Multiple solutions, illegal parameter values, local minima of the sum of squares, and anomalous parameter estimates in least-squares fitting of the two-compartment pharmacokinetic model with absorption

1996 ◽  
Vol 24 (1) ◽  
pp. 79-101 ◽  
Author(s):  
R. D. Purves
Keyword(s):  
Least Squares ◽  
Multiple Solutions ◽  
Pharmacokinetic Model ◽  
Sum Of Squares ◽  
Parameter Estimates ◽  
Local Minima ◽  
Least Squares Fitting ◽  
Parameter Values

Mixing efficiency as estimated by nonlinear least squares

1983 ◽  
Vol 10 (4) ◽  
pp. 703-712
Author(s):  
David T. Chapman
Keyword(s):  
Least Squares ◽  
Dispersion Model ◽  
Nonlinear Least Squares ◽  
Statistical Technique ◽  
Sum Of Squares ◽  
Axial Dispersion ◽  
Mixing Efficiency ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Washout Curves

The suitability of a statistical technique known as nonlinear least squares for use in estimating mixing coefficients was evaluated by fitting models to residence time distribution curves. The washout curves were generated by adding slug inputs of tracers to three different reactors. Each of the reactors, used to treat wastewaters, was a different size and represented a different degree of mixing.Three models, described in the paper, were examined for use in conjuction with the nonlinear least squares technique. They included the axial dispersion, N-tanks-in-series, and Cholette–Cloutier models. The form of the equation for the axial dispersion model depends on the boundary conditions for the reactor being studied. For reactors which cannot be classified as "open" vessels, the required analytical solutions either do not exist or are not suitable for use with the nonlinear least squares technique.Mixing coefficients for the N-tanks and Chollette–Cloutier models were obtained from the tracer washout curves for the three reactors. The residual sum of squares based on nonlinear least squares estimates for the model parameters was compared with the sum of squares obtained using more conventional methods for estimating the parameters. The existence of trailing tails on the tracer curves resulted in misleading parameter estimates for the two models using conventional techniques. Keywords: mixing, least squares, tracer, dispersion, short-circuiting, deadspace.

Identification of Biodegradation models under model and data uncertainty

1996 ◽  
Vol 33 (2) ◽  
pp. 91-105 ◽  
Author(s):  
Peter A. Vanrolleghem ◽  
Karel J. Keesman
Keyword(s):  
Parameter Estimation ◽  
Real World ◽  
Data Uncertainty ◽  
Estimation Methods ◽  
Parameter Estimates ◽  
Local Minima ◽  
Real World Data ◽  
Correlation Matrices ◽  
Parameter Estimation Methods ◽  
Parameter Values

In this paper a number of nonlinear parameter estimation methods are evaluated with respect to their ability to identify biodegradation models from “real-world” data. Important aspects are then the sensitivity to local minima, rate of convergence, required prior knowledge and direct or indirect availability of parameter estimates uncertainty. Furthermore, it is important whether a method is robust against invalid assumptions. In addition to the final parameter values, covariance and correlation matrices, confidence intervals and residual sequences are presented to obtain information about the validity of the models and noise assumptions. Finally, recommendations on the method's applicability range are provided.

scholarly journals An Error-Reduction Method for Temporal Phase Unwrapping Based on Double Three-Step Phase-Shifting and Least-Squares Fitting

2012 ◽  
Vol 6-7 ◽  
pp. 76-81
Author(s):  
Yong Liu ◽  
Ding Fa Huang ◽  
Yong Jiang
Keyword(s):  
Least Squares ◽  
Reduction Method ◽  
Intermediate Phase ◽  
Error Reduction ◽  
Phase Unwrapping ◽  
Phase Shifting ◽  
Surface Measurement ◽  
Acquisition Time ◽  
Least Squares Fitting ◽  
Temporal Phase

Phase-shifting interferometry on structured light projection is widely used in 3-D surface measurement. An investigation shows that least-squares fitting can significantly decrease random error by incorporating data from the intermediate phase values, but it cannot completely eliminate nonlinear error. This paper proposes an error-reduction method based on double three-step phase-shifting algorithm and least-squares fitting, and applies it on the temporal phase unwrapping algorithm using three-frequency heterodyne principle. Theoretical analyses and experiment results show that this method can greatly save data acquisition time and improve the precision.

Generalised DOPs with Consideration of the Influence Function of Signal-in-Space Errors

2011 ◽  
Vol 64 (S1) ◽  
pp. S3-S18 ◽  
Author(s):  
Yuanxi Yang ◽  
Jinlong Li ◽  
Junyi Xu ◽  
Jing Tang
Keyword(s):  
Least Squares ◽  
Stochastic Models ◽  
Influence Function ◽  
A Priori ◽  
Functional Model ◽  
Global Navigation Satellite Systems ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Integrated Navigation ◽  
Satellite Systems

Integrated navigation using multiple Global Navigation Satellite Systems (GNSS) is beneficial to increase the number of observable satellites, alleviate the effects of systematic errors and improve the accuracy of positioning, navigation and timing (PNT). When multiple constellations and multiple frequency measurements are employed, the functional and stochastic models as well as the estimation principle for PNT may be different. Therefore, the commonly used definition of “dilution of precision (DOP)” based on the least squares (LS) estimation and unified functional and stochastic models will be not applicable anymore. In this paper, three types of generalised DOPs are defined. The first type of generalised DOP is based on the error influence function (IF) of pseudo-ranges that reflects the geometry strength of the measurements, error magnitude and the estimation risk criteria. When the least squares estimation is used, the first type of generalised DOP is identical to the one commonly used. In order to define the first type of generalised DOP, an IF of signal–in-space (SIS) errors on the parameter estimates of PNT is derived. The second type of generalised DOP is defined based on the functional model with additional systematic parameters induced by the compatibility and interoperability problems among different GNSS systems. The third type of generalised DOP is defined based on Bayesian estimation in which the a priori information of the model parameters is taken into account. This is suitable for evaluating the precision of kinematic positioning or navigation. Different types of generalised DOPs are suitable for different PNT scenarios and an example for the calculation of these DOPs for multi-GNSS systems including GPS, GLONASS, Compass and Galileo is given. New observation equations of Compass and GLONASS that may contain additional parameters for interoperability are specifically investigated. It shows that if the interoperability of multi-GNSS is not fulfilled, the increased number of satellites will not significantly reduce the generalised DOP value. Furthermore, the outlying measurements will not change the original DOP, but will change the first type of generalised DOP which includes a robust error IF. A priori information of the model parameters will also reduce the DOP.

Fitting Weighted Total Least-Squares Planes and Parallel Planes to Support Tolerancing Standards

2012 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Point Contact ◽  
Coordinate Measuring Machine ◽  
Total Least Squares ◽  
Point Sampling ◽  
Least Squares Fitting ◽  
Weighted Total Least Squares ◽  
Measuring Machine ◽  
Fitting In ◽  
Value Decomposition

We present elegant algorithms for fitting a plane, two parallel planes (corresponding to a slot or a slab) or many parallel planes in a total (orthogonal) least-squares sense to coordinate data that is weighted. Each of these problems is reduced to a simple 3×3 matrix eigenvalue/eigenvector problem or an equivalent singular value decomposition problem, which can be solved using reliable and readily available commercial software. These methods were numerically verified by comparing them with brute-force minimization searches. We demonstrate the need for such weighted total least-squares fitting in coordinate metrology to support new and emerging tolerancing standards, for instance, ISO 14405-1:2010. The widespread practice of unweighted fitting works well enough when point sampling is controlled and can be made uniform (e.g., using a discrete point contact Coordinate Measuring Machine). However, we demonstrate that nonuniformly sampled points (arising from many new measurement technologies) coupled with unweighted least-squares fitting can lead to erroneous results. When needed, the algorithms presented also solve the unweighted cases simply by assigning the value one to each weight. We additionally prove convergence from the discrete to continuous cases of least-squares fitting as the point sampling becomes dense.

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