## 3 Data Analysis Setup
### Evaluation for Deterministic Inputs (Regular Waves)  
Although wave elevation patterns possess random characteristics, they can be broken down to the superposition of many numbers of deterministic sinusoidal waves [31; 32]. Unlike free response, which reflects the system characteristics (sphere), the deterministic input determines the forced response of the system excited by inputs [9]. As mentioned, the new space projection enables monitoring and investigating features produced by different data generation processes. To examine this, the test ran first with a deterministic input-output relationship known as regular waves based on one DOF, heave (\(z\)). It basically contains an exciting sphere in the wave basin; further, the simulation in ANSYS AQWA using a sinusoidal wave and (response amplitude operator) RAO-based response. The test was carried out on the heave response in a range of wave periods, starting with close periods of 1.1 and 1.0 s. Figure 9 pictures the comparison of experimental and simulation results for sphere heave response for the respective waves. As can be observed, the simulation well met the experimental results apart from minor discrepancies. Figure 10 indicates the results of feature spread projection for a heave period of 1.1 s. Figure 10 indicates the projection of features from experimental tests and simulations in 3D space, and Figure 10 shows the projection of vectors in white and black planes. According to Figure 10b, the difference between features accumulated more in high-frequency regions with less signal amplitude (blackish pixels), whereas the difference in low-frequency regions with higher amplitude is less pronounced with quite a similar pattern. Figure 8: 3D view of feature spread projection (**a**), projection to black plane (**b**), and white plane (**c**). Examining the planes in Figure 10b, it is evident that the low-frequency traits, depicted in the white plane, remain consistent and nearly identical between experiment and simulation. However, notable disparities emerge in the projection onto the black plane, despite overall spread similarities. As noted earlier, the presence of noise amplifies the variance in feature distribution, a trend also observed in black plane values below 0.5, mirroring the behavior seen in Figure 7. This discrepancy can be attributed to inherent uncertainties in the experiment, which are challenging to replicate in numerical simulations. Revisiting primary objectives, integrating entropy is vital for ML applications, as demonstrated in Figure 11, showcasing the proposed method for enhancing simulation outcomes through uncertainty integration. In essence, the features of simulation data with a period of 1.1 s are subtracted from the experimental data in each plane, and the  
Figure 10: (**a**) 3D view of projection of features in new space. (**b**) Features spread on black planes (**left**) and white planes (**right**). Figure 9: Validation and comparison of simulation results with experiment. residual features are combined with those obtained from simulation with (another period) a period of 1.0 s. To this end, the vector \(\mathrm{K_{a,b}}\) as the sphere response from the experiment, which includes only \(\mathrm{y_{z}[k]}\) for have, is reconstructed in two vectors of \(\mathrm{w_{a,b}}\) and \(\mathrm{q_{a,b}}\), where \(a=1\) and \(b=256\), and they are projected on white and black planes, respectively. The same operation has been carried out for vector \(\mathrm{K_{a,b}^{\prime}}\) as the result of simulation as well as projection vectors \(\mathrm{w_{a,b}^{\prime}}\) and \(\mathrm{q_{a,b}^{\prime}}\). Equation (8) shows the simple algebraic subtraction that yields the residual features \(W_{a,b}\) and \(Q_{a,b}\).