Near infrared (NIR) spectroscopy is a widely used analytical method with the advantages of rapidness,non-destructiveness, non-pretreatment,and cost-effectiveness ^{[1, 2, 3, 4, 5, 6, 7, 8, 9]}. In routine NIR analyses,the spectra should be pretreated to enhance informative signals of the interested components and reduce uninformative signals as much as possible in advance of modeling. Savitzky- Golay differentiation ^{[10, 11, 12]} is a commonly used spectral pretreatment method that can eliminate baseline interference and improve spectral resolution. The ordinary differentiation method used in NIR is integral order Savitzky-Golay differentiation (IOSGD),and in practice 1st and 2nd derivatives often exhibit significant improvement over the calibration models. Recently, due to the development of fractional calculus,fractional order differentiation is becoming more and more prominent in many fields of applied sciences especially in signal pretreatments [13- 18]. Compared with IOSGD,fractional order Savitzky-Golay differentiation (FOSGD) can extract more details from signals. Furthermore,in contrast to other fractional order differentiation computation methods including Fourier transformation ^{[19, 20]} and wavelet transformation ^{[21, 22]},the FOSGD can apply a window to extract the local details of signals. Meanwhile,FOSGD is easy to apply,since it only constructs a band matrix for differentiation ^{[23, 24]}. In this paper,we tried to apply FOSGD to process NIR spectra in order to improve the multivariate calibration model of NIR spectral analysis. 2. Methods 2.1. Definition of fraction order differentiation

A polynomial function f(j),which is applied to fit in FOSGD,is a linear combination of power functions that could be expressed as:

In this paper,Riemann-Liouville differentiation definition is utilized to generate the fractional order differentiator. The definition of Riemann-Liouville differentiation is shown as follows:

Here,r is differentiation order,l is a natural number with l-1n ,Eq. (2) can be simplified as:

Here,Γ(.) is Gamma function,which is the generation of factorial function. When fixing n and r,(Γ(n+1))/(Γ(n+1-r)) could be obtained. Thus the fractional order differentiation is another power function multiplying a constant (Γ(n+1))/(Γ(n+1-r)). Specially,in the case of n=0,the polynomial function f(x)=x⁰ = 1,then:

Thus,unlike integral order differentiation,the fractional order differentiation of a constant is not zero. 2.2. Savitzky-Golay fractional order differentiation

Same as IOSGD,FOSGD also needs to obtain the coefficients of polynomial function by fitting the spectral intensities with a polynomial function. 2.2.1. Polynomial function

With a polynomial function defined in Eq. (1),a window of data withm(m>n) points can be applied to fit,andmequations could be obtained that can be rewritten as

After obtaining the coefficients of polynomial function,the FOSGD can be computed with Eq. (3). For thejth point in the window,the value of f(j) can be calculated with the estimated coefficients

Then by combining Eqs. (3) and (7),the differentiation ofrth order could be obtained:

As only the middle point in the window is concerned,with the middle pointjand the related d_j^T Savitzky-Golay differentiation of rth order could be conveniently calculated with the Eq. (9).

In fact,the Eq. (9) is a generalized formula to calculate Savitzky- Golay derivatives including integral ones. Additionally,the conventional Savitzky-Golay smoothing is also a special case of the differentiation calculation of Eq. (9) at the order of zero. The detailed examples of using Eq. (9) for smoothing and integral order differentiation are shown in supplementary materials. 3. Datasets 3.1. Diesel dataset

The diesel dataset downloaded from the Internet at http:// www.eigenvector.com/data/SWRI/index.html contained 401 NIR spectral points with a range from 750 nm to 1550 nm. Two properties of diesel including density with 395 samples and cetane number (CN) with 381 samples,respectively,were set as vectors of y. For each property,the samples were sorted by y values,then the third one in each contiguous five samples were set aside as the prediction set while the remaining samples as the calibration set. Thus for the properties of density and cetane number,the size ratios of the calibration set to the prediction set were 316-79 and 305-76,respectively. 3.2. Wheat dataset

The wheat dataset with 150 data points (1004-2494 nm) was downloaded from the Internet at http://www.wiley.com/legacy/ wileychi/chemometrics/datasets.html. The values of the protein concentration with 183 samples and the hardness with 180 samples,respectively,were chosen and set as vectors of y. Similar to the diesel dataset,for each property,one fifth samples were selected as the prediction set whereas the remaining samples as the calibration set. Thus,the calibration sets and prediction sets were separated in proportions of 147-36 and 144-36,respectively. 3.3. Corn dataset

The corn dataset (m5) at the range of 1100-2498 nm (700 points) was downloaded from the Internet at http:// www.eigenvector.com/data/Corn/index.html. The oil and starch concentrations both containing 80 samples were set as vectors of y. Hence,for the two properties,the sizes of the calibration set and the prediction set were both in a ratio of 64:16 that was constructed with the method described in Section 3.1. 4. Results and discussion 4.1. Results of the diesel dataset 4.1.1. Modeling to the property of density

In order to investigate the utility of FOSGD,PLS models established using differentrvalues (with an interval of 0.01) were evaluated in point of root mean square error of cross validation (RMSECV),where a 5-fold cross validation was used and the number of latent variable s(LVs) was set as 7 because the RMSECV values at LVs number of 7 were the smallest in most cases. Meanwhile,the effects ofp and mon RMSECV were also investigated. In the study it is found that the case of n= 2 and m= 9 often yields smaller RMSECV values,thus,for simplicity,n andmwere set as 2 and 9,respectively. After optimizingnandm, the value ofr,which is a key factor,should also be optimized and the corresponding RMSECV values are shown in Fig. 1.

Fig. 1

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Fig. 1. Plots of RMSECVvs.r using fractional order Savitzky-Golay differentiation (FOSGD) (n=2 and m= 9).

From Fig. 1,one can see that at the point ofr= 1.8,the RMSECV reaches the lowest value,therefore it can be concluded that FOSGD produces better results with smaller RMSECV than IOSGD. The reason may be that the density of diesel is related to many components rather than only one or a few components. Absorption bands of a number of components must be severely overlapped, thus the relationship between density and spectra must be very complex. For IOSGD,smallr valuescannot identify the informative signals related to the density from the uninformative signals, whereas,the overlarge values ofrcan impair both informative and uninformative spectra. However,FOSGD can use a decimal number between two adjacent integral numbers as the order,which may provide a better chance to identify (or resolve) informative and uninformative signals due to density than IOSGD. Moreover, fractional order calculus is always utilized to build models for the density of liquid and semisolid ^{[25, 26]}.

In order to further compare model performance between FOSGD and IOSGD,an independent prediction set was used to calculate the root mean error of the prediction (RMSEP) that was listed in Table 1.]

Table 1

Table 1 Performance comparison between FOSGD and IOSGD for the diesel dataset

Table 1 Performance comparison between FOSGD and IOSGD for the diesel dataset

Table 1 shows that the values of RMSECV and RMSEP at fractional order are both smaller than those at integral orders. The RMSEP being 0.00228 at r= 1.8 is the lowest. The results further confirm that for the density of diesel,FOSGD can achieve small prediction errors compare to IOSGD. 4.1.2. Modeling to the property of cetane number

Similar to density,for the cetane number of diesel,the number of latent variable was selected as nine,and values ofnandmwere set as 2 and 11. A series of calculations with varying values ofr from 0 to 2 revealed that the PLS model shows the best results at r= 0.85. The results are given in Table 1. From Table 1 it is clear that for the cetane property,fractional order differentiation still produces smaller calibration and prediction errors than integral order differentiation. 4.2. Results of the wheat dataset

With the same method to that used for the diesel dataset,the parameters ofn,mandrwere optimized to be 3,11 and 2.8 for the property of hardness,respectively,and to be 2,9 and 1.02, respectively for protein concentration. The modeling results obtained with the optimized parameters are shown in Table 2.

Table 2

Table 2 Performance comparison between FOSGD and IOSGD for the wheat dataset

Table 2 Performance comparison between FOSGD and IOSGD for the wheat dataset

Table 2 shows that,for hardness as the property,the values of RMSECV and RMSEPusing fractional order of 2.8 are the smallest, but they are quite close to those using an integral number of three. Whereas,for protein concentration as the property,the optimal fractional order with the lowest RMSECV appears at r= 1.02,but it is close to integer of one very much,and the RMSECV values corresponding tor= 1.02 andr= 1 are quite close. Meanwhile, RMSEP atr= 1 shows the smaller value comparing with any other fractional orders. Therefore,for the property of protein concentration we think that IOSGD produces the better results than FOSGD. 4.3. Results of the corn dataset

In the corn dataset,the concentrations of oil and starch were considered using FOSGD as the spectral pretreatment method. Similarly,parameters ofnandmof oil were firstly optimized as 2 and 13,respectively,for the oil concentration while as 2 and 7, respectively,for the starch concentration. The optimized parameters and the modeling results are shown in Table 3.

Table 3

Table 3 Performance comparison between FOSGD and IOSGD for the corn dataset

Table 3 Performance comparison between FOSGD and IOSGD for the corn dataset

The results listed in Table 3 are similar to those of protein concentration of the wheat dataset. For both properties of oil and starch concentrations in the corn dataset,the optimal orders are integer numbers of 2 and 1,respectively,althoughr values corresponding to the lowest RMSECV and RMSEP are 1.98 and 1.01, which are quite close to the integer numbers.

In summary,FOSGD provides more choices to pretreat NIR spectra with differentiation than the ordinary derivatives,and sometimes FOSGD clearly shows better performance of calibration model than that of IOSGD. Comparing models of the three datasets, it is very interesting that if the property of interest is a physical property,such as density,hardness,cetane number,fractional order Savitzki-Golay derivatives should be used,while if the property is mainly related to chemical compositions,such as content or concentration of chemical components,integral order Savitzki- Golay derivatives are often better. The reason for these observations is unclear. We hypothesize that when modeling to chemical components,the related signals (informative signals) in the NIR spectra to the chemical components is relatively simple and could be identified with IOSGD,while to the physical properties,the informative signals are severely overlapped with the uninformative ones and could not be identified easily,and FOSGD may offer a better chance to resolve them with a fractional number as the order between two adjacent integral numbers (see Section 4.1.1). 5. Conclusion

Fractional order Savitzky-Golay differentiation (FOSGD) is the generalization of integral order Savitzky-Golay differentiation (IOSGD) while IOSGD is a special case of FOSGD. The FOSGD can also be used to pretreat NIR spectra. Fractional order differentiation computation of the NIR spectra often improves the performance of the PLS model with smaller RMSECV and RMSEP numbers than integral order ones,especially for physical properties,such as density,cetane number and hardness. Furthermore,FOSGD can be easily computed with the definition of Riemann-Liouville differentiation and applied as conveniently as IOSGD. Thus,FOSGD has strong application potentials in spectral analyses.

This work was supported by Science and Technology Commission of Shanghai Municipality (No. 14142201400). Appendix A. Supplementary data Supplementary data associated with this article can be found,in the online version,at http://dx.doi.org/10.1016/j.cclet. 2014.10.023.