## 3 Results and Discussion
### Mathematical Models
\begin{table}
\begin{tabular}{l l l l} \hline
**Frequency** & \(R^{2}\) & **Adjus. \(R^{2}\)** & **RMSE** \\ \hline
10 Hz & 0.9957 & 0.9950 & 15.0282 \\
1 kHz & 0.9969 & 0.9965 & 11.2313 \\
1 MHz & 0.9949 & 0.9941 & 12.2881 \\ \hline \end{tabular} \[\begin{split}&|Z_{L}|~{}=~{}414.73-12.0732\cdot T-4.81235\cdot D+0.149 0\cdot T^{2}+0.434274\cdot D^{2}\\ &~{}~{}~{}~{}-0.0092\cdot D^{3}+7.04196\times 10^{-5}\cdot D^{4}-1. 79283\times 10^{-7}\cdot T\cdot D^{4}\\ &|Z_{L}|~{}=~{}298.425-8.38551\cdot T-4.74573\cdot D+0.103552\cdot T ^{2}+0.413078\cdot D^{2}\\ &-0.0086\cdot D^{3}+6.47784\times 10^{-5}\cdot D^{4}-1.44913\times 10^{-7} \cdot T\cdot D^{4}\\ &|Z_{L}|~{}=~{}233.488-6.47223\cdot T-3.50796\cdot D+0.0750184\cdot T ^{2}+0.30982\cdot D^{2}\\ &-0.0065\cdot D^{3}+5.02774\times 10^{-5}\cdot D^{4}-1.31422\times 10^{-7} \cdot T\cdot D^{4}\\ \end{split}\] (4)  
It should be noted that the equation models for \(10^{1}\) Hz, \(10^{3}\) Hz, and \(10^{6}\) Hz were similar to the significance of the terms and the interactions between the variables. In all of them, the linear, quadratic, cubic, and fourth-order factors of dilution (% water) have statistical significance (\(p<0.05\)), as well as the temperature (T) \(\times\) dilution (D) interaction for the analysis of milk adulterated with water. However, the high-order temperature terms showed no significance in any of the models, leaving only their linear effect. The adjustment of impedance values in the regression models resulted in a coefficient of determination (\(R^{2}\)) of approximately 0.99 for all models. The same value (0.99) was found for the adjusted \(R^{2}\). [PERSON] et al. [18] presented a coefficient of determination for their models with values greater than 0.94 for milk adulterated with ethanol, sodium chloride, and sodium bicarbonate. The value of \(R^{2}\) indicates the ratio between the variation explained and the total variation, representing a measure of the degree of adjustment [47], and values close to unity imply that most of the variability in the magnitude of \(Z\) is explained by the regression model, which indicates that the model has a good degree of adjustment [37]. Given this, the model showed a good fit, which can be seen in the regression surface graph (Figure 11). The probability values were lower than 0.05, indicating that the terms of the model have significant effects on the impedance response variable, i.e., the model is adequate at the 95% confidence level (Table 5).  
\end{table} 
Table 5: Statistical parameters used to evaluate the effects of temperature and dilution on the bioimpedance of milk for different frequencies. Figure 12A-C show the relationship between the measured impedance values and the values predicted by the proposed models, considering the frequencies \(10^{1}\), \(10^{3}\), and \(10^{6}\) Hz, for the samples in the simulation of the fraudulent addition of distilled water. Also, to extend the validation of the model, samples were analysed at random temperatures (between 16 and 37\({}^{\circ}\)C) and frequency variation. It is observed that the measured impedance values compare adequately with the predicted values in all situations. Therefore, the model provides a valid description of the data used to generate them. Also, because of the similarity between the measured and predicted impedance values, the model obtained can be used directly to estimate the percentage of water in adulterated milk samples. However, despite the good results, we consider it important to test the model at other temperatures, close to 4\({}^{\circ}\)C because, according to current legislation, this is the temperature at which the product is stored and transported between the farm and the processing plants. Figure 11: Regression surfaces considering the frequencies (**A**) \(10^{1}\) Hz, (**B**) \(10^{3}\) Hz, and (**C**) \(10^{6}\) Hz.