## Materials and Methods  
### Ripening Stages Classification Based on Color CIELab Chromaticity  
A color space allows the representation of color at a visible light range in a three-dimensional system [28]. The CIELab is a uniform color space whose representation is composed by the axes \(L*\), \(a*\), and \(b*\)[30]. \(a*\) is the gradient from green to red colors, and \(b*\) is the gradient from blue to yellow and color as depicted in Figure 1. Moreover, \(L*\) represents the lightness gradient. For each ripening stage, a domain was computed through Gaussian distribution. This domain embodies the probability that the \(a*\) and \(b*\) values can be inside the interval defined by cluster separation according to the following distribution:  
\[N(\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}} \,, \tag{1}\]  
Here, \(\mu\) is the mean of color data, \(\sigma\) is the standard deviation, \(a*\) and \(b*\) are values of data represented by \(x\)[31]. The distribution functions are calculated as \(N_{a}\) and \(N_{b}\) in each color direction. The covariance matrix correlates this as follows  
\[\mathbf{C}=\left[ \begin{array}{cc}\sigma_{a}^{2}&\sigma_{ab}\\ \sigma_{ba}&\sigma_{b}^{2}\end{array} \right]. \tag{2}\]  
Figure 1: Color measurement on coffee fruits with a colorimeter. \(\sigma_{a}^{2}\) and \(\sigma_{b}^{2}\) are variances; \(\sigma_{ab}\) and \(\sigma_{ba}\) are crossed covariances. This matrix allows setting up an eigenvalues problem as  
\[\mathbf{C}x=\lambda x, \tag{3}\]  
where \(\lambda_{\text{max}}\) and \(\lambda_{\text{min}}\) are the maximum amplitudes of data variation in each ripening stage represented by the eigenvalue. The data clusters are delimited by ellipses oriented at the principal direction depending on the eigenvectors \(\boldsymbol{\lambda}=\lambda_{a}\mathbf{i}+\lambda_{b}\mathbf{j}\), and \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors. Hence, the principal direction of each data is determined by:  
\[\beta=\tan^{-1}\biggl{(}\frac{\lambda_{b}}{\lambda_{a}}\biggr{)}. \tag{4}\]  
Scaling the principal axes by a factor of \(\alpha=2.4477\) (chi-square), a confidence domain of 95% is reached. Each ripening stage is represented by the following expression, which describes a rotated ellipse \(\beta\) with maximum amplitudes \(\lambda_{\text{max}}\) and \(\lambda_{\text{min}}\)  
\[ \begin{array}{l}\frac{((a-\mu_{a})\cos(\beta)+(b-\mu_{b})\sin(\beta))^{2}}{ \text{max}^{2}}+\\ \frac{(-(a-\mu_{a})\sin(\beta)+(b-\mu_{b})\cos(\beta))^{2}}{\lambda_{\text{min }}}=1\end{array} . \tag{5}\]  
In this study, the ripeness stages of coffee fruits (_Coffea arabica var_. Castillo) were classified through measurements in the CIELab color space. Samples were detached from a 2-year-old coffee tree from a farm located in Belalcazar, Caldas (Colombia). The environmental conditions of the farm were 25.2 \({}^{\circ}\)C temperature, 75.9 kPa of atmospheric pressure, 81% of relative humidity, 3 km/h west-northwest direction of wind speed, 1352 m of altitude, with coordinates 5\({}^{\circ}\)0\({}^{\prime}\)40\({}^{\prime\prime}\), 75\({}^{\circ}\)47\({}^{\prime}\)44\({}^{\prime\prime}\). The color of each fruit was measured with a PCE CSM 4 colorimeter, as shown in Figure 1.  
After carrying color measurements out in 84 fruits, three data clusters were established over \(a*\) and \(b*\) values; then, a classification by data covariance was applied. Three ripening stages represented by three ellipses domains resulted from the covariance calculations depicted in Figure 2. For the present study, two groups of fruits of three ripening stages (unripe, semi-ripe, and ripe) were chosen as study samples for a vibrations analysis. These samples were initially classified by color and located in their corresponding ellipse space, as detailed in Figure 2. Unripe fruits were labeled as UR, semi-ripe were labeled as SMR, and ripe were labeled as RP.