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Fuzzy and Dempster-Shafer Theory based Techniques in Finance, Management and Economics. Malcolm J. Beynon. Cardiff Business School BeynonMJ@cardiff.ac.uk. Uncertain Reasoning. Uncertain Reasoning (Soft Computing)

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malcolm j beynon

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

Malcolm J. Beynon

Cardiff Business School

BeynonMJ@cardiff.ac.uk

uncertain reasoning
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
Talk Direction

Rough Set Theory (Briefly)

VPRS – Competition Commission

Fuzzy Set Theory

Fuzzy Queuing Fuzzy Ecological Footprint

Fuzzy Decision Trees – Strategic Management

Antonym-based Fuzzy Hyper-Resolution (AFHR)

Dempster-Shafer Theory

Example Connection with AFHR

Classification and Ranking Belief Simplex (CaRBS)

rough set theory rst
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 b

Dominance Based Rough Set Approach (DBRSA)

Based on dominance relation

slide5
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

slide6
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

slide7
VPRS

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

slide8
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 = 1

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

vprs competition commission
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.

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

slide10

VPRS Competition Commission

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

vprs rules
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
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

fuzzy set theory1
Fuzzy Set Theory
  • Triangular and piecewise membership functions
  • Series of membership functions (linguistic terms) – forming linguistic variable
slide15

Fuzzy Set Theory (Example)

  • Fuzzy Statistical Analysis

.

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

slide16

Fuzzy Queuing (Example)

  • A fuzzy queuing model with priority discipline (2)

Arrival rate

= [26, 30, 32]

Service rate

= [38, 40, 45]

Costs of waiting (2 groups)

= [15, 20, 22]

= [2.5, 3, 5]

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

slide17

Fuzzy Queuing (Example)

  • A fuzzy queuing model with priority discipline

Arrival rate = [26, 30, 32]

Service rate = [38, 40, 45]

CL

CU

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

slide18

Fuzzy Queuing (Example)

C1,L

C1,U

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

slide19

Fuzzy Queuing (Example)

  • Fuzzy Statistical Analysis

.

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

numbers, FSS

slide20

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

slide21

Fuzzy Ecological Footprint

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

Transactions matrix for three sector economy $m except Land input

,

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

slide22

Fuzzy Ecological Footprint

A =

li,j = 0

ui,j = 2mi,j

,

.

=

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

slide23

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

slide24

Fuzzy Decision Trees

Analyzing Public Service Strategy

[0.000, 0.154, 0.846]

Likelihood of Strategic Stance of State ‘Long

Term Care Systems’

Using 13 Experts

Assignment

,

.

.

Kitchener and Beynon (2008) Analysing Public Service Strategy: A Fuzzy

Decision Tree Approach, BAM

slide26

Fuzzification of State Characteristics II

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

Kitchener and Beynon (2008) Analysing Public Service Strategy: A Fuzzy

Decision Tree Approach, BAM

constructed fuzzy decision tree
Constructed Fuzzy Decision Tree

Kitchener and Beynon (2008) Analysing Public Service Strategy: A Fuzzy

Decision Tree Approach, BAM

example decision rules
Example Decision Rules

R4: “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
Fuzzy Resolution Principle

Antonym-based fuzzy hyper-resolution (AFHR)

Antonym Small

Large

Negation Small

Not-small

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

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.

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

fuzzy resolution principle1
Fuzzy Resolution Principle

Examples of AFHR

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.

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

slide31
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

slide32

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}) = 0.2W2; 60% sure Peter on a plane, so BOE m2(), m2({Paul, Mary}) = 0.6, m2({Peter, Paul, Mary}) = 0.4Combining 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

slide33

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}) = 0.2W2; 60% sure Peter on a plane, so BOE m2(), m2({Paul, Mary}) = 0.6, m2({Peter, Paul, Mary}) = 0.4Combining 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

slide34

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}) = 0.2W2; 60% sure Peter on a plane, so BOE m2(), m2({Paul, Mary}) = 0.6, m2({Peter, Paul, Mary}) = 0.4Combining 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

slide35

AFHR and DST

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.

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

Paradis and Willners (2006) Antonymy and negation - The boundedness

hypothesis, Journal of Pragmatics

slide36

AFHR and DST

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

slide37

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

slide38

Stages of CaRBS (Graphical)

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

slide39

Classification with CaRBS

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

slide40

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

slide41

Objective Functions with CaRBS

Beynon (2005) A Novel Approach to the Credit Rating Problem: Object

Classification Under Ignorance, IJISAFM

slide45

Osteoarthritic Knee Analysis

Experiments to Measure Gait

Beynon et al. (2006) Classification of Osteoarthritic and Normal Knee Function

using Three Dimensional Motion Analysis and the DST, IEEE TSMC

slide46

Osteoarthritic Knee Analysis

Evaluation of Gait Characteristic Values

Beynon et al. (2006) Classification of Osteoarthritic and Normal Knee Function

using Three Dimensional Motion Analysis and the DST, IEEE TSMC

slide47

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

slide48

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

slide50

RCaRBS (Graphical)

Figure 6. Simplex plot based representation of final respondent BOEs,

and subsequent mappings, using configuration of RCaRBS system

slide51

CaRBS (Missing)

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