Malcolm J. Beynon
Cardiff Business School
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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• Fuzzy Statistical Analysis
1
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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
...
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)( 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