## 5 Conclusions  
We developed an extinction-based multi-objective evolutionary approach to solve QTL mapping problem. The Genetic Algorithm searches the powerset of marker pairs along with the powerset of  
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.  
individual markers in order to identify both additive and epistatic genes that control a quantitative trait. Effectively searching such a large space and identifying biologically relevant results is challenging for many reasons. First, there is a need to avoid the premature convergence to local minima solutions by the GA. In this work the method of population extinction is used to ensure continued population diversity. Second, in order to obtain solutions that can be of use for biologists and truly represent the underlying genetic mechanisms, parsimonious solutions must be found. Here, another minimization objective is introduced into the search that measures the number of parameters of solutions, and multi-objective optimization performed. Third, the search engine must not only identify the markers that contribute to the quantitative trait, but also their magnitude and whether they have negative or positive influence. To solve this problem and identify the parameter values, the search engine applies PLS to fit the best predictive function over only those markers in each potential GA solution.  
Using the methodology from [35] experiments were performed demonstrating the effectiveness of the new method in solving realistic QTL problem instances with hundreds of samples and complex gene interactions. In each case the results obtained were compared with results from the Bayesian Interval Mapping method using Windows QTL Cartographer (WQTLC) Ver. 2.5 [36]. The GA algorithm consistently identified additional correct marker locations which were missed by the Bayesian Interval Mapping approach even when the data set was relatively large. The problem of false positives where markers that do not exist in the problem are identified incorrectly is minimized in the new approach. Unlinked markers that exist away from the correct markers were never found once the data set exceeded 100 samples. Linked markers that lie nearby were still incorrectly identified, but at a lower rate than the Bayesian Interval Mapping approach employed by WQTLC. These errors are of less consequence since they can still provide useful information to the scientists.  
Significantly, exploring the trade off between model complexity and accuracy using the multi-optimization approach provides insights into the QTL problem not available in many other approaches. Having the Pareto optimal frontier available to the scientists studying the QTL data enables them to select the solution that best matches their biological objective. As discussed in the results, solutions that under-fit the data often identify the most influential markers for the trait and point the scientists to those genes whose modification could produce the most benefit for the least cost. Additionally, the most accurate solution complexity (between over and under fitting the data) is apparent from the Pareto optimal frontier by observing the point where increases in solution complexity lead to insignificant improvements in accuracy.  
An important and almost universal problem in modeling biological data is selecting the true features from the multitude of irrelevant features masking the desired model. This work presents a hybrid approach to solving this problem where a GA searching the powerset of features is combined with a statistical approach that measures the significance and utility of each generated feature set.