dintegra: Calculating the definite integral of a spline.

In Splinets: Functional Data Analysis using Splines and Orthogonal Spline Bases

Calculating the definite integral of a spline.

Description

The function calculates the definite integrals of the splines in an input Splinets-object.

Usage

dintegra(object, sID = NULL)

Arguments

object	Splinets-object;
sID	vector of integers, the indicies specifying for which splines in the Splinets-object the definite integral is to be evaluated; If sID=NULL, then the definite integral of all splines in the object are calculated. The default is NULL.

Value

A length(sID) x 2 matrix, with the first column holding the id of splines and the second column holding the corresponding definite integrals.

References

Liu, X., Nassar, H., Podgorski, 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>.

Podgorski, K. (2021) "Splinets – splines through the Taylor expansion, their support sets and orthogonal bases." <arXiv:2102.00733>.

Nassar, H., Podgorski, K. (2023) "Splinets 1.5.0 – Periodic Splinets." <arXiv:2302.07552>

See Also

integra for generating the indefinite integral; deriva for generating derivative functions of splines;

Examples

#------------------------------------------#
#--- Example with common support ranges ---#
#------------------------------------------#
n=23; k=4
set.seed(5)
xi=sort(runif(n+2)); xi[1]=0; xi[n+2]=1
# generate a random matrix S
S=matrix(rnorm((n+2)*(k+1)),ncol=(k+1))
# construct the spline
spl=construct(xi,k,S) #constructing the first correct spline
spl=gather(spl,construct(xi,k,S,mthd='CRFC')) #the second and the first ones
spl=gather(spl,construct(xi,k,S,mthd='CRLC')) #the third is added

plot(spl)
dintegra(spl, sID = c(1,3))
dintegra(spl)
plot(spl,sID=c(1,3))

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

n=25; k=2
xi=seq(0,1,by=1/(n+1))
#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
set.seed(5)
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

plot(spl)
dintegra(spl, sID = 1)
dintegra(spl)

#The third order case
n=40; xi=seq(0,1,by=1/(n+1)); k=3; 
support=list(matrix(c(2,12,15,27,30,40),ncol=2,byrow = TRUE))
sp=new("Splinets",knots=xi,smorder=k,supp=support) 
m=sum(sp@supp[[1]][,2]-sp@supp[[1]][,1]+1) #the number of knots in the support
sp@der=list(matrix(rnorm(m*(k+1)),ncol=(k+1))); sp1 = is.splinets(sp)[[2]] 

support=list(matrix(c(2,13,17,30),ncol=2,byrow = TRUE))
sp=new("Splinets",knots=xi,smorder=k,supp=support) 
m=sum(sp@supp[[1]][,2]-sp@supp[[1]][,1]+1) #the number of knots in the support
sp@der=list(matrix(rnorm(m*(k+1)),ncol=(k+1))); sp2 = is.splinets(sp)[[2]] 

sp = gather(sp1,sp2)
dintegra(sp)
plot(sp)

lcsp=lincomb(sp,matrix(c(-1,1),ncol=2))
dintegra(lcsp)                  #linearity of the integral
dintegra(sp2)-dintegra(sp1)

Splinets documentation built on March 7, 2023, 8:24 p.m.