A new scheme for calculating weights and describing correlations in nonlinear least-squares fits

Hessler, Jan P.; Current, David H.; Ogren, Paul J.

doi:10.1063/1.168569

Cited by 9 publications

(17 citation statements)

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“…To extract the cross section and rate coefficients from an absorption profile, we perform a nonlinear least-squares analysis to minimize the χ 2 merit function. 52 To perform this analysis, we have introduced a dimensionless curVature matrix 53 into the equations of standard nonlinear least-squares analysis. One advantage of this reformulation is that the dimensionless curvature matrix used to reduce the data is mathematically equivalent to the discrimination matrix of the correlation analysis discussed above.…”

Section: Experimental Design and Methods Of Data Reductionmentioning

confidence: 99%

Recombination of Methyl Radicals. 1. New Data between 1175 and 1750 K in the Falloff Region

1996

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The rate coefficient for the recombination of methyl radicals, CH 3 + CH 3 f C 2 H 6 , was measured in incident shock-wave experiments on azomethane (0.12-1.3%) in argon. Methyl radicals were detected at 46 294 cm -1 (215.94 nm in air) by the tunable-laser flash-absorption technique. Postshock pressures between 114 and 230 kPa (1.13-2.27 atm) produced buffer gas concentrations between 5.4 × 10 18 and 1.

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Section: Experimental Design and Methods Of Data Reductionmentioning

confidence: 99%

Recombination of Methyl Radicals. 1. New Data between 1175 and 1750 K in the Falloff Region

1996

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“…Two other good sources are Meyer's book [12] and the recent classic Numerical Recipes [10]. Recently, we introduced a dimensionless formulation of least-squares analysis [13]. An advantage of this reformulation is that the relative weights or importance of parameters may be determined before the fitting procedure is executed.…”

Section: Review Of Techniques Used To Model Datamentioning

confidence: 99%

“…Before Monte Carlo simulations are performed it is instructive to test the parabolic approximation to the chi-squared surface. This may be accomplished by calculating the eigenvalues and eigenvectors of the dimensionless curvature matrix [13]. Then, the actual surface is calculated along each ei-If the distribution of the fractional deviations is normal both the skewness and the kurtosis will be zero.…”

Section: Monte Carlo Simulationsmentioning

confidence: 99%

The use of Monte Carlo simulations to evaluate kinetic data and analytic approximations

Hessler

1997

Int. J. Chem. Kinet.

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Experimental kineticists are always faced with the problem of reducing kinetic data to extract physically meaningful information. A particularly vexing problem arises when different models reproduce the data but yield different values for the physical parameters. For over forty-five years Monte Carlo simulation techniques have been used to study the statistical behavior of parameters extracted from data. Not only do these simulations provide realistic uncertainties, correlation coefficients, and confidence envelopes, but they also provide insight into the nature of the model. These insights may be obtained by viewing two-dimensional scatter plots of the fractional changes of the parameters and one-dimensional histograms of the distributions of the changes in the parameters. Monte Carlo simulations are illustrated with examples from and the high-pressure rate coefficient for OH ϩ CH : CH ϩ H O 4 3 2 methyl-methyl association. A more complex problem involves models for pressure-dependent rate coefficients in the falloff region. We have modeled methyl-methyl association with five of the most current analytic approximations for behavior in the falloff region. All of these reproduce the data to within their uncertainties. However, when Monte Carlo techniques are applied the correlations between the parameters and the nonlinear nature of their behavior become evident. We postulate that the statistical behavior of the parameters of a model may be used to distinguish one model from another and, thereby, identify those analytic approximations that hold promise for further investigation and utilization. Finally, the recent advent of highspeed workstations implies that Monte Carlo simulations should become a routine part of the analysis of kinetic data.

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“…Therefore, their data will not be considered. All of the remaining data below 1400 K were reduced with a dimensionless reformulation of nonlinear least-squares analysis, 32 where the uncertainty in each point was assumed to be proportional to the value of the rate coefficient. From this reduction we assigned an 8% uncertainty to each of the points below 1400 K. Above 1400 K, where data is obtained from experiments in shock tubes, the scatter is significantly larger and it is not possible to select one set over the other.…”

Section: ͑29͒mentioning

confidence: 99%

New empirical rate expression for reactions without a barrier: Analysis of the reaction of CN with O2

Hessler

1999

The Journal of Chemical Physics

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A quantum reactive scattering study of the spin-forbidden CH (X 2 Π)+ N 2 (X 1 Σ g + )→ HCN (X 1 Σ + )+ N ( 4 S) reactionThe rate coefficients of reactions that occur on potential energy surfaces without a barrier often exhibit a negative temperature dependence at low temperatures. Generally, this behavior is modeled with either the Harcourt-Essen equation, k(T)ϭAT Ϫm , or a ''negative'' activation energy, k(T) ϭAT m exp͕⌬E/k B T͖. Neither of these expressions is consistent with the Wigner threshold law. The general expression k(where the relative angular momentum of the reacting species is l, T W and m are independent parameters to be extracted from the data, and the amplitude of each partial wave is A l . This expression may be approximated by k(T)ϭA 0 (1ϩT/T W ) Ϫm exp͓(T/T W )/(1ϩT/T W )͔. For CNϩO 2˜N COϩO and COϩNO the above expression reproduces the rate data, the branching ratio to the COϩNO channel, and the reactive cross section for the NCOϩO channel. The rate coefficient for the NCOϩO channel is given by k(cm 3 s Ϫ1 )ϭ1.79ϫ10 Ϫ10 (ϩT/21.7) Ϫ1.38 ͕exp͓(T/21.7)/(1ϩT/21.7)͔Ϫ1͖ϩ4.62 ϫ10 Ϫ12 exp͓(T/21.7)/(1ϩT/21.7)͔ while for COϩNO we obtain k(cm 3 s Ϫ1 )ϭ1.79ϫ10 Ϫ10 (1 ϩT/21.7) Ϫ1.38 . An analytic form of the C-O bonding potential and the electric dipole-quadrupole interaction is used to show that the quantum threshold region extends up to 7 K. These results demonstrate the need of a complete quantum treatment for reactions that proceed on potential surfaces without a barrier.

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A new scheme for calculating weights and describing correlations in nonlinear least-squares fits

Cited by 9 publications

References 0 publications

Recombination of Methyl Radicals. 1. New Data between 1175 and 1750 K in the Falloff Region

Recombination of Methyl Radicals. 1. New Data between 1175 and 1750 K in the Falloff Region

The use of Monte Carlo simulations to evaluate kinetic data and analytic approximations

New empirical rate expression for reactions without a barrier: Analysis of the reaction of CN with O2

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