in this way (16, 13,even before the usual methods of identification had been used. With the exception of sterols isomeric at C-24 in the side chain, each sterol in Table I can be distinguished from other sterols in Table I after analysis on the four GLC systems used in this work. Sometimes there were slight apparent differences in the retention times of sterols isomeric at C-24, but these differences were never as great as 1 %. Much GLC work has been done using only one column such as SE-30. If GLC is to be used as a tool in the identification of sterols, use of at least three columns seems to be required. In plants the most common sterol mixture is one which contains a C-28 sterol with a saturated side chain, a C-29 sterol with a saturated side chain, and a C-29 sterol with a A Z 2double bond (for example, campesterol, stigmasterol, sitosterol). Of the GLC systems used here, only the SE-30 column resolves this mixture. The QF-I column is unique with respect to 14a-methyl sterols. In scanning the list of sterols in Table I, a sterol with a higher relative retention time on QF-1 than on any other column, with very few exceptions, will be a 14a-methyl sterol. The Hi-Eff 8BP and PMPE columns differ from both the SE-30 and QF-1 in that they have a much stronger affinity for A24(25),424(28), and A5a7double bond systems, and less affinity for sterols with methyl groups at positions 4 or 14. These liquid phases differ from each other primarily in the degree of their affinities for A5f7 sterols and 14a-methyl sterols. Neither the Hi-Eff 8BP nor the PMPE can resolve A5 sterols from stanols as can an NGS column, but both columns have the advantage over most polar columns in their thermal stability. The PMPE column has been in this work (16) P. J. Doyle, Ph.D. Thesis, University of Maryland, 1970. (17) L. G. Dickson, Ph.D. Thesis, University of Maryland, 1971.
for extended periods at 250 "C without significant deterioration. Gas chromatography cannot replace any analytical method now used in the identification of sterols, but using available data, we can gain much more information about the structure of a sterol from gas chromatography than has been possible in the past. ACKNOWLEDGMENT
The author is grateful to: M. J. Thompson for samples of 47-coprostenol, coprostanol, A8(14)-coprostenol,cis-22dehydrocholesterol, trans-22-dehydrocholesterol, A59 2 , 4 - ~ h o lestatrienol, A58 25-cholestadienol,24S-A5,7-ergostadienol,and A6~7-stigmastadienol; to M. Barbier and A. Alcaide for pollinastanol, 31-nor-cycloartanol, 24-methyl pollinastanol, 24-methylene lophenol, and cyclolaudenol; to F. B. Mallory for 45~7~Z2-cholestatrienol; to I. A. Watkinson for A8(9), 14-cholestadienol; to D. H. R. Barton and D. N. Kirk for zymosterol, obtusifoliol, cycloartenol, and A75 2e-ergostadienol; to G. R. Pettit for 14a-methyl-A7-cholestenol; to Merck and Co. for 4 7 , 9 ( 1 1 ) , 22-ergostatrienol; to Carl Djerassi for lophenol, peniocerol, and macdougallin; to Wolfgang Sucrow for 45325-stigmastadienol,A7~2S-~tigma~tadienol, 47,22,25-~tig28 isomastatrienol ; to J. A. Fioriti for A8(14)-stigmastenol, 47324(2s)-stigmastadienol,and A8!9) 4(28)-stigmastatrienol ; to G. J. Schroepfer for As-cholestenol; to Guy Ourisson for cycloeucalenol, and to F. J. Schmitz for gorgosterol and 23demethyl gorgosterol.
for review March 5 , 1971, Accepted May 4, 1971, This work was supported in part by Grant N ~ G - 7 0 - 3 from the National Aeronautics and Space Administration. This is Scientfic Article No. A1687, Contribution No. 4439 of the Maryland Agricultural Experimental Station. RECEIVED
Estimation of Chromatographic Peaks with Particular Consideration of Effects of Base-Line Noise P. C . Kelly' and W. E. Harris Department of Chemistry, University of Alberta, Edmonton 7, Alberta, Canada The precision of a measurement of a gas chromatographic peak depends on the intensity and character of the base-line noise, the shape and size of the peak, the prior information, and the method of estimation. An estimation method is presented that approaches the highest precision possible of any method used with a given gas chromatographic system. Unlike most other estimation methods, it can use all the information available to estimate the peak parameters, including prior information as well as information from the chromatogram. The base-line noise of many detectors can be statistically characterized by a powerdensity spectrum. In such systems, the estimation of peak parameters is most easily carried out in the frequency domain on the Fourier transform of the chromatogram. Although any mathematical model that can completely describe a peak is suitable for estimation purposes, a peak model that uses cumulants, which are related to moments, has certain mathematical advantages. 1 Present address, Dean and Kelly Analytical Chemists Ltd., 5920-149 Ave., Edmonton, Alberta.
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THEBASIC FUNCTION of an analytical instrument is to transform information about composition into a form that can be interpreted by the analyst. Many instruments present information in the form of a chart recording. This paper is concerned with translating the information contained in a recording into a comprehensible form. Specifically, techniques for the analysis of peaks obtained from an isothermal gas chromatographic system with a thermal conductivity detector are reported. The basic approach is applicable as well to all instrumental systems, particularly those giving the desired information in the form of a peak on a noisy base line or background. A true chromatographic peak may be defined as the mass or concentration of a solute flowing across a plane at the end of a column as a function of time. The size of a peak gives information about quantity ; the position and shape give information about the identity of the solute. In gas chromatography, size and position are usually given in terms of area and retention time. Area and retention time are
ANALYTICAL CHEMISTRY, VOL. 43, NO. 10, AUGUST 1971
parameters of a peak.
Parameters concentrate the pertinent information contained in a chromatogram into numbers that are more comprehensible than the chromatogram itself. The conversion of data into numbers is called estimation, and the set of operations used to make the conversion, an estimator.
The theory of estimation and also the mathematical techniques used to describe peaks and noise that are presented here have been successfully applied in many fields other than gas chromatography : economics, communications, radar, and seismology, among others. Since these fields are outside the usual range of contact of workers in gas chromatography, we include a general, though incomplete, discussion of these methods and a brief guide to the literature. Most of the general material is discussed more completely in References ( I ) to ( 4 ) . Figure 1 depicts the flow of information from the material being analyzed through the instrument to a chromatogram. Figure 2 shows the flow continuing from the chromatogram to the estimator and then to the action based on the interpretation of the estimates. An ideal system would represent a peak as a smooth curve identical to the true peak. In reality, however, determinate and indeterminate errors enter at every step of the process (Figures 1 and 2). Our object is to find an estimator that will minimize estimation errors. Before discussing the theory of estimation, let us define the problem and discuss the sources and types of errors arising in a gas chromatographic system, using Figures 1 and 2 as guides. Sampling errors occurring when material being analyzed is heterogeneous can cause both determinate and random errors in analytical results. Errors can be introduced in transferring a sample into the separation system (column in a gas chromatograph). A sample may decompose or incompletely volatilize in the injection port. Sample size is, of course, random to a certain extent. Errors introduced by the separation system may result from incompletely resolved peaks or from a reaction of the solute with the column packing. An important class of errors in the separation process is caused by variations in experimental conditions, such as column temperature and carrier gas flow rate. When identification is based on retention time, gross errors may occur where several compounds have the same retention time. The odds on the correctness of the identification of a peak can be improved with external or prior information about the material being analyzed. The true peak in Figure 1 appears at the end of the chromatographic column. Because of sampling and separation errors, however, the true peak may not have an area corresponding to the composition of the material being analyzed. Nevertheless, it is the true peak for the given experiment when sampling and separation errors are disregarded. A detector, as shown on Figure 1, converts information about mass or concentration into an observable signal such as voltage. Since the output of the detector is not in units of mass or concentration, the necessary calibration may introduce errors. The sensing element of any detector has a (1) R. M. Bracewell, “The Fourier Transform and Its Applica-
tions,” McGraw-Hill, New York, N. Y., 1965. (2) H. L. Van Trees, “Detection, Estimation, and Modulation Theory, Part I,” Wiley, New York, N. Y., 1968. (3) C. W. Helstrom, “Statistical Theory of Signal Detection,” 2nd ed., Pergamon Press, Oxford, 1968. (4) D. V. Lindley, “Introduction to Probability and Statistics,” Cambridge University Press, Cambridge, 1965.
t t 1 -1
L r
Separation
+5?L2 True Peak
Detection
Recording
v Chromatogram
Figure 1. Flow of information through an instrument
1
G Error
1r
Estimator
L r
r L
& Action
Figure 2. Flow of information to and from an estimator
finite volume and therefore will respond not to a concentration at a plane, but to an average concentration between two planes. The output of a detector, then, is a broadened and distorted version of the true peak. Similar determinate errors arise from the distortion caused by dead space between column and detector, the finite response time of the detector, and nonlinearity of response of the detector.
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The most obvious source of random error in a gas chromatographic system is random fluctuation in the base line. Base-line fluctuation has many sources: an electrical or mechanical disturbance within the detector, such as shot noise, thermal noise, contact noise, or vibration; fluctuation in ambient conditions, in the temperature of the detector, in the flow rate of carrier gas, etc. Also, the average value of the base line is arbitrarily set by the operator, and must be measured; the measurement introduces random errors. We define base-line noise as the parts of a chromatogram not caused by a peak. If the signal from a detector is recorded, the recording process may introduce errors. Potentiometric strip-chart recorders introduce distortion through response-time and nonlinear effects. Dead-band effects and amplifier noise cause both determinate and random errors. If a chromatogram is digitized for subsequent computer calculation, errors will be introduced by the digitization process. It can be shown that determinate errors caused by sampling can be controlled by making the interval between samples small enough and by taking samples over a sufficient period of time (5). Fluctuations in sampling intervals also will cause errors. It can be shown also that the quantization of a signal into numbers introduces random errors proportional to the size of the quantization interval (5). The errors originating in the instrumental part of a chromatographic system discussed above are well known and understood. The effect of random errors can be decreased by carefully designing the system and controlling the experimental parameters. Goedert and Guiochon (6)determined the degree of control of flow rate, pressure, and temperature required for a given precision of peak area or height for a gas chromatographic system employing a thermal conductivity detector. In principle the effects of determinate errors, including distortion, can be measured and the final results corrected. Alternatively, the determinate errors can be minimized experimentally. No matter how well designed the instrumental system may be, or how accurate and noise free the chromatogram, the quality of final results will be affected by the estimator. As illustrated in Figure 2, an estimator has four inputs: the noise model, the peak model, the chromatogram, and the prior information. The peak model defines the analyst's concept of a true peak. We will take the peak model to be a continuous function of time, f ( t , a ) , and a set of parameters, a = [cy", cyl, cy2 . . .]. The parameters normally include area, retention time, width, and possibly others, which collectively account for the shape of the peak. (We are interested here primarily in single isolated peaks. Multiple overlapping peaks can be handled by simply extending the peak model and introducing additional parameters.) Being concerned primarily with estimation errors, we will postulate a true peak and true values for the parameters for each chromatogram. The true values will be different in different chromatograms because of random errors resulting from non-base-line noise. Nonbase-line noise includes sampling errors and fluctuations in the experimental conditions. A chromatogram will be taken to be a continuous function of time, x ( t , a). It can be shown that, when a chromatogram is digitized, the sampled data can be treated as continuous if the sampling interval ( 5 ) P. C. Kelly, W. E. Harris, and G . Horlick, University of
Alberta, Edmonton, in preparation. ( 6 ) M. Goedert and G. Guiochon, J. Chromatogr. Sci.,7, 323 (1969). 1172
is small enough. A chromatogram is also, of course, a function of the parameters of the peak being measured. The true peak and the chromatogram are ordinarily related through the equation x(t, a> =
f(t, a )
+ n(t>
(1)
where n(t) is base-line noise. An appropriate model for base-line noise is discussed below. Prior information enters into the peak model in the choice of the mathematical form of the peak and the choice of the parameters to be measured. Prior information about the values of the parameters may be available-for example, in a routine analysis the position and shape of a peak may be known. Prior information is involved in the choice of the method of estimation and in the assumption of Equation 1. The interpretation step requires prior information to relate the area of a peak to concentration and the retention time to identity. Estimators commonly used in chromatography include planimetry, cutting and weighing, height times width, triangulation, electronic digital integration, and digital computer methods. It is well known that the performance of these estimators differs in terms of precision and accuracy, depending on the shape and size of the peak as well as the intensity and type of base-line noise. It might be said that the principal difference in these methods is the extent to which they utilize the available information including prior information. The perimeter and ruler-and-pencil methods make good use of prior information. The analyst scans the base line before and after the peak-that is, he considers prior information about the base line to interpolate a base line under the peak. He may smooth bumps on the profile of the peak caused by noise, using his concept of what a peak should look like (prior information about the peak model). The precision of these methods, however, is limited by quantization errors in the final step where the human eye distinguishes between the divisions on a ruler or a dial. Electronic integrators are less affected by quantization errors, but make poor use of prior information. A horizontal or sloping straight line is usually used for the base line under a peak, and a peak is defined as a signal that goes up, stays up, and then comes down again. Many computer methods are merely digital models of electronic integrators. Computer methods using least-square fitting techniques are better, but may suffer from inadequately defined peak and noise models, What is needed is an estimator that utilizes all available information and is not influenced by the shape and size of a peak or by the type of base-line noise. Such an estimator, which we would term ideal, undoubtedly would be valuable, but might be too complicated and costly for routine use. Nevertheless, an investigation of the performance of an ideal estimator could serve as a valuable basis for the comparison of the performance of other estimators. The axiomatic Bayesian approach to the treatment of data shows how information can be expressed in an intuitively reasonable manner ( 4 ) . In the presence of random errors, the exact values of the parameters of a peak cannot be determined. For example, a peak in a noisy chromatogram could be the result of a true peak with any one area out of a more or less broad range of areas. It is intuitively reasonable, however, that the information obtained from a chromatogram will indicate some ranges of area are more likely or probable than others. Therefore, experimental evidence about a parameter is best expressed as a probability density function giving the probability densities of all values of area.
ANALYTICAL CHEMISTRY, VOL. 43, NO. 10, AUGUST 1971
Such a probability density function is called a posterior pdf. The posterior pdf for a given chromatogram can be derived from the chromatogram and prior information by use of the basic rules or axioms of probability theory. Posterior pdf‘s that might be obtained from a chromatogram with decreasing intensities of base-line noise are shown in Figure 3. The arrow represents the infinitely narrow posterior pdf obtained with no base-line noise. Curve 3 in Figure 3 would be obtained with base-line noise intense enough to nearly obliterate the peak. Obviously, the chromatogram giving Curve 3 contains little infoimation about the area of the peak. The more narrow Curve 2 certainly contains more information, and Curve 1 would give perfect information. This intuitive connection between the spread or dispersion of a posterior pdf and the information content or informative value of an experiment can be used to define useful quantitative measures of information. One such measure is defined below. Note that a posterior pdf in the Bayesian sense used here is a distribution of true values in contrast to the classical definition of a pdf as a frequency distribution of observed or estimated values. A Bayesian posterior pdf is a function of the data from a given experiment; the posterior pdf’s of duplicate experiments generally will differ. A posterior pdf concentrates the information from a chromatogram. It contains all the available information about the parameters of a peak and excludes the irrelevant information about base-line noise. Therefore, the ideal estimator we are looking for should be based directly on the posterior pdf. An obvious and relatively simple estimator is one that uses the most probable parameter values (MP method). The MP method is discussed in detail below. Another method, the mean of a posterior pdf, is usually more difficult to calculate than a most probable value. The pcsterior pdf itself could, of course, be used to determine detection limits or to give the probability that a parameter exceeds a certain value. Results obtained by the MP method will be reasonable and self-consistent because the axioms used to derive them are self-consistent. But a reasonable method alone does not guarantee an accurate description of reality ; the accuracy of an estimator cannot exceed the accuracy of the models used. The principal symbols are listed at the end of the paper. The notational conventions of Bracewell ( I ) are followed whenever possible: one of these is that the Fourier transform of a function is capitalized; another is the use of the symbol * for convolution. Two different time and frequency scales are used. Time in seconds is represented by the symbol t , and frequency in the corresponding scale (hertz) by v. Time in peak-width units is represented by the symbol (, and frequency in the corresponding scale by TJ. PEAK MODELS
Peak shapes can be most easily compared when peaks are reduced to unit area and unit width, and are centered at the origin. Defining area, A , retention time, to,and width, by
to = u* =
A A
Jm
A
sm
-- tf(t)dt
(t
- t o ) * f ( t- to) dr
t’
Peak Area , A
Figure 3. Posterior probability density functions for peak area a dimensionless standardized peak-shape function, y ( ( ) , can be derived from a peak on a noise-free chromatogram, f ( t ) , through the formula Y(S7 = f ( d a / A
(5)
where f is dimensionless reduced time:
r = ( t - to>/u
(6)
The above definitions of the retention time and width of a peak are the mean and the standard deviation. Other definitions, such as the time of the maximum or the width at a certain fraction of the height, may be used. We prefer the mean and standard deviation because they simplify the mathematical form of the peak models discussed below. The time of the maximum is greater than the mean for a tailed peak but is equal to the mean for a Gaussian or any other symmetrical peak. With a noise model applicable to most detectors (described in the next section) the parameter estimates are best calculated on the Fourier transform of the chromatogram. Moreover, the mathematical expression for the peak-shape model defined below is simpler as a function of frequency than as a function of time. Even though the Fourier transform operator is complicated, advances in computer algorithms for numerical Fourier transforms make the use of the operator of little concern (7). The Fourier transform of a standardized peak-shape function is defined by ~ ( 7 )=
exp1--2vi1 Y(u d l
(7)
where i is the square root of -1 and 7 is frequency in a dimensionless standardized scale (multiplying TJ by u gives frequency in hertz). The original function y ( l ) can be obtained from Y(Q) through an inverse Fourier transform; basically, then, the Fourier transform is merely a different view of the same subject. For example, the area or D C component of a function is the value of the Fourier transform of the function at zero frequency. Parameters. The three parameters area, A , retention time, to, and peak width, u, along with a peak-shape function, completely and uniquely describe a peak. Knowing the values of the three parameters is enough to distinguish a particular peak from any other peak only when the peak-
(3) (4)
(7) W. M. Gentleman and G. Sande, “1966 Fall Joint Computer Conf., AFIPS Proc.,” Vol. 29, Spartan, Washington, D. C., 1966, pp 563-78.
ANALYTICAL CHEMISTRY, VOL. 43, NO. 10, AUGUST 1971
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function are defined as derivatives of the logarithm of the Fourier transform of the peak-shape function:
Table I. Standardized Moments and Cumulants of Common Functions
Moments ( r > 2)
Function
( r > 2)
(2r)!
Gaussian
K,
pzi = __ 2rr! PC?T+l =
=
0
0
r-2
Poisson
fir
(r - l)! j = o (r - 1 - j ) ! j !
pj
K, = 1
Exponential
K, = ( r - I)! =
0, x
