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

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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)

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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)

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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

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VPRS X 1 = {o 1 }, X 2 = {o 2, o 5, o 7 }, X 3 = {o 3 }, X 4 = {o 4 } and X 5 = {o 6 } Y M = {o 1, o 2, o 3 } and Y F = {o 4, o 5, o 6, o 7 } Beynon (2001) Reducts within the Variable Precision Rough Set Model: A Further Investigation, EJOR objsc1c1 c2c2 c3c3 c4c4 c5c5 c6c6 d1d1 o1o1 111111M o2o2 101011M o3o3 001100M o4o4 111001F o5o5 101011F o6o6 000110F o7o7 101011F

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VPRS X 1 = {o 1 }, X 2 = {o 2, o 5, o 7 }, X 3 = {o 3 }, X 4 = {o 4 } and X 5 = {o 6 } Y M = {o 1, o 2, o 3 } and Y F = {o 4, o 5, o 6, o 7 } Beynon (2001) Reducts within the Variable Precision Rough Set Model: A Further Investigation, EJOR objsc1c1 c2c2 c3c3 c4c4 c5c5 c6c6 d1d1 o1o1 111111M o2o2 101011M o3o3 001100M o4o4 111001F o5o5 101011F o6o6 000110F o7o7 101011F

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

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VPRS R1: If c 4 = 0 and c 5 = 0 then d 1 = F, S = 1 C = 1 P = 1 R2: If c 5 = 1 then d 1 = F, S = 5 C = 3 P = 0.6 R3: If c 4 = 1 then d 1 = M, S = 1 C = 1 P = 1 Beynon (2001) Reducts within the Variable Precision Rough Set Model: A Further Investigation, EJOR objsc1c1 c2c2 c3c3 c4c4 c5c5 c6c6 d1d1 o1o1 111111M o2o2 101011M o3o3 001100M o4o4 111001F o5o5 101011F o6o6 000110F o7o7 101011F

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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

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VPRS Competition Commission

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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

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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

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Triangular and piecewise membership functions Series of membership functions (linguistic terms) – forming linguistic variable Fuzzy Set Theory

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Membership function and Inverse Graphical Representation Fuzzy Set Theory (Example)

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Fuzzy Statistical Analysis. Carlsson and Fuller (2001) On possibilistic mean value and variance of fuzzy numbers, FSS Fuzzy Set Theory (Example)

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Fuzzy Queuing (Example) A fuzzy queuing model with priority discipline (2) Arrival rate = [26, 30, 32] Service rate = [38, 40, 45] = [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

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Fuzzy Queuing (Example) A fuzzy queuing model with priority discipline Arrival rate = [26, 30, 32]Service rate = [38, 40, 45] CLCL CUCU Pardo and Fuente (2007) Optimizing a priority-discipline queueing model using fuzzy set theory, CaMwA

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Fuzzy Queuing (Example) C 1,L C 1,U Pardo and Fuente (2007) Optimizing a priority-discipline queueing model using fuzzy set theory, CaMwA

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Fuzzy Queuing (Example) Fuzzy Statistical Analysis. Carlsson and Fuller (2001) On possibilistic mean value and variance of fuzzy numbers, FSS

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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

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Fuzzy Ecological Footprint Transactions matrix for three sector economy $m except Land input AgricManufServFDExportsTotal Output Agriculture451585525148 Manufacturing2330422520140 Services15251040595 Value added45553020 Imports2015510 Total inputs14814095 Land input (ha)140002000100, 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

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Fuzzy Ecological Footprint A =, =. l i,j = 0 u i,j = 2m i,j Beynon and Munday (2008) Considering the Effects of Imprecision and Uncertainty in Ecological Footprint Estimation: An Approach in a Fuzzy Environment, EE

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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

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.., 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

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Fuzzification of State Characteristics I

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StateKYMN CharValueFuzzy ValuesTermValueFuzzy ValuesTerm C12[0.788, 0.212, 0.000]Low7[0.000, 0.000, 1.000]High C26[0.000, 1.000, 0.000]Medium5[0.762, 0.238, 0.000]Low C310[0.966, 0.034, 0.000]Low45[0.000, 0.880, 0.120]Medium C4 7.5 [0.143, 0.857, 0.000]Medium 33.1 [0.847, 0.153, 0.000]Low C511.7[0.179, 0.821, 0.000]Medium11.1[0.589, 0.411, 0.000]Low C617.76[0.000, 0.000, 1.000]High10.52[1.000, 0.000, 0.000]Low C718587[1.000, 0.000, 0.000]Low25579[0.000, 0.151, 0.849]High C85.89[0.000, 0.823, 0.177]Medium6.76[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

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

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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)”

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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.

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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.

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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

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Mr Jones killed by assassin, = {Peter, Paul, Mary} W1; 80% sure it was a man, body of evidence (BOE), m 1 ( ), has m 1 ({Peter, Paul}) = 0.8. Remaining value to ignorance, m 1 ({Peter, Paul, Mary}) = 0.2 W2; 60% sure Peter on a plane, so BOE m 2 ( ), m 2 ({Paul, Mary}) = 0.6, m 2 ({Peter, Paul, Mary}) = 0.4 Combining evidence, create a BOE m 3 ( ); m 3 ({Paul}) = 0.48, m 3 ({Peter, Paul}) = 0.32, m 3 ({Paul, Mary}) = 0.12, m 3 ({Peter, Paul, Mary}) = 0.08 DST (Example)

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Mr Jones killed by assassin, = {Peter, Paul, Mary} W1; 80% sure it was a man, body of evidence (BOE), m 1 ( ), has m 1 ({Peter, Paul}) = 0.8. Remaining value to ignorance, m 1 ({Peter, Paul, Mary}) = 0.2 W2; 60% sure Peter on a plane, so BOE m 2 ( ), m 2 ({Paul, Mary}) = 0.6, m 2 ({Peter, Paul, Mary}) = 0.4 Combining evidence, create a BOE m 3 ( ); m 3 ({Paul}) = 0.48, m 3 ({Peter, Paul}) = 0.32, m 3 ({Paul, Mary}) = 0.12, m 3 ({Peter, Paul, Mary}) = 0.08 DST (Example)

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Mr Jones killed by assassin, = {Peter, Paul, Mary} W1; 80% sure it was a man, body of evidence (BOE), m 1 ( ), has m 1 ({Peter, Paul}) = 0.8. Remaining value to ignorance, m 1 ({Peter, Paul, Mary}) = 0.2 W2; 60% sure Peter on a plane, so BOE m 2 ( ), m 2 ({Paul, Mary}) = 0.6, m 2 ({Peter, Paul, Mary}) = 0.4 Combining evidence, create a BOE m 3 ( ); m 3 ({Paul}) = 0.48, m 3 ({Peter, Paul}) = 0.32, m 3 ({Paul, Mary}) = 0.12, m 3 ({Peter, Paul, Mary}) = 0.08 DST (Example)

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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

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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

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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

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

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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

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

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Objective Functions with CaRBS OB1 OB2 ¬x x x OB2

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Objective Functions with CaRBS OB1 OB2 ¬x x x

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Ranking Results with CaRBS

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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

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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

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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

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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

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RCaRBS (Graphical)

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

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

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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

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WELCOME TO THE WORLD OF FUZZY SYSTEMS. DEFINITION Fuzzy logic is a superset of conventional (Boolean) logic that has been extended to handle the concept.

WELCOME TO THE WORLD OF FUZZY SYSTEMS. DEFINITION Fuzzy logic is a superset of conventional (Boolean) logic that has been extended to handle the concept.

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