Use of lp norms in fitting curves and surfaces to data

Atieg, A.; Watson, G. A.

doi:10.21914/anziamj.v45i0.882

Cited by 15 publications

(21 citation statements)

References 10 publications

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“…We build simple models to address this question, investigate their properties and apply a variant of the Weierstrass theorem to prove the existence of a solution. Our results extend and improve some other comparable results of the author [2,8,20,21,22,23,24,25]. …”

supporting

confidence: 92%

“…If appropriate probabilistic assumptions about underlying error distributions are made, least squares produces what is known as the maximum-likelihood estimate parameters. Even if the probabilistic assumptions are not satisfied, research in this areas has shown that least squares produces useful results [1,2,5,6,7,8,9,10,11,14,15,19,21,22,23,24,25,26] A very common source of least squares is curve fitting. Let x be the independent variable and let y  x denote an unknown function of x that we want to approximate.…”

Section: Optimal Designmentioning

confidence: 99%

“…A large body of literature is devoted to the algorithms development for the solution of the highlighted problems [2,7,8,11,15,19,23] and it is our wish to apply a simple and more general results to resolve(solve) the problem.…”

Section: Introductionmentioning

confidence: 99%

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A Note on Convex Programming in Practical Problems

Osinuga

Agunbiade

2013

BSMaSS

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Abstract. In recent years, convex programming has become a sophisticated tool of central importance in engineering, finance, operations research, statistics etc. The goal of this paper is to emphasize modeling and present several convex programming problem formulations especially in optimal design and location theory. We build simple models to address this question, investigate their properties and apply a variant of the Weierstrass theorem to prove the existence of a solution. Our results extend and improve some other comparable results of the author [2,8,20,21,22,23,24,25].

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supporting

confidence: 92%

Section: Optimal Designmentioning

confidence: 99%

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A Note on Convex Programming in Practical Problems

Osinuga

Agunbiade

2013

BSMaSS

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“…During the last few decades an increased interest in alternative l s -norm has become apparent (see, e.g., Atieg and Watson, 2004;Gonin and Money, 1989). For example, l 1 -norm criteria are more suitable if there are wild points (outliers) in the data.…”

Section: Satisfy the Conditions (5) And (6) Then Functional F Defimentioning

confidence: 99%

On parameter estimation in the bass model by nonlinear least squares fitting the adoption curve

Marković¹,

Jukić²

2013

International Journal of Applied Mathematics and Computer Science

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The Bass model is one of the most well-known and widely used first-purchase diffusion models in marketing research. Estimation of its parameters has been approached in the literature by various techniques. In this paper, we consider the parameter estimation approach for the Bass model based on nonlinear weighted least squares fitting of its derivative known as the adoption curve. We show that it is possible that the least squares estimate does not exist. As a main result, two theorems on the existence of the least squares estimate are obtained, as well as their generalization in the ls norm (1 ≤ s < ∞). One of them gives necessary and sufficient conditions which guarantee the existence of the least squares estimate. Several illustrative numerical examples are given to support the theoretical work.

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“…[6] use the ℓ 1 and ℓ ∞ for fitting parametric curves and surfaces. The case of general ℓ p norms is described in [8]. In a recent manuscript [3], we study the relation between Gauss-Newton-type methods for approximation with respect to general normlike functions and the technique of iteratively re-weighted least squares.…”

Section: Surface Fittingmentioning

confidence: 99%

Robust fitting of implicitly defined surfaces using Gauss–Newton-type techniques

Aigner

Jüttler

2009

Vis Comput

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We describe Gauss-Newton type methods for fitting implicitly defined curves and surfaces to given unorganized data points. The methods are suitable not only for leastsquares approximation, but they can also deal with general error functions, such as approximations to the ℓ 1 or ℓ ∞ norm of the vector of residuals. Two different definitions of the residuals will be discussed, which lead to two different classes of methods: direct methods and data-based ones. In addition we discuss the continuous versions of the methods, which furnish geometric interpretations as evolution processes.It is shown that the data-based methods -which are less costly, as they work without the computation of the closest points -can efficiently deal with error functions that are adapted to noisy and uncertain data. In addition, we observe that the interpretation as evolution process allows to deal with the issues of regularization and with additional constraints.

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Use of lp norms in fitting curves and surfaces to data

Cited by 15 publications

References 10 publications

A Note on Convex Programming in Practical Problems

A Note on Convex Programming in Practical Problems

On parameter estimation in the bass model by nonlinear least squares fitting the adoption curve

Robust fitting of implicitly defined surfaces using Gauss–Newton-type techniques

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