Fourier based partial least squares algorithm: new insight into influence of spectral shift in &#x201C;frequency domain&#x201D;

H. Y. Bian; Y. L. Zhang; W. R. Gao; J. Gao

doi:10.1364/OE.27.002926

1. Introduction

Partial least squares discriminant analysis is a traditional chemometrics analysis method [1–4]. Recent years, it is combined with spectroscopic technology and stands out in the analysis of chemical components because of its advantages of non-invasive, non-contact and high sensitivity [5–8]. In analytical chemistry, PLS models are nowadays widely used to predict analyte concentrations and discriminate similar analytes from spectral measurements [9,10]. For example, C. J. Meunier used PLS model to discriminate Hydrogen peroxide (H₂O₂) and _{$Δ pH$}signals [11]. H. Khajehsharifi used PLS to simultaneously determine the concentration of ascorbic acid, dopamine and uric acid and satisfactory results were achieved [12]. In addition, it has also been widely used by chemometricians in many other applications such as forensic science [13,14], food safety [15,16], clinical diagnosing [17,18] and public safety [19,20]. The most common approach of the detection is: firstly, a series of spectra is first measured of the sample; secondly, signal pre-processing algorithm is used to select the proper training set; finally, PLS model is built with the training data and is validated by cross-validation.

Even though such great advances have been achieved in lab experiments by many research groups, the practical application of this approach is not satisfied because of spectral shift induced by the change of the environment, the shift of the laser wavelength and the performance of the spectroscopy, which has been recognized by many researchers [21-24]. To decrease the influence of spectral shift, the simplest method is to strictly control the environment such as the temperature and humidity and using high quality laser. However, these requirements mean the increasing cost and inconvenience for the end user.

To utilize the full power of spectroscopic analyzer, many efforts have been made to avoid the influence of the spectral shift. Among these reported methods, the methods can be divided into two kinds: (i) to calibrate the original spectra; Brown and Stoyanova proposed using principal-component analysis (PCA) as an automatic method of spectral quantification to determine the frequency misalignment and phase misadjustment in a single resonant peak across a series of spectra [25]. Their results demonstrated that it has been possible to use PCA methodology to determine and correct the spectra. Westad et al. proposed using Horn and Schunck algorithm to give estimates that can move the spectra with respect to some selected reference spectrum back to reference position [26]. (ii) to derived the relationship between the spectral shift and prediction value and then compensate the prediction value; Bian et al. analyzed the relationship between the spectral shift and the prediction errors and the formula was derived [27]. When the spectral shift is known, the prediction error can be calculated with the formula. The disadvantages of these methods include: (i) all these methods can only decrease the influence of spectral shift and most of them will no longer valid when the spectral shift is large enough; (ii) the parameters to decrease the influence of spectral shift need to be modified according to the value of spectral shift which is difficult to end user; (iii) it is trouble and time-consuming to select the spectra without spectral shift as the training data. All these disadvantages are fatal and unwanted to end users. Actually, a perfect instrument for end user should be easily operated for everyone who has no background about the principle of the instrument.

Here, we investigated the possibility to remove the term of spectral shift in the spectra completely using Fourier partial least squares method. The spectra were first done Fourier transform and as a result, the spectra shift in “time domain” is transferred to the change of the phase in “frequency domain” which means that the modules of the Fourier transformed spectra in “frequency domain” have no relationship to the spectral shift. When the modules are used to build the PLS model, the model will be insensitive to the spectral shift. Different from all the methods until now, the method proposed here provides a new insight to the influence of spectral shift instead of compensation of the spectra or the prediction error. The discrimination of blood species based on Raman spectroscopy was used to demonstrate the effectiveness of FPLS. By validating 33 spectra with different spectral shift, the results indicated that FPLS compared with PLS actually removed the term of spectral shift completely and the algorithm is really simple.

2. Principle

The experimental setup and the preparation of the blood samples are similar to that in our previous work [27,28]. 31 Raman spectra of human blood and 41 Raman spectra of nonhuman blood originating from chicken, duck, dove, rabbit, rat, monkey, cat, dog, sheep and pig were used to build the PLS model. 8 Raman spectra of human blood and 25 Raman spectra of nonhuman blood originating from the same species were used to validate the PLS model. The Raman spectra of the same sample with a spectral shift of 2.82, 6.36, 9.18 and 19.77 cm⁻¹ were obtained and were used to demonstrate that FPLS is insensitive to spectral shift.

The Raman shift _$k$ is defined as

k = \frac{1}{λ_{e x c i t a t i o n}} - \frac{1}{λ}

where _{$λ_{e x c i t a t i o n}$}is the wavelength of the laser, _$λ$is the wavelength of the scattered light.

From the above equation, we can see if the laser wavelength is changed, the wavelength of scattered light will change accordingly since Raman shift _$k$ is a constant value for a given functional group. For a given spectroscopy, each pixel is corresponding to a particular wavelength, as a result, the Raman peaks will have a shift in pixel number domain as shown in Fig. 1. When the spectral shift is smaller, the error of the prediction value can be calculated with the following equation [27]

Fig. 1 The Raman spectra of blood samples with and without spectral shift.

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\begin{array}{l} Δ y = (a (k_{1}) I^{'} (k_{1}) + a (k_{2}) I^{'} (k_{2}) + ... + a (k_{n}) I^{'} (k_{n})) δ k \\ + (a (k_{1}) \frac{I^{'} (k_{1})}{2!} + a (k_{2}) \frac{I^{'} (k_{2})}{2!} + ... + a (k_{n}) \frac{I^{'} (k_{n})}{2!}) δ k^{2} \\ + ... \\ + (a (k_{1}) \frac{I^{'} (k_{1})}{n!} + a (k_{2}) \frac{I^{'} (k_{2})}{n!} + ... + a (k_{n}) \frac{I^{'} (k_{n})}{n!}) δ k^{n} \end{array}

However, when the spectral shift is large, the equation is not suitable to calculate the error. As a result, these Raman spectra with large spectral shift should be treated as outlier data for the PLS model. This means the repeat measurements should be performed after the maintenance of Raman spectroscopy or the environment. To make the measurements easier and time-saving, here, Fourier based PLS method is proposed.

Assume that the Raman spectrum for blood is _{$I (k)$}, the actual Raman spectrum sampled by CCD is

I_{s a m p l e d} (k) = I (k) \otimes Π_{δ ξ / 2} (ξ)

where _$Π$ is the rectangular function (Rect function) which is defined as _{$Π_{δ ξ / 2} (ξ) = {\begin{cases} 1, - δ ξ / 2 < ξ < δ ξ / 2 \\ 0, others \end{cases}$},_{$δ ξ$} is the size of the pixel of CCD._$ξ$ is the coordinate in the pixel direction.

To remove the term of the spectral shift, the Fourier transform of the Raman spectrum is first performed. Assume the Fourier transform of the theoretical Raman spectrum _{$I (k)$} is _{$G (ω)$} and considering that the Fourier transform of Rect function is Sinc function, the Fourier transform of the sampled Raman spectrum can be calculated as

\begin{array}{l} F F T (I_{s a m p l e d} (k)) = F F T (I (k) \otimes Π_{δ ξ / 2} (ξ)) \\ = F F T (I (k)) \times F F T (Π_{δ ξ / 2} (ξ)) \\ = G (ω) \times \sin c (ω δ ξ / 2) \end{array}

The module of the Fourier transformed Raman spectrum is

\begin{array}{l} | F F T (I_{s a m p l e d} (k)) | = | G (ω) \times \sin c (ω δ ξ / 2) | \\ = A (ω) \times | \sin c (ω δ ξ / 2) | \end{array}

where _{$A (ω)$} is the amplitude-frequency characteristic

For the Raman spectrum with a spectral shift _{$δ k$}, the module of the Fourier transform is

\begin{array}{l} | F F T (I_{s a m p l e d} (k + δ k)) | = | F F T (I (k + δ k) \otimes Π_{δ ξ / 2} (ξ)) | \\ = | F F T (I (k + δ k)) \times F F T (Π_{δ ξ / 2} (ξ)) | \\ = | G (ω) \exp (- j ω δ k) \times \sin c (ω δ ξ / 2) | \\ = A (ω) \times | \exp (- j ω δ k) | \times | \sin c (ω δ ξ / 2) | \\ = A (ω) \times | \sin c (ω δ ξ / 2) | \end{array}

In Eq. (6), we find that the term of spectral shift is removed by calculating the module. Compared Eq. (5) and (6), we can find that

| F F T (I (k)) | = | F F T (I (k + δ k)) | = A (ω) \times | \sin c (ω δ ξ / 2) |

It can be seen that the modules of the Fourier transform of Raman spectra has no relationship to the spectral shift. To demonstrate the result, we do Fourier transform of the Raman spectra in Fig. 1 and the results were shown in Fig. 2. The three lines in Fig. 2 were completely overlapped which demonstrated the accuracy of Eq. (7). However, there is small difference of the amplitude of the Fourier transformed Raman spectra. The main reason is the non-uniform sampling in the wave-number domain which means the spectral shift _{$δ k$} is not a constant value for each Raman peaks. Assume that the Raman spectrum has n Raman peaks, then

Fig. 2 The modules of the Fourier transform of the Raman spectra in Fig. 1.

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I_{s a m p l e d} (k) = \sum_{i = 1}^{n} I_{s a m p l e d} (k_{i})

Thus, Eq. (5) should be revised as

\begin{array}{l} | F F T (I_{s a m p l e d} (k)) | = | \sum_{i = 1}^{n} G_{i} (ω) \times \sin c (ω δ ξ / 2) | \\ = | \sum_{i = 1}^{n} G_{i} (ω) | \times | \sin c (ω δ ξ / 2) | \end{array}

For the Raman spectrum with spectral shift, Eq. (6) should be revised as

\begin{array}{l} | F F T (I_{s a m p l e d}^{'} (k)) | = | F F T (\sum_{i = 1}^{n} I_{s a m p l e d} (k_{i} + δ k_{i}) \otimes Π_{δ ξ / 2} (ξ)) | \\ = | \sum_{i = 1}^{n} (G_{i} (ω) \exp (- j ω δ k_{i})) \times \sin c (ω δ ξ / 2) | \\ = | \sum_{i = 1}^{n} (G_{i} (ω) \exp (- j ω δ k_{i})) | \times | \sin c (ω δ ξ / 2) | \end{array}

When _{$δ k_{i} = constant$} or _{$δ k_{i} = 0$}, _{$\sum_{i = 1}^{n} G_{i} (ω) = \sum_{i = 1}^{n} (G_{i} (ω) \exp (- j ω δ k_{i}))$}which means that the value calculated by Eq. (9) and (10) is equal. Even though the amplitude of the Fourier transformed Raman spectrum is not equal for the Raman spectral with and without spectral shift, the influence of the non-uniform sampling to PLS model is small because the amplitude will be normalized first and the difference of spectral shift induced by non-uniform sampling is small.

3. Results

To avoid the influence of different measurement which may be induced by the temperature, the mixture of the blood or the focus position, we use the same Raman spectra to validate the performance of FPLS model. The spectral shift was introduced by shifting the pixel mathematically. Figure 3 presented the results obtained by PLS and FPLS for the same Raman spectra. The prediction value obtained by PLS distributed discretely and when the shift of the pixel is larger, the prediction value which is much larger than 1.5 is meaningless. However, the prediction value obtained by FPLS was distributed around 0.9 which means FPLS is insensitive to the shift of the pixel as analyzed in the theory section. The prediction value was not a constant value as analyzed because of the influence of noise. Nevertheless, the influence of the noise can be neglected even when the shift of the pixel is 100. In practical application, the Raman spectra of different measurements are more complex.

Fig. 3 The prediction value. (a) is obtained by PLS when the Raman spectrum was shifted the pixel mathematically; (b) is obtained by FPLS when the Raman spectrum was shifted the pixel mathematically.

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To demonstrate the effectiveness of FPLS in the practical application, we measured the Raman spectra of the same blood samples with different spectral shift. The spectral shift was realized by displacing the CCD in the wavelength direction which can be easily controlled using a motor. The Raman spectra with a spectral shift of 2.82, 6.36, 9.18 and 19.77 cm⁻¹ were obtained. The training data set included 72 Raman spectra without spectral shift. The prediction value for the Raman spectral with different spectral shift was shown in Fig. 4. The circular sign in Fig. 4 represented human blood, while the asterisk sign represented nonhuman blood. Since in the model 1 represents human blood and 2 represents nonhuman blood, the prediction value of human and nonhuman blood should be distributed around 1 and 2. Usually, to achieve the higher discrimination accuracy, algorithm such as three sigma [29] and posterioti probability [30–32] can be used to establish the threshold value to discriminate blood species. In this paper, to simply explain the problem, the threshold value was set as 1.5 [33,34]. The discrimination accuracy of the PLS model is 100% for the Raman spectra without spectral shift. When small spectral shift occurred (Fig. 4(b)), the prediction value was still distributed around 1 and 2. But the discrimination accuracy decreased and this small spectral shift was usually inconspicuous for the end user. This is a disaster for them because they will obtain the wrong results while they are not aware of it. Compared with small spectral shift, the large spectral shift can be easily observed. Since the prediction value of the Raman spectra with large spectral shift was obviously larger than 1 and 2 for human and nonhuman blood. The remedial method such as compensation or repeat measurement can be done to achieve the right results. However, no matter the compensation or the repeat measurement is undesired for the end user. Figure 5 is the average prediction value for the Raman spectra with different spectral shift. In Fig. 3, the prediction value became smaller when the shift of the pixel (in the range between 1 and 10) increased (each pixel represents 2.6 cm⁻¹ if the nonlinearity between the Raman shift and pixel number is not considered). While in Fig. 5, the prediction value became larger when the spectral shift increased. The difference was induced by the spectral shift direction (for Fig. 3, the wavelength was shifted in the direction of short wavelength, while that for Fig. 5 was shifted in the direction of long wavelength).

Fig. 4 The prediction value obtained by PLS when the Raman spectra have a spectral shift. (a) shown prediction value of the Raman spectra without spectral shift; (b)-(e) were corresponding to prediction value of the Raman spectra with a shift of 2.82 cm⁻¹, 6.36 cm⁻¹, 9.18 cm⁻¹ and 19.77 cm^−1.

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Fig. 5 The average prediction value with different spectral shift when PLS model was used.

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The same training data set and validation data set were used to build FPLS and validate the FPLS model. The results were shown in Fig. 6. The prediction value for the Raman spectra with and without spectral shift were all distributed around 1 and 2. The discrimination accuracy for the Raman spectra without spectral shift was 100% which was the same as that of the PLS model. The discrimination accuracy is 94% for the Raman spectra with spectral shifts of both 2.82 and 6.36 cm⁻¹. For the Raman spectra with a spectral shift of 2.82 cm⁻¹, one human blood sample and one nonhuman blood sample were mis-discriminated while for the Raman spectra with a spectral shift of 6.36 cm⁻¹, two nonhuman blood samples were mis-discriminated. For the Raman spectra with a spectral shift of 9.18 cm⁻¹, only one nonhuman blood sample was mis-discriminated and the discrimination accuracy was 97%. For the Raman spectra with a spectral shift of 19.77 cm⁻¹, the discrimination accuracy was 100%. The difference for the results of different spectral shifts was mainly induced by different focus position and the homogeneity of the blood (the blood would gradually subside with the time) in the repeat measurements. Comparing the results in Fig. 4 with that in Fig. 6, it seems that spectral shift have greater influence on the prediction value than other factors such as noise, intensity or the homogeneity of the blood. Even though the discrimination accuracy decreases, it is still larger than 94% which may be acceptable in practical application. The discrimination accuracy can be improved by establishing the threshold value with three sigma method or using dual-model method to avoid the influence of the human-like nonhuman blood samples [28]. Figure 7 was the averaged prediction value, compared with that in Fig. 5, the averaged prediction value of FPLS was found more stable and not change with the spectral shift which means that FPLS method is insensitive to the spectral shift.

Fig. 6 The prediction value obtained by FPLS when the Raman spectra have a spectral shift. (a) shown prediction value of the Raman spectra without spectral shift; (b)-(e) were corresponding to prediction value of the Raman spectra with a shift of 2.82 cm⁻¹, 6.36 cm⁻¹, 9.18 cm⁻¹ and 19.77 cm⁻¹

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Fig. 7 The average prediction value with different spectral shift when FPLS model was used

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Figure 8 compared the results obtained by PLS and FPLS model when the Raman spectra in the training data set have spectral shift. PLS model cannot discriminate blood species any more, while FPLS model’s discrimination accuracy is 100%. The spectral shift in the training data set made the PLS model more complex. As the spectral shift is one of the loading factors, the Raman spectra would be predicted by the combination of the Raman peaks and spectral shift. In Fig. 8 the prediction value was distributed around 2, because the spectral shift in the training data set is the same as that in the validation set. However, in the practical application, the training data set is impossible to include all the spectral shift. The result may be worse than that obtained in Fig. 8.

Fig. 8 The prediction value. (a) is obtained by PLS when the Raman spectra in the training data set have spectral shift; (b) is obtained by FPLS model when the Raman spectra in the training data set have spectral shift

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4. Conclusion

A new approach aims to remove the term of spectral shift, instead of compensating the spectral shift, based on the combination of Fourier transform and PLS algorithm has been illustrated in this study. This method treated the influence of the spectral shift in a new direction-“frequency domain” which made the problem easier. To demonstrate the effectiveness of FPLS method, a total of 237 Raman spectra of blood originating from 105 human or nonhuman samples were collected. Among 237 Raman spectra, 132 Raman spectra have spectral shift including 33 Raman spectra with a spectral shift of 2.82 cm⁻¹, 6.36 cm⁻¹, 9.18 cm⁻¹ and 19.77 cm⁻¹. PLS and FPLS methods were compared in three aspects. First, the mathematically shifted Raman spectra which aim to decrease the influence of the noise and repeat measurements were predicted by the two models while the Raman spectra in the training data set had no spectral shift. The results indicated that FPLS method was insensitive to spectral shift; Second, the Raman spectra with a spectral shift of 2.82 cm⁻¹, 6.36 cm⁻¹, 9.18 cm⁻¹ and 19.77 cm⁻¹ induced by displacing CCD were predicted by the two models while the Raman spectra in the training data set had no spectral shift. The results indicated two facts: (1) the influence of the spectral shift for PLS models is more obvious compared with the factors such as noise and intensity which means that it is necessary to avoid the influence of the spectral shift; (2) the Fourier transform based on partial least squares is indeed insensitive to spectral shift in the validating data set; Third, the Raman spectra in training data set and validating data set both had a random spectral shift, and the results also indicated FPLS method was insensitive to spectral shift.

The three experiments demonstrated the effectiveness of FPLS method. Compared with other methods reported until now, FPLS method is simpler and can solve the influence of spectral shift once and for all by removing the term of spectral shift in “frequency domain”. This method is important for Raman or infrared spectroscopy in practical application and readers in many applications such as forensic science, food safety, drug testing, chemistry analysis or clinical diagnosis would be interested in this method.

Funding

National Key R&D Program of China (2018YFF01011104, 2016YFC1000701); National High Technology Research and Development Program of China (863 Program 2015AA021105); the Natural Science Foundation of Jiangsu Province (BK20180220); Key Project of Jiangsu Province (Grant No. BE2016090 and BE2016005-2); Jiangsu Postdoctoral Research Foundation (Grant No. 1701045B, 1701046B).

Acknowledgments

The authors gratefully acknowledge Y. B. Tian, J. Wang and N. Wang for the help of contacting the supplier of the blood samples and Wanrong Gao for improving the English of the manuscript.

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Fourier based partial least squares algorithm: new insight into influence of spectral shift in “frequency domain”

Abstract

1. Introduction

2. Principle

3. Results

4. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (8)

Equations (10)

Optics Express