Calibration in non-linear near infrared reflectance spectroscopy: a comparison of several methods

Analytica Chimica Acta 384 (1999) 207±214

Calibration in non-linear near infrared re¯ectance spectroscopy: a comparison of several methods M. Blanco*, J. Coello, H. Iturriaga, S. Maspoch, J. PageÁs Departament de QuõÂmica, Unitat de QuõÂmica AnalõÂtica, Facultat de CieÁncies, Universitat AutoÁnoma de Barcelona, E-08193 Bellaterra, Barcelona, Spain Received 17 July 1998; received in revised form 16 November 1998; accepted 16 November 1998

Abstract Principal component regression (PCR) and partial least-squares regression (PLSR) are the two calibration procedures most frequently used in quantitative applications of near infrared diffuse re¯ectance spectroscopy (NIRRS). Some systems, however, exhibit a non-linear relationship that neither methodology can model. Frequently, the main culprit of such nonlinearity is the multiplicative effect arising from non-uniform particle sizes or diameters in the samples. In this work, we tested various approaches to minimizing the non-linearity resulting from the multiplicative effect of differences in particle size or sample thickness, using the determination of linear density in acrylic ®bres as physical model. The approaches tested involve the prior linearizing of data by logarithmic conversion and/or the use of non-linear calibration systems; in this context, the results of applying stepwise polynomial PCR (SWP-PCR) and PLSR (SWP-PLSR), and those provided by a neural network based on the scores of the PCR model (PC-ANN), were compared. The PC-ANN approach was found to provide the best results with linear density data. On the other hand, the SWP-PLSR approach performed on par with the previous one when the variable was linearized by conversion of its values into decimal logarithms. # 1999 Elsevier Science B.V. All rights reserved. Keywords: Stepwise polynomial PLS; NIR spectroscopy; Arti®cial neural networks; Non-linearity

1. Introduction In recent years, qualitative and quantitative applications of near infrared spectroscopy (NIRS) in various chemical ®elds including the pharmaceutical [1,2], food [3,4] and textile industries [5,6] have grown dramatically. Such rapid expansion of the NIRS technique has largely been favoured by its simplicity, the expeditiousness with which spectra can be *Corresponding author. Fax: +34-3-5812477; e-mail: [email protected]

recorded and the ability to derive information from samples containing complex matrices. The NIR spectrum contains information about not only the chemical composition of the sample but also some physical properties such as particle size or crystallinity. This relationship to physical properties of the sample have made NIRS a suitable choice for quantifying not only chemical but also physical parameters. NIR spectra consist of uncharacteristic, highly overlapping, broad, low absorption bands that hindered their widespread use until multivariate calibration

0003-2670/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S0003-2670(98)00814-9

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methods for withdrawing their analytically relevant information were developed. Principal component regression (PCR) [7,8] and partial least-squares regression (PLSR) [9,10], are two of the calibration methods most frequently used in this context. Multivariate calibration techniques for NIRS assume a linear relationship between the target parameter and the intensity of spectral absorption bands; while a strictly linear relationship rarely exists, both PCR and PLSR provide correct results as long as deviations from linearity are not too large. With high non-linearity, however, both calibration techniques lead to substantial errors so an alternative chemometric tool must be used instead. Non-linearity in NIR spectra may arise from various factors, namely: deviations from the Lambert±Beer law, which are typical of highly absorbing samples; non-linear detector responses; interactions between analytes, etc. The most frequent source of non-linearity in NIR diffuse re¯ectance spectroscopy is non-compliance with Beer's law as a result of the multiplicative effect of differences in effective pathlength arising from differences in particle size in the sample. This type of non-linearity can be modelled by linearizing the original data, altering them with non-linear relations or using mathematical models capable of modelling a non-linear relationship. The last include locally weighted regression (LWR) [11±13] and multivariate adaptive regression splines (MARS) [14]. In addition, non-linearity here can be addressed by using polynomial or quadratic versions or PCR and PLS [15±20], as well as arti®cial neural networks (ANNs) [21±26]. The expressions ``polynomial PCR'' and ``polynomial PLS'' can be misleading as they apply equally to non-linear internal relations between the scores of matrices X and Y [27] and other cases where only transformed variables are introduced in matrix X [28] or quadratic polynomials are ®tted after the PCR or PLS model is constructed. In order to avoid potential confusion, Wold et al. [27] have distinguished the different existing polynomial versions of PCR and PLSR. Thus, they recommend the designations ``polynomial PCR'' and ``polynomial PLS'' to refer to those cases where a non-linear internal relation between matrices X and Y is established. In most reported cases, such a relation is a quadratic polynomial function and the model is referred to as ``quadratic PLS'' (QPLS).

On the other hand, as suggested by Gnanadesikan [28], non-linearity can be corrected by expanding matrix X with values of its variables, their squares and their binary products. Although this approach provides good results, it uses a signi®cantly larger number of variables, involves more cumbersome computations and provides data that are dif®cult to interpret. This type of regression is called ``linear quadratic PCR'' (LIN-QPCR) or ``linear quadratic PLS'' (LINQPLS) as it maintains a linear relationship between the scores of matrices X and Y. One other approach involves constructing a preliminary PCR or PLSR model and subsequently ®tting a polynomial by least squares from the scores, their squares and their binary products: y ˆ b0 ‡ b1  t1 ‡ b2  t2 ‡ b3  t12 ‡ b4  t22 ‡ b5  t1  t2 ‡


‡ e;

(1)

where ti are the scores of the calibration matrix for the PCR or PLS model, bi the regression coef®cients to be determined and e is the error term, which is ®tted by least-squares regression. As with LIN-QPLS, a linear internal relationship between the scores of matrices X and Y is maintained. Virtually all reported examples use the ®rst few principal components, usually not more than 4 [20]. Including additional PCs in the polynomial model complicates the establishment of the regression equation and does not lead to signi®cantly improved results. We used these PCR and PLS polynomial versions in this work. Polynomial terms were selected by using an ascending stepwise procedure; new terms were successively added if they resulted in a signi®cantly improved multiple correlation coef®cient for the samples in the calibration set. The combinations of the PCR and PLS models with the quadratic polynomial constructed from their scores were designated stepwise polynomial PCR (SWP-PCR) and stepwise polynomial PLS (SWP-PLS), according to whether PCR or PLS scores were used. Arti®cial neural networks (ANNs) are also among the most widely used mathematical algorithms for overcoming non-linearity. The networks are straightforward mathematical descriptions of what is currently known about the physical structure and mechanism of biological learning and knowledge. Neurons (nodes) form layers that constitute the socalled ``network architecture''. The node layer

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through which data are fed to the network is designated the ``input layer'' and is used exclusively for this purpose. On the other hand, the layer that contains nodes with the answer or answers ± depending on whether one or more parameters are quanti®ed ± is called the ``output layer''. The number of ``hidden layers'' and that of nodes they contain vary with the particular problem. In this work, we used ``feed-forward networks'', which are based on the back-propagation algorithm [29]. These networks transfer the information on the input layer to one or several hidden layers; subsequently, the information held by the neurons in the hidden layers is combined via non-linear functions ± frequently of the sigmoidal type ± in order to obtain the output data, i.e. the target parameter(s). This type of algorithm is highly suitable as it lends itself readily to supervised learning, i.e. to learn from data with known responses and to use the acquired knowledge to predict the answers for other problems. The use of a non-linear transfer function allows one to model nonlinear relationships between the analytical signal and the analyte or physical parameter of interest. Linear density (a mass per unit length) is a physical parameter used by the textile industry to measure the diameter of acrylic ®bres. Its typical units are the denier (viz. the mass, in grams, of 9000 linear metres of ®bre), and the decitex or dtex (viz. the mass, in decimilligrams, of one linear metre of ®bre). The linear density of ®bres is highly signi®cant for textile as it determines their aesthetics and feel, and hence their uses. Differences in ®bre linear density results in spectral shifts similar to those produced by non-uniformity in particle size in powdered or granulated products. In this work, we critically assessed various approaches to minimizing the non-linearity derived from the multiplicative effect of such non-uniformity, using the determination of linear density in acrylic ®bres as physical model. 2. Experimental 2.1. Samples Samples consisted of cut ®laments of acrylic ®bres composed of a 90:10 (w/w) acrylonitrile±vinyl acetate

209

copolymer. Fibres were produced by passing a concentrated solution of the copolymer through a drawplate to obtain a ®lament that was subsequently subjected to various stretching and drying processes. Depending on the extent of stretching of the ®lament, ®bres of variable linear density were obtained. Overall 125 samples were studied. They were obtained from a production plant and varied in several physical properties such as colour, linear density, brightness, moisture content, etc. In order to facilitate comparison among the different methods, all models were constructed by using a test set for external validation. The calibration set consisted of 43 samples and the test set of 26. An external set comprising 56 samples was used to assess the predictive ability of the models. Samples were distributed among the sets in such a way as to ensure representativeness of the whole set in terms of colour, nuance and other physical properties. They spanned a linear density range from 1 to 18 dtex, equivalent to ®bre diameters between about 10 and 44 mm. 2.2. Apparatus An NIRSystems 6500 near infrared spectrophotometer equipped with a re¯ectance detector and a spinning module were used. The module allowed spectra to be recorded while the cuvette was being spun. After a preset number of scans, an average spectrum was obtained. In this way, the potential effects of sample non-uniformity were lessened. 2.3. Software We used the following software packages:  NSAS (Near Infrared Spectral Analysis Software) v. 3.52, from NIRSystems, which enables recording of spectra and their mathematical processing (averaging, derivation).  UNSCRAMBLER v. 5.03, from Camo A/S, for principal component regression (PCR) and partial least-squares regression (PLSR).  Neural-UNSC v. 1.02, from Camo A/S, for constructing ANN models.  SPSS v. 6.1.2, which allowed the linear regression of scores as a function of linear density in stepwise

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polynomial PCR (SWP-PCR) and stepwise polynomial PLS (SWP-PLS). 2.4. Recording of NIR spectra Each spectrum was the average of 32 scans over the wavelength range 1100±2500 nm. Prior to the spectrum for each sample, a reference spectrum was obtained for a porcelain plate. All sample spectra were recorded in triplicate, using three different portions of the same sample. The spectra subsequently used were the averages of the three spectra for the three aliquots. 2.5. Data processing All the regression models constructed were based on the absorbance mode, viz. log (1/re¯ectance), and the entire NIR wavelength range (1100±2500 nm). PCR and PLS calibration models were constructed with mean centred data. To compare the results obtained by different calibration techniques, an external test set was used to ®nd the optimal model, instead of cross-validation, so training samples and validation process were the same for all the systems assayed. The smallest number of principal components that provided the lowest error of prediction for the samples in the test set, % RSETS, was chosen. % RSETS is de®ned as  sP n D ÿD †2 iˆ1 … PnNIRi 2 REFi  100; (2) % RSETS ˆ iˆ1 DREFi where n is the number of samples included in the test set, DREF the linear density of the samples as determined by the reference method and DNIR is that calculated from the NIR spectrum, using the selected regression method. The SWP-PCR and SWP-PLS results were compared at the signi®cance levels ˆ0.05 and ˆ0.01. While SWP-PCR exhibited no differences in the terms included in the equation regression, SWP-PLS over®tted the system and resulted in large % RSEP values at ˆ0.05. The signi®cance level ˆ0.01 was thus adopted for both. In order to expedite neural network calculations, only the scores of the previously constructed model rather than all 700 variables in each NIR spectrum

were used. The algorithm employed was thus designated PC-ANN since the neural network was constructed from the scores of the previously established PCR model. It should be noted that, in applying PC-ANN, the optimum model was taken to be that leading to the smallest root mean square error of prediction for the body of samples in the test set, s n X …DNIRi ÿ DREFi †2 RMSETS ˆ (3) n iˆ1 because the programme Neural-UNSC optimizes neural networks on the basis of this parameter. The architecture of a neural network is de®ned as (i, h1, h2, o), where i denotes the number of neurons or nodes in the input layer; h1 and h2 the numbers of nodes in the ®rst and second hidden layer; and o the number of nodes in the output layer. No bias weights were used. We tested variable numbers of nodes in the different layers: 3±5 in the input layer, 0±9 in the ®rst hidden layer and 0±2 in the second. The output layer consisted of a single node as only one parameter was determined. During the learning process, the learning rate and update of the neural network must be optimized [30]. The ``learning rate'' is the rate at which the weights of the input and output data for each layer in the neural network change; the ``update'' is the number of samples in the calibration set that are introduced by each iteration. Both parameters can be altered while the neural network is being optimized. No universal procedure for optimizing both parameters exists, however. In our case, the best results were obtained in 15 min by starting at a high learning rate of one and gradually decreasing it to 0.2 as the minimum RMSETS was approached. Starting with learning rates higher than 1 made the network oscillate and it was impossible to ®nd the correct path to the minimum; the use of learning rates lower than 0.1 increased enormously the computation time. Changing the update over the range 5±15 had little in¯uence on RMSETS, so a value of 10 was adopted. The predictive ability of the PCR, PLS, SWP-PCR, SWP-PLS and PC-ANN models was compared in terms of the relative standard error of the test set, % RSETS.

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Fig. 1. NIR diffuse reflectance spectra for acrylic fibres: (1) 1.0; (2) 3.3; (3) 7.8; (4) 12.0; (5) 18.0 dtex.

The goodness of the previously constructed models was assessed by quantifying the samples in the external validation set.

Fig. 2. Plot of log(1/R) versus linear density for the calibration samples.

3. Results and discussion Fig. 1 shows the NIR spectra for ®bres of variable linear density. As can be seen, scattering resulting from differences in linear density had a strong effect on baseline shifts. Fig. 2 shows the absorbance of the samples in the calibration matrix at one of the absorption maxima in the NIR spectrum (2270 nm). A very similar behaviour was observed at all other wavelengths, which testi®es to the marked non-linearity of the system. It should be noted that acrylic ®bres were highly uniform as regards the target parameter: linear density. Such a high uniformity resulted in a clearly non-linear signal±analyte relation in the plots of absorbance against linear density (Fig. 2) and, irrespective of the speci®c problem, the relation was representative of a non-linear response for which no theoretical model for predicting a linear analytical solution exists. Fig. 3 shows the relationship between the absorbance and the decimal logarithm of the linear density at the same wavelength. As can be seen, linearity was much better and the correlation coef®cient much greater than in Fig. 2.

Fig. 3. Plot of log(1/R) versus logarithmic linear density for the calibration samples.

Table 1 gives the % RSE values obtained for the calibration, test and external validation samples, using the different algorithms tested and linear density values (or their decimal logarithms). Table 2 shows the ®gures of merit for the calculated curves against

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Table 1 Figures of merit for the different models constructed from linear density data and their decimal logarithms Model

PCR PLSR SWP-PCR SWP-PLS PC-ANN

Linear density

Log (linear density)

Parameters

RSEC (%)

RSETS (%)

RSEEV (%)

Parameters

RSEC (%)

RSETS (%)

RSEEV (%)

5 6 t1 ; t12 ; t3 t1 ; t12 ; t4 3,3,0,1

10.64 8.99 2.97 3.21 2.17

10.68 9.08 3.67 4.29 2.55

15.95 12.85 3.61 4.80 2.58

5 5 t1 ; t12 ; t3 t1 ; t12 ; t2 3,5,0,1

3.11 3.07 3.37 3.20 2.24

4.80 4.73 2.89 2.71 2.49

4.20 4.24 2.74 2.52 2.30

Table 2 Figures of merit of the regression lines for the external validation set Model

PCR PLSR SWP-PCR SWP-PLS PC-ANN a

Log (linear density)a

Linear density Intercept

Slope

r2

Intercept

Slope

r2

ÿ0.190.36 ÿ0.130.29 0.090.07 0.030.09 0.080.06

0.9450.057 0.9560.047 0.9670.011 0.9630.014 0.9810.009

0.953 0.969 0.998 0.997 0.999

ÿ0.100.08 ÿ0.090.08 0.010.07 ÿ0.010.06 0.010.06

1.0380.013 1.0380.013 0.9880.010 0.9930.010 0.9950.009

0.998 0.998 0.999 0.999 0.999

Values obtained in the original data scale.

the reference values for the external validation samples and different models used. 3.1. Models based on linear density The ®rst principal component in the PCR model accounted for 94.7% of the variance of matrix Y. The lowest % RSETS value was obtained with 5 PCs, which accounted for 97.1% of the variance in matrix Y. Similarly, the ®rst PC in the PLS model accounted for 94.7% of the variance in Y and the lowest % RSETS

value was provided by 6 PCs, which accounted for 97.4% of the variance. Fig. 4 shows a plot of predicted against reference values for PLS calibration with 1 and 6 PLS components. The use of a single PLS component revealed non-linearity in the system, which can be partly corrected by using additional components [31,32], the usual choice of number being that which minimizes % RSETS. However, the increased dispersion in the results arising from the increased noise produced by the additional PLS components led to similarly high errors.

Fig. 4. Predictions for the external sample set obtained with: (A) 1 PLS component and (B) 6 PLS components (the optimum number).

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In applying SWP-PCR and SWP-PLS, data were subjected to stepwise least-squares regression of the scores for the previously constructed PCR and PLSR models. Thus, a polynomial containing the scores for the samples in the calibration matrix, their squares and their binary products was ®tted. The optimum regression equation was taken to be that obtained at the signi®cance level ˆ0.01. The best regressions equations were as follows: linear density ˆ 4:630 ‡ 2:745  t1 ‡ 0:620  t12 ÿ 1:987  t3 with SWP-PCR and linear density ˆ 4:797 ‡ 2:821  t1 ‡ 0:546  t12 ‡ 1:600  t4 with SWP-PLS. The results were much better than those provided by the PCR and PLSR models. However, the regression line of calculated versus reference values never included 0 among intercept values or 1 among slope values at the 95% con®dence level in predicting these samples (Table 2). This suggests that the calibration models corrected non-linearity only partly. A (3,3,0,1) architecture, with 12 weights, based on the scores of the PCR model provided the best results with PC-ANN. Feeding the input later with more than three PCs or including more than three nodes in either hidden layer resulted in no improvement in RMSTS; rather, it lengthened computations. The errors obtained (Table 1) were slightly better than those provided by SWP-PCR and SWP-PLS. However, the intercept was never zero, nor was the slope unity, at the 95% con®dence level, so PC-ANN based on direct linear density measurements also failed to effectively correct non-linearity in the system. 3.2. Models based on the logarithmic linear density Converting linear density values into their logarithms led to a minimum in RSETS with ®ve PCR or PLSR components. Because virtually all data were thus linearized, 99.7% of the variance in matrix Y was accounted for by both methods. Also, the predictive ability was signi®cantly better (Table 1), even though systematic deviations between the calculated and reference values persisted (Table 2).

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SWP-PCR and SWP-PLS provided very similar regression equations in the ®rst two terms …t1 ; t12 † as a result of non-linearity in the system being corrected to a great extent (Fig. 3). The third term completed the modelling, even though the ®rst two provided more than acceptable % RSE values. The third term introduced in the SWP-PLS model was the second score, which is logical taking into account that PLS gradually incorporates those components that account for the greatest variability in matrix Y. The best regression equations were log …linear density† ˆ 0:668 ‡ 0:242  t1 ÿ 0:013  t12 ÿ 0:130  t3 with SWP-PCR and log …linear density† ˆ 0:667 ‡ 0:241  t1 ÿ 0:013  t12 ‡ 0:140  t2 with SWP-PLS. SWP-PLS provided a regression line of zero intercept and unity slope at the 95% con®dence level. Also, although the slope of the regression line obtained with SWP-PCR was not unity, it was very close; in addition, the ®gures of merit of the line were very similar to those of the SWP-PLS line. The best PC-ANN results were obtained by using the ®rst three scores as input for the neural network. The lowest RMSETS was provided by a (3,5,0,1) architecture with 20 weights. While using the decimal logarithm of the linear density resulted in improved linearity, it did not allow us to decrease the number of nodes to be included in each neural layer. The calculated % RSE values were very similar to those obtained with SWP-PCR and SWP-PLS. 4. Conclusions Scattering in NIR diffuse re¯ectance spectra causes variable shifts that result from changes in the path travelled by NIR light; this ultimately leads to deviations from linearity that are quite signi®cant in many cases. The traditional chemometric methods PCR and PLSR fail to correct this effect. On the other hand, SWP-PCR and SWP-PLS correct major deviations from linearity and provide signi®cantly better quantitations (decreased % RSE values); however, they are

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still subject to small systematic errors that suggest incomplete correction of non-linearity. The use of an intrinsically non-linear technique subject to none of the previous restrictions (e.g. PC-ANN) provides very good results (even smaller RSEEV values); however, calculated values also exhibit slight systematic deviations from reference values. The most effective modelling of non-linear systems is provided by an essentially non-linear calibration method. PC-ANN is an excellent choice for this purpose. Even so, data should be previously converted (e.g. into their decimal logarithms) in order to ensure complete linearization. The chief shortcoming of PC-ANN calibration is the need to use a relatively large number of calibration samples. If only a few are available, SWP-PCR and SWP-PLS are two ef®cient alternatives. Acknowledgements This work was carried out in cooperation with Cortaulds EspanÄa, SA, which provided the authors with the samples and valuable technical information. It was funded by Spain's DireccioÂn General de InvestigacioÂn Cientõ®ca y TeÂcnica (DGICyT) in the framework of Project PB96-1180. References [1] S. Sekulic, H.W. Ward, D. Brannegan, E. Stanley, C. Evans, S. Sciavolino, P. Hailey, P. Aldridge, Anal. Chem. 68 (1996) 509. [2] E. Dreassi, G. Ceramelli, P. Corti, S. Lonardi, P.L. Perruccio, Analyst 129 (1995) 1005. [3] Y. Ozaki, R. Cho, K. Ikegaya, S. Muraishi, K. Kawauchi, Appl. Spectrosc. 46 (1992) 1503.

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