Stochastic Lanczos estimation of genomic variance components for linear mixed-effects models

BMC Bioinformatics

Table 1 Time complexity of stochastic algorithms

Method	Overhead	Objective function evaluation
SLDF_REML\(\left \{\begin {array}{l}\text {with precomputed GRM}\\ \text {with genotype matrix}\end {array}\right.\)	\(\mathcal {O}{\left (n^{2}\cdot (n_{\text {rand}}+c)\cdot n_\kappa \right)}\)	\(\mathcal {O}{(n\cdot c\cdot n_\kappa)}\)
	\(\mathcal {O}(2m\cdot n\cdot (n_{\text {rand}}+c)\cdot n_\kappa)\)	\(\mathcal {O}(n\cdot c\cdot n_\kappa)\)
L_FOMC_REML	\(\mathcal {O}(4m\cdot n\cdot n_{\text {rand}}\cdot n_\kappa)\)	\(\mathcal {O}(m\cdot n\cdot n_{\text {rand}})\)
BOLT_LMM	\(\mathcal {O}\left (n\cdot c^{2}+m\cdot c\right)\)	\(\mathcal {O}(4m\cdot n\cdot n_{\text {rand}}\cdot n_\kappa)\)

n denotes the number of individuals, m the number of markers, and c the number of covariates. n_rand indicates the number of random probing vectors and is fixed at 15 in all numerical experiments. n_κ reflects the number of conjugate gradient iterations required to achieve convergence at a specified tolerance and can be bounded in terms of the spectral condition number of H₀. As noted in [8], implicit preconditioning of H₀ can be achieved by including the first few right singular vectors of the genotype matrix (or eigenvectors of the GRM) as covariates

ISSN: 1471-2105