# Virtual Sensor for Estimating the Strain-Hardening Rate of
## 3 Results  
The results of the LR models, which establish the linear relationship between \(Rm\) and the cold thickness reduction for each analysed steel sample, are shown in Figure 2. This includes the coefficient \(a\) in Figure 2a, representing quantitatively the slope of each regression line and serving as a dependent variable in the MLR model pursued in this work, as well as the values of \(R\) in Figure 2b. A higher \(R\) value indicates a better linear fit of the regression line between \(Rm\) and cold thickness reduction. The minimum \(R\)-value determined for the LR models is 0.9850 and maximum one is 0.9999. The MLR model used to define the new parameter \(HRHS\), which represents the strain-hardening rate of hot-rolled and annealed ASS sheets as a function of the chemical composition uses the coefficients \(a\) obtained from the LR models as the output variable and the chemical composition as the input variables. The Sum of Squares (SOS) [44], \(RMSE\), \(MSE\) and \(R\) are some of the most extensively mathematical parameters used to measure the quality of an MLR model. This study employs the \(R\) and \(MSE\) as quality indices, resulting in values of 0.9983 and 0.0307, respectively. The expression for the \(HRHS\) parameter obtained by the MLR model is shown in Equation (6). The obtained results are displayed in Figure 3.  
\[ \begin{split} HRHS=33.560-5.046\cdot C-1.473\cdot Cr-0.389\cdot Cu+0.19 5\cdot Mn-2.406\cdot Mo+8.044\cdot N\\ +6.935\cdot Si+0.446\cdot Ni+133.524\cdot S-41.093\cdot Nb-27.245\cdot Ti \end{split}  \tag{6}\]  
where \(HRHS\) is an estimated value of the strain-hardening rate for ASS, expressed in dimensionless form, and each alloying element is included in the equation with its chemical symbol and unit of weight percentage (%). Figure 3a illustrates the outcomes of the MLR model for the \(HRHS\) parameter. The \(x\)-axis categorizes each analysed sample, with the initial two digits specifying the ASS grade and the subsequent one denoting the sheet number within that steel type. The primary \(y\)-axis quantifies the coefficient \(a\) values derived from individual LR models, represented by blue circles. In contrast, the secondary \(y\)-axis displays the \(HRHS\) values predicted by the MLR model across all austenitic grades, depicted as orange crosses. A tighter convergence of the crosses and circles signifies a better fit of the MLR model in predicting the \(HRHS\) parameter, as evidenced by higher correlation coefficients and diminished \(MSE\). To discern the most impactful variables within the model, an evaluation of the \(p\)-values corresponding to the alloying elements is undertaken. The findings of this analysis are portrayed in Figure 3b, highlighting the alloying elements with the most significant influence on the \(HRHS\) estimation. Figure 2: (a) Coefficient \(a\); (b) correlation coefficient (\(R\)) of the LR models of each sample. To validate the _HRHS_ parameter's formulation, the LOOCV method is employed as per the protocol outlined in Table 3. The results obtained from this validation procedure are illustrated in Figure 4. Specifically, Figure 4a plots the identification of each LOOCV model along the \(x\)-axis. The primary \(y\)-axis displays the coefficient \(a\) values obtained from the LR model for each sample, symbolized by blue circles. Concurrently, the secondary \(y\)-axis shows the corresponding _HRHS_ estimates for the same sample when used as validation data in the LOOCV strategy, represented by orange crosses. A graphical representation of the \(R\) and \(MSE\) for each LOOCV model is presented in Figure 4b, plotted against the primary and secondary \(y\)-axes, respectively. The MLR model is recalculated considering only the most significant variables determined by analysing the \(p\)-values from the initial MLR model.