# Quantitative Trait Loci Mapping Problem: An Extinction-Based Multi-Objective Evolutionary Algorithm Approach

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## Abstract

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## 1. Introduction

## 2. QTL Mapping Problem Description

#### 2.1. Statistical PLS Methods

#### 2.2. Genetic Algorithms QTL Solution Methods

## 3. Solving QTL by Multi-Criteria Optimization

#### 3.1. QTL Solution Representation

#### 3.2. Multi-objective Optimization Methods

- The solution S1 is no worse than S2 in all objectives.
- The solution S1 is strictly better than S2 in at least one objective.

#### 3.3. Extinction-Based Evolutionary Algorithm

#### 3.4. Hybrid Genetic Algorithm for QTL

**Table 1.**

**Algorithm Parameters:**Parameters used for the Modified Extinction Evolutionary Algorithm (MEEA).

population size | 200 |

mutation probability | 0.05 |

crossover operator | uniform crossover |

crossover probability | 0.90 |

genome representation | binary vectors |

stopping criterion | max iterations (200) |

- Generate the initial population as uniformly random binary vectors.
- Evaluate each solution for both objectives (accuracy and complexity).
- Rank each solution based on the number of dominating solution in the population.
- Compute ($Fi{t}^{\prime}$) according to Equation (7) for each solution.
- Eliminate individuals from the population when $Fi{t}^{\prime}$ less than the stress factor.
- For each elimination apply tournament selection to identify two extant solutions.Crossover these solutions with crossover probability. Replace the elimination with the offspring if created, or one of the extant solutions.Mutate the new individual with probability mutation probability.Evaluate the new individual.
- If the termination condition is not satisfied go to the step 3.

## 4. Experimental Results and Analysis

#### 4.1. Experiment Case I

**Figure 1.**

**Algorithm Sample Run:**The $-{R}^{2}$ value as a function of algorithm iteration of the best individual for an arbitrary simulation.

**Figure 2.**

**Algorithm Sample Run:**The complexity as a function of algorithm iteration of the best individual for an arbitrary simulation.

**Figure 3.**

**Case I Pareto Front:**showing the relation between $-{R}^{2}$ and complexity for Case 1. The values are average of 10 runs of 100 simulation replicates.

**Figure 4.**

**Case I Identified Markers:**Identified markers found in different complexity Pareto optimal solutions for Case I. The color is the proportion of random executions that the specific marker was found. Bright red signifies that in each run the marker was consistently found. Blue means that the marker did not appear in any solutions.

**Table 2.**

**Case I: MOEA vs. Bayesian Interval Mapping:**Comparative results of the MOEA approach vs. Bayesian Interval Mapping method using Windows QTL Cartographer version 2.5 with complexity of 7. The number in the tables is the average of the count of parameters found. So the best result for the Correct column would be 7.0. Ex-L is the number of extraneous linked QTL and Ex-UL is the number of extraneous unlinked QTLs. The best result for these columns would be 0.0.

DataSet Size | Bayesian Interval Mapping | MOEA Method | ||||
---|---|---|---|---|---|---|

Correct | Ex-L | Ex-UL | Correct | Ex-L | Ex-UL | |

100 | 2.16 ± 1.06 | 0.82 ± 0.44 | 0.41 ± 0.37 | 4.54 ± 1.77 | 1.08 ± 0.52 | 0.08 ± 0.05 |

200 | 5.21 ± 0.80 | 0.36 ± 0.15 | 0.15 ± 0.12 | 6.77 ± 0.93 | 0.21 ± 0.17 | 0.0 ± 0.0 |

300 | 6.69 ± 0.28 | 0.33 ± 0.24 | 0.02 ± 0.02 | 6.92 ± 0.11 | 0.06 ± 0.06 | 0.0 ± 0.0 |

#### 4.2. Experiment Case II

**Figure 5.**

**Case II Pareto Front:**Pareto front showing the relation between $-{R}^{2}$ and complexity for Case II. The values are average of 10 runs of 100 simulation replicates.

**Figure 6.**

**Case II Markers Identified:**in different complexity Pareto optimal solutions for Case II. The color is the proportion of random executions that the specific marker was found. Bright red signifies that in each run the marker was consistently found. Blue means that the marker did not appear in any solutions.

**Table 3.**

**Case II: MOEA vs. Bayesian Interval Mapping**: Comparative results of the MOEA approach vs. Bayesian Interval Mapping method using Windows QTL Cartographer version 2.5 with complexity of 7.

DataSet Size | Bayesian Interval Mapping | MOEA Method | ||||
---|---|---|---|---|---|---|

Correct | Ex-L | Ex-UL | Correct | Ex-L | Ex-UL | |

100 | 2.65 ± 0.99 | 1.23 ± 0.84 | 0.29 ± 0.22 | 4.06 ± 0.75 | 1.15 ± 0.40 | 0.11 ± 0.10 |

200 | 4.90 ± 1.24 | 0.70 ± 0.63 | 0.13 ± 0.15 | 6.26 ± 0.81 | 0.44 ± 0.33 | 0.03 ± 0.06 |

300 | 6.36 ± 0.30 | 0.29 ± 0.16 | 0.07 ± 0.10 | 6.84 ± 0.18 | 0.17 ± 0.09 | 0.0 ± 0.0 |

#### 4.3. Experiment Case III

**Figure 7.**

**Case II Pareto Front:**showing the relation between $-{R}^{2}$ and complexity for Case III. The values are average of 10 runs of 100 simulation replicates.

**Figure 8.**

**Case III: Markers Identified:**in different complexity Pareto optimal solutions for Case III. The color is the proportion of random executions that the specific marker was found. Bright red signifies that in each run the marker was consistently found. Blue means that the marker did not appear in any solutions.

## 5. Conclusions

## Acknowledgements

## Conflicts of Interest

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Ghaffarizadeh, A.; Eftekhari, M.; Esmailizadeh, A.K.; Flann, N.S.
Quantitative Trait Loci Mapping Problem: An Extinction-Based Multi-Objective Evolutionary Algorithm Approach. *Algorithms* **2013**, *6*, 546-564.
https://doi.org/10.3390/a6030546

**AMA Style**

Ghaffarizadeh A, Eftekhari M, Esmailizadeh AK, Flann NS.
Quantitative Trait Loci Mapping Problem: An Extinction-Based Multi-Objective Evolutionary Algorithm Approach. *Algorithms*. 2013; 6(3):546-564.
https://doi.org/10.3390/a6030546

**Chicago/Turabian Style**

Ghaffarizadeh, Ahmadreza, Mehdi Eftekhari, Ali K. Esmailizadeh, and Nicholas S. Flann.
2013. "Quantitative Trait Loci Mapping Problem: An Extinction-Based Multi-Objective Evolutionary Algorithm Approach" *Algorithms* 6, no. 3: 546-564.
https://doi.org/10.3390/a6030546