Anal. Chem. 1993, 65, 2270-2275
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Resolution Parameter for Multidetection Chromatography Jostein Toft and Olav M. Kvalheim’ Department of Chemistry, University of Bergen, N-5007 Bergen, Norway
A resolution parameter to be used for multidetection chromatography is derived. Unlike traditional resolution parameters, the parameter developed in this work can be obtained prior to curve resolution. A rank map providing the number of analytes in every local region of the profile is first estimated. The overall resolution is defined as the ratio between single-analyte regions and regions with one or more analytes. A resolution parameter for each analyte separately is obtained by the same kind of approach. The performance of the resolution parameter for suggesting successful/failed resolution, using alternating regression (AR), is assessed on simulated GC-IR and LC-DAD profiles. The influence of noise, number and relative concentration of analytes, and spectral and chromatographicoverlap on the resolution parameter is determined by means of orthogonal design and partial leastsquares regression. INTRODUCTION Multidetection chromatography has provided new possibilities in the field of multicomponent analysis. In case of poorly resolved analytes, interpretation has to be aided by curve-resolution techniques.’-$ The success or failure of the resolution depends on many factors, e.g., to which extent the specific curve-resolution technique utilizes possible selective information, the noise level of the data, and how well the analytes are resolved in the chromatographic and spectroscopic directions. The term “resolution parameter” is well-known from traditional single-detectionchromatography. The commonly used resolution parameter, R,? based on peak widths and (1) Maeder, M.; Zuberbuehler, A. D. The resolution of overlapping chromatographic peaks by evolving factor analysis. Anal. Chim. Acta 1986,181,287-291. (2) Maeder, M. Evolvingfactor analysiafor theresolution of overlapping chromatographic peaks. Anal. Chem. 1987,59,527-530. (3) Vandeginste, B. G. M.; D e r h W.; Kateman, G. Multicomponent
self-modelling curve resolution in high-performance liquid chromatography by iterative target transformation analysis. Anal. Chim.Acta 1986, 173, 253-264. (4) Vandeginste,
B. G. M.; Leyten, F.; Gerritsen, M.; Noor, J. W.; Kateman, G.;Frank, J. Evaluation of curve resolution and iterative target transformation factor analysis in quantitative analysis by liquid chromatography. J. Chemom. 1987,1,57-71. (5) Karjalainen, E. J. The spectrum reconstruction problem. Use of alternating regreasion for unexpected spectral components in twodimensional spectroscopies. Chemom. Zntell. Lab. Syst. 1989,1,31-38. ( 6 ) Karjalainen, E. J. Spectrum Reconstruction in GC/MS. The Robustness of the Solution Found by Alternating Regression. Scientific Computing and Automation; Elsevier Science Publisher B. V.: Amsterdam, 1990. (7) Kvalheim, 0. M. Liang, Y.-z. Heuristic evolving latent projections: resolvingtwo-way multicomponent data. 1. Selectivity,latent-projective graph, datascope, local rank, and unique resolution. Anal. Chem. 1992,
separation between peak maxima, has ita obvious limitations. Thus, R, is well-defined for single-detection chromatographic profiles only, and only for two overlapping peaks at a time. It further assumes either that the pure chromatographic profiles already are resolved or at least that the peak characteristics (position of peak maxima, width) are known. The resolution in multidetection chromatographic profiles can to some extent be quantified by R,. The use of R,alone, however, may give a poor picture of the resolution as all the information in the spectral direction is lost. An approach for revealing an analyte’s resolution in onedimensional spectral profiles is based upon the analyte’s net analytical signal (NAS).l0 The NAS gives an analyte’s unique information relative to the others, without any restrictions on the number of analytes. A preassumption is that the spectra of all the analytes are known a priori. Although developed for spectral profiles, NAS can just as well be used for other kinds of analytical profiles. In order to quantify the ease of resolving profiles from multidetection chromatography, Karjalainen has suggested an overlap index (OI)sp6using the covariance between each analyte’s resolved profile and the variance of the resolved multicomponent profile. The 01 is an overall resolution parameter for the multicomponent sample. A major drawback for all the above-mentioned resolution parameters, is that they assume that some curve-resolution technique already has been successfully performed on the multicomponent profiles and, thus, that the chromatographic and/or spectral profiles of the pure analytes are available. In order to be really useful however, the resolution parameter should be calculated prior to curve resolution, so that it might be used to assess the possibility for a successful resolution of a multicomponent profile with the battery of methods available. The aim of this work is to develop and check the usefulness of such a parameter. Resolution is defined as the ratio between the number of selective (one analyte present) and nonselective sampling points for singledetection chromatography as well as multidetection chromatography. The use of selective information makes it possible to estimate an overall resolution parameter without demanding that curve resolution already has been performed. The resolution parameter is also provided for each analyte separately by means of the same kind of approach. The required information for multidetection chromatography may be achieved by use of latent-projective graphs (LPG)798JlJ2 or by eigenstructure tracking analysis (ETA).13J4 The results of these procedures can be expressed as local rank maps.1P16 The local rank maps for multidetection chromatography correspond to Lorber’slo vectors of NAS for one-dimensional
(10) Lorber, A. Error propagation and f v e s o f merit for quantification by solving matrix equations. Anal. Chem. 1986,58,1167-1172. (11) Kvalheim, 0. M. Latent-structure decompositions (projections) of multivariate data. Chemom. Zntell. Lab. Syst. 1987,2, 283-290. (12) Kvalheim,0.M. Interpretationon direct latent-variable projection methods and their a i m s and use in the analysis of multicomponent 64,936-946. spectroscopic and chromatographic data. Chemom. Zntell. Lab. Syst. (8)Kvalheim,O.M.;Liang,Y.-z.;Keller,H.R.;Massart,D.L.;Kiechle, 1988,4, 11-25. P.; Emi, F. Heuristic evolving latent projections: resolving two-way (13) Toft, J.; Kvalheim, 0. M. Eigenstructure tracking analysis multicomponent data. 2. Detection and resolution of minor constituents. for revealing noise pattern and local rank in instrumental Anal. Chem. 1992,64,946-953. profiles-applications to transmittance and absorbance Et. Chemom. (9) Skoog,D. A.; West, D. M.;Holler,F. J. FundamentalsojAnalytical Zntell. Lab. Syst. 1993,19, 65-73. Chemistry; Saunders College Publishing: Philadelphia, PA, 1988. 0003-2700/93/0365-2270$04.00/0
0 1993 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 65, NO. 17, SEPTEMBER 1, 1993 2271 V
profiles. The highest rank in the local rank map gives the minimum number of analytes present in the profile and thus, for curve-resolution methods such as alternating regression ( A R ) 6 S and iterative target transformation factor analysis (ITTFA),994 the minimum number of factors. Thiswork starta with a discussion of resolution in traditional single-detection chromatography. Thereafter, a resolution parameter for multidetection chromatography is defined for the overall analytical system and for each analyte separately. Subsequently, a procedure for determining these parameters is developed. In order to evaluate the effect of noise and other factors on the proposed resolution parameter, ita performance is assessed on simulated profiles from liquid chromatography with diode array detection (LC-DAD) and gas chromatography interfaced to an infrared spectrometer (GC-IR).
wh
half height
2.4 u
THEORETICAL SECTION Resolution in Single-Detection Chromatography. The standard way of defining resolution in single-detection chromatography is by means of R,, which compares two analytes at a time:
R, = 2AZJ(Wa+ w b )
R ,
-6 0 + 60 6 0 + 60
=
1.0
(1)
R, is defined by using the distance between the peak maxima (AZ)andpeakwidths(W.and Wbforpeaksaandb,respectively). Peakwidths (Figure la) at the inflectionpoint (Wi,corresponding to 2a for a Gaussian peak), at half-height (Wh 2.4~4,or as estimated by drawing lines tangent to the peak at the inflection points and then extrapolating to the baseline ( W b = 4a), are used.9 Usually the widths used for estimation of R, are scaled relative to the 4a width. Assuming Gaussian peaks, an R, of 1.5 corresponds to approximately 0.15% overlap between two peaks and thus suggests full chromatographic resolution (Figure lb) for most practical applications. Two completely resolved peaks give different R, values, depending on the distance between the peak maxima. Thus, R, values larger than 1.5indicate excessive resolution. The calculation of R, aasumes symmetric peaks and that the elution profiles of the pure analytes or at least their most important peak characteristics (width, position) are known. Due to the arbitrary definition of the peaks’ “true” start and end points at the baseline, as a reliable estimate of the noise is missing (see however ref 17), the above-mentioned peak widths are reasonable in single-detection chromatography. On the other hand, for multidetection chromatography, retention time regions where no analytes are eluting (zero-component regions’s8) may be used to estimate the noise level and to obtain a reliable detection limit.18 Such a procedure may reveal improved estimates of the peak widths (the method is described under Local Rank Map by Eigenstructure Tracking Analysis). Table I shows detected peak widths obtained for a simulated multidetection chromatographic profile with varying added noise. From the figures in Table I, it is clear that, for Gaussian peaks, the method detects chromatographic peak widths ranging from 8.6~to 5 . 0 ~for profiles containing from 0.01 to 5% noise. In preparative chromatography,purity better than99 % is commonly required. A width of 60 covers 99.7% of a Gaussian peak, while a width of 4a describes only 95.5% of the peak. For the noise our methods will detect level typically encountered in LC-DAD, peak widths of 6-7u, depending on the net analytical signalloof the analytes.
-
(14)Liang, Y.-2.; Kvalheim, 0. M.;Rahmani, A.; Brereton, R. G. Heuristic evolving latent projections: Resolution of strongly overlapping two-way multicomponent data by means of heuristic evolving latent projections. J. Chemom. 1993, 7, 15-43. (15) Geladi, P.; Wold, S. Local principalcomponent models, rank maps and contextuality for curve resolution and multi-waycalibrationinference. Chemom. Zntell. Lab. Syst. 1987,2,273-281. (16) Keller, H. R.; Maseart, D. L. Peak purity control in liquid chromatographywith photodiode-array detection by a fixed size moving window evolving factor analysis. Anal. Chim. Acta 1991,246,379-390. (17) HBmiliinen, M.D.; Liang, Y. 2.; Kvalheim, 0.M.;Andereeon, R. Deconvolution in one-dimensional chromatographyby heuristic evolving latent projections of whole profiles retention time shifted by simplex optimization of cross-correlationbetween peaks. Anal. Chim. Acta 1993, 271,101-114.
I,*
T
,
t2,n 40
‘
11
tl,b
, ,
30
’. 3 0
40
,
t2,b
*
Flgure 1. (a, top) Peak wldths In traditional one-dlmenslonal chromatography: width at Inflection point ( W; being 20. for Gausslan peaks),width at half-height ( W, 2 . 4 ~and ) width at the base ( W k = 4a) revealed by drawing tangent llnes through the Inflection polnts. (b, bottom) Chromatographic resolution of two (a and b) equal Chromatographlcpeaks. WW a separation of 6, between peak maxima (0.15Y0overlap of each peaks total area),R. = 1.5 while RloIs close to 1. Retention time Is t. Subscripts 1 and 2 represents start and end of the peaks.
-
Table I. DeteotedPeak Widths Using Local Rank Maps Generated by Eigenstructure Tracking Analysis for a Simulated Multidetection Chromatographic Peak (Gaussian), as a Function of Normally Distributed Noise Added % detected peak % detected peak added noise width, u added noise width, u 0.01 0.05 0.10
0.50 1.00
8.6 7.8
7.4 6.6 6.2
2.00 3.00 4.00 5.00
5.6 5.4 5.4 5.0
Rephrasing eq 1in terms of the peaks retention times ( t ) for start (subscript 1)and end (subscript 2) of elution for analyte a
and b (Figure lb), R, becomes
Equation 2a can be rearranged to obtain (2b) The two terms in the numerator represent the selective chromatographic regions for analytes a and b. The first term in the denominator describes the total time elapsed for elution of the two peaks. The second term corresponds to coelution if the peaks are overlapping and gives a positive contribution to the denominator in this case (Figure 2a). If, however, the two peaks are completely separated (Figure 2b), the second term in the
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time and each column representing the chromatographic profile at a particular wavelength. A resolution parameter R=Danalogous to RLD,but taking care of both directions in profiles from multidetection chromatography, can be expressed as Rz-D= XoneeomdXmdtimmp (4) In eq 4, xOnampis the summed single-analyte regions in the is the summed regions multicomponent profiles while xmdtiwmp with at least one analyte. Zero-component regions are excluded:
t1,a
T
XmUltimmp = %tal - ‘mroeomp (5) In eq 5, xt0t.l is the total area of the profile and x~rocomp represents the summed zero-component regions. A comparison of eqs 3a and b with eq 4 shows that R ~ D defines the represents a direct generalization Of R1.D. Thus, R~.D overall resolution uniquely on a scale between 0 and 1.0 independent of the number of analytes present. TheR2.D is related to Karjalainen’s overlap index616 in the sense that it is an overall resolution parameter for the entire profile. Contrary to the 01, the RZ.Dcan be obtained prior to doing any curve resolution. In case of completely overlapping profiles in either direction, the resolution parameter is calculated by eq 6a or b
t2,aT
Ttl,b
ItZ,b
Flgurs 2. Elution profiles for two peaks. (a, top) The peaks coelute partially and the second term in the denominator of eq 2b becomes negative. (b, bottom) Fully resolved chromatographic peaks. The second term In the denominator of eq 2b corresponds to separation
between peaks (zero-componenet region) and becomes positive.
denominator provides a negative contribution proportional to the separation between end of elution of analyte a and start of elution of analyte b. Thus, the second term improves the sensitivity of the resolution parameter but adds no new information to R,. The resolution parameter R, ratios the time of single-analyte elution (selective information) to the total elution time plus an additional term, depending on whether the analytes are resolved or not. This term is redundant, as the separation of the two peaks depends on the amount of selectivity, which is already described by the numerator (eq 2b). This points to the possibility of defining a resolution parameter as the ratio between the number of selective retention times and the total retention times containing signals for one or both analytes. In case of overlap, the resolution is less than 1 and is defined as
Note that the numerators are equal for R~.D and R, (eq 2b), while the denominators differ in the sense that the overlapping region (tzs - t1,b) contributes twice to R,. For resolved peaks (R, 2 1.5) R~.Dbecomes 1.0 (Figure lb):
The parameter R~.D has several advantages compared to R,. First, R~.Ddefines the resolution uniquely on a scale between 0 and 1.0. Thus, resolved peaks have R~.Dequal to 1.0, independent of the distance between the peaks. Second, R1.o can handle more than two analytes at a time (see below). Finally, R~.Dcan be generalized to define resolution in profiles from multidetection chromatography. The parameter R~.D lacks information of excessive chromatographic resolution, but this is easily detected from the chromatogram. By incorporating a term in the numerator describing zero-component regions between peaks, R~.Dcan be modified to describe excessive resolution similarly to R,. Resolution in Multidetection Chromatography Using Selectivity. Multidetection chromatography,for instance,liquid chromatography with diode array detection and gas chromatography inter faced to an infrared spectrometer, gives a spectrum for each retention time and a chromatographic profile for each spectral variable. The profiles are acquired as a matrix X with each row corresponding to the spectrum at a particular retention
Rz-D= t o n a a m d trndtimmp (6a) (6b) Rz-D= AonecomJAmultimmp Here, ton,,, and tm~tiwmp are the number of selective retention times and the total number of retention times with signals above the detection limit, respectively. Similarly, Leeompand Amdticomp are the number of selective wavenumbers and the total number of wavenumbers containing signals above the detection limit. Equation 6a describes the situation where spectral overlap is complete, while 6b shows a similar situation in the chromatographic direction. A resolution parameter for a particular analyte a can be obtained by ratioing the selective regions for that analyte to the whole elution region of the same analyte: %-D+
-- X o n m m p J ~ m ~ t i m m p , a
(7)
In eq 7, x--pa is the selective region for analyte a while xmdd-,, is the total region for component a. Note that this definition of the resolution parameter for a single analyte is analogous to the overall resolution, with 0 corresponding to complete overlap and 1.0 implying a resolved analyte. For a system with all analytes resolved, the R~.D+’ssum to the total number of analytes. Local Rank Map by Eigenstructure Tracking Analysis. The resolution parameters described above take care of both directions in profiles from multidetection chromatography. While the true start and end points of the peaks may not be possible to determine uniquely for single-detection chromatography, good estimates (Table I) can be obtained for multidetection chromatography using the information in the zero-component region. As noise and thus the detection ability in profiles from multidetection chromatography varies from sample to sample and from run to run, a detection limit for each run is required. Since each retention time is represented by a spectrum, a multivariate detection limit based on the zero-component regions can be established for each run independently.18 Rank mapslC16 and estimates of each chromatographic peaks “true” start and end pointa are subsequently revealed. Excessive chromatographic resolution is revealed as large zero-component regions between peaks. A procedure will now be developed (Figure 3) that provides a rank map, which describes the number of analytes in local regions of profiles from multidetection chromatography. This rank map corresponds to Lorber’slo vector of orthogonal signals (Appendix I) for one-dimensional profiles. The ratio between the single-analyte area and the total area containing signal from one or more analytes is calculated by the following dynamic and evolving procedure called eigenstructure tracking analysis:laJ4 Establish detection limits for each window size by using the spectra in the zero-component regions7.8J8 as repeated analytical blanks. The detection limit is determined from the distribution of the first eigenvalues obtained from singular-value decomposition (SVD) of all possible combinations of spectra in the zero-
ANAL.YTICAL CHEMISTRY, VOL. 65, NO. 17, SEPTEMBER 1, 1993 2273
Figure 3. Generatlnga local rank map for profiles from multldetection chromatography using elgenstructure tracking a n a l y ~ i s . ~ The ~~~‘
procedure goes in two dlstinct steps: (I) An evolvingsize moving window is used to define subgroups containing the Same number of analytes In chromatographic direction. Thus, each subgroup embraces ail neighboring spectra with the Same number of analytes present. (11) A similar wlndow technique is used in the spectral direction on each subgroup found in step I.
component regions. The number of spectra in each calculation should correspond to the number included in the following examination of the local retention time regions. Reveal a rank map showing the number of analytes present in local regions in the chromatographic direction, by use of an evolving size moving window. In each subwindow, the number of SVD eigenvalues larger than the detection limit determines the number of analytes contributing to the signals. The procedure starts with a window size of 2 (i = 2) and is repeated by increasing the window size by 1until the size just exceeds the maximum number of coeluting analytes. Starting from the first retention time, the first window contains the i first spectral profiles. The second window contains the spectra for retention time 2 until retention time i + 1. The procedure is continued until the spectrum of the last retention time is included. The profile from multidetection chromatography is divided into subgroups based on the rank map in the chromatographic direction, each subgroup embracing neighboring spectra describing a region of the same number of analytes present. For any of the local chromatographic regions defied by the procedure above, the number of analytes in wavelength direction may vary from zero to the number defined by the local chromatographic region, depending on how many analytes are contributing to the signal at the different wavelengths. This is revealed by local rank analysis using an evolving size moving window in the wavelength direction. The rank analysis is performed for each subgroup defiied in the chromatographic direction with number of analytes larger than 1. Previously Geladi and Woldi6 derived a procedure for generating rank maps for profiles from multidetection chromatography that determined the local rank in a moving quadratic window. ETA is expected to provide better resolution than that approach when selectivity is unequally distributed in the two directions. Selectivity and Curve-Resolution Methods. Several techniques for resolving multicomponent profiies from multidetection chromatography into their pure constituents, i.e., their spectral and chromatographicprofiles, have been The extent to which the methods recognize and utilize selective information varies greatly. Evolving factor analysis (EFA)ls2identifies the chromatographic selectivity, but does not utilize the selective information in the curve resolution. Iterative target transformation factor analysis (ITTFA)3v4uses only chromatographic selectivity in the curve resolution. Indirectly, however, some chromatographic overlap and spectral selectivity is necessary to obtain the correct number of analytes. Heuristic evolving latent (18) Liang, Y. z.; Kvalheim, 0.M.; HBskuldsson, A. Determination of a multivariate detection limit and local chemical rank by designing a nonparametrictest from the zero-component regions. J. Chcmom. 1993, 7,277-289.
Flgure 4. (a,top) Simulated L O A D proflles contalnlngthree analytes. (b, bottom) Simulated G G I R profiles contalnlng three analytes.
projections (HELP)’a reveals and uses selectivity in both directions. Alternating regression (AR)5a6uses selectivityin both directions to obtain the spectral and chromatographic profiles of the pure constituents.
EXPERIMENTAL SECTION Sample Description. Pure analytes were generated by simulating profiles with a varying amount of noise, spectral, and chromatographic resolution. The profiles were multiplied together to give pure multidetection chromatographic profiles for each analyte. Profiles of pure analytes were summed to give multicomponent profiles. Figure 4shows examples of simulated LC-DAD and GC-IR profiles. In the simulated LC-DAD profiles, the spectral profile of each pure analyte was composed of two Gaussian peaks. The sample preparation was performed by use of a zc1 fractional factorial designtQthe levela being defiied as either 2or 3 chemical analytes, 5 or 20% relative concentrations for the minor analyte, 0.01 or 0.03% noise relative to the maximum peak intensity, and an RNM(eq A-2) between chromatographic profiles of 0.36or 0.98 (corresponding to an R,value of 0.31 or 0.63,e9 1). The spectral profiles were kept constant, always overlapping completely. In the simulated GC-IR profiles, the spectral profile of each pure analyte was composed of five Gaussian peaks. The sample preparation was performed by use of a 2”’ fractional factorial design with levels chosen as 2 or 3 analytes, 5 or 20% relative concentrations for the minor analyte, 0.01 or 0.03% noise relative to the maximum peak intensity, p R ~ l gbetween ~ chromatographic profiles of 0.36 or 0.98, and an R N between spectral profiles of 0.71 or 0.95. Evaluation. To evaluate the relationship between the res) the design variables, the regression olution parameter ( R ~ Dand coefficients20 of partial least-squares (PLS)21~22models were (19) Morgan, E. Chemometrics: Ezperimental Design; John Wiley & Sons: London, 1991.
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0.30
0.20 0.10 0.00
-0.10 -0.20
-0.30 I
Flguro 5. Regression coefficients for PLS model obtained for Simulated LC-DAD profiles. The regression coefficients show the Importance of the predictor Variables to the part of the resolution parameter R24, accounted for by the PLS model. Predictors negatively correlated to increased R20 plot negatively in the graph.
inspected. In the PLS model, c r ~ s s - t e r m were s ~ ~ included and variables were standardized to unit variance. R~.Dwas compared with Karjalainen’s overlap index.5~~The overlap index was determined using AR to resolve the pure constituent’s multidetection chromatographic profiles. The choice of using AR as the curve-resolution technique is based on is utilizing selectivityin both directions. the fact that AR, like R~.D, Software. The multidetection chromatographic simulations were written in VAX FORTRAN, version 5.0. The software for the resolution parameters were written in VAX PASCAL version 3.5. Latent-projective graphs and PLS modelingwere performed by use of the Sirius%program. Three-dimensional plots, 01,and AR were performed/programmed by use of 386-MATLAB.
RESULTS AND DISCUSSION Liquid Chromatography with Diode Array Detection (LC-DAD). Chromatographic overlap was defined by use of R N (eq ~ A-2) between the simulated single-detection chromatographic profiles of the pure analytes. Since LC-DAD profiles often have poorly resolved spectral profiles (no spectral selectively) for real-world cases, the resolution was calculated by use of eq 6a. In this work the resolution ranged from 0.32 to 0.62. The PLS analysis gave one significant PLS component, describing 79.0% of the variance in € 2 2 . ~ . The regression coefficients20 were calculated to find the most important predictors for R z . ~ .Figure 5 shows that, with a decreasing number of analytes present, with an increasing amount of noise added, and with increasing R N ~the, resolution increases. The two major interactions are, with resolution, an increasing interaction between RNASand noise and a decreasing interaction between the number of chemical analytes and the noise level. Furthermore, the concentration of the minor analyte is found to have little effect on the resolution. This is as expected, as the noise is rather small compared to the net analytical signal of the minor analyte. The relationship between added noise and resolution is surprising but explainable. With an increasing noise, the detected part of the peak width is narrowed (Table I). The selective regions have start and end positions depending on (20) Kvalheim, 0.M.; Karstang, T. V. Interpretation of latent-variable regression models. Chemom. Intell. Lab. Syst. 1987, 7, 39-51. (21) Wolde, S.; Ruhe, A,; Wold, H.; Dunn, W. J., I11 The collinearity problem in linear regression. The partial least squares (PLS) approach to the generalized inverses. SIAM J.Sci. Stat. Comput. 1984,5, 735743. (22) Hbkuldsson, A. PLS regression methods. J. Chemom. 1988,2, 211-228. (23) Kvalheim, 0.M. Model building in chemistry, a unified approach. Anal. Chim. Acta 1989,223,53-73. (24) Kvalheim, 0. M.; Karstang, T. V. A general-purpose program for multivariate data analysis. Chemom. Intell. Lab. Syst. 1987,2,235-237.
the added noise, but their size will be virtually unaffected. The effect of increased noise is to displace the selective regions of analyte a to slightly later and analyte b to slightly earlier retention times. However, the total detected elution time will be shorter since the start of analyte a is detected later and the end of analyte b is detected earlier with increased noise. The overall result is a slightly increased resolution with increasing noise (see eq 3a). For LC-DAD profiles, alternating regression5.6 failed to resolve the profiles from multidetection chromatography into their pure analytes, due to lack of selectivity in the spectral direction. The “resolved” spectra obtained were linear combinations of the true spectra. Replicated AR curve resolutions did not provide stable solutions, thus excluding the possibility of calculating a reliable 01.
Gas Chromatography Interfaced to an Infrared Spectrometer. Chromatographic and spectra overlap was defined by use of R N between ~ simulated one-dimensional profiles, chromatographic and spectral, respectively, of the pure analytes. GC-IR profiles are assumed to contain some selective information in both directions. Accordingly, the resolution was calculated by use of eq 4. Here, the resolution ranged from 0.46 to 0.85. The PLS analysis gave one significant PLS component, describing 76.6% of the variance in &.D. The regression coefficients (Figure 6) reveal a similar picture as obtained for LC-DAD profiles. The resolution parameter increases with increasing added noise, chromatographic and spectral R N ~and , decreasing number of analytes. The observation that chromatographic R N seem ~ to have exceeded the importance of the spectral R Nis ~due to the larger variation for chromatographic R N(see ~ Experimental Section). The concentration of the minor analyte has little effect on the resolution. The major interactions are noise interacting with both chromatographic and spectral resolutions, the interaction between the R N in~the two directions, and the interaction between the number of analytes and the spectral RNM. Due to selectivity in both spectral and chromatographic directions, AR resolved to some extent the GC-IR profiles. A visual inspection of the resolved IR spectra and GC elution profiles showed that the quality of the AR solution decreased with increasing complexity in the profiles from multicomponent chromatography. Thus, for the most complex profiles, 01 was calculated with a higher uncertainty. The correlation between 01 and R~.Dwas -0.69. This negative correlation is expected as 01 increases with an increasing overlap, while the opposite holds for &.D. A PLS analysis similar to that for R~.Dgave one significant PLS component describing 80.0% of the variance in 01. As
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0.10 0.03 -0.04 -0.11
-0 18 a
Fburr 6. Regression coefficlents for PLS model obtained for simulated G G I R proflles. The regression coefficients show the Importance of the predictor variables to the part of the resolutlon parameter accounted for by the PLS model. Predictors negathrely correlated to Increased plot negatively in the graph.
expected, the regression coefficients for 01 showed the opposite picture (due to the negative correlation between 01 and &.D) of what they did for &.D. Except for this, the main difference in regression coefficients for 01 and R~.Dwas the increased importance of noise and its interactions for R~.D.
CONCLUSION An overall resolution parameter, &-D, for multicomponent profiles from multidetection chromatography has been developed, ratioing the selective regions to the total regions with eluting analytes. The R~.Dis not restricted to analytical profiles of a particular kind. The value of R2.D is in the range of 0-1 for completely overlapped and fully resolved analytes, respectively. The new resolution parameter can be used prior to curve resolution and should be a valuable tool when the performances of different methods for curve resolution are being compared. Finally, the parameter and the procedure to obtain the parameter can be generalized to other multicomponent profiles and higher-order data arrays.
R. RI-D RNAS RZ-D
The Norwegian Research Council for Natural Science and Humanities (NAVF) are thanked for a Ph.D. grant to J.T.
A
sk* =
(I - S k s k ’ )
sk
Az W
X n
(A-1)
Here s k + is the generalized inverse25of s k and 1 is the unit matrix. A resolution parameter, R N ~representing , the systems’ overall resolution is defined as the ratio between the norms of the A analytes’ NASs (sk*) and original profiles ( s k h
k=l
k=l
Thus, a net analytical signal to the total signal is obtained in the same manner as the more familiar signal-to-noise (S/N) ratio. It is worth noticing that RNa is general and is not restricted to profiles containing a specified number of peaks or having a specific peak shape.
t
Resolution for One-Dimensional Profiles. A onedimensional resolution parameter can be defined by using the analytes’ orthogonal part of the profilelo relative to the other analytes’ profiles. As the amount of noise is expressed as a scalar, the profiles of unique information are represented by their norms. The profiles (sk) of the andytes are collected in a matrix s. S k is the matrix containing profiles for d l analytes but k. Each analyte’s profile of orthogonal signals, unique information/net analytical signals (NAS, sk*) relative to the other analytes ( s k ) , is expressed as
A
(A-2)
ACKNOWLEDGMENT
APPENDIX I
A
i S S
s*
A
APPENDIX I1 NOTATION chromatographicresolution, one-dimensional using separation between peakmaximaand peak widths chromatographicresolution, one-dimensional using selectivity spectral resolution, one-dimensionalusing net analytical signal resolution in multidetection chromatography using selectivity retention time distance between two peak maxima wavenumber peak width profile from multidetection chromatography size of subarea, number of data points window size in moving window (ETA) matrix of spectral profiles spectral profiles net analytical signal number of analytes
Subscripts
a, b onecomp multicomp total zerocomp 1 2
i h base
chemical analytes single-analyte variable/area variable/area containing signal above detection limit from one or more analytes total area, with and without absorption selective/no elution/noise variable or area start chromatographic peak end chromatographic peak inflection point half-height baseline
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(25) Massart, D. L.; Vandeginste, B. G. M.; Deming, S. N.; Michotte, Y .;Kaufman,L. Chemometrics: A Textbook;Elsevier: New York,1988.
RECEIVED for review April 2, 1993. Accepted May 4, 1993.
