Parameter Transformations for Improved Approximate Confidence Regions in Nonlinear Least Squares

Bates, Douglas M.; Watts, Donald G.

doi:10.1214/aos/1176345633

Cited by 104 publications

(69 citation statements)

References 11 publications

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“…Our approach is in the spirit of Bates and Watts' treatment of the subject [15][16][17][18]. However, the intrinsic properties of the model manifold can be calculated in an alternative way without reference to the embedding through the methods of Jeffreys, Rao and others [9][10][11][12][13].…”

Section: Figmentioning

confidence: 99%

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Geometry of nonlinear least squares with applications to sloppy models and optimization

2011

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Parameter estimation by nonlinear least squares minimization is a common problem that has an elegant geometric interpretation: the possible parameter values of a model induce a manifold within the space of data predictions. The minimization problem is then to find the point on the manifold closest to the experimental data. We show that the model manifolds of a large class of models, known as sloppy models, have many universal features; they are characterized by a geometric series of widths, extrinsic curvatures, and parameter-effects curvatures, which we describe as a hyper-ribbon. A number of common difficulties in optimizing least squares problems are due to this common geometric structure. First, algorithms tend to run into the boundaries of the model manifold, causing parameters to diverge or become unphysical before they have been optimized. We introduce the model graph as an extension of the model manifold to remedy this problem. We argue that appropriate priors can remove the boundaries and further improve the convergence rates. We show that typical fits will have many evaporated parameters unless the data are very accurately known. Second, 'bare' model parameters are usually ill-suited to describing model behavior; cost contours in parameter space tend to form hierarchies of plateaus and long narrow canyons. Geometrically, we understand this inconvenient parameterization as an extremely skewed coordinate basis and show that it induces a large parameter-effects curvature on the manifold. By constructing alternative coordinates based on geodesic motion, we show that these long narrow canyons are transformed in many cases into a single quadratic, isotropic basin. We interpret the modified Gauss-Newton and Levenberg-Marquardt fitting algorithms as an Euler approximation to geodesic motion in these natural coordinates on the model manifold and the model graph respectively. By adding a geodesic acceleration adjustment to these algorithms, we alleviate the difficulties from parameter-effects curvature, improving both efficiency and success rates at finding good fits.

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Section: Figmentioning

confidence: 99%

“…Curvature has also been used to study confidence regions [16,20,[47][48][49], kurtosis (deviations from normality) in parameter estimation [50], and criteria for determining if a minimum is the global minimizer [51]. We will see below that the large anisotropy in the metric produces FIG.…”

Section: Curvaturementioning

confidence: 99%

Geometry of nonlinear least squares with applications to sloppy models and optimization

2011

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“…Bates and Watts (1981) schlagen als Optimalit atskriterium z.B. vor, jenen Versuchsplan zu w ahlen, der die Parameter-E ekt-Nichtlinearit at minimiert.…”

Section: E Ekte Des Versuchsplansunclassified

Zur Quantfizierung und Analyse der Nichtlinearität von Regressionsmodellen

Sedlacek

2016

AJS

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“…For a small parameter-effects curvature, straight parallel equispaced lines in the parameter space map onto ones in the expectation surface, as happens with the tangent plane. The parameter-effects curvature can be decreased by suitable parameterizations (Bates & Watts, 1981). On the other hand, the intrinsic curvature measures the degree of nonlinearity inherent in the model itself.…”

Section: The Formula For the Relative Curvature Measurementioning

confidence: 99%

A Note on Parameterizations in Pharmacokinetic Compartment Models

Daimon

Goto²

2008

Behaviormetrika

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In pharmacokinetics, compartment models often play an important role in the description of the concentration of the drug in the blood over time after its administration to a subject. In the inference in the compartment models in pharmacokinetics, unlike that of the usual nonlinear regression models, parameterizations with respect to the parameters related to pharmacokinetic indices are frequently applied, for some reasons of the constraint that the parameter takes only the positive value, facilitation of the compartment models, or interpretation of the relationships between the parameters and the demographic or physiological individual attributes. However, in practice no attention is paid to checking if the utilized parameterization is valid. In this article, the relative curvature measure that enables us to assess the intrinsic and parameter-effects nonlinearity in the model is applied for checking it from the latter nonlinearity. From a viewpoint of the relative curvature measure, we reconsider several well-known examples and demonstrate the parameterization checking.

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Parameter Transformations for Improved Approximate Confidence Regions in Nonlinear Least Squares

Cited by 104 publications

References 11 publications

Geometry of nonlinear least squares with applications to sloppy models and optimization

Geometry of nonlinear least squares with applications to sloppy models and optimization

Zur Quantfizierung und Analyse der Nichtlinearität von Regressionsmodellen

A Note on Parameterizations in Pharmacokinetic Compartment Models

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