## 4 Discussion  
In flavor science, the combination of GC-O-MS and AEDA has proven a valuable tool for identifying and ranking key odors in various foods. The aldehydes were most abundant among the 30 volatile components identified by GC-O-MS after the fat was oxidized to produce beef tallow residue such as (E,E)-2,4-decadien [25]. Moreover, the threshold of volatile aldehyde components was low. They contributed substantially to flavor due to their additive effect [26]. The significant decrease of (E,E)-2,4-decadienal in MRPs E might be due to its degradation by the reverse aldol condensation reactions with the prolongation of the hydrolysis time [27] and consumption by the Maillard reaction [28]. Benzene acetaldehyde \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Aroma-Active Compounds**} & \multicolumn{4}{c}{**Relative Content**} & \multicolumn{2}{c}{**Error**} \\ \cline{2-7}  & **Actual Measured** & **PCF** & **CSI** & **CPM** & **PCF** & **CSI** & **CPM** \\ \hline (E)-2-Octenal & 1.682 & 2.263 & 1.895 & 1.638 & 0.581 & 0.213 & 0.044 \\ Octanal & 3.972 & 5.361 & 3.971 & 3.972 & 1.389 & 0.001 & 0 \\ (E,E)-2,4-Nonadienal & 1.032 & 4.265 & \(-\)21.173 & 0.971 & 3.233 & 22.205 & 0.061 \\ (E,E)-2,4-Decadienal & 0.843 & 1.966 & \(-\)6.910 & 0.803 & 1.123 & 7.753 & 0.040 \\
1-Octen-3-ol & 0.826 & 1.568 & \(-\)3.742 & 0.791 & 0.742 & 4.568 & 0.035 \\ (E)-2-Nonanal & 20.535 & 28.317 & \(-\)45.821 & 22.492 & 7.782 & 66.356 & 1.957 \\ Heptanal & 1.576 & 1.646 & 1.394 & 1.575 & 0.070 & 0.182 & 0.001 \\ Benzene acetaldehyde & 1.994 & 2.254 & 0.042 & 2.061 & 0.260 & 1.952 & 0.067 \\ Nonanal & 2.295 & 2.722 & 1.421 & 2.344 & 0.427 & 0.874 & 0.049 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Prediction error of three proxy models in verification experiment. Figure 3: Fitting curves by different proxy models. Simulation of volatile component curves by three models: (**1**) Cubic Spline Interpolation; (**2**) Polynomial Curve Fitting; (**3**) Curve Prediction Model. produced a plant-like odor at low concentrations. However, high concentrations produce a waxy odor, decreasing sensory evaluation scores [29]. The lowest benzene acetaldehyde was present in MRPs C, contributing to the lowest rancidity. Some researchers suggested that unsaturated aldehydes during lipid oxidation cause phenylalanine degradation in response to hydrolysis time [30], resulting in increased benzene acetaldehyde content in MRPs D and E. This high benzene acetaldehyde content in our results might be due to beef's high phenylalanine content in beef tallow residues. The (E)-2- octanal provides a fatty flavor primarily with a low FD value and a high content in MRPs C. Through oxidation, Oleic, linoleic and linolenic acids produce 9-hydroperoxide, which is capable of generating (E)-2-octenal via \(\beta\)-shearing [31; 32]. There are three isomers of ethyl dimethyl pyrazine, among which 2-ethyl-3,5-dimethylpyrazine has been identified as an essential flavor component of roasted beef, roasted coffee, popcorn and roasted sesame [33]. Among the 30 volatile components, 2-ethyl-3,5-dimethylpyrazine had the highest FD values, and excessive or insufficient hydrolysis time resulted in a decrease in the 2- ethyl-3,5-dimethylpyrazine content, which was responsible for the most potent roast flavor found in MRPs C. Alanine was an essential precursor of fragrance-related trialkyl pyrazines such as 2-ethyl-3,5-dimethylpyrazine and 2,3-diethyl-5-methylpyrazine [33]. In addition, threonine and reducing sugar can also generate 2-ethyl-3,5-dimethylpyrazine and 2-ethyl-3,6-dimethylpyrazine through the Maillard reaction under specific temperature and pressure conditions [34]. In oxidative degradation of unsaturated fatty acids, alcohols are formed by positional isomerization of double bonds in the carbon chain [35]. The content of 1-octen-3-ol increased initially then decreased but then increased again due to prolonging hydrolysis time, similar to the findings of the research by other scholars [6].