Relative Curvature Measures of Nonlinearity

Bates, Douglas M.; Watts, Donald G.

doi:10.1111/j.2517-6161.1980.tb01094.x

Cited by 413 publications

(314 citation statements)

References 13 publications

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“…It is obvious that for any flat estimation subspace ϕ = 0. Note that a similar measure of intrinsic nonlinearity is proposed before in Beale (1960) and it is actually expressed by the angle ϕ as sin 2 ϕ (Bates and Watts 1980).…”

Section: Indices Of Nonlinearitymentioning

confidence: 97%

“…According to the geometrical interpretation of linear LS-problem (3) (Fig. 1) the vector of the measured values specifies a point p O in the N -dimensional space of observations p whereas the K -dimensional subspace p C called estimation subspace (Bates and Watts 1980;Draper and Smith 1981) is an image Aq of the parametric space q in the observation space p. Obviously, since the vector δp O represents only measurement errors, the precise observationsp corresponding to the true parameter valuesq must belong to that subspace, in other words,p = p C (q) = Aq. Besides, it is important to note that in the linear case the estimation subspace is a flat, i.e.…”

Section: Ls-problem Its Geometry and Confidence Regionsmentioning

confidence: 99%

“…A set like that can be constructed if the K -dimensional estimation subspace specified by the model p C (q) in the N -dimensional space p is flat (Beale 1960;Bates and Watts 1980). In the nonlinear LS-problem the estimation subspace p C is generally not flat.…”

Section: S/2003 J04mentioning

confidence: 99%

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Nonlinear methods of statistic simulation of virtual parameter values for investigating uncertainties in orbits determined from observations

Avdyushev

2011

Celest Mech Dyn Astr

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Determination of orbital parameters from observations is formally a nonlinear inverse problem for solving which evidently nonlinear methods are required. Meanwhile, an accompanying stage in solving the inverse problem is the evaluation of parametric accuracy to which, however, linear methods are conventionally applied. This is quite justified if parametric errors caused by observation errors are rather small, otherwise this is not at all since the nonlinearity of the inverse problem can be considerable to influence on the evaluations of parametric accuracy especially when the observations are very few. With the advent of quick-operating and multiprocessor computers, recently one tends to employ statistic simulation of virtual parameter values for investigating uncertainties in orbits determined from observations. In the paper are just discussed the methods designed specially for nonlinear statistic simulation of virtual parameter values. Their efficiency is investigated in application to estimating uncertainties in the orbit of Jovian satellite S/2003 J04 whose orbital parameters are ill-determined owing to scanty available observations. Indices of nonlinearity are introduced for making decision in the choice between linear and nonlinear methods.

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Section: Indices Of Nonlinearitymentioning

confidence: 97%

Section: Ls-problem Its Geometry and Confidence Regionsmentioning

confidence: 99%

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Nonlinear methods of statistic simulation of virtual parameter values for investigating uncertainties in orbits determined from observations

Avdyushev

2011

Celest Mech Dyn Astr

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“…Although several measures of nonlinearity have been proposed [3,11,26], for this investigation, it is more useful to quantify the nonlinearity of a problem based on the effect of a linearity assumption on the magnitude of the likelihood function. We therefore define the nonlinearity measure for g at x 0 as the value of the function…”

Section: Single-parameter Test Problemmentioning

confidence: 99%

Ensemble filter methods with perturbed observations applied to nonlinear problems

2009

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The ensemble Kalman filter (EnKF) has been shown repeatedly to be an effective method for data assimilation in large-scale problems, including those in petroleum engineering. Data assimilation for multiphase flow in porous media is particularly difficult, however, because the relationships between model variables (e.g., permeability and porosity) and observations (e.g., water cut and gas-oil ratio) are highly nonlinear. Because of the linear approximation in the update step and the use of a limited number of realizations in an ensemble, the EnKF has a tendency to systematically underestimate the variance of the model variables. Various approaches have been suggested to reduce the magnitude of this problem, including the application of ensemble filter methods that do not require perturbations to the observed data. On the other hand, iterative least-squares data assimilation methods with perturbations of the observations have been shown to be fairly robust to nonlinearity in the data relationship. In this paper, we present EnKF with perturbed observations as a square root filter in an enlarged state space. By imposing second-order-exact sampling of the observation errors and independence constraints to eliminate the cross-covariance with predicted observation perturbations, we show that it is possible in linear problems to obtain results from EnKF with observation perturbations that are equivalent to ensemble square-root filter results. Results from a standard EnKF, EnKF with second-order-exact sampling of measurement errors that satisfy independence constraints (EnKF (SIC)), and an ensemble square-root filter (ETKF) are compared on various test problems with varying degrees of nonlinearity and dimensions. The first test problem is a simple one-variable quadratic model in which the nonlinearity of the observation operator is varied over a wide range by adjusting the magnitude of the coefficient of the quadratic term. The second problem has increased observation and model dimensions to test the EnKF (SIC) algorithm. The third test problem is a two-dimensional, two-phase reservoir flow problem in which permeability and porosity of every grid cell (5,000 model parameters) are unknown. The EnKF (SIC) and the mean-preserving ETKF (SRF) give similar results when applied to linear problems, and both are better than the standard EnKF. Although the ensemble methods are expected to handle the forecast step well in nonlinear problems, the estimates of the mean and the variance from the analysis step for all variants of ensemble filters are also surprisingly good, with little difference between ensemble methods when applied to nonlinear problems.

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“…Box (1971) showed that the normalized bias is smaller than a specified function of the measure of non-linearity. Bates and Watts (1980) defined curvature measures for each direction in parameter space, and suggested to use the maximum over all directions. Hougaard (1985b) showed that the normalized bias and the skewness are smaller than specified functions of both Beales and Bates and Watts measures of non-linearity.…”

Section: Evaluation Of Non-normalitymentioning

confidence: 99%

Nonlinear regression and curved exponential families. Improvement of the approximation to the asymptotic distribution

Hougaard¹

1995

Metrika

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Inference in nonlinear models is usually based on the asymptotic normal distribution, based on linearizing the model. The accuracy of this approximation can in many cases be improved by a reparametrization. Systematic methods for doing this will be described. Sometimes a saddlepoint approximation can be used, and this offers several advantages compared to the asymptotic distribution and the Edgeworth expansion. The improved methods are unfortunately not commonly used. It will be discussed why this is so. The methods will be illustrated by a series of examples.

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Relative Curvature Measures of Nonlinearity

Cited by 413 publications

References 13 publications

Nonlinear methods of statistic simulation of virtual parameter values for investigating uncertainties in orbits determined from observations

Nonlinear methods of statistic simulation of virtual parameter values for investigating uncertainties in orbits determined from observations

Ensemble filter methods with perturbed observations applied to nonlinear problems

Nonlinear regression and curved exponential families. Improvement of the approximation to the asymptotic distribution

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