A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread

Trutschnig, Wolfgang; González‐Rodríguez, Gil; Colubi, Ana; Gil, María Ángeles

doi:10.1016/j.ins.2009.06.023

Cited by 125 publications

(57 citation statements)

References 24 publications

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“…This is the variance suggested by Fréchet for random elements defined on a complete metric space. In fact, it can be extended to random (fuzzy) sets of any dimension by considering, for instance, the families of distances introduced in [41] or [71].…”

Section: Scalvar(d) Measures the Variability Of The Intervals In D Bmentioning

confidence: 99%

“…This line of research has been considerably extended so as to adapt classical statistical methods to functional data [7,39]. The main issue is to define a space of functions equipped with a suitable metric structure [23,71]. In this theory of random fuzzy sets, a scalar distance between fuzzy sets is instrumental when defining variance viewed as a mean squared deviation from the fuzzy mean value [41], in the spirit of Fréchet.…”

Section: Various Notions Of Random Fuzzy Setsmentioning

confidence: 99%

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Statistical reasoning with set-valued information: Ontic vs. epistemic views

Couso

Dubois

2014

International Journal of Approximate Reasoning

159

107

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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. In information processing tasks, sets may have a conjunctive or a disjunctive reading. In the conjunctive reading, a set represents an object of interest and its elements are subparts of the object, forming a composite description. In the disjunctive reading, a set contains mutually exclusive elements and refers to the representation of incomplete knowledge. It does not model an actual object or quantity, but partial information about an underlying object or a precise quantity. This distinction between what we call ontic vs. epistemic sets remains valid for fuzzy sets, whose membership functions, in the disjunctive reading are possibility distributions, over deterministic or random values. This paper examines the impact of this distinction in statistics. We show its importance because there is a risk of misusing basic notions and tools, such as conditioning, distance between sets, variance, regression, etc. when data are set-valued. We discuss several examples where the ontic and epistemic points of view yield different approaches to these concepts.

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Section: Scalvar(d) Measures the Variability Of The Intervals In D Bmentioning

confidence: 99%

Section: Various Notions Of Random Fuzzy Setsmentioning

confidence: 99%

Statistical reasoning with set-valued information: Ontic vs. epistemic views

Couso

Dubois

2014

International Journal of Approximate Reasoning

159

107

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“…The mid-spread metric D ϕ θ , introduced by [11] and extended in [5], is defined as follow. Let f ∈ L, it can be decomposed as f = mid f + spr f where…”

Section: For Anymentioning

confidence: 99%

“…For the "only if" part, in order to obtain Equation (12), it is sufficient to compute the expectation in Equation (11). "If" part.…”

Section: Hukuhara Decompositionmentioning

confidence: 99%

“…In particular, we start to consider a P-almost surely centered FRV X ∈ L 2 [F] and the family ΘX of all centered deterministic η ∈ F for which the Hukuhara difference X ⊖H η exists almost surely. Then, we prove that ΘX is not empty, convex and closed in F with respect to ρ, where ρ can be chosen between d2 (that corresponds to the L 2 metric in the space of support functions) and D θ (the mid-spread metric; see [5,11]). Further,…”

Section: Introductionmentioning

confidence: 98%

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A decomposition theorem for fuzzy set-valued random variables

Aletti

Bongiorno

2013

Fuzzy Sets and Systems

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In this paper, a decomposition theorem for a (square integrable) fuzzy random variable FRV is proposed. The paper is mainly divided in two parts. In the first part, for any FRV X, we define the Hukuhara set as the family of (deterministic) fuzzy sets C for which the Hukuhara difference X ⊖H C exists almost surely; in particular, we prove that such a family is a closed (with respect to different well known metrics) convex subset of the family of all fuzzy sets. In the second part, we prove that any square integrable FRV can be decomposed, up to a random translation, as the sum of a FRV Y and an element C ′ chosen uniquely (thanks to a minimization argument) in the Hukuhara set. This decomposition allows us to characterize fuzzy random translations; in particular, a FRV is a fuzzy random translation if and only if its Aumann expectation equals C ′ (given by the above decomposition) up to a deterministic translation. Examples and open problems are also presented.

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The sign test and the signed‐rank test for interval‐valued data

Grzegorzewski

Śpiewak

2019

Int J Intell Syst

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Two versions of the generalized sign test and the signed‐rank test for interval‐valued data (both for one‐sample and paired two‐sample problem) are proposed. These two versions correspond to different possible views on the interval outcomes of the experiment—either the epistemic or the ontic one. Each view yields its own approach to data analysis which results in different test construction and the way of carrying on the statistical inference.

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A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread

Cited by 125 publications

References 24 publications

Statistical reasoning with set-valued information: Ontic vs. epistemic views

Statistical reasoning with set-valued information: Ontic vs. epistemic views

A decomposition theorem for fuzzy set-valued random variables

The sign test and the signed‐rank test for interval‐valued data

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