Edge Detection Using Maximum Likelihood Estimate Of Change Point: The One-Dimensional Case

This chapter develops a Euclidean, two-dimensional, regions-based theory. As with the semi-Aristotelian account in Chapter 2, the goal here is to recover the now orthodox Dedekind–Cantor continuum on a point-free basis. The chapter derives the Archimedean property for a class of readily postulated orientations of certain special regions, what are called “generalized quadrilaterals” (intended as parallelograms), by which the entire space is covered. Then the chapter generalizes this to arbitrary orientations, and then establishes an isomorphism between the space and the usual point-based one. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit “extremal clause”, and we have no axiom of induction other than ordinary numerical (mathematical) induction.

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Box-Cox Transformations

10.1093/oso/9780198505044.003.0010 ◽

2017 ◽

Author(s):

Russell Cheng

Keyword(s):

Maximum Likelihood ◽

Normal Form ◽

Maximum Likelihood Estimate ◽

Likelihood Estimate ◽

Grouped Data ◽

Numerical Example ◽

Alternative Method ◽

Modified Likelihood

This chapter examines the well-known Box-Cox method, which transforms a sample of non-normal observations into approximately normal form. Two non-standard aspects are highlighted. First, the likelihood of the transformed sample has an unbounded maximum, so that the maximum likelihood estimate is not consistent. The usually suggested remedy is to assume grouped data so that the sample becomes multinomial. An alternative method is described that uses a modified likelihood similar to the spacings function. This eliminates the infinite likelihood problem. The second problem is that the power transform used in the Box-Cox method is left-bounded so that the transformed observations cannot be exactly normal. This biases estimates of observational probabilities in an uncertain way. Moreover, the distributions fitted to the observations are not necessarily unimodal. A simple remedy is to assume the transformed observations have a left-bounded distribution, like the exponential; this is discussed in detail, and a numerical example given.

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Preserving the Shape of Functions by Applying Multidimensional Schoenberg-Type Operators

Symmetry ◽

10.3390/sym13061016 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1016

Author(s):

Camelia Liliana Moldovan ◽

Radu Păltănea

Keyword(s):

Applied Sciences ◽

Degree Of Approximation ◽

Higher Order ◽

Second Order ◽

Dimensional Case ◽

Moduli Of Continuity ◽

One Dimensional ◽

Multidimensional Generalization ◽

The One

The paper presents a multidimensional generalization of the Schoenberg operators of higher order. The new operators are powerful tools that can be used for approximation processes in many fields of applied sciences. The construction of these operators uses a symmetry regarding the domain of definition. The degree of approximation by sequences of such operators is given in terms of the first and the second order moduli of continuity. Extending certain results obtained by Marsden in the one-dimensional case, the property of preservation of monotonicity and convexity is proved.

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Limit of 𝑝-Laplacian obstacle problems

Advances in Calculus of Variations ◽

10.1515/acv-2019-0058 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Raffaela Capitanelli ◽

Maria Agostina Vivaldi

Keyword(s):

Asymptotic Behavior ◽

Uniform Convergence ◽

Explicit Expression ◽

Positive Measure ◽

Sufficient Conditions ◽

Obstacle Problems ◽

Dimensional Case ◽

Asymptotic Behavior Of Solutions ◽

One Dimensional ◽

The One

AbstractIn this paper, we study asymptotic behavior of solutions to obstacle problems for p-Laplacians as {p\to\infty}. For the one-dimensional case and for the radial case, we give an explicit expression of the limit. In the n-dimensional case, we provide sufficient conditions to assure the uniform convergence of the whole family of the solutions of obstacle problems either for data f that change sign in Ω or for data f (that do not change sign in Ω) possibly vanishing in a set of positive measure.

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