Fuzzy if-then rules Associates a condition described using linguistic variables and fuzzy sets to a conclusion A scheme for capturing knowledge that involves imprecision 23.11.2010 1 fuzzy database modeling
Two types of fuzzy rules Fuzzy mapping rules Af functional mapping relationship between inputs and an output using linguistic terms Function approximation in system identification and artificial neural network Fuzzy implication rules A generalized logic implication relationship between two logic formula Related to classical two-valued logic and multivalued logic 23.11.2010 2
Fuzzy implication rules A generalization of implication in two value logic. To reason with ideas and statements that are imprecise. 23.11.2010 3
Fuzzy mapping rules Function approximation techniques Global modeling Using one mathematical structure Linear, second-order d polynomial l Local modeling Fuzzy rule-based function approximation Superimposition functions (B-splines, Taylor expansions) Partition-based approximate Partitioning the input space of the function and approximate the function in each partitioned region 23.11.2010 4
Fuzzy rule-based function Fuzzy partition approximation Allowing a sub region to partially overlap with neighboring sub region Partitioning the input space of the function 23.11.2010 5
Partial matching Calculate the matching degree between a fuzzy input A and a fuzzy condition i A 23.11.2010 6
Fuzzy rule-based models Fuzzy partition Mapping of fuzzy sub regions to local model Fusing of multiple local models Defuzzification 23.11.2010 7
Classical partition A classical partition of a space is a collection of disjoint i subspaces whose union is the entire space. 23.11.2010 8
Fuzzy partition Generalizes classical partition so that the transition from one subspace into a neighbor one is smooth. 23.11.2010 Fuzzy partition 9
Piecewise linear approximation A nonlinear global model can often be approximated dby a set of flinear local lmodels. 23.11.2010 10
Mapping of fuzzy space to local General form model Four different types of local model Crisp constant Fuzzy constant Linear model Nonlinear model 23.11.2010 11
Fusion of local models through interpolative reasoning Interpolative reasoning If-then rule1 w 1 w 2 Overall opinion If-then rule2 w 3 w 4 If-then rule3 If-then rule4 23.11.2010 12
Interpret a possibility distribution Linguistic approximation A qualitative interpretation Defuzzification A quantitative summary Mean of maximum (MOM) Center of area (COA) The height method 23.11.2010 13
Mean of maximum (MOM) Calculates the average of those output values that have the highest h possibility degrees 23.11.2010 14
Center of area (COA) Calculate the center-of-gravity (the weighted sum of the results) 23.11.2010 15
The height method 1. Convert the consequent membership function C i into crisp consequent y = c i 2. Apply the centroid defuzzification w i is the degree to which the ith rule matches the input data 23.11.2010 16
Fuzzy rule-based models Takagi-Sugeno-Kang Standard Additive Model 23.11.2010 17
If speed is med and distance is small then force is negative If speed is zero and distance is large then force is positive Mamdani model Linguistic rules Input form output membership of rule i Matching degree of rule i, condition j Aggregation of outputs from all rules 23.11.2010 18
TSK model Linguistic rules To reduce the number of rules output 23.11.2010 19
TSK model (example) 23.11.2010 20
Standard Additive Model Linguistic rules Output 23.11.2010 21
Comparison of SAM and Mamdani SAM Mamdani inputs crisp Crisp & fuzzy Composition scaling Clipping operator (min) Fusion method addition max Defuzzification Centroids Not insist 23.11.2010 22
Multiconditional Approximate Reasoning 23.11.2010 23
Step 1. Step 2. 23.11.2010 24
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