## 3 Fatigue Life Prediction Modeling
### Evaluation Indexes  
In this paper, three evaluation indexes are used to evaluate the prediction accuracy of the model. They are correlation coefficient (R\({}^{2}\)), root mean square error (RMSE), and mean absolute error (MAE). Among them, R\({}^{2}\) indicates the similarity between the predicted and actual values given by the model, and the closer the value is to one, the closer the predicted value is to the actual value; RMSE and MAE are the average error magnitude between the predicted and actual values of fatigue life derived from the model using the two calculation methods, and the smaller the value is, the smaller the error magnitude between the predicted and actual values is. Therefore, the closer the values of RMSE and MAE are to zero and the closer the value of R\({}^{2}\) is to one, the higher the prediction accuracy of the model. The three evaluation indicators are defined by Equations (8)-(10).  
\[\text{R}^{2}=\frac{\sum\limits_{i=1}^{n}{(y_{i}-\overline{y}_{i})(\hat{y}_{i} -\overline{\hat{y}}_{i})}}{\sqrt{\sum\limits_{i=1}^{n}{(y_{i}-\overline{\hat{y} }_{i})}^{2}}}\sqrt{\sum\limits_{i=1}^{n}{(\hat{y}_{i}-\overline{\hat{y}}_{i})} ^{2}}} \tag{8}\]  
\begin{table}
\begin{tabular}{c c c c} \hline
**Datasets** & **Number** & **Input Variables** & **Explanation** \\ \hline  & 1 & \(P1\%\) & Pore size of 0.1–27.98 nm \\  & 2 & \(P2\%\) & Pore size 27.98–524.26 nm \\  & 3 & \(P3\%\) & Pore size 524.26–6463.30 nm \\  & 4 & \(PM1\%\) & Porosity of pore size corresponding to highest peak of \(P1\) part of the curve \\  & 5 & \(PM3\%\) & Porosity of pore size corresponding to highest peak of \(P3\) part of the curve \\  & 6 & \(S1\) & Rate of change of \(P1\) pore size \\  & 7 & \(S3\) & \(P3\) rate of pore size change \\  & 8 & \(Dna\) & \(P1\) pore fractal dimension \\  & 9 & \(Dnb\) & \(P3\) pore fractal dimension \\  & 10 & \(P1\)Q\% & \(P1\) part of the 0.1–7.5 nm pore size porosity \\ Dataset 1 & 11 & \(P1b\%\) & \(P1\) part less than 5 nm pore size porosity \\  & 12 & \(P1h\) & \(P1\) part 5–27.98 nm pore size porosity \\  & 13 & \(Sz\) & Pore size integrated change rate \\  & 14 & \(Cz\) & Pore structure complexity factor \\  & 15 & _Large capillaries_ & Pore size of 50–10,000 nm \\  & 16 & _Small capillaries_ & Pore size 10–50 nm \\  & 17 & _Inter-colloidal pores_ & Pore size 2.5–10 nm \\  & 18 & _Microopres_ & Pore size 0.5–2.5 nm \\  & 19 & _Interlayer pores_ & Aperture size is less than 0.5 nm \\  & 20 & _Non-harmful pores_ & Pore size is less than 20 nm \\  & 21 & _Less harmful pores_ & Pore size is 20–100 nm \\  & 22 & _Harmitt pores_ & Pore size is 100–200 nm \\  & 23 & _Multi-harmful holes_ & Pore size is greater than 200 nm \\  & 24 & \(P_{total}\) & Total pore size \\ \hline  & 1 & _fc_ & Compressive strength of concrete \\  & 2 & _h/w_ & Height-to-width ratio \\  & 3 & _Shape_ & Shape of the test specimens \\  & 4 & _Smax_ & Maximum stress level \\  & 5 & \(R\) & Minimum stress to maximum stress ratio \\  & 6 & \(f(HZ)\) & Loading frequency \\ \hline \end{tabular}
\end{table}
Table 1: Explanation of the meaning of the input variables in the datasets.  
\[\text{RMSE}=\sqrt{\frac{1}{n}{\sum_{i=1}^{n}{(y_{i}-\hat{y}_{i})}^{2}}} \tag{9}\]  
MAE =\(\frac{1}{n}{\sum_{i=1}^{n}{|(y_{i}-\hat{y}_{i})|}}\) (10)  
where \(n\) is the total number of values, \(\hat{y}\) denotes the actual value, \(y\) is the predicted result calculated by the model, \(\overline{\hat{y}}\) is the average of the actual value, and \(\overline{\hat{y}}\) denotes the average of the predicted result.