Approximate distances, pointless geometry and incomplete information

Coppola, Cristina; Pacelli, Tiziana

doi:10.1016/j.fss.2006.03.019

Cited by 7 publications

(5 citation statements)

References 11 publications

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“…Another metric approach to pointfree geometry is based on the ideas of interval analysis. Indeed in [4] C. Coppola and T. Pacelli argue that in several cases while it is impossible to determinate the precise value of a distance, it is possible to individuate an interval containing such a value. This leads to consider an "approximate distance" function whose values are intervals and to propose the following definition where the functions σ and Σ represent the lower bounds and the upper bounds of the intervals.…”

Section: A Comparison With the Literaturementioning

confidence: 99%

“…For example in [5] and [8] the notions of distance between regions and diameter are assumed as primitives. Moreover, in [4] one refers to the interval-distance. Now, a very interesting series of researches for representation theorems for lattices with a diameter can be interpreted in the framework of pointfree geometry (see the papers of B. Banaschewski and A.…”

Section: Introductionmentioning

confidence: 99%

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Whitehead's pointfree geometry and diametric posets

Gerla

Paolillo

2010

LLP

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This note is motivated by Whitehead's researches in inclusion-based point-free geometry as exposed in An Inquiry Concerning the Principles of Natural Knowledge and in The concept of Nature. More precisely, we observe that Whitehead's definition of point, based on the notions of abstractive class and covering, is not adequate. Indeed, if we admit such a definition it is also questionable that a point exists. On the contrary our approach, in which the diameter is a further primitive, enables us to avoid such a drawback. Moreover, since such a notion enables us to define a metric in the set of points, our proposal looks to be a good starting point for a foundation of the geometry metrical in nature (as proposed, for example, by L. M. Blumenthal).

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Section: A Comparison With the Literaturementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Whitehead's pointfree geometry and diametric posets

Gerla

Paolillo

2010

LLP

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show abstract

“…In her survey [2], Bloch recalls the three types of fuzzy distances already defined: between two points in a fuzzy set, from a point to a fuzzy set and between two fuzzy sets. Many works tackle this problem, among them [7], [5], [20], [9], [4]. Bloch [2] does not mention the distance between individuals within a fuzzy partition (FP).…”

Section: Introductionmentioning

confidence: 99%

Fuzzy partition-based distance practical use and implementation

Guillaume¹,

Charnomordic²

2013

2013 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE)

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This work discusses the implementation of a semidistance based on fuzzy partitions, that allows to introduce expert knowledge into distance computations done on numerical data. It can be used in various kinds of statistical clustering or other applications. The semi-distance univariate behaviour is first studied, then a multivariate clustering case study is presented, that includes an optimization procedure. All calculations are done using open source softwares: FisPro for fuzzy modelling and R for clustering. Emphasis is given to a number of software functionalities, which are targeted at the practical use of the semi-distance.

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“…Many works address this question, among them [6,4,13,7,3]. Some proposals paid attention to the fulfillment of the triangle inequality [2].…”

Section: Introductionmentioning

confidence: 99%

A numerical distance based on fuzzy partitions

Guillaume¹,

Charnomordic²,

Loisel³

2011

Proceedings of the 7th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011)

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This work studies a new distance function which takes into account expert knowledge by making use of fuzzy partitions. It considers the symbolic distances between concepts and is equivalent to the Euclidean distance for regular partitions made of triangular membership functions. Its behaviour is investigated in comparison with that of the Euclidean distance and its interest is shown for clustering applications.

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Approximate distances, pointless geometry and incomplete information

Cited by 7 publications

References 11 publications

Whitehead's pointfree geometry and diametric posets

Whitehead's pointfree geometry and diametric posets

Fuzzy partition-based distance practical use and implementation

A numerical distance based on fuzzy partitions

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