Statistical Aspects of Near-Infrared Spectroscopy for the Characterization of Errors and Model Building

Abstract

Due to the complex nature of near-infrared (NIR) spectra, it is usually very difficult to provide quantitative interpretations of spectral data. As a consequence, careful building and validation of calibration models are of fundamental importance prior to development of useful applications of NIR technologies. For this reason, this work presents a statistical study about the NIR spectroscopy, analyzing the NIR behavior when the experimental conditions are changed. Near-infrared spectra were measured at different temperatures and stirring velocities for systems containing a pure solvent and a suspension of polymer powder in order to perform the error analysis. Then, mixtures of xylene and toluene were analyzed through NIR at different temperatures and stirring velocities and the obtained data were used to build calibration models with multivariate techniques. The results showed that the precision of the NIR measurements depends on the analytical conditions and that unavoidable fluctuations of spectral data (or spectral data variability) are strongly correlated, leading to full covariance matrices of spectral fluctuations, which has been surprisingly neglected during quantitative analyses. In particular, modeling of the xylene/toluene NIR data performed with different multivariate techniques revealed that the principal directions are not preserved when the real covariance matrix of measurement errors is taken into account.

Keywords

Near-infrared spectroscopy NIR multivariate calibration principal component regression PCR partial least squares PLS statistical error analysis

Introduction

Near-infrared (NIR) spectroscopy is a very robust technique for measurement of process variables and, consequently, for on-line monitoring and control of industrial processes, finding, therefore, application in many different areas,¹ such as agriculture,² petrochemistry,³ pharmaceutics,⁴ medicine,⁵ process control,⁶ among many others. In particular, on-line monitoring techniques have improved significantly with the recent development of spectroscopic methods and fiber optics technology, which allow for the in situ acquisition and interpretation of process data. As a consequence, the long time delays normally involved with sample preparation and laboratory analyses have been reduced in many applications.⁷ Santos et al.⁷ have recently reviewed the area and the interested reader should refer to this publication for more detailed description of NIR fundamentals and additional applications.

Experimental measurements are subject to perturbations that are not necessarily independent from each other. Perturbations (or experimental fluctuations) can be generated by a large number of factors, including sample preparation, modification of operation procedures, uncontrolled variations of the measurement environment, among many other causes. It is not different with NIR measurements, as light source fluctuations, temperature oscillations, mechanical vibrations (especially at real industrial sites), among other factors, can potentially affect the performance of NIR spectrometers.⁸ Besides, fluctuations of certain variables can affect the behavior of other variables, as one can easily understand when temperature and pressure are measured simultaneously in a pressurized vessel. (As temperature increases, pressure is also expected to increase, meaning that temperature and pressure fluctuations cannot be independent in a closed vessel.) Similarly, one can also imagine that fluctuations of light absorbance in neighboring wavelengths must be strongly correlated (as shown in the following sections) in NIR experiments, as it seems reasonable to assume that uncontrolled fluctuations that lead to the random increase of absorbance at λ = 1000 nm, for instance, also lead to the random increase of absorbance at λ = 1001 nm. Despite that, the importance of measurement errors and measurement correlations has been largely overlooked in the literature. This is particularly true in the NIR field, as the detailed statistical analysis of NIR data are still missing in the technical literature.

The main problem related to the existence of varying measurement fluctuations and measurement error correlations is that correlations can affect the model calibration step and the final model performance, although the effects of varying measurement fluctuations and measurement error correlations on model building procedures and obtained model prediction errors have been systematically neglected in NIR applications. In spite of that, it is well-known that calibration models are of fundamental importance for development and implementation of NIR applications.⁷ In order to illustrate this point, let us assume that a set of measurement values y^exp is available and that a calibration model y^mod is proposed. Then, the usual maximum-likelihood procedure requires that the following objective function be minimized for estimation of the model calibration parameters:⁹

F = (y^{exp} - y^{mod})^{T} V^{- 1} (y^{exp} - y^{mod})

(1)

where it is assumed that measurement errors follow the Gaussian distribution and V is the covariance matrix of measurement errors, usually assumed to be diagonal in the form:

V = σ^{2} I

(2)

where

σ^{2}

is the variance of unavoidable experimental fluctuations (assumed to be the same for all measured values) and I is the identity matrix. In this case, Eq. 1 leads to the least squares calibration technique.

In the most general case, V is not diagonal. In this particular case, the diagonal term $σ_{i}^{2}$ of V represents the variance of the measurement fluctuations of the i^th measured variable, while the non-diagonal term $σ_{ij}^{2}$ of V is the covariance between measurement fluctuations of variables i and j. If V is not diagonal (and most times V is not diagonal, as shown in the following sections),¹⁰ model calibration and final model performance can be seriously affected by erroneously assuming the validity of Eq. 2.¹¹ In order to make the scenario even more complex, V can vary along the experimental grid and depend on the measurement conditions. In this case, the system is named heteroscedastic, as V is not constant and can depend on the particular analyzed experimental conditions.

Illustrative Example

In order to illustrate the importance of the previous remarks for model calibration and model building, let us consider the very simple linear model, commonly used for model calibration of simple spectroscopic data (the well-known Beer–Lambert Law):

y = a x

(3)

where x represents an independent measurement (for instance, the solution concentration), y represents a dependent measurement (for instance, the light absorption at the wavelength λ), a is a parameter that must be estimated from available experimental data (absorptivity), which lumps the intrinsic absorptivity of the medium and the light path length. In practical applications, when NIR is used for process monitoring, the inverse formulation of Eq. 3 is normally employed and x becomes the measured light absorption at the wavelength λ and y becomes the calculated solution concentration. From a mathematical stand point, however, both problems may be regarded as similar, although the analyst must be careful for proper definition of independent and dependent variables.

It is assumed for simplicity and without loss of generality that a pair of data points is available for model calibration: (x₁,y₁) and (x₂,y₂). If the independent variable x is free of error, if the dependent variable y is subject to independent fluctuations that follow the Gaussian distribution with zero mean and constant variance σ ² , then it is possible to formulate the estimation problem through minimization of the following objective function:⁹

F_{Obj} = \sum_{i = 1}^{2} \frac{(y_{i}^{mod} - y_{i}^{exp})^{2}}{σ^{2}} = \sum_{i = 1}^{2} \frac{(a x_{i}^{exp} - y_{i}^{exp})^{2}}{σ^{2}}

(4)

where the superscript mod refers to the calibration model, while the superscript exp refers to the experimental data. Eq. 4 constitutes the well-known least squares estimation problem (when Eqs. 1 and 2 are assumed to be valid) and leads to the solution

a = \frac{\sum_{i = 1}^{2} (x_{i}^{exp} y_{i}^{exp})}{\sum_{i = 1}^{2} (x_{i}^{e xp})^{2}}

(5)

One must observe that the value of the model calibration parameter a does not depend on the variance of experimental errors σ ² , which explains why σ ² is frequently neglected during model development and parameter estimation. However, if the measurement fluctuations change along the experimental grid, when the system is heteroscedastic and V is not diagonal, then the variance cannot be regarded as constant and Eq. 4 must be rewritten as:⁹

F_{Obj} = \sum_{i = 1}^{2} \frac{(y_{i}^{mod} - y_{i}^{exp})^{2}}{σ_{i}^{2}} = \sum_{i = 1}^{2} \frac{(a x_{i}^{exp} - y_{i}^{exp})^{2}}{σ_{i}^{2}}

(6)

where

σ_{i}^{2}

represents the variance of the measurement errors of y at experimental condition i. Therefore, the solution of the proposed calibration problem becomes:

a = \frac{\sum_{i = 1}^{2} \frac{(x_{i}^{exp} y_{i}^{exp})}{σ_{i}^{2}}}{\sum_{i = 1}^{2} \frac{(x_{i}^{exp})^{2}}{σ_{i}^{2}}}

(7)

so that the value of the model calibration parameter a depends on the variances of the measurement fluctuations, normally neglected in most calibration problems. According to Eq. 7, the importance of the analyzed experimental point on the final calibration results diminishes when the variance of the experimental error increases, as precise measurements contain more information than measurements that are subject to large random fluctuations. Based on Eq. 7, it can be concluded that the calibration problem should not be performed without evaluation of variances of the experimental fluctuations (

σ_{i}^{2}

) when experimental errors are expected to vary.

More interesting yet, according to Eq. 1, if the measurement fluctuations are not independent, Eq. 4 becomes:⁹

F_{Obj} = [(y_{1}^{mod} - y_{1}^{exp}) (y_{2}^{mod} - y_{2}^{exp})] \times [σ_{1}^{2} {ρ σ}_{1} σ_{2} {ρ σ}_{1} σ_{2} σ_{2}^{2}]^{- 1} [(y_{1}^{mod} - y_{1}^{exp}) (y_{2}^{mod} - y_{2}^{exp})] = [(a x_{1}^{exp} - y_{1}^{exp}) (a x_{2}^{exp} - y_{2}^{exp})] \times [σ_{1}^{2} {ρ σ}_{1} σ_{2} {ρ σ}_{1} σ_{2} σ_{2}^{2}]^{- 1} [(a x_{1}^{exp} - y_{1}^{exp}) (a x_{2}^{exp} - y_{2}^{exp})]

(8)

where ρ is the correlation coefficient between measurement fluctuations in conditions 1 and 2 and

σ_{12}^{2} = {ρ σ}_{1} σ_{2}

is the covariance between measurement fluctuations in conditions 1 and 2. In this case, the solution of the proposed calibration problem becomes:

a = \frac{{σ_{2}^{2} y_{1}^{exp} x_{1}^{exp} - {ρ σ}_{1} σ_{2} y_{1}^{exp} x_{2}^{exp} - {ρ σ}_{1} σ_{2} y_{2}^{exp} x_{1}^{exp} + σ_{1}^{2} y_{2}^{exp} x_{2}^{exp}}}{σ_{2}^{2} (x_{1}^{exp})^{2} - 2 {ρ σ}_{1} σ_{2} x_{1}^{exp} x_{2}^{exp} + σ_{1}^{2} (x_{2}^{exp})^{2}}

(9)

so that the value of the model calibration parameter a also depends on the correlation ρ between measurement fluctuations, almost always neglected in calibration problems. Based on Eq. 9, it can be concluded that the calibration problem should not be performed without evaluation of correlations between experimental fluctuations when correlations are expected to occur. It becomes clear, based on Eqs. 7 and 9, that rigorous calibration of spectral data should take into consideration the evaluation of variances of measurement errors (

σ_{i}^{2}

) and of correlations between measurement fluctuations (ρ).

Problem Proposition

Little efforts have been made to evaluate the covariances of measurement uncertainties through experiments in the literature and to introduce such covariances in the modeling step, as shown in Eqs. 1 and 8. In spite of that, as illustrated in the previous section, variances and correlations of measurement errors can affect the calibration problem significantly.

Numerical procedures have been proposed for estimation of covariance matrixes of measurement errors and allow for the simultaneous estimation of covariances of measurement fluctuations and calibration model parameters when sufficiently large sets of industrial data are available.^10–12 To the best of our knowledge, these procedures have not been used for NIR model calibrations and characterization of measurement error correlations in NIR experiments. As a consequence, the effects of varying covariance matrixes of measurement fluctuations on model calibration and model performance have not been analyzed yet in NIR problems. Despite that, it is important to recognize that previous studies have attempted to reduce the sensitivity of calibration models to unknown measurement perturbations using different pretreatment techniques,^13–16 although not based on the detailed statistical characterization of the measurement fluctuations, as described in terms of variance spectra and covariance matrixes of measurement fluctuations calculated with replicates at distinct experimental conditions.

Based on the previous paragraphs, the present work presents a statistical study about some simple NIR experiments, in order to characterize the importance of measurement error variances and covariances for quantitative NIR analyses. For this reason, NIR spectra are measured at different experimental conditions and the covariance matrices of error fluctuations are computed through experimental replication at each analyzed condition. Afterwards, covariance matrices are used for qualitative and quantitative analyses and for model building.

For illustrative purposes, NIR spectra are measured first at different temperatures and stirring velocities for systems containing a pure solvent and a polymer powder suspension in order to perform the error analyses. Despite the experimental simplicity, these experimental systems are important for analysis of polymerization reactions. The obtained data are used to show that V is not diagonal and can change with the experimental conditions. Afterwards, xylene and toluene mixtures are analyzed through NIR at different temperatures and stirring velocities. The obtained data, without any preprocessing technique, are used to build calibration models with different multivariate techniques: multiple linear regression (MLR),¹⁷ principal component regression (PCR),¹⁸ and partial least squares (PLS).¹⁹ Calibration models are built with help of standard least squares procedures and more involving heteroscedastic maximum-likelihood procedures.²⁰ The obtained results indicate that the precision of the NIR measurements depends very strongly on the analytical conditions, so that variances of experimental fluctuations are not constant. Additionally, it is shown that spectral data are strongly correlated, so that measurement fluctuations are not independent. Particularly, modeling of the xylene/toluene NIR data performed with different multivariate techniques revealed that the principal directions of standard PCR and PLS calibrations are not preserved when the real covariance matrix of measurement errors is taken into account.

Experimental

The experimental work was divided into two parts. The first part (Part 1) included experiments for standard error analyses, using NIR spectra obtained at different experimental conditions. Near-infrared spectra were collected for a pure solvent and a polymer powder suspension at different stirring velocities and temperatures. These variables were manipulated because they are subject to frequent perturbations in real polymerization reactions. The second part (Part 2) included experiments for model calibration in a simple chemical system. Mixtures of xylene and toluene were prepared with different concentrations and analyzed at different temperatures and stirring velocities. Xylene and toluene were selected because they are used very frequently in most chemical labs, are miscible in the full concentration range and present similar NIR spectra, which make the calibration process more difficult. It is important to note that equipment dimensions, solvents, temperature, and stirring velocities used in the present manuscript do not limit the scope of this study, as the main objective pursued here is to report variations of measurement error covariances and of how calibration models respond to these variations in NIR experiments.

Part 1: Experiments for Error Analyses

The experimental setup used to perform the experiments comprises a 1.0 L borosilicate tank, whose cover has orifices for sample removal and for introduction of any components (e.g., probe, stirrer, thermocouple, reflux condenser), a microcomputer, a mechanical stirrer with magnetic seal equipped with an impeller of spades, a reflux condenser, a heating/cooling bath connected to the reactor jacket and a NIR-6500 spectrophotometer connected in situ to collect spectra. For more details, the Experimental Unit Schema is presented in the Supplemental Material. Near-infrared spectra were measured in regular intervals of 2 min, using an in situ spectrophotometer NIR-6500 (NIRSystems, Inc., Silver Spring, MD, USA), working in the transflectance mode in the spectral region of 400–2500 nm. The spectra were collected using a stainless steel transflectance probe with a constant path length of 34 cm and diameter of 19 mm, connected to the instrument through a fiber optics cable of 3 m. The fiber optics cable comprised three bundles of fibers. The light bulb contained a filament of tungsten and the light detector was based on the standard PbS technology. Data acquisition was performed with NIR Spectral Analysis Software version 3.30, a software provided by the manufacturer of the NIR spectrometer (Vision(R)). Spectra were recorded as averages of 32 readings with precision of 0.1 nm, according to spectrophotometer specification (NIRSystem Process Analytics Manual version 1.0 NIRSystems Inc., Silver Spring, MD, USA). Spectral bandwidth was chosen in the equipment as 10 ± 1 nm and the dynamic range is 2–3 AU.

Near-infrared analyses were performed with (1,2,4)–Trichlorobenzene (TCB) and polypropylene (PP) powder. TCB is a solvent that is frequently used to carry out polypropylene analyses. TCB was provided by TEDIA Brasil (Rio de Janeiro, Brazil) as a high-purity HPLC grade and used as received. The polypropylene powder was produced in the lab with minimum purity of 99.5% and presented weight-average molecular weight of 982 × 10³g/gmol and average particle diameter of 365 µm.

Near-infrared readings were performed only after temperature stabilization. Eight replicates were obtained in all conditions and saved for standard error analyses. Initially, NIR spectra were collected for pure TCB at distinct temperatures, from 30 ℃ to 100 ℃ with a 10 ℃ interval. The temperature range was defined because PP analyses are usually performed in this temperature range. Then a full three-level factorial design was used for the simultaneous manipulation of temperature and stirring velocity in pure solvent experiments, at temperature levels of 30 ℃, 60 ℃, and 90 ℃ and stirring speed levels of 250, 350, and 450 rpm. Finally, a full three-level factorial design was also used for the simultaneous manipulation of temperature and stirring velocity in suspension experiments (2 g of PP in 400 mL of TCB), at temperature levels of 30 ℃, 60 ℃, and 90 ℃ and stirring speed levels of 250, 350, and 450 rpm.

Part 2: Experiments for Model Calibrations

The same experimental setup and operation procedures described previously were used in this section. Xylene and toluene were provided by TEDIA Brasil (Rio de Janeiro, Brazil) as a high-purity HPLC grades and used as received. Near-infrared readings were performed only after temperature stabilization and ten replicates were obtained in all conditions and saved for standard error analyses. Near-infrared spectra were collected for distinct xylene/toluene compositions, from 0 to 100 wt% with intervals of 10 wt%. A full three-level factorial design was used for the simultaneous manipulation of temperature and stirring velocity for each xylene/toluene solution, at temperature levels of 30 ℃, 60 ℃, and 90 ℃ and stirring speed levels of 250, 350, and 450 rpm. Table 1 summarizes the experiments performed for calibration purposes. It is important to emphasize that ten replicates were obtained for each condition of the experimental plan, leading to a set of 11 (concentrations) × 3 (temperatures) × 3 (stirring speeds) × 10 (replicates) = 990 NIR spectra, as described previously.

Table 1.

Experiments performed for model calibration.

Xylene concentration (wt%)	Toluene concentration (wt%)	Temperature (℃)	Stirring speed (rpm)
0	100	30	250	350	450
		60	250	350	450
		90	250	350	450
10	90	30	250	350	450
		60	250	350	450
		90	250	350	450
20	80	30	250	350	450
		60	250	350	450
		90	250	350	450
30	70	30	250	350	450
		60	250	350	450
		90	250	350	450
40	60	30	250	350	450
		60	250	350	450
		90	250	350	450
50	50	30	250	350	450
		60	250	350	450
		90	250	350	450
60	40	30	250	350	450
		60	250	350	450
		90	250	350	450
70	30	30	250	350	450
		60	250	350	450
		90	250	350	450
80	20	30	250	350	450
		60	250	350	450
		90	250	350	450
90	10	30	250	350	450
		60	250	350	450
		90	250	350	450
100	0	30	250	350	450
		60	250	350	450
		90	250	350	450

Results

Part 1 Variances

Near-infrared spectra for pure TCB were measured in the temperature range of 30–100 ℃ allowing for verification of the reproducibility of the experimental data. Near-infrared spectra for pure TCB can be found in the Supplemental Material. Absorption peaks at 800 nm, 1130 nm, and 1680 nm related to the third, second, and first overtones of the C–H stretching, respectively, could be observed. The spectral region of 2100–2300 nm was very noisy, containing information about combinations of overtones of the C–H stretching.^21,22 It was noticed that the NIR spectra of pure TCB were not very sensitive to variations of stirring velocities and temperature, as one might already expect as the NIR spectra contains information about chemical bonds and molecular dynamics.^1,21

Solid powder was introduced in the reaction media and its effect on spectral data was analyzed using NIR measurements. Near-infrared spectra for PP suspensions in TCB can be found in the Supplemental Material. Polypropylene (PP) suspensions in TCB increased measurement variability, which seems to depend on the particular analyzed spectral region. Besides, it also seems clear that measurement variability depends on the experimental condition, when spectra obtained at different temperatures and stirring speeds are compared to each other. Modification of spectral responses for suspensions of PP in TCB might already be expected, as NIR responses are sensitive to modification of the system composition and to particle sizes in heterogeneous polymer systems, due to scattering of light.⁷ Variations of stirring velocity and temperature were imposed on PP suspensions in TCB media and analyses of the NIR spectra showed that stirring velocity affects measurement variability more significantly than temperature, which can be related to the fact that the stirring velocity changes the suspension characteristics and the frequency that particles cross the sampling window.

Variance spectra for pure TCB from 30 ℃ to 100 ℃ (variances of measured absorbances at distinct wavelengths) were calculated in the form:

σ_{i}^{2} = \frac{\sum_{j = 1}^{N} (A_{i, j} - \bar{A_{i}})^{2}}{N - 1}

(10)

where

σ_{i}^{2}

is the variance of the absorbance measurement fluctuations at the wavelength λ_i, N is the number of replicates,

A_{i, j}

is the j^th replicate at condition i, and

\bar{A_{i}}

is the average absorbance signal at condition i. It can be observed in the variance spectra of Figure 1 (and in the Supplemental Material) that variances are not the same and depend strongly on both the wavelength and temperature, despite the good reproducibility. Modification of spectral variances with the wavelength are probably related to the characteristics of the analyzed chemical system, but also to the characteristics of the measuring device, as one can certainly speculate that the measuring performance is not uniform in the whole spectral range. Modification of spectral variances with temperature is probably related to modification of the rotational energy of the molecules and of the bulk properties of the system, such as density and viscosity.

Variations of stirring velocity and temperature were imposed on the media for pure TCB and for PP suspension in TCB. Figure 1a and b shows the results for absorbance variances of pure TCB at different stirring velocities at T = 30 ℃ and absorbance variances of the TCB/PP suspension (400 mL/2 g) at different temperatures at w = 450 rpm, respectively. It can be noticed that the stirring speed and temperature also affect the reproducibility of the spectral measurement. Stirring velocity provides unavoidable mechanical vibration of the measuring system and formation of small air bubbles, as the stirring speed increases, which possibly justifies this behavior. In the presence of the solid, it has been verified that variances increase significantly.

As the information content of available measurements depend on the variances of measurement errors,²³ the obtained data clearly indicate that the information content of absorbance values measured at different wavelengths and experimental conditions depend on the particular conditions considered. Therefore, the matrix V in Eq. 1, which contains the variances and covariances of measurement fluctuations, is expected to change along the experimental range, including the wavelength. Variations of the covariance matrix of measurement fluctuations in the experimental range have been largely overlooked in previous modeling studies and have not been taken into account for building of NIR calibration models.

Correlations

Figure 2 illustrates NIR measurement error correlations for some experimental conditions, as obtained for pure TCB and PP suspension in TCB and calculated as:

σ_{ik}^{2} = \frac{\sum_{j = 1}^{N} (A_{i, j} - \bar{A_{i}}) (A_{k, j} - \bar{A_{k}})}{N - 1}

(11)

ρ_{i, k} = \frac{σ_{ik}^{2}}{σ_{i} σ_{k}}

(12)

It can be noted that correlations between spectral measurements can be very high and very often essentially equal to 1, even when the NIR signals are very distant from each other in the wavelength scale. This means that V is a full matrix in Eq. 1 and that the independent spectral error hypothesis can be very poor. As observed in Figure 2, measurement fluctuations seem to respond to few common sources of error, as spectral data vary along the same direction. These common sources of error can include modification of the base line, vibration of the sampling device, oscillation of the electric and light sources, among others. It is not intended here to identify the sources of fluctuation in the analyzed experimental setup, but it seems clear that NIR measurement fluctuations cannot be represented by diagonal V matrixes, as usually assumed during calibration. Besides, correlations change with the experimental conditions, as shown in Figure 2, showing once more that matrix V in Eq. 1 is expected to change along the experimental range, as pointed out previously and usually neglected in calibration studies.

It can be also observed that strong positive correlations are related mostly to small unavoidable baseline variations. Besides, the strong positive correlations are also related do the slow modification of the absorbance spectra with the modification of the wavelength (or wave number), as deviations of the absorbance values at neighboring wavelengths do not change independently. On the other hand, strong negative correlations are observed mostly between absorbance values places at distinct regions of the spectrum (absorbance values at low and high wavelengths), which suggests fluctuations of the lamp or transmittance performance due to, for instance, small variations of the local temperature (especially at the lamp).

Part 2 Variances

Near-infrared spectra of pure xylene and of pure toluene at different temperatures and stirring speeds were obtained, showing absorption peaks placed at 870 nm, 1130 nm, 1680 nm, and 1767 nm, related, respectively, to the third and second overtones of the aromatic C–H stretching, combinations of overtones of aromatic C–H stretching, first overtone of aromatic C–H stretching, and first overtone of C–H stretching of the CH₃ group.^21,22 Near-infrared spectra of pure xylene and toluene can be found in the Supplemental Material. As observed previously for TCB, the spectral region of 2100–2300 nm is very noisy, containing information about combinations of overtones of the C–H stretching.^21,22 It can be noticed that xylene and toluene spectra are very similar, making model calibration more difficult.

Solutions containing 40 wt% of xylene and 60 wt% of toluene have been chosen to analyze stirring and temperature effects. It could be observed that noise increases with temperature and stirring speeds, as observed previously for TCB, suggesting that similar effects can possibly be observed for other chemical systems. The effect of stirring speed and temperature in variance spectra of measurement errors has also been verified. As previously discussed, variances are sensitive to wavelength, temperature and stirring speeds and respond similarly to modifications of these variables, when compared to the results presented previously for TCB. Therefore, as observed before, the matrix V in Eq. 1 is also expected to change along the experimental range for xylene/toluene solutions, including the wavelength, and should be taken into consideration during model calibration.

Covariances

Near-infrared measurement error correlations for some experimental conditions have been obtained and the results are presented in the Supplemental Material. Once more, it can be noted that correlations between spectral measurements can be very high and very often essentially equal to 1, even when the NIR signals are very distant from each other in the wavelength scale. As explained before, this means that V is a full matrix in Eq. 1 and that the independent spectral error hypothesis can be very poor. Measurement fluctuations seem to respond to few common sources of error, as described before. As the experimental noise increases with temperature and the stirring speed, observed correlations also decrease slightly with temperature and stirring speeds, due to the higher noise levels. However, it seems clear once more that NIR measurement fluctuations cannot be represented by diagonal V matrixes, as usually assumed during calibration. Besides, correlations change with the experimental conditions showing once more that matrix V in Eq. 1 is expected to change along the experimental range, as pointed out previously and usually neglected in calibration studies. For more details, the figure showing the correlation between the absorbance and the others spectral measurements for xylene 40% and toluene 60% in volume at 828 nm, at 30 ℃ and 250 rpm; at 760 nm, 60 ℃ and 450 rpm; and at 612 nm, 90 ℃ and 350 rpm can be found in the Supplemental Material.

Standard Calibration Procedures

In order to analyze how model calibrations respond to the measurement errors, calibration models were built to provide the xylene content of the analyzed organic solution. As usual in this field, first and second derivative spectra were calculated in order to magnify the differences between toluene and xylene measurements and facilitate the calibration process, since this procedure can remove base line variations, remove measurement noise and discriminate overlapping bands.²⁴ Derivatives were computed with second-degree interpolating polynomials and five neighboring data points, simultaneously providing a smoothing effect. It is important to emphasize that the qualitative behavior of first-derivative spectra is very similar to the behavior of the crude spectra shown before, being sensitive to changes of temperature and stirring speeds and presenting variances and covariances that respond to changes of wavelengths, temperature, and stirring speeds as described before. In order to show that the use of standard pretreatment techniques do not change the overall behavior of the spectral variability, as discussed in the previous sections, available spectral data were treated with different standard pre-treatment techniques and used to compute the variance spectra, as shown in the Supplemental Material. As one can see, differences of at least one order of magnitude can be observed for computed variances in all cases.

Model calibration was performed with the help of MLR procedures implemented in the software Statistica 6.0.²⁵ In this case, the calibration model is defined as:

C_{x} = D_{1} P_{1} + \dots + D_{N} P_{N} + P_{N + 1}

(13)

where C_x is the xylene concentration, D_i is the intensity of the first-derivative of the measured absorbance NIR spectra, and P_i are the calibration model parameters. Calibration models were built with different data sets, using all available data for specific wavelength values or grouping the data in terms of temperature, stirring speed or both for specific wavelength values. Stirring speed and temperature variations significantly affect the performances of the calibration models, because of the modification of the statistical behavior of the collected spectral data. For this reason, most models were also built for constant temperature and stirring speed conditions. Model parameters were estimated through minimization of the least squares function, which assumes constant error in the experimental grid (Eq. 14):

assuming that variances of concentration measurement errors (

σ_{i}^{2}

) are independent and essentially constant throughout the experimental grid. Additional modeling and numerical information are provided by Pinto et al.²⁶

The performances of the calibration models were compared to each other considering four aspects: wavelength range, measurement conditions, input signal and objective function. Wavelengths were selected arbitrarily in the spectral regions where absorptions were more intense. Input signals included first-derivatives, second-derivatives, and ratios of signals. As an example, the effect of stirring velocity variation at 60 ℃ is presented in Figure 3 for a model built with first-derivatives at a single wavelength value of 2052 nm. Additional results for calibration model performances can be seen in the Supplemental Material.

It can be observed that model parameters (and model performances) changed considerably with the analyzed measurement conditions, due to variation of the NIR measurements with temperature and stirring velocity, indicating that these effects must be considered during the quantitative analysis. Moreover, it can be noted that model performances also changed with the spectral region, due to the different information contents of the distinct wavelength ranges, although the input signal did not affect the calibration performance significantly, as similar model performances could be obtained with different inputs. Additionally, model performances did not change with the objective functions in the analyzed case, due to the similar fluctuations of xylene and toluene concentration measurements in the analyzed concentration ranges.

Principal Directions

In the previous sections, it was observed that the spectral measurements in the NIR region are subject to measurement errors that are not uniform and depend on the measurement conditions. Besides, it was observed that measurement variability depends on the measurement conditions and on the spectral region and that fluctuations seem to respond to few common sources of error, since measurement correlations can be very high. For this reason, these effects were also taken into consideration during the quantitative calibration analysis of the xylene / toluene mixtures, in order to determine the xylene content based on the NIR readings.

The detailed presentation of heteroscedastic PCR and PLS procedures is beyond the scope of the present manuscript and can be found elsewhere.²⁶ However, it is important to say that homoscedastic procedures assume that measurements errors are uniform and independent in the experimental grid, so that the principal components are calculated as the eigenvectors of the matrix ( $X^{T} X$ ), where $x_{i, j}$ represents the measurement of variable j in the experiment i. In the case of NIR calibrations, $x_{i, j}$ normally is the absorption for the wavelength λ_j at experiment i. In a certain sense, matrix ( $X^{T} X$ ) also represents a covariance matrix of variable measurements, although matrix ( $X^{T} X$ ) is related to the variations of measurements in the full experimental grid and is not related to the local measurement errors of each variable. In accordance with Eq. 1, when measurement errors are correlated and change along the experimental grid, the model calibration should take the full matrix V into consideration, so that the principal components should be calculated as the eigenvectors of the matrix ( $X^{T} V^{- 1} X$ ), as the relative importance of each available measurement depends on the respective error content. Obviously, when V is diagonal (independent and constant error fluctuations), heteroscedastic and homoscedastic procedures become completely equivalent. However, as shown in the previous sections, V is a full matrix in real NIR experiments, so that the use of homoscedastic procedures cannot be justified on firm statistical grounds.

For the sake of quantitative analyses, principal directions were calculated with help of homoscedastic PCR (used as benchmark for comparison), heteroscedastic PCR and heteroscedastic PLS methods.²⁶ It is important to stress once more that homoscedastic procedures project the measured spectral data onto the subspace formed by the principal directions calculated with the covariance matrix of the full set of measured spectral data ( $X^{T} X$ ), while heteroscedastic procedures project the spectral data locally, using information provided by the local covariance matrix calculated with replicates ( $X^{T} V^{- 1} X$ ).

In order to analyze the V_x matrix structure, 80 measurements of 20% xylene and 80% toluene solution at 30 ℃ and 250 rpm were performed. Eigenvalues and eigenvectors were calculated. According to the accumulated variability presented in the Supplemental Material, no more than four eigenvalues are sufficient to express 99% of the problem variability. It can be observed, therefore, that spectral measurements do not present 1050 error sources, but only four. The four eigenvectors are presented in the Supplemental Material. As a consequence, these results indicate that there are few sources of variability in the analyzed experimental problem (for instance, baseline variation, probe trepidation, bubbles formation, and temperature fluctuations).

Principal directions with homoscedastic and heteroscedastic PCR procedures have been obtained, allowing verifying that calculated principal directions can be very different in both cases and that they change with the experimental conditions. This is not surprising, given the very different structures of the V matrix used in both cases. In general, the principal directions become more different as more information is extracted from the data; in other words, differences are higher for the fifth direction than for the first. As a consequence, calibration models built in each particular case are expected to be different and to present different performances.

Calibration Results

Based on the previous results, calibration models were built for xylene (and toluene) concentrations with homoscedastic PCR, heteroscedastic PCR, and PLS techniques using first-derivative and second-derivative spectra, as reported previously. Illustrative examples for models built with first-derivative spectra are presented in Figure 4.

It can be observed that the performances of calibration models built with homoscedastic and heteroscedastic PCR procedures are usually similar in the analyzed cases, despite the use of different principal components, although it seems that experimental fluctuations are more evident when model predictions are performed with the heteroscedastic PCR models. (It must be emphasized, though, that the use of the principal directions for modeling purposes with PCR techniques is based on heuristical assumptions, as it is not possible to assure formally that the directions that contain the highest variations of the independent variables are also the most influential ones in respect to the dependent variables.) On the other hand, calibrations performed with the heteroscedastic PLS technique led to the best model calibration performances. (This might already be expected, as PLS techniques are designed to provide the principal directions that allow for maximization of the correlation coefficients between measured and calculated results during model calibration). Perhaps more important than that, as experimental fluctuations were taken into consideration during model formulation, as required by Eq. 1, the obtained model performances are also compatible with the observed data variability. In this sense, one must observe in Figure 4c that all calibrations performed with the heteroscedastic PLS procedure provide predictions that fall inside the variation interval of available experimental measurements (as obtained with replicates). As a matter of fact, the heteroscedastic procedure is expected to provide more accurate predictions and simultaneously respect the statistical structure of the real measurement environment.

Conclusion

The present work presented a statistical study regarding NIR spectroscopy in two simple model problems: suspensions of polypropylene powder in (1,2,4)-trichlorobenzene and solutions of xylene and toluene. Near-infrared spectra were collected at distinct experimental conditions, through manipulation of compositions, temperature and stirring velocities. Replicates were used for computation of covariance matrices, showing that measurement variability changed with the experimental conditions and that NIR signals measured at distinct wavelengths were strongly correlated. For this reason, heteroscedastic modeling techniques based on principal components analysis (principal components regression and partial least squares) were used for calibration of model performances in the xylene / toluene problem, aiming at predicting the analyzed xylene concentrations. The results showed, therefore, that the assumption that the new set of linear variables is independent and uncorrelated is not correct when the spectral information (described by the variance spectra) is heteroscedastic and correlated. In other words, the intrinsic assumption of uniform experimental variability is not supported by the real experimental data. As a matter of fact, spectral measurements are subject to non-uniform disturbances, which vary with wavelength and measurement conditions, and we believe we have shown this clearly in the manuscript. As a consequence, model performance should not be expected to present uniform performance, as model deviations can be higher where experimental variations are also higher, which is neglected by standard procedures, that assume that model performances should be uniform in the measurement space. Experimental data show that spectral measurements at different wavelengths are strongly correlated, also suggesting the existence of few common error sources. In other words, NIR spectral measurements are subject to fluctuations and measurement errors that are not independent. It has been demonstrated in the present study that these effects can be considered during quantitative NIR data analyses.

The obtained results revealed that the principal directions were not preserved when the real covariance matrix of measurement errors was taken into account, affecting the calibration step significantly. Besides, calibration results obtained with the heteroscedastic PLS procedure provided more accurate predictions and simultaneously respected the statistical structure of the real measurement environment. Based on the obtained results, it seems clear that the intrinsic variability of NIR spectra must be characterized and used for modeling purposes in real NIR calibration problems.

Footnotes

Conflict of Interest

The authors report there are no conflicts of interest.

Funding

The authors thank CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) and FAPERJ (Fundação Carlos Chagas Filho de Apoio à Pesquisa do Estado do Rio de Janeiro) for financial support and scholarships.

Supplemental Material

All supplemental material mentioned in the text, consisting of nine figures, is available in the online version of the journal.

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Xylene concentration (wt%)	Toluene concentration (wt%)	Temperature (℃)	Stirring speed (rpm)
0	100	30	250	350	450
		60	250	350	450
		90	250	350	450
10	90	30	250	350	450
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		90	250	350	450
20	80	30	250	350	450
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30	70	30	250	350	450
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40	60	30	250	350	450
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50	50	30	250	350	450
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60	40	30	250	350	450
		60	250	350	450
		90	250	350	450
70	30	30	250	350	450
		60	250	350	450
		90	250	350	450
80	20	30	250	350	450
		60	250	350	450
		90	250	350	450
90	10	30	250	350	450
		60	250	350	450
		90	250	350	450
100	0	30	250	350	450
		60	250	350	450
		90	250	350	450

Xylene concentration (wt%)	Toluene concentration (wt%)	Temperature (℃)	Stirring speed (rpm)
0	100	30	250	350	450
		60	250	350	450
		90	250	350	450
10	90	30	250	350	450
		60	250	350	450
		90	250	350	450
20	80	30	250	350	450
		60	250	350	450
		90	250	350	450
30	70	30	250	350	450
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40	60	30	250	350	450
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50	50	30	250	350	450
		60	250	350	450
		90	250	350	450
60	40	30	250	350	450
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		90	250	350	450
70	30	30	250	350	450
		60	250	350	450
		90	250	350	450
80	20	30	250	350	450
		60	250	350	450
		90	250	350	450
90	10	30	250	350	450
		60	250	350	450
		90	250	350	450
100	0	30	250	350	450
		60	250	350	450
		90	250	350	450

Xylene concentration (wt%)	Toluene concentration (wt%)	Temperature (℃)	Stirring speed (rpm)
0	100	30	250	350	450
		60	250	350	450
		90	250	350	450
10	90	30	250	350	450
		60	250	350	450
		90	250	350	450
20	80	30	250	350	450
		60	250	350	450
		90	250	350	450
30	70	30	250	350	450
		60	250	350	450
		90	250	350	450
40	60	30	250	350	450
		60	250	350	450
		90	250	350	450
50	50	30	250	350	450
		60	250	350	450
		90	250	350	450
60	40	30	250	350	450
		60	250	350	450
		90	250	350	450
70	30	30	250	350	450
		60	250	350	450
		90	250	350	450
80	20	30	250	350	450
		60	250	350	450
		90	250	350	450
90	10	30	250	350	450
		60	250	350	450
		90	250	350	450
100	0	30	250	350	450
		60	250	350	450
		90	250	350	450