Showing papers on "Non-linear least squares published in 1986"

A semiautomatic algorithm for rutherford backscattering analysis

Abstract: Nonlinear least squares techniques have been applied to Rutherford backscattering spectrometry (RBS), allowing routine multivariable fits of simulated spectra to experimental data. Once a qualitatively correct simulation is made, this algorithm varies parameters of the simulation to obtain quantitative results. Optimization is done according to a maximum likelihood chi-squared definition, so that the best fit values of parameters and their uncertainties can be determined. Convergence of the algorithm is rapid for practical problems, allowing a typical four-variable fit to be accomplished in 30 seconds on a VAX 11/750. This algorithm allows confident treatment of spectra which might otherwise be considered too complex. An implementation of the algorithm is incorporated as part of an RBS analysis and simulation package, making it readily available for routine RBS analysis.

Multilevel mixed linear model analysis using iterative generalized least squares

Abstract: SUMMARY Models for the analysis of hierarchically structured data are discussed. An iterative generalized least squares estimation procedure is given and shown to be equivalent to maximum likelihood in the normal case. There is a discussion of applications to complex surveys, longitudinal data, and estimation in multivariate models with missing responses. An example is given using educational data.

Technical Note---Nonlinear Least Squares Estimation of New Product Diffusion Models

Abstract: Schmittlein and Mahajan Schmittlein, D. C., V. Mahajan. 1982. Maximum likelihood estimation for an innovation diffusion model of new product acceptance. Marketing Sci.1 Winter 57--78. made an important improvement in the estimation of the Bass Bass, F. M. 1969. A new product growth model for consumer durables. Management Sci.15 January 215--227. diffusion model by appropriately aggregating the continuous time model over the time intervals represented by the data. However, by restricting consideration to only sampling errors and ignoring all other errors such as the effects of excluded marketing variables, their Maximum Likelihood Estimation MLE seriously underestimates the standard errors of the estimated parameters. This note uses an additive error term to model sampling and other errors in the Schmittlein and Mahajan formulation. The proposed Nonlinear Least Squares NLS approach produces valid standard error estimates. The fit and the predictive validity are roughly comparable for the two approaches. Although the empirical applications reported in this paper are in the context of the Bass diffusion model, the NLS approach is also applicable to other diffusion models for which cumulative adoption can be expressed as an explicit function of time.

Measuring Abnormal Performance: The Event Parameter Approach Using Joint Generalized Least Squares

Abstract: Event studies generally seek to measure abnormal security performance associated with firm-specific events. In principle, estimators of and tests for abnormal performance should appropriately reflect cross-sectional dependence between abnormal returns to different se? curities. Joint generalized least squares provides a natural framework for developing such estimators and tests. This paper derives a joint generalized least squares estimator and related test statistic applicable in the typical event study context. Simulation techniques comparable to those of Brown and Warner [2] are used to assess the frequency distribution of the estimator and power of the test statistic. Several simpler procedures are simulated for comparison. The results provide no evidence that joint generalized least squares is superior to simpler procedures.

Nonlinear Least Squares

Abstract: One of the most common problems encountered in practical data analysis involves the fitting of a theoretical model to experimental data. Frequently the model takes the form of a dependent variable expressed as a function of several independent variables. Often the model will contain one or more parameters which have to be estimated. This estimation takes place on the basis of fitting the model to observations using the least-squares concept.

Transfer function synthesis using frequency response data

Abstract: Various linear least squares methods for transfer function synthesis from frequency response data are presented in a unified format. Solutions are derived from Householder transformations and recursive least squares. An alternative formulation derived from a time domain error criterion is also shown to be of the linear least squares type. The comparative performance of the various methods is illustrated by several examples.

A comparison of forecasts from least absolute value and least squares regression

Abstract: A Monte Carlo simulation is used to compare forecasts from least absolute value and least squares estimated regression equations. When outliers are present, the least absolute value forecasts are shown to be superior to least squares forecasts. The results emphasize the importance of exercising caution when using forecasts from least squares estimated regressions. Use of least absolute value regression (or some other robust regression method) instead of, or as an adjunct to, least squares is recommended. The comparison of forecasts from the two methods provides one way of assessing whether the least squares forecasts have been adversely affected by outliers. If outliers are present, the least absolute value regression forecasts can be used.

When, why, and how to use weighted least squares

Abstract: This article features examples that illustrate how straightforward it is to use weighted least squares.

An efficient line search for nonlinear least squares

Abstract: The line search subproblem in unconstrained optimization is concerned with finding an acceptable steplength which satisfies certain standard conditions. Prototype algorithms are described which guarantee finding such a step in a finite number of operations. This is achieved by first bracketing an interval of acceptable values and then reducing this bracket uniformly by the repeated use of sectioning in a systematic way. Some new theorems about convergence and termination of the line search are presented.

An analysis program MULTI(ELS) based on extended nonlinear least squares method for microcomputers.

Abstract: An analysis program MULTI(ELS) was developed for population pharmacokinetics on a microcomputer. The program based on the extended least squares (ELS) is written in the Microsoft minimum BASIC command alone. ELS simultaneously estimates not only the population pharmacokinetic parameters but also the variances of inter-individual variabilities around the population parameters and of intra-individual variabilities for the plural time courses, whereas the ordinary least squares estimates the pharmacokinetic parameters of each time course. Two least squares algorithms (i.e. quasi-Newton and simplex methods) are provided in MULTI(ELS). MULTI(ELS) was compared with NONMEM (Version I, Level 3) developed by Sheiner and Beal for several time course data. It was shown that MULTI(ELS) gave the same results as NONMEM. MULTI(ELS) calculates the Akaike's information criterion (AIC) for the extended least squares. It was also shown that the AIC is effective in comparing the population characteristics of time courses in a group with those in another group and in verifying the model structures of population means, inter-individual variations and intra-individual variations.

Linear least squares with bounds and liner constraints

Abstract: An algorithm is given for solving linear least squares systems of algebraic equations subject to simple bounds on the unknowns and (more general) linear equality and inequality constraints.The method used is a penalty function approach wherein the linear constraints are (effectively) heavily weighted. The resulting system is then solved as an ordinary bounded least squares system except for some important numerical and algorithmic details.This report is a revision of an earlier work. It reflects some hard-won experience gained while using the resulting software to solve nonlinear constrained least squares problems.

An Exact Maximum Likelihood Estimation Procedure for Regression- ARMA Time Series Models with Possibly Nonconsecutive Data

Abstract: SUMMARY An explicit procedure is given to obtain the exact maximum likelihood estimates of the parameters in a regression model with ARMA time series errors with possibly nonconsecutive data. The method is based on an innovation transformation approach from which an explicit recursive procedure is derived for the efficient calculation of the exact likelihood function and associated derivatives. The innovations and associated derivatives are used to develop a modified Newton-Raphson procedure for computation of the estimates. A weighted nonlinear least squares interpretation of the estimator is also given. A numerical example is provided to illustrate the method.

Fitting linear regression models to censored data by least squares and maximum likelihood methods

Abstract: Several approaches have been suggested for fitting linear regression models to censored data. These include Cox's propor­tional hazard models based on quasi-likelihoods. Methods of fitting based on least squares and maximum likelihoods have also been proposed. The methods proposed so far all require special purpose optimization routines. We describe an approach here which requires only a modified standard least squares routine. We present methods for fitting a linear regression model to censored data by least squares and method of maximum likelihood. In the least squares method, the censored values are replaced by their expectations, and the residual sum of squares is minimized. Several variants are suggested in the ways in which the expect­ation is calculated. A parametric (assuming a normal error model) and two non-parametric approaches are described. We also present a method for solving the maximum likelihood equations in the estimation of the regression parameters in the censored regression situation....

The equation of state of the two-dimensional Lennard―Jones fluid

Abstract: By combining pressure and energy data from the virial equation of state, through fifth virial coefficients, with the second and third virial coefficients themselves and the results of computer-simulation calculations, we have constructed an equation of state for the two-dimensional Lennard–Jones fluid for 0.45 ≤ T* ≤ 5 and 0.01 ≤ ρ* ≤ 0.8. The fitted data include some in the metastable region, and, therefore, the equation of state also describes "van der Waals loops" including unstable regions. The form used is a modified Benedict–Webb–Rubin equation having 33 parameters including one nonlinear one. The fitting was done using a nonlinear least squares algorithm based on a Levenberg–Marquardt method. A total of 211 simulation points, 97 reported here for the first time, were used in the fitting, and the overall standard deviation is less than 2% for both energy and pressure. Second and third virial coefficients derived from the fit in the supercritical region are in excellent agreement with exact values. T...

Quadratic forms between spheres and the non-existence of sums of squares formulae

Abstract: Hurwitz [6] posed in 1898 the problem of determining, for given integers r and s, the least integer n, denoted by r s, for which there exists an [r, s, n] formula, namely a sums of squares formula of the typewhere are bilinear forms with real coefficients in and . Such an [r, s, n] formula is equivalent to a normed bilinear map satisfying . We shall, therefore, speak of sums of squares formulae and normed bilinear maps interchangeably.

An Alternative Approach to Fitting Models When Re-expression of the Response is Useful

Abstract: It is common practice to mathematically transform or re-express data (e.g., logs, reciprocals, square roots) to ensure that the assumptions of normal distribution statistics are more closely satisfied. Re-expression is a problem in some instances becaus..

Orthogonal least squares fitting with linear manifolds

Abstract: The problem is considered of fitting a linear manifold of dimensions with 1≦s≦n−1 to a given set of points in ℝ n such that the sum of orthogonal squared distances attains a minimum.

An Algorithm for Exponential Fitting Revisited

Abstract: An algorithm for exponential fitting is presented which exploits the separable regression structure and a reparametrization. The algorithm has proved very satisfactory, and theoretical reasons for this are developed.

SIMPAR: A Versatile Technology Independent Parameter Extraction Program Using a New Optimized Fit-Strategy

Abstract: For custom integrated circuit design, the use of a reliable device model parameter set as input for the circuit simulator is very important. The curve fitting algorithm in our parameter extraction program SIMPAR is based on a modified Marquardt fitting routine for nonlinear least squares. Classical parameter extraction programs rely only on the minimization of the relative current deviation. But, especially in analog applications (e.g., gain), a correct value for the slope of the I /sub ds/ - V /sub ds/ curves in the saturation region is at least equally important. The new generalized error criterion implemented in SIMPAR extracts a parameter set which minimizes both the current and slope residuals in every point. In doing this, the accuracy of the output conductance modeling has improved considerably, and in some cases even the current residual itself decreases. Although any analytical current-voltage relationship can be used, the SPICE MOS level 3 was chosen as a test vehicle. To illustrate the effect of this new fitting strategy, the gain of a CMOS invertor was calculated.

Analysis of dielectric or conductive system frequency response data using the Williams–Watts function

Abstract: The electrical, optical, and mechanical behavior of many materials, particularly polymers and glasses, have been analyzed using the Kohlrausch–Williams–Watts stretched exponential relaxation function in both the time and frequency domains This function is currently of considerable experimental and theoretical interest Unfortunately, no relatively simple and accurate approximation representing the small‐signal frequency response of stretched exponential relaxation has been available Thus it has been impractical to obtain accurate parameter estimates from fitting of frequency response data or to discriminate well between Williams–Watts response and that of other similar response models Here we develop such an approximation for both dielectric systems and for intrinsically conducting ones (eg, defect hopping materials) It is in complex form and allows fitting of both real and imaginary parts of all the data simultaneously (eg, by complex nonlinear least squares) or of either part separately For app

A comparison between some direct and iterative methods for certian large scale godetic least squares problems

Abstract: The purpose of this paper is to describe and compare some numerical methods for solving large dimensional linear least squares problems that arise in geodesy and, more specially, from Doppler positioning. The methods that are considered are the direct orthogonal decomposition, and the combination of conjugate gradient type algorithms with projections as well as the exploitation of “Property A”. Numerical results are given and the respective advantage of the methods are discussed with respect to such parameters as CPU time, input/output and storage requirements. Extensions of the results to more general problemsare also discussed.

Kinetic analysis by the method of nonlinear least squares: A reaction involving consecutive steps

Abstract: This paper describes the reaction of 2-metacaptoethanol with 3-chlorophenyl dinitrile, a potent uncoupler of oxidative phosphorylation in cells.

On the consistency of least squares estimators for a threshold ar(1) model

Abstract: . For the SETAR (2; 1,1) model where {at(i)} are i.i.d. random variables with mean 0 and variance σ2(i), i = 1,2, and {at(l)} is independent of {at(2)}, we consider estimators of φ1, φ2 and r which minimize weighted sums of the sum of squares functions for σ2(1) and σ2(2). These include as a special case the usual least squares estimators. It is shown that the usual least squares estimators of φ1, φ2 and r are consistent. If σ2(1) ≠σ2(2) conditions on the weights are found under which the estimators of r and φ1 or φ2 are not consistent.