## 1 Introduction
Data-driven estimation approaches are gaining broad interest at the industrial level because of their practicability, robustness, and flexibility to be developed and applied to a wide range of processes, in addition to their independence from a mathematical process model [17; 27]. Examples of data-driven methods used in AD processes include artificial neural networks (ANN) [28; 29], regression models [30], machine learning [16; 31], and fuzzy inference systems [32]. Data-driven methods based on multiscale approaches have been introduced due to the superior robustness and accuracy concerning single-scale analysis [33]. Indeed, the motivation for using multiscale methods for feature extraction is that the data from almost all practical processes are of a multiscale nature. For instance, the events can occur at different locations and with different localization in time and frequency. Moreover, conventional statistical methods based on a single scale are not ideally suited for separating the deterministic component from the normal process variation. [PERSON] et al. [34] applied a multiscale approach with principal component analysis and wavelets to a full-scale biological anaerobic process. Data-driven methods have two main disadvantages: (i) High implementation costs in terms of design effort (architecture selection, training stage, extensive multivariable data sets) and computational cost. (ii) The lack of phenomenological interpretability since they are based on black-box models to predict the desired estimated states. In the last few years, multiscale data-driven methods based on fractal analysis have shown great potential to infer the key variables of the process. [PERSON] et al. [35] proposed the rescaled range (\(R/S\)) analysis of pH time series obtained from an up-flow reactor for the anaerobic digestion of requila viruses, where high correlations between the multiscale indices and critical variables of the process (i.e., COD, VFA, and biogasproduction) were determined. Similar results were found by [PERSON] et al. [36] applying multiscale analysis in pH time series obtained from anaerobic treatment of tequila vinasses in continuous digesters (FBR and CSTR). They concluded that the \(R/S\) analysis could qualitatively estimate essential variables for the process evolution. Finally, [PERSON] et al. [37] studied the anaerobic digestion of cheese whey, determining that multiscale analysis can be used for monitoring the anaerobic treatment of complex substrates. Hence, the above works have shown that through multiscale analysis of time series, it is possible to dynamically characterize the key variables of the anaerobic digestion process using different substrates. Multiscale fractal analysis has two nice features for its practical applicability. The multiscale analysis allows a better understanding of the underlying process phenomena occurring at different scales. Furthermore, a reduction of the implementational costs is obtained (in terms of both design effort and computational cost) since it is based on using a single easy-to-measure and low-cost variables, such as pH and electrical conductivity. In this paper, the application of the multiscale fractal analysis is presented for diagnosis purposes of the anaerobic treatment of raw cheese whey. It should be noted that previous contributions on the multiscale fractal analysis of anaerobic digestion have been applied to stable operating conditions favoring methane production. However, multiscale analysis in situations where the process is affected by inhibition or problems that do not favor methane production has not been evaluated. An efficient monitoring system must assess the dynamic behavior of the variables in a broad region of operation, especially in the region where the process tends to destabilize since, in these conditions, it is when action is required to control the process. Indeed, the need for a proper diagnosis and monitoring system is more significant when the process is in critical conditions.