FastScanner

class glimix_core.lmm.FastScanner(y, X, QS, v)

Approximated fast inference over several covariates.

Specifically, it maximizes the marginal likelihood

p(𝐲)ⱼ = 𝓝(𝐲 | 𝚇𝜷ⱼ + 𝙼ⱼ𝜶ⱼ, 𝑠ⱼ(𝙺 + 𝑣𝙸)),

over 𝜷ⱼ, 𝜶ⱼ, and sⱼ. Matrix Mⱼ is the candidate set defined by the user. Variance 𝑣 is not optimised for performance reasons. The method assumes the user has provided a reasonable value for it.

Parameters

• y – Real-valued outcome.

• X – Matrix of covariates.

• QS – Economic eigendecomposition ((Q0, Q1), S0) of K.

• v – Variance due to iid effect.

Notes

The implementation requires further explanation as it is somehow obscure. Let 𝚀𝚂𝚀ᵀ = 𝙺, where 𝚀𝚂𝚀ᵀ is the eigendecomposition of 𝙺. Let 𝙳 = (𝚂 + 𝑣𝙸) and 𝙳₀ = (𝚂₀ + 𝑣𝙸₀), where 𝚂₀ is the part of 𝚂 with positive values. Therefore, solving

(𝙺 + 𝑣𝙸)𝐱 = 𝐲

for 𝐱 is equivalent to solving

𝚀₀𝙳₀𝚀₀ᵀ𝐱 + 𝑣𝚀₁𝚀₁ᵀ𝐱 = 𝚀₀𝙳₀𝚀₀ᵀ𝐱 + 𝑣(𝙸 - 𝚀₀𝚀₀ᵀ)𝐱 = 𝐲.

for 𝐱. Let

𝙱 = 𝚀₀𝙳₀⁻¹𝚀₀ᵀ if 𝑣=0, and 𝙱 = 𝚀₀𝙳₀⁻¹𝚀₀ᵀ + 𝑣⁻¹(𝙸 - 𝚀₀𝚀₀ᵀ) if 𝑣>0.

We therefore have

𝐱 = 𝙱𝐲

as the solution of (𝙺 + 𝑣𝙸)𝐱 = 𝐲.

Let 𝐛ⱼ = [𝜷ⱼᵀ 𝜶ⱼᵀ]ᵀ and 𝙴ⱼ = [𝚇 𝙼ⱼ]. The optimal parameters according to the marginal likelihood are given by

(𝙴ⱼᵀ𝙱𝙴ⱼ)𝐛ⱼ = 𝙴ⱼᵀ𝙱𝐲

and

𝑠 = 𝑛⁻¹𝐲ᵀ𝙱(𝐲 - 𝙴ⱼ𝐛ⱼ).

__init__(y, X, QS, v)

Initialize self. See help(type(self)) for accurate signature.

Methods

__init__(y, X, QS, v)	Initialize self.
fast_scan(M[, verbose])	LMLs, fixed-effect sizes, and scales for single-marker scan.
null_lml()
scan(M)	LML, fixed-effect sizes, and scale of the candidate set.

Attributes

null_beta	Optimal 𝜷 according to the marginal likelihood.
null_beta_covariance	Covariance of the optimal 𝜷 according to the marginal likelihood.
null_beta_se	Standard errors of the optimal 𝜷.
null_scale	Optimal s according to the marginal likelihood.