# limix.qc.normalise_covarianceΒΆ

limix.qc.normalise_covariance(K, out=None)[source]ΒΆ

Variance rescaling of covariance matrix πΊ.

Let n be the number of rows (or columns) of πΊ and let mα΅’ be the average of the values in the i-th column. Gower rescaling is defined as

$πΊ(n - 1)/(πππππ(πΊ) - βmα΅’).$

Notes

The reasoning of the scaling is as follows. Let π  be a vector of n independent samples and let π² be the Gowerβs centering matrix. The unbiased variance estimator is

$v = β (gα΅’-αΈ‘)Β²/(n-1) = πππππ((π -αΈ‘π)α΅(π -αΈ‘π))/(n-1) = πππππ(π²π π α΅π²)/(n-1)$

Let πΊ be the covariance matrix of π . The expectation of the unbiased variance estimator is

$π[v] = πππππ(π²π[π π α΅]π²)/(n-1) = πππππ(π²πΊπ²)/(n-1),$

assuming that π[gα΅’]=0. We thus divide πΊ by π[v] to achieve an unbiased normalisation on the random variable gα΅’.

Parameters
• K (array_like) β Covariance matrix to be normalised.

• out (array_like, optional) β Result destination. Defaults to None.

Examples

>>> from numpy import dot, mean, zeros
>>> from numpy.random import RandomState
>>> from limix.qc import normalise_covariance
>>>
>>> random = RandomState(0)
>>> X = random.randn(10, 10)
>>> K = dot(X, X.T)
>>> Z = random.multivariate_normal(zeros(10), K, 500)
>>> print("%.3f" % mean(Z.var(1, ddof=1)))
9.824
>>> Kn = normalise_covariance(K)
>>> Zn = random.multivariate_normal(zeros(10), Kn, 500)
>>> print("%.3f" % mean(Zn.var(1, ddof=1)))
1.008