## 2 Materials and Methods
### Multiscale Time Series Analysis  
In the literature, several methodologies are available for multiscale time-series analysis. However, due to its simplicity and easy implementation, rescaled range (\(R/S\)) [42] is widely applied for time-series analysis obtained from physical, chemical, and biological processes. The \(R/S\) statistical measures the range of the deviations of the partial sums in a time series about its average, rescaled by the series standard deviation. For a time series \(Z_{N}\) = (\(Zi\)) with length \(N\), it is considered a subsequence \(X_{NS}\) = (\(Xii\)) of length \(N_{S}\), where \(N_{S}\) < \(N\). The \(R/S\) statistic for \(X_{NS}\) is calculated as  
\[(R/S)=\frac{1}{\sigma_{S}(N_{S})}\left\{\max_{1\leq i\leq M_{k=1}}\sum_{(X_{k} -\overline{X}_{Ns})}-\min_{1\leq i\leq M_{k=1}}\sum_{(X_{k}-\overline{X}_{Ns} )}\right\} \tag{1}\]  
where \(\overline{X}_{Ns}\) is the subsequence mean and \(\sigma_{S}(N_{S})\) is the sample standard deviation, which are defined as  
\[\overline{X}_{Ns}=\frac{1}{N_{S}}\sum_{i=1}^{Ns}X_{i}\text{ and }\sigma(N_{S})=\sqrt{\frac{1}{N_{S}}\sum_{k=1}^{N_{S}}\left(X_{k}- \overline{X}_{Ns}\right)^{2}} \tag{2}\]  
The \(R/S\) statistic follows a power law, \((R/S)=aN_{S}^{H}\) where \(a\) is a constant and \(H\) is the Hurst exponent. A log-log plot of \((R/S)\) as a function of \(N_{S}\in(N_{S,\text{min}},N_{S,\text{max}})\), gives a straight line with slope \(H\). If the series data is independent (e.g., white-noise process), the plot is roughly a straight line with slope \(H\) = 0.5. If \(H\) > 0.5, the time series \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Whey** & **Inoculum** \\ \hline COD (g L\({}^{-1}\)) & 74.24 \(\pm\) 1.35 & 32.52 \(\pm\) 4.27 \\ Carbohydrates (CH) (gglucose L\({}^{-1}\)) & 35.04 \(\pm\) 1.56 & 0.94 \(\pm\) 0.07 \\ Total Solids (g L\({}^{-1}\)) & 51.21 \(\pm\) 1.364 & 49.87 \(\pm\) 0.95 \\ Volatile Solids (g L\({}^{-1}\)) & 36.68 \(\pm\) 4.68 & 22.46 \(\pm\) 4.30 \\ Volatile Sedimented solids (g L\({}^{-1}\)) & 1.20 \(\pm\) 0.24 & 2.15 \(\pm\) 0.41 \\ VFA (g\({}_{\text{COD}}\) L\({}^{-1}\)) & 0.57 \(\pm\) 0.08 & 28.56 \(\pm\) 1.90 \\ Protein (mg L\({}^{-1}\)) & 27.32 \(\pm\) 2.17 & 7.22 \(\pm\) 1.09 \\ pH & 4.72 \(\pm\) 0.04 & 6.85 \(\pm\) 0.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characterization of whey and inoculum. is persistent, indicating the presence of long-term autocorrelations. Conversely, if \(H<0.5\) autocorrelations in the signal are antipersistent [43,44]. Multifractality is a valuable tool for explaining many patterns seen in nature. Particularly, multifractal analysis allows for the investigation of a mixture of fractal dimensions characterizing the inherent complexity in some data series. Based on the thoughts of [PERSON] and [PERSON] [45] and [PERSON] and [PERSON] [46], the multifractal analysis can be done through the calculation of the rescaled range through the q-norm of \(\mathfrak{os}\), that is,  
\[\sigma_{q}(N_{S})=\left[\frac{1}{N_{S}}\underset{k=1}{\overset{Ns}{\sum}}\left( X_{k}-\overline{X}_{Ns}\right)^{q}\right]^{\frac{1}{q}} \tag{3}\]  
In this case, the \(R/S\) statistical is given by \(\left(R/S\right)_{q}=\left(R/S\right)\sigma(N_{S})/\sigma_{q}(N_{S})\), and the average range is expected to follow the scaling behavior \(\left(R/S\right)_{q}=aN_{S}^{2H_{q}}\), where \(H_{q}\) is the q-th Hurst exponent. If \(H_{q}\) is constant for all \(q\) then the underlying phenomena are monofractal. A nontrivial dependence of \(H_{q}\) vs. \(q\) indicates that the process is multifractal. It should be recalled that a multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics. In general, multifractality also indicates the nonlinear nature of the mechanisms that generated the series [47,48].