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Sisteme de programe pentru timp real Universitatea “Politehnica” din Bucuresti 2004 -2005 Adina Magda Sisteme de programe pentru timp real Universitatea “Politehnica” din Bucuresti 2004 -2005 Adina Magda Florea http: //turing. cs. pub. ro/sptr_05

Curs Nr. 11 Learning decision rules by evolutionary algorithms 1. 2. 3. 4. Decision Curs Nr. 11 Learning decision rules by evolutionary algorithms 1. 2. 3. 4. Decision rules Representation Genetic operators Fitness function 2

1. Decision rules • Both discrete and continuous values attributes • Rules of the 1. Decision rules • Both discrete and continuous values attributes • Rules of the form: if A 1=v 1 and x 1

Decision rules A class ck C Positive examples E+(ck) = {e E: C(e)=ck} Negative Decision rules A class ck C Positive examples E+(ck) = {e E: C(e)=ck} Negative examples – E-(ck) = E – E+(ck) A decision rule R if t 1 and t 2. . and tr then ck LHS RHS – class membership of an example Rule set RS – disjunctive set of decision rules with the asme RHS • CRS C – denotes the class on the RHS of an RS • • 4

Decision rules • The EA is called for each class ck C to find Decision rules • The EA is called for each class ck C to find the RS separating E+(ck) from E-(ck) • Search criteria – fitness function -> prefers rules consisting of fewer conditions, which cover many ex+ and very few ex- 5

2. Representation • One chromosome encodes an RS • Use variable length chromosomes (as 2. Representation • One chromosome encodes an RS • Use variable length chromosomes (as the no. of rules in RS is not knwon) + provide operators which change the no. Of rules • 1 chromosome = concatenation of strings • Each fixed-length string = the LHS of one decision rule (no need for RHS) • 1 string is composed of N sub-strings (LHS) – a condition for each attribute 6

Representation • • Discrete value attributes – binary flags Continous value attributes – li, Representation • • Discrete value attributes – binary flags Continous value attributes – li, ui, li

Representation Example • 2 cont val attr: Salary, Amount • 1 disc val attr Representation Example • 2 cont val attr: Salary, Amount • 1 disc val attr – Purpose (car, house, school) • Class - Accept Salary - + 100 250 Amount - 250 - 500 750 + - + Purpose 1 1 1 if Amount<250 then ACCEPT 1 1 1 if 100750 then ACCEPT and Purpose = (car or house) then ACCEPT 8

3. Genetic operators 4 operators applied to a single sule set: • Changing condition 3. Genetic operators 4 operators applied to a single sule set: • Changing condition • Positive example insertion • Negative example removal • Rule drop 2 operators applied with 2 arguments to RS 1 and RS 2: • Crossover • Rule copy 9

Genetic operators Changing condition • A mutation like operator – alters a single condition Genetic operators Changing condition • A mutation like operator – alters a single condition related to an attribute Ai • If Ai disc – randomly chooses a flag and flip • If Ai cont – randomly replaces a threshold (li or ui) by a boundary threshold Pos ex insertion • Modifies a single dec rule R in RS to allow to cover a new random e+ E+(CRS) currently uncovered by R • All conditions in the rule, which conflict with e+ have to be altered. • Ai disc – flag set • Ai cont – li=Ai; similar if li>= Ai(ex+) 10

Genetic operators Negative ex removal • The negative example removal operator alters a single Genetic operators Negative ex removal • The negative example removal operator alters a single rule R from the ruleset RS. • It selects at random a negative example e- from the set of all the negative examples covered by R. • Then it alters a random condition in R in such a way, that the modied rule does not cover e-. • If the chosen condition concerns a discrete attribute Ai the flag which corresponds to Ai(e-) is cleared. • If Ai is a continuous-valued attribute then the condition li < Ai ui is narrowed down either to li’ < Ai <= ui or to li < Ai <= ui’, where li is the smallest boundary threshold such that Ai(e-) >= li’ and ui’ is the largest boundary threshold such that ui’ < Ai(e-). 11

Genetic operators Rule drop and rule copy operators are the only ones capable of Genetic operators Rule drop and rule copy operators are the only ones capable of changing the number of rules in a ruleset. Rule drop • The single argument rule drop removes a random rule from a ruleset RS. Rule copy • Rule copy adds to one of its arguments RS 1, a copy of a rule selected at random from RS 2, provided that the number of rules in RS 1 is lower than max. R. • max. R is an user-supplied parameter, which limits 12

Genetic operators Crossover • The crossover operator selects at random two rules R 1 Genetic operators Crossover • The crossover operator selects at random two rules R 1 and R 2 from the respective arguments RS 1 and RS 2. Then it applies an uniform crossover to the strings representing R 1 and R 2. • Uses rank-based fitness asignment 13

4. Fitness • • • Goal = reduction of the no. of errors ERS 4. Fitness • • • Goal = reduction of the no. of errors ERS – the set of ex covered by the RS (class of RHS – CRS) E+RS = ERS E+(CRS) – set of ex+ correctly classified by RS E-RS = ERS E-(CRS) – set of ex- covered by RS The total no. of ex+ and ex. POS = |E+(CRS)| NEG=|E-(CRS)| = M – POS • The RS correctly classifies: - pos = |E+RS| ex+ and - NEG-neg ex- where neg=|E-RS| 14

Fitness • • • Ferror = Pr(RS) / Compl(RS) Pr(RS) = the probability of Fitness • • • Ferror = Pr(RS) / Compl(RS) Pr(RS) = the probability of classifying correctly an example from the learning set by RS Compl(RS) = the complexity of RS Pr(RS) = (pos + NEG – neg) / (POS+NEG) Compl(RS) = (L/N+1) L – total no. of conditions in RS N – no. of attributes - user supplied in [0. 001. . 0. 1] We are interested in maximizing the probability and minimizing the complexity – to obtain a compact rule set and acorrect classification 15