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Prediction Modeling for the Estimation of Dynamic Elastic Young’s Modulus of Thermally Treated Sedimentary Rocks Using Linear–Nonlinear Regression Analysis, Regularization, and ANFIS

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Abstract

Thermal cracking significantly affects the dynamic and mechanical stability of rock mass. This study first focuses on the evaluation of dynamic-mechanical behavior of thermally deteriorated rocks in terms of their dynamic elastic Young’s modulus (Ed), quality factor (Q-factor), resonance frequency (Fr), unconfined compressive strength (UCS) and tensile strength (BTS). Secondly, it focuses on the comparison of the performance of different statistical data modeling techniques. The overall reduction in the values of Ed, Q-factor, Fr, UCS, and BTS for all thermally treated rock samples was recorded as 23–49%, 6–28%, 7–21%, 10–38%, and 14–56%, respectively. The previous studies do not show any significant correlation between the strength parameters of thermally deteriorated rocks. In this study, a total of 7 predictive models were developed to estimate Ed for thermally deteriorated rocks using linear-nonlinear regression analysis, regularization, and adaptive-neuro fuzzy inference system (ANFIS). Results of hypothesis testing showed that the linear-nonlinear regression equations were statistically significant. Similarly, outcomes of neuro-fuzzy logic analysis, based on the degree of thermal cracking of rocks satisfied the statistical significance of the ANFIS model. Among all prediction models, the ANFIS model has the lowest value of root mean square error (RMSE) and the highest value of the Nash–Sutcliffe coefficient (E). Based on the results of model quality indices, these statistical modeling techniques are arranged in the following order; ANFIS > Nonlinear Regression > Regularization > Linear Regression. The outcomes of this study can provide references to solve complex problems in geostatistics.

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Abbreviations

E d :

Dynamic elastic Young’s modulus

Q-factor:

Quality factor in the longitudinal direction

F r :

Resonance frequency n the longitudinal direction

UCS:

Unconfined compressive strength

BTS:

Brazillian tensile strength

β n :

Assigned weights to predictors in the regression equation

ε :

Error term in the regression equation

VIF:

Variance inflation factor

LF:

Loss function

L, R, and EN:

Lasso regression, Ridge regression, and Elastic Net regression respectively

λ and α :

Tuning parameters for the regularized regression equation

ANOVA:

Analysis of variance

F-sig and t-sig:

Significance value to check the acceptance or rejection of a hypothesis test

ANFIS:

Adaptive neuro-fuzzy inference system

RMSE:

Root mean square error

E :

Nash–Sutcliffe coefficient

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  1. Department of Geological Engineering, University of Engineering and Technology Lahore, Lahore, Pakistan

    Umer Waqas & Muhammad Farooq Ahmed

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  1. Umer Waqas
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Waqas, U., Ahmed, M.F. Prediction Modeling for the Estimation of Dynamic Elastic Young’s Modulus of Thermally Treated Sedimentary Rocks Using Linear–Nonlinear Regression Analysis, Regularization, and ANFIS. Rock Mech Rock Eng 53, 5411–5428 (2020). https://doi.org/10.1007/s00603-020-02219-8

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