## 3 Data Analysis Setup
### The Mathematical Framework
\[\Psi_{p,\gamma}\ =\mathrm{U}(\omega)\ a_{\beta,\gamma}\ \omega^{\frac{p^{2}}{ \gamma}}e^{-\omega\gamma} \tag{4}\]  
The 2D matrix of \(\Gamma_{\mathrm{m,n}}\ (\mathrm{k})\) as the result of wavelet coefficients can be represented by an image, which helps for better visual perception of the spectral constitution of a signal and features amplitude over a fixed temporal direction (40 s). The image can be formed in the RGB channel; however, it has turned into grayscale with a normalized value between 0 and 1 for all pixels due to computational efficiency and less sensitivity to noise [24]. The corresponding 2D matrix, \(\Gamma_{\mathrm{m,n}}\ (\mathrm{k})\) can be transformed into a [256 \(\times\) 256] grayscale image using a linear conversion described in Equations (5) and (6). Here, the pixel value is determined by \(Y\), which serves as a scale factor adjusting the pixel intensity between 0 and 255, and \(\xi\), the shift parameter, represents a constant that can be adjusted to control brightness, which is taken as 0 here. \[\Gamma_{\mathrm{m,n}}\ (\mathrm{k})=\Psi_{p,\gamma}\times\xi(\mathrm{k}) \tag{5}\]  
\[\mathrm{P}_{\mathrm{m,n}}=Y\times\Gamma_{\mathrm{m,n}}(\mathrm{k})+\xi \tag{6}\]  
Therefore, through sliding Equation (6) to all \(m\) and \(n\) values of \(\Gamma_{\mathrm{m,n}}\ (\mathrm{k})\) results in matrix \(\mathrm{P}_{\mathrm{m,n}}\), where \(\mathrm{P}\) represents the pixel intensity values ranging between 0 and 255, and \(\mathrm{m}\) and \(\mathrm{n}\) are both 256. Subsequently, the pixel values are normalized to a range between 0 and 1 using Equation (7), resulting in the vector \(\mathrm{K}_{\mathrm{a,b}}\), where \(a\) equals 256 and \(b\) equals 1, turning the matrix into a column vector. To have a better visual representation, Figure 7a indicates the cumulative scalogram of \(\xi(\mathrm{k})\) and subsequent transformation of vector \(\mathrm{K}\) on a diagonal plane within the new space in Figure 7b. This depicts the result of six DOFs accumulation for a single observation of 40 s. As can be witnessed, the features stand out differently for each image, corresponding to different time stamps of the sphere's response to the wave. \[\mathrm{K}_{\mathrm{a,b}}=\frac{\sum_{\mathrm{m=1}}^{256}P}{max\sum_{\mathrm{ m=1}}^{256}P} \tag{7}\]  
In Figure 7a (left image), the scalogram is depicted in grayscale, overlaid with white Gaussian noise with a variance of \(\sigma^{2}=0.01\), corresponding to a signal-to-noise ratio (SNR) of 8.99. This noise is visually represented as a scattered pattern of white pixels across the scalogram. As can be seen in Figure 7a, the noise manifested within new feature space in higher variation and feature spread for the range of pixel intensity below 0.5. Apart from that, the pixels shifted in the range below 0.5 due to the higher average value of the vector. Thus, the noise variance can be correlated with the features spread in new space qualitatively and quantitatively. Zooming in on the primary low-frequency components of Figure 7a, prominently visible as a broad white region in Figure 7a (left image) and (right image), reveals their consistent influence across the diagonal spread in Figure 7b above 0.5, akin to the significant features. Acknowledging the reverse spectral direction in the new feature space, noting that upper regions in image height direction attribute to lower frequency constitution, while lower regions represent higher frequencies. As such, the lower spectral elements, white pixels in the bottom of the left image in Figure 7a, show up at the higher height of Figure 7b. By and large, the new operation compressed the features in the temporal direction while preserving the spectral constitution through normalized wavelet coefficients not only in the temporal direction but also given the intensity of pixels amplitude in the spectral direction. This densifies the nuanced information within the 40 s of the time series, shrunk into a vector as a single information packet, either for further classification or regression purposes.