Nonlinear least squares method

Data can be fit to this equation by the nonlinear least-squares method. As it turns out, the Guggenheim approach for first-order kinetics is valid, even though the reaction... 

Kinetic curves were analyzed and the further correlations were determined with a nonlinear least-square-method PC program, working with the Gauss-Newton method. 

This system of i + 2 equations is nonlinear, and for this reason probably has not received attention in the least-squares method (207). We are able to give an explicit solution (163) for the particular case when Xy = xj and m,- = m for all values of i that is, when all reactions of the series are studied at a set of temperatures, not necessarily equidistant, but the same for all reactions. Let us introduce... 

Parameter estimation. Integral reactor behavior was used for the interpretation of the experimental data, using N2O conversion levels up to 70%. The temperature dependency of the rate parameters was expressed in the Arrhenius form. The apparent rate parameters have been estimated by nonlinear least-squares methods, minimizing the sum of squares of the residual N2O conversion. Transport limitations could be neglected. 

Pigure 8.8a and b, respectively, show fluorescence autocorrelation curves of R6G in ethylene glycol and R123 in water at 294.4 K. The solid lines in these traces are curves analyzed by the nonlinear least square method with Eq. (8.1). Residuals plotted on top of the traces clearly indicate that the experimental results were well reproduced by the... 

Nonlinear Least-Squares Methods of Data Analysis.174... 

Parameter estimation including nonlinear least-squares methods... 

Kinetic analysis usually employs concentration as the independent variable in equations that express the relationships between the parameter being measured and initial concentrations of the components. Such is the case with simultaneous determinations based on the use of the classical least-squares method but not for nonlinear multicomponent analyses. However, the problem is simplified if the measured parameter is used as the independent variable also, this method resolves for the concentration of the components of interest being measured as a function of a measurable quantity. This model, which can be used to fit data that are far from linear, has been used for the resolution of mixtures of protocatechuic... 

J. Rudzki Small, L. J. Libertini, E. W. Small. Analysis of Photoacoustic Waveforms Using the Nonlinear Least Squares Method. Biophys. Chem. 1992, 42, 29—48. 

The constants needed to obtain a value of ifl as a function of m can be obtained by fitting the enthalpy of dilution data to Equation (18.67) by a nonlinear least-squares method, (see Section A.l). 

Selected entries from Methods in Enzymology [vol, page(s)] Computer programs, 240, 312 infrared S-H stretch bands for hemoglobin A, 232, 159-160 determination of enzyme kinetic parameter, 240, 314-319 kinetics program, in finite element analysis of hemoglobin-CO reaction, 232, 523-524, 538-558 nonlinear least-squares method, 240, 3-5, 10 to oxygen equilibrium curve, 232, 559, 563 parameter estimation with Jacobians, 240, 187-191. 

Only in the simplest cases—a single Gaussian component, for example— may conventional linear least-squares method be employed to solve for u. More commonly, either approximate linearized methods or nonlinear methods are employed. 

In a strict sense parameter estimation is the procedure of computing the estimates by localizing the extremum point of an objective function. A further advantage of the least squares method is that this step is well supported by efficient numerical techniques. Its use is particularly simple if the response function (3.1) is linear in the parameters, since then the estimates are found by linear regression without the inherent iteration in nonlinear optimization problems. 

Least-squares methods are usually used for fitting a model to experimental data. They may be used for functions consisting of square sums of nonlinear functions. The well-known Gauss-Newton method often leads to instabilities in the minimization process since the steps are too large. The Marquardt algorithm [9 1 is better in this respect but it is computationally expensive. 

At modest SNR, nonlinear least-squares methods (either in the time or the frequency domain) have been found to give more accurate results than linear... 

Application of a least-squares method to the linearized plots (e.g., Scatchard and Hames) is not reasonable for analysis of drug-protein binding or other similar cases (e.g., adsorption) to obtain the parameters because the experimental errors are not parallel to the y-axis. In other words, because the original data have been transformed into the linear form, the experimental errors appear on both axes (i.e., independent and dependent variables). The errors are parallel to the y-axis at low levels of saturation and to the x-axis at high levels of saturation. The use of a double reciprocal plot to determine the binding parameters is recommended because the experimental errors are parallel to the y-axis. The best approach to this type of experimental data is to carry out nonlinear regression analysis on the original equation and untransformed data. 

Therefore it is better to use the nonlinear model directly in a nonlinear regression of the observed variable, the nonlinear least-squares method. Because of the nonlinearity minimization is an iterative process. 

The problem consists in seeking such a combination of the values of constants k which gives the minimum value of Q ( >mm). Before computers became commonly available, the kinetic equations had usually been transformed into a linear form and the linear regression ( least-squares method ) had been applied to find the best set of constants. This procedure is not statistically correct in most cases. Therefore, only the nonlinear regression method can be recommended to optimize constants in kinetic equations that have a nonlinear form [48-51]. 

The crystal structures of 4-(6-nitro-2-benzoxazolyl)phenyl 4-(acryloyloxyhexyloxy) benzoate, 2-[4-A,iV-bis(2-hydroxyethyl)] phenyl-6-nitrobenzoxazole monohydrate (A) and 2- 4-/V-(6-hydroxyhexyl)-A-mcthyl] phenyl-6-nitrobenzoxazole (B) have been determined at room temperature by direct methods and refined by full-matrix least-squares method [210], These compounds are monomer precursors of polymers with nonlinear optical properties of the second order. 

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