495–1506. CrossRefGoogle Scholar
  30. 30.
    Kang F, Liu X, Li J (2019) Concrete Dam Behavior Prediction Using Multivariate Adaptive Regression Splines with Measured Air Temperature. Arabian Journal for Science and Engineering. CrossRefGoogle Scholar
  31. 31.
    Lin CN, Li TC, Chen SY, Liu XQ, Lin C, Liang SL (2019) Gaussian process regression-based forecasting model of dam deformation. Neural Comput Appl 31(12):8503–8518. CrossRefGoogle Scholar
  32. 32.
    Kang F, Li JJ (2019) Displacement Model for Concrete Dam Safety Monitoring via Gaussian Process Regression Considering Extreme Air Temperature. Journal of Structural Engineering 146(1):05019001CrossRefGoogle Scholar
  33. 33.
    Tipping ME (2001) Sparse Bayesian learning and the relevance vector machine. Journal of Machine Learning Research 1(3):211–244. MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Imani M, Kao HC, Lan WH, Kuo CY (2018) Daily sea level prediction at Chiayi coast, Taiwan using extreme learning machine and relevance vector machine. Global Planet Change 161:211–221. CrossRefGoogle Scholar
  35. 35.
    Zhang ZF, Liu ZB, Zheng LF, Zhang Y (2014) Development of an adaptive relevance vector machine approach for slope stability inference. Neural Comput Appl 25(7–8):2025–2035. CrossRefGoogle Scholar
  36. 36.
    Wang TZ, Xu H, Han JG, Elbouchikhi E, Benbouzid MEH (2015) Cascaded H-Bridge Multilevel Inverter System Fault Diagnosis Using a PCA and Multiclass Relevance Vector Machine Approach. Ieee T Power Electr 30(12):7006–7018. CrossRefGoogle Scholar
  37. 37.
    Kong DD, Chen YJ, Li N, Duan CQ, Lu LX, Chen DX (2019) Relevance vector machine for tool wear prediction. Mechanical Systems and Signal Processing 127:573–594. CrossRefGoogle Scholar
  38. 38.
    Rao R (2016) Jaya: A simple and new optimization algorithm for solving constrained and unconstrained optimization problems. International Journal of Industrial Engineering Computations 7(1):19–34Google Scholar
  39. 39.
    Holland JH (1975) Adaptation in natural and artificial systems : an introductory analysis with applications to biology, control, and artificial intelligence. University of Michigan Press, Ann ArborzbMATHGoogle Scholar
  40. 40.
    Farmer JD, Packard NH, Perelson AS (1986) The Immune-System, Adaptation, and Machine Learning. Physica D 22(1–3):187–204. MathSciNetCrossRefGoogle Scholar
  41. 41.
    Eberhart R, Kennedy J A new optimizer using particle swarm theory. In: MHS’95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, 1995. Ieee, pp 39-43Google Scholar
  42. 42.
    Li XL (2003) A new intelligent optimization-artificial fish swarm algorithm. PhD Dissertation, Zhejiang UniversityGoogle Scholar
  43. 43.
    Karaboga D, Basturk B (2007) A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm. Journal of global optimization 39(3):459–471MathSciNetCrossRefGoogle Scholar
  44. 44.
    Ding ZH, Li J, Hao H (2019) Structural damage identification using improved Jaya algorithm based on sparse regularization and Bayesian inference. Mechanical Systems and Signal Processing 132:211–231. CrossRefGoogle Scholar
  45. 45.
    Abhishek K, Kumar VR, Datta S, Mahapatra SS (2017) Application of JAYA algorithm for the optimization of machining performance characteristics during the turning of CFRP (epoxy) composites: comparison with TLBO, GA, and ICA. Engineering with Computers 33(3):457–475. CrossRefGoogle Scholar
  46. 46.
    Berger JO (2013) Statistical decision theory and Bayesian analysis. Springer Science & Business Media,Google Scholar
  47. 47.
    MacKay DJJNc (1992) Bayesian interpolation. 4 (3):415-447CrossRefGoogle Scholar
  48. 48.
    Rao R, Waghmare GG (2017) A new optimization algorithm for solving complex constrained design optimization problems. Engineering Optimization 49(1):60–83CrossRefGoogle Scholar
  49. 49.
    Migallon H, Jimeno-Morenilla A, Sanchez-Romero JL, Rico H, Rao RV (2019) Multipopulation-based multi-level parallel enhanced Jaya algorithms. J Supercomput 75(3):1697–1716. CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2020

Authors and Affiliations

  1. 1.State Key Laboratory of Hydrology-Water Resources and Hydraulic EngineeringHohai UniversityNanjingChina
  2. 2.College of Water Conservancy and Hydropower EngineeringHohai UniversityNanjingChina
  3. 3.National Engineering Research Center of Water Resources Efficient Utilization and Engineering SafetyHohai UniversityNanjingChina

Personalised recommendations