Fig. 1.

Schematic illustration of matrices and tensors of the permGWAS architecture. (A) Commonly used matrix representation when computing sequential univariate tests, where $\mathit{y}\in {ℝ}^n$ is the phenotypic vector for n samples and ${\mathit{X}}_j\in {ℝ}^{n\times c}$ denotes the matrix of fixed effects, including a column of ones for the intercept, the covariates and the jth SNP $x_j\in {ℝ}^n$. (B) 3D-tensor representation of a LMM to compute univariate tests batch-wise. The phenotype is represented as a 3D tensor containing b copies of the phenotype vector $\mathit{y}\in {ℝ}^n$ and ${\mathit{X}}_j^b\in {ℝ}^{b\times n\times c}$ is a 3D tensor containing the matrices ${\mathit{X}}_j$ to ${\mathit{X}}_{j+b-1}$. (C) 4D-tensor representation of a permutation-based batch-wise LMM. The phenotype is represented as a 4D tensor containing for each permutation ${}^{(k)}\mathit{y}$ the 3D tensor ${}^{(k)}{\mathit{Y}}^b$ for all q permutations and ${}^q{\mathit{X}}_j^b\in {ℝ}^{q\times b\times n\times c}$ is a 4D tensor containing q copies of ${\mathit{X}}_j^b$