Malcolm J. Beynon

Cardiff Business School

BeynonMJ@cardiff.ac.uk

Fuzzy and Dempster-Shafer Theory based Techniques in Finance, Management and

Economics

Uncertain Reasoning• Uncertain Reasoning (Soft Computing)

“the process of analyzing problems utilizing evidence from unreliable, ambiguous and incomplete data sources”

• Associated methodologies (include)

Fuzzy Set Theory (Zadeh, 1965)Dempster-Shafer Theory (Dempster, 1967; Shafer, 1976)

Rough Set Theory (Pawlak, 1981)

Talk Direction• Rough Set Theory (Briefly)

VPRS – Competition Commission

• Fuzzy Set TheoryFuzzy Queuing Fuzzy Ecological Footprint

Fuzzy Decision Trees – Strategic Management

Antonym-based Fuzzy Hyper-Resolution (AFHR)

• Dempster-Shafer TheoryExample Connection with AFHR

Classification and Ranking Belief Simplex (CaRBS)

Rough Set Theory (RST)• Rough Set Theory (RST)

Based on indiscernibility relation

Objects classified with certainty

• Variable Precision Rough Sets (VPRS)Objects classified with at least certainty

• Dominance Based Rough Set Approach (DBRSA) Based on dominance relation

VPRS

X1 = {o1}, X2 = {o2, o5, o7}, X3 = {o3}, X4 = {o4} and X5 = {o6}

YM = {o1, o2, o3} and YF = {o4, o5, o6, o7}

Beynon (2001) Reducts within the Variable Precision Rough Set Model: A Further Investigation, EJOR

objs c1 c2 c3 c4 c5 c6 d1

o1 1 1 1 1 1 1 M

o2 1 0 1 0 1 1 M

o3 0 0 1 1 0 0 M

o4 1 1 1 0 0 1 F

o5 1 0 1 0 1 1 F

o6 0 0 0 1 1 0 F

o7 1 0 1 0 1 1 F

VPRS

X1 = {o1}, X2 = {o2, o5, o7}, X3 = {o3}, X4 = {o4} and X5 = {o6}

YM = {o1, o2, o3} and YF = {o4, o5, o6, o7}

Beynon (2001) Reducts within the Variable Precision Rough Set Model: A Further Investigation, EJOR

objs c1 c2 c3 c4 c5 c6 d1

o1 1 1 1 1 1 1 M

o2 1 0 1 0 1 1 M

o3 0 0 1 1 0 0 M

o4 1 1 1 0 0 1 F

o5 1 0 1 0 1 1 F

o6 0 0 0 1 1 0 F

o7 1 0 1 0 1 1 F

VPRS

Beynon (2001) Reducts within the Variable Precision Rough Set Model: A Further Investigation, EJOR

VPRS

R1: If c4 = 0 and c5 = 0 then d1 = F , S = 1 C = 1 P = 1

R2: If c5 = 1 then d1 = F , S = 5 C = 3 P = 0.6

R3: If c4 = 1 then d1 = M , S = 1 C = 1 P = 1Beynon (2001) Reducts within the Variable Precision Rough Set Model: A Further Investigation, EJOR

objs c1 c2 c3 c4 c5 c6 d1

o1 1 1 1 1 1 1 M

o2 1 0 1 0 1 1 M

o3 0 0 1 1 0 0 M

o4 1 1 1 0 0 1 F

o5 1 0 1 0 1 1 F

o6 0 0 0 1 1 0 F

o7 1 0 1 0 1 1 F

VPRS Competition Commission

• Findings of the monopolies and mergers commission (competition commission).

• Whether an industry was found to be acting against the public interest.

• No precedent or case law allowed for within the deliberations of the MMC.

otherwise0

MMC by the findings adversein results case theif1Remedy

Beynon and Driffield (2005) An Illustration of VPRS Theory: An Analysis of the Findings of the UK Monopolies and Mergers Commission, C&OR

Beynon and Driffield (2005) An Illustration of VPRS Theory: An Analysis of the Findings of the UK Monopolies and Mergers Commission, C&OR

VPRS Competition Commission

VPRS Rules

Beynon and Driffield (2005) An Illustration of VPRS Theory: An Analysis of the Findings of the UK Monopolies and Mergers Commission, C&OR

Fuzzy Set Theory• Its introduction enabled the practical analysis of

problems with non-random imprecision

• Well known techniques which have been developed in a fuzzy environment, include:

Fuzzy Queuing Fuzzy Decision Trees

Fuzzy Regression Fuzzy Clustering

Fuzzy Ranking

• Triangular and piecewise membership functions

• Series of membership functions (linguistic terms) – forming linguistic variable

Fuzzy Set Theory

• Membership function and Inverse

• Graphical Representation

Fuzzy Set Theory (Example)

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xx

x

otherwise.0

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• Fuzzy Statistical Analysis

1

0

1U

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1

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1U d))()((

2

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.

Carlsson and Fuller (2001) On possibilistic mean value and variance of fuzzy numbers, FSS

Fuzzy Set Theory (Example)

333.2)(M̂ 125.1)( VAR

Fuzzy Queuing (Example)• A fuzzy queuing model with priority discipline (2) 1/

~/

~MM i

Arrival rate = [26, 30, 32] ~

Service rate = [38, 40, 45] ~

1C~

2C~

= [15, 20, 22] = [2.5, 3, 5]

Costs of waiting (2 groups)

Pardo and Fuente (2007) Optimizing a priority-discipline queueing model using fuzzy set theory, CaMwA

Fuzzy Queuing (Example)• A fuzzy queuing model with priority discipline 1/

~/

~MM i

Arrival rate = [26, 30, 32] ~ Service rate = [38, 40, 45] ~

)~~(~

with~

)~~~~(

~ 12211 WWCCC

CL CU

Pardo and Fuente (2007) Optimizing a priority-discipline queueing model using fuzzy set theory, CaMwA

Fuzzy Queuing (Example)

C1,L C1,U

11L,C

11U,C

Pardo and Fuente (2007) Optimizing a priority-discipline queueing model using fuzzy set theory, CaMwA

Fuzzy Queuing (Example)• Fuzzy Statistical Analysis

1

0

11 d))()(()(M̂ U,L, CCC

1

0

211

2

1 d))()(()( L,U, CCCVAR

.

Carlsson and Fuller (2001) On possibilistic mean value and variance of fuzzy numbers, FSS

Fuzzy Ecological Footprint

,

Footprint provides estimate of the demands on global bio-capacity and the supply of that bio-capacity.

Bicknell et al. (1998) New methodology for the ecological footprint with an application to the New Zealand economy, EE

Fuzzy Ecological Footprint

Transactions matrix for three sector economy $m except Land input

Agric Manuf Serv FD Exports Total OutputAgriculture 45 15 8 55 25 148

Manufacturing 23 30 42 25 20 140Services 15 25 10 40 5 95

Value added 45 55 30 20Imports 20 15 5 10

Total inputs 148 140 95Land input (ha) 14000 2000 100

,

Footprint provides estimate of the demands on global bio-capacity and the supply of that bio-capacity.

Reference population is a nation, but can be applied to individual industries and organizations

Bicknell et al. (1998) New methodology for the ecological footprint with an application to the New Zealand economy, EE

Fuzzy Ecological FootprintA =

105.0179.0101.0

442.0214.0155.0

084.0107.0304.0

,

A

]210.0,105.0,000.0[]358.0,179.0,000.0[]202.0,101.0,000.0[

]884.0,442.0,000.0[]428.0,214.0,000.0[]310.0,155.0,000.0[

]168.0,084.0,000.0[]214.0,107.0,000.0[]608.0,304.0,000.0[

=

.

li,j = 0 ui,j = 2mi,j

307133302640

791051414530

2800273053911

...

...

...

)( AI

Beynon and Munday (2008) Considering the Effects of Imprecision and Uncertainty in Ecological Footprint Estimation: An Approach in a Fuzzy Environment, EE

Fuzzy Ecological Footprint

Beynon and Munday (2008) Considering the Effects of Imprecision and Uncertainty in Ecological Footprint Estimation: An Approach in a Fuzzy Environment, EE

..

,

Likelihood of Strategic Stance of State ‘Long Term Care Systems’ Using 13 Experts Assignment

Analyzing Public Service Strategy

Fuzzy Decision Trees

[0.000, 0.154, 0.846]

Kitchener and Beynon (2008) Analysing Public Service Strategy: A Fuzzy Decision Tree Approach, BAM

Fuzzification of State Characteristics I

State KY MNChar Value Fuzzy Values Term Value Fuzzy Values Term

C1 2 [0.788, 0.212, 0.000] Low 7 [0.000, 0.000, 1.000] High

C2 6 [0.000, 1.000, 0.000] Medium 5 [0.762, 0.238, 0.000] Low

C3 10 [0.966, 0.034, 0.000] Low 45 [0.000, 0.880, 0.120] MediumC4 7.5 [0.143, 0.857, 0.000] Medium 33.1 [0.847, 0.153, 0.000] LowC5 11.7 [0.179, 0.821, 0.000] Medium 11.1 [0.589, 0.411, 0.000] Low

C6 17.76 [0.000, 0.000, 1.000] High 10.52 [1.000, 0.000, 0.000] Low

C7 18587 [1.000, 0.000, 0.000] Low 25579 [0.000, 0.151, 0.849] HighC8 5.89 [0.000, 0.823, 0.177] Medium 6.76 [0.000, 0.500, 0.500] Medium/High

Stance [0.000, 0.154, 0.846] Reactor [0.923, 0.077, 0.000] Prospector

Fuzzification of State Characteristics II

Kitchener and Beynon (2008) Analysing Public Service Strategy: A Fuzzy Decision Tree Approach, BAM

Yuan and Shaw (1995) Induction of fuzzy decision trees, FSS

Constructed Fuzzy Decision Tree

Kitchener and Beynon (2008) Analysing Public Service Strategy: A Fuzzy Decision Tree Approach, BAM

Example Decision RulesR4: “If C1 is Low and C7 is Medium then LTC

Strategic Stance of a state is Prospector (0.248), Defender (0.907) and Reactor (0.571)”

R4: “If a state LTC system has a low number of innovative home care programs & medium state wealth then its LTC Strategic Stance is Prospector (0.248), Defender (0.907) and Reactor (0.571)”

Fuzzy Resolution Principle• Antonym-based fuzzy hyper-resolution (AFHR)

Kim et al. (2000) A new fuzzy resolution principle based on the antonym, FSS

The meaningless range is a special set, unknown, that is not true and also that is not false. This range should not be considered in reasoning.

Negation Small Not-small

Antonym Small Large

Fuzzy logic is divided into fuzzy valued logic and fuzzy linguistic valued logic.

Fuzzy Resolution Principle• Examples of AFHR

Kim et al. (2000) A new fuzzy resolution principle based on the antonym, FSS

The meaningless range is a special set, unknown, that is not true and also that is not false. This range should not be considered in reasoning.

• Methodology associated with uncertain reasoning

• Considered a generalisation of the Bayesian formulisation

• Obtaining degrees of belief for one question from subjective probabilities describing the evidence from others.

• Described in terms of mass values (belief), bodies of evidence and frames of discernment

Dempster-Shafer Theory

Mr Jones killed by assassin, = {Peter, Paul, Mary}

W1; 80% sure it was a man, body of evidence (BOE), m1(), has m1({Peter, Paul}) = 0.8. Remaining value to ignorance, m1({Peter, Paul, Mary}) = 0.2

W2; 60% sure Peter on a plane, so BOE m2(), m2({Paul, Mary}) = 0.6, m2({Peter, Paul, Mary}) = 0.4

Combining evidence, create a BOE m3();

m3({Paul}) = 0.48, m3({Peter, Paul}) = 0.32, m3({Paul, Mary}) = 0.12, m3({Peter, Paul, Mary}) = 0.08

DST (Example)

Mr Jones killed by assassin, = {Peter, Paul, Mary}

W1; 80% sure it was a man, body of evidence (BOE), m1(), has m1({Peter, Paul}) = 0.8. Remaining value to ignorance, m1({Peter, Paul, Mary}) = 0.2

W2; 60% sure Peter on a plane, so BOE m2(), m2({Paul, Mary}) = 0.6, m2({Peter, Paul, Mary}) = 0.4

Combining evidence, create a BOE m3();

m3({Paul}) = 0.48, m3({Peter, Paul}) = 0.32, m3({Paul, Mary}) = 0.12, m3({Peter, Paul, Mary}) = 0.08

DST (Example)

Mr Jones killed by assassin, = {Peter, Paul, Mary}

W1; 80% sure it was a man, body of evidence (BOE), m1(), has m1({Peter, Paul}) = 0.8. Remaining value to ignorance, m1({Peter, Paul, Mary}) = 0.2

W2; 60% sure Peter on a plane, so BOE m2(), m2({Paul, Mary}) = 0.6, m2({Peter, Paul, Mary}) = 0.4

Combining evidence, create a BOE m3();

m3({Paul}) = 0.48, m3({Peter, Paul}) = 0.32, m3({Paul, Mary}) = 0.12, m3({Peter, Paul, Mary}) = 0.08

DST (Example)

AFHR and DST

Kim et al. (2000) A new fuzzy resolution principle based on the antonym, FSS

The meaningless range is a special set, unknown, that is not true and also that is not false. This range should not be considered in reasoning.

Paradis and Willners (2006) Antonymy and negation - The boundedness hypothesis, Journal of Pragmatics

AFHR and DST

Safranek et al. (1990) Evidence Accumulation Using Binary Frames of Discernment for Verification Vision, IEEE Transactions on Robotics and Automation

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Classification and Ranking Belief Simplex (CaRBS)

• CaRBS introduced in Beynon (2005)– Operates using DST– Binary classification, discerning objects (and evidence)

between a hypothesis ({x}), not-hypothesis ({¬x}) and ignorance ({x, ¬x})

– RCaRBS to replicate regression analysis– CaRBS with Missing Values– FCaRBS moving towards fuzzy CaRBS

Beynon (2005) A Novel Technique of Object Ranking and Classification under Ignorance: An Application to the Corporate Failure Risk Problem, EJOR

Stages of CaRBS (Graphical)

)( ii vke 1

1

Beynon (2005) A Novel Technique of Object Ranking and Classification under Ignorance: An Application to the Corporate Failure Risk Problem, EJOR

Classification with CaRBS

Beynon (2005) A Novel Technique of Object Ranking and Classification under Ignorance: An Application to the Corporate Failure Risk Problem, EJOR

Classification with CaRBS

Beynon (2005) A Novel Approach to the Credit Rating Problem: Object Classification Under Ignorance, IJISAFM

Beynon (2005) A Novel Technique of Object Ranking and Classification under Ignorance: An Application to the Corporate Failure Risk Problem, EJOR

Objective Functions with CaRBS

Beynon (2005) A Novel Approach to the Credit Rating Problem: Object Classification Under Ignorance, IJISAFM

Objective Functions with CaRBS

OB1

OB2

¬x x

¬x x

OB2

Objective Functions with CaRBS

OB1

OB2

¬x x

¬x x

Ranking Results with CaRBS

Osteoarthritic Knee Analysis

Experiments to Measure Gait

Beynon et al. (2006) Classification of Osteoarthritic and Normal Knee Functionusing Three Dimensional Motion Analysis and the DST, IEEE TSMC

Osteoarthritic Knee Analysis

Evaluation of Gait Characteristic Values

Beynon et al. (2006) Classification of Osteoarthritic and Normal Knee Functionusing Three Dimensional Motion Analysis and the DST, IEEE TSMC

Osteoarthritic Knee Analysis

Classification of OA and NL subjects

Jones et al. (2006) A novel approach to the exposition of the temporal development of post-op osteoarthritic knee subjects, JoB

Osteoarthritic Knee Analysis

Progress of Total Knee Replacement Patients

Jones et al. (2006) A novel approach to the exposition of the temporal development of post-op osteoarthritic knee subjects, JoB

RCaRBS (Graphical)

RCaRBS (Graphical)

Figure 6. Simplex plot based representation of final respondent BOEs, and subsequent mappings, using configuration of RCaRBS system

CaRBS (Missing)• CaRBS allows analysis of Incomplete Data Sets – Retaining the Missing Values

Conclusions• Fuzzy Set Theory (FST)

– Existing techniques developed using FST– Techniques still need to be developed using FST

• Dempster-Shafer Theory (DST)– Less used in developing existing techniques (??)

• Soft Computing