deriva {Splinets}

Derivatives of splines

Description

The function generates a Splinets-object which contains the first order derivatives of all the splines from the input Splinets-object. The function also verifies the support set of the output to provide the accurate information about the support sets by excluding regions over which the original function is constant.

Usage

deriva(object, epsilon = 1e-07)

Arguments

object

Splinets object of the smoothness order k;

epsilon

positive number, controls removal of knots from the support; If the derivative is smaller than this number, it is considered to be zero and the corresponding knots are removed from the support.The default value is 1e-7.

Value

A Splinets-object of the order k-1 that also contains the updated information about the support set.

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

integra for generating the indefinite integral of a spline that can be viewed as the inverse operation to deriva; dintegra for the definite integral of a spline;

Examples

#-------------------------------------------------------#
#--- Generating the deriviative functions of splines ---#
#-------------------------------------------------------#
n=13; k=4
set.seed(5)
xi=sort(runif(n+2)); xi[1]=0; xi[n+2]=1
spl=construct(xi,k,matrix(rnorm((n+2)*(k+1)),ncol=(k+1))) #constructing three splines
spl=gather(spl, construct(xi,k,matrix(rnorm((n+2)*(k+1)),ncol=(k+1)))) 
spl=gather(spl, construct(xi,k,matrix(rnorm((n+2)*(k+1)),ncol=(k+1)))) 
# calculate the derivative of splines
dspl = deriva(spl)
plot(spl)
plot(dspl)

#----------------------------------------------#
#--- Examples with different support ranges ---#
#----------------------------------------------#

n=25; k=3
xi=seq(0,1,by=1/(n+1));
set.seed(5)
#Defining support ranges for three splines
supp=matrix(c(2,12,4,20,6,25),byrow=TRUE,ncol=2)
#Initial random matrices of the derivative for each spline
SS1=matrix(rnorm((supp[1,2]-supp[1,1]+1)*(k+1)),ncol=(k+1)) 
SS2=matrix(rnorm((supp[2,2]-supp[2,1]+1)*(k+1)),ncol=(k+1)) 
SS3=matrix(rnorm((supp[3,2]-supp[3,1]+1)*(k+1)),ncol=(k+1)) 
spl=construct(xi,k,SS1,supp[1,]) #constructing the first correct spline
nspl=construct(xi,k,SS2,supp[2,])
spl=gather(spl,nspl) #the second and the first ones
nspl=construct(xi,k,SS3,supp[3,])
spl=gather(spl,nspl) #the third is added

der_spl = deriva(spl)
par(mar=c(1,1,1,1))
par(mfrow=c(2,1))
plot(der_spl)
plot(spl)
par(mfrow=c(1,1))