Wednesday, April 3, 2019

Genetic Algorithms II: Operators

Target audience: Advanced
Estimated reading time: 6'

In the realms of science and engineering, genetic algorithms serve dual purposes: as adaptable algorithms addressing real-world challenges and as computational representations of natural evolutionary mechanisms. 

This article stands as the subsequent chapter in our series on genetic algorithms. Within, we detail the Scala-based implementation of genetic operators, encompassing selection, cross-over, and mutation, acting upon a chromosome population.

Table of contents
Follow me on LinkedIn
 

Introduction

In the first article on genetic algorithms, you learned about the key elements of genetic algorithms.Genetic Algorithms I: Foundation
  • Chromosomes
  • Genes
  • Quantization
  • Population
This second part introduces the concept and implements of genetic operations (cross-over, mutation and selection) on a population of chromosomes. These operators are applied recursively on each chromosome and each of the genes it contains.
The 3rd and final post on genetic algorithms, explores the application of genetic algorithm as a solver or optimizer Genetic Algorithms III: Solver

Note: For the sake of readability of the implementation of algorithms, all non-essential code such as error checking, comments, exception, validation of class and method arguments, scoping qualifiers or import is omitted


Selection

The first genetic operator of the reproduction cycle is the selection process. The select method of the class Population implements the steps of the selection of the fittest chromosomes in the population in the most efficient manner, as follows:

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
def select(score: Chromosome[T] => Unit, cutOff: Double) = {
   val cumul = chromosomes./:(0.0)(
      (s,x) =>{ score(xy); s + xy. unfitness} 
   ) 

   chromosomes foreach( _ /= cumul) 
   val newChromosomes = chromosomes.sortWith(_.unfitness < _.unfitness)
   val cutOffSize = (cutOff*newChromosomes.size).floor.toInt
   val newPopSize = if(limit<cutOffSize) limit else cutOffSize

   chromosomes.clear
   chromosomes ++= newChromosomes.take(newPopSize)
}

The select method computes the cumulative sum of an unfitness value, cumul, for the entire population (lines 2 -3). It normalizes the unfitness of each chromosome (line 6), orders the population by decreasing value (line 7), and applies a soft limit function on population growth, cutOff (line 8). The last step reduces the size of the population to the lowest of the two limits: the hard limit, limit, or the soft limit,cutOffSize (line 9).

Cross-over

There are several options to select pairs of chromosomes for crossover. This implementation ranks the chromosomes by their fitness value and then divides the population into two halves. Finally, it pairs the chromosomes of identical rank from each half as illustrated in the following diagram: 


Population cross-over
The crossover implementation, +- , selects the parent chromosome candidates for crossover using the pairing scheme described earlier.

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
def +- (xOver: Double): Unit = {
 if( size > 1) {
   val mid = size>>1
   val bottom = chromosomes.slice(mid, size) 
   
   val gIdx = geneticIndices(xOver)
   val offSprings = chromosomes.take(mid)
      .zip(bottom)
      .map(p => p._1 +-(p._2, gIdx))
      .unzip

   chromosomes ++= offSprings._1 ++ offSprings._2
 }
}

The implementation of the cross-over on the population of chromosomes ranked by their unfitness consists of
  • Get the mid point of the list of ranked chromosomes (line 3)
  • Get the least fit half of the chromosome (line 4)
  • Retrieve the position of the bit in the chromosome the cross-over applies (line 6)
  • Retrieve the two offspring by crossing over pairs of chromosomes from each half of the ranked population (lines 7 - 10)
  • Add the two off-springs to the current population (line 12)

Chromosome cross-over
The implementation of the crossover for a pair of chromosomes using hierarchical encoding follows two steps:
  • Find the gene on each chromosome that corresponds to the crossover index, gIdx.chOpIdx, and then swap the remaining genes (line 6)
  • Split and splice the gene crossover at xoverIdx (lines 8 & 12)

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
def +-(
   that: Chromosome[T], 
   gIdx: GeneticIndices
): (Chromosome[T], Chromosome[T]) = {
 
    val xoverIdx = gIdx.chOpIdx
    val xGenes = spliceGene(gIdx, that.code(xoverIdx) )
 
    val offSprng1 = code.slice(0, xoverIdx) ::: xGenes._1 
        :: that.code.drop(xoverIdx+1)

    val offSprng2 = that.code.slice(0, xoverIdx) ::: xGenes._2 
        :: code.drop(xoverIdx+1)
 
   (Chromosome[T](offSprng1), Chromosome[T](offSprng2)
}

The crossover method computes the index of the bit that defies the crossover xoverIdx in each parent chromosome. The genes code(xoverIdx) and that.code(xoverIdx) are swapped and spliced by the spliceGene method to generate a spliced gene (lines 9 - 13).

The method spliceGene is implemented below.

 
def spliceGene(gIdx: GeneticIndices, thatCode: T): (T, T) = {
     ((this.code(gIdx.chOpIdx) +- (thatCode, gIdx)),
      (thatCode +- (code(gIdx.chOpIdx), gIdx)) )
}

Gene cross-over
The crossover is applied to a gene through the +- method of the Gene class. The exchange of bits between the two genes this and that uses the BitSet Java class to rearrange the bits after the permutation (lines 4 - 6). The bit string is then decoded to produce the predicate or logical representation of the gene (line 8).

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
def +- (that: Gene, idx: GeneticIndices): Gene = {
    val clonedBits = cloneBits(bits)
  
    Range(gIdx.geneOpIdx, bits.size).foreach(n =>
        if( that.bits.get(n) ) clonedBits.set(n)
        else clonedBits.clear(n)
    ) 
    val valOp = decode(clonedBits)

    Gene(id, valOp._1, valOp._2)
}

Mutation 

Population mutation
The mutation operator ^ invokes the same operator for all the chromosomes in the population and then adds the mutated chromosomes to the existing population, so that they can compete with the original chromosomes. 
The mutation parameter mu is used to compute the absolute index of the mutating gene, geneticIndices(mu). We use the notation ^ to define the mutation operator to remind the reader that the mutation is implemented by flipping one bit:

 
def ^ (mu: Double): Unit =
     chromosomes ++= chromosomes.map(_ ^ geneticIndices(mu)) 
 

Chromosome mutation
The implementation of the mutation operator ^ on a chromosome consists of mutating the gene of the index gIdx.chOpIdx and then updating the list xs of genes in the chromosome. The method returns a new chromosome with this new generic code that is added to the population.

def ^ (gIdx: GeneticIndices): Chromosome[T] = {
    val mutated = code(gIdx.chOpIdx) ^ gIdx

    val xs = Range(0, code.size).map(
        i => if(i==gIdx.chOpIdx) mutated else code(i)
    ).toList

    Chromosome[T](xs)
}

Gene mutation
Finally, the mutation operator flips (XOR) the bit at the index gIdx.geneOpIdx
The ^ method mutates the cloned bit string, clonedBits (line 2) by flipping the bit at the index gIdx.geneOpIdx (line 3). It decodes line 5) and converts the mutated bit string by converting it into a (target value, operator) tuple (line 7). The last step creates a new gene from the target-operator tuple.

1
2
3
4
5
6
7
8
def ^ (gIdx: GeneticIndices): Gene = {
     val clonedBits = cloneBits(bits)
     clonedBits.flip(gIdx.geneOpIdx)

     val valOp = decode(clonedBits)

     Gene(id, valOp._1, valOp._2)
}

This concludes the second part of the implementation of genetic algorithms in Scala, dedicated to genetic operators.

Thank you for reading this article. For more information ...

References


---------------------------
Patrick Nicolas has over 25 years of experience in software and data engineering, architecture design and end-to-end deployment and support with extensive knowledge in machine learning. 
He has been director of data engineering at Aideo Technologies since 2017 and he is the author of "Scala for Machine Learning" Packt Publishing ISBN 978-1-78712-238-3

No comments:

Post a Comment

Note: Only a member of this blog may post a comment.