For each choice from a pair of edges, we assign a binary variable called a selector whose values are 0 if the first edge from a pair is chosen and 1 otherwise.
As a result, for the case where g i, j is a heterozygous SNP site, then the model requires that the following is satisfied: (5) (h k, j ∨ ¬ g i, j l ∨ ¬ s k, i l ) ∧ (¬ h k, j ∨ g i, j l ∨ ¬ s k, i l ), where 1 ≤ k ≤ r and 1 ≤ l ≤ p. For each i and l, it is necessary that exactly one haplotype is used, and so exactly one selector variable be assigned value 1.
In this paper, we study a variable selection method via the Dantzig selector, proposed by Candes and Tao [2007. The Dantzig selector: statistical estimation when p is much larger than n. Annals of Statistics 35, 2313 2351], to screen important effects.
As a result, for each genotype g i, the model uses selector variables for selecting which haplotypes are used for explaining g i.
Since the genotype is to be explained by p haplotypes, the model uses p sets of r selector variables, s k, i l.
The general idea of collapsing states into equivalence classes is a natural one, yet its realization in genetic linkage analysis is far less obvious because the standard way to represent linkage problems uses redundant selector variables.
In such chains, it is possible to cluster together selector variables that control the inheritance from a founder allele in the pedigree, indicating only if the allele is transmitted to the individual at the end of the chain or, if not, what is the number of selectors that block its inheritance.
To improve the stability and robustness of proposed method, an ensemble variable selection framework is designed which ensemble the results of many L1 regularization selectors.
For simplicity of notation, we assume that H i = {0, 1} and so each S i j is a binary variable, which we call a selector, and the state-space size is 2 k.
