## 3 Data Analysis Setup
### The Mathematical Framework  
In terms of wave surface elevation, the most common statistical models are formed by years of measurement, represented by a distribution such as JONSWAP [25]. While this approach describes the system's overall behaviors, the average quantity fails to provide deterministic insights or, more precisely, reliable insights within short observation windows. This fact basically stems from the lack of temporal labels used in statistical distribution, or, in other words, the statistical distribution is essentially set up by spectral information. Therefore, the studies carried out until recently rest upon these fundamental properties, such as [26; 27; 28]. Considering all this, the new approach can be conceptually simplified in the flow diagram in Figure 6.  
Figure 5: Numerical results validation with the heave decay test results (**a**) and heave response to regular waves (**b**). Starting from the left side of the diagram, wave as the input excites the ship as the system, resulting in six DOF responses in the form of time series, the system output. The purpose of applying wavelet on the short pieces of time series is to extract as much information as possible from time series data in the form of detailed features representation. Further, these features are reorganized through numerical operation to facilitate eventual manipulation and classification based on the categorical wave classes. The final grey blocks indicated the outcomes of feature reorganization that could facilitate feature manipulation, or, in other words, direct control over the densified features. Additionally, feature classification can lighten the training cost, which is aimed at being the focus of part B of the current issue. The data generation process up to the output has been explained in Section 2. Nevertheless, six DOFS must be represented as a specific form for further mathematical operation. To include all features in the data, an additive approach has been adopted according to [15], representing every three DOFs of translational \((x,y,z)\) and angular motions \((\varphi,\theta,\psi)\) to a condensed vector resulting from the summation of DOFs. The pertinent reason is simply due to the complex interaction of the mooring system's impact on floating object responses and the mutual effects of responses. The vectors obtained from the summation of each three DOFs are normalized between (0,1) and represent a unique vector of \(\varsigma(k)\) by summation of \(y_{d}[k]\) where \(d=1,2\) denoting each group of normalized DOFs. The matrix given in Equation (3), indicating summation of the six DOFs. For a vector \(\varsigma(k)\) representing the sphere/ship response, k denotes the discrete sample instant, and \(l\) the length of the vector, taken in this study as 40 s, corresponding to 8000 samples. However, noting that for the first part of the study addressing the deterministic input-output, only have response is considered. \[\varsigma(k)_{l}=\sum_{d=1}^{6}y_{d}[k]_{l} \tag{3}\]  
To capture detailed spectral-temporal features from sphere responses corresponding to various waves, we employed the Morse wavelet. This wavelet, formed by combining complex exponential and Gaussian windows, offers distinct advantages as outlined in [29]. Specifically, the generalized Morse wavelet is utilized with parameters, such as symmetry (\(\gamma\)), set to be 3, and time-bandwidth \(p^{2}\) set to 60. This analytical wavelet, characterized by a scaling factor, acts as a filter bank, allowing to systematically extract detailed spectral-temporal features, convolving over the observation window. The benefits of representing these features via a scalogram, discussed extensively in [24], surpass alternative Fourier and wavelet-based methods. Equation (4) illustrates the wavelet function, where \(U(\omega)\)  
Figure 6: Data analysis flow diagram. represents the unit step, \(a_{\beta,\gamma}\) is a normalizing constant, and \(\omega\) denotes frequency, with \(m\) and \(n\) determining the matrix arguments, resulting in \(\Psi\) as a wavelet.