| lincomb {Splinets} | R Documentation |
Linear transformation of splines.
Description
A linear combination of the splines S_j in the input object is computed according to
R_i=\sum_{j=0}^{d} a_{i j} S_j,\, i=1,\dots, l.
and returned as a Splinet-object.
Usage
lincomb(object, A, reduced = TRUE, SuppExtr = TRUE)
Arguments
object |
|
A |
|
reduced |
logical; If |
SuppExtr |
logical; If |
Value
A Splinet-object that contains l splines obtained by linear combinations of
using coefficients in rows of A. The SLOT type of the output splinet objects is sp.
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
exsupp for extracting the correct support;
construct for building a valid spline;
rspline for random generation of splines;
Examples
#-------------------------------------------------------------#
#------------Simple linear operations on Splinets-------------#
#-------------------------------------------------------------#
#Gathering three 'Splinets' objects
n=53; k=4; xi=sort(runif(n+2)); xi[1]=0; xi[n+2]=1;Nspl=10
set.seed(5)
S=matrix(rnorm((n+2)*(k+1)),ncol=(k+1))
spl=construct(xi,k,S) #constructing the first proper spline
spl@epsilon=1.0e-5 #to avoid FALSE in the next function due to inaccuracies
is.splinets(spl)
RS=rspline(spl,Nspl) #Random splines
plot(RS)
A = matrix(rnorm(5*Nspl, mean = 2, sd = 100), ncol = Nspl)
new_sp1 = lincomb(RS, A)
plot(new_sp1)
new_sp2 = lincomb(RS, A, reduced = FALSE)
plot(new_sp2)
#---------------------------------------------#
#--- Example with different support ranges ---#
#---------------------------------------------#
n=25; k=3
set.seed(5)
xi=sort(runif(n+2)); xi[1]=0; xi[n+2]=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
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
A = matrix(rnorm(3*2, mean = 2, sd = 100), ncol = 3)
new_sp1 = lincomb(spl, A) # based on reduced supports
plot(new_sp1)
new_sp2 = lincomb(spl, A, reduced = FALSE) # based on full support
plot(new_sp2) # new_sp1 and new_sp2 are same
#-----------------------------------------#
#--- Example with varying support sets ---#
#-----------------------------------------#
n=40; xi=seq(0,1,by=1/(n+1)); k=2;
support=list(matrix(c(2,9,15,24,30,37),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))) #the derivative matrix at random
sp1 = is.splinets(sp)[[2]] #the corrected vs. the original 'der' matrices
support=list(matrix(c(5,12,17,29),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))) #the derivative matrix at random
sp2 = is.splinets(sp)[[2]] #building a valid spline
spp = gather(sp1,sp2)
support=list(matrix(c(3,10,14,21,27,34),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))) #the derivative matrix at random
sp3 = is.splinets(sp)[[2]] #building a valid spline
spp = gather(spp, sp3)
plot(spp)
spp@supp #the supports
set.seed(5)
A = matrix(rnorm(3*4, mean = 2, sd = 100), ncol = 3)
new_sp1 = lincomb(spp, A) # based on reduced supports
plot(new_sp1)
new_sp1@supp #the support of the output from 'lincomb'
new_sp2 = lincomb(spp, A, reduced = FALSE) # based on full support
plot(new_sp2) # new_sp1 and new_sp2 are same
new_sp2@supp #the support of the output from 'lincomb' with full support computations
#-------------------------------------#
#--- Support needs some extra care ---#
#-------------------------------------#
set.seed(5)
n=53; k=4; xi=sort(runif(n+2)); xi[1]=0; xi[n+2]=1
supp1 = matrix(c(1, ceiling(n/2)+1), ncol = 2)
supp2 = matrix(c(ceiling(n/2)+1, n+2), ncol = 2)
S = matrix(rnorm(5*(ceiling(n/2)+1)), ncol = k+1)
a = construct(xi,k,S,supp = supp1) #constructing the first proper spline
S = matrix(rnorm(5*(ceiling(n/2)+1)), ncol = k+1)
b = construct(xi,k,S,supp = supp2) #constructing the first proper spline
sp = gather(a,b)
plot(sp)
# create a+b and a-b
s = lincomb(sp, matrix(c(1,1,1,-1), byrow = TRUE, nrow = 2))
plot(s)
s@supp
# Sum has smaller support than its terms
s1 = lincomb(s, matrix(c(1,1), nrow = 1), reduced = TRUE)
plot(s1)
s1@supp # lincomb based on support, the full support is reported
s2 = lincomb(s, matrix(c(1,1), nrow = 1), reduced = FALSE)
plot(s2)
s2@supp # lincomb using full der matrix
s3=lincomb(s, matrix(c(1,1), nrow = 1), reduced = FALSE, SuppExtr=FALSE)
s3@supp #the full range is reported as support
ES=exsupp(s1@der[[1]]) #correcting the matrix and the support
s1@der[[1]]=ES[[1]]
s1@supp[[1]]=ES[[2]]
plot(s1)
s1@supp[[1]]