Concrete Compressive Strength Prediction Using Combined NonDestructive Methods: A Calibration Procedure Using Preexisting Conversion Models Based on Gaussian Process Regression
Abstract
:1. Introduction
2. Materials and Methods
2.1. Conversion Models
2.2. Calibration Methods
 Use ${\mathcal{M}}_{p}$ to evaluate the set of predicted values ${\left\{C{S}_{i}^{\prime}\right\}}_{i=1}^{n}$ for the set of cores ${\left\{{c}_{i}\right\}}_{i=1}^{n}$.
 Use DT to evaluate the set of exact $CS$ values ${\left\{C{S}_{i}\right\}}_{i=1}^{n}$ for the set of cores ${\left\{{c}_{i}\right\}}_{i=1}^{n}$.
 Compute ${\mathsf{\Delta}}_{\sigma}$ as:$${\mathsf{\Delta}}_{\sigma}={\displaystyle \frac{1}{n}}\sum _{i=1}^{n}(C{S}_{i}C{S}_{i}^{\prime}).$$
 Compute ${\mathsf{\Delta}}_{\mu}$ as:$${\mathsf{\Delta}}_{\mu}={\displaystyle \frac{{\sum}_{i=1}^{n}C{S}_{i}}{{\sum}_{i=1}^{n}C{S}_{i}^{\prime}}}.$$
2.3. Gaussian Process Regression
2.4. GPR Calibration Procedure and Model Validation
2.4.1. GPR Calibration
2.4.2. Error Metrics
3. Numerical Results
Algorithm 1 Matlab GPR calibration procedure pseudocode. 

4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
${V}_{p}$  ultrasonic pulse velocity; 
R  Schmidt rebound index; 
$CS$  compressive strength; 
RMSE  rootmeansquared error; 
MAPE  mean absolute percentage error; 
MAE  mean absolute error; 
A20  a20index error; 
${R}^{2}$  correlation coefficient; 
Appendix A. Data
#  R  ${\mathit{V}}_{\mathit{p}}$  $\mathit{CS}$  #  R  ${\mathit{V}}_{\mathit{p}}$  $\mathit{CS}$ 

(km/s)  (MPa)  (km/s)  (Mpa)  
1  $26.4$  $4.61$  $20.2$  56  $28.6$  $4.74$  $23.63$ 
2  $28.5$  $4.44$  $19.19$  57  $30.7$  $4.7$  $25.69$ 
3  27  $4.55$  $20.3$  58  $28.6$  $4.48$  $23.05$ 
4  $26.6$  $4.58$  $22.16$  59  $28.5$  $4.51$  $25.2$ 
5  27  $4.62$  $20.89$  60  $29.4$  $4.55$  $24.03$ 
6  $28.8$  $4.51$  $21.67$  61  $29.2$  $4.54$  $25.5$ 
7  26  $4.23$  $18.63$  62  30  $4.73$  $25.3$ 
8  27  $4.35$  $20.4$  63  $30.6$  $4.69$  $27.85$ 
9  26  $4.37$  $19.22$  64  $30.3$  $4.77$  $27.36$ 
10  $26.1$  $4.37$  $21.57$  65  $25.1$  $4.65$  $17.16$ 
11  $22.8$  $4.18$  15  66  26  $4.48$  $19.61$ 
12  25  $4.25$  $17.36$  67  $24.4$  $4.51$  $16.67$ 
13  23  $4.19$  $15.3$  68  $24.4$  $4.58$  $16.67$ 
14  $24.5$  $4.2$  $15.89$  69  26  $4.54$  $18.34$ 
15  $23.8$  $4.13$  $14.22$  70  26  $4.54$  $19.12$ 
16  24  $4.26$  $16.67$  71  $28.6$  $4.48$  $22.16$ 
17  $32.2$  $4.76$  $32.36$  72  28  $4.52$  $23.14$ 
18  32  $4.76$  $30.6$  73  $28.3$  $4.55$  $22.95$ 
19  30  $4.88$  $35.79$  74  $26.3$  $4.48$  $23.05$ 
20  $31.8$  $4.78$  $31.58$  75  $29.1$  $4.48$  $22.36$ 
21  $32.5$  $4.73$  $30.89$  76  $26.6$  $4.45$  $19.61$ 
22  32  $4.77$  $32.36$  77  $24.6$  $4.45$  $16.67$ 
23  $32.8$  $4.88$  $34.62$  78  $25.4$  $4.44$  $18.34$ 
24  $33.6$  $4.8$  $34.62$  79  $26.4$  $4.46$  $17.85$ 
25  33  $4.88$  $35.5$  80  $25.4$  $4.53$  $20.1$ 
26  $33.8$  $4.85$  $35.5$  81  $29.7$  $4.63$  $24.32$ 
27  $33.5$  $4.8$  $34.32$  82  $29.3$  $4.58$  $24.52$ 
28  34  $4.85$  $37.46$  83  $30.8$  $4.77$  $25.2$ 
29  30  $4.61$  $28.14$  84  29  $4.62$  $26.87$ 
30  $30.5$  $4.65$  $27.07$  85  30  $4.69$  $28.18$ 
31  31  $4.66$  $30.11$  86  23  $4.17$  $14.51$ 
32  $30.8$  $4.66$  $28.14$  87  $23.5$  $4.15$  $16.18$ 
33  29  $4.64$  $28.44$  88  $23.5$  $4.19$  $14.42$ 
34  31  $4.64$  $28.44$  89  $23.1$  $4.08$  $14.22$ 
35  $30.5$  $4.55$  $27.46$  90  24  $4.12$  $14.91$ 
36  30  $4.65$  $28.64$  91  22  $4.17$  $15.2$ 
37  $31.9$  $4.62$  $28.14$  92  $21.6$  $4.03$  $12.85$ 
38  $30.8$  $4.62$  $28.93$  93  $21.8$  $3.9$  $12.45$ 
39  $31.3$  $4.65$  $29.62$  94  $21.1$  $3.93$  $12.45$ 
40  $30.8$  $4.62$  $29.52$  95  20  $3.95$  $12.65$ 
41  $30.1$  $4.55$  $29.62$  96  22  $3.95$  $14.32$ 
42  $32.4$  $4.55$  $34.81$  97  $25.8$  $4.36$  $16.18$ 
43  33  $4.55$  $36.19$  98  $24.4$  $4.31$  $15.3$ 
44  $32.6$  $4.52$  $30.3$  99  $22.2$  $4.2$  $15.4$ 
45  $33.8$  $4.55$  $35.3$  100  22  $4.17$  $14.42$ 
46  28  $4.6$  $22.16$  101  $21.9$  $4.22$  $14.32$ 
47  30  $4.55$  $23.63$  102  $22.1$  $4.2$  $15.1$ 
48  28  $4.5$  $22.26$  103  $21.9$  $4.14$  15 
49  $27.5$  $4.51$  $22.16$  104  $27.2$  $4.35$  $20.1$ 
50  31  $4.62$  $27.85$  105  $28.4$  $4.35$  $19.81$ 
51  $31.2$  $4.67$  $27.26$  106  $27.9$  $4.45$  $20.89$ 
52  $30.2$  $4.6$  $25.1$  107  $28.6$  $4.48$  $22.85$ 
53  30  $4.62$  $28.44$  108  $29.2$  $4.45$  $22.56$ 
54  $30.8$  $4.65$  $26.77$  109  29  $4.41$  $21.08$ 
55  31  $4.54$  $29.03$  110  $28.7$  $4.62$  $23.05$ 
111  28  $4.55$  $24.22$  144  $38.1$  $4.8$  $40.4$ 
112  $29.8$  $4.58$  $23.54$  145  $37.2$  $4.78$  $39.72$ 
113  $28.4$  $4.62$  $24.52$  146  $36.1$  $4.77$  $38.83$ 
114  29  $4.65$  $23.83$  147  $37.3$  $4.77$  $41.19$ 
115  $29.4$  $4.55$  $23.73$  148  $37.1$  $4.77$  $39.72$ 
116  $28.1$  $4.38$  $21.38$  149  $36.8$  $4.73$  $38.05$ 
117  $28.6$  $4.4$  $21.67$  150  $37.1$  $4.77$  $39.42$ 
118  28  $4.45$  $24.42$  151  $36.1$  $4.88$  $40.21$ 
119  $26.2$  $4.1$  $14.42$  152  33  $4.84$  $31.87$ 
120  $22.9$  $3.26$  $14.81$  153  $33.9$  $4.75$  $33.05$ 
121  $38.2$  $5.05$  $43.25$  154  $33.8$  $4.72$  $33.93$ 
122  $37.2$  $5.05$  $40.31$  155  $34.6$  $4.75$  $33.93$ 
123  39  $5.1$  $43.44$  156  $33.3$  $4.72$  $34.81$ 
124  38  $5.1$  $39.32$  157  30  $4.42$  $25.01$ 
125  $36.8$  $5.05$  $42.47$  158  $30.9$  $4.45$  $26.97$ 
126  $37.1$  $5.05$  $41.58$  159  $30.9$  $4.45$  $26.58$ 
127  36  $4.8$  $47.56$  160  $30.5$  $4.45$  $25.99$ 
128  $34.4$  $4.88$  $42.95$  161  $28.3$  $4.52$  $26.09$ 
129  $35.4$  $4.93$  $40.7$  162  30  $4.48$  $26.58$ 
130  $34.6$  $4.81$  $41.97$  163  30  $4.45$  $27.46$ 
131  $35.2$  $4.92$  $44.33$  164  28  $4.26$  $22.85$ 
132  $35.5$  $4.92$  $43.64$  165  $28.3$  $4.28$  $23.14$ 
133  $41.5$  $5.22$  $50.21$  166  27  $4.28$  $22.36$ 
134  $40.6$  $5.18$  $49.33$  167  $28.7$  $4.32$  $23.54$ 
135  42  $5.22$  $50.01$  168  $28.2$  $4.32$  $23.34$ 
136  $41.4$  $5.22$  $50.01$  169  29  $4.11$  $24.52$ 
137  41  5  $50.41$  170  27  $4.29$  $22.46$ 
138  41  $4.95$  $52.17$  171  $28.5$  $4.26$  $22.16$ 
139  $40.8$  $4.92$  $50.8$  172  $28.2$  $4.2$  $21.38$ 
140  41  5  $50.01$  173  $29.6$  $4.29$  $22.85$ 
141  40  $4.98$  $49.13$  174  28  $4.29$  $23.54$ 
142  $41.2$  $4.99$  $50.8$  175  $29.2$  $4.45$  $27.46$ 
143  $37.3$  $4.8$  $38.25$  176  $30.9$  $4.45$  $27.46$ 
177  31  $4.45$  $27.46$ 
#  R  ${\mathit{V}}_{\mathit{p}}$  $\mathit{CS}$  #  R  ${\mathit{V}}_{\mathit{p}}$  $\mathit{CS}$ 

(km/s)  (MPa)  (km/s)  (Mpa)  
1  $4.35$  $25.5$  $20.1$  12  $3.94$  $20.5$  $12.16$ 
2  $4.32$  $27.3$  $19.22$  13  $4.35$  $24.9$  $15.79$ 
3  $4.55$  $30.7$  $35.5$  14  $4.35$  $24.3$  $15.89$ 
4  $4.52$  28  $22.26$  15  $4.4$  $25.6$  $16.38$ 
5  $4.51$  $28.9$  $22.36$  16  $4.17$  $21.8$  $15.49$ 
6  $4.69$  30  $28.44$  17  $4.55$  $28.2$  $20.99$ 
7  $4.77$  30  $28.05$  18  $5.22$  40  $45.4$ 
8  $4.73$  $29.4$  $28.14$  19  $4.52$  30  $26.38$ 
9  $4.55$  $27.4$  $22.36$  20  $4.45$  $30.4$  $26.28$ 
References
 Breysse, D.; Balayssac, J.P. (Eds.) NonDestructive In Situ Strength Assessment of Concrete; Springer International Publishing: Berlin/Heidelberg, Germany, 2021. [Google Scholar]
 Maierhofer, C.; Reinhardt, H.W.; Dobmann, G. (Eds.) NonDestructive Evaluation of Reinforced Concrete Structures: NonDestructive Testing Methods; Elsevier: Amsterdam, The Netherlands, 2010. [Google Scholar]
 Kouddane, B.; Sbartaï, Z.M.; Alwash, M.; AliBenyahia, K.; Elachachi, S.M.; Lamdouar, N.; Kenai, S. Assessment of concrete strength using the combination of NDT—Review and performance analysis. Appl. Sci. 2022, 12, 12190. [Google Scholar] [CrossRef]
 Yoon, H.; Kim, Y.J.; Kim, H.S.; Kang, J.W.; Koh, H.M. Evaluation of earlyage concrete compressive strength with ultrasonic sensors. Sensors 2017, 17, 1817. [Google Scholar] [CrossRef]
 Fadiel, A.A.; Mohammed, N.S.; AbuLebdeh, T.; Munteanu, I.S.; Niculae, E.; Petrescu, F.I.T. A Comprehensive Evaluation of the Mechanical Properties of Rubberized Concrete. J. Compos. Sci. 2023, 7, 129. [Google Scholar] [CrossRef]
 Breysse, D. Nondestructive evaluation of concrete strength: An historical review and a new perspective by combining NDT methods. Constr. Build. Mater. 2012, 33, 139–163. [Google Scholar] [CrossRef]
 Moein, M.M.; Saradar, A.; Rahmati, K.; Mousavinejad, S.H.G.; Bristow, J.; Aramali, V.; Karakouzian, M. Predictive models for concrete properties using machine learning and deep learning approaches: A review. J. Build. Eng. 2023, 63, 105444. [Google Scholar] [CrossRef]
 Li, D.; Tang, Z.; Kang, Q.; Zhang, X.; Li, Y. Machine learningbased method for predicting compressive strength of concrete. Processes 2023, 11, 390. [Google Scholar] [CrossRef]
 Sah, A.K.; Hong, Y.M. Performance Comparison of Machine Learning Models for Concrete Compressive Strength Prediction. Materials 2024, 17, 2075. [Google Scholar] [CrossRef]
 Asteris, P.G.; Mokos, V.G. Concrete compressive strength using artificial neural networks. Neural Comput. Appl. 2020, 32, 11807. [Google Scholar] [CrossRef]
 Bonagura, M.; Nobile, L. Artificial neural network (ANN) approach for predicting concrete compressive strength by SonReb. Struct. Durab. Health Monit. 2021, 15, 125. [Google Scholar] [CrossRef]
 Almeida, T.A.D.C.; Felix, E.F.; de Sousa, C.M.A.; Pedroso, G.O.M.; Motta, M.F.B.; Prado, L.P. Influence of the ANN Hyperparameters on the Forecast Accuracy of RAC’s Compressive Strength. Materials 2023, 16, 7683. [Google Scholar] [CrossRef]
 Ngo, T.Q.L.; Wang, Y.R.; Chiang, D.L. Applying artificial intelligence to improve onsite nondestructive concrete compressive strength tests. Crystals 2021, 11, 1157. [Google Scholar] [CrossRef]
 Arora, H.C.; Bhushan, B.; Kumar, A.; Kumar, P.; HadzimaNyarko, M.; Radu, D.; Cazacu, C.E.; Kapoor, N.R. Ensemble learning based compressive strength prediction of concrete structures through realtime nondestructive testing. Sci. Rep. 2024, 14, 1824. [Google Scholar] [CrossRef] [PubMed]
 Li, Q.F.; Song, Z.M. Highperformance concrete strength prediction based on ensemble learning. Constr. Build. Mater. 2022, 324, 126694. [Google Scholar] [CrossRef]
 Chandak, N.R.; Kumavat, H.R. SonReb method for evaluation of compressive strength of concrete. IOP Conf. Ser. Mater. Sci. Eng. 2020, 810, 012071. [Google Scholar] [CrossRef]
 Gramacy, R.B. Surrogates: Gaussian Process Modeling, Design, and Optimization for the Applied Sciences; Chapman and Hall/CRC: Boca Raton, FL, USA, 2020. [Google Scholar]
 Hoang, N.D.; Pham, A.D.; Nguyen, Q.L.; Pham, Q.N. Estimating compressive strength of high performance concrete with Gaussian process regression model. Adv. Civ. Eng. 2016, 1, 2861380. [Google Scholar] [CrossRef]
 Ly, H.B.; Nguyen, T.A.; Pham, B.T. Investigation on factors affecting early strength of highperformance concrete by Gaussian Process Regression. PLoS ONE 2022, 17, e0262930. [Google Scholar] [CrossRef] [PubMed]
 Gupta, S.; Sihag, P. Prediction of the compressive strength of concrete using various predictive modeling techniques. Neural Comput. Appl. 2022, 34, 6535. [Google Scholar] [CrossRef]
 FernándezGodino, M.G. Review of multifidelity models. arXiv 2016, arXiv:1609.07196. [Google Scholar]
 Angiulli, G.; Versaci, M.; Calcagno, S.; Di Barba, P. Quick retrieval of effective electromagnetic metamaterial parameters by using a Multifidelity Surrogate Modelling approach. Eur. Phys. J. Appl. Phys. 2021, 90, 20901. [Google Scholar] [CrossRef]
 Li, M.; Jia, G. Multifidelity Gaussian process model integrating lowand highfidelity data considering censoring. J. Struct. Eng. 2020, 146, 04019215. [Google Scholar] [CrossRef]
 Liu, H.; Ong, Y.S.; Cai, J.; Wang, Y. Cope with diverse data structures in multifidelity modeling: A Gaussian process method. Eng. Appl. Artif. Intell. 2018, 67, 211. [Google Scholar] [CrossRef]
 Wong, E. Introduction to Random Processes; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
 RILEM Draft Recommendation. Draft Recommendation for in situ concrete strength determination by combined nondestructive methods. Mater. Struct. 1993, 26, 43–49. [Google Scholar] [CrossRef]
 Fasshauer, G.E.; McCourt, M.J. KernelBased Approximation Methods Using Matlab; World Scientific Publishing Company: Singapore, 2015. [Google Scholar]
 MATLAB Team. Statistics and Machine Learning Toolbox; The Mathworks Inc.: Natick, MA, USA, 2019. [Google Scholar]
 Logothetis, L.A. Combination of Three NonDestructive Methods for the Determination of the Strength of Concrete. Ph.D. Thesis, National Technical University of Athens, Athens, Greece, 1978. [Google Scholar]
 Amini, K.; Jalalpour, M.; Delatte, N. Advancing concrete strength prediction using nondestructive testing: Development and verification of a generalizable model. Constr. Build. Mater. 2016, 102, 762–768. [Google Scholar] [CrossRef]
 Arioglu, E.; Manzak, O. Application of ‘sonreb’ method to concrete samples produced in yedpa construction site. Prefabr. Union 1991. [Google Scholar]
 Bellander, U. NDT testing methods for estimating compressive strength in finished structures–evaluation of accuracy and testing system. In RILEM Symposium Proceedings on Quality Control of Concrete Structures; CRC Press: Boca Raton, FL, USA, 1979. [Google Scholar]
 Dolce, M.; Masi, A.; Ferrini, M. Estimation of the actual inplace concrete strength in assessing existing RC structures. In Proceedings of the Second International Fib Congress, Naples, Italy, 5–8 June 2006. [Google Scholar]
 Erdal, M. Prediction of the compressive strength of vacuum processed concretes using artificial neural network and regression techniques. Sci. Res. Essay 2009, 4, 1057. [Google Scholar]
 Huang, Q.; Gardoni, P.; Hurlebaus, S. Predicting Concrete Compressive Strength Using Ultrasonic Pulse Velocity and Rebound Number. ACI Mater. J. 2011, 108, 403. [Google Scholar]
 Kheder, G.F. A two stage procedure for assessment of in situ concrete strength using combined nondestructive testing. Mater. Struct. 1999, 32, 410. [Google Scholar] [CrossRef]
 Nash’t, I.H.; A’bour, S.H.; Sadoon, A.A. Finding an unified relationship between crushing strength of concrete and nondestructive tests. In Proceedings of the Middle East Nondestructive Testing Conference & Exhibition, Manama, Bahrain, 27–30 November 2005. [Google Scholar]
 Nikhil, M.; Minal, B.R.; Deep, C.S.; Vijay, G.D.; Vishal, T.S.; Shweta, P. The use of combined non destructive testing in the concrete strength assessment from laboratory specimens and existing buildings. Int. J. Curr. Eng. Sci. Res. 2015, 2, 55–59. [Google Scholar]
 Shariati, M.; RamliSulong, N.H.; Arabnejad, M.M.; Shafigh, P.; Sinaei, H. Assessing the strength of reinforced concrete structures through Ultrasonic Pulse Velocity and Schmidt Rebound Hammer tests. Sci. Res. Essays 2011, 6, 213. [Google Scholar]
 Yasuo Tanigawa, Y.K.B.; Hiroshi, M. Estimation of concrete strength by combined nondestructive testing method. ACI Symp. Publ. 1984, 82, 57–76. [Google Scholar]
 Turgut, P.; Kucuk, O.F. Comparative relationships of direct, indirect, and semidirect ultrasonic pulse velocity measurements in concrete. Russ. J. Nondestruct. Test. 2006, 42, 745. [Google Scholar] [CrossRef]
 Chai, T.; Draxler, R.R. Root mean square error (RMSE) or mean absolute error (MAE)?—Arguments against avoiding RMSE in the literature. Geosci. Model Dev. 2014, 7, 1247. [Google Scholar] [CrossRef]
 De Myttenaere, A.; Golden, B.; Le Grand, B.; Rossi, F. Mean absolute percentage error for regression models. Neurocomputing 2016, 192, 38. [Google Scholar] [CrossRef]
 Li, G.; Shi, J. On comparing three artificial neural networks for wind speed forecasting. Appl. Energy 2010, 87, 2313. [Google Scholar]
 Kanagawa, M.; Hennig, P.; Sejdinovic, D.; Sriperumbudur, B.K. Gaussian processes and kernel methods: A review on connections and equivalences. arXiv 2018, arXiv:1807.02582. [Google Scholar]
 Bilotta, G.; Calcagno, S.; Bonfa, S. Wildfires: An application of remote sensing and OBIA. WSEAS Trans. Environ. Dev. 2021, 17, 282. [Google Scholar] [CrossRef]
 Angiulli, G. Design of square substrate integrated waveguide cavity resonators: Compensation of modelling errors by support vector regression machines. Am. J. Appl. Sci. 2012, 9, 1872. [Google Scholar]
#  Model  Equation  Ref. 

1  $\mathsf{Amini}$  $CS=109.83+1.57R7.9315{V}_{p}2{R}^{2}+0.129261{V}_{p}^{2}$  [30] 
2  $\mathsf{Arioglu}$  $CS=18.6{e}^{0.515{V}_{p}+0.019R}0.0981$  [31] 
3  $\mathsf{Bellander}$  $CS=25.568+0.000635{R}^{3}+8.397{V}_{p}$  [32] 
4  $\mathsf{Dolce}$  $CS=8.925\times {10}^{11}{\left({10}^{3}{V}_{p}\right)}^{2.6}{R}^{1.4}$  [33] 
5  $\mathsf{Erdal}$  $CS=0.42R+13.166{V}_{p}40.255$  [34] 
6  $\mathsf{Huang}$  $CS={(1.26+0.00015{R}^{2}+0.035{{V}_{p}}^{3}+0.8024)}^{2}$  [35] 
7  $\mathsf{Kheder}$  $CS=0.0158{\left(1000{V}_{p}\right)}^{0.4254}{R}^{1.1171}$  [36] 
8  $\mathsf{Logothetis}$  $CS=0.0981{e}^{1.78ln\left({V}_{p}\right)}+0.85ln\left(R\right)0.02$  [29] 
9  $\mathsf{Nash\u2019t}$  $CS=0.356{R}^{0.866}{e}^{0.302{V}_{p}}$  [37] 
10  $\mathsf{Nikhil}$  $CS=1.6411\times {10}^{9}{\left(1000{V}_{p}\right)}^{2.29366}{R}^{1.30768}$  [38] 
11  $\mathsf{Shariati}$  $CS=0.0981(173.04+131R+57.96{V}_{p}+{4.07{V}_{p}}^{2})$  [39] 
12  $\mathsf{Tanigawa}$  $CS=0.544+0.745R+0.951{V}_{p}$  [40] 
13  $\mathsf{Turgut}$  $CS=194+0.77R+44.8{V}_{p}$  [41] 
#  Model  RMSE′ (MPa)  RMSE″ (MPa)  

CM  #4  #8  #10  #12  #14  #16  #20  
1  $\mathsf{Amini}$  $20.06$  ${\overline{y}}_{\mathbf{x}}$  $2.544$  $3.651$  $2.861$  $2.686$  $2.849$  $2.675$  $3.052$ 
${\mathsf{\Delta}}_{\sigma}$  $5.427$  $5.408$  $5.44$  $5.55$  $5.775$  $5.421$  $5.409$  
${\mathsf{\Delta}}_{\mu}$  $4.476$  $4.387$  $4.385$  $4.445$  $4.678$  $4.439$  $5.266$  
2  $\mathsf{Arioglu}$  $8.262$  ${\overline{y}}_{\mathbf{x}}$  $2.726$  $2.871$  $2.434$  $2.407$  $2.402$  $2.385$  $2.727$ 
${\mathsf{\Delta}}_{\sigma}$  $3.497$  $3.296$  $3.28$  $3.33$  $3.494$  $3.403$  $3.593$  
${\mathsf{\Delta}}_{\mu}$  $4.28$  $4.111$  $4.11$  $4.121$  $4.138$  $4.137$  $5.034$  
3  $\mathsf{Bellander}$  $5.012$  ${\overline{y}}_{\mathbf{x}}$  $3.764$  $2.647$  $2.313$  $2.316$  $2.505$  $2.364$  $2.403$ 
${\mathsf{\Delta}}_{\sigma}$  $3.062$  $3.084$  $3.216$  $3.051$  $2.948$  $3.042$  $3.047$  
${\mathsf{\Delta}}_{\mu}$  $2.483$  $2.413$  $2.425$  $2.473$  $2.365$  $2.535$  $2.355$  
4  $\mathsf{Dolce}$  $9.672$  ${\overline{y}}_{\mathbf{x}}$  $3.153$  $3.3$  $2.596$  $2.731$  $2.684$  $2.704$  $2.97$ 
${\mathsf{\Delta}}_{\sigma}$  $5.031$  $4.999$  $5.357$  $4.912$  $4.855$  $4.933$  $4.964$  
${\mathsf{\Delta}}_{\mu}$  $2.915$  $2.846$  $2.797$  $2.915$  $2.888$  $2.959$  $2.721$  
5  $\mathsf{Erdal}$  $7.321$  ${\overline{y}}_{\mathbf{x}}$  $2.939$  $3.102$  $2.491$  $2.39$  $2.302$  $2.394$  $2.442$ 
${\mathsf{\Delta}}_{\sigma}$  $5.234$  $4.996$  $5.195$  5  $5.316$  $5.148$  $5.558$  
${\mathsf{\Delta}}_{\mu}$  $5.725$  $5.606$  $5.648$  $5.553$  $5.665$  $5.602$  $6.639$  
6  $\mathsf{Huang}$  $6.127$  ${\overline{y}}_{\mathbf{x}}$  $2.552$  $3.127$  $2.56$  $2.499$  $2.521$  $2.47$  $2.87$ 
${\mathsf{\Delta}}_{\sigma}$  $4.684$  $4.456$  $4.395$  $4.441$  $4.776$  $4.481$  $4.975$  
${\mathsf{\Delta}}_{\mu}$  $4.946$  $4.745$  $4.734$  $4.757$  $4.928$  $4.758$  $5.778$  
7  $\mathsf{Kheder}$  $4.995$  ${\overline{y}}_{\mathbf{x}}$  $2.886$  $2.99$  $2.45$  $2.362$  $2.277$  $2.358$  $2.425$ 
${\mathsf{\Delta}}_{\sigma}$  $5.117$  $4.774$  $5.064$  $4.766$  $4.742$  $5.056$  $5.144$  
${\mathsf{\Delta}}_{\mu}$  $4.518$  $4.504$  $4.469$  $4.398$  $4.328$  $4.498$  $5.175$  
8  $\mathsf{Logothetis}$  $4.152$  ${\overline{y}}_{\mathbf{x}}$  $2.697$  $2.852$  $2.425$  $2.327$  $2.247$  $2.318$  $2.461$ 
${\mathsf{\Delta}}_{\sigma}$  $4.341$  $4.046$  $4.191$  $4.075$  $4.172$  $4.26$  $4.466$  
${\mathsf{\Delta}}_{\mu}$  $3.904$  $3.835$  $3.817$  $3.817$  $3.809$  $3.863$  $4.622$  
9  $\mathsf{Nash\u2019t}$  $4.232$  ${\overline{y}}_{\mathbf{x}}$  $2.71$  $2.876$  $2.435$  $2.337$  $2.25$  $2.324$  $2.475$ 
${\mathsf{\Delta}}_{\sigma}$  $4.549$  $4.222$  $4.4$  $4.242$  $4.293$  $4.458$  $4.638$  
${\mathsf{\Delta}}_{\mu}$  $4.273$  $4.198$  $4.174$  $4.157$  $4.114$  $4.212$  $5.014$  
10  $\mathsf{Nikhil}$  $9.473$  ${\overline{y}}_{\mathbf{x}}$  $2.911$  $3.016$  $2.476$  $2.572$  $2.566$  $2.52$  $2.788$ 
${\mathsf{\Delta}}_{\sigma}$  $4.045$  $4.02$  $4.252$  $4.002$  $3.919$  $3.992$  $3.961$  
${\mathsf{\Delta}}_{\mu}$  $2.652$  $2.556$  $2.573$  $2.636$  $2.612$  $2.699$  $2.626$  
11  $\mathsf{Shariati}$  $13.72$  ${\overline{y}}_{\mathbf{x}}$  $2.935$  $3.101$  $2.49$  $2.389$  $2.3$  $2.393$  $2.442$ 
${\mathsf{\Delta}}_{\sigma}$  $3.976$  $3.789$  $4.056$  $3.843$  $3.734$  $4.13$  $3.988$  
${\mathsf{\Delta}}_{\mu}$  $5.682$  $5.596$  $5.635$  $5.481$  $5.402$  $5.609$  $6.36$  
12  $\mathsf{Tanigawa}$  $6.099$  ${\overline{y}}_{\mathbf{x}}$  $2.939$  $3.102$  $2.491$  $2.39$  $2.302$  $2.394$  $2.442$ 
${\mathsf{\Delta}}_{\sigma}$  $6.485$  $6.045$  $6.43$  $5.993$  $5.985$  $6.313$  $6.53$  
${\mathsf{\Delta}}_{\mu}$  $6.041$  $5.915$  $5.979$  $5.806$  $5.732$  $5.935$  $6.729$  
13  $\mathsf{Turgut}$  $10.39$  ${\overline{y}}_{\mathbf{x}}$  $2.939$  $3.102$  $2.491$  $2.39$  $2.301$  $2.394$  $2.442$ 
${\mathsf{\Delta}}_{\sigma}$  $9.004$  $8.509$  $8.782$  $8.66$  $8.729$  $8.619$  $8.541$  
${\mathsf{\Delta}}_{\mu}$  $5.848$  $6.342$  $5.881$  $6.846$  $7.859$  $6.209$  $5.949$ 
#  Model  MAPE′ (%)  MAPE″ (%)  

CM  #4  #8  #10  #12  #14  #16  #20  
1  $\mathsf{Amini}$  $78.91$  ${\overline{y}}_{\mathbf{x}}$  $7.744$  $11.12$  $8.569$  $8.315$  $9.375$  $8.024$  $7.632$ 
${\mathsf{\Delta}}_{\sigma}$  $19.12$  $18.7$  $19.11$  $19.07$  $19.46$  $18.95$  $18.98$  
${\mathsf{\Delta}}_{\mu}$  $14.91$  $14.02$  $14.44$  $14.33$  $16.6$  $14.87$  $12.61$  
2  $\mathsf{Arioglu}$  $34.78$  ${\overline{y}}_{\mathbf{x}}$  $8.266$  $8.792$  $7.07$  $7.26$  $6.884$  $6.996$  $6.803$ 
${\mathsf{\Delta}}_{\sigma}$  $12.25$  $10.83$  $10.43$  $10.95$  $12.43$  $11.77$  $9.4$  
${\mathsf{\Delta}}_{\mu}$  $16.02$  $13.96$  $14.55$  $13.88$  $15.24$  $15.13$  $13.16$  
3  $\mathsf{Bellander}$  $15.94$  ${\overline{y}}_{\mathbf{x}}$  $11.25$  $7.15$  $6.868$  $7.189$  $7.207$  $6.95$  $6.373$ 
${\mathsf{\Delta}}_{\sigma}$  $8.361$  $8.456$  $9.886$  $8.049$  $7.949$  $8.056$  $8.111$  
${\mathsf{\Delta}}_{\mu}$  $6.947$  $6.66$  $6.749$  $7.013$  $6.617$  $7.251$  $6.409$  
4  $\mathsf{Dolce}$  $30.94$  ${\overline{y}}_{\mathbf{x}}$  $8.494$  $9.661$  $7.312$  $7.677$  $6.563$  $7.88$  $7.426$ 
${\mathsf{\Delta}}_{\sigma}$  $14.84$  $13.59$  $18.55$  13  $13.15$  $13.65$  $13.38$  
${\mathsf{\Delta}}_{\mu}$  $8.069$  $7.856$  $8.023$  $8.123$  $8.011$  $8.177$  $7.656$  
5  $\mathsf{Erdal}$  $31.02$  ${\overline{y}}_{\mathbf{x}}$  $9.966$  $9.548$  $8.133$  $7.572$  $7.318$  $7.482$  $6.525$ 
${\mathsf{\Delta}}_{\sigma}$  $19.14$  $16.1$  $19.38$  $16.63$  $20.23$  $18.86$  $13.43$  
${\mathsf{\Delta}}_{\mu}$  $21.41$  $18.32$  $21.47$  $18.5$  $21.53$  $20.77$  $16.31$  
6  $\mathsf{Huang}$  $24.69$  ${\overline{y}}_{\mathbf{x}}$  $7.498$  $9.678$  $7.445$  $7.598$  $7.3$  $7.359$  $7.175$ 
${\mathsf{\Delta}}_{\sigma}$  $16.82$  $15.12$  $14.54$  $14.53$  $17.59$  $15.47$  $12.59$  
${\mathsf{\Delta}}_{\mu}$  $18.04$  $16.26$  $16.43$  $15.78$  $18.18$  $16.85$  $14.69$  
7  $\mathsf{Kheder}$  $13.18$  ${\overline{y}}_{\mathbf{x}}$  $9.768$  $9.219$  $7.905$  $7.486$  $7.173$  $7.297$  $6.412$ 
${\mathsf{\Delta}}_{\sigma}$  $19.79$  $15.27$  $19.87$  $16.45$  17  $19.69$  $12.97$  
${\mathsf{\Delta}}_{\mu}$  $17.19$  $14.26$  $17.29$  $15.22$  $15.43$  $17.51$  $13.05$  
8  $\mathsf{Logothetis}$  $11.76$  ${\overline{y}}_{\mathbf{x}}$  $8.98$  $8.79$  $7.575$  $7.318$  $6.932$  $7.054$  $6.285$ 
${\mathsf{\Delta}}_{\sigma}$  $16.25$  $13.07$  $15.4$  $13.75$  $15.42$  $15.98$  $11.28$  
${\mathsf{\Delta}}_{\mu}$  $14.26$  $12.33$  $13.64$  $12.84$  $13.9$  $14.29$  $11.7$  
9  $\mathsf{Nash\u2019t}$  $13.57$  ${\overline{y}}_{\mathbf{x}}$  $9.053$  $8.877$  $7.64$  $7.35$  $6.974$  $7.089$  $6.319$ 
${\mathsf{\Delta}}_{\sigma}$  $17.28$  $13.62$  $16.48$  $14.34$  $15.77$  $16.93$  $11.72$  
${\mathsf{\Delta}}_{\mu}$  $16.05$  $13.53$  $15.45$  $14.05$  $15.07$  $15.91$  $12.68$  
10  $\mathsf{Nikhil}$  $33.36$  ${\overline{y}}_{\mathbf{x}}$  $8.04$  $9.022$  $7.004$  $7.513$  $6.934$  $7.324$  $7.024$ 
${\mathsf{\Delta}}_{\sigma}$  $11.99$  $11.36$  14  $11.08$  $10.99$  $11.26$  $11.36$  
${\mathsf{\Delta}}_{\mu}$  $7.848$  $7.472$  $7.466$  $7.826$  $7.8$  $8.092$  $7.273$  
11  $\mathsf{Shariati}$  $60.08$  ${\overline{y}}_{\mathbf{x}}$  $9.954$  $9.547$  $8.128$  $7.57$  $7.313$  $7.477$  $6.522$ 
${\mathsf{\Delta}}_{\sigma}$  $14.49$  $11.11$  $15.48$  $13.35$  $12.74$  $15.99$  $9.738$  
${\mathsf{\Delta}}_{\mu}$  $22.01$  $18.01$  $22.36$  $18.82$  $19.2$  $21.85$  $16.01$  
12  $\mathsf{Tanigawa}$  $18.07$  ${\overline{y}}_{\mathbf{x}}$  $9.966$  $9.548$  $8.133$  $7.572$  $7.318$  $7.482$  $6.525$ 
${\mathsf{\Delta}}_{\sigma}$  $25.37$  $19.64$  $25.64$  $20.4$  $21.4$  $24.45$  $16.41$  
${\mathsf{\Delta}}_{\mu}$  $23.51$  $19.17$  $23.78$  $19.81$  $20.48$  $23.04$  $16.94$  
13  $\mathsf{Turgut}$  $34.49$  ${\overline{y}}_{\mathbf{x}}$  $9.966$  $9.548$  $8.133$  $7.572$  $7.317$  $7.482$  $6.525$ 
${\mathsf{\Delta}}_{\sigma}$  $31.01$  $28.57$  $29.85$  $29.28$  $29.98$  $29.2$  $28.91$  
${\mathsf{\Delta}}_{\mu}$  $20.06$  $21.67$  $20.08$  $23.49$  $27.3$  $21.3$  $20.25$ 
#  Model  MAE′ (MPa)  MAE″ (MPa)  

CM  #4  #8  #10  #12  #14  #16  #20  
1  $\mathsf{Amini}$  $19.33$  ${\overline{y}}_{\mathbf{x}}$  $1.903$  $2.562$  $2.195$  $2.066$  $2.269$  $1.921$  $2.059$ 
${\mathsf{\Delta}}_{\sigma}$  $4.372$  $4.411$  $4.309$  $4.539$  $4.754$  $4.399$  $4.347$  
${\mathsf{\Delta}}_{\mu}$  $3.583$  $3.428$  $3.49$  $3.489$  $3.845$  $3.548$  $3.773$  
2  $\mathsf{Arioglu}$  $7.672$  ${\overline{y}}_{\mathbf{x}}$  $2.062$  $2.171$  $1.894$  $1.906$  $1.782$  $1.756$  $1.927$ 
${\mathsf{\Delta}}_{\sigma}$  $2.85$  $2.611$  $2.574$  $2.636$  $2.85$  $2.737$  $2.636$  
${\mathsf{\Delta}}_{\mu}$  $3.525$  $3.279$  $3.324$  $3.276$  $3.359$  $3.359$  $3.805$  
3  $\mathsf{Bellander}$  $4.172$  ${\overline{y}}_{\mathbf{x}}$  $2.713$  $1.953$  $1.844$  $1.838$  $1.89$  $1.733$  $1.739$ 
${\mathsf{\Delta}}_{\sigma}$  $2.211$  $2.236$  $2.475$  $2.179$  $2.136$  $2.155$  $2.215$  
${\mathsf{\Delta}}_{\mu}$  $1.829$  $1.754$  $1.773$  $1.833$  $1.753$  $1.882$  $1.717$  
4  $\mathsf{Dolce}$  $8.343$  ${\overline{y}}_{\mathbf{x}}$  $2.248$  $2.467$  $1.968$  $2.082$  $1.836$  $2.012$  $2.124$ 
${\mathsf{\Delta}}_{\sigma}$  $3.633$  $3.488$  $4.213$  $3.416$  $3.421$  $3.449$  $3.522$  
${\mathsf{\Delta}}_{\mu}$  $2.108$  $2.044$  $2.074$  $2.115$  $2.112$  $2.124$  $2.007$  
5  $\mathsf{Erdal}$  $6.584$  ${\overline{y}}_{\mathbf{x}}$  $2.195$  $2.412$  $2.045$  $1.887$  $1.762$  $1.813$  $1.797$ 
${\mathsf{\Delta}}_{\sigma}$  $4.408$  $3.946$  $4.401$  $4.003$  $4.553$  $4.316$  $4.029$  
${\mathsf{\Delta}}_{\mu}$  $4.88$  $4.467$  $4.837$  $4.457$  $4.851$  $4.728$  $4.836$  
6  $\mathsf{Huang}$  $5.39$  ${\overline{y}}_{\mathbf{x}}$  $1.914$  $2.337$  $1.968$  $1.974$  $1.887$  $1.823$  $2.01$ 
${\mathsf{\Delta}}_{\sigma}$  $3.851$  $3.55$  $3.492$  $3.491$  $3.952$  $3.587$  $3.622$  
${\mathsf{\Delta}}_{\mu}$  $4.053$  $3.794$  $3.823$  $3.763$  $4.028$  $3.843$  $4.302$  
7  $\mathsf{Kheder}$  $3.678$  ${\overline{y}}_{\mathbf{x}}$  $2.185$  $2.312$  $2.006$  $1.866$  $1.743$  $1.779$  $1.768$ 
${\mathsf{\Delta}}_{\sigma}$  $4.427$  $3.797$  $4.397$  $3.914$  $3.952$  $4.365$  $3.762$  
${\mathsf{\Delta}}_{\mu}$  $3.879$  $3.567$  $3.855$  $3.616$  $3.6$  $3.885$  $3.785$  
8  $\mathsf{Logothetis}$  $3.158$  ${\overline{y}}_{\mathbf{x}}$  $2.089$  $2.157$  $1.965$  $1.835$  $1.713$  $1.739$  $1.757$ 
${\mathsf{\Delta}}_{\sigma}$  $3.654$  $3.185$  $3.487$  $3.262$  $3.476$  $3.571$  $3.313$  
${\mathsf{\Delta}}_{\mu}$  $3.232$  $3.012$  $3.121$  $3.042$  $3.13$  $3.203$  $3.432$  
9  $\mathsf{Nash\u2019t}$  $3.337$  ${\overline{y}}_{\mathbf{x}}$  $2.096$  $2.179$  $1.977$  $1.844$  $1.719$  $1.746$  $1.768$ 
${\mathsf{\Delta}}_{\sigma}$  $3.863$  $3.332$  $3.704$  $3.405$  $3.583$  $3.766$  $3.434$  
${\mathsf{\Delta}}_{\mu}$  $3.606$  $3.311$  $3.489$  $3.336$  $3.424$  $3.546$  $3.711$  
10  $\mathsf{Nikhil}$  $8.613$  ${\overline{y}}_{\mathbf{x}}$  $2.108$  $2.281$  $1.894$  $2.006$  $1.864$  $1.855$  $1.994$ 
${\mathsf{\Delta}}_{\sigma}$  $2.936$  $2.875$  $3.219$  $2.877$  $2.853$  $2.846$  $2.912$  
${\mathsf{\Delta}}_{\mu}$  $1.994$  $1.921$  $1.945$  $1.97$  $1.969$  $2.023$  $1.974$  
11  $\mathsf{Shariati}$  $13.19$  ${\overline{y}}_{\mathbf{x}}$  $2.193$  $2.411$  $2.044$  $1.887$  $1.761$  $1.812$  $1.797$ 
${\mathsf{\Delta}}_{\sigma}$  $3.365$  $2.906$  $3.493$  $3.174$  $3.061$  $3.584$  $2.86$  
${\mathsf{\Delta}}_{\mu}$  $4.934$  $4.466$  $4.934$  $4.495$  $4.489$  $4.858$  $4.62$  
12  $\mathsf{Tanigawa}$  $4.672$  ${\overline{y}}_{\mathbf{x}}$  $2.195$  $2.412$  $2.045$  $1.887$  $1.762$  $1.813$  $1.797$ 
${\mathsf{\Delta}}_{\sigma}$  $5.643$  $4.832$  $5.631$  $4.88$  $4.969$  $5.434$  $4.741$  
${\mathsf{\Delta}}_{\mu}$  $5.252$  $4.726$  $5.239$  $4.738$  $4.764$  $5.125$  $4.889$  
13  $\mathsf{Turgut}$  $8.615$  ${\overline{y}}_{\mathbf{x}}$  $2.195$  $2.412$  $2.045$  $1.887$  $1.761$  $1.813$  $1.797$ 
${\mathsf{\Delta}}_{\sigma}$  $6.323$  $6.399$  $6.114$  $6.757$  $7.365$  $6.267$  $6.366$  
${\mathsf{\Delta}}_{\mu}$  $4.262$  $4.786$  $4.251$  $5.345$  $6.685$  $4.598$  $4.344$ 
#  Model  A20′ (%)  A20″ (%)  

CM  #4  #8  #10  #12  #14  #16  #20  
1  $\mathsf{Amini}$  $0.5181$  ${\overline{y}}_{\mathbf{x}}$  $95.34$  $86.24$  $96.26$  $97.3$  $90.71$  $95.03$  $97.18$ 
${\mathsf{\Delta}}_{\sigma}$  $65.28$  $62.96$  $65.78$  $62.16$  $57.38$  $65.75$  $65.54$  
${\mathsf{\Delta}}_{\mu}$  $74.61$  $76.19$  $74.87$  $74.05$  $67.21$  $74.59$  $80.79$  
2  $\mathsf{Arioglu}$  $25.91$  ${\overline{y}}_{\mathbf{x}}$  $94.82$  $93.12$  $98.4$  $98.38$  $98.91$  $95.03$  $97.74$ 
${\mathsf{\Delta}}_{\sigma}$  $82.38$  $85.19$  $86.63$  $84.32$  $80.87$  $84.53$  $93.79$  
${\mathsf{\Delta}}_{\mu}$  $68.39$  $73.54$  $72.73$  $74.05$  $70.49$  $69.06$  $79.1$  
3  $\mathsf{Bellander}$  $68.39$  ${\overline{y}}_{\mathbf{x}}$  $84.97$  $98.41$  $97.86$  $98.38$  $97.27$  $95.03$  $98.31$ 
${\mathsf{\Delta}}_{\sigma}$  $93.26$  $92.59$  $89.3$  $96.22$  $94.54$  $95.03$  $96.61$  
${\mathsf{\Delta}}_{\mu}$  $97.41$  $97.88$  $97.33$  $96.76$  $97.27$  $96.13$  $98.31$  
4  $\mathsf{Dolce}$  $20.21$  ${\overline{y}}_{\mathbf{x}}$  $91.71$  $90.48$  $98.4$  $98.92$  $97.81$  $93.92$  $97.74$ 
${\mathsf{\Delta}}_{\sigma}$  $73.58$  $76.72$  $66.84$  $80.54$  $78.14$  $76.8$  $79.66$  
${\mathsf{\Delta}}_{\mu}$  $95.85$  $96.83$  $95.72$  $94.59$  $95.63$  $94.48$  $96.05$  
5  $\mathsf{Erdal}$  $39.9$  ${\overline{y}}_{\mathbf{x}}$  $88.6$  $88.89$  $96.26$  $96.22$  $95.63$  $95.03$  $97.18$ 
${\mathsf{\Delta}}_{\sigma}$  $57.51$  $65.61$  $57.75$  $65.95$  $54.64$  $58.01$  $77.97$  
${\mathsf{\Delta}}_{\mu}$  $53.37$  $61.38$  $54.55$  $61.62$  $54.64$  $55.8$  $64.97$  
6  $\mathsf{Huang}$  $48.19$  ${\overline{y}}_{\mathbf{x}}$  $96.37$  $88.89$  $97.33$  $97.84$  $97.81$  $95.03$  $96.61$ 
${\mathsf{\Delta}}_{\sigma}$  $65.8$  $73.02$  $74.87$  $75.14$  $62.84$  $69.61$  $75.71$  
${\mathsf{\Delta}}_{\mu}$  $63.21$  $67.2$  $65.78$  $66.49$  $62.3$  $65.75$  $77.4$  
7  $\mathsf{Kheder}$  $75.65$  ${\overline{y}}_{\mathbf{x}}$  $89.64$  $89.95$  $96.79$  $96.76$  $96.17$  $95.03$  $97.74$ 
${\mathsf{\Delta}}_{\sigma}$  $61.14$  $69.31$  $60.43$  $67.03$  $66.12$  $60.77$  $77.4$  
${\mathsf{\Delta}}_{\mu}$  $64.77$  $72.49$  $64.71$  $71.35$  $71.04$  $63.54$  $76.27$  
8  $\mathsf{Logothetis}$  $82.38$  ${\overline{y}}_{\mathbf{x}}$  $92.23$  $91.01$  $97.86$  $97.3$  $96.72$  $95.58$  $98.87$ 
${\mathsf{\Delta}}_{\sigma}$  $67.36$  $77.25$  $69.52$  $74.05$  $70.49$  $67.4$  $85.31$  
${\mathsf{\Delta}}_{\mu}$  $73.06$  $78.84$  $74.33$  $77.84$  $73.77$  $72.38$  $84.75$  
9  $\mathsf{Nash\u2019t}$  $74.09$  ${\overline{y}}_{\mathbf{x}}$  $91.71$  $90.48$  $97.86$  $97.3$  $96.72$  $95.58$  $98.31$ 
${\mathsf{\Delta}}_{\sigma}$  $64.77$  $71.96$  $68.98$  $71.35$  $69.4$  $64.64$  $84.75$  
${\mathsf{\Delta}}_{\mu}$  $66.84$  $73.02$  $70.59$  $72.43$  $71.04$  $66.3$  $80.23$  
10  $\mathsf{Nikhil}$  $11.4$  ${\overline{y}}_{\mathbf{x}}$  $94.3$  $91.01$  $98.93$  $98.92$  $98.36$  $94.48$  $98.31$ 
${\mathsf{\Delta}}_{\sigma}$  $80.83$  $84.13$  $75.94$  $85.41$  $86.34$  $84.53$  $84.75$  
${\mathsf{\Delta}}_{\mu}$  $94.82$  $95.24$  $96.79$  $93.51$  $93.99$  $92.82$  $97.18$  
11  $\mathsf{Shariati}$  $11.4$  ${\overline{y}}_{\mathbf{x}}$  $88.6$  $88.89$  $96.26$  $96.22$  $95.63$  $95.03$  $97.18$ 
${\mathsf{\Delta}}_{\sigma}$  $70.98$  $86.24$  $68.98$  $79.46$  $82.51$  $65.75$  $90.4$  
${\mathsf{\Delta}}_{\mu}$  $54.4$  $62.43$  $55.61$  $62.7$  $60.11$  $55.25$  $61.58$  
12  $\mathsf{Tanigawa}$  $61.14$  ${\overline{y}}_{\mathbf{x}}$  $88.6$  $88.89$  $96.26$  $96.22$  $95.63$  $95.03$  $97.18$ 
${\mathsf{\Delta}}_{\sigma}$  $47.67$  $58.73$  $45.99$  $60.54$  $56.28$  $52.49$  $59.32$  
${\mathsf{\Delta}}_{\mu}$  $51.3$  $59.26$  $51.87$  $60.54$  $59.02$  $53.04$  $58.76$  
13  $\mathsf{Turgut}$  $27.46$  ${\overline{y}}_{\mathbf{x}}$  $88.6$  $88.89$  $96.26$  $96.22$  $95.63$  $95.03$  $97.18$ 
${\mathsf{\Delta}}_{\sigma}$  $54.92$  $53.44$  $56.68$  $48.11$  $33.33$  $57.46$  $56.5$  
${\mathsf{\Delta}}_{\mu}$  $69.43$  $64.02$  $70.59$  $52.97$  $34.97$  $67.4$  $67.8$ 
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. 
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Angiulli, G.; Calcagno, S.; La Foresta, F.; Versaci, M. Concrete Compressive Strength Prediction Using Combined NonDestructive Methods: A Calibration Procedure Using Preexisting Conversion Models Based on Gaussian Process Regression. J. Compos. Sci. 2024, 8, 300. https://doi.org/10.3390/jcs8080300
Angiulli G, Calcagno S, La Foresta F, Versaci M. Concrete Compressive Strength Prediction Using Combined NonDestructive Methods: A Calibration Procedure Using Preexisting Conversion Models Based on Gaussian Process Regression. Journal of Composites Science. 2024; 8(8):300. https://doi.org/10.3390/jcs8080300
Chicago/Turabian StyleAngiulli, Giovanni, Salvatore Calcagno, Fabio La Foresta, and Mario Versaci. 2024. "Concrete Compressive Strength Prediction Using Combined NonDestructive Methods: A Calibration Procedure Using Preexisting Conversion Models Based on Gaussian Process Regression" Journal of Composites Science 8, no. 8: 300. https://doi.org/10.3390/jcs8080300