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Probabilistic and Possibilistic Graphical Models in Complex Applications

Probabilistic and Possibilistic Graphical Models in Complex Applications. Research Group „Computational Intelligence“ in Magdeburg. Main Research Topic: Intelligent Data Analysis Intelligent Data Analysis with different methods such as Neuronal Networks, Fuzzy Systems und Bayes-Methods

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Probabilistic and Possibilistic Graphical Models in Complex Applications

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  1. Probabilistic and Possibilistic Graphical Modelsin Complex Applications

  2. Research Group „Computational Intelligence“ in Magdeburg • Main Research Topic: Intelligent Data Analysis • Intelligent Data Analysis with different methods such as Neuronal Networks, Fuzzy Systems und Bayes-Methods • Development of the Data Mining Platform „InformationMiner“ • Current Industrial Projects • Item Planning with Markov-Networks (Volkswagen) • Information Mining (BMW, Daimler Chrysler) • Bayes-Methods in Finance (Several German Banks) • Implementation of new Data Analysis Methods (British Telecom)

  3. Marketing Strategies in Automotive Industry STRATEGY OF VW GROUP prefer individual vehicle specifications by customers bestseller-oriented vehicle specifications by car maker Marketing strategy very large number of possible variants low number of possible variants Complexity Vehicle specification 2,8L 150 kW spark Item short back Type alpha leather, Type L3 ...... yes 4 body variant door layouts seat covering vanity mirror ...... Item family engine radio

  4. Example: „Golf“ Class of Vehicles • approximately 200 item families (variables) • from 2 to 50 items in each family • i.e. more than possible vehicle specifications • choice of valid specifications is restricted by RULE SYSTEMS (10.000technical rules, even more marketing-, and production-oriented) Example (technical rules that restrict validity of item combinations) if then if and then

  5. Problem Representation Rules for the validity of item combinations (specified for a vehicle class and a planning interval) Sample of produced vehicle specifications (representative choice, context-dependent, f.e. Golf) System of rules Historical Data If engine = e1 and auxiliary heater = h2 then generator in {g3,g4,g5} ... (Golf, short back, 2.8 L 150 kW spark engine, radio alpha, ...) ... ? predicted / assigned planning data (production program, demands, installation rates, capacity restrictions, ... bills of material, ...) Prediction Planning

  6. Result of Problem Analysis • Handling rules: Modelling Constraints • Handling historical data: Learning from Data • Combining the different sources: Fusion of Models • Supporting planning: Belief Change These types of problems were treated in the three big EC Projects: DRUMS 1, DRUMS 2 and FUSION Our recommendation was to use Probabilistic Graphical Models, see e.g.

  7. Software Environment? 2. SPSS + Information Miner 1. Statistic Package 3. Specialized Bayesian Networks Software +Consultants Decision at VW: Version 3

  8. Problem Representation Rules for the validity of item combinations (specified for a vehicle class and a planning interval) Sample of produced vehicle specifications (representative choice, context-dependent, f.e. Golf) System of rules Historical Data If engine = e1 and auxiliary heater = h2 then generator in {g3,g4,g5} ... (Golf, short back, 2.8 L 150 kW spark engine, radio alpha, ...) ... ? predicted / assigned planning data (production program, demands, installation rates, capacity restrictions, ... bills of material, ...) Prediction Planning

  9. Basic Ideas: A Toy Example Example World Relation color shape size                   small medium small medium medium large medium medium medium large 3 variables, 36 item combinations here: 10 simple geometric objects

  10. Item Combinations Geometric Interpretation Relation color shape size                    small medium small medium medium large medium medium medium large    large medium small Each cube represents one tuple

  11. Projections         large large medium medium small small         large large medium medium small small

  12. Rule Systems: Use Material Implication Rules : Rule scheme : (A,B)

  13. Cylindrical Extensions and Their Intersection  Intersecting the cylindrical extensions of the projection to the subspace formed by color and shape and of the projection to the subspace formed by shape and size yields the original three-dimensional relation.    large medium small         large large medium medium small small

  14. Focussing Let it be known (e.g. from an observation) that the given object is green. This information considerably reduces the space of possible valuecombinations. From the prior knowledge it follows that the given object must be- either a triangle or a square and- either medium or large         large large medium medium small small

  15. Focussing with Projections The same result can be obtained using only the projections to the subspaces without reconstructing the original three-dimensional space:   s m l color size extend shape project  project extend        s m l This justifies a network representation color shape size

  16. Graphical Models • Relational Graphical Model  Decomposition + Local Model Example size size colour shape colour shape graph hypergraph Operations: Focussing ,…

  17. Transformation into Hypertree Structure B D A A D B E C C E H G F J G H F J Hypergraph Undirected Graph Interpretation as a conditional independence graph:

  18. Transformation into Hypertree Structure A B C Triangulation B D E B D A B C E E C H G F C E G J C G H J E F G Loss of some information: Sceleton of Tree of Cliques (hypertree structure)

  19. Tree Of Cliques ( VW Bora ) 186 variables 174 cliques max. 9 dimensions

  20. Problem Representation Rules for the validity of item combinations (specified for a vehicle class and a planning interval) Sample of produced vehicle specifications (representative choice, context-dependent, f.e. Golf) System of rules Historical Data If engine = e1 and auxiliary heater = h2 then generator in {g3,g4,g5} ... (Golf, short back, 2.8 L 150 kW spark engine, radio alpha, ...) ... ? predicted / assigned planning data (production program, demands, installation rates, capacity restrictions, ... bills of material, ...) Prediction Planning

  21. Planning Problem: Prediction of Parts Demand Variants-related bill of material root of vehicle class specification tree ... ... ... intermediate structuring levels ... ... ... installation point variants of parts Installation condition: disjunction of item combinations Installation rates at installation point sum up to 1 EXAMPLE : > 100.000 item combinations needed in „Golf“ class

  22. Choice of the Uncertainty Calculus single-valued set-valued crisp relational probabilistic random sets uncertain Approximation by aggregation One-point-coverage possibilistic

  23. Probabilistic Graphical Model Probabilistic Graphical Model : Decomposition + Local Models Decomposition : Hypergraph on Variables C B A Local Models: Marginal Distributions of A,B and B,C that „fit together“

  24. Bayes Networks

  25. Graphical Model Historical data System of rules context-dependent sample of produced vehicle specifications context-dependent rules for the validity of item combinations context : vehicle class, planning interval Composition Decomposition Learning Probabilistic Graphical Model Modify representation Relational Graphical Model Fusion fused consistent Markov network Graphical Model

  26. Learning Graphical Models

  27. Application at the DaimlerChrysler AG • Improvement of Product Quality by Finding Weaknesses • Learn decision trees or inference network for vehicle properties and faults. • Look for unusual conditional fault frequencies. • Find causes for these unusual frequencies. • Improve construction of vehicle. • Improvement of Error Diagnosis in Garages • Learn decision trees or inference network for vehicle properties and faults. • Record properties of new faulty vehicle. • Test for the most probable faults.

  28. Analysis of Daimler/Chrysler Database Database: ~ 18.500 passenger cars > 100 attributes per car Analysis of dependencies between special equipment and faults. Results used as a starting point for technical experts looking for causes.

  29. Analysis of Daimler/Chrysler Database electrical roof top air con- ditioning type of engine type of tyres slippage control faulty battery faulty compressor faulty brakes Fictitious example: There are significantly more faulty batteries, if both air conditioningandelectrical roof top are built into the car.

  30. Example Subnet Influence of special equipment on battery faults: significant deviation from independent distribution hints to possible causes and improvements here: larger battery may be required, if an air conditioning system and an electrical sliding roof are built in (The dependencies and frequencies of this example are fictitious)

  31. Problems in Structure Learning of PGM Complexity of learning problem Exhaustive graph search in „poor“ classes Greedy search (heuristics) in „richer“ classes Dependency analysis (CI-Tests) probability maximization (Bayesian-Dirichlet) Unsufficient quality of results, need for controllable search strategies Handling „soft“ dependencies Integrability of structure knowledge

  32. Information Fusion Historical data System of rules context-dependent sample of produced vehicle specifications context-dependent rules for the validity of item combinations context : vehicle class, planning interval Use cond. independencies (Composition) Estimate prior distribution of installation rates Quantitative Learning PGM (Markov network) having the structure of the relational network Modify representation Transformation into a relational network with hypertree structure Fusion

  33. Planning Models Typical complexity: • 200 item families • 150 cliques • 5 to 7 dimensions (typical) • max. dimensions: 11 to 14 • 100 vehicle model groups • 20 to 40 planning intervals (i.e. 2000 to 4000 networks)

  34. Planning Operation : Conditioning ( Focussing) • Input Data : item combination (set of variable instantiations) • Operation : Calculate the conditioned network distribution and the probability of the given item combination (propagation). • Application : Calculation of part demands Compute the installation rate of item combination . Simulation Analyze customers‘ preferences with respect to those persons who buy a navigation system in a VW Polo.

  35. Knowledge Propagation in Trees of Cliques 1. Local computations w.r.t. cliques A B C A B C B D E B D E B C E B C E Local Operation: Conditioning Lauritzen, Spiegelhalter, 1988 Shafer, Shenoy, 1988 C E G C E G C G H J E F G C G H J E F G 2. Collect information 3. Distribute information

  36. Planning Model based on Belief Change Historical data System of rules context-dependent sample of produced vehicle specifications context-dependent rules for the validity of item combinations context : vehicle class, planning interval Use cond. independencies (Composition) Estimate prior distribution of installation rates Quantitative Learning PGM (Markov network) having the structure of the relational network Modify representation Transformation into a relational network with hypertree structure Revision Adaption of installation rates of item combinations that change from valid to invalid Updating Find referential for item combinations that change from invalid to valid Fusion fused consistent Markov network for item planning Planning Model

  37. Effiziency gain with HUGIN Example: Markov Net for VW „Bora“ • Installation Rates for 460.000 attribute combinations • Reduction of RAM from 600 MB to 16 MB(Divisor = 38) • Reduction of computing time from infeasible to 250 sec(Divisor = 80.000)

  38. Gain with efficient operations Example : Markov Net for Volkswagen Sharan First Prototyping (HUGIN) ... today ... Max. number dimensions 7 11 Number tuples 500.000 20.000.000.000 Valid Tuples 1.000.000 100.000 Full Network Propagation (Pentium 1,5 GHz) 1 sec 30 ms 14

  39. Planning Operation : Updating • Input Data : Set of item combinations that will change from invalid to valid; set of valid referential combinations • Operation : Copy dependency structure (cross-product ratios) from referential combination to input combination and initialize with -probabilties. • Application : Technical modifications The combination of engine and transmission changes from invalid to valid, and it adapts the quantitative dependencies from .

  40. Planning Operation : Revision • Input Data : Family of marginal / conditional probability distributions • Operation : Calculate Markov network with same structure that satisfies all input distributions and is conform to the principle of minimal change. • Application : Marketing stipulations Installation probability of item air condition increases by 10 % in case of Golf all-wheel drive in France. Logistic restrictions The maximum availability of engine in week 32/05 is 1.000 .

  41. Specification of Planning Data Name: Golf - No. 02/07/05 - 17 Vehicle class: Market: Germany Planning interval: 36/05 Golf Revision scheme: Engines Revision context: Short back Comfort Context scheme: Body Equipment Restriction Partitioning: Installation rates (%) estimated assigned 5,79 Group of 1,8L spark engines 9,00 ≤ 500 2,13 3,00 Diesel engine X1 (single item) 21,07 [18,20] Diesel engine X2 (single item) 71,01 Rest 6

  42. Current State of Software Development • Client-Server System (current state: software implementation and test environment for users) • Server on 6-8 Machines (16 GB each) • 4-Processor AMD Opteron system • Terabyte storage device • Operating System: Linux • up to 15 system developers • Programming language: JAVA • WebSphere Application Developer, Eclipse • DB-System: Oracle • Worldwide rollout: now 18

  43. Need for Theory / Efficient Algorithms Efficient transformation of logical rule systems into a relational network, techniques for complexity reduction and inconsistency management Consistent quantitative fusion of a prior Markov network with a dependencies modifying relational network to a new Markov network Handling generalized constraints Efficient algorithms for revision and updating Quantitative Learning PGM (Markov network) having the structure of the relational network Modify representation Transformation into a relational network with hypertree structure Revision Adaption of installation rates of item combinations that change from valid to invalid Updating Find referential for item combinations that change from invalid to valid Fusion

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