On the granularity of summative kernels

Loquin, Kevin; Strauss, Olivier

doi:10.1016/j.fss.2008.02.021

Cited by 29 publications

(50 citation statements)

References 30 publications

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“…A summative kernel [19] is an R + -valued Lebesgue measurable function , defined on an interval Q ⊆ R, satisfying the summative normalization condition:…”

Section: Summative and Maxitive Kernelsmentioning

confidence: 99%

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Quasi-continuous histograms

Strauss

2009

Fuzzy Sets and Systems

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Histograms are very useful for summarizing statistical information associated with a set of observed data. They are one of the most frequently used density estimators due to their ease of implementation and interpretation. However, histograms suffer from a high sensitivity to the choice of both reference interval and bin width. This paper addresses this difficulty by means of a fuzzy partition. We propose a new density estimator based on transferring the counts associated with each cell of the fuzzy partition to any subset of the reference interval. We introduce three different methods of achieving this transfer. The properties of each method are illustrated with a classic real observation set. The density estimator obtained relates to the Parzen-Rosenblatt kernel density estimation technique. In this paper, we only consider the monovariate case with precise and imprecise observations.

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“…A summative kernel [19] is an R + -valued Lebesgue measurable function , defined on an interval Q ⊆ R, satisfying the summative normalization condition:…”

Section: Summative and Maxitive Kernelsmentioning

confidence: 99%

“…A maxitive kernel is another way of defining a weighted neighborhood. A maxitive kernel [19] is a normalized fuzzy subset E, defined on a domain , satisfying the maxitive normalization condition:…”

Section: Summative and Maxitive Kernelsmentioning

confidence: 99%

Quasi-continuous histograms

Strauss

2009

Fuzzy Sets and Systems

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“…According to Corollary 1, an estimateĥ(x) of h obtained with a summative kernel κ, such that P κ belongs to core(Π π x ∆ ), belongs to the estimated interval (8). Besides, the estimation bounds are attained, i.e.…”

Section: Corollary 1 Imprecise Functional Estimationmentioning

confidence: 97%

“…Replacing a summative kernel by a maxitive kernel for estimating a function h aims at taking into account the imperfect knowledge of the modeler to choose a particular κ. The specificity [16,8] of the maxitive kernel chosen by the modeler for performing this imprecise estimation reflects his knowledge. The most specific is the maxitive neighborhood, the smallest is the encoded set.…”

Section: Corollary 1 Imprecise Functional Estimationmentioning

confidence: 99%

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Imprecise Functional Estimation: The Cumulative Distribution Case

Loquin

Strauss

Soft Methods for Handling Variability and Imprecision

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In this paper, we propose an adaptation of the Parzen Rosenblatt cumulative distribution function estimator that uses maxitive kernels. The result of this estimator, on every point of the domain of F, the cumulative distribution to be estimated, is interval valued instead of punctual valued. We prove the consistency of our approach with the classical Parzen Rosenblatt estimator, since, according to consistency conditions between the maxitive kernel involved in the imprecise estimator and the summative kernel involved in the precise estimator, our imprecise estimate contains the precise Parzen Rosenblatt estimate.

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