Introduction Principal options Dealing with granulation hierarchies Conclusions Basic modelling choices to add roughness to granular levels C. Maria Keet Faculty of Computer Science, Free University of Bozen-Bolzano, Italy keet@inf.unibz.it First International Workshop on Rough Set Theory (RST’09) 25-27 May 2009, Milano, Italy 1 / 33
Introduction Principal options Dealing with granulation hierarchies Conclusions Outline Introduction Principal options Orthogonal issues Ontological commitments Dealing with granulation hierarchies Defining perspectives Examples Conclusions 2 / 33
Introduction Principal options Dealing with granulation hierarchies Conclusions Motivation • Usage of rough sets for granularity • Left implicit or ambiguous where exactly rough sets are used for granulation and what, if any, the ontological commitments are • Aim to analyse and disambiguate this and, if necessary, find a way how this can be made explicit so as to facilitate implementations and integration of applications 3 / 33
Introduction Principal options Dealing with granulation hierarchies Conclusions First distinctions • ‘vertical’ vs. ‘horizontal’ aspects with rough sets: • Computing granules at different levels of granularity to create a granulation hierarchy • One does have a basic hierarchy of levels of detail, but one tweaks the spaces that apply to a given level, be it thanks to a variation in the rough set approach [HY09] or the values themselves. • Vertical axis involves the scale relevant for a particular granulation hierarchy, i.e, for defining the levels • Horizontal axis for the refinement, amount of impreciseness, that is given in any of the finer-grained quantities at each level, i.e., precision in the definition of a level 4 / 33
Introduction Principal options Dealing with granulation hierarchies Conclusions First distinctions • ‘vertical’ vs. ‘horizontal’ aspects with rough sets: • Computing granules at different levels of granularity to create a granulation hierarchy • One does have a basic hierarchy of levels of detail, but one tweaks the spaces that apply to a given level, be it thanks to a variation in the rough set approach [HY09] or the values themselves. • Vertical axis involves the scale relevant for a particular granulation hierarchy, i.e, for defining the levels • Horizontal axis for the refinement, amount of impreciseness, that is given in any of the finer-grained quantities at each level, i.e., precision in the definition of a level 5 / 33
Introduction Principal options Dealing with granulation hierarchies Conclusions First distinctions • Latter implicitly deals with distinction between “enforced” and “intended” indistinguishability [KK04] : • Enforced because of limited precision due to noisy data, the equipments itself and indirect measurement taking • Intended regarding chosen impreciseness because the measurement-taker does not care about more precise measurements • Linking approximation spaces to levels of granularity [Kee08, SS07, Yao04] with variations in ways of specifying or dressing up approximation spaces [HY09, SS07] . • Can one have multiple approximation spaces defined for each level of granularity. How to manage levels and boundary regions computationally? 6 / 33
Introduction Principal options Dealing with granulation hierarchies Conclusions First distinctions • Latter implicitly deals with distinction between “enforced” and “intended” indistinguishability [KK04] : • Enforced because of limited precision due to noisy data, the equipments itself and indirect measurement taking • Intended regarding chosen impreciseness because the measurement-taker does not care about more precise measurements • Linking approximation spaces to levels of granularity [Kee08, SS07, Yao04] with variations in ways of specifying or dressing up approximation spaces [HY09, SS07] . • Can one have multiple approximation spaces defined for each level of granularity. How to manage levels and boundary regions computationally? 7 / 33
Introduction Principal options Dealing with granulation hierarchies Conclusions First distinctions • Latter implicitly deals with distinction between “enforced” and “intended” indistinguishability [KK04] : • Enforced because of limited precision due to noisy data, the equipments itself and indirect measurement taking • Intended regarding chosen impreciseness because the measurement-taker does not care about more precise measurements • Linking approximation spaces to levels of granularity [Kee08, SS07, Yao04] with variations in ways of specifying or dressing up approximation spaces [HY09, SS07] . • Can one have multiple approximation spaces defined for each level of granularity. How to manage levels and boundary regions computationally? 8 / 33
Introduction Principal options Dealing with granulation hierarchies Conclusions Preliminaries • Lift rough sets’ approximation spaces up to the knowledge representation layer to conceptualise various possibilities • As a start, for the moment, recognise level, space, and bounds: • ∀ x , y ( has roughness ( x , y ) → GranularLevel ( x ) ∧ ApproxSpace ( y )) • ∀ x , y ( bound of ( x , y ) → ApproxSpace ( x ) ∧ LowerBound ( y )) • ∀ x , y ( bound of ( x , y ) → ApproxSpace ( x ) ∧ UppoerBound ( y )) • Granulation mechanisms 9 / 33
Introduction Principal options Dealing with granulation hierarchies Conclusions Preliminaries • Lift rough sets’ approximation spaces up to the knowledge representation layer to conceptualise various possibilities • As a start, for the moment, recognise level, space, and bounds: • ∀ x , y ( has roughness ( x , y ) → GranularLevel ( x ) ∧ ApproxSpace ( y )) • ∀ x , y ( bound of ( x , y ) → ApproxSpace ( x ) ∧ LowerBound ( y )) • ∀ x , y ( bound of ( x , y ) → ApproxSpace ( x ) ∧ UppoerBound ( y )) • Granulation mechanisms 10 / 33
Introduction Principal options Dealing with granulation hierarchies Conclusions Preliminaries cG sG nG sgG saG nrG nfG naG part_of, ER Clustering containment sgrG sgpG saoG samG nacG nasG cell wall as coin Map of second, Team as aggregate collection of line, lipid bi- separator earth with minute, hour of its players phone points, layer, 3-D more/less landline, structure isotherms mobile [Kee08] 11 / 33
Introduction Principal options Dealing with granulation hierarchies Conclusions Basic ontological commitments A. For each ApproximationSpace , that ApproximationSpace has some LowerBound if and only if that ApproximationSpace has some UpperBound. B. Each ApproximationSpace has at most one LowerBound. Each ApproximationSpace has at most one UpperBound. Each ApproximationSpace has some LowerBound or has some UpperBound. C. 12 / 33 D. ∅
Introduction Principal options Dealing with granulation hierarchies Conclusions A. This is common practice for scale-based granularity, ∀ x , y , z ( ApproxSpace ( x ) → (( bound of ( y , x ) ∧ LowerBound ( y )) ↔ ( bound of ( z , x ) ∧ UpperBound ( z )))) But one can also have a method of granulation where specifying just one bound suffices; e.g., sgG type of granularity based on physical sizes B. To fix problem with option A leads, we get option B: if one has an approximation space, then at least one of the bounds must be specified ∀ x ( ApproxSpace ( x ) → ∃ y , z (( bound of ( y , x ) ∧ LowerBound ( y )) ∨ ( bound of ( z , x ) ∧ UpperBound ( z )))) 13 / 33
Introduction Principal options Dealing with granulation hierarchies Conclusions C. Combine A & B and demand that for each level there must be exactly one fully defined approximations space ∀ x ( GanularLevel ( x ) → ∃ ! y has roughness ( x , y )) Obviously does not hold for all types of granularity because one can identify granularity also for non-rough crisp data, information, and knowledge D. Enforce that the space is there and that it is not empty, thus excluding the option that the target set may be crisp 14 / 33
Introduction Principal options Dealing with granulation hierarchies Conclusions Basic ontological commitments 15 / 33
Introduction Principal options Dealing with granulation hierarchies Conclusions Basic ontological commitments A. GranularLevel contains Set. For each Set, at most one GranularLevel contains that Set. It is possible that the same GranularLevel contains more than one Set. Set has ApproximationSpace. Each Set has at most one ApproximationSpace. For each ApproximationSpace, at most one Set has that ApproximationSpace. B. For each Set, at most one of the following holds: that Set is some ApproximationSpace; that Set is some GrSet; that Set is some LowerBound; that Set is some UpperBound. 16 / 33
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