Bi-Gaussian Fitting of Skewed Peaks T. S. Buys and K. de Clerk Chromatographic Research Unit of the South African Council for Scientific and Industrial Research, Department of Physical and Theoretical Chemistry, Unicersity of Pretoria, Pretoria, Republic of South Africa THELOWER STATISTICAL MOMENTS are particularly convenient parameters by means of which the component peaks in a chromatogram may be characterized. The zero'th moment corresponds t o the mass and is thus fundamental in quantitative work, while the first moment is related t o the thermodynamics of the system and can be used as a n indexing parameter for identification purposes. The second and third moments, respectively, measure peak width and asymmetry. In the case of Gaussian peaks, the first moment coincides with the peak maximum, but chromatographic peaks are frequently observed t o be asymmetric in which cases the peak maximum loses its significance. The calculation of peak area is also complicated since there is n o easy way of measuring it, as is possible in the analysis of Gaussian peaks where a simple height and width measurement would suffice. The situation becomes increasingly complex for overlapping peaks and, even in the more sophisticated computer analysis, some sort of guiding function is required in the first approximation.
Figure 1. Parameters measured for bi-Gaussian fitting of asymmetrical peaks UI =
THEORY
Peak asymmetry can be the result of a variety of causes; there is therefore n o unique analytical functional form which can be used for the description of such peaks. This circumstance gives rise t o a fundamental problem in the evaluation of possible analytical fitting functions since success or failure in the fitting of a specific chromatogram in no way guarantees the result t o be generally applicable. Instead of using experimental data as a reference, we have therefore preferred to use a function which is known t o qualify as a suitable fitting-function because of its inherent flexibility. In this respect the Poisson distribution (normalized) 1
F(z)
=
I zn exp(-z/b) bn+' ( n ! )
is particularly convenient since peaks varying progressively in skewness from the limiting symmetrical Gaussian case t o the extremely skewed exponential case may be generated by varying the parameter n between m and zero. Moreover, the use of this distribution function extends the usefulness of the present results t o the analysis of skewed peaks generated by means other than chromatographic. As a fitting function itself, the Poisson distribution is not very suitable, however, since the parameter values and consequently the statistical moments are not simply measured from a n experimentally recorded distribution. The bi-Gaussian function,
(~ip)!/1.177
(3)
4%u1 C,
(4)
and ml
=
with similar expressions for u2 and i n 2 . This type of function has already been used by Grushka et al. ( I ) in their analysis of the significance of excess and skewness for unresolved peaks. An additional feature in favor of the bi-Gaussian function is the fact that its use allows the mathematical description of the relationship between impurity fraction and resolution, R , to be readily extended to skewed peaks. This topic is considered in a separate paper ( 2 ) . The primary aim of the present note is t o investigate the applicability of the bi-Gaussian function for statistical moment predictions and t o define its limitations. The first three moments are defined by (5)
and (z - a',)' Cdz
mo
=
2, 3
(6)
where
is the zero'th moment or total area under the peak C(z). The first moment a'1 is the mean, ( z } , while the second central moment, cy2, is the total variance, u 2 ,of C(z). For the C(z) of Equation 2, it then follows directly that mo = !/2!m1
on the other hand, is particularly convenient in this respect; the relevant parameters are simply determined by measuring C,, (w1p)l and (wl;& (see Figure 1) from which u1 and ml are obtained as
i
+ m?)
(8)
(1) E. Grushka, M. N. Meyers, and J. C. Giddings. ANAL. CHEM., 42, 21 (1970). (2) T. S. Buys and K. de Clerk, accepted for publication in Separ. Sci., 7, No. 5 (1972). ANALYTICAL CHEMISTRY, VOL. 44, NO. 7 , JUNE 1972
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1.0
.-.-.-.--.-
-----__ - ----------------------- -----____-----____ 0.6 -I-.
0.8
Y
:::h
0.4
--*-_
- ----c---_ 0.2 -
8
------_-_______
’
ol,
’ 012 ’ 013 ’
d4
’
d.5 ’ d.6 ’ d.7 ’ ole ’ 019 ’
o:
0.I
0.4
O
,
00
z
110
The moment predictions of a bi-Gaussian peak fitted a t the peak maximum and half-height of a Poisson distribution are representcd in Figure 2. A number of actual peaks are also included in Figure 3 t o illustrate the effect of the parameters n and b. From these results, several conclusions regarding the ability of the bi-Gaussian model in predicting the various statistical moments may be made. In all cases the ratios of the predicted moments t o the actual (Poisson) moments were found to be independent of b. As regards the dependence on n, the results may be summarized as follows: (i) The prediction of the mass is generally good and it is only for very asymmetrical peaks that the accuracy decreases; e.g., for n = 1, cf, Figure 2, the relative error is only 4z. (ii) In the case of the first moment, the ratio ( ( z ) ~)z, is found to be practically constant (-0.625) for all n and b values. Although the position of the mean is therefore incorrectly predicted, the constancy of the ratio (less than i1%) suggests that the bi-Gaussian approximation might still qualify as a method for the empirical prediction of retention times. Since the skewness of component peaks of a particular chromatogram are probably due t o a common origin, one would therefore expect a unique proportionality constant for every particular chromatogram. Once this constant is determined, ( z ) can be predicted by using Equation 9 which merely requires the measurement of z,, ul, and uz. (iii) The second moment is surprisingly sensitive t o the
Figure 3. Poisson distribution, F(z) (solid curve), and fitted bi-Gaussian C(z) (X), us. z for various values of n and b peak details and is not satisfactorily predicted by the biGaussian model. If plate height predictions are therefore required, some other approximate method, e.g., that of Sternberg (3),would be more appropriate. (iv) Various reduced forms of the third moment are used to characterize asymmetry. Of these, the skewness S defined by [e.g.Ref. ( 4 ) ]
is the most generally used. The absolute prediction of the third moment is not very successful as is evident from Figure 2 although it can be regarded as a fair empirical model. An important deduction follows, however, from a n analysis of the functional form of the skewness. A number of different asymmetric peak functions were analyzed and in all cases S was found to be a function of the parameter
4
=
((4 - z m ) / ( 4 1 ’ z
= ( ( z ) - z,)/u
which is also a standard measure of asymmetry [e.g.Ref. ( 4 ) ]
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ANALYTICAL CHEMISTRY, VOL. 44, NO. 7,JUNE 1972
-
Poisson distribution: Sp = 2 Bi-Gaussian distribution: SG
(3) J. C. Sternberg, “Advances in Chromatography, Vol. 11,” J. C. Giddings and R. A. Keller, Ed., Marcel Dekker, New York, N.Y., 1966, p 205.
(13)
+P
4~
(14) (15)
(4) A. E. Waugh, “Elements of Statistical Methods,” McGrawHill, New York, N.Y., 1943.
~~
Table I. Bi-Gaussian (subscript G ) Moment Predictions Compared with Computer Simulation (subscript S) Results ( k l = 20, inlet variance = 3.667, effective diffusion coefficient = 0.01, mobile phase velocity = 1) Time,
mOG -
(Z)G
-
SeC.
mQS
(2)s
- zm
200 400 600 800 lo00 200 400 600 800
0.8721 0.8771 0.8479 0.8249 0.8389 0.9825 0.9713 0.9410 0.9478 0.9390
0.9808 0.9873 0.9782 0.9838 0.9828 1.0202 1.0090 1.0129 1.0100 1.0120
lo00
_OG
ffLG
us2
a35
ss
0.8743 0.8725 0.8524 0.8657 0.8582 1.0561 1.0163 1.0164 1.0078 1.0057
0.6821 0,6637 0.6204 0,6396 0,6086 1.1521 1.0549 0.9661 0.9607 0.9512
0.8343 0.8143 0.7883 0.7941 0.7656 1.0617 1,0294 0.9428 0.9496 0.9431
zm
Pearson’s curves (Type 111): SpE= 2
(16)
+ P ~
(The approximate relationship for SGis obtained from Equation 11 by neglect of the second term on the right hand side). This implies that the third moment, which is not easily measured, need not be included in the set of basis parameters for skewed peak characterization. The position of the peak maximum, zm, may be used instead. Whether S or is the significant measure of asymmetry in chromatography is irrelevant at this stage and can only be decided upon by investigating which of the parameters is linked t o a chromatographically important criterion of merit. Such a criterion is always a function of the specific form of the overlapping peaks which in turn establishes a definite functional relationship beThe significance of skewness and its role in tween S and chromatography will be discussed in more detail in a separate Paper (5). As an illustration of the use of the bi-Gaussian approximation, consider peaks that are skewed as a result of a nonlinear distribution isotherm
+
+.
c, =
tklC,O
+ EkZ(Cm0)2
C, C O ,
C,
0.5 0.5 -0.5 -0.5 -0.5 -0.5 -0.5
LIST OF SYMBOLS parameter in F(z) bi-Gaussian distribution function, Equation 2 initial solute concentration in the mobile phase at the inlet = maximum value of C(z) = solute concentration in mobile phase (moles per unit volume of mobile phase) = solute concentration in stationary phase (moles per unit column volume) = Poisson distribution function, Equation 1 = (i = 1, 2) parameters in nonlinear distribution isotherm, Equation 17 = (i = 1, 2) mass parameters in C(z) = zero’th moment = parameter in F ( z ) = skewness, Equation 12 = S for bi-Gaussian distribution = S for Poisson distribution = S for Pearson’s Type 111 Curve = (i = 1, 2) width-at-halfheight parameters (see Figure 1) = = = = =
+
( 5 ) T. S. Buys and K. de Clerk, accepted for publication in Separ. Sci., 7, No. 4 (1972). (6) K. de Clerk and T. S. Buys, J. Cliranzamgr., 63, 193 (1971).
XC, 0.5 0.5 0.5
= = =
(17)
CONCLUSIONS The above analysis should be seen as a first statement of the applicability of the bi-Gaussian function in the prediction of the moments of skewed peaks. It appears that its use will be primarily restricted to the approximate prediction of the
99 9s 0.9324 0.9390 0.9184 0.8912 0.9071 0,9588 0.9635 0.9335 0.9441 0.9363
zero’th and first moments. In this respect, two uses are foreseen: It can serve as the basis for a more refined computer analysis in the case of overlapping peaks; and it can be used directly where high accuracy is not imperative. The latter is of special significance in work where the retention time is related to thermodynamic properties since it is the first moment, ( z ) , which is of importance here and even an approximate estimate of ( z ) is preferred above the commonly used peak maximum, z,,‘.
A computer simulation for this purpose is available [ e . g . Ref. (6)]and the results of a moments analysis of the resulting peak shapes by means of the bi-Gaussian model are compared in Table 1 with the actual moments. The required parameter values (zm, ul,and uz) were obtained by manual fitting a t the maximum and half-height. These are in accordance with the general predictions of the analysis outlined above. The zero’th moment is closely approximated while the ratio of the relative first moments are practically constant and are seen to depend on the functional form of the peaks which in this case is controlled by the sign of the nonlinearity parameter XC1, where (18) = 2 kz/(l kl) and Ca is the initial solute concentration in the mobile phase at the inlet. It is difficult to assess the significance of the variation in the proportionality constant (for the first moments) since the computer output was in digital form and the subsequent manual plotting of the peaks gave rise to additional errors in the width and height measurements.
SG
Ql ’1 Ql2
Ql3 Qli
E
x
distance coordinate coordinate of peak’s maximum coordinate of peak’s mean mean of bi-Gaussian distribution mean of Poisson distribution
Greek Symbols first moment relative to z = 0, Equation 5 second central moment, Equation 6 third central moment, Equation 6 a2(i = 2 ) , a3(i. = 3) void fraction parameter in model for non-linear chromatography, Equation 18. = variance = (i = 1, 2) standard deviation parameters in C(z) = skewness parameter, Equation 13 = for bi-Gaussian distribution = for Poisson distribution = for Pearson’s Type I11 Curve = = = = = =
+ + +
RECEIVED for review Semptember 16, 1971. Accepted December 21, 1971. ANALYTICAL CHEMISTRY, VOL. 44, NO. 7, JUNE 1972
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