A total of 210 serum samples were collected from a hospital, the clinical actual values of serum urea nitrogen of the samples were obtained. The SUN values ranged from 2.1 to 41.6 (mmol∙L⁻¹), and the mean and standard deviation (SD) were 8.0 and 7.0 (mmol∙L⁻¹), respectively. Since the serum samples were collected and used in this study, the informed consent of all individual participants was obtained. The experiment was carried out according to relevant laws and institutional guidelines and approved by local medical institutions, which obtained the informed consent of all participants.

The instrument was an XDS Rapid Content™ Liquid Grating Spectrometer (FOSS, Denmark) equipped with a 2 mm. The spectra spanned 780 to 2498 nm with a 2 nm wavelength gap, including the overall NIR region. The detectors on wavebands of 780 - 1100 and 1100 - 2498 nm were Si and PbS detectors, respectively. Every sample was measured thrice. And the average spectrum of each sample was calculated and used for modeling. The spectra were measured at 25˚C ± 1˚C and 46% ± 1% relative humidity.

The 75 samples were randomly selected from 210 samples as independent validation set (not involved in modeling). The remaining 135 samples were used as the modeling set, which was further divided into calibration (80 samples) and prediction (55 samples) sets for 10 times to achieve the parameter stability. The root-mean-square errors (SEP) and correlation coefficients (R_P) for prediction were determined for each partition. The mean values (SEP_Ave, R_P,Ave) and standard deviations (SEP_SD, R_P,SD) for all the partitions were further determined, respectively. The comprehensive indicator SEP⁺ = SEP_Ave + SEP_SD was used to select the parameters with stability. The SEP⁺ takes the modeling prediction accuracy (SEP_Ave) and stability (SEP_SD) both into account. The selected models were validated using the validation set. The corresponding SEP, R_P, and ratio of performance to deviation (RPD) were further determined, respectively. R P D = C S D S E P , where C_SD was SD of actual values for the 75 samples (Mean: 8.6 mmol∙L⁻¹; SD: 7.8 mmol∙L⁻¹).

The NDF algorithm uses the symmetrical window of wavelengths to perform moving average smoothing on spectra firstly. The number of wavelengths in the smoothing-window is called the number of smoothing points (s, odd), which is set to s = 1 , 3 , ⋯ , S ; S can take the maximum odd number of the total number of wavelengths (N₀) in the entire spectral waveband. Since the low correlation, it is unreasonable to use a too wide smoothing-window. In this study, the upper limit S was set as 99. In addition, when s = 1, the moving average smoothing cannot be performed for that the original spectra were not pretreated.

The absorbance of the k^th wavelength was x_k. And the absorbance of the smoothing-window center x_k were x_i, i ∈ [ k − s − 1 2 , k + s − 1 2 ] . The smoothing value of x_k was as follows:

x ˜ k = ∑ i = k − s − 1 2 k + s − 1 2 x i s (1)

It’s worth noting that for the leftmost s − 1 2 and rightmost wavelengths, the symmetrically smoothing cannot be performed. Based on an idea of data continuity, for the leftmost s − 1 2 wavelengths, the smoothing values of the absorbance x_k were as follows:

x ˜ k = ∑ i = 1 k + s − 1 2 x i k + s − 1 2 , k = 1 , 2 , ⋯ , s − 1 2 (2)

For the rightmost s − 1 2 wavelengths, the smoothing method was similar and omitted.

Difference derivation: The spectra pretreated by moving average smoothing were then used for derivation. The 1^st derivative of absorbance was calculated using the center difference method. Since the NIR spectra are relatively flat and the spectral resolution of different objects is usually different, the original spectral data gap is not necessarily suitable for the differential gap of derivative. The Norris derivative uses variable number of wavelength gaps as the number of differential gaps (g) for the derivative, g = 1 , 2 , ⋯ , G . Large g is unreasonable due to the correlation is low. In this study, the upper limit G was set as 50.

For x_k, the 1^st derivative of absorbance was calculated using the following center difference:

D x k = x ˜ k + g − x ˜ k − g 2 g (3)

Similarly, for the leftmost and the rightmost g wavelengths respectively, the center difference cannot be performed. Based on the idea of data continuity, for the g leftmost wavelengths, the 1^st derivative value of the absorbance was calculated by the forward difference method as follows:

D x k = x ˜ k + g − x ˜ k g , k = 1 , 2 , ⋯ , g (4)

For the rightmost g wavelengths, the calculation method for 1^st derivative value of the absorbance was similar and omitted.

The 2^nd derivative value can be obtained by derivation on the basis of the above 1^st derivative value, and the process was not described again.

Optimization: Considering that the absolute value of the derivative of the 3^rd or more is small and the spectral information is low, it is generally not recommended to use the derivative of the 3^rd or more. In this study, the derivative order is set as d = 0, 1, 2. In particular, when d = 0, only the previous moving average smoothing was required.

As above, based on parameters combinations (d, s, g), d = 0, 1, 2; s = 1 , 3 , ⋯ , 99 ; g = 1 , 2 , ⋯ , 50 , a total of 5050 NDF modes were obtained. They were used to pretreat the sample spectra separately. The corrected spectra were then used to establish PLS models, named Norris-PLS models. Finally, according to the predicted effect, the optimal parameters were preferred as follows:

S E P ∗ + = min d ∈ { 0 , 1 , 2 } s ∈ { 1 , 3 , ⋯ , 99 } g ∈ { 1 , 2 , ⋯ , 50 } S E P ( d , s , g ) (5)

The s and the g are both important parameters of NDF. When d = 0, there is only one variable parameter s. The corresponding prediction error is SEP⁺(0, s), s = 1 , 3 , ⋯ , 99 . When d = 1, 2, there are two variable parameters (s, g). The corresponding single-parameter local optimal models were as follows:

S E P + ( d , s ) = m i n g ∈ { 1 , 2 , ⋯ , 50 } S E P + ( d , s , g ) , d = 1 , 2 ; s = 1 , 3 , ⋯ , 9 9 (6)

S E P + ( d , g ) = m i n s ∈ { 1 , 3 , ⋯ , 99 } S E P + ( d , s , g ) , d = 1 , 2 ; g = 1 , 2 , ⋯ , 5 0 (7)

The NIR spectra of the 210 serum samples in the entire scanning region (780 - 2498 nm) are illustrated in “Figure 1(a)”. As comparison, the PLS model in the whole region (780 - 2498 nm) without spectral pretreatment was established. The modeling effect (SEP_Ave, R_P,Ave, SEP_SD, R_P,SD, and SEP⁺) are summarized in “Table 1”. The results showed that high prediction error (SEP⁺ = 7.07 mmol∙L⁻¹) and low correlation (R_P = 0.535).

All Norris-PLS models corresponding to 5050 NDF modes also were established. For different derivative order, the optimal SEP⁺ corresponding to each single-parameter (number of smoothing points s or number of differential gaps g) are shown in “Figure 2”. It can be observed that the global optimal NDF mode was 2^nd derivative, 33 smoothing points and 15 differential gaps (d = 2, s = 33 and g = 15). The corresponding SEP⁺ and R_P were obviously improved to 2.62 mmol∙L⁻¹ and 0.930, respectively. In particular, the prediction error (SEP⁺) is only 37% of the original. From “Figure 2”, it is observed that the prediction effects corresponding to different parameters (d, s, g) are significantly different. Thus, the ergodic choice of the parameters was significantly necessary to achieve

better modeling performance. The prediction effects of the global optimal Norris-PLS model are also summarized in “Table 1”. The spectra used the optimal NDF mode (d = 2, s = 33, g = 15) are shown in “Figure 1(b)”.

In fact, there was significant saturate absorbance around 1900nm and 2400 nm in raw spectra, see in “Figure 1(a)”. They introduce noise and have an impact on modeling. Therefore, the absorbance higher than 3 (corresponded 99.9% absorption rate) was further excluded, and the remaining was 780 - 1880 & 2082 - 2344 nm. On the basis of the NDF spectra, the PLS model was established in the unsaturated region (780 - 1880 & 2082 - 2344 nm). The modeling effect is also summarized in Table 1. The SEP⁺ and was R_P further improved to 1.87 mmol∙L⁻¹ and 0.966, respectively. Among then, the SEP⁺ further declined 29%.

Therefore, eliminating the interference waveband with high-absorption does help to improve the modeling effect.

The 75 validation samples not incorporated in modeling were used to evaluate the Norris-PLS model that were established in the unsaturated region (780 - 1880 & 2082 - 2344 nm) with the optimal NDF mode. The PLS regression coefficients were determined using the spectra and actual SUN values of all modeling samples. The predicted values were then determined using the resulting regression coefficients and the spectra of the validation samples.

For the model, the relationship between the predicted and actual SUN values is shown in “Figure 3”. The evaluation values for validation (SEP, R_P, and RPD) are 1.66 mmol∙L⁻¹, 0.977 and 4.7, respectively. This model has achieved good prediction effect. The prediction values were close to the actual values with high precision and correlation. The results showed that the prediction effects corresponding to different parameters were significantly different. The parameters can’t be pre-set by experience. The global optimization of the parameters was significantly necessary to achieve better modeling performance.