## 2 Materials and Methods
### Development and Cross Validation of a Gaussian Process Model to Predict Unit Load Segment Bending
\[\hat{y}(x)=\hat{\mu}+r^{\prime}(x)R(\hat{\theta})^{-1}(y-j\hat{\mu}) \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**ID**} & \multirow{2}{*}{**Height** **(mm)**} & \multirow{2}{*}{**Pallet** **Friction**} & \multicolumn{2}{c}{**Box**} & \multicolumn{4}{c}{**Box**} & \multicolumn{4}{c}{**Deflection Ratio by Model**} \\ \cline{4-9}  & & & **Vertical** & **Horizontal** & **3** & **4** & **5** & **6** \\  & & & **Friction** & **Friction** & **Columns** & **Columns** & **Columns** & **Columns** \\ \hline Run 001 & 508.00 & 0.66 & 0.48 & 0.41 & 0.38 & 0.36 & 0.33 & 0.39 \\ Run 002 & 547.08 & 0.47 & 0.26 & 0.64 & 0.35 & 0.42 & 0.56 & 0.64 \\ Run 003 & 586.15 & 0.61 & 0.24 & 0.34 & 0.35 & 0.37 & 0.53 & 0.64 \\ Run 004 & 625.23 & 0.43 & 0.39 & 0.43 & 0.37 & 0.42 & 0.61 & 0.64 \\ Run 005 & 664.31 & 0.33 & 0.67 & 0.61 & 0.20 & 0.44 & 0.48 & 0.65 \\ Run 006 & 703.38 & 0.46 & 0.52 & 0.23 & 0.48 & 0.41 & 0.46 & 0.65 \\ Run 007 & 742.46 & 0.28 & 0.30 & 0.30 & 0.65 & 0.43 & 0.58 & 0.62 \\ Run 008 & 781.54 & 0.48 & 0.61 & 0.44 & 0.15 & 0.33 & 0.44 & 0.49 \\ Run 009 & 820.62 & 0.25 & 0.34 & 0.55 & 0.36 & 0.42 & 0.56 & 0.65 \\ Run 010 & 859.69 & 0.70 & 0.29 & 0.53 & 0.26 & 0.47 & 0.55 & 0.64 \\ Run 011 & 898.77 & 0.24 & 0.57 & 0.25 & 0.60 & 0.38 & 0.55 & 0.65 \\ Run 012 & 937.85 & 0.65 & 0.64 & 0.29 & 0.10 & 0.20 & 0.30 & 0.34 \\ Run 013 & 976.92 & 0.56 & 0.44 & 0.60 & 0.22 & 0.43 & 0.55 & 0.62 \\ Run 014 & 1016.00 & 0.69 & 0.62 & 0.52 & 0.14 & 0.24 & 0.30 & 0.34 \\ Run 015 & 1055.08 & 0.35 & 0.49 & 0.69 & 0.29 & 0.43 & 0.62 & 0.63 \\ Run 016 & 1094.15 & 0.60 & 0.41 & 0.33 & 0.20 & 0.45 & 0.55 & 0.66 \\ Run 017 & 1133.23 & 0.52 & 0.66 & 0.67 & 0.17 & 0.31 & 0.41 & 0.57 \\ Run 018 & 1172.31 & 0.29 & 0.65 & 0.46 & 0.27 & 0.44 & 0.57 & 0.66 \\ Run 019 & 1211.38 & 0.42 & 0.25 & 0.56 & 0.32 & 0.43 & 0.58 & 0.64 \\ Run 020 & 1250.46 & 0.20 & 0.43 & 0.39 & 0.34 & 0.46 & 0.59 & 0.64 \\ Run 021 & 1289.54 & 0.38 & 0.42 & 0.24 & 0.29 & 0.42 & 0.63 & 0.63 \\ Run 022 & 1328.62 & 0.41 & 0.47 & 0.48 & 0.28 & 0.43 & 0.57 & 0.64 \\ Run 023 & 1367.69 & 0.30 & 0.23 & 0.37 & 0.32 & 0.45 & 0.59 & 0.64 \\ Run 024 & 1406.77 & 0.53 & 0.20 & 0.32 & 0.34 & 0.45 & 0.60 & 0.64 \\ Run 025 & 1445.85 & 0.49 & 0.70 & 0.38 & 0.14 & 0.32 & 0.42 & 0.66 \\ Run 026 & 1484.92 & 0.64 & 0.32 & 0.49 & 0.27 & 0.43 & 0.59 & 0.65 \\ Run 027 & 1524.00 & 0.23 & 0.33 & 0.58 & 0.32 & 0.44 & 0.58 & 0.64 \\ Run 028 & 1563.08 & 0.67 & 0.51 & 0.20 & 0.18 & 0.32 & 0.40 & 0.43 \\ Run 029 & 1602.15 & 0.34 & 0.69 In order to generate the most accurate model and to validate the prediction results from the Gaussian Process model, a \(k\)-fold cross validation was conducted. The response data set for each per column model was divided into five equal parts. The GP model was run five separate times with only four-fifths of the data. The remaining data points were used to conduct the cross validations. The model with the data set that provided the highest accuracy was selected as the resulting statistical model. Goodness of fit and predictive performance was evaluated by comparing the Root Mean Square Error (RMSE) of the predictions against the finite element model simulation results, as well as the Mean Absolute Percent Error (MAPE). Both error measurements together provided enough confidence to determine model accuracy.