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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 (Soft Computing)

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Malcolm J. Beynon

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

  2. 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)

  3. 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)

  4. 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

  5. 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

  6. 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

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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

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

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

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

  16. 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

  17. 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

  18. Fuzzy Queuing (Example) C1,L C1,U Pardo and Fuente (2007) Optimizing a priority-discipline queueing model using fuzzy set theory, CaMwA

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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. Fuzzification of State Characteristics I

  26. 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

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

  28. 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)”

  29. 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

  30. 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

  31. 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

  32. 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

  33. 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

  34. 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

  35. 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

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

  37. 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

  38. 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

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

  40. 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

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

  42. Objective Functions with CaRBS x ¬x OB1 x ¬x OB2 OB2

  43. Objective Functions with CaRBS x ¬x OB1 x ¬x OB2

  44. Ranking Results with CaRBS

  45. 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

  46. 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

  47. 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

  48. 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

  49. RCaRBS (Graphical)

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

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