One single nucleotide of A, C, G or T in the DNA sequence is replace by any other 3 nucleotides, e.g., from CCCTAC to CCTTAC, we call this variation (C à T) as single nucleotide polymorphism (SNP). It has the following three characteristics: 1) very common in the human genome (it is estimated that the number of SNPs in the human genome is about 12 million); 2) among the SNPs, two out of every three SNPs are the variation from cytosine (C) to thymine (T); 3) very stable from generation to generation.

Disease association study is to identify factors that may contribute to a medical condition by comparing the cases (subjects who have that condition) with controls (subjects who do not have the condition).

Assume that we have m samples (each sample has n SNPs/features and the disease status).

Let denote the value of each SNP, where 0 and 1 stand for homozygous sites with major and minor allele, respectively, and 2 stands for heterozygous sites.

We use to represent an individual sample, where, (−1 stands for case and +1 stands for control), ,.

Let represent the feature set, where correspond to the n SNPs.

The goal of this study is to select a least redundant feature subset with predicting power for disease status. Then the problem can be described as the following model.

EDAs have been proposed for the purpose of overcoming some drawbacks that classical EAs presented. The advantage of EDAs over classical EAs is the reduction in the number of parameters to be tuned. Univariate Marginal Distribution Algorithm (UMDA) presented for the first time by Mühlenbein is one of the simplest algorithms in the EDAs family [7]. The UMDA sorts M individuals by their fitness value from high to low, and then selects N best ones to estimate the probability distribution at each generation. The probability distribution for the l-th iteration is defined as follow. The pseudocode of UMDA is presented in Figure 1.

where n is the number of variables, and

In UMDA, lack of diversity, particularly during the latter stages of the optimization, is the dominant factor for converging to local optimum solutions. To improve the capability of UMDA, multi-population UMDA (MPUMDA) is proposed in this paper. MPUMDA is an extension of traditional UMDA by dividing the population into three sub-groups (best, middle and worst) according to their performances. Each sub-group executes UMDA separately. At the end of each generation, sub-groups are reassigned to ensure information sharing.

The largest problems encountered in setting up the SVM model are how to select the kernel function and its parameters [8]. However, optimizing feature subsets and SVM model parameters respectively may not be reached the good results. Therefore, we use the MPUMDA to optimize the SVM model parameters and feature subsets simultaneously.

1) Individual representation

An individual of MPUMDA comprises two parts: the features mask and the SVM model parameters. The part of SVM model parameters consists of two sub-parts: kernel function type and other parameters. Figure 2 shows the representation of an individual with N dimensions.

In Figure 2, represent features mask. is the number of SNPs. For the feature mask, the bit with value ‘1’ indicates the SNP is selected and “0” indicates not.

represent kernel function types.

And represent the related kernel parameter and the penalty parameter C.

2) Fitness definition

Classification accuracy is often used as a criterion to design fitness function. However, due to the difficulty in obtaining clinical information on patients, the data used in disease association studies are often imbalanced. The traditional accuracy measure may be misleading since the all-negative classifier may achieve a good measure value. Thus, the is used as the fitness function which is able to deal with imbalanced dataset.

In this subsection, we focus on the selection of the best combination of SNPs with maximal difference between case and control groups. Three datasets used in this study are summarized in Table 1.

Firstly, in order to verify the effectiveness of MPUMDA-SVM, we compared it with sequential forward search (SFS), UMDA with SVM and the method proposed in [8] (Table 2). The numbers of individuals and iterations were set to 60 and 100, respectively. It is clearly that, the accuracy of SVM with feature selection outperformed that without feature selection. This means that not all features are necessary to achieve total classification accuracy. The classification accuracy on Crohn’s disease data set obtained by MPUMDA-SVM was 93.3%, whereas, only 81.4%, 84.8% and 89.6% was obtained by SFS, UMDA or PSO with SVM, respectively. For the Lung cancer and Tick-borne

encephalitis datasets, the classification accuracy obtained by our method (87.9% and 96.4%) was also better than that of other algorithms. Therefore, we may draw the conclusion that the MPUMDA is able to select more relevant SNPs.

Figure 3 shows that the number of SNPs can be reduced by MPUMDA-SVM. The percentage of SNPs selected was reduced to 0.068 and 0.171 for the Crohn’s disease and Tick-borne encephalitis datasets, respectively. However, for the Lung cancer dataset, the percentage of SNPs selected was only reduced to 0.588. That can be explained by the fact that Lung cancer is a more complex disease than the other two.

Finally, we did some further analysis using the selected SNPs. Figure 4 and Figure 5 shows the frequencies of 0, 1 and 2 in the selected SNPs for above three data sets. From Figure 4 we can see that at 57, 86 and 95 the frequency of 2 in the controls group is higher than that in cases group. 1’s frequency is almost equal to zero at 9 and 85 in case and control group respectively. Those 7 SNPs combination is mostly association with Crohn’s disease. We can see from the Figure 5 that almost in all the sites 0’s frequencies are higher in case group than controls. In addition, 1’s frequency is zero in SNP 1, which can be regarded as a risk factor of tick-borne encephalitis.