Nonlinear least-squares analysis

Lieb, S. G. Simplex Method of Nonlinear Least-Squares—A Logical Complementary Method to Linear Least-Squares Analysis ofData, /. Chem. Educ. 1997, 74, 1008-1011. 

This reaction was studied spectrophotometrically by monitoring the absorbance at 830 nm, where Pu02+ absorbs. The paired values of time and absorbance are presented for one experiment in Table 2-4. Figure 2-5 shows the data treatment according to Eq. (2-35). Nonlinear least-squares analysis gives k = (9.49 0.22) X 102 L mol-1 s-1 and a calculated end point absorbance of 0.025 0.003. 

These equations contain only one unknown parameter, kiAj. Assume values for it and solve Equations (11.20) and (11.21) simultaneously. Compare the calculated results with the experimental measurements using nonlinear least-squares analysis as in Equation (7.8). This is the preferred, modern approach, but the precomputer literature relied on computationally simpler methods for fitting kiAj. 

Kittrell, J.R., W.G. Hunter, and C.C. Watson, "Nonlinear Least Squares Analysis of Catalytic Rate Models", AIChEJ., 11(6), 1051-1057 (1965a). 

Provided that a value for KL is available, it is possible to use this equation to obtain a value for Kh the dissociation equilibrium constant for the inhibitor, by nonlinear least-squares analysis of the displacement curve. Alternatively, K can be calculated from the IC50, which may be obtained by simple interpolation by eye from a Hill plot or by fitting a curve to an equation of the type ... 

These problems are provided to afford an opportunity for the reader to analyze binding data of different sorts. The problems do not require nonlinear least squares analysis, but this would be recommended to those with access to appropriate facilities. It must be emphasized that, while linearizing transformations allow binding data to be clearly visualized, parameter estimation should... 

These plots can also provide information about the assumption of constant error variance (Section III) made in the unweighted linear or nonlinear least-squares analyses. If the residuals continually increase or continually decrease in such plots, a nonconstant error variance would be evident. Here, either a weighted least-squares analysis should be conducted (Section III,A,2) or a transformation should be found to stabilize the error variance (Section VI). 

Two experimental systems have been used to illustrate the theory for two-step surface electrode mechanism. O Dea et al. [90] studied the reduction of Dimethyl Yellow (4-(dimethylamino)azobenzene) adsorbed on a mercury electrode using the theory for two-step surface process in which the second redox step is totally irreversible. The thermodynamic and kinetic parameters have been derived from a pool of 11 experimental voltammograms with the aid of COOL algorithm for nonlinear least-squares analysis. In Britton-Robinson buffer at pH 6.0 and for a surface concentration of 1.73 X 10 molcm, the parameters of the two-step reduction of Dimethyl Yellow are iff = —0.397 0.001 V vs. SCE, Oc,i = 0.43 0.02, A sur,i =... 

Selected entries from Methods in Enzymology [vol, page(s)] Association constant determination, 259, 444-445 buoyant mass determination, 259, 432-433, 438, 441, 443, 444 cell handling, 259, 436-437 centerpiece selection, 259, 433-434, 436 centrifuge operation, 259, 437-438 concentration distribution, 259, 431 equilibration time, estimation, 259, 438-439 molecular weight calculation, 259, 431-432, 444 nonlinear least-squares analysis of primary data, 259, 449-451 oligomerization state of proteins [determination, 259, 439-441, 443 heterogeneous association, 259, 447-448 reversibility of association, 259, 445-447] optical systems, 259, 434-435 protein denaturants, 259, 439-440 retroviral protease, analysis, 241, 123-124 sample preparation, 259, 435-436 second virial coefficient [determination, 259, 443, 448-449 nonideality contribution, 259, 448-449] sensitivity, 259, 427 stoichiometry of reaction, determination, 259, 444-445 terms and symbols, 259, 429-431 thermodynamic parameter determination, 259, 427, 443-444, 449-451. 

ABORTIVE COMPLEX Nonlinear kinetics of drug bioavailability, PHARMACOKINETICS NONLINEAR LEAST SQUARES ANALYSIS Nonmetal oxides,... 

Tavare and Garside ( ) developed a method to employ the time evolution of the CSD in a seeded isothermal batch crystallizer to estimate both growth and nucleation kinetics. In this method, a distinction is made between the seed (S) crystals and those which have nucleated (N crystals). The moment transformation of the population balance model is used to represent the N crystals. A supersaturation balance is written in terms of both the N and S crystals. Experimental size distribution data is used along with a parameter estimation technique to obtain the kinetic constants. The parameter estimation involves a Laplace transform of the experimentally determined size distribution data followed a linear least square analysis. Depending on the form of the nucleation equation employed four, six or eight parameters will be estimated. A nonlinear method of parameter estimation employing desupersaturation curve data has been developed by Witkowki et al (S5). 

An Excel spreadsheet illustrating the use of the Solver tool for nonlinear least-squares analysis of a fluorescent decay curve of a ruby crystal. The sum of the squares of residuals is calculated in cell C14 and is minimized in Solver by iterative variation of the parameters in cells CIO, Cll, and C12. 

Optimization of predietions ean be made utilizing linear as well as nonlinear relationships by means of statistieal methods to correlate chemical and physiological descriptors to experimental datasets. These statistical methods inelude multilinear partial least square analysis, principal component analysis, and neural networking. Many of these tools are included in QSPR/QSAR packages through companies sueh as Advanced Chemistry Development, SemiChem, EduSoft, BioByte, TOPKAT, MDL, ChemSilico, Pallas, Pharma Algorithms, and others. 

The most generally useful methods and the only statistically correct procedures for calculating reactivity ratios from binary copolymerization data involve nonlinear least squares analysis of the data or application of the error in variables (EVM) method. Effective use of either procedure requires more iterations than can be performed by manual calculations. An cfHcicnl computer program for nonlinear least squares estimates of reactivity ratios has been published by Tidwell and Mortimer [13]. The EVM procedure has been reported by O Driscoll and Reilly [14]. 

If an independent measure of [ ]q is available from chemisorption, the constants k2, Ki, and K2 can be obtained from linear regression. It should be noted that many kineticists no longer use the linearized form of the rate equation to obtain rate constants. Inverting the rate expression places greater statistical emphasis on the lowest measured rates in a data set. Since the lowest rates are usually the least precise, a nonlinear least squares analysis of the entire data set using the normal rate expression is preferred. 

Therefore, regardless of the pathway that the reaction follows, fitting the data to equation (5) will yield the correct Kapp. Dissociation constants were estimated by nonlinear least squares analysis of me data using Kaleidagraph 3.0.2 (Abelbeck Software, Reading, PA). 

Lagtime, 75 Laplace transform, 82 Larmor precessional frequency, 155, 165 Laser pulse absorption, 144 Lattice energy, 403 Law of mass action, 60, 125 Least-squares analysis linear, 41 nonlinear, 49 univariate, 44 unweighted, 44, 51 weighted, 46, 51, 247 Leaving group, 9, 340, 349, 357 Lennard-Jones potential, 393 Lewis acid-base adduct, 425 Lewis acid catalysis, 265 Lewis acidity, 426... 

In nonlinear least squares analysis we search for those parameter values I that mirumize the sum of squares of the differences between the measured values and the calculated values for all the data points. Many software programs are available to find these parameter values and all one has to do is to enter the data. The PO LYMATH software will be used to illustrate this technique. In order to carry out the search efficiently, in some cases one has to enter initial estimates of the parameter values close to the actual values. These estimates can be obtained using the linear-least-squares technique just discussed. 

We will now apply nonlinear least-squares analysis to reaction rate data to determine the rate law parameters. Here we make estimates of the parameter values (e.g., reaction order, specific rate constants) in order to calculate the rate of reaction, We then search for those values that will minimize the sum of the squared differences of the measured reaction rates, r , and the calculated reaction rates, r. That is, we want the sum of (r — for all data points to be minimum. If we carried out N experiments, we would want to find the parameter values (e.g., E, activation energy, reaction orders) that would minimize the quantity... 

Both the linear and nonlinear least-squares analyses presented above assume that the variance is constant throughout the range of the measured variables. If this is not the case, a weighted least-squares analysis must be used to obtain better estimates of the rate law parameters. If the error in measurement is at a fixed level, the relative error in the dependent variable will increase as the independent variable increases (decreases). For example, in a first-order decay reaction (Ca = if the error in concentration measurement is... 

Integrate the equation for the combined mole balance and rate law and then use nonlinear least-squares analysis to determine a and k. 

Kinetic experiments were carried out by measuring the formation of ADP and GDP with a coupled assay as previously described (Agarwal et al., 1978 Li et al., 1996). Kinetic parameters were obtained by nonlinear least square analysis of the data according to Cleland (1986). 

One ean use the linearized least squares analysis to obtain initial estimates of the parameters k, Kj, K, in order to obtain convergence in nonlinear regression. However, in many eases it is possible to use a nonlinear regression analysis direetly as deseribed m Section 5.4 and in Example 5-6. 

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