Basic modelling choices to add roughness to granular levels C. - - PowerPoint PPT Presentation

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

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Introduction Principal options Dealing with granulation hierarchies Conclusions

Outline

Introduction Principal options Orthogonal issues Ontological commitments Dealing with granulation hierarchies Defining perspectives Examples Conclusions

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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

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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

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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

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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?

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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?

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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?

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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

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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

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Introduction Principal options Dealing with granulation hierarchies Conclusions

Preliminaries

cG nG sG nrG

part_of, containment

saG sgG naG nfG

ER Clustering

samG

second, minute, hour

saoG

Map of earth with more/less isotherms

nasG

collection of phone points, landline, mobile

nacG

Team as aggregate

• f its players

sgrG

cell wall as line, lipid bi- layer, 3-D structure

sgpG

coin separator

[Kee08] 11 / 33

Introduction Principal options Dealing with granulation hierarchies Conclusions

Basic ontological commitments

For each ApproximationSpace, that ApproximationSpace has some LowerBound if and

• nly if that ApproximationSpace has some UpperBound.

Each ApproximationSpace has at most one LowerBound. Each ApproximationSpace has at most one UpperBound. Each ApproximationSpace has some LowerBound or has some UpperBound. ∅

A. B. C. D.

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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))))

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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

• ne 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

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Introduction Principal options Dealing with granulation hierarchies Conclusions

Basic ontological commitments

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Introduction Principal options Dealing with granulation hierarchies Conclusions

Basic ontological commitments

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. 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.

A. B.

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Introduction Principal options Dealing with granulation hierarchies Conclusions

Other questions

• (Can we identify approximation space (e.g., by its lower and

upper bound)? Could it affect ontological nature of granular level?)

• For information systems, can we add the mandatory

constraint to approximation space? i.e. ∀y(ApproxSpace(y) → ∃x has roughness(x, y)) ∀y(ApproxSpace(y) → ∃x hasSpace(x, y))

• Should each level be allowed to have defined more than one

approximation space? (now permitted in options A and B)

• If so, what effect does it have on consistency with other levels

and granulation hierarchies?

• Changing space results in a different level (because of

different data in the granules that make up the level)

• Create different granulation hierarchies for each way of

computing approximation spaces

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Introduction Principal options Dealing with granulation hierarchies Conclusions

Other questions

• (Can we identify approximation space (e.g., by its lower and

upper bound)? Could it affect ontological nature of granular level?)

• For information systems, can we add the mandatory

constraint to approximation space? i.e. ∀y(ApproxSpace(y) → ∃x has roughness(x, y)) ∀y(ApproxSpace(y) → ∃x hasSpace(x, y))

• Should each level be allowed to have defined more than one

approximation space? (now permitted in options A and B)

• If so, what effect does it have on consistency with other levels

and granulation hierarchies?

• Changing space results in a different level (because of

different data in the granules that make up the level)

• Create different granulation hierarchies for each way of

computing approximation spaces

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Introduction Principal options Dealing with granulation hierarchies Conclusions

Other questions

• (Can we identify approximation space (e.g., by its lower and

upper bound)? Could it affect ontological nature of granular level?)

• For information systems, can we add the mandatory

constraint to approximation space? i.e. ∀y(ApproxSpace(y) → ∃x has roughness(x, y)) ∀y(ApproxSpace(y) → ∃x hasSpace(x, y))

• Should each level be allowed to have defined more than one

approximation space? (now permitted in options A and B)

• If so, what effect does it have on consistency with other levels

and granulation hierarchies?

• Changing space results in a different level (because of

different data in the granules that make up the level)

• Create different granulation hierarchies for each way of

computing approximation spaces

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Introduction Principal options Dealing with granulation hierarchies Conclusions

From granulation hierarchy to granular perspective

• Informally, a granular perspective is a ‘granulation hierarchy

with additional properties’, i.e. explicating and adding ‘metadata’ to taxonomies, lattices and similar

[QCLH07, CY06, SW98, BS03]

• Uniquely identify, hence, distinguish, such perspectives based
• n their semantics by using a criterion and type of granularity

used for granulation

• Link the perspectives
• Add roughness

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Introduction Principal options Dealing with granulation hierarchies Conclusions GranularPerspective GranularLevel contains precedes links TypeOfGranularity has granulation adheres to Criterion has criterion ≥2 cG sG nG Property combines ≥2 QualityProperty

….

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Introduction Principal options Dealing with granulation hierarchies Conclusions

A note on ‘attributes’ and properties

• as a relation A ⊆ U × Ud, e.g. Colour ⊆ Eye × String
• as a [total/partial] function A : U → Va (relation +

participation constraint);

• unary property; e.g. quality property with qualia (DOLCE),

the ‘dependent continuant’ Colour ‘inheres in’ the ‘independent continuant’ Eye

• not all properties are alike (primary, secondary, sortal, mass,

quality, essential, ....), and have properties themselves

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Introduction Principal options Dealing with granulation hierarchies Conclusions

Early definition

Definition (Granular perspective [Kee07])

∀x∃!w, y, z, φ such that GP(x) is a concept CN(x), has a definition DF(x, y), relates to its criterion C(z) through the relation RC(x, z), has granulation, RGp, of type TG(φ) and is contained in a domain Df (w).

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Introduction Principal options Dealing with granulation hierarchies Conclusions

Distinguishing perspectives

Lemma

A criterion C can be used with more than one perspective GP, provided the perspectives have distinct granulation types TG: ∀x1, x2, y, φ1, φ2(RC(x1, y) ∧ RC(x2, y) ∧ RGp(x1, φ1) ∧ RGp(x2, φ2) ∧ ¬(x1 = x2) → ¬(φ1 = φ2)).

Theorem

The combination of some C(y) with a TG(φ) determines uniqueness of each GP(x).

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Introduction Principal options Dealing with granulation hierarchies Conclusions

Relating perspective

• Mereology-based approach ontologically better, but not nice

for computation

• Simpler ‘linking’ of the elements, i.e., level to each other, to

perspective, and perspective to each other through RP:

Definition (RP)

RP relates two distinct perspectives: ∀x, y(RP(x, y) → GP(x) ∧ GP(y) ∧ ¬(x = y)).

Lemma

RP is irreflexive, ¬RP(x, x), and symmetric, RP(x, y) ↔ RP(y, x).

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Introduction Principal options Dealing with granulation hierarchies Conclusions

Examples

• saoG with straight-forward raster over the map: precision of

the scale of each specific level, i.e., the refinement of measurements, such as “km2 ± 1 m2” or “km2 ± 1 cm2”, where the choice for m2 or cm2 is provided by the approximation space

• sgpG with pore sizes, flux, or permeability coefficients of

semi-permeable membranes: only one bound for each level, e.g., flux, Ji, in [kmol m−2 s−1], [kg/m−2 s−1] or [m3 / m−2 s−1] as number of moles, volume, or mass of a component i passing per unit time through a unit of membrane surface area (IUPAC)

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Introduction Principal options Dealing with granulation hierarchies Conclusions

Examples: one or two bounds

Varying scales at different levels of regions as well as within-scale variations (values populating the levels are taken from maps in the Dutch “Grote Bos Atlas”): sorting

• ut the hierarchies and (in)consistency of the bounds
• Avg. July temperature (π5)
• Avg. Yearly Precipitation (π6)

(◦C) (in mm) λ1 World 0 – 10 – 20 – 30 <250 – 250-500 – 500-1000 – 1000-2000 – ≥2000 ↑ ↑ λ2 Europe (EU) <10 – 10-15 – 15-17.5 – 17.5-20 – 20-25 – ≥25 <200 – 200-400 – 400-600 – 600-800 – 800-1200 – 1200-2000 – ≥2000 ↑ ↑ λ3 Nether- lands (coun- try) 16 – 16.5 – 17 – 17.5 <750 – 750-800 – 800-850 – 850-900 – ≥900

[Kee09] 27 / 33

Introduction Principal options Dealing with granulation hierarchies Conclusions

Examples: one or two bounds

Varying scales at different levels of regions as well as within-scale variations (values populating the levels are taken from maps in the Dutch “Grote Bos Atlas”): sorting

• ut the hierarchies and (in)consistency of the bounds
• Avg. July temperature (π5)
• Avg. Yearly Precipitation (π6)

(◦C) (in mm) λ1 World 0 – 10 – 20 – 30 <250 – 250-500 – 500-1000 – 1000-2000 – ≥2000 ↑ ↑ λ2 Europe (EU) <10 – 10-15 – 15-17.5 – 17.5-20 – 20-25 – ≥25 <200 – 200-400 – 400-600 – 600-800 – 800-1200 – 1200-2000 – ≥2000 ↑ ↑ λ3 Nether- lands (coun- try) 16 – 16.5 – 17 – 17.5 <750 – 750-800 – 800-850 – 850-900 – ≥900

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Introduction Principal options Dealing with granulation hierarchies Conclusions

Examples: no(?) bounds and one bound

Conditional perspectives Admin (π3) Hydro (π4) (river with flow ≥) Country ⇔ 100 000 litres/min ↑ ↑ Province ⇔ 10 000 litres/min ↑ ↑ Region ⇔ 2500 litres/min ↑ ↑ Municipality ⇔ 1000 litres/min ↑ ↑ Municipality ⇔ 250 litres/min district

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Introduction Principal options Dealing with granulation hierarchies Conclusions

Conclusions

• Distinction between defining levels and precision of levels
• Within refinement, there are several ontological commitments
• ApproxSpace attribute to GranularLevel vs Set contained in

GranularLevel

• Changing refinement parameter or values generate a different

hierarchy

• Granular perspectives as first step to manage the hierarchies

created with varying rough sets

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Introduction Principal options Dealing with granulation hierarchies Conclusions

References

• T. Bittner and B. Smith.

Foundations of Geographic Information Science, chapter A Theory of Granular Partitions, pages 117–151. London: Taylor & Francis Books, 2003. Yao Hua Chen and Yi Yu Yao. Multiview intelligent data analysis based on granular computing. In IEEE International Conference on Granular Computing (GrC2006), volume 1, pages 281–286. IEEE Xplore, 2006. 1. Joseph P. Herbert and Jing Tao Yao. Criteria for choosing a rough set model. Computers and Mathematics with Applications, 57:908–918, 2009.

• C. Maria Keet.

Granulation with indistinguishability, equivalence or similarity. In IEEE International Conference on Granular Computing (GrC2007), volume 2, pages 11–16. IEEE Computer Society, 2007. San Francisco, November 2-4, 2007.

• C. Maria Keet.

A Formal Theory of Granularity. Phd thesis, KRDB Research Centre, Faculty of Computer Science, Free University of Bozen-Bolzano, Italy, April 2008.

• C. Maria Keet.

Structuring GIS information with types of granularity: a case study. In Proceedings of the 6th International Conference on Geomatics, 2009. La Habana, Cuba, Feb 10-12, 2009. 31 / 33

Introduction Principal options Dealing with granulation hierarchies Conclusions

References

• F. Klawonn and R. Kruse.

The inherent indistinguishability in fuzzy systems. In W. Lenski, editor, Logic versus Approximation: Essays Dedicated to Michael M. Richter on the Occasion

• f his 65th Birthday, volume 3075 of LNCS. Springer: Berlin / Heidelberg, 2004.
• T. Qiu, X. Chen, Q. Liu, and H. Huang.

A granular space model for ontology learning. In IEEE International Conference on Granular Computing (GrC’07), pages 61–65. IEEE Computer Society, 2007. San Francisco, November 2-4, 2007. Andrzej Skowron and Jaroslaw Stepaniuk. Modeling of high quality granules. In Proc. of RSEISP 2007, volume 4585 of LNAI, pages 300–309. Springer, 2007.

• J. Stell and M. Worboys.

Stratified map spaces: a formal basis for multi-resolution spatial databases. In Proceedings of the 8th International Symposium on Spatial Data Handling (SDH’98), pages 180–189. International Geographical Union, 1998.

• Y. Y. Yao.

A partition model of granular computing. LNCS Transactions on Rough Sets, 1:232–253, 2004. 32 / 33

Introduction Principal options Dealing with granulation hierarchies Conclusions

Thank you for your attention

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