distributions.chisq

Module: distributions.chisq

Functions

selectinf.distributions.chisq.quadratic_bounds(y, operator, affine_constraints)

Given a set specified by an affine constraint

\[C = \{z \in \mathbb{R}^q: Az \leq b \}\]

and \(A_{p \times q}\) given by operator, this function determines the slice

\[\{t: A(y + t \eta) \leq b\}\]

where

\[\eta = \frac{A^TAy}{\|A^TAy\|_2}\]

This is used to create a truncated \(\chi\) test, as described for the group LASSO in `Kac Rice`_ and implemented in the function quadratic_test.

Parameters

y : np.float((q,))

operator : np.float((p,q))

affine_constraints : `selection.constraints.constraints`_

Returns

lower_bound : float

observed : float

upper_bound : float

sd : float

Notes

The test is based on the fact that, conditional on \(\eta\) and the constraints, \(Ay\) is a truncated \(\chi\) random variable.

selectinf.distributions.chisq.quadratic_test(y, operator, affine_constraints)

Test the null hypothesis $$H_0:A_{p times q}mu_{q times 1} = 0$$ based on \(y \sim N(\mu,\Sigma)\) with \(\Sigma\) given by affine_constraints.covariance where affine_constraints represents the set

\[C = \{z \in \mathbb{R}^q: Az \leq b \}\]

Parameters

y : np.float((q,))

operator : np.float((p,q))

affine_constraints : `selection.constraints.constraints`_

Returns

lower_bound : float

selectinf.distributions.chisq.tangent_space(operator, y)

Return the unit vector \(\eta(y)=Ay/\|Ay\|_2\) where \(A\) is operator. It also forms a basis for the tangent space of the unit sphere at \(\eta(y)\).

Parameters

operator : np.float((p,q))

y : np.float((q,)

Returns

eta : np.float(p)

Unit vector that achieves the norm of \(Ay\) with \(A\) as operator.

tangent_space : np.float((p-1,p))

An array whose rows form a basis for the tangent space of the sphere at eta.

Notes

Implicitly assumes \(A\) is of rank p and any (p-1) rows of the identity can be used to find a basis of the tangent space after projection off of \(Ay\).