application to two-dimensional time domain NMR data

Analyst, April 1998, Vol. 123 (551–559)

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Method for detecting information in signals: application to two-dimensional time domain NMR data† D. N. Rutledge* and A. S. Barros Laboratoire de Chimie Analytique, Institut National Agronomique, 16 Rue Claude Bernard, 75231 Paris Cedex 05, France. E-mail: [email protected]

Time domain (TD) NMR is used in industry for quality control. Like near-infrared (NIR) spectrometry, it has many advantages over wet chemistry including speed, ease of use and versatility. Unlike NIR, TD-NMR can generate a wide range of responses depending on the particular pulse sequences used. The resulting relaxation curves may vary as a function of the physico-chemical properties or even the biological and geographical origin of the product. The curves are usually decomposed into sums of exponentials and the relaxation parameters are then used in regression models to predict water content, iodine number, etc. The diversity of possible signals is both an advantage and disadvantage for TD-NMR as it broadens the range of potential applications of the technique but also complicates the development and optimisation of new analytical procedures. It is shown that univariate statistical techniques, such as analysis of variance or chi-squared, may be used to determine whether a signal contains any information relevant to a particular application. These techniques are applied to 2D TD-NMR signals acquired for a series of traditional and ‘light’ spreads. Once it has been demonstrated that the signals contain relevant information, partial least-squares (PLS) regression is applied directly to the signals to create a predictive model. The Durbin–Watson function is shown to be a means characterising the signal-to-noise ratio of the vectors calculated by PLS to select the components to be used in PLS regression. Keywords: Time domain NMR; chemometrics; analysis of variance; partial least squares; Durbin–Watson

Time domain nuclear magnetic resonance (TD-NMR) is often used to quantify major proton-containing constituents in agrofood products or to monitor their evolution during processing. Like Raman, FT-IR and NIR spectrometry, it has many advantages over wet chemistry, including speed, ease of use and versatility. TD-NMR has two major advantages over these other instrumental techniques: it is possible to obtain non-invasively a signal from the whole of the sample, not just a superficial layer; and the nature of the signal observed depends on the particular radiofrequency pulse sequences applied to the sample to excite the protons. One can generate very different signals depending on the particular pulse sequences, Carr–Purcell– Meiboom–Gill (CPMG), inversion–recovery (I-R), progressive saturation (PS), Hahn spin echo (HSE), free induction decay (FID), Goldmann–Shen (GS), spin locking, Jeener–Broekhaert, etc., used to excite the sample.1–3 The resulting relaxation curves may vary depending on the water content, hydration state, solid fat content, iodine number or even biological or geographical origin of the studied product. This diversity of responses is both an advantage and a disadvantage for NMR compared with other instrumental techniques, as it broadens the †

Presented at the XXX Colloquium Spectroscopicum Internationale (CSI), Melbourne, Australia, September 21–26, 1997.

range of potential applications of the technique but also complicates the development and optimisation of new analytical procedures. As far as TD-NMR is concerned, where the objective is usually the development of a rapid instrumental method of quantification or characterisation,4 this apparently unlimited number of possible signals means that it is often difficult, or at least time consuming, to determine which pulse sequence, if any, produces a signal with information content. The objective of this study was to demonstrate that chemometric techniques, which are already widely used in NIR spectrometry,5 can facilitate the detection of information in a signal. These techniques may also be used for the analysis of other signals, such as spectra,6 chromatograms or sensor responses. There has been a small number of recent examples of the application of chemometrics to the analysis of TD-NMR signals. Rutledge et al.7 used ANOVA and factor analysis to examine the effects of factors such as fat and moisture content, pH and temperature on the one-dimensional relaxation curves of spreads and gelatines, and to determine the stoichiometry of a complexation reaction. Davenel et al.8 applied multivariate statistics to the relaxation curves of doughs during cooking, Gerbanowski et al.9 compared partial least squares (PLS) regression applied to relaxation curves with PLS and multiple linear regression (MLR) applied to estimated relaxation parameters. Vackier and Rutledge10,11 applied univariate and multivariate statistical techniques to relaxation curves to study the influence of the physico-chemical characteristics of gelatines on their relaxation properties. Clayden et al.12 applied factor analysis to 19F TD-NMR FID signals acquired for a set of PTFE samples of varying crystallinity. In this study, the univariate techniques ANOVA and chisquared were used to determine whether the two-dimensional TD-NMR relaxation surfaces acquired for a series of traditional and ‘light’ spreads contain information on their moisture content. Once it is demonstrated that the signals contain relevant information, the multivariate regression technique PLS is applied directly to the signals to create a predictive model. The Durbin–Watson function is shown to be a means for characterising the signal-to-noise ratio of the loadings and B-coefficient vectors calculated by PLS to select the components to be used in PLS regression. As the term indicates, univariate techniques analyse only one variable at a time—each of the N points in a set of M TD-NMR relaxation curves, NIR spectra or gas chromatograms is analysed separately. These univariate techniques therefore require little computer memory and can be very rapid. However, they have the disadvantage of neglecting any correlations or interactions between the variables and only revealing proximities between samples within the uni-directional space of the individual variable. Multivariate techniques, such as factor analysis or PLS regression, analyse the total N 3 M data matrix at one time, with the advantage of giving information about interactions between variables and similarities between samples within the multi-dimensional space of all the variables. This, however, makes much greater demands on computer memory and calculation time.

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Experimental Samples Twelve samples of different traditional and ‘light’ butters and margarines were studied. The moisture contents, determined in triplicate by Karl Fischer titration, ranged from 12 to 25% for the traditional spreads and from 31 to 60% for the light spreads. The fat content given by the manufacturers ranged from 25 to 82% for the butters and from 60 to 80% for the margarines. The number of samples used is small, but the objective here is to demonstrate the methods of detecting information in signals, not to develop a robust multivariate predictive model. The very strong anticorrelation between the fat and moisture content of the samples (Fig. 1) makes it impossible to separate their effects on the signal. Any characterisation based on moisture content necessarily parallels a characterisation due to fat content. NMR measurements NMR tubes of 10 mm od were filled to a height of 10 mm. The samples were stabilised at 20 °C before being transferred to the NMR apparatus thermostated at 20 °C. The measurements were performed in triplicate on a 20 MHz TD-NMR apparatus (QP20+, Oxford Instruments, Oxford, UK) with phase quadrature detection and a maximum acquisition frequency of 10 MHz. A VISUAL BASIC program, developed in collaboration with Oxford Instruments, was used to generate the pulse sequences and to acquire the data. This program can be used to acquire two-dimensional relaxation signals and a set of relaxation curves was acquired by inserting a CPMG sequence into an I–R sequence: [(180°)x–tvar–CPMG–RD]n where CPMG = (90°)x–t–[(180°)y–2t–(180°)y–2t– (180°)y–2t–(180°)y–t–Acq–t]m tvar(i) = 2000 3 1.55(i21) ms for i = 1 to n; n = 20; t = 1000 ms; m = 100; RD = 3 s; and four scans are performed with phase cycling. This sequence gave a 2000 variable vector of 20 T1-weighted 100-point CPMG curves (Fig. 2). This vector can be folded back, by putting each CPMG curve on a separate line, to produce a 20 3 100 matrix corresponding

Fig. 1

Correlation between fat and moisture contents of the spreads.

to a T1–T2 relaxation surface made of transverse relaxation curves, weighted by longitudinal relaxation (Fig. 3). It should be noted that the exponential form of the I–R curves is distorted by the equal spacing of the data points in the graphic, resulting in a sigmoid shape. Different statistical techniques were then applied to the set of 36 vectors, using a program developed in the laboratory, and validated with Matlab routines.13 Statistical analyses Analysis of variance If the samples being examined can be classed into definite groups, it is possible to calculate the amount of the variation in the signal intensity due to the samples belonging to particular groups.14 By difference from the total variability of the measurement, it is possible to calculate the amount of variability that is not due to the groups. For each group one can then calculate the group variance: g

VG =

∑ j =1

(

nj x j − x

)

2

(1)

(g − 1)

and the residual variance: g

VR =

nj

∑∑ j =1

i =1 g

(

n j x ji − x

∑(

)

2

(2)

)

nj − 1

j =1

where g = number of groups, nj = number of samples in group j, xij = value for sample i in group j, x¯ j = mean value for group j, x= = grand mean value and N = total number of samples. If the variables studied are in fact points in a signal, such as a TD-NMR relaxation curve, it is interesting to plot these variance values as a function of their position in the signal. Regions in the signal that vary systematically from one group to another will give high VG values. In addition, if there are no important differences between the samples, other than those due to the groups, the VR will be low. A high VG associated with a low VR indicates a significant influence of the factor used to create the a priori grouping on the signal intensity. Fisher’s F value is commonly used to estimate the level of significance of the influence of the factor studied on the variable x. The F value for the variable x may be calculated as

Fig. 2 Series of T1 weighted CPMG relaxation curve for a ‘light’ butter.

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Fx =

VG VR

(3)

Other statistical functions may also be of interest to characterise the effect of the factors on the signal intensity. The group standard deviation (SDG), given by g

SDG = VG =

∑ j =1

(

nj x j − x

)

2

(g − 1)

∑

(

j

−x

)

nj x j − x

)

2

(g − 1) SDG (5) = x x has the added advantages of being signed and of correcting for the distortion introduced by differences in grand mean signal intensities. CVG =

The absolute value of this statistic may then be used, like Fisher’s F, to give a level of significance for the groupings. It also has the advantage of being in the same units as the signal.

For most, if not all, instrumental techniques, the intensities of adjacent points in signals containing information are strongly correlated. Therefore, when the variables being studied are points in a signal, it is interesting to plot the above statistics as a function of their position. Regions in the signal that vary systematically from one group to another will give high absolute values for SDG, F, VG, c2G and CVG. In addition, the values of SDG, F, VG, c2G and CVG will have a structured distribution as a function of position of the point, as they are not random but evolve as a function of the information in the signal. It is possible to have an objective measure of this structure or non-randomness. The Durbin–Watson D statistic is commonly used to determine whether the residuals after a regression are randomly distributed.15 This statistic is given by

∑( n

This test is commonly used to determine whether the observed frequencies for a particular set of events differ significantly from their expected frequencies:

∑ j =1

(

Oj − E j

)

(7)

x

j =1

Chi-squared criterion

g

2

Durbin–Watson criterion

j =1

χ2 =

∑

(x

(4)

has the advantage of giving values which are in the same units as the signal, making comparisons easier. Similarly, the group coefficient of variation (CVG) (relative standard deviation), given by g

g

χ2G =

553

2

(6)

Ej

(c2)

Chi-squared may also be used to test the significance of the grouping by using the grand mean as the expected value and the means for each group as the observed values:

Fig. 3

D=

δxi − δxi − 1

)

2

i=2

∑( n

δxi δxi )

(8)

i=2

where dxi and dxi21 are the residuals for successive points in a series. For n > 100, the distribution is random with a 95% confidence interval for D between 1.7 and 2.3.

T1–T2 relaxation surface for a ‘light’ butter.

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Partial least-squares regression PLS regression is a least-squares regression procedure based on regressing a reduced set of uncorrelated, linear combinations, T, of the original independent variables, X, on to the dependent variable, Y. It is very similar to principal components regression, where the T are simply the principal components, but in PLS the T are calculated iteratively, maximising their covariance with Y.16 This procedure helps to avoid or reduce the collinearity problems associated with strongly correlated variables and, as a consequence, to have a better predictive model. This predictive regression model is of the form : Y = X·B + e (9) where B, the vector of B coefficients, is calculated using the loadings of the X variables on the T vectors. As before in the case of the univariate statistics, the structure of both the loadings vectors and B coefficient vectors may be used to give indications on the information content of the signal analysed. The Durbin–Watson statistic can help in determining the optimum number of components to be used in the model. Results and discussion Univariate analyses Effect of moisture content The ANOVA procedure was first applied to the set of 36 I–R weighted CPMG curves for the butters and margarines using the factor ‘light’/‘traditional’ as a grouping criterion. To eliminate signal intensity variations due to differences in sample size, the vectors were first normalised to unit maximum intensity. Fig. 4(a) presents the group variance surface based on this grouping of the samples and Fig. 4(b) presents the corresponding residual variances. The values close to zero at the beginning of the CPMG and at the ends of the I–R in Fig. 4(a) are due to the normalisation of the signals. As these points usually have the highest intensity, they are all set close to unity and their variability is therefore almost zero. It is clear that both the VG and VR surfaces are ‘structured’. The VG surface indicates that the relaxation surfaces contain information on the moisture content, whereas the VR surface shows that there is another factor which influences the relaxation to a certain extent. The regions which are most sensitive to the ‘Light’/‘Traditional’ nature of the samples are, on the one hand, the beginning of the CPMG curves and the

Fig. 4

middle of the I–R, near its null point, and on the other hand, the extremities of the I–R and all along the CPMG. One could therefore suppose that the discrimination was based on a variation in the properties of the constituents of the spreads with a fast T2 relaxation rate (early part of the CPMG) and whose T1 varies (null point of the I–R). This hypothesis will be confronted below with the results of the other calculations. From these figures, it can be seen that the highest VG is 100 times greater than the highest VR. The Fisher’s F values calculated from these two sets of values are plotted in Fig. 5(a). It is clear from this graph that the F plot is much ‘noisier’ than the VG plot because of the contribution from the VR, and that the F value is high in some areas simply because the corresponding VR values are small. This problem will become all the more important when the factor used to group the samples explains more of the variability in the signal, resulting in small, ‘noisy’ VR values. For this reason, despite its usual interest as a measure of the level of significance of the effect of a factor, the Fisher’s F plot will rarely be of use as a means to detect information in a signal. The group standard deviation surface was plotted [Fig. 5(b)] in order to be able to compare more easily the evolution of the variability due to the factor with the evolution of the original relaxation surface. As expected, the topography of this surface is less pronounced than that of VG and the evolution of the surface in the CPMG direction is more like that of the relaxation surface (Fig. 3). The coefficient of variation is not only on the same scale as the original relaxation surface, but has also been corrected for differences in the grand average signal intensity at each point on the relaxation surface. If the variability in the signal intensity due to the factor is proportional to the average signal intensity, then the CVG surface should remain almost constant, making it easier to pinpoint regions of unusual variability. However, this plot introduces a major distortion when the signal changes sign or the signal intensity approaches zero, as can be seen here when the I–R signal goes through the null point. In this figure it is possible to see that the position of this passage through the null point shifts to a later position in the I–R direction, going from point 11 to point 15, at the same time as it moves down the CPMG direction. Given the fact that for a monoexponential T1 relaxation, the null point corresponds to T1ln2, and given the delays used in the I–R pulse sequence, the observed shift would correspond to T1 changing from about 230 to 1300 ms.

(a) Group variance and (b) residual variance surfaces based on the factor ‘light’/‘traditional’.

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These approximate values are not incompatible with the T2 values calculated for these samples. A detailed analysis of the CPMG relaxation curves was performed using CONTIN17 to decompose them into a continuous distribution of relaxation times, and MARQT18,19 to have a discrete sum of exponentials. The whole butters and margarines, and also the separated aqueous and lipid phases, were studied (results not shown). The T2 values of the two relaxation components in the butters were about 30 and 200 ms, and those in the margarines were close to 50 and 500 ms. In the light spreads, the values were close to 100 and 1000 ms for the margarines and largely unchanged at about 30 and 200 ms for the butters. An expansion of the plot [Fig. 5(c)] gives information on the zones of the relaxation surface which are most sensitive to the factor ‘light’/‘traditional’. Except for a small region at the beginning of the CPMG and the zone where the I–R passes through zero, the CVG is almost constant, meaning that the effect of the factor on the relaxation surface is proportional to the intensity of the signal. The low CVG values at the beginning of the CPMGs are due to the sample sizes being very similar. When the grand mean signal intensity approaches zero at the I– R null point, the CVG becomes undefined, making difficult the

555

interpretation of the surface. Because of the change in sign of the signal intensity, the CVG values change sign at the null point. The chi-squared test also introduces a distortion when the signal intensity approaches zero. The expanded chi-squared plot [Fig. 5(d)], like the CVG plot, shows a uniform effect of the factor on the rest of the relaxation surface. The CVG and chisquared plots show that the discrimination ‘light’/‘traditional’ is in fact due to relaxation components present throughout the CPMG curve, and not just the fast relaxing component as appeared to be indicated by the VG plot. By calculating the Durbin–Watson D from successive values in the vectors of SDG, F, VG, VR, c2G and CVG values, it is possible to have an indication of the non-randomness of the values or, in other words, of the information content of these vectors. For each of these statistics, the D values were calculated for the vectors in the I–R direction and plotted as a function of the corresponding CPMG point [Fig. 6(a)]. The same calculation was done in the CPMG direction and the results were plotted as a function of the I–R point [Fig. 6(b)]. Because of the null point in the I–R curve, the c2G and CVG vectors give D values close to 2, as can be seen in Fig. 6. The VR

Fig. 5 (a) Fisher’s F, (b) group standard deviation, (c) expanded plot of the coefficient of variation and (d) expanded plot of chi-squared for the factor ‘light’/ ‘traditional’.

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values are higher than SDG, F and VG for the I–R vectors and higher than SDG and VG for the CPMG vectors, outside the nullpoint zone, confirming the lower signal-to-noise ratio of this vector. Effect of type of fat The ANOVA procedure was applied to the same set of normalised T1–T2 relaxation surfaces, this time using the factor ‘butter’/‘margarine’ as the grouping criterion. Fig. 7(a) presents the group variance surface and Fig. 7(b) the corresponding residual variance, based on this grouping of the samples. It can be seen from these figures that the highest VG is only three times greater than the highest VR. Although this VG surface is also ‘structured’, the signal-to-noise ratio is much lower than in the previous case, indicating that the relaxation surface contains less information about this factor. The VR plot for this factor has a higher signal-to-noise ratio and has a structure similar to the VG for ‘light’/‘traditional’. This confirms the predominant effect of this latter factor in determining the relaxation properties of the spreads. The general complementarity of the two VG and VR plots indicates that the two factors ‘light’/‘traditional’ and ‘butter’/‘margarine’ are responsible for nearly all the variability in the relaxation surfaces. The Fisher’s F values are plotted in Fig. 7(c), and confirm that the grouping based on ‘butter’/‘margarine’ is much less significant than that based on ‘light’/‘traditional’. The F plot is, once again, much ‘noisier’ than the VG plot, confirming its limited utility in the case of signal intensities as a measure of the level of significance of the effect of a factor. The group standard deviation surface plotted in Fig. 7(d) shows once again that irregularities in the surface are less pronounced than for VG and its evolution is similar to that of the original relaxation surface. The coefficient of variation plot shows a major distortion again when the signal intensity passes through the null point of the I–R signal. An expansion of the plot [Fig. 8(a)] shows that the effect of this factor is not proportional to the intensity of the relaxation surface. The CPMG curves vary more towards the end. This is also the case for the chi-squared plot [Fig. 8(b)]. This greater variability in the final part of the CPMGs implies a change in the proportions or relaxation properties of the slower relaxing T2 components. One would expect the nature of the lipid phase to be responsible for this ‘butter’/‘margarine’ differences between the samples, but as the region of greatest variance is at the end of the CPMG curves, this would imply that

Fig. 6

the lipid phase has a longer transverse relaxation time than the aqueous phase. The analysis of the CPMG relaxation curves using CONTIN17 and MARQT18,19 showed that the major differences were in the proportion of aqueous phase between ‘light’ and traditional, and in the longer relaxation times of the aqueous phase of the margarine samples. As before, the Durbin–Watson D was calculated for the vectors of SDG, F, VG, VR, c2G and CVG values in the I–R and CPMG directions. Once again, the c2G and CVG vectors gave D values close to 2, as can be seen in Fig. 9. However, outside the null-point zone, the VR values are lower than the others, confirming the higher signal-to-noise ratio of this vector. Multivariate analysis Having shown that the relaxation surfaces contain information on the moisture content of the samples, it should be possible to use them to develop a predictive model. PLS regressions were performed using the 36 2000-point vectors as the X-matrix and the corresponding moisture contents, determined by Karl– Fischer titration, as the Y-matrix. The major danger in using multivariate regression techniques on signals, where there are usually many more variables than samples, is over-fitting, in which case noise present in the signal is included in the model to adjust it to the data. This problem may be partly overcome by limiting the number of principal components included in the model. The choice of the cut-off point is sometimes made based on the evolution of the X-matrix eigenvalues, or more usually by examining the root mean squared error of prediction (RMSEP) values calculated using test-set validation or cross-validation. As will be seen below, the Durbin–Watson criterion may be used as a means of detecting which principal components contain most information. Fig. 10 clearly shows the increase in noise intensity and decrease in ‘structure’ with increasing number of the principal component. As the B coefficient vectors are derived from these loadings, if more than the optimal number are used, excessive noise will be included. The evolution of the Durbin–Watson values for both the B coefficient vectors and the X-matrix loadings vectors for the first 10 principal components are plotted in Fig. 11, along with the corresponding eigenvalues. An abrupt increase in the Durbin–Watson values occurs after three principal components, confirming the evolution of the eigenvalues. Although both criteria indicated the same maximum number of components, the Durbin–Watson criterion has the advantage over the eigenvalue of being a measure of the structured variability in

Durbin–Watson D values of the vectors in (a) the I–R direction and (b) the CPMG direction for the factor ‘light’/‘traditional’.

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Fig. 7

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(a) Group variance, (b) residual variance, (c) Fisher’s F and (d) group standard deviation surface based on the factor ‘butter’/‘margarine’.

Fig. 8

Expanded plot of (a) the coefficient of variation and (b) chi-squared for the factor ‘butter’/‘margarine’.

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the eigenvectors derived from the X-matrix, and not just of the proportion of the total variability extracted from the Xmatrix. Fig. 12 presents the B coefficient surface for the threecomponent PLS regression model between the T1–T2 relaxation surfaces and the measured moisture content. There is a definite similarity between the B coefficient surface and the group variance surface, as both contain information on the variability of the X-matrix, the set of relaxation surfaces. The regression line for the prediction of the moisture content of the butters and margarines based on the T1–T2 relaxation surfaces is plotted in Fig. 13. The statistics for the model are root mean squared error of calibration (RMSEC) = 4.06% and R2 = 0.967. However, as was seen in Fig. 1, the fat and moisture contents of these samples are strongly anti-correlated so it is impossible to distinguish their effects on the relaxation surfaces. The PLS regression model for the moisture content will necessarily

Fig. 9

Fig. 10 clarity.

Fig. 11 Durbin–Watson values for the B coefficients and X-matrix loading vectors and the eigenvalues of the first 10 PLS principal components.

Durbin–Watson D values of the vectors in (a) the I–R direction and (b) the CPMG direction for the factor ‘butter’/‘margarine’.

Loadings of the first 10 PLS principal components (from the bottom up) presented as vectors. The curves have been shifted vertically for

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Fig. 12 B coefficient surface for the PLS regression model between the T1–T2 relaxation surface and the moisture content.

References 1 2 3 4 5 6 7 8

Fig. 13 Regression line and 95% confidence limits for the threecomponent PLS regression model for moisture in the butters and margarines.

9 10 11

include a contribution from the effect of the anti-correlated variation in the fat content. Nevertheless, the ANOVA and chi-squared procedures have demonstrated their usefulness for the detection of information in the signals. The Durbin–Watson criterion has been shown to be of use to quantify the signal-to-noise ratio of the variability estimators and to determine the optimal number of principal components for the PLS regression. These statistical estimators all have the advantage of being very quick to calculate and requiring little computer memory. However, they have the disadvantage of being based on supervised methods, requiring the division of the data set into groups. We thank the French Institut National de la Recherche Agronomique for partial financing of the TD-NMR instrument and Stephen Provencher for the CONTIN source code.

12 13 14 15 16 17 18 19

Atta-Ur-Rahman, Nuclear Magnetic Resonance. Basic Principles, Springer, Berlin, 1986. Canet, D., La RMN: Concepts et M`ethodes, InterEditions, Paris, 1992. Farrar, T. C., and Becker, E. D. Pulse and Fourier Transform NMR, Academic Press, New York , 1971. Rutledge, D. N., J. Chim. Phys., 1992, 89, 273. Osborn, B. G., and Fearn, T., Near Infrared Spectroscopy in Food Analysis, Longman, New York, 1986. Vercauteren, J., Forveille, L., and Rutledge, D. N., Food Chem., 1996, 57(3), 441. Rutledge, D. N., Barros, A. S., and Gaudard, F., Magn. Reson. Chem., 1997, 35, 13. Davenel, A., Marchal, P., and Guillement, J. P., in Magnetic Resonance in Food Science, ed. Belton, P. S., Delgadillo, I., Gil, A. M., and Webb, G. A., Royal Society of Chemistry, Cambridge, 1995, pp. 146–155. Gerbanowski, A., Rutledge, D. N., Feinberg, M., and Ducauze, C., Sci. Aliments, 1997, 17, 309. Vackier, M. C., and Rutledge, D. N., J. Magn. Reson. Anal., 1996, 2(4), 321. Vackier, M. C., and Rutledge, D. N., J. Magn. Reson. Anal., 1996, 2(4) 311. Clayden, N. J., Lehnert, R. J., and Turnock, S., Anal. Chim. Acta, 1997, 344, 261. Matlab, Mathworks, S. Natick, MA, 1992. Czerminski, J., Iwasiewicz, A., Paszk, Z., and Sikorski, A., in Statistical Methods in Applied Chemistry, Elsevier, Amsterdam, 1990, pp. 186–207. Durbin, J., and Watson, G. S., Biometrika, 1950, 37, 409. H¨oskuldson, A., J. Chemom., 1988, 2, 211. Provencher, S., Comput. Phys. Commun., 1982, 27, 229. Marquardt, D. W., J. Soc. Ind. Appl. Math., 1963, 11, 431. Rutledge, D. N., in Signal Treatment and Signal Analysis in NMR, ed. Rutledge, D. N., Elsevier, Amsterdam, 1996, pp. 191–217.

Paper 7/07058F Received September 30, 1997 Accepted November 7, 1997