LMM

class glimix_core.lmm.LMM(y, X, QS=None, restricted=False)

Fast Linear Mixed Models inference via maximum likelihood.

Examples

>>> from numpy import array
>>> from numpy_sugar.linalg import economic_qs_linear
>>> from glimix_core.lmm import LMM
>>>
>>> X = array([[1, 2], [3, -1]], float)
>>> QS = economic_qs_linear(X)
>>> covariates = array([[1], [1]])
>>> y = array([-1, 2], float)
>>> lmm = LMM(y, covariates, QS)
>>> lmm.fit(verbose=False)
>>> print('%.3f' % lmm.lml())
-3.649

One can also specify which parameters should be fitted:

>>> from numpy import array
>>> from numpy_sugar.linalg import economic_qs_linear
>>> from glimix_core.lmm import LMM
>>>
>>> X = array([[1, 2], [3, -1]], float)
>>> QS = economic_qs_linear(X)
>>> covariates = array([[1], [1]])
>>> y = array([-1, 2], float)
>>> lmm = LMM(y, covariates, QS)
>>> lmm.fix('delta')
>>> lmm.fix('scale')
>>> lmm.delta = 0.5
>>> lmm.scale = 1
>>> lmm.fit(verbose=False)
>>> print('%.3f' % lmm.lml())
-3.832
>>> lmm.unfix('delta')
>>> lmm.fit(verbose=False)
>>> print('%.3f' % lmm.lml())
-3.713

Notes

The LMM model can be equivalently written as

𝐲 ∼ 𝓝(𝚇𝜷, 𝑠((1-𝛿)𝙺 + 𝛿𝙸)),

and we thus have v₀ = s (1 - 𝛿) and v₁ = s 𝛿. Consider the economic eigendecomposition of 𝙺:

𝙺 = [𝚀₀  𝚀₁] [𝚂₀  𝟎] [𝚀₀ᵀ]
             [ 𝟎  𝟎] [𝚀₁ᵀ]

and let

𝙳 = [(1-𝛿)𝚂₀ + 𝛿𝙸₀ 𝟎 ]

[ 𝟎 𝛿𝙸₁].

In order to eliminate the need of 𝚀₁, note that 𝚀𝚀ᵀ = 𝙸 implies that

𝚀₁𝚀₁ᵀ = 𝙸 - 𝚀₀𝚀₀ᵀ.

We will need to solve ((1-𝛿)𝙺 + 𝛿𝙸)𝐱 = 𝐮 for 𝐱. Let 𝙳₀ = ((1-𝛿)𝚂₀ + 𝛿𝙸₀) and let us define

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

We therefore have

𝐱 = 𝙱𝐮.

__init__(y, X, QS=None, restricted=False)

Constructor.

Parameters

• y (array_like) – Outcome.

• X (array_like) – Covariates as a two-dimensional array.

• QS (tuple) – Economic eigendecompositon in form of ((Q0, ), S0) of a covariance matrix K.

• restricted (bool) – True for restricted maximum likelihood optimization; False otherwise. Defaults to False.

Methods

__init__(y, X[, QS, restricted])	Constructor.
covariance()	Covariance of the prior.
fit([verbose])	Maximise the marginal likelihood.
fix(param)	Disable parameter optimization.
get_fast_scanner()	Return FastScanner for association scan.
gradient()	Not implemented.
lml()	Log of the marginal likelihood.
mean()	Mean of the prior.
unfix(param)	Enable parameter optimization.
value()	Internal use only.

Attributes

X	Covariates matrix.
beta	Fixed-effect sizes.
beta_covariance	Estimates the covariance-matrix of the optimal beta.
delta	Variance ratio between K and I.
name	Name of this function.
ncovariates	Number of covariates.
nsamples	Number of samples.
scale	Scaling factor.
v0	First variance.
v1	Second variance.