## 3 Data Analysis Setup
### Inspiration and General Framework
The rationale behind this is primarily attributed to the complex interaction between the mooring system's impact on floating object responses and the mutual effects of these responses as given in [26]. Therefore, the \(\varsigma(\mathrm{k})\) as a vector in Equation (6) represents the summation of the 6 DOFs. For a vector \(\varsigma(\mathrm{k})\), representing the response of the sphere/vessel, k stands for the discrete sample instance, and l the length of the vector, set in this setup to 40 s, equating to 8000 samples. \[\varsigma(\mathrm{k})_{\mathrm{l}}=\sum_{\mathrm{d=1}}^{6}\mathrm{y}_{\mathrm{ d}}[\mathrm{k}]_{\mathrm{l}} \tag{6}\]  
To capture intricate spectral-temporal features of sphere responses to waves, we implemented the Morse Wavelet applied on \(\varsigma(\mathrm{k})_{\mathrm{l}}\). This wavelet, constructed by amalgamating complex exponential and Gaussian windows, offers distinct advantages as highlighted in prior literature [23; 27]. Specifically, the Generalized Morse wavelet has been utilized with parameters such as symmetry (\(\gamma\)) set to be 3, and time-bandwidth p\({}^{2}\) set to 60 that ends up with the most symmetric shape of the wavelet. This analytical wavelet, characterized by a scaling factor, functions as a filter bank, systematically extracting detailed spectral-temporal features by convolving it over \(\varsigma(\mathrm{k})_{\mathrm{l}}\). Equation (7) delineates the wavelet function, wherein \(\mathrm{U}(\omega)\) signifies the unit step, \(\mathrm{a}_{\beta,\gamma}\) is a normalizing constant, and \(\omega\) denotes frequency, with m and n determining the matrix arguments. For more details, please read [28]. \[\Psi_{\mathrm{P},\gamma}=\mathrm{U}(\omega)\mathrm{a}_{\beta,\gamma}\omega^{ \frac{\mathrm{p}^{2}}{\gamma}}\mathrm{e}^{-\omega\gamma} \tag{7}\]  
Equation (8), as the result of wavelet operation on the data, denotes 2D matrix of \(\Gamma_{\mathrm{m,n}}(\mathrm{k})\), representing the wavelet coefficients. This matrix can be visualized as an image, aiding in a clearer comprehension of the spectral composition of a signal and the amplitude of its features along a fixed temporal axis. Each pixel in the image correspond to the normalized wavelet coefficient between 0 and 1. This 2D array \(\Gamma_{\mathrm{m,n}}(\mathrm{k})\) can then be converted into a grayscale image of size \([\mathrm{M}\times\mathrm{M}]\) using a linear transformation outlined in Equations (8) and (9). In this transformation, the Pixel Value is determined by Y, which functions as a scaling factor adjusting the pixel intensity between 0 and 255 for picture size of [256 \(\times\) 256], while \(\xi\), the shift parameter, represents a constant that can be adjusted to control brightness; for our purposes, it is set to 0. The benefit of the normalization setup presented in Equations (8) and (9) lies in establishing a global normalization approach right from the outset for all subsequent values. The range of the global normalization can be adjusted by m and n which leads to the extension of feature space capacity. \[\Gamma_{\text{m,n}}(\text{k})=\Psi_{\text{p,}\gamma}\times\varsigma(\text{k}) \tag{8}\]  
\[\text{P}_{\text{m,n}}=\text{Y}\times\Gamma_{\text{m,n}}(\text{k})+\xi. \tag{9}\]  
Applying Equation (7) iteratively to all m and n values of \(\Gamma_{\text{m,n}}(\text{k})\), we obtain matrix \(\text{P}_{\text{m,n}}\) that represents the pixel intensity values or range of normalization bin ranging from 0 to 255, where both are set to 256. Subsequently, the pixel values are normalized to a range between 0 and 1 using Equation (10), resulting in the vector \(\text{K}_{\text{a,b}}\), where \(\text{a,b}=(256,1)\), thereby transforming the matrix into a column vector. \[\text{K}_{\text{a,b}}=\frac{\sum_{\text{m=1}}^{256}\text{P}}{\text{max}\sum_{ \text{m=1}}^{256}\text{P}} \tag{10}\]  
To have a visual representation, the normalized coefficient can be captured in the form of an image, so-called scalogram. Nevertheless, the vector K as the feature vector can be reflected in 3D space on a diagonal plane.