Spatial Vagueness, Uncertainty, Granularity

A Special Double Issue of spatial Cognition and Computation

Brandon Bennett, Matteo Cristani

• Publisher: Psychology Press
• Year: 2017
• ISBN:
  9781135066246
• Edition: 1
• Topic: Psicología
• Subtopic: Historia y teoría en psicología

This special issue collects enhanced and extended versions of papers that were presented at the Symposium on Spatial Vagueness, Uncertainty, and Granularity held in October 2001. The contributions examine fundamental problems in the analysis of spatial vagueness and uncertainty, and the editors hope this selection stimulates further investigation in this growing subfield of the theory of spatial information.

Granularity-sensitive Spatial Attributes

Antony Galton
University of Exeter

We investigate the effect of granulation on the spatial attributes of regions it is applied to. An informal illustrative treatment, using a single specific granulation and a variety of regions illustrating some key spatial attributes, is followed by a more rigorous theoretical development in which granulations are defined in terms of equivalence relations on the source space. It is found that many spatial attributes are both-ways granularity-sensitive, meaning that the attributes may be gained by some regions, but lost by others, under the same granulation.

Keywords: granulation, spatial attributes

Correspondence concerning this article should be addressed to Antony Galton, School of Engineering and Computer Science, University of Exeter, Exeter EX4 4QF, UK email: [email protected]>

1 Introduction

A granulation, in the sense that the term is used in this paper, is a mapping from a continuous source space into a discrete target space. Typically, this will be a many-to-one mapping, with the result that information present in the source space is lost under granulation. The particular concern of this paper is the behaviour of various spatial attributes under granulation. The attributes in question are qualitative properties and relations of regions in the source and target spaces. The formai theory will need to specify what is meant by a ‘region’ in each space, and what is meant by the ‘same’ property or relation for these two classes of regions.

A spatial attribute A may be lost under a given granulation g if there is a region R in the source space such that R possesses attribute A but its image g(R) under the granulation does not. Likewise, A may be gained under g if there is a region R which does not possess A although its image g(R) does. In the case of an attribute expressed by a binary relation, we must refer to two regions in the source space, but the essential idea is the same. We shall describe an attribute as upward granularity sensitive if it can be lost under granulation, and downward granularity sensitive if it can be gained. Some attributes may be both gained and lost under the same granulation, and these are referred to as both-ways granularity-sensitive.

Granularity-sensitivity matters because its presence is indicative of falsification: a granular representation is not veridical with respect to those attributes which are sensitive to the granulation which produced it. Granularity-sensitivity is a special case of the more general phenomenon of resolution-sensitivity, where by resolution is meant some measure of the level of discriminable detail present in an image or representation. The spatial attributes that an object is portrayed as having are heavily dependent on the resolution of the representation. À binary star may appear as a single point of light when looked at with the naked eye, but as two separate points of light when looked at through a telescope. From a distance a swarm of bees looks like a wisp of smoke, homogeneous in texture, but from closer up it resolves into many discrete individuals. These may be called resolution effects; they arise when one and the same object is ascribed contradictory attributes when represented at different resolutions. When the difference in resolution is attributable to granulation, we may refer to granularity effects.

The purpose of this paper is to investigate which spatial attributes are granularity-sensitive, i.e., susceptible to granularity effects. In Section 2, a range of spatial attributes is surveyed to investigate the extent of their granularity-sensitivity under one specific granulation. It turns out that surprisingly many spatial attributes are both-ways granularity-sensitive. The treatment here is discursive and informal. In Section 3, a more formal theory of granulations is developed, and the granularity-sensitivity of certain attributes is investigated in this more general setting. This will not be a comprehensive investigation, but is intended to establish a framework for further researches.

2 Some Examples of Granularity-sensitivity

This section systematically ...

Table of contents

• Cover
• Title
• Contents
• Copyright
• Editorial
• Articles
• Research Notes