## 2 Materials and Methods
### Bootstrap Methodology  
Bootstrapping is a resampling technique used to determine the parameters of an unknown distribution of an estimator through an empirical distribution [46]. The idea underlying bootstrapping is that the empirical distribution is a good approximation of the theoretical distribution of the estimator [31]. The empirical distribution can be used to calculate the error, bias, percentiles, and confidence intervals of the estimator. In this study, BCa (Bias-Corrected and accelerated) bootstrapping is applied. This produces good confidence intervals based on the percentiles of the bootstrap distribution of the estimator. Described by [PERSON] and [PERSON] [46], this methodology has been used to obtain robust confidence intervals for various properties of wood-based products [27; 30; 31]. When applying bootstrapping to ANN modelling, we start from n ANN developed from various training, validation, and testing sets obtained at random. The modelling output will be the mean of n outputs [32]. Let \(\mathbf{x}=(\mathbf{x}_{1},\mathbf{x}_{2},\ldots\mathbf{x}_{\text{n}})\) be the vector formed by the outputs of the n networks developed. Let the bootstrap sample \(\mathbf{x}^{\text{*b}}=\left(\mathbf{x}_{1}^{\text{*b}},\mathbf{x}_{2}^{\text {*b}},...\mathbf{x}_{\text{n}}^{\text{*b}}\right)\) be the sample obtained through random replacement sampling B number of times. And let (\(\hat{\Theta}\)) be the real distribution of the estimator we want to obtain from the sample, in this case, the mean. Using the mean of every bth bootstrap sample (\(\hat{\Theta}^{\text{*b}}\)), the empirical distribution of the bootstrap estimator for the mean is calculated (\(\hat{\Theta}^{\text{*}}\)). Following [PERSON] and [PERSON] [38] the distribution (\(\hat{\Theta}^{\text{*}}\)) can be used as the estimator of the real distribution (\(\hat{\Theta}\)) of the mean. One of the major advantages of this resampling method is that is not necessary to assume any hypotheses about the underlying distribution (\(\hat{\Theta}\)) of the statistic studied, in this case, the mean [27]. Figure 1: Internal bond test following EN 319 [3]. Of all the methods for determining confidence intervals described by [PERSON] and [PERSON] [46], BCa is the only one that is second-order accurate and transformation respecting and produces better results because they are closest to the real confidence intervals. The BCa method is based on the percentiles of the bootstrap distribution (\(\hat{\Theta}^{*}\)), but the percentiles are corrected using two parameters (\(\hat{\alpha}\) and \(\hat{2}_{0}\)) known as acceleration and bias correction (Equation (1)). \[\hat{2}_{0}=\phi^{-1}\left(\frac{\#\left(\hat{\Theta}^{*\text{b}}<\hat{\Theta }\right)}{\text{B}}\right) \tag{1}\]  
where \(\phi^{-1}\) is the inverse function of a standard normal cumulative distribution function. In other words, \(\hat{2}_{0}\) measures the bias of the median of \(\hat{\Theta}^{*}\); i.e., the discrepancy between the median of \(\hat{\Theta}^{*}\) and \(\hat{\Theta}\). Therefore, a BCa confidence interval with coverage of 100(1 \(-\)\(\alpha\)) percent for our estimator \(\hat{\Theta}\) will have the form (Equation (2)):  
\[\left[\hat{\Theta}^{*(\hat{\alpha}_{1})},\hat{\Theta}^{*(\alpha_{2})}\right] \tag{2}\]  
where \(\alpha_{1}\) and \(\alpha_{2}\) are the acceleration (Equation (3)):  
\[ \begin{array}{l}\alpha_{1}=\phi\left(2\hat{2}_{0}+\text{z}^{(\frac{\phi}{2} )}\right)\\ \alpha_{2}=\phi\left(2\hat{2}_{0}+\text{z}^{(1-\frac{\phi}{2})}\right)\end{array}  \tag{3}\]  
where \(\phi\) is the standard normal cumulative function and z\({}^{(\alpha)}\) is the 100\(\alpha\)th percentile of the standard normal distribution. To assess the quality of the confidence intervals, the prediction interval coverage probability (PICP) was used. This indicator has been widely applied to determine confidence interval quality [25; 33; 47; 48; 49; 50; 51; 52] (Equation (4)). It assesses the number of experimental data included in the confidence interval. Following [PERSON] et al.