Sisteme de programe pentru timp real

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Sisteme de programepentru timp real

• Universitatea Politehnica din Bucuresti
• 2004-2005
• Adina Magda Florea
• http//turing.cs.pub.ro/sptr_05

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Curs Nr. 11

• Learning decision rules by evolutionary
  algorithms
• Decision rules
• Representation
• Genetic operators
• Fitness function

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1. Decision rules

• Both discrete and continuous values attributes
• Rules of the form
• if A1v1
• and x1ltA2ltv2
• then Class C1
• Learning set Ee1,..eM
• e?E is described by N atributes A1, ..AN and
  labeled by a class c(e) ?C
• Discrete value attributes finite set V(Ai)
• Continuous value attributes an interval V(Ai)
  li,ui

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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 t1 and t2 .. 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

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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 -gt prefers
  rules consisting of fewer conditions, which cover
  many ex and very few ex-

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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

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Representation

• Discrete value attributes binary flags
• Continous value attributes li, ui, liltAiltui
  (?, - ?)
• li, ui are selected from a finite set of boundary
  thresholds
• Boundary threshold midpoint between a
  successive apir of examples in the sequence
  sorted by increasing value of Ai, that one is ex
  and the other ex-
• ex ex ex ex- ex- ex- ex ex ex- ex- Ai
• thik-1 thik thik1
• If a condition is not present li - ?, ui ?

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Representation

• Example
• 2 cont val attr Salary, Amount
• 1 disc val attr Purpose (car, house, school)
• Class - Accept
• Salary Amount Purpose
• - ? ? - ? 250 1 1 1 if Amountlt250
  then ACCEPT
• 100 250 - ? 500 1 1 1 if 100ltsalary250
• and Amount lt500
• then ACCEPT
• 750 ? - ? ? 1 1 0 if Salarygt750
  then ACCEPT
• and Purpose (car or house)
• then ACCEPT

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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 RS1 and
  RS2
• Crossover
• Rule copy

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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 liltAiltui because uiltAi(ex) smallest
  ui such as uigtAi similar if ligt Ai(ex)

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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 lt Ai ui is narrowed down either to
  li lt Ai lt ui or to li lt Ai lt ui, where li is
  the smallest boundary threshold such that Ai(e-)
  gt li and ui is the largest boundary threshold
  such that ui lt Ai(e-).

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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 RS1, a
  copy of a rule selected at random from RS2,
  provided that the number of rules in RS1 is lower
  than maxR.
• maxR is an user-supplied parameter, which limits
  the maximal number of rules in the ruleset.

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Genetic operators

• Crossover
• The crossover operator selects at random two
  rules R1 and R2 from the respective arguments RS1
  and RS2. Then it applies an uniform crossover to
  the strings representing R1 and R2.
• Uses rank-based fitness asignment

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4. Fitness

• Goal reduction of the no. of errors
• ERS the set of ex covered by the RS (class of
  RHS CRS)
• ERS 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)
• NEGE-(CRS) M POS
• The RS correctly classifies
• - pos ERS ex
• and
• - NEG-neg ex- where negE-RS

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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) / (POSNEG)
• Compl(RS) (L/N1)?
• 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

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