Accurate Initial Estimates Research Articles

Numerical ellipsometry: Applications of a new algorithm for real-time, in situ film growth monitoring

The equations associated with the popular models of film deposition of a material on a substrate tend not to be invertible in practical situations. Thus development of numerical techniques for obtaining approximate solutions to these equations has become necessary. At present, real-time, in situ thin film growth ellipsometry data are limited to a single angle of incidence due to the deposition system windows. The 2C algorithm, developed by the authors, represents a new approach to solve this problem. This algorithm is fast enough for reliable real-time solutions without requiring either accurate initial estimates or any off-line processing. Algorithm 2C employs the techniques of dynamic step adjustment, overrelaxation, and reduction in dimension, as well as curve following and curve intersection location techniques developed by the authors. The algorithm was originally developed for nickel deposited on BK7 glass, and in this article it is applied to silver, silicon, and gold depositions on BK7 glass. Gold films could be solved readily with no changes, while problems with convergence were encountered with some silver measurements and silicon data required changes in the ‘‘standard’’ initial estimate.

Spectral curve fitting of infrared spectra obtained from semi-crystalline polyester yarns

Infrared spectra of polymer yarns exhibit a large number of strongly overlapping peaks. One of the methods to study the physical yarn structure quantitatively is to resolve the spectra into separate bell-shaped peaks. Due to the overlapping peak patterns, curve fitting with steepest descent like procedures depend on accurate initial estimates of the peak parameters. We developed a non-local curve fitting procedure, based on a genetic algorithm, which proved to be less sensitive to initial estimates. The combination of a genetic algorithm with a nonlinear estimator, based on Marquardt's method, gave the best curve fitting results.

Modelling velocity profiles in inclined multiphase flow to provide a priori information for flow imaging

Velocity profile measurements obtained by cross corelating between two axially separated image planes are becoming increasingly important in multiphase flow measurement systems. At present, considerable computing power is required to implement this technique because no use is made of the constraints imposed by the physics of the flow. This paper describes a mathematical model which, when combined with a simple auxiliary measurement, provides an accurate initial estimate of the velocity profile of an inclined multiphase flow. The paper goes on to describe how this a priori information can be used to cross correlating between image planes. The a prior information can also be used to provide on-line validation of velocity profile measurements.

Numerical ellipsometry: enhancement of new algorithm for real-time, in situ film growth monitoring

The equations associated with the popular models of film deposition of a material on a substrate tend not to be invertible in practical situations. Thus, development of numerical techniques for obtaining approximate solutions to these equations has become necessary. The variably damped least squares (VDLS) algorithm of Levenberg and Marquardt is commonly used, although it requires accurate initial estimates and is typically to slow for real-time applications. Another algorithm combines the speed of an artificial neural network (ANN) for accurate initial estimates, and the real-time refinement capability of VDLS for real-time performance; however, substantial off-line processing (from hours to days) is necessary to train ANNs. A third algorithm (called 2C) has been developed, which is fast enough for reliable real-time solutions without requiring either accurate initial estimates or any off-line processing. The algorithm 2C employs the techniques of dynamic step adjustment, overrelaxation and reduction in dimensions, as well as curve following and curve intersection location techniques developed by the authors. The performance of the new 2C algorithm in terms of speed and convergence is comparable with the ANN-VDLS algorithms for Ni and BK7 glass.

Curve fitting using natural computation

In curve fitting the most commonly used technique is an iterative hill-climbing procedure that makes use of partial derivatives to calculate the steepest path to an optimum in solution space. Howver, reliable and accurate initial estimates of the number of peaks, individual peak positions, heights, and widths are necessary to find acceptable solutions. One of the main drawbacks unvolved is that as the number of overlapping peaks increases, the problem becomes progressively more ill-conditioned. Consequently, small errors in the data (e.g., noise or baseline distortions), errors in the mathematical model or errors in the estimates can be magnified, leading to large errors in the parameters of the final model

The DISCON Algorithm, an Accurate and Robust Alternative to an Eigenvalue Solution for Extracting Experimental Distances from NOESY Data

The DISCON algorithm is a novel, computationally direct, iterative solution for cross-relaxation rates that can be applied to a single NOESY data set without prior knowledge of the applicable motional regime or a structural model. Tests with simulated NOESY data demonstrate that it is less subject to error propagation from incorrect autopeak values than an eigenvalue solution. The resulting cross rates are fully corrected for secondary NOE contributions. The algorithm has been incorporated into a software suite that provides distance estimates and their experimental precision from raw NOESY data. Missing values in the data matrix are approximated as class averages of existing data rather than calculated values based on a starting conformational model; thus, more accurate initial distance estimates for modeling can be derived without a starting model bias. The accuracy of the program is demonstrated using NOESY data for a rigid steroid, Its application for conformational analysis is illustrated with a flexible prostaglandin analog.

Obtaining initial parameter estimates for nonlinear systems using multicriteria associative memories

Parameter estimation problems for nonlinear systems are typically formulated as nonlinear optimization problems. For such problems, one has the usual difficulty that standard successive approximation schemes require good initial estimates for the parameter vector. This paper develops a simple multicriteria associative memory (MAM) procedure for obtaining useful initial parameter estimates for nonlinear systems. An easily calculated one-parameter family of associative memory matrices is developed; see Equation (25). Each memory matrix is efficient with respect to two criteria: accurate recovery of parameter-output training case associations; and small matrix norm to guard against noise arising from imprecise calculations and observations. For illustration, the MAM procedure is used to obtain initial parameter estimates for a well-known nonlinear economic model, the Solow-Swan growth model. Surprisingly accurate initial parameter estimates are obtained over broad ranges of the family of MAM memory matrices, even when observations are corrupted by i.i.d. or correlated noise.

Optimal Designs for Binary Data

Abstract This article is concerned with the problem of selecting the values of the explanatory variable in logistic regression to obtain likelihood-based confidence regions of minimum area. One way of finding analytic solutions is by replacing the log-likelihood by a quadratic surface around the maximum, approximating in this way the likelihood regions by ellipses. The optimal allocation for this local approximation is derived without unnecessary restrictions. Since the implementation of the optimal allocation requires accurate initial estimates, a two-stage procedure that uses the second stage to complement the first is recommended. An alternative approach is to deal directly with the likelihood regions. In this case, the identification of the globally optimal allocation calls for numerical integration and optimization. The results presented here attempt to strengthen those introduced by Abdelbasit and Plackett (1983), who derived optimal allocation for the local criterion under the restriction of symmetry and suggested a two-stage procedure. Here it is shown that although for even sample size the optimal allocation is in fact symmetric, for odd sample size the restriction of symmetry leads to suboptimal designs (one important case is only 93% efficient). For the two-stage procedure, their proposal does not attempt to compensate in the second stage. This may lead to a serious loss of efficiency (below 80% in extreme cases). The numerical results corresponding to the global criterion (i.e., using the likelihood regions) indicate that the globally best allocation corresponds to less extreme probabilities than those prescribed by the local criterion.

Optimal Designs for Binary Data

Abstract This article is concerned with the problem of selecting the values of the explanatory variable in logistic regression to obtain likelihood-based confidence regions of minimum area. One way of finding analytic solutions is by replacing the log-likelihood by a quadratic surface around the maximum, approximating in this way the likelihood regions by ellipses. The optimal allocation for this local approximation is derived without unnecessary restrictions. Since the implementation of the optimal allocation requires accurate initial estimates, a two-stage procedure that uses the second stage to complement the first is recommended. An alternative approach is to deal directly with the likelihood regions. In this case, the identification of the globally optimal allocation calls for numerical integration and optimization. The results presented here attempt to strengthen those introduced by Abdelbasit and Plackett (1983), who derived optimal allocation for the local criterion under the restriction of symmetry and suggested a two-stage procedure. Here it is shown that although for even sample size the optimal allocation is in fact symmetric, for odd sample size the restriction of symmetry leads to suboptimal designs (one important case is only 93% efficient). For the two-stage procedure, their proposal does not attempt to compensate in the second stage. This may lead to a serious loss of efficiency (below 80% in extreme cases). The numerical results corresponding to the global criterion (i.e., using the likelihood regions) indicate that the globally best allocation corresponds to less extreme probabilities than those prescribed by the local criterion.

A Fortran program for solving a nonlinear equation by Muller's method

Computational experience with Muller's method indicates that it is very efficient algorithm for computing real, complex, and multiple zeros of arbitrary functions. Furthermore, it does not require accurate initial estimates of the zeros, nor does it require the evaluation of the derivative of the function. We present a Fortran program for Muller's method and, using numerical examples, we compare its efficiency to that of a good bracketing algorithm.

Discussion of parameter estimation in biological modelling: Algorithms for estimation and evaluation of the estimates.

This paper is concerned with the estimation of parameters when mathematical models are fitted to data. Two new algorithms are presented. The first is fast (economical in computation time), requires no initial estimates, but is not so accurate. The second requires more computation time, and fairly accurate initial estimates, but achieves high accuracy. The models discussed consist of sets of coupled, non-linear differential equations, but the second algorithm is applicable to wider classes of models as well.The accuracy of the computed values of the parameters depends on the number of data points, and the errors in the data. The sensitivity of the different parameters to errors may differ by orders of magnitude. A method of calculating the expected errors in the parameters is described, and the results of some applications of the method are presented.

Timing Recovery in Synchronous Equalized Data Communication

The best estimation of sampler timing in adaptively equalized systems is examined. First, two previous approaches to the estimation problem are reviewed and are found to have fundamental shortcomings. The first technique sets the timing instants coincident with the peak of the unequalized system impulse response. Under certain circumstances this technique can produce nulls in the spectrum of the sampled system, thus preventing efficient operation of the equalizer. The second technique sets the timing instants so as to minimize the residual distortion of the equalized system. This does assure proper operation but requires a second equalizer structure and also requires an accurate initial estimate of sampler phase which it cannot provide itself. A timing recovery technique based on band-edge component maximization which circumvents these problems is presented. Although the scheme is not optimal, it efficiently and effectively recovers timing in synchronous equalized data communication systems.

THE USE OF AUTOREGRESSION IN FITTING AN EXPONENTIAL CURVE

is being increasingly used in statistical work in many branches of biology. Various numerical methods of fitting this curve have been suggested in recent years for cases in which the y's are independent and have equal variances and the x's take the n values 0, 1, 2, ..., n-1. Stevens (1951) described a fully efficient least squares method for estimating a, ,8 and p, and the errors of these estimates. The method is one of successive approximation and requires a reasonably accurate initial estimate of p. It works well on an electronic computer. He also provided tables for n = 5, 6 and 7, since extended by S. Lipton, which are very useful when the computations are carried out on a desk machine. Pimentel Gomes (1953) provided further tables for n = 4 and n = 5 which permit the least squares estimates of p to be obtained more rapidly than by the iterative procedure. In a previous paper I pointed out that the least squares estimates of p, r' say, are of the form n-1 EWX (A)YX r