The GOS content in the infant formula powder samples was determined using an enzymatic method (26). The GOS raw materials from the same batch were subjected to preprocessing analysis. The GOS content was measured using an ion chromatography-electrochemical pulse amperometric detector, which has high sensitivity. GOS raw materials usually contain other components such as lactose, glucose, and galactose. Based on the principle of consistent ratio of low-degree oligosaccharides in the raw materials and infant formula powder, a set of characteristic peaks for GOS was selected. The GOS content in the infant formula powder was indirectly determined using the same batch of raw material syrup as the reference. The content range of GOS in the test samples was determined to be 5–29 mg·kg⁻¹ through chemical analysis.

The FOS content in the milk powder samples was determined according to the national standard GB 5009.255–2016 “determination of fructosan in food.” The milk powder samples were extracted with hot water. The sucrose in the sample solution was hydrolyzed into glucose and fructose by sucrase. Glucose and fructose were then reduced to their corresponding sugar alcohols by sodium borohydride, and the excess sodium borohydride was neutralized with acetic acid. The fructosan in the sample solution were hydrolyzed into fructose and glucose by fructan hydrolase. The fructose content was determined using ion chromatography with pulsed amperometric detector. The content of fructosan was calculated based on conversion factors. The content range of FOS in the test samples was determined to be 4.4–26.6 mg·kg⁻¹ through chemical analysis.

The Ca content in the milk powder samples was determined according to the national standard GB 5009.92–2016 “determination of calcium in food.” Flame atomic absorption spectroscopy was used to measure the Ca content in the milk powder samples after digestion. Lanthanum solution was added as a releasing agent, and the absorbance values measured at 422.7 nm were proportional to the Ca concentration within a certain concentration range. The Ca content was quantitatively determined by comparing with a standard series. The content range of Ca in the test samples was determined to be 3.1–7.12 mg·kg⁻¹ through chemical analysis.

The Vc content in the milk powder samples was determined according to the national standard GB 5413.18–2010 “Determination of Vitamin C in Infant Food and Dairy Products.” Vc was oxidized to dehydroascorbic acid in the presence of activated carbon. It reacted with o-phenylenediamine to form a fluorescent substance, and the fluorescence intensity was measured using a fluorescence spectrophotometer. The fluorescence intensity was proportional to the concentration of Vc, and the content was quantified using an external standard method. The content range of Vc in the test samples was determined to be 0.54–1.82 mg·kg⁻¹ through chemical analysis.

The CARS algorithm is a feature wavelength extraction method based on the theory of Darwinian evolution (27, 28). Absolute values of regression coefficients and the weights corresponding to each wavelength are calculated. It retains the wavelength points with the highest absolute weight coefficients and removes those with smaller weights. The feature wavelengths were determined based on the lowest Root Mean Square Error of Cross Validation (RMSECV). The parameters for the CARS algorithm were set as follows: maximum number of principal components = 10, number of cross-validations = 10, and number of Monte Carlo runs = 40.

The RF algorithm is an efficient method for selecting variables from high-dimensional data (29, 30). It calculates the probability of each wavelength being selected after N iterations and sorts them accordingly. The wavelengths with higher probabilities are selected for model building. To ensure convergence, the iteration parameter N was set to 10,000, and the number of selected feature wavelengths was set to 40.

The PLSR algorithm is used to establish prediction models (31, 32). The optimal number of latent variables for the PLSR model was also determined. After 10 rounds of training, the performance of the 10 models was evaluated, and the best hyperparameters were selected as the optimal number of latent variables for the PLS model.

The SVR algorithm uses a nonlinear kernel function to map low-dimensional input to a high-dimensional feature space and performs linear regression in the high-dimensional feature space (33, 34). It is suitable for handling problems with a small number of samples, nonlinearity, and high dimensionality. A Gaussian function was selected as the kernel function, and the values of the parameters c and g were set within the range of [−10, 10] with a step size of 0.5.

Figure 4A depicts the process of feature wavelength extraction using CARS for GOS. From Figures 4Aa, it can be observed that as the number of samples increases, the sampling ratio of the variable subset starts to decrease and gradually stabilizes. Figures 4Ab shows that as the number of samples increases, the RMSECV of the eliminated unimportant wavelengths decreases slowly. However, it starts to increase when important wavelengths are eliminated. The lowest RMSECV is achieved when the Monte Carlo run reaches 23, resulting in the extraction of 26 feature wavelengths by CARS. The distribution of wavelengths is shown in Figure 4B. Figure 5A illustrates the process of feature wavelength extraction using CARS for FOS. The lowest RMSECV is achieved when the Monte Carlo run reaches 24, resulting in the extraction of 31 feature wavelengths. The distribution of wavelengths is shown in Figure 5B. Figure 6A shows the process of feature wavelength extraction using CARS for Ca. The lowest RMSECV is achieved when the Monte Carlo run reaches 17, resulting in the extraction of 101 feature wavelengths. The distribution of wavelengths is shown in Figure 6B. Figure 7A presents the process of feature wavelength extraction using CARS for Vc. The lowest RMSECV is achieved when the Monte Carlo run reaches 25, resulting in the extraction of 26 feature wavelengths. The distribution of wavelengths is shown in Figure 7B. Comparing the feature wavelengths selected by the CARS algorithm with the original full wavelengths, GOS, FOS, Ca, and Vc achieved reductions of 98.33, 98.01, 93.51, and 98.33%, respectively. This demonstrates that the CARS algorithm is capable of significantly reducing the number of wavelengths effectively (27).

Figure 8A illustrates the probability of each wavelength being selected after 10,000 iterations during the process of feature wavelength extraction using the Random Frog (RF) algorithm for GOS. The x-axis represents the number of wavelengths, and the y-axis represents the probability of a wavelength being selected. From the figure, it can be observed that wavelengths near the absorption peaks at 4800 cm⁻¹, 6,000 cm⁻¹, 6,800 cm⁻¹, and 8,100 cm⁻¹ have a higher probability of being selected. Although wavelengths near other feature peaks are also selected, the probability of selection is relatively low. Ultimately, the top 40 wavelengths with the highest selection probabilities are chosen as feature wavelengths, as shown in Figure 8B. Figures 9A, 10A, 11A represent the probability of wavelength selection for FOS, Ca, and Vc, respectively. The distributions of the 40 selected feature wavelengths are shown in Figures 9B, 10B, 11B respectively. It can be observed that the selected feature wavelengths are distributed near the feature peaks. The feature wavelengths selected by the RF algorithm for GOS, FOS, Ca, and Vc achieved a reduction of 97.43% compared to the original full wavelengths. The RF algorithm has demonstrated effective feature wavelength extraction performance. The advantages of RF algorithm have also been verified in other literature, which is consistent with the conclusion of this study (29).

PLSR prediction models for GOS, FOS, Ca, and Vc in infant formula milk were established based on feature wavelength extraction using the CARS and RF algorithms (Table 2). Compared to the full spectral range PLSR models, both the CARS-PLSR models and RF-PLSR models showed improved prediction performance for the four nutritional components. Considering the comprehensive evaluation of Rc and Rp, the prediction performance of the CARS-PLSR models was superior. The CARS-PLSR models for GOS, FOS, Ca, and Vc showed an increase in Rc by 0.0998, 0.0447, 0.0351, and 0.07, a decrease in RMSEC by 0.1146, 0.0406, 0.0055, and 0.001, an increase in Rp by 0.1362, 0.0521, 0.0440, and 0.0799, and a decrease in RMSEP by 0.1201, 0.0302, 0.0074, and 0.0071, respectively. This indicates that the CARS algorithm effectively reduces the interference of irrelevant spectral variables in modeling and improves the performance of the prediction models (28).

Compared to the full spectral range PLS models, the CARS-SVR models and RF-SVR models showed significant improvements in the prediction performance for GOS, FOS, Ca, and Vc (Table 3). Considering the comprehensive evaluation of Rc and Rp, the prediction performance of the CARS-SVR models for all four nutritional components was superior to the RF-SVR models and CARS-PLS models. The CARS-SVR models showed an increase in Rc by 0.1480, 0.1104, 0.1187, and 0.3722, a decrease in RMSEC by 0.1267, 0.1063, 0.0193, and 0.0127, an increase in Rp by 0.1923, 0.0951, 0.0869, and 0.3502, and a decrease in RMSEP by 0.1857, 0.0616, 0.0007, and 0.0132 for GOS, FOS, Ca, and Vc, respectively.

After establishing the SVR models based on feature wavelength extraction, it was found that the SVR models outperformed the PLSR models in both the calibration and prediction sets. This improvement can be attributed to the complex composition of infant formula milk powder, which contains multiple nutritional components. The interactions between different functional groups and absorption peaks of different categories contribute to the existence of complex nonlinear relationships between near-infrared spectroscopic data and the content of micronutrients in infant formula milk powder. The ability of PLSR to handle nonlinearity is significantly inferior to SVR. SVR, with its core utilization of nonlinear kernel functions, effectively enhances the correlation between spectroscopic data and the physicochemical content of the components (33, 34, 38).

Therefore, utilizing feature wavelength extraction algorithms and establishing nonlinear SVR models is a more effective approach for the rapid detection of GOS, FOS, Ca, and Vc contents in infant formula milk powder. The comparison between the predicted values and true values of the CARS-SVR models for the four nutritional components is illustrated in Figures 12, 13, 14, 15. It can be observed from the figures that the deviation between the predicted and true values is low, indicating good calibration and prediction performance (39). To meet the requirements of online detection and optimization control in the milk powder production process, future research can focus on expanding the range of sample content, improving the applicability of the models, and investigating the feasibility of this method for rapid prediction in liquid milk powder ingredients (40–43). These efforts will provide valuable insights for online optimization control.