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Anal. Chem. 1988, 60, 1083-1084
Comment on Rigorous Convergence Algorithm for Fitting a Monoexponential Function with a Background Term Using the Least-Squares Method Sir: Recently, Smith et al. (1)described an algorithm for analyzing data that can be described by a single exponential ae-kt plus a background term b, claiming among others guaranteed convergence to the global minimum according to the least-squares criterion
the derivative of eq 1with respect to k . Obviously, eq 4 and 6 are not identical and therefore their roots will be different in the general case. In order to clarify the difference between the two approaches, eq 10 of (1)and ita correct analogue can be reformulated as 7 and 8, respectively. Equation 7 is slightly
n
Cai2 = Min; ai = y i - b - ae-kt,
s =
n
(1)
i=l
f ( k ) = CAiti
In eq 1, n is the number of data pairs y L ,ti and b, a , and k are adjustable parameters. Weights are omitted throughout this comment for simplicity. The algorithm proposed in ref 1seems to be fast, new, and attractive. Because this might induce widespread application, it is mandatory to point out that the underlying mathematics are incorrect in a nonobvious way and will not yield the global minimum as postulated by the authors, who propose to find this minimum by solving the equations n
n
i=l
i=l
C(yi- b - ae-ktt)e-ktz = Caie-kta
0=
n
n
i=l
i=l
0 = C(yi- b - ae-kti)ti=
(2)
(4)
Equations 2 and 3 are the derivatives of eq 1with respect to a and b , respectively, and eq 4 pertains to a hypothetical second exponential term be-kOtwhere ko is zero. In eq 2-4 a and b are linear parameters and therefore may be elimated, leaving a single equation f ( k ) = 0 with k as the only adjustable parameter. The optimum value of 12 is found iteratively by the Newton-Raphson method
kn = k - f ( k ) / f ' ( k )
(5)
According to the authors, cf. ref 1,this algorithm is about 20 times faster than a classical least-squares approach as described by Wentworth (2). As already stated, the use of eq 2-4 does not yield the global minimum, however. The global minimum is uniquely defined by eq 2 , 3 , and 6 where eq 6 is n
0=
i=l
(7)
i=l
n
(yi - b - ae-kti)tie-kti =
(6)
Caitie-ktc i=l
n
g(k) =
(8)
CAitie-kti i=l
simpler than 8 but the residuals are not properly weighted; the use of ti instead of overweights the residuals near the end of the measurement relative to the least-squares solution. Unfortunately, the authors have been mislead by an invalid test to believe that the global minimum can be reached this way. Rather than discussing the errors in the involved test, we present the results of some calculations that show that the incriminated algorithm, while giving rather close estimates, never actually reaches the global minimum. Computer data were generated at 20 equidistant times ti = 1, 2, ..., 20 with a = 1, b = 0.5, and k = 0.2 or k = 0.4, and random noise (3) of 0.005-0.08 absorbance unit was superimposed. Rate constants were calculated with the algorithm proposed by Smith et al. (I),eq 5, with a program based on the correct normal equations and with the noniterative method of Guggenheim ( 4 ) . By use of these rate constants, the best estimates of a and b were obtained by linear regression and subsequently the s u m s of squared residuals, s, were calculated. The results are compiled in Table I. Several points are obvious. (i) The fit based on the correct normal equations always gives the lowest value of s, as postulated by theory. (ii) differences between the results based on a given set of data are generally small, 1%in either k or s is observed only at relatively high noise levels. (iii) With small or moderate noise levels the noniterative method of
Table I. Sum of Squared Residuals, s, and Rate Constants, k,Obtained with Different Algorithms
model data entry
k
noisea
6 7 8 9 10
0.2 0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4 0.4
0.5 1 2 4 8 0.5 1 2 4 8
In percent of total change in absorbance.
least-squares estimates SLS
2.43 x 9-76 x 3.90 x 1.56 X 6.24 X 2.43 x 9.72 x 3.89 x 1.56 x 6.22 X
kLS
10-4 10-4 10-3
lo-* 10-4 10-4
10-4 10-3
0.200 0.200 0.200 0.201 0.201 0.399 0.398 0.396 0.393 0.386
Smith et al. ( I ) As Akb 0.04 0.04 0.04 0.04 0.05 0.25 0.25 0.24 0.24 0.23
0.07 0.14 0.27 0.54 1.08 0.14 0.29 0.57 1.13 2.20
Guggenheim (3) ilkb
As
0.01
0.01 0.01 0.03 0.07 0.13 0.15 0.17 0.24 0.43
0.03 0.06 0.16 0.43 1.34 0.11 0.22 0.48 1.13 2.96
percent of least-squares estimates: As = (s - sLs).lOO/sLs,Ak = Ik - kLsl.lOO/kLs.
0003-2700/88/0380-1083$01.50/00 1988 American Chemical Society
Anal. Chem. 1988, 6 0 , 1084-1086
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Guggenheim ( 4 ) may be equal to or better than the result based on eq 2-5.
CONCLUSIONS The use of the algorithm described in ref 1 should not be advocated, since it does not converge to the global minimum as defined by the least-squares criterion. If for any reason finding the global minimum is thought to be superfluous, noniterative approaches, e.g. the method of Guggenheim ( 4 ) , will give results of comparable quality in still much less computing time. If the global minimum is wanted, and we surely support this approach, the elimination of linear parameters and the use of analytical instead of numerical derivatives is indeed the method of choice. An algorithm based on the correct normal equations (eq 2 , 3, and 6 in this comment) and otherwise following the ideas presented in ref 1 will find the global minimum in a one-dimensional parameter space. Elimination of linear parameters (5-7) and analytical derivatives ( 4 9 )have been suggested before, however, and, contrary to an explicit statement in ref 1, have been used in combination (8, 9). An algorithm based on the correct normal equations combined with the ideas presented in ref 1 then is rather close to the approach described in ref 9. As the main remaining difference the implicit changes of the best estimates for a and b as a function of k, i.e. the partial derivatives aalak and abldk are included in ref 9, but not in ref 1. As a consequence, the time needed per iteration is reduced, but the number of it-
erations is increased. More important, a much better initial estimate for k is needed using eq 2, 3, and 6 rather than following the ideas described in ref 5-9 where any estimate between lo* and 10 gave unproblematic convergence.
LITERATURE CITED Jericevic, Z.; Benson, D. M.; Bryan, J.: Smith, L. C. Anal. Chem. 1987, 5 9 , 658-662. Wentworth. W. E. J. Chem. Educ. 1965, 42, 96-103. Abramowltz, M.; Stegun, I. A. Haandbook of Mathematical Functions; NBS Appl. Math. Ser. 55; Natlonal Bureau of Standards: Washington, DC, 1964; p 952. Guggenhelm, E. A. Philos. Mag. 1926, 2, 538. Richards, F. S. G. J . R . Stat. Soc. 1961, 23, 469-475. Lawton, W. H.; Sylvestre, E. A. Technometrics 1871, 75, 461-467. Gampp, H.; Maeder, M.; Zuberbuhler, A. D. Talanta 1980, 2 7 , 1037-1045. Golub, G. H.; Pereyra, V. SIAM J. Numer. Anal. 1973. 10, 413-432. Gampp, H.; Maeder, M.; Meyer, C.: Zuberbuhler, A. D. Talanta 1985, 3 2 , 95-101.
Harald Gampp Marcel Maeder* Andreas D. Zuberbuhler* Institute of Inorganic Chemistry University of Basel CH-4056 Basel, Switzerland
RECEIVED for review April 14, 1987. Accepted December 9, 1987. We thank one of the referees for the suggestion to specifically include 7-9 into this work. The work was supported by the Swiss National Science Foundation, Grant No. 2.851-0.85.
TECHNICAL NOTES Fluorlnation of Sulfur Tetrafluoride, Pentafluorosulfur Chloride, and Disulfur Decafluoride to Sulfur Hexafluoride for Mass Spectrometric Isotope Ratio Analysis Swroop K. Bains-Sahota and Mark H. Thiemens* Department of Chemistry, B-017, University of California-Sun
Measurements of isotopic ratio variations have been employed to study a wide range of processes, e.g. paleoclimatology, rock formation temperatures, and quantum chemical mechanisms. Hulston and Thode ( I ) were the first to show how stable isotopic variations in meteoritic material could be used to distinguish nuclear and chemical processes. Since chemical processes were thought to obey mass-dependent fractionations, any departure from this fractionation law would then reflect a nuclear process such as nucleosynthesis, radiogenic decay, or spallation. A mass-dependent isotope effect between species of mass M and (M + 2 ) is approximately twice as large as the effect between species of mass M and (M + 1). Therefore, in order to distinguish between mass-dependent and mass-independent processes, at least three isotopes of the same element must be simultaneously measured. Clayton et al. ( 2 )observed a non-mass-dependent oxygen isotope distribution in the high-temperature minerals within the carbonaceous chondrite Allende. Since chemical processes were assumed to produce mass-dependent fractionations, the observed effect was attributed to a nuclear process. Since that time, mass-independent isotopic fractionations have been observed in chemical processes ( 3 ) . For oxygen a chemically produced mass-independent effect has been observed which
Diego, La Jolla, California 92093
produces an isotopic fractionation identical with that observed in Allende. The mass-independent effect has been suggested as involving isotopic symmetry effects in the production of ozone ( 4 ) . A mass-independent sulfur isotope effect has also been reported in the formation of S2FIo(5). The study of mass-independent fractionations involving symmetry and quantum effects is of considerable importance and necessitates multiisotopic measurements. Employment of multiisotopic measurements allows the detection of physical-chemical processes which might not be observed by other spectroscopic techniques, such as those reported for oxygen and sulfur (3-5). Development of the analytical capability to simultaneously measure 3aS/32S,33S/32S,at high precision, may also eventually permit measurement of these ratios in atmospheric samples such as sulfate and H,S. These gas-phase reactions may produce mass-independent isotopic fractionations which might ultimately be of interest in identifying specific atmospheric transformations. At present, only 34S/32S ratios have been measured in atmospheric sulfur species. Isotopic analyses of sulfur minerals and native sulfur were first performed by conversion to sulfur dioxide for mass spectrometric analysis. As discussed by Puchett et al. (6), sulfur hexafluoride is a superior gas for mass spectrometric
0003-2700/68/0360-1084$01.50/0 0 1988 American Chemical Society
