exsupp {Splinets}

Correcting support sets and reshaping the matrix of derivatives at the knots.

Description

The function is adjusting for a potential reduction in the support sets due to negligibly small values of rows in the derivative matrix. If the derivative matrix has a row equal to zero (or smaller than a neglible positive value) in the one-sided representation of it (see the references and sym2one), then the corresponding knot should be removed from the support set. The function can be used to eliminate the neglible support components from a Splinets-object.

Usage

exsupp(S, supp = NULL, epsilon = 1e-07)

Arguments

S

(m+2)x(k+1) matrix, the values of the derivatives at the knots over some input support set which has the cardinality m+2; The matrix is assumed to be in the symmetric around center form for each component of the support.

supp

NULL or Nsupp x2 matrix of integers, the endpoints indices for the input support intervals, where Nsupp is the number of the components in the support set; If the parameter is NULL, than the full support is assumed.

epsilon

small positive number, threshold value of the norm of rows of S; If the norm of a row of S is less than epsilon, then it will be viewed as a neglible and the knot is excluded from the inside of the support set.

Details

This function typically would be applied to an element in the list given by SLOT der of a Splinets-object. It eliminates from the support sets regions of negligible values of a corresponding spline and its derivatives.

Value

The list of two elements: exsupp$rS is the reduced derivative matrix from which the neglible rows, if any, have been removed and exsupp$rsupp is the corresponding reduced support. The output matrix has all the support components in the symmetric around the center form, which is how the derivatives are kept in the Splinets-objects.

References

Liu, X., Nassar, H., Podg\mbox{\'o}rski, K. "Dyadic diagonalization of positive definite band matrices and efficient B-spline orthogonalization." Journal of Computational and Applied Mathematics (2022) <https://doi.org/10.1016/j.cam.2022.114444>.

Podg\mbox{\'o}rski, K. (2021) "Splinets – splines through the Taylor expansion, their support sets and orthogonal bases." <arXiv:2102.00733>.

Nassar, H., Podg\mbox{\'o}rski, K. (2023) "Splinets 1.5.0 – Periodic Splinets." <arXiv:2302.07552>

See Also

Splinets-class for the description of the Splinets-class; sym2one for switching between the representations of a derivative matrix over a general support set; lincomb for evaluating a linear transformation of splines in a Splinets-object; is.splinets for a diagnostic tool of the Splinets-objects;

Examples

#----------------------------------------------------#
#---Correcting support sets in a derivative matrix---#
#----------------------------------------------------#
n=20; k=3; xi=seq(0,1,by=1/(n+1)) #an even number of equally spaced knots 
set.seed(5)
S=matrix(rnorm((n+2)*(k+1)),ncol=(k+1))
spl=construct(xi,k,S) #this spline will be used below to construct a 'sparse' spline
is.splinets(spl) #verification
plot(spl)


xxi=seq(0,20,by=1/(n+1)) #large set of knots for construction of a sparse spline 
nn=length(xxi)-2
spspl=new('Splinets',knots=xxi,smorder=k) #generic object from the 'Splinets'-class
spspl@der[[1]]=matrix(0,ncol=(k+1),nrow=(nn+2)) #starting with zeros everywhere

spspl@der[[1]][1:(n+2),]=sym2one(spl@der[[1]]) #assigning local spline to a sparse spline at 
spspl@der[[1]][nn+3-(1:(n+2)),]=spspl@der[[1]][(n+2):1,] #the beginning and the same at the end
spspl@der[[1]]=sym2one(spspl@der[[1]],inv=TRUE) 
                                #at this point the object does not account for the sparsity

is.splinets(spspl) #a sparse spline on 421 knots with a non-zero terms at the first 22 
                   #and at the last 22 knots, the actual support set is not yet reported
plot(spspl)
plot(spspl,xlim=c(0,1)) #the local part of the sparse spline

exsupp(spspl@der[[1]]) #the actual support of the spline given the sparse derivative matrix

#Expanding the previous spline by building a slightly more complex support set
spspl@der[[1]][(n+1)+(1:(n+2)),]=sym2one(spl@der[[1]]) #double the first component of the
                                              #support because these are tangent supports 
spspl@der[[1]][(2*n+3)+(1:(n+2)),]=sym2one(spl@der[[1]]) #tdetect a single component of
                                                #the support with no internal knots removed
is.splinets(spspl)
plot(spspl)


es=exsupp(spspl@der[[1]])
es[[2]]   #the new support made of three components with the two first ones
          #separated by an interval with no knots in it 
          

spspl@der[[1]]=es[[1]]         #defining the spline on the evaluated actual support
spspl@supp[[1]]=es[[2]]
#Example with reduction of not a full support. 

xi1=seq(0,14/(n+1),by=1/(n+1)); n1=13; #the odd number of equally spaced knots 
S1=matrix(rnorm((n1+2)*(k+1)),ncol=(k+1))
spl1=construct(xi1,k,S1) #construction of a local spline
xi2=seq(16/(n+1),42/(n+1),by=1/(n+1)); n2=25; #the odd number of equally spaced knots 

S2=matrix(rnorm((n2+2)*(k+1)),ncol=(k+1))
spl2=construct(xi2,k,S2) #construction of a local spline

spspl@der[[1]][1:15,]=sym2one(spl1@der[[1]])
spspl@der[[1]][16,]=rep(0,k+1)
spspl@der[[1]][17:43,]=sym2one(spl2@der[[1]])
spspl@der[[1]][1:43,]=sym2one(spspl@der[[1]][1:43,],inv=TRUE)

is.splinets(spspl) #three intervals in the support are repported 

exsupp(spspl@der[[1]],spspl@supp[[1]])