Granular KnowledgeRepresentation and Inference using

Labels and Label Expressions

Jonathan Lawry

j.lawry@bris.ac.uk

Dept. Engineering Mathematics,

University of Bristol,

Bristol, BS8 1TR, UK

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 1/37

Overview

Information granules and granular models.

Representing imprecise labels.

A prototype interpretation of labels.

Imprecise probabilities from imprecise descriptions.

Applications: Case Studies.

Conclusion.

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 2/37

Information Granules

Fundamental to human communication is the ability toeffectively describe the continuous domain of sensoryperception in terms of a finite set of description labels.

Granular modelling permits us to process and transmitinformation efficiently at an appropriate level of detail.

Information Granule [Zadeh] A granule is a clump ofobjects (points) which are drawn together byindistinguishability, similarity, proximity and functionality.

Information Granule [Lawry] An information granule is acharacterisation of the relationship between a discretelabel or expression and elements of the underlying(often continuous) domain which it describes.

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 3/37

Granular Models

Granular Modelling aims to develop formalrepresentations on information granules and embedthese into computational models (intelligent systems).

Granular models (Zadeh) are high-level (oftenrule-based) models which incorporate concepts asrepresented by information granules.

Mechanisms for representing and processinguncertainty and linguistic vagueness are fundamental.

Model transparency and accuracy are dual goals.

Traceability of decision processes is vital in manyapplications.

Key problem areas are: learning, fusion and reasoning.

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 4/37

Deciding What to Say...

Describing the world requires us to make decisionsabout what is the best choice of words when referring toobjects and instances, in order to convey theinformation we intend.

Suppose you are witness to a robbery, how should youdescribe the robber so that police on patrol in thestreets will have the best chance of spotting him?

You will have certain labels that can be applied, forexample tall, short, medium, fat, thin, blonde, etc, but whichare more appropriate (according to convention)?

We contend that this is fundamentally an epistemicproblem (Williamson).

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 5/37

The Epistemic Stance

Within a population of communicating agents, individuals assumethe existence of a set of labelling conventions for the populationgoverning what linguistic labels and expression can be appropriatelyused to describe particular instances.

Making an assertion to describe an object or instance x

involves making a decision as to what labels can beappropriately used to describe x.

An individual’s knowledge of labelling conventions ispartial and uncertain.

This will result in uncertainty about the appropriatenessof labels to describe instance x.

In accordance with De Finetti it is reasonable to modelthis uncertainty using subjective probabilities.

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 6/37

Measures of Appropriateness

As a (big) simplification...

Assume that there is a finite set of labelsLA = L1, . . . , Ln for describing elements of theuniverse Ω.

LE is the set of expressions generated from LA throughrecursive application of the connectives ∧,∨ and ¬.

LA = red, blue, green, yellow . . .LE = red&blue, not yellow, green or not yellow . . .

For θ ∈ LE, x ∈ Ω, µθ (x) = the subjective probabilitythat θ is appropriate to describe x.

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 7/37

Mass Functions

Dx is the complete set of labels appropriate to describex.

Dx = red, pink means that both red and pink can beappropriately used to describe x, and no other labelsare appropriate.

mx : 2LA → [0, 1] is a probability mass function onsubsets of labels.

For F ⊆ LA mx(F ) is the subjective probability thatDx = F .

The mass function mx and the appropriatenessmeasure µ are strongly related...

µθ(x) is the sum of mx over those values for Dx

consistent with θ.Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 8/37

General Relationships

‘x is θ’ requires that Dx ∈ λ (θ) where ∀θ, ϕ ∈ LE

∀L ∈ LA λ(L) = F ⊆ LA : L ∈ F

λ(θ ∧ ϕ) = λ(θ) ∩ λ(ϕ)

λ(θ ∨ ϕ) = λ(θ) ∪ λ(ϕ)

λ(¬θ) = λ(θ)c

This results in the following equation relating mx as µ:

µθ(x) = P (Dx ∈ λ(θ)) =∑

S∈λ(θ) mx(S)

λ(red ∧ ¬pink) = F ⊆ LA : red ∈ F, pink 6∈ F

µred∧¬pink(x) =∑

F :red∈F,pink 6∈F mx(F )

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 9/37

Some Links ...

...to probability For a fixed x µθ(x) forms a probabilitymeasure on LE as θ varies.

If |= θ then ∀x, µθ(x) = 1

If θ ≡ ϕ then ∀x, µθ(x) = µϕ(x)

∀θ,∀x, µ¬θ(x) = 1 − µθ(x)

If |= ¬(θ ∧ ϕ) then µθ∨ϕ(x) = µθ(x) + µϕ(x)

...to DS Theory For conjunctions and disjunctions oflabels appropriateness measure can be interpreted asCommonality and Plausibility functions.

µL1∧...∧Lk(x) =

∑F :L1,...,Lk⊆F mx(F ) = Q(L1, . . . , Lk|x)

µL1∨...∨Lk(x) =

∑F :F∩L1,...,Lk6=∅ mx(F ) = Pl(L1, . . . , Lk|x)

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 10/37

Assuming Consonance

Focal Sets Fx = F ⊆ LA : mx(F ) > 0.

Consonance For F, F ′ ∈ Fx then either F ⊆ F ′ orF ′ ⊆ F .

Ordering Labels Assume that for each x ∈ Ω an agentfirst identifies a total ordering on the appropriateness oflabels. They then evaluate their belief values mx aboutwhich labels are appropriate to describe x in such away so as to be consistent with this ordering.

LE∧,∨ = expressions only involving the connectives ∧or ∨.

Under Consonance ∀θ, ϕ ∈ LE∧,∨, ∀x ∈ Ω it holds thatµθ∧ϕ(x) = min(µθ(x), µϕ(x)) andµθ∨ϕ(x) = max(µθ(x), µϕ(x))

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 11/37

Consonance and Functionality

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

small medium large

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

µL1(x), . . . , µLn(x) ordered so that

µLi(x) ≥ µLi+1

(x) for i = 1, . . . , n − 1

mx (L1, . . . , Ln) = µLn(x)

mx (L1, . . . , Li) = µLi(x) − µLi+1

(x)

and mx (∅) = 1 − µL1(x)

mx (s)

mx (s, m)

mx (m)

mx (m, l)

mx (l)

µm∧¬l (x)

min (µm (x) , 1 − µl (x))

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 12/37

Prototype Theory

The membership of elements in a concept isdetermined by their similarity to certain prototypicalcases (Rosch 1973).

Prototypes may be actual exemplars of the concept(case-based reasoning) or abstractions or aggregations(clustering).

Voronoi Model Neighbourhood Model

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 13/37

Conceptual Spaces: NCS Colour SpindleHue

Chromaticness

Orange

Red

Violet Blue

Green

Yellow

Grey

B

W

Bl R

YG

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 14/37

A Prototype Interpretation

Let d : Ω2 → [0,∞) be a distance function satisfyingd(x, x) = 0 and d(x, y) = d(y, x).

For S, T ⊆ Ω let d(T, S) = infd(x, y) : x ∈ S, y ∈ T.

For Li ∈ Ω let Pi ⊆ Ω be a set of prototypical elementsfor Li.

Let ǫ be a random variable into [0,∞) with densityfunction δ.

Li is appropriate to describe x iff d(x, Pi) ≤ ǫ.

Dǫx = Li : d(x, Pi) ≤ ǫ.

∀F ⊆ Ω mx(F ) = δ(ǫ : Dǫx = F)

Naturally satisfies consonance.

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 15/37

GeneratingDǫx

x

ǫ3ǫ2

ǫ1p1

p2

p3

p4

p5p6

p7

Identifying Dǫx; Dǫ1

x = ∅, Dǫ2x = L1, L2, Dǫ3

x = L1, L2, L3, L4

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 16/37

Appropriateness as Subsets ofǫ

∀x ∈ Ω and ∀θ ∈ LE, I(θ, x) ⊆ [0,∞) is definedrecursively by:

∀Li ∈ LA I(Li, x) = [d(x, Pi),∞)

∀θ ∈ LE I(¬θ, x) = I(θ, x)c

∀θ, ϕ ∈ LE I(θ ∨ ϕ, x) = I(θ, x) ∪ I(ϕ, x)

∀θ, ϕ ∈ LE I(θ ∧ ϕ, x) = I(θ, x) ∩ I(ϕ, x)

Theorem: ∀θ ∈ LE,∀x ∈ Ω I(θ, x) = ǫ : Dǫx ∈ λ(θ)

Corollary: ∀θ ∈ LE,∀x ∈ Ω µθ(x) = δ(I(θ, x))

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 17/37

Area under δ: Appropriateness

I(Li ∧ ¬Lj , x) = [d(x, Pi), d(x, Pj))

ǫd(x, Pi) d(x, Pj)

µLi∧¬Lj(x)

δ(ǫ)

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 18/37

Area under δ: Mass

mx(F ) = δ([maxd(x, Pi) : Li ∈ F, mind(x, Pi) : Li 6∈ F))

ǫ

δ(ǫ)

d(x, P1) d(x, P2) d(x, P3) d(x, P4)

mx(∅)

mx(L1)

mx(L1, L2)

mx(L1, L2, L3)

mx(L1, L2, L3, L4)

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 19/37

Neighbourhood Representation

Let N ǫLi

= x ∈ Ω : d(x, Pi) ≤ ǫ, more generallyN ǫ

θ = x : Dǫx ∈ λ(θ)

Satisfies N ǫθ∧ϕ = N ǫ

θ ∩N ǫϕ, N ǫ

θ∨ϕ = N ǫθ ∪N ǫ

ϕ,N ǫ

¬θ = (N ǫθ )c.

For θ ∈ LE∧,∨ N ǫθ is nested, otherwise not in general.

Alternative characterisation: µθ(x) = δ(ǫ : x ∈ N ǫθ)

Labels represent sets of points sufficiently similar toprototypes (Information Granules).

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 20/37

Example (Uniform δ)Let Ω = R and d(x, y) = ||x − y||. Let Li = about [a, b]where a ≤ b, so that Pi = [a, b].

N ǫLi

= [a − ǫ, b + ǫ]

Let δ be the uniform distribution on [k, r] for 0 ≤ k < r:

a b

r r

k k

1

µLi(x)

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 21/37

Random Sets

A random set is simply a set valued variable.

Random sets are known to be a unifying concept inuncertainty theory e.g. DS theory, possibility theory,fuzzy theory all have random set interpretations.

∀x ∈ Ω Dx is a random set into 2LA (i.e. taking sets oflabels as values).

∀θ ∈ LE N ǫθ is a random set into 2Ω (i.e. taking subsets

of the underlying universe Ω as values).

Single Point Coverage (of Dx) P (Li ∈ Dx) = µLi(x)

(of N ǫθ ) P (x ∈ N ǫ

θ ) = µθ(x)

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 22/37

Imprecise Probabilities

Suppose an agent receives the information ‘x is θ’ whatinformation does this tell them about x?

According to our prototype model, the agent can inferx ∈ N ǫ

θ

From this they can define upper and lower probabilities:For S ⊆ Ω

P (S|θ) = P (N ǫθ ∩ S 6= ∅) = δθ(ǫ : N ǫ

θ ∩ S 6= ∅)

P (S|θ) = P (N ǫθ ⊆ S) = δθ(ǫ : N ǫ

θ ⊆ S)

Single point coverage:spc(x|θ) = P (x ∈ N ǫ

θ ) = δθ(ǫ : x ∈ N ǫθ) ∝ µθ(x)

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 23/37

Possibility Measures

If θ ∈ LE∧,∨ then for 0 ≤ ǫ ≤ ǫ′ we have that N ǫθ ⊆ N ǫ′

θ .

In this case the resulting upper and lower probabilitiesare possibility and necessity measures (Dubois, Prade).

P (S|θ) = Pos(S|θ) = supµθ(x) : x ∈ S

P (S|θ) = Nec(S|θ) = 1 − supµθ(x) : x ∈ Sc

With properties...

Pos(S ∪ T |θ) = max(Pos(S|θ), Pos(T |θ))

Nec(S ∩ T |θ) = min(Nec(S|θ), Nec(T |θ))

Hence spc(x|θ) = µθ(x) is a possibility distribution(Zadeh)

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 24/37

Given a Prior...

Suppose that agent also has knowledge in the form of aprior distribution on Ω.

Given ‘x is θ’ they should determine the posteriorp(x|N ǫ

θ ).

But since ǫ is uncertain this results in second orderprobabilities.

If precise probabilities are requires one solution wouldbe to take an expected value:

p(x|θ) =∫ ∞0 p(x|N ǫ

θ )dǫ.

This satisfies p(S|θ) ∈ [P (S|θ), P (S|θ)]

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 25/37

Example (Fuzzy Numbers)

Let Ω = R and d(x, y) = ||x − y||. Let Li = about 2 so thatPi = 2. In this case N ǫ

Li= [2 − ǫ, 2 + ǫ]

Let δ be the uniform distribution on [0, 1].

Let the prior p on Ω correspond to the uniformdistribution on [0, 10].

0 1 2 3 4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

µLi(x)

Appropriateness

0 1 2 3 4

0

1

2

3

p(x|Li)

Posterior

0 1 2 3 4

0

0.2

0.4

0.6

0.8

1.0

F (y|Li) F (y|Li)

F (y|Li)

Envelope

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 26/37

Prototype-Based Rule-Learning

Joint work with Yongchuan Tang from ZhejiangUniversity.

Given a database of pairs (~x, y) where y = f(~x) learn aset of rules to represent mapping f .

For label Lj on y with prototypes Pj learn a ruleIF(~x is Li) THEN (y is Lj) where Pi is determined fromthe set of points ~x : (~x, y) ∈ DB and y ∈ Pj.

For input ~x then µLj(y) = µLi

(~x) = δ([d(~x, Pi),∞)).

Determine BetPy(Lj) =∑

F :Lj∈Fmy(F )|F |

Evaluate y =∑

LjBetPy(Lj)cj where cj is the average

of Pj.

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 27/37

Sunspot Database

15 20 25 30 35 40 45 50 55 605

10

15

20

25

30

The widht σ of density function δ(0,σ)

RM

SE

of s

unsp

ot tr

aini

ng d

ata Two Clusters

Three Clusters

One Cluster

|DBi| Clusters

Four Clusters

15 20 25 30 35 40 45 50 55 6020

22

24

26

28

30

32

34

The widht σ of density function δ(0,σ)

RM

SE

of s

unsp

ot te

st d

ata

Two Clusters

Three Clusters

One Cluster

|DBi| Clusters

Four Clusters

1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 19800

20

40

60

80

100

120

140

160

180

200

Year

Num

ber

of S

unsp

ots

Predicted OutputActual Output

RMSE =21.91

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

160

180

200

Actual Output

Pre

dict

ed O

utpu

t

RMSE = 21.91

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 28/37

Decision Tree ModelsDx1

Dx2Dx3

LF1 LF2 LF3

LF4

LF5 LF6 LF7

ss ∧ ¬l s, l

s ∧ l l ∧ ¬s

l

ss ∧ ¬l

s, ls ∧ l

ll ∧ ¬s

ss ∧ ¬l

s, ls ∧ l

ll ∧ ¬s

m(G|s, s) m(G|s, s, l) m(G|s, l)

m(G|s, l)

m(G|l, s) m(G|l, s, l) m(G|l, l)

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 29/37

Classification of Weather-Radar Images

The use of weather radars to estimate precipitation isan increasingly important area in hydrology.

Brightband= region of enhanced reflexivity cause byamplified echoes due to scattering of microwaves frommelting snow

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 30/37

Rules for Radar Classification

Attributes: Zh = Reflectivity factor, Zdr = Differentialreflectivity, Ldr = Linear Depolarisation Ration, H =Height of the measurement.

Rules: (DLdr= low,medium) ∧ (DH =

med., high)∧ (DZh= high)∧ (DZdr

= med., high) →Rain : 0.998, Snow : 0.002, Brightband : 0

(DLdr= medium) ∧ (DH = med., high) ∧ (DZh

=med., high) ∧ (DZdr

= low) →Rain : 0.03, Snow : 0.97, Brightband : 0

(DLdr= high) ∧ (DH = med.) ∧ (DZh

=high) ∧ (DZdr

= high) →Rain : 0.02, Snow : 0, Brightband : 0.98

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 31/37

River Severn ForecastingThe River Severn is situated in the South West ofEngland with a catchment area that spans from theCambrian Mountains in Mid Wales to the BristolChannel in England.

We focus on the Upper Severn from Abermule in Powysand its tributaries, down to Buildwas in Shropshire.

The Data consists of 13120 training examples from1/1/1998 to 2/7/1999 and 2760 test examples from8/9/2000 to 1/1/2001, recorded hourly.

Each example has 19 continuous attributes falling intotwo categories; station (water) level measurements andrain fall gauge measurements.

The forecasting problem is to predict the river level atBuildwas at time t + δ (δ = lead time).

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 32/37

Severn Catchment Map

EPNEY

FINHAM

LYDNEY

LUDLOW (TEME)

MUNLYN

LUDLOW (CORVE)

MEIFOD

WELFORD

LONGDON

KEMPSEY BIDFORD

BERRIEW

CAERSWSSWINDON

HASFIELD

KNIGHTON

PERSHORE

STOURTON

CAE HOWEL

MAESBROOK

MALEHURST

BARBOURNE

MARLBROOK

SANDHURST SANDHURST

SHARPNESS

LLANERFYLPOOL QUAY

BRANSFORD

PERDISWELL

KENILWORTH

SHUTHONGER

LLANIDLOES

PONT-Y-GAER

WYRE PIDDLE

STOURBRIDGE

HURCOTT WOOD

HENLEY RIVER

PUXTONMARSH

LEINTWARDINE

MINSTERWORTH

MYTHE BRIDGE

TRIMPLEY RES.

HEWELL GRANGEBURFORD

BRIDGE

STANFORD

DIGLIS

HAMPTON LOADE

LAKEVYRNWY

BRIDGNORTH

BUTTINGTON

CREWGREEN

WELSHBRIDGE

ASHFORD

PLYNLIMON

TEWKS. UPPER POUND

ONIBURY

ACRE LANE

STRATFORD

BRIDGNORTHBUTTINGTON

WELSHBRIDGE

LLANDRINIO

PONTROBERT

ASHLEWORTHSLATE MILL

LEIGH COURT

SEVERN STOKE

SHELLEY CLOSE

CHARLTON KINGS

NEWNHAM BRIDGE

GLOUCESTERDOCKS

DOWDESWELL WATER

CLYWEDOG RES.

TEWKSBURY

CAM P/S

SARN WEN

BROOM

RUGBY

BREDON

DOLWEN

YEATON

WALCOT

SOUTHAM

STUDLEY WARWICK

COSFORD

BURCOTE

TENBURY

EVESHAM

BEWDLEY

CHALFORD

BUILDWAS

COVENTRY

BRYNTAIL

SHIPSTON

TERNHILL

STARETON

ABERMULE

MONTFORD

FECKENHAM

DEERHURST

CAMBRIDGE

LILBOURNE

PRESTWOOD

TIBBERTON

HOOKAGATE

COLEY MILL

CHURCHOVER

HAW BRIDGE

EBLEY MILL

BROMSBERROW

COWNWY WEIR

LITTLE ALNE

OAK COTTAGE

SAXONS LODE

LLANYMYNECH

ALSCOT PARK

VYRNWYWEIR

CHUCHOVER EM

WARDS BRIDGE

WELLESBOURNE

LLANYBLODWEL

HARFORD HILL

KITES HARDWICK

BORETONBRIDGE

RHOS-Y-PENTREF

BESFORD BDGE

BURLINGTONWEIR

KIDDERMINSTER

WEDDERBURN BRIDGE

KNIGHTSFORD BRIDGE

ALNE

EATHORPE

STONELEIGH

EATHORPE

OFFENHAM

RODINGTON

STONELEIGH

CHILDSERCALL

EATON ON TERN

CRUDGINGTON

LEAMINGTON PDW

HINTON ON GREEN

SANDYFORD BRIDGEWHITLEYFORDBRIDGE

BUCKLEY FARM

BARBOURNE LOCK

10 0 10 20 30 405Kilometers

LOCATION OF PERMANENT RIVER LEVELAND FLOW MEASUREMENT STATIONS

IN THE SEVERN CATCHMENT

SOME MINOR SITES HAVE BEEN OMITTEDFOR REASONS OF CLARITY

Tidal level sites not show:AvonmouthIlfracombe

LEVEL MEASUREMENT STATIONS WITHOUT TELEMETRY

LEVEL MEASUREMENT STATIONS WITH TELEMETRY

FLOW MEASUREMENT STATIONS WITH TELEMETRY

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 33/37

36 Hours Ahead Prediction

0 500 1000 1500 2000 2500 30000

1

2

3

4

5

6

7

8

Time (t)

Riv

er L

evel

(m

)

ULID3 PredictionRiver Level at Buildwas

Test Data Set

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

Actual Level (m)

Pre

dict

ed L

evel

(m

)

ULID3 PredictionZero Error Prediction

Scatter Plot

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 34/37

Severn Rules

xAt = time t level at Abermule, xB

t = time t level at Buildwas, xMt = time t level at Meifod

Focal set low: low ∈ DxAt

∧ low ∈ DxBt

→ DxBt+36

= low

Focal set low, medium:low ∈ DxA

t

∧ medium ∈ DxBt

∧ low ∈ DxMt

→ DxBt+36

= low, medium

low ∈ DxAt

∧ medium ∈ DxBt

∧ medium ∈ DxMt

→ DxBt+36

= low, medium

medium ∈ DxAt

∧ low ∈ DxBt

∧ medium ∈ DxMt

→ DxBt+36

= low, medium

Focal set medium:medium ∈ DxA

t

∧ medium ∈ DxBt

∧ medium ∈ DxMt

→ DxBt+36

= medium

Focal set medium, high: medium:high ∈ DxA

t

∧ medium ∈ DxBt

∧ medium ∈ DxMt

→ DxBt+36

= medium, high

high ∈ DxAt

∧ medium ∈ DxBt

∧ high ∈ DxMt

→ DxBt+36

= medium, high

high ∈ DxAt

∧ high ∈ DxBt

∧ medium ∈ DxMt

→ DxBt+36

= medium, high

medium ∈ DxAt

∧ high ∈ DxBt

∧ high ∈ DxMt

→ DxBt+36

= medium, high

Focal set high: high ∈ DxAt

∧ high ∈ DxBt

∧ high ∈ DxMt

→ DxBt+36

= high

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 35/37

Conclusions

Granular modelling aims to provide high-level linguistic(rule-based) models for complex systems.

The approach balances the requirements of predictiveaccuracy and model transparency.

We have proposed random set approach to modelimprecise labels consistent with an epistemic theory ofvagueness.

A prototype theory interpretation has been alsointroduced.

A number of case study applications of granularmodelling have been described.

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 36/37

Shameless Advertising

Modelling and Reasoning with Vague Concepts byJonathan Lawry, Springer 2006

Granular Knowledge Representation and Inference using Labels and Label Expressions – p. 37/37