Nonlinear Data Fitting Research Articles

Off-line data reduction

Presented is a novel, heuristic algorithm, based on fuzzy set theory, allowing for significant off-line data reduction. Given the equidistant data, the algorithm discards some points while retaining others with their original values. The fraction of original data points retained is typically 1 6 of the initial value. The reduced data set preserves all the essential features of the input curve. It is possible to reconstruct the original information to high degree of precision by means of natural cubic splines, rational cubic splines or even linear interpolation. Main fields of application should be non-linear data fitting (substantial savings in CPU time) and graphics (storage space savings).

Genetic algorithms in seismic waveform inversion

SUMMARY Recently a new class of methods, to solve non-linear optimization problems, has generated considerable interest in the field of Artificial Intelligence. These methods, known as genetic algorithms, are able to solve highly non-linear and non-local optimization problems and belong to the class of global optimization techniques, which includes Monte Carlo and Simulated Annealing methods. Unlike local techniques, such as damped least squares or conjugate gradients, genetic algorithms avoid all use of curvature information on the objective function. This means that they do not require any derivative information and therefore one can use any type of misfit function equally well. Most iterative methods work with a single model and find improvements by perturbing it in some fashion. Genetic algorithms, however, work with a group of models simultaneously and use stochastic processes to guide the search for an optimal solution. Both Simulated Annealing and genetic algorithms are modelled on natural optimization systems. Simulated Annealing uses an analogy with thermodynamics; genetic algorithms have an analogy with biological evolution. This evolution leads to an efficient exchange of information between all models encountered, and allows the algorithm to rapidly assimilate and exploit the information gained to find better data fitting models. To illustrate the power of genetic algorithms compared to Monte Carlo, we consider a simple multidimensional quadratic optimization problem and show that its relative efficiency increases dramatically as the number of unknowns is increased. As an example of their use in a geophysical problem with real data we consider the non-linear inversion of marine seismic refraction waveforms. The results show that genetic algorithms are inherently superior to random search techniques and can also perform better than iterative matrix inversion which requires a good starting model. This is primarily because genetic algorithms are able to combine both local and global search mechanisms into a single efficient method. Since many forward and inverse problems involve solving an optimization problem, we expect that the genetic approach will find applications in many other geophysical problems; these include seismic ray tracing, earthquake location, non-linear data fitting and, possibly seismic tomography.

Interaction parameters for blends containing polycarbonates: 1. Tetramethyl bisphenol-A polycarbonate/polystyrene

Abstract The interaction parameters for polystyrene and tetramethyl bisphenol-A polycarbonate blends were evaluated by fitting experimental PVT behaviour and a lower critical solution temperature (LCST)-type phase boundary to the lattice fluid theory of Sanchez and Lacombe. The phase boundaries of these blends were obtained by using a differential scanning calorimetry (d.s.c.) method, and were compared with those obtained by light transmission and visual methods. The characteristic properties for these polymers were determined by the non-linear least-squares fitting of specific volume data as a function of temperature and pressure to the equation of state. The net bare interaction parameters from the Sanchez-Lacombe theory, ΔP∗ , calculated at several compositions yielded negative values ( ΔP∗ = −0.17 ± 0.01 cal cm −3 ) and did not depend on composition. Flory-Huggins type interaction parameters reported in the literature for this system determined by a small-angle neutron scattering technique compare favourably with the results of this study except in terms of composition dependence.

PHARMFIT—a Nonlinear Fitting Program for Pharmacology

A new program is presented for nonlinear fitting of data from pharmacological and chronobiological investigations. It contains functions for calculating data from ligand-binding studies and competition experiments, for the analysis of dose-response curves, for pharmacokinetic calculations, and for cosine analysis of harmonic and overlapping rhythms. In addition, it is possible to implement general equations by the user. The program allows data exchange with most spreadsheet, database, and graphics presentation programs, and accepts data from two widely used ambulatory 24-h blood-pressure monitoring systems. The fitting procedure uses the Marquardt-Levenberg algorithm. It calculates the weighted or the unweighted fit together with a great variety of statistics for estimation of goodness of fit. A graphics module permits graphical presentation of the fitted curve. Moreover, fitting of data to different models can be compared for the most likely fit and model discrimination statistics for improvement of further experiments are provided. To demonstrate the chronobiological application of the fitting program PHARMFIT, the analysis of telemetric heart rate data from rats is presented.

Nonlinear least-squares fitting of multivariate absorption data

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTNonlinear least-squares fitting of multivariate absorption dataMarcel. Maeder and Andreas D. ZuberbuehlerCite this: Anal. Chem. 1990, 62, 20, 2220–2224Publication Date (Print):October 15, 1990Publication History Published online1 May 2002Published inissue 15 October 1990https://doi.org/10.1021/ac00219a013RIGHTS & PERMISSIONSArticle Views1091Altmetric-Citations365LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (625 KB) Get e-Alerts Get e-Alerts

Liquid Analysis by Dry-Extract Near-Infrared Reflectance on Fiberglass

Nonvolatile residues of liquid samples were retained on fiberglass filters, and near-infrared reflectance spectroscopy was used for a rapid nondestructive analysis of the residue. However, neither log(1/R) nor (1 – R)2/2 R computations gave results that varied in a linear manner with solute concentration. Nonlinear curve fitting of experimental data to a mathematical model [log 1/ R = h + s(1 – exp(– kC))] has shown a good correlation with the model. On the other hand, NIR analysis by multiple linear regression has been evaluated for rapid determination of both total and individual sugars in aqueous solutions containing glucose, fructose, and sucrose mixtures. The concentration ranges of glucose, fructose, and sucrose are, respectively, 20–50 g/L, 20–50 g/L, and 0–50 g/L. Standard errors of performance for prediction are 1.8 g/L for sucrose, 3.5 g/L for glucose, 3.0 g/L for fructose, and 5.3 g/L for total sugars (range 0–150 g/L). The results of this study show its potential application in beverage analysis (fruit juices, lemonade) and other liquid systems.

A critical comparison of least absolute deviation fitting (robust) and least squares fitting: The importance of error distributions

Non-linear least absolute deviation fitting of data has been shown to be superior to least squares where the data errors are unevenly distributed about the function. The methods give insignificantly different results for evenly-distributed errors. Criteria are given for choosing between least absolute deviation and least squares error minimization. Optical density measurements have non-Gaussian error distributions and are better fitted with minimization of the absolute deviation.

Potentiometric Determination of Thermodynamic and Apparent Dissociation Constants by Nonlinear Least Squares Fitting

AbstractPotentiometric titration curve models have been developed for mono-, di- and triprotic weak electrolytes. Apparent dissociation constants or thermodynamic dissociation constants, corrected for ionic strengths of up to O.5M, can be determined by nonlinear least squares fitting of potentiometric data to the models. A commercially available program, PCNONLIN (an IBM PC version of NONLlN84), was utilized for the fitting, allowing the models to be portable. The models have been tested with theoretical data containing random or systematic error, and data for several model compounds of various weak electrolyte class.The method in this report has several advantages over traditional 'half neutralization' or graphical methods. First, the models are based on the analytic solution of the charge balance equation, with no simplifying approximations. Second, the method can determine dissociation constants in cases where no inflection point is present, such as for overlaping ionization processes or low concentrat...

Bioconcentration kinetics of some chlorinated benzenes and chlorinated phenols in American flagfish, [formula omitted][formula omitted] (Goode and Bean)

The bioconcentration of three chlorobenzenes: 1,4-dichlorobenzene, 1,2,4-trichlorobenzene, and 1,2,4,5-tetrachlorobenzene, and three chlorophenols: 2,4,6-trichlorophenol, 2,3,5,6-tetrachlorophenol, and pentachlorophenol, was examined in a flow-through exposure system using juvenile American flagfish ( Jordanella floridae ) as the test organism. Exposure to the each chemical lasted for 28 days followed by a depuration period of 5 to 7 days. Uptake and elimination rate constants, as well as the steady state bioconcentration factor were determined for each chemical tested. The elimination rate constants determined from these bioconcentration tests compared favourably with elimination rate constants derived from non-linear curve fitting of time-toxicity data obtained in acute toxicity testing carried out with the same chemicals.

Ponalrestat: A potent and specific inhibitor of aldose reductase

Many of the complications of diabetes appear to be closely linked to increased conversion of tissue glucose to sorbitol which is catalysed by aldose reductase (aldehyde reductase 2, ALR2). Inhibition ofALR2 could, therefore, lead to a reduction in the development of diabetic complications. Ponalrestat [“Statil” (a trademark, the property of Imperical Chemical Industries PLC), “Prodiax” (a trademark, the property of Merck, Sharp and Dohme), ICI 128436, MK538] inhibits ALR2 from a number of sources. Until now, the mechanism of this inhibition has not been fully elucidated. In this paper, we present a detailed mechanism for inhibition of bovine lens ALR2 by ponalrestat. Treatment of humans with some ALR2 inhibitors leads to side-effects, some of whcih may result from interactions with other enzymes. Aldehyde reductase (ALR1) is probably the most closely related enzyme to ALR2. Inhibition of ALR1 from bovine kidney was, therefore, investigated in order to assess the specificity of ponalrestat. The values of K i , and K ies (apparent dissociation constants for inhibitor from enzyme-inhibitor and enzyme-inhibitor-substrate complexes, respectively) for the interactions of ponalrestat with ALR1 and ALR2 hae been calculated by non-linear fitting of kinetic data. These values indicate that ponalrestat does not compete with binding of glucose of NADPH to ALR2, nor with binding of glucuronate or NADPH to ALR1. Lack of competition and the structural dissimilarity of substrates and inhibitor make it unlikely that ponalrestat will utilize substrate binding sites on other enzymes, and so produce undesirable side-effects via such a mechanism. Ponalrestat is a potent inhibitor ( K i = K ies = 7.7 nM) of ALR2 and follows a pure noncompetitive mechanism with respect to glucose. Efficacy, therefore, will not be decreased by development of hyperglycaemia. The compound is a mixed noncompetitive inhibitor of ALR1 when glucuronate is varied. The values of K i , and K ies are 60 μM and 3 μM, respectively, so that inhibition tends towards uncompetitive. The selectivity of ponalrestat in favour of ALR2, therefore, lies in the range 390 to 7,800-fold, being higher at lower concentrations of glucuronate. The high selectivity of ponalrestat in favour of ALR2 rather than ALR1 suggests that the compound is unlikely to inhibit other enzymes which have less homology with ALR2.

A flexible procedure for analyzing impedance spectroscopy results: Description and illustrations

General considerations in nonlinear least squares fitting of small-signal ac frequency response data (conductive or dielectric systems) are first discussed. A very general and flexible complex nonlinear least squares fitting (CNLS) program (LOMFP) is described which runs on microcomputers of the PC and AT type. Advantages and disadvantages of LOMFP are discussed and compared to those of a recent CNLS program developed by Boukamp. LOMFP incorporates many thousands of equivalent circuits fitting possibilities and ten different distributed circuit elements which can be used in these circuits. Its fitting is both speedy and accurate, and it provides considerable insurance against confusing local minima fits with true least squares fits. It is optimized for fitting real, imaginary, or complex data of either partly conducting or purely dielectric character, and it incorporates great flexibility in handling and fitting data in different forms (different immittance levels and rectangular or polar representation). Seven different data weighting choices are provided. Exact synthetic data were generated from a typical Impedance Spectroscopy equivalent circuit, and the effects on CNLS fitting were investigated of rounding the exact data to 4, 3, and 2 decimal places. We present the results of an extensive study of the effects of transforming exact data to different immittance levels and representations and then rounding to two places or of transforming rounded 2-place data directly. We conclude that real data should not be transformed from its original measured form before carrying out CNLS fitting. Finally, the effects were investigated of various kinds of weightings on CNLS fitting of 2-place data, leading to the conclusion that in spite of its advocacy and use by Zoltowski and Boukamp, modulus weighting generally yields misleading and appreciably worse CNLS fitting results than does proportional weighting. The LOMFP program (source and executable files) is available from the first author.

Binding of lipoproteins to inert materials revisited with computer-assisted analysis

A number of studies conducted in the last decade showed that saturable (‘specific’) binding, by itself, does not necessarily imply biological significance. That is, biological ligands were shown to bind to inert materials as well as to biological receptors in a saturable manner. In these studies specific binding was operationally defined as binding that was displaceable by excess concentrations of unlabeled ligand. This method of measuring specific binding is now no longer considered optimal. To investigate whether optimal (computerassisted) techniques of measuring specific binding — namely, nonlinear least-squares curve fitting of total binding data, with mathematical separation of the total binding into its various components — might ensure biological significance of measured specific binding, we studied the binding of high-density lipoproteins (HDL 3) to tissue culture dishes as an example of binding without biological significance. This binding closely followed the paradigm of a ligand interacting with a class of homogeneous, saturable sites and with a class of relatively unsaturable sites, just as it would have if the HDL 3 were interacting with an unpurified biological receptor. This finding indicates that computer-assisted analysis, while most accurately describing binding data, nevertheless does not ensure that measured specific binding has biological significance. Saturability is such a nonselective feature of equilibrium binding data that it should probably no longer be considered one of the criteria for deciding whether or not a defined binding site is a receptor.

Infant temperament as a predictor of preschool behavior problems

Stereoselectivity in protein binding can have a significant effect on the pharmacokinetic and pharmacodynamic properties of chiral drugs. In this paper, the enantioselective binding of propanocaine (PRO) enantiomers to human serum albumin (HSA), the most relevant plasmatic protein in view of stereoselectivity, has been evaluated by incubation and ultrafiltration of racemic PRO–HSA mixtures and chiral analysis of the bound and unbound fractions by electrokinetic chromatography using HSA as chiral selector. Experimental conditions for the separation of PRO enantiomers using HSA as chiral selector and electrokinetic chromatography have been optimised. Affinity constants and protein binding in percentage (PB) were obtained for both enantiomers of PRO, as well as the enantioselectivity (ES) to HSA. Data were obtained in two independent working sessions (days). The influence of the session and fraction processed factors were examined. A univariate direct-estimation approach was used facilitating outliers’ identification and statistical comparison. Non-linear fitting of data was used to verify the stoichiometry and affinity estimations obtained by the direct approach. Robust statistics were applied to obtain reliable estimations of uncertainty, accounting for the factors (day and processed fraction), thus representing intermediate precision conditions. Mimicking in vivo experimental conditions, information unapproachable by in vivo experiments was obtained for PRO enantiomers interacting with HSA. For the first (E1) and the second (E2) eluted PRO enantiomers the results were: 1:1 stoichiometry, medium affinity constants, log KE1 = 3.20 ± 0.16 and log KE2 = 3.40 ± 0.14, medium protein binding percentage, PB = 48.7 and 60.1% for E1 and E2, respectively, and moderate but significant enantioselectivity, ES = KE2/KE1 = 1.5 ± 0.3.

Theophylline and Caffeine Disposition in the Neonatal Piglet

A pharmacokinetic study to evaluate the piglet as a model for caffeine and theophylline disposition was undertaken in 28 animals who received a 10.0-mg/kg intravenous dose of either methylxanthine, followed by multiple blood sampling over a 24-hour period. Theophylline and caffeine concentrations were quantitated from serum using a microanalytical HPLC technique (coefficient of variability less than 7% at 0.1-100 mg/l). The influence of postnatal development on drug disposition was examined by dividing the experimental animals into three age groups (newborn, 2 days; perinatal, 4-5 days; infants or young, 15-22 days). Pharmacokinetic parameters were calculated from nonlinear curve fitting of serum concentration vs time data. On the basis of the terminal elimination rate constant (lambda z) and total clearance (CL) for both theophylline and caffeine, differences were found among the age groups. The apparent volume of distribution (Vz) for both methylxanthines was not found to be different when compared between age groups. Covariance determinations between age and pharmacokinetic parameters revealed linear correlations for Vz of caffeine and CL for both theophylline and caffeine. In all piglets receiving theophylline, no caffeine was detected in serum during a 24 hr period. Theophylline, however, was detected (0.4 +/- 0.2 mg/l at 11.5 hr) in the four oldest piglets who received caffeine. Similarities between pharmacokinetic parameters for caffeine and theophylline reported for human neonates were found only for newborn piglets. Furthermore, the biotransformation of theophylline to caffeine reported for human neonates was not observed in neonatal piglets.

Multifit: a flexible non-linear least squares regression program in BASIC

MULTIFIT, a program in BASIC for implementation on microcomputers, has been developed for non-linear least squares regression fitting of enzyme kinetic, pharmacokinetic and other data to specific models. The program contains a simple procedure for insertion of model equations with up to five parameters (to be fitted) up to 3 independent variables and 1 dependent variable, and can be used to generate a family of programs with pre-set model functions.

13C NMR measurement of flux through alanine aminotransferase by inversion- and saturation-transfer methods

Rate constants for the reaction catalyzed by alanine aminotransferase were measured by 13C saturation transfer and inversion transfer between the enriched methyl groups of pyruvate and alanine. At a high enzyme concentration four techniques provided a reasonable estimate of parameters. (1) saturation transfer, (2) nonlinear least-squares fitting of inversion-transfer data, (3) linear least-squares fitting of the initial slopes of inversion-transfer data, and (4) a combination of initial slope and saturation-transfer data. At a lower concentration of enzyme, the saturation-transfer method provided the most accurate determination of rate constants. An inversion-transfer experiment produces a series of measurements of magnetization at various delays after inversion from which the parameters of interest may be determined by nonlinear least-squares fitting. Error in the determined parameters is influenced by the delays at which data are acquired, the number of data points acquired, and error in the measurement of magnetization. In the presence of significant noise in the data, the complete inversion-transfer experiment (inversion of both sites, sequentially) must be analyzed for an accurate determination of the parameters. If the inversion-transfer experiment is performed, an experimental protocol that minimizes error in the parameters may be designed to efficiently acquire inversion-transfer data if an estimate of the parameters and error in measurement of magnetization is available. The most efficient approach is the combination of initial slope data from the inversion-transfer experiment with measurement of equilibrium magnetization during saturation.

Zum Einfluß der totalen Hormonkonzentration auf die Ermittlung von Bindungskenngrößen bei der Hormon/Receptor-Wechselwirkung zwischen Erythrocyten und Insulin

The interaction of erythrocytes and [125I]insulin/insulin were studied up to a total insulin concentration of 409 mumol/l. Assuming a single class receptor model the evaluation of receptor affinity Ka and concentration R0 may be performed either by non-linear regression analyses with iteration procedures of R0, Ka and U (nonspecific binding), or by a linear regression analysis of the initial part of the Scatchard plot. Nonlinear fitting of data to a two class receptor model gives results that are reliable only for the high affinity receptor site. The non-definable step of R0 determination leads to uncertainties in results determined by the negative cooperativity model. From the results of this investigation and from considerations of signal modulation by receptor occupancy, some recommendations have been formulated for the evaluation of binding parameters; these should contribute to an improvement in the comparability of studies on the interactions of erythrocytes and insulin.

Electrical and structural characterization of implantation doped silicon by infrared reflection

Abstract A physical model is presented for calculating infrared reflection interference spectra from ion implanted and annealed crystalline materials. The utility of the method is illustrated by presenting best fit spectra for a <111> silicon sample implanted with 2.7 MeV phosphorous to a fluence of 1.74 × 1016 ions/cm2 and isothermally annealed at 500°C. Non-linear least-squares fitting of reflection data yields structural and electrical information about the implanted region with reasonable precision. The physical quantities determined are (i) the depth of the amorphous layer produced by implantation both before and during isothermal annealing, the thickness of the recrystallized material, and the widths of any transition regions, (ii) the dielectric properties of the amorphous and recrystallized material, and (iii) the characteristics of the free carrier plasma which yield the carrier density profile, the mobility near the carrier density maximum, and the carrier activation efficiency. Up to nine fitting parameters are necessary to describe these physical quantities. A critical discussion of the sensitivity of data fit to variation in the parameters is given to establish the uniqueness of fitted parameters. The infrared method is non-destructive, is applicable to other dopants and semiconductors, and provides information complementary to both ion channeling and resistivity profiling techniques.

Simultaneous Modeling of Pharmacokinetics and Pharmacodynamics

We propose a model of drug pharmacodynamic response that when integrated with a pharmacokinetic model allows characterization of the temporal aspects of pharmacodynamics as well as the time-independent sensitivity component. The total model can accommodate extremes of effect. It allows fitting of simultaneous plasma concentration (Cp) and effect data from the initial distribution phase of drug administration, or from any non-equilibrium phase. The model postulates a hypothetical effect compartment, the dynamics of which are adjusted to reflect the temporal dynamics of drug effect. The effect compartment is modeled as an additional compartment linked to the plasma compartment by a first-order process, but whose exponential does not enter into the pharmacokinetic solution for the mass of drug in the body. The hypothetical amount of drug in the effect compartment is then related to the observed effect by the Hill equation, a nonlinear sigmoid form. Nonlinear least-squares data fitting is used for parameter estimation. The model is demonstrated on two different sets of Cp and effect data for the drug d-tubocurarine (dTC). In 7 normal subjects, the (mean +/- SD) rate constant for equilibration of dTC effect (paralysis) and Cp is 0.13 +/- 0.04 min-1 and the (mean +/- SD) steady-state Cp required to produce 50% paralysis is 0.37 +/- 0.05 microgram/ml.

Simultaneous modeling of pharmacokinetics and pharmacodynamics: application to d-tubocurarine.

We propose a model of drug pharmacodynamic response that when integrated with a pharmacokinetic model allows characterization of the temporal aspects of pharmacodynamics as well as the time-independent sensitivity component. The total model can accommodate extremes of effect. It allows fitting of simultaneous plasma concentration (Cp) and effect data from the initial distribution phase of drug administration, or from any non-equilibrium phase. The model postulates a hypothetical effect compartment, the dynamics of which are adjusted to reflect the temporal dynamics of drug effect. The effect compartment is modeled as an additional compartment linked to the plasma compartment by a first-order process, but whose exponential does not enter into the pharmacokinetic solution for the mass of drug in the body. The hypothetical amount of drug in the effect compartment is then related to the observed effect by the Hill equation, a nonlinear sigmoid form. Nonlinear least-squares data fitting is used for parameter estimation. The model is demonstrated on two different sets of Cp and effect data for the drug d-tubocurarine (dTC). In 7 normal subjects, the (mean +/- SD) rate constant for equilibration of dTC effect (paralysis) and Cp is 0.13 +/- 0.04 min-1 and the (mean +/- SD) steady-state Cp required to produce 50% paralysis is 0.37 +/- 0.05 microgram/ml.