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

Abstract

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.

Appendix

$$R^{2} = \frac{{\left[ {\sum\nolimits_{i = 1}^{N} {\left( {y_{S} (i) - \bar{y}_{S} } \right)} \left( {y(i) - \bar{y}} \right)} \right]^{2} }}{{\sum\nolimits_{i = 1}^{N} {\left( {y_{S} (i) - \bar{y}_{S} } \right)^{2} } \sum\nolimits_{i = 1}^{N} {\left( {y(i) - \bar{y}} \right)^{2} } }}$$

(A.1)

$${\text{RMSE}} = \sqrt {\frac{1}{N}\sum\limits_{i = 1}^{N} {\left( {y_{S} (i) - y(i)} \right)^{2} } }$$

(A.2)

$${\text{MAE}} = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left| {y_{S} (i) - y(i)} \right|}$$

(A.3)

$${\text{ME}} = \hbox{max} \left| {y_{S} (i) - y(i)} \right|$$

(A.4)

$${\text{AWCI}} = \frac{1}{N}\sum\limits_{i = 1}^{N} {2 \times 1.96 \times \sqrt {\sigma_{i}^{2} } }$$

(A.5)

$${\text{AVCI}} = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left| {2 \times 1.96 \times \sqrt {\sigma_{i}^{2} } - {\text{AWCI}}} \right|}$$

(A.6)

where $y_{S} (i)$ and $y(i)$ denote the model output and measured values of the radial displacement, respectively ($i = 1,2, \ldots ,N$); $\bar{y}_{S}$ and $\bar{y}$ represent the average of the model output and measured values, respectively; $N$ represents the number of observations. $\sigma_{i}^{{}}$ is the predicted variance of ORVM-based prediction model at output point $y_{S} (i)$.