lincomb: Linear transformation of splines.
In Splinets: Functional Data Analysis using Splines and Orthogonal Spline Bases
lincomb 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=∑_{j=0}^{d} a_{i j} S_j,\, i=1,…, l.
and returned as a Splinet-object.
Usage
lincomb(object, A, reduced = TRUE, SuppExtr = TRUE)
Arguments
object
Splinets object containing d splines;
A
l x d matrix; coefficients of the linear transformation,
reduced
logical; If TRUE (default), then the linear combination is
calculated accounting for the actual support sets (recommended for sparse splines),
if FALSE, then the
full support computations are used (can be faster for lower dimension or non-sparse cases).
SuppExtr
logical; If TRUE (default), the true support is extracted, otherwise, full range
is reported as the support. Applies only to the case when reduced=FALSE.
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., 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
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]]
Splinets documentation built on March 7, 2023, 8:24 p.m.
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| lincomb | 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=∑_{j=0}^{d} a_{i j} S_j,\, i=1,…, 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., 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
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]]
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