make_summation_matrix {sparseFLMM}

Construct symmetry constraint matrix for bivariate symmetric smoothing.

Description

This function can be used to construct a symmetry constraint matrix that imposes a (skew-)symmetry constraint on (cyclic) spline coefficients in symmetric bivariate smoothing problems and is especially designed for constructing objects of the class "symm.smooth", see smooth.construct.symm.smooth.spec.

Usage

make_summation_matrix(F, skew = FALSE, cyclic.degree = 0)

Arguments

F

number of marginal basis functions.

skew

logical, should the basis be constraint to skew-symmetry instead of symmetry.

cyclic.degree

integer, specifying the number of basis functions identified with each other at the boundaries in order to implement periodicity. Should be specified to match the degree of the utilized B-spline basis.

Details

Imposing a symmetry constraint to the spline coefficients in order to obtain a reduced coefficient vector is equivalent to right multiplication of the bivariate design matrix with the symmetry constraint matrix obtained with function make_summation_matrix. The penalty matrix of the bivariate smooth needs to be adjusted to the reduced coefficient vector by left and right multiplication with the symmetry constraint matrix. This function is used in the constructor function smooth.construct.symm.smooth.spec.

Value

A basis transformation matrix of dimension F^2 \times G with G<F^2 depending on the specified constraint.

Author(s)

Jona Cederbaum, Almond Stoecker

References

Cederbaum, Scheipl, Greven (2016): Fast symmetric additive covariance smoothing. Submitted on arXiv.

See Also

smooth.construct and smoothCon for details on constructors