Effects of Resolution on Quantification in Open-Path Fourier Transform

Environ. Sci. Technol. 2000, 34, 1346-1351

Effects of Resolution on Quantification in Open-Path Fourier Transform Infrared Spectrometry under Conditions of Low Detector Noise. 2. Partial Least Squares Regression BRIAN K. HART, R. JAMES BERRY, AND PETER R. GRIFFITHS* Department of Chemistry, University of Idaho, Moscow, Idaho 83844-2343

The effects of resolution, spectral window, and background type on the predictive capability of partial least squares regression (PLS) on spectra measured by an open-path Fourier transform (OP/FT-IR) spectrometer were tested with spectra of mixtures of alkanes and chlorinated hydrocarbons. The results were compared with the results obtained with the identical data sets using classical least squares regression (CLS). It is shown that the most accurate predictions are obtained using the same conditions that were optimal for CLS, namely spectra measured at low resolution and ratioed to background spectra over the same path length, with the calculations made over limited spectral windows. However, good predictions could be achieved with background spectra measured over a very short path. Even in the worst cases, the relative error of predictions made by PLS was usually less than 5%. On average, the predicted concentrations of the components of mixtures containing up to five chemically similar analytes made using the PLS algorithm are 120 times more accurate than the predicted concentrations of the components of the identical data sets made using CLS.

Introduction The most commonly used multivariate calibration routine used today for open-path Fourier transform (OP/FT-IR) spectrometry is classical least squares regression (CLS). CLS assumes a causal (Beer’s law) relationship between C and A, namely A ) f(C), where C is the concentration matrix and A is the absorbance spectra matrix, bold denoting matrix format (1). This type of model is adversely affected by any information in A that cannot be modeled as a function of concentration. For OP/FT-IR measurements, the most important factor leading to errors is the effect of absorption features caused by atmospheric water vapor and carbon dioxide. As seen in the first paper of this series (2), the presence of atmospheric absorption features reduces and in many circumstances eliminates the capacity of CLS to adequately model OP/FT-IR spectra of mixtures of chemically similar compounds. An alternative approach to modeling is to use the predictive relationship, C ) f(A), also known as the inverse * Corresponding author phone: (208)885-5807; fax: (208)885-6173; e-mail: [email protected]. 1346

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Beer’s law model or, when data at only a few wavelengths are used, multiple linear regression. This type of model is less sensitive to baseline shifts and the presence of analytes not accounted for in the calibration matrix. The inverse Beer’s law predictive model is the basis for partial least squares regression (PLS). The predictive ability of PLS is further improved by regressing C on only a few linear combinations of A. By using only a few components from a linear recombination of A, as opposed to the full A matrix used in CLS, information not relevant to the calibration problem is excluded from the calibration and prediction step. To determine accurately what information in A is relevant to the prediction of C by PLS, A is decomposed into a number of orthogonal eigenvectors using an algorithm that simultaneously balances both A and C to reduce the effect of large, irrelevant variations to the calibration model in A. Certainly, OP/FT-IR spectra contain large, irrelevant variations caused by the presence of atmospheric H2O and CO2. In this paper we describe the effects of resolution, spectral window, and type of background spectra used in OP/FT-IR on the predictive ability of PLS. The effect of resolution on the performance of PLS was tested on two sets of five vapor phase compounds. The first set, composed of the C1 to C5 straight-chain alkanes, is intended to show the effect of resolution on a group of compounds with strongly overlapping bands. The width of the rotational lines in the spectra of methane and ethane is typically about 0.2 cm-1, which is significantly narrower than the resolution of 1 cm-1 at which OP/FT-IR spectra are typically measured, while the spacing of these lines is greater than 1 cm-1. On the other hand, the separation of the lines in the spectra of n-butane and n-pentane is smaller than their full-width at half-height (fwhh), and, therefore, these lines cannot be resolved at any resolution when broadened by air at a pressure of 1 atm. Thus the spectra of the alkanes exhibit significant changes in peak height and width as a function of resolution. These effects can be seen in Paper 1 (2), Figure 3 and Table 1. The second set of analytes was a group of five chlorinated hydrocarbons taken from the list of hazardous air pollutants (HAPs) as defined by the Clean Air Act (trichloromethane, 1,4-dichlorobenzene, 1,2dichlorethane, dichloromethane, and 1,1,2-trichloroethane). These compounds are representative of many HAPs that have unresolvable rotational fine structure and a band contour with a width of between 10 and 25 cm-1. They were selected so that their bands overlapped (see Paper 1, Figure 4).

Experimental Section The spectra for our previous study on CLS (2) were used for this study without any alteration. A total of 108 separate calibration/validation sets were created to test the PLS algorithm, which is a singular value decomposition algorithm written in-house using MATLAB 5.1 (The Math Works Inc., Natick MA) by and is based on the approach of Wang et al. (3). As mentioned earlier, one aspect of PLS is its decomposition of the A matrix into a linear combination of orthogonal components, and its consequent use of only a subset of these components in the calibration and prediction steps. The determination of the optimum number of components to use is one key aspect of using PLS. In previous work (4) we established that for OP/FT-IR spectra the optimal number of components to use was N+10, where N is the number of compounds in the calibration model. This conclusion was rechecked and verified on the data sets used in this study by cross validation. All PLS data in this study 10.1021/es990439v CCC: $19.00

 2000 American Chemical Society Published on Web 02/29/2000

FIGURE 1. Plot of the effect of resolution on RMSEP for the five alkanes using EQDI0 backgrounds and TO-16, Type 3, spectral windows.

FIGURE 2. Plot of the effect of resolution on RMSEP for the five chlorinated hydrocarbons using EQDI0 backgrounds and TO-16, Type 3, spectral windows. were calculated using 15 factors (10 background factors and one for each of the five compounds).

Results and Discussion “Reality Check”. As in the previous article, an extra data set was created to verify the accuracy of the PLS algorithm and to demonstrate that any prediction errors are a direct consequence of using OP/FT-IR atmospheric backgrounds for each set of different test conditions. These data sets were synthesized by adding scaled reference spectra of the alkanes or the chlorinated hydrocarbons without the added OP/ FT-IR atmospheric backgrounds. Analysis of these data sets

using the PLS algorithm shown above resulted in a rootmean-square error of prediction (RMSEP) of less than 1 × 10-7 in all cases. Any prediction errors from the synthesized spectra must, therefore, be attributed to changes in interferometer alignment, uncompensated absorption of water vapor and CO2, and for the case of spectra created using short-path reference spectra, sloping nonzero baselines. Effect of Resolution. The effect of resolution, ∆ν˜ , on the accuracy to which the concentration of the chlorinated hydrocarbons can be predicted is shown in Figure 1. It can be seen that the RMSEP improves significantly as ∆ν˜ is degraded from 1 to 4 cm-1. The detector noise on all the VOL. 34, NO. 7, 2000 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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TABLE 1. RMSEP (ACU) of PLS Calibrations Using Equidistant Backgrounds as a Function of Resolution and Spectral Window full atmospheric window, EQDI0 resolution (cm-1) 1

2

4

8

16

32

0.0040 0.0028 0.0093 0.0096 0.0087 0.0069 0.0076 0.0074 0.0116 0.0082

0.0035 0.0014 0.0074 0.0068 0.0052 0.0048 0.0065 0.0092 0.0122 0.0085

0.0039 0.0018 0.0059 0.0061 0.0047 0.0071 0.0058 0.0097 0.0127 0.0072

0.0055 0.0022 0.0066 0.0062 0.0047 0.0138 0.0064 0.0130 0.0167 0.0088

0.0027 0.0066 0.0059 0.0059 0.0238 0.0061 0.0196 0.0299 0.0153

trichloromethane0.0057 1,4-dichlorobenzene 1,2-dichloroethane dichloromethane 1,1,2-trichloroethane methane ethane propane butane pentane

0.0049 0.0035 0.0103 0.0088 0.0077 0.0075 0.0075 0.0077 0.0156 0.0096

trichloromethane 1,4-dichlorobenzene 1,2-dichloroethane dichloromethane 1,1,2-trichloroethane methane ethane propane butane pentane

0.0087 0.0033 0.0094 0.0076 0.0083 0.0084 0.0015 0.0028 0.0046 0.0054

0.0045 0.0026 0.0074 0.0071 0.0084 0.0026 0.0020 0.0029 0.0029 0.0041


0.0079 0.0031 0.0078 0.0084 0.0079 0.0098 0.0016 0.0033 0.0050 0.0058

0.0054 0.0025 0.0076 0.0074 0.0080 0.0029 0.0020 0.0031 0.0033 0.0044

analytical windows corresponding to absorption features, EQDI0 0.0024 0.0020 0.0070 0.0033 0.0042 0.0027 0.0022 0.0028 0.0035 0.0035

0.0017 0.0022 0.0028 0.0031 0.0033 0.0013 0.0032 0.0026 0.0033 0.0029

0.0022 0.002 0.003 0.0032 0.0032 0.0030 0.0033 0.0040 0.0037 0.0040

0.0034 0.0017 0.0031 0.0033 0.0031 0.0044 0.0035 0.0031 0.0057 0.0041

0.0026 0.0017 0.0035 0.0044 0.0036 0.0031 0.0048 0.0037 0.0032 0.0039

0.0041 0.0019 0.0036 0.0033 0.0033 0.0042 0.0033 0.0030 0.0062 0.0045

TO-16 analytical windows, EQDI0 0.0022 0.0014 0.0049 0.0049 0.0045 0.0029 0.0027 0.0031 0.0035 0.0040

0.0018 0.0016 0.0039 0.0038 0.0047 0.0014 0.0040 0.0027 0.0033 0.0037

OP/FT-IR background spectra used in this study was low. Had detector noise been the dominant noise source, the RMSEP would be expected to drop by a factor of x2 every time that ∆ν˜ is doubled, since all interferograms were acquired with the same number of scans. This is obviously not the case.

alkanes) are similar to the widths of the analytical bands of the chlorinated hydrocarbons. Methane and ethane might be expected to exhibit different behavior than propane, n-butane, and n-pentane because their rotational fine structure is resolved when ∆ν˜ < 4 cm-1 (8 cm-1 in the case of methane).

We believe that the behavior exhibited in Figure 1 can be ascribed to two causes. First, for each of the analytes, the fwhh of the rotational contour of each band is greater than ∆ν˜ for 1 < ∆ν˜ < 8 cm-1. Thus the band contour is scarcely affected when the resolution is changed from 1 cm-1 to 8 cm-1. When ∆ν˜ > 8 cm-1, the peak absorbance of the analyte bands starts to decrease slowly (see Paper 1, Figure 1). On the other hand, the air-broadened width of the lines of atmospheric water vapor is about 0.2 cm-1, so the intensity of these lines is always reduced significantly each time ∆ν˜ is increased (i.e., the resolution is degraded). Thus the intensity of the interfering water lines is decreased by a greater factor than the analyte bands as the resolution is degraded. This effect would lead us to expect that the greatest improvement in RMSEP would be found when the resolution is degraded from 1 to 4 cm-1. Second, as the resolution becomes very low (∆ν˜ g 16 cm-1), the analyte spectra are broadened to the point that they start to overlap significantly. This effect would lead us to expect that the RMSEP would start to increase when the resolution is ∆ν˜ > 8 cm-1. In summary, therefore, for measurement of many HAPs by OP/ FT-IR, measurement at a resolution of ∼8 cm-1 is indicated.

Surprisingly, the RMSEP for methane drops significantly when ∆ν˜ is degraded from 1 to 8 cm-1. Methane absorbs at a higher frequency than the other alkanes and is, therefore, even more susceptible to interference by water lines than the other alkanes. The RMSEP for ethane shows a steady decrease as the resolution is degraded from 1 to 16 cm-1. Ethane absorption features are located in a region where they strongly overlap with the bands of propane, n-butane, and n-pentane. We believe that the reason that ethane behaves so differently from all the other analytes is because the lines in its vibration-rotation spectrum are so easily distinguished from water and the other alkanes when the spectrum is measured at fairly high resolution. As the resolution is degraded, the band profile of ethane begins to look more like that of propane, n-butane, and n-pentane in the same region so that RMSEP starts to increase.

This trend can also be seen for propane, n-butane, and n-pentane, which is not unexpected because the widths of the rotational contours of the C-H stretching bands of these analytes (which are by far the strongest in the spectra of the 1348

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Effect of Background Type. As in the first paper in this series (2), three types of background spectra were compared. Data sets denoted as EQDI0 were measured over a path length equal to the sample spectrum, whereas the SPI0 and SPBI0 data sets were measured with a short-path background, i.e., with the retroreflector held within a meter of the telescope. Spectra in the SPI0 data set were not baseline corrected, whereas the specta in the SPBI0 set were. The RMSEPs found for the EQDI0, SPBI0, and SPI0, data sets are given in Tables 1-3, respectively. Even for the EQDI0 set, the effect of

TABLE 2. RMSEP (ACU) of PLS-2 Calibrations Using Baseline Corrected Short Path Reference Backgrounds as a Function of Resolution and Spectral Window full atmospheric window, SPBI0 resolution (cm-1) 1

2

trichloromethane0.0157 1,4-dichlorobenzene 1,2-dichloroethane dichloromethane 1,1,2-trichloroethane methane ethane propane butane pentane

0.0144 0.0050 0.0269 0.0170 0.0229 0.0219 0.0142 0.0295 0.0287 0.0236

0.0136 0.0049 0.0250 0.0133 0.0128 0.0118 0.0164 0.0230 0.0334 0.0210


0.0177 0.0049 0.0222 0.0150 0.0203 0.0123 0.0027 0.0050 0.0074 0.0070

0.0149 0.0049 0.0186 0.0119 0.0149 0.0041 0.0034 0.0037 0.0070 0.0068


0.0174 0.0047 0.0217 0.0144 0.0203 0.0131 0.0027 0.0050 0.0079 0.0074

0.0153 0.0049 0.0210 0.0117 0.0134 0.0046 0.0033 0.0039 0.0072 0.0068

4 0.0080 0.0048 0.0242 0.0119 0.0094 0.0161 0.0198 0.0170 0.0200 0.0158

8 0.0067 0.0047 0.0140 0.0100 0.0125 0.0199 0.0152 0.0138 0.0226 0.0143

16 0.0113 0.0046 0.0115 0.0092 0.0118 0.0170 0.0138 0.0163 0.0301 0.0183

32 0.0063 0.0192 0.0083 0.0094 0.0251 0.0151 0.0308 0.0436 0.0248

analytical windows corresponding to absorption features, SPBI0 0.0104 0.0043 0.0183 0.0096 0.0119 0.0042 0.0039 0.0046 0.0052 0.0051

0.0093 0.0044 0.0147 0.0064 0.0082 0.0028 0.0049 0.0042 0.0048 0.0047

0.0056 0.0047 0.0109 0.0078 0.0063 0.0040 0.0067 0.0055 0.0051 0.0050

0.0063 0.0047 0.0110 0.0072 0.0050 0.0056 0.0056 0.0042 0.0078 0.0047

0.0052 0.0039 0.0134 0.0091 0.0079 0.0040 0.0065 0.0054 0.0051 0.0047

0.0065 0.0028 0.0100 0.0074 0.0071 0.0077 0.0044 0.0042 0.0081 0.0053

TO-16 analytical windows, SPBI0

interference by atmospheric water lines is immediately obvious. With the addition of OP/FT-IR backgrounds the prediction errors increase from ∼10-7 ACU that was found when the spectra were synthesized using noise-free baselines to values between 0.0027 and 0.0569 ACU for the EQDI0 set. These results show that, even though the presence of incompletely compensated atmospheric spectral features and detector noise undeniably degrade the performance of PLS, they induce an average relative error of less than 2%, which is an acceptable accuracy for OP/FT-IR spectrometry. It should be noted that a RMSEP of 0.5 is equivalent to a relative error of 100%. Thus an RMSEP of less than 0.01 is required if the average prediction error is to be less than 2% of full scale (1 ACU). These results are typically between 1 and 2 orders of magnitude better than the corresponding results found by CLS. A more detailed comparison is given at the end of this paper. The effect of not compensating for the effect of the atmospheric bands at all is readily seen by comparing the results obtained with the SPBI0 and EQDI0 data sets. The spectra in both data sets are characterized by level baselines centered at zero absorbance units (AU). Therefore, any differences in prediction errors may be attributed primarily to the number and strength of atmospheric absorption features present in the spectra. The improvement in RMSEP on changing from a short-path to an equal-path background spectrum is consistent but remarkably small. For example, the RMSEPs for ethane using TO-16 analytical windows are 0.0027 for the SPBI0 set and 0.0016 for the EQDI0 set. In most cases the improvements vary from less than a factor of 2 to a factor of 3. When the corresponding computations were performed by CLS, the average RMSEP obtained for the SPBI0

0.0110 0.0040 0.0186 0.0091 0.0108 0.0043 0.0041 0.0047 0.0052 0.0052

0.0092 0.0040 0.0154 0.0068 0.0086 0.0028 0.0049 0.0042 0.0050 0.0048

data set was greater than 0.5 ACU, i.e., the results were completely random. This result demonstrates the capability of PLS to compensate for the effect of completely uncompensated atmospheric absorption lines with far better accuracy than CLS. A second type of spectral aberration found in absorption spectra created from short-path reference spectra is baseline drift. The baseline drift is manifested as both nonzero and sloping baselines. The similarity between the results obtained with the SPI0 and SPBI0 data sets with the same number of factors shows that PLS is well able to model this type of variation. Effect of Spectral Window Selection. The effect of the size of the spectral window used in the PLS model can be seen by comparing the RMSEP values obtained with the normal atmospheric windows (700-1300, 2000-2150, and 2400-3000 cm-1), referred to as Type 1 spectra in Paper 1 of this series, analytical windows where at least one of the analytes of interest had an absorbance of 0.015 AU or higher at a concentration of 1 ACU at 1-cm-1 resolution (Type 2 spectra), and by using the spectral region that corresponds to the top 90% of the most useful single band for each analyte (usually the strongest and in one of the atmospheric windows), called Type 3 spectra in the previous paper. The last window is the one recommended in U.S. Environmental Protection Agency (EPA) Compendium Method TO-16 (5). The results for the three types of windows are tabulated for each type of background in Tables 1-3. The results for dichloromethane are representative of most of the analytes studied and are summarized in Figure 3. It can be seen that the RMSEP is much less strongly dependent on the spectral window selected than with CLS, but in general the best results VOL. 34, NO. 7, 2000 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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TABLE 3. RMSEP (ACU) of PLS-2 Calibrations Using Short Path Reference Backgrounds as a Function of Resolution and Spectral Window full atmospheric window, SPI0 resolution (cm-1) 1

2

4

8

16

32

0.0159 0.0053 0.0287 0.0170 0.0224 0.0186 0.0150 0.0359 0.0304 0.0198

0.0170 0.0047 0.0219 0.0149 0.0162 0.0135 0.0171 0.0287 0.0380 0.0210

0.0126 0.0047 0.0244 0.0112 0.0126 0.0162 0.0185 0.0194 0.0313 0.0175

0.0090 0.0054 0.0181 0.0122 0.0114 0.0291 0.0244 0.0182 0.0364 0.0208

0.0124 0.0079 0.0199 0.0102 0.0141 0.0400 0.0189 0.0341 0.0569 0.0299


0.0197 0.0057 0.0289 0.0191 0.0215 0.0304 0.0139 0.0269 0.0434 0.0248


0.0152 0.0128 0.0271 0.0201 0.0191 0.0139 0.0027 0.0051 0.0075 0.0071

0.0137 0.0122 0.0242 0.0154 0.0180 0.0044 0.0036 0.0035 0.0071 0.0067


0.0166 0.0077 0.0259 0.0205 0.0192 0.0100 0.0028 0.0055 0.0085 0.0078

0.0148 0.0058 0.0224 0.0126 0.0202 0.0050 0.0037 0.0036 0.0071 0.0067

analytical windows corresponding to absorption features, SPI0 0.0108 0.0069 0.0215 0.0136 0.0170 0.0049 0.0041 0.0047 0.0046 0.0045

0.0112 0.0045 0.0145 0.0102 0.0147 0.0029 0.0047 0.0043 0.0048 0.0048

0.0079 0.0045 0.0093 0.0098 0.0127 0.0041 0.0056 0.0050 0.0054 0.0053

0.0062 0.0072 0.0160 0.0069 0.0083 0.0065 0.0054 0.0046 0.0080 0.0049

0.0080 0.0039 0.0137 0.0098 0.0120 0.0040 0.0067 0.0055 0.0058 0.0050

0.0088 0.0043 0.0159 0.0078 0.0095 0.0077 0.0068 0.0049 0.0080 0.0051

TO-16 analytical windows, SPI0 0.0127 0.0033 0.0191 0.0132 0.0179 0.0048 0.0042 0.0048 0.0049 0.0048

0.0122 0.0029 0.0173 0.0109 0.0132 0.0029 0.0046 0.0042 0.0050 0.0048

FIGURE 3. Comparison of the effect of spectral window type on the RMSEP for dichloromethane using EQDI0 backgrounds.

FIGURE 4. Comparison of the effect of spectral window type on the RMSEP for ethane using EQDI0 backgrounds.

are found with the Type 2 spectra and the poorest with the Type 1 spectra. The results for ethane are illustrated in Figure 4. While the effect of resolution on the accuracy to which the concentration of ethane can be predicted is qualitatively different from dichloromethane, the same trends seen in Figure 3 are observed, with Type 2 and 3 spectra giving very similar results. The performance of PLS with OP/FT-IR is best when the spectral windows have the greatest amount of information for the analyte while still allowing other sources of variance to be compensated. Using the full atmospheric windows apparently does not allow the PLS algorithm to measure the analyte to a high enough accuracy. On the other hand, reducing the spectral window to a single band reduces the predictive ability of PLS by limiting the correlation between

the analyte concentration and its absorbance in the presence of baseline drift and uncompensated atmospheric absorption. Often the absorption features that provide a chemometric technique with the most correlated information are very different from the ones that would be selected by a trained spectroscopist. Comparison with CLS. On average the PLS results are 120 times better than the results from CLS on the identical calibration and validation matrices. This is illustrated by the histogram in Figure 5, showing the results for all PLS and CLS predictions performed in this project. The improvement factor for predictions by PLS versus CLS ranged from 12.7 (methane, 1-cm-1 resolution, EQDI0 spectra, TO-16 analytical window) to 747 (propane, 32-cm-1 resolution, SPI0 spectra, TO-16 analytical window). The EQDI0 spectra showed average

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types were 0.446, 0.692, and 1.426, respectively, for CLS and 0.006, 0.011, and 0.012, respectively, for PLS. These numbers again show the superior predictive ability of PLS when working with OP/FT-IR spectra. It can also readily be seen that while proper spectral window selection and the use of equidistant reference spectra are paramount in a valid CLS calibration, PLS works very well using larger spectral windows and short-path reference spectra. This reduces the need for a trained spectroscopist and decreases the logistical problems involved in obtaining an equidistant background.

Literature Cited

FIGURE 5. Histogram of the RMSEP results for all data sets using PLS (gray) and CLS (black). improvements of 73.6, while the SPBI0 and SPI0 showed improvements factors of 88.5 and 199, respectively. This indicates the very robust nature of PLS, with its superior ability to model atmospheric absorption features and once more emphasizes the need to use equidistant reference spectra when using CLS. A comparison of the RMSEPs obtained by the two multivariate calibration systems for the three spectra types, EQDI0, SPBI0, and SPI0, for all window

(1) Martens, H.; Naes, T. Multivariate Calibration; John Wiley & Sons: New York, 1994. (2) Hart, B. K.; Griffiths, P. R. Environ. Sci. Technol. 2000, 34, 13371345. (3) Wang, T. W.; Batra, J.; Sarma, P.; Khettry, A.; Berry, M.; Hansen, M. The First International Chemometrics InterNet Conference, InCINC′94; 1994; http://www.emsl.pnl.gov:2080/docs/incinc/ homepage.html. (4) Hart, B. K.; Griffiths, P. R. Proc. 11th Int. Conf. Fourier Transform Spectrosc. 1998, 430, 241. (5) Compendium Method TO-16 Long-Path Open-Path Fourier Transform Infrared Monitoring of Atmospheric Gases; EPA/625/ R-96/010b; U.S. Environmental Protection Agency: Research Triangle Park, NC, 1999.

Received for review April 19, 1999. Revised manuscript received January 3, 2000. Accepted January 3, 2000. ES990439V

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Effects of Resolution on Quantification in Open-Path Fourier Transform

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