A Fast Multiparametric Least-Squares Adjustment of GC Data.pdf

A Fast Multiparametric Least-Squares Adjustment of GC Data
M. Maeck / A. Y. Badjah Hadj Ahmed / A. Touabet / B. Y. Meklati*
U.S.T.H.B., Institut de Chimie, Laboratoire d'analyse organique fonctionnelle syst~matique, EI-Alia BP 31, Bab-Ezzouar, Alger,
Algeria
Key Words
Multiparametric least-squares adjustment
Kovats indices
Corrected retention times
Summary
Multiparametric least-squares regressions are well known
statistical methods which have not been fully applied to
the treatment of GC data. We describe here such a
weighted method. The calculation times are short, con-
vergence is straightforward and there is no need for a
series of consecutive homologous compounds. Equations
are obtained for estimation of standard deviations on
parameters and derived quantities. Confidence intervals
so constructed can save much time in the analytical use
of GC data.
Introduction
For comparative inter-laboratory studies, the most widely
used presentations of gas chromatographic data 'are the
relative retention method [1] and the Retention Index
system of Kov&ts [2]. To calculate the adjusted retention
time (t'R) the determination of the column dead time (t M)
is necessary.
For the experimental determination of the gas hold up time
[3], air and other non-retarded substances (hydrogen, argon,
helium, methane, etc.... ) have been used. However, for
example with the flame ionization detector (FID), certain
substances do not give a signal (e.g. air) or show a signifi-
cant retention which poses unexpected difficulties [4].
An alternative approach to the above is to take advantage
of the well known assumption, at least for higher homol-
ogues, of a linear relationship between the logarithm of
the adjusted retention time and the number of carbon
atoms for normal paraffins:
t
In tRi = In (tRi -- t M ) = b" Z i + c (1)
where Zi and tRi are the carbon number and the uncorrect-
ed retention time of the i'th n-alkane examined.
ChromatographiaVol. 22, No. 7--12, December1986 Originals
In addition to the dead time calculation, equation (1)
permits the determination of the slope (b) and the inter-
cept (c) of the n-alkane retention time curve.
Many mathematical methods to calculate tM havebeen de-
scribed [5-8] and this subject is still of considerable
interest [9, 10] because of computer facilities now available.
Most of the methods make use of equation (1) and differ
in the mathematical or statistical method adopted (optimi-
zation of tai; Kovfits indices, etc.... ).
In a recent review [11] Smith, Haken and Wainwright point
out the lack of a fast "simultaneous non-linear least-squares
estimation of tM, b and c by the use of numerical minimisa-
tion". The aim of this paper is to present such a method
with an expected improvement for weighting of the data
as discussed by the same authors [12] about their choice
of an objective function to minimize.
The procedure detailed below and routinely used in our
laboratory [13, 14} is an adaptation of the statistical
method of Wentworth [15] to the Kov~ts's Retention Index
system [2, 16] under the assumption of linearity as de-
scribed by equation (1).
For a proper weighting of the data, equation (1) is best
transformed to equation (2) where the measured gross
retention times (tRi) appear explicitly:
tRi= tM+A'e B'zi (B = b;A=eC) (2)
In our opinion, a major feature of the method is the op-
portunity to obtain reliable estimations of standard devia-
tions for the parameters (tM, A and B) and derived quanti-
ties (corrected retention times, indices.... ) by usual error
propagation formulae.
Moreover some problems relative to the extrapolation
involved in the determination of the dead time [17] are
much reduced and the relevance to a linear relationship
such as equation (1) can be tested.
Description of the Method
The flow chart of Fig. 1 is fully explained as follows:
(a) The starting point is either a matrix of N • NI reten-
tion times (tRi j) obtained from NI injections of N references
or a N • 2 matrix containing N mean retention times
(tRi) and their standard deviations (si). In the first case,
245
0009-5893/86/7--12 0245-04 ~ 03.00/0 9 1986 Friedr. Vieweg& SohnVerlagsgesellschaftmbH

PRINT
t H , s(t N
tRij
N re f e r o l l c u s
NI injections
tlli , si
N references
I C
A
L
C
U
L
A
T
E
tNi , si
CALCULATE FIRST
GUESS : t~ , A e , B e
1
CALCULATE EI-EHENTSI
~ N O ~ of matrix H J
of" vecknr V J
1
SOLVE SYSTEM
H.X = V
x(At M , AA , &B)
a , s(A)
8 , s(O)
,l
'INVERSE [
matrix H
l YES
Fig. 1
1
tH A - &A [
Flow chart for the proposed method (listing for HP 71B or Sinclair
ZX-81 will be sent on request)9
tRij
N
NI
tRi
si
tM,A, B
t ~ , AO,BO
AtM, AA, AB
gross retention time of the ith reference compound
in the jth duplicate
number of reference compounds
number of injections (duplicates)
mean retention time of the ith reference compound
standard deviation of tRi
parameters of equation (2)
initial guessesfor parameters of equation (2)
corrections on the parameters issued from the last
iteration
S(tM),S(A),s(B) standard deviations of the adjusted parameters
M 3 X 3 matrix of coefficients
V 3 X 1 vector of independent terms
M-1 3 X 3 matrix of variances/covariances
the reduction of data follows the usual pattern of statistical
data processing:
NI N//Z
tRi = ~, tRij/NI Si = (tRij--tRi) 2/NI
j=l j=l
(b) A satisfactory first guess for the parameters is obtained
by any simple method but the iterative process is robust
enough to support a very crude approximation:
t ~ = (tRi)min, the lowest retention time
Be = In [(tRN -- t0M)/(tRN -1 -- tOM)]
ZN -- ZN - 1
AO= tRN.e-B'ZN
where N denotes the reference compound with the longer
retention time and N - 1 the one with the shorter retention
time.
(c) According to Wentworth [15] a 3 • 3 matrix system is
now solved for the corrections At M, AA and AB:
~] (FtM 9FtM/L ) 9AtM + ~ (FtM" FA/L) 9A A
+ ~;(F~M ' FB/L)' AB= X(F' FtM/L)
~;(FA" FtM/L)"AtM +~(FA
+%(FA
Y' (FB " FtM/L) " ~tM +%(FB
+~](F B
with:
F= tRi--tM-- A'e B'Z; L = s2
9FA/L ) 9AA
' FB/L)" z~B = ~](F' FA/L)
9FA/L ) 9AA
9FB/L)-AB=%(F" FB/L)
(the weight of a piece of the data is inversely proportional
to its variance and the indices are exact by definition)
FtM = (~)F/CqtM)=-1 FA = (GqF/OA) =-e B'Zi
FB = (;)F/cqB) = -A 9 9EB'zi
All the summations are taken over i = 1 to i = N, where N
is the number of references used in the adjustment. There
is no need for consecutive homologues.
(d) The 3 x 3 system has never been found ill-conditioned
and is solved by a Crout method (HP 71B microcomputer
with Math Pac module) or by direct expansion of de-
terminants (Cramer's method) on a ZX-81 Sinclair micro-
computer.
(e) After updating the parameters, a test for convergence is
made. Only five or six iterations are usually needed to
stabilize the dead time.
(f) The M-1 inverse matrix of variances/covariances is
easily obtained by classical methods. We use a threefold
resolution of the M" X = V system on the ZX-81 Sinclair.
(g) Outputs include parameters (tM, A and B) with their
standard deviations and the calculated retention times as
well as calculated corrected retention times (t'Ri (calc.)):
tM+ M~/-~11,1) A_+ M~/MEI(2,2) B+ M~'M--113,3)
tai(calc.) = t M+A'e B'zi
s2(calc.) =M-1(1,1)+ F2-M-l(2,2)+F2 9 1(3,3)
- 2' [FA" M-1 (1,2) + FB" M-1 (1,3)
- FA 9FB 9M 1 (2, 3)]
tRi (calc.) = A' eB" zi
si2(calc.) = F~,'M -1(2,2)+F 2.M -1(3,3)
+ 2' FA ' FB ' M-1 (2,3)
(F A and Fg as defined in step (c) above).
The process (a)to (f) is completed in less than 30 seconds
when applied to 10 injections of 10 references. For normal
on-line use about half this time is enough to achieve con-
vergence.
Application
The experimental data of Table l were obtained from
reference [18] and used in the following regression9
246 Chromatographia Vol. 22, No. 7-12, December 1986 Originals

Table I. Raw data from reference [18] Table V. Kov,4ts Indices: back-calculations and interpolations, data
n-alkanes tRi/S si/s
n-C9 H20
n-C10H22
nC11 H24
n-C12H26
n-C13H28
115.7
200.7
370.0
706
1371
0.21
0.32
0.66
3.20
6.90
Table II. Convergence of the regression
Iteration B s/s
t M/s A/s
115.7000 0.075382
31.4134 0.142408
29.3836 0.179286
29.3810 0.179340
29.3810 0.179343
29.3810 0.1 79343
carculation time: 13.O8s (HP 71B); 13s
0.754499
0.684302
0.686818
0.686253
0.686250
0.686250
ZX-81 Sinclair
s =residualerror= J%(tRi(ex p)-tRi(cal))2/N (N= 5)
72.86
154.5
5.364
0.735
0.723
0.723
Table III. Quality of the regression: calculated retention times in
seconds.
Z
9 115.7
10 200.8
11 369.8
12 705.4
13 1372.1
s 0.55
(GB) (TZ)
115.7
200.9
369.9
705.4
1371.4
0.35
115.5
200.9
370.2
705.8
1371.0
0.18
115.7
200.8
369.8
705.6
1372.6
0.72
(W) (WW)
{0.2)
(0.3)
(O.5)
(1.5)
(5.7)
s= residual error (seeTable II)
(GB) method of GroblerandBalizs[6]
(TZ) method of TothandZala [18]
(W) present method without weighting
(WW) present weighted method; standard deviations in brackets
Table IV. Back-calculation of Kov&ts Indices
Z (GB)
9 900.0
10 999.9
11 1100.1
12 1200.1
13 1299.9
(TZ)
899.9
999.8
1100.0
1200.1
1300.0
(W)
900.3
999.8
1099.9
1200.0
1300.0
(ww)
900.3 (0.5)
999.9 (0.4)
1100.1 (0.4)
1200.1 (1.0)
1299.8 (1.1)
(GB), (TZ), (W) and (WW): see table Ill.
Progress to convergence is shown in Table II and the good-
ness of fit and the influence of weighting can be appreciated
in Table III.
With the same objective function, i.e. without weighting,
the method presented here finds the best fitting with a
residual error (s) half that of Toth's and one third that of
Grobler's. A modified simplex method [19] leads to the
same parameters within about fifteen minutes depending
strongly on the initial guess for tM, A and B.
Use of experimental variances (s2) as weighting functions
recognizes the fact that an error of about one second on the
from reference 14]
Compounds
n-Hexane
2-Pentanone
n-Heptane
n-Butanol
1,4-Dioxane
2-Methyl-2-pentanol
1-nitropropane
n-octane
1-Iodobutane
2-Octyne
n-Nonane
n-Decane
cis-Hydrindane
n-Undecane
Retention I(GD)
0.00 600.0
1.24 685.2
154 700.1
2.09 724.2
2.28 731.7
2.82 751.2
4.09 789.0
4.51 799.7
6.64 844.6
7.45 858.7
10.32 900.0
21.61 1000.3
23.65 1013.0
43.36 1099.8
I(w) I(ww)
601.2 600.0 (1.3)
685.4 685.1 (0.8)
700.3 700.1 (0.9)
724.2 724.2 (1.0)
731.7 731.7 (1.1)
751.1 751.2 (1.1)
788.8 789.0 (1.3)
799.5 799.7 {1.3)
844.3 844.6 (1.2)
858.4 858.7 (1.4)
899.8 900.0 (1.4)
1000.2 1000.3 (1.6)
1012.9 1013.0 (1.6)
1100.0 1099.9(1.9)
(GD) Exact Calculator method of reference [4]
(W) present method without weighting
(WW) present weighted method, standard deviations in brackets
calculated retention time is of little importance for Z = 13
(si = 5.7) but is dramatic for Z = 9 (si= 0.2). This often
makes the residual error (s) greater (Table III, last column).
Calculation of Kov;tts Indices
It is common to use the model function (eq. (2)) to back-
calculate the Kov~ts Indices of the reference compounds
[20] and such results are included in Table IV.
Here again a proper weighting minimizes the impact of
residual errors by loosing the adjustment near the higher
boiling reference compounds. Standard deviations on
indices (si (I)) are calculated from error propagation formula:
100
I i = ~--' In [(tRi- tM)/A], the ith index
s2(I) = I~" Is2 + M-1(1, 1)] + 12" M-'(2,2)
+ 1239M-1 (3,3) + 2'[1112' M-1 (1,2)
+1113"M 1(1,3)+1213'M 1(2,3)]
11 = (~)l/(3tM) = -- (al/atRi) = --(100/B)/(tRi-- tM )
12 = (al/~)A) = - 1/(A" B) 13 = (~)I/~)B) =- I/B
A final illustration of the versatility of the proposed method
is obtained from the experimental data of Dominguez et al.
(reference [4] and Table V) where an n-alkane peak is used
as origin for measurement. The paper gives no explicit
standard deviation and a constant relative error of 1% on
retention time is assumed.
Indices in column (GD) and (WW) are quite similar but the
opportunity to define confidence intervals from the square
deviations may be of uppermost importance in analytical
gas chromatography.
Conclusion
The adaptation of Wentworth's statistical method to GC
data leads, in short calculation times, to high quality least-
squares fitted parameters. If weighting of data is included,
square deviations will be obtained.
Chromatographia Vol. 22, No. 7--12, December 1986 Originals 247

There is no need to use consecutive homologues and the
calculated values are identical with those obtained by the
modified simplex method but the iteration is rarely, if ever,
lost on a local minimum.
Extensive use of the proposed method confirms a standard
deviation of one or two units on Kov&ts Indices for 1%
scattered experimental data.
References
[11 A.T. James, A.J.P. Martin, Biochem. J.,50,679(1952).
[2] E. Kov#ts, Helv. Chim. Acta, 41, 1915 (1958).
[3] R.E. Kaiser, Chromatographia, 2, 215 (1969).
[4] J.A.G. Oominguez, J.G. Munoz, E. F. Sanchez, M.J. Mole-
ra, J. Chromatogr. Sci., 15, 520 (1977).
[5] R.E. Kaiser, Chromatographia, 7,251 (1974).
[6] A. Grobler, G. Balizs, J. Chromatogr. Sci., 12, 57 (1974).
[7] L.S. Ettre, Chromatographia, 13, 73 (1980),
[8] M.S. Wainwright, J. K. Haken, J. Chromatogr., 184, 1 (1980).
[9] A. Toth, E. Zala, J. Chromatogr., 284, 53 (1984).
[10] L. Ambrus, J. Chromatogr., 294, 328 (1984).
[11] R. J. Smith, J. K. Haken, M. S. Wainwright, J. Chromatogr.,
334, 95 (1985).
[12] R.J. Smith, J. K. Haken, M.S. Wainwright, J. Chromatogr.,
147, 65 (1978).
[13] A. Touabet, M. Maeck, B. Y. Meklati, IIe S(~minaireNational
sur la Chimie, Oran (Nov. 1985).
[14] A. Touabet, A. Y. Badjah Hadj Ahmed, M. Maeck, B.Y.
Meklati, J.H.R.C. & C.C., in press6 (86).
[15] W.E. Wentworth, J. Chem. Educ.,42,96(1965).
[16] M. V. Budahegyi, E. R. Lombosi, T.S. Lombosi, S. Y. Mesza-
ros, Sz. Nyiredy, G. Tarjan, I. Timar, J.M. Takacs, J. Chro-
matogr., 271,213 (1983).
[17] M.S. Wainwright, J.K. Haken, D. Srisukh, J. Chromatogr.,
179, 160 (1979).
[18] A. Toth, E. Zala, J. Chromatogr., 298, 381 (1984).
[19] S. IV, Deming, S. L. Morgan, Anal. Chem., 45,278A (1973).
[20] R. J. Smith, J. K. Haken, M. S. Wainwright, B, G. Madden, J.
Chromatogr. $ci., 15, 520 (1977).
Received: Jan. 2, 1986
Revisedmanuscript
received: June 3, 1986
Accepted: June 17, 1986
E
248 Chromatographia VoL 22, No. 7--12, December 1986 Originals

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