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Prediction of Uniaxial Compressive Strength Using Fully Bayesian Gaussian Process Regression (fB-GPR) with Model Class Selection

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Abstract

In rock, mining, and/or tunneling engineering, determination of uniaxial compressive strength (UCS) of rocks is an important and crucial task, which is often estimated from readily available index properties of rocks in practice, such as porosity (n), Schmidt hammer rebound number (Rn), P-wave velocity (Vp), point load index (Is(50)). This is especially true for projects with medium- or small-size, as well as for rocks with a high degree of fragility and porosity. While numerous methods have been proposed for predicting UCS indirectly, linear assumptions are frequently made during model development, despite the possibility of nonlinear relationships between UCS and the abovementioned indices. Furthermore, many established methods often struggle to strike a balance between model complexity and performance, resulting in models that are either over- or under-fitted. As a result, constructing an optimal UCS model with minimal variables while maintaining a high level of performance remains a great challenge in rock engineering practice. This paper proposes a fully Bayesian Gaussian process regression (fB-GPR) approach to develop an optimal model for UCS prediction which strikes a balance between prediction accuracy and model complexity. Both real-world and numerical examples are used to illustrate the proposed method. Results show that the optimal model of predicting UCS for the database from Malaysia is constructed by only n and Vp, with the same coefficient of determination of around 0.9 as the more complex model involving n, Rn, Vp and Is(50). A sensitivity study is also performed to systematically examine its robustness and accuracy of the proposed method in developing optimal model for UCS prediction.

Highlights

  • A fully Bayesian Gaussian process regression (fB-GPR) method is proposed for constructing an optimal model for UCS prediction from rock indices.

  • The optimal model for UCS prediction has minimal variables but maintains a high level of performance.

  • The fB-GPR approach is data-driven and non-parametric, and can automatically strike a balance between model complexity and performance.

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Abbreviations

ANN:

Artificial neural network

DT:

Decision tree

fB-GPR:

Fully Bayesian Gaussian process regression

GP:

Gaussian process

GPR:

Gaussian process regression

IQR:

Interquartile range

MAPE:

Mean absolute percentage error

Max:

Maxima

MCMC:

Markov Chain Monte Carlo

Min:

Minima

MLE:

Maximum likelihood estimation method

PDF:

Probability density function

RF:

Random forest

RMSE:

Root mean squared error

SD:

Standard deviation

SVM:

Support vector machine

UCS:

Uniaxial compressive strength (MPa)

C(·):

Copular function

d :

Number of input variables (i.e., index properties of rocks)

f(X):

Mapping function from X to UCS, and UCS = f(X)

I :

Identity matrix

I s (50) :

Point load index (MPa)

K :

Covariance (kernel) matrix

l k :

Correlation length corresponding to Xk

M k :

The kth GPR candidate model

n :

Porosity (%)

N(·):

Normal distribution

n M :

Number of candidate GPR models

N s :

Number of MCMC samples

n t :

Number of available measurements

p(·):

Probability density function

P r(·):

Probability

R 2 :

Coefficient of determination

R n :

Schmidt hammer rebound number

V p :

P-wave velocity (m/s)

X :

Readily available engineering indices (or input variables)

X k :

The kth index property of interest (i.e., input variable)

x :

Measurements of X corresponding to measured UCS data

y :

Available UCS data corresponding to x

y i :

The ith measurement of UCS

ε :

Measurement error associated with UCS

Θ:

GPR hyper-parameters

\(\Theta_{i}^{L}\), \(\Theta_{i}^{U}\) :

Lower and upper bounds of the ith parameter Θi

μ * :

The most probable UCS predicted from new measurements of index properties x* by GPR in theory

\(\mu_{q}^{*}\) :

The most probable UCS predicted using the qth set of \(\Theta_{q}\) MCMC samples

μ UCS :

The most probable UCS predicted from fB-GPR with consideration of statistical uncertainty of hyper-parameters of GPR

μ y :

Mean of the nM measurements of UCS

ρ :

Pearson correlation coefficient

\({{\varvec{\Sigma}}}^{ * }\) :

Estimated covariance associated with predicted UCS data

\(\Sigma_{q}^{*}\) :

Covariance of predicted UCS using the qth set of \(\Theta_{q}\) MCMC samples

σ f :

Magnitude of the kernel function

\((\sigma_{q}^{*} )^{2}\) :

Diagonal elements of \(\Sigma_{q}^{*}\)

σ ε :

Standard deviation of measurement error

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Acknowledgements

The authors acknowledge the computational resources provided by HPC platform, Xi’an Jiaotong University, China.

Funding

This work described in this paper was supported by grants from the National Natural Science Foundation of China (Project No. 42107204), and the Fundamental Research Funds for the Central Universities (xjh012020046). The financial supports are gratefully acknowledged.

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Authors and Affiliations

  1. School of Human Settlements and Civil Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi Province, China

    Tengyuan Zhao, Chao Song, Shifeng Lu & Ling Xu

Authors
  1. Tengyuan Zhao
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  2. Chao Song
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  3. Shifeng Lu
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  4. Ling Xu
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Contributions

TZ: supervision, technical discussion, and writing—reviewing and editing; CS: conceptualization, modeling/simulation, validation, and original draft; SL: technical discussion and reviewing, manuscript editing; LX: supervision, technical discussion, and writing—reviewing and editing.

Corresponding author

Correspondence to Ling Xu.

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Cite this article

Zhao, T., Song, C., Lu, S. et al. Prediction of Uniaxial Compressive Strength Using Fully Bayesian Gaussian Process Regression (fB-GPR) with Model Class Selection. Rock Mech Rock Eng 55, 6301–6319 (2022). https://doi.org/10.1007/s00603-022-02964-y

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  • DOI: https://doi.org/10.1007/s00603-022-02964-y

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