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- 1. Q-Aggregate Based Gene Expression Programming 2006-03-16 Magdi Mohamed (PRR CoE, Labs) Weimin Xiao (PRR CoE, Labs) Chi Zhou (PRR CoE, Labs) Method for Constructing Compact Infinite-Valued Logical Forms Using Gene Expression Programming with Q-Aggregate Operators 2006:03:16 Magdi A. Mohamed 1/9
- 2. Q-Aggregate Operator Impacts on soft computing and computational intelligence paradigms 1. Neural Networks • avoids “sigmoid-like” function limitations • simplifies implementations using per-unit values 2. Genetic Computing • reduces complexity of Gene Expression Programming (GEP) • provides natural interpretations of rules 3. Crisp and Soft Rule-Based Systems • opens a new paradigm shift in rule-based systems (Beyond Fuzzy Logic) • provides higher degrees of automation 2006:03:16 Magdi A. Mohamed 2/9
- 3. Closest Prior Art on GEP IEEE-TEC 12/2003, “Evolving Accurate & Compact Classification Rules with GEP”, by Chi Zhou et al. 7−S • A genetic-algorithm-based / symbolic regression algorithm F ( P + 3) (Ferreira 2001, Zhou 2003). - * • Simpler and faster than 7 S + F conventional Genetic Programming approaches. 3 • An expression tree is flattened to form a chromosome. P / - * 7 S + F 3 P 2006:03:16 Magdi A. Mohamed 3/9
- 4. The GEP Algorithm Probabilistic Search Create Chromosomes of Initial Population Create Chromosomes of Initial Population Express Chromosomes as ETs Express Chromosomes as ETs Evaluate Fitness of Expression Trees Evaluate Fitness of Expression Trees Select New Population Probabilistically Select New Population Probabilistically Crossover, Mutation, and Rotation Crossover, Mutation, and Rotation Chromosomes for New Generation Chromosomes for New Generation Yes Termination Criterion Terminate Satisfied? No 2006:03:16 Magdi A. Mohamed 4/9
- 5. Innovation Gene Expression Programming with Q-Aggregate Operators • A binary operation expressed by a unique Q-Aggregate operator characterizing disjunction, compensation, and conjunction operations: A(x,y) = (x + y + lambda * x * y) / (2 + lambda) • A unary operation expressed by a formula which generates value in the unit interval characterizing Negation operations: N(x) = (1 – x) / (1 + gamma * x) • The training task is performed as an optimization process where flexible model parameters (lambda/gamma values) are adjusted for each internal node in a general non-fixed tree structure to minimize an overall risk criterion using Gene Expression Programming and Differential Evolution techniques. 2006:03:16 Magdi A. Mohamed 5/9
- 6. Innovation Gene Expression Programming with Q-Aggregate Operators y1 y2 A A A A A N x1 x2 A 0.6 N x2 x3 0.3 N A x3 0.7 N x1 2006:03:16 Magdi A. Mohamed 6/9
- 7. Experimental Results Nonlinear Regression Case //XOR problem a=0; b=0; Y = 0; a=1; b=0; Y = 1; a=0; b=1; Y = 1; a=1; b=1; Y = 0; //Best Tree QA 0 |QA 0 ||a 1.0 ||N 0 |||b 1.0 |QA 0 ||b 1.0 ||N 0 |||a 1.0 ;PGEP Formula: ;QA(QA(a, N(b, gamma = 0), lambda= 164015206785.06702), QA(b, N(a, gamma = 0), lambda = 231379355057.6994 ), lambda = -0.99998 ) 2006:03:16 Magdi A. Mohamed 7/9
- 8. Experimental Results Nonlinear Classification Case Test Accuracy on Monk's Problems M1 M2 M3 C4.5 75.70% 65.00% 97.20% C4.5Rules 100% 66.20% 96.30% GEP 100% 99.07% 100% QAB-GEP 100% …..% 100% QAB-GEP is able to generate accurate, compact, and noise tolerant rules Rule 1: QA(a5_1, N(a2_3, gamma= 0.050061), lambda= 0.548952) Rule 2: N(QA(a2_3, a5_4, lambda= 0.221941), gamma= 0.535057) Rule 3: QA(a4_1, a5_3, lambda= 0.602964) The average size of Monks-3 rules is 4, while GEP has 14 The set of operations {QA, N} can replace the list of pre-selected functions or operators, e.g., {IF, AND, OR, +, -, *, /, …} 2006:03:16 Magdi A. Mohamed 8/9
- 9. Advantages of QAB-GEP Characteristics and Promises 1. compact set of operations { Aggregate (x,y), Negate (x) } 2. numerical stability 3. per unit calculations simplify implementations (software and hardware) 4. suitability for massive parallel implementations 5. automatic discovery of complex logical expressions 6. consistent handling of unary, binary, and n-ary operations 7. improvement over conventional GEP techniques in terms of average rule size 8. usability for both classification and regression applications 9. ease of use and interpretability 2006:03:16 Magdi A. Mohamed 9/9

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