rspline {Splinets}R Documentation

Random splines

Description

The function simulates a random Splinets-object that is made of random splines with the center at the input spline and the matrix of derivatives has the added error term of the form

\boldsymbol \Sigma^{1/2}\mathbf Z \boldsymbol \Theta^{1/2},

where \mathbf Z is a (n+2)\times (k+1) matrix having iid standard normal variables as its entries, while \boldsymbol \Sigma and \boldsymbol \Theta are matrix parameters. This matrix error term is then corrected by one of the methods and thus resulting in a matrix of derivatives at knots corresponding to a valid spline.

Usage

rspline(S, N = 1, Sigma = NULL, Theta = NULL, mthd = "RRM")

Arguments

S

Splinets-object with n+2 knots and of the order of smoothness k, representing the center of randomly simulated splines; When the number of splines in the object is bigger than one, only the first spline in the object is used.

N

positive integer, size of the sample;

Sigma

matrix;

  • If (n+2)x(n+2) matrix, it controls correlations between derivatives of the same order at different knots.

  • If a positive number, it represents a diagonal (n+2)x(n+2) matrix with this number on the diagonal.

  • If a n+2 vector, it represents a diagonal (n+2)x(n+2) matrix with the vector entries on the diagonal.

  • If NULL (default) represents the identity matrix.

Theta

matrix;

  • If (k+1)x(k+1), this controls correlations between different derivatives at each knot.

  • If a positive number, it represents a diagonal matrix with this number on the diagonal.

  • If a k+1 vector, it represents a diagonal matrix with the vector entries on the diagonal.

  • If NULL (default), it represents the k+1 identity matrix;

mthd

string, one of the three methods: RCC, CR-LC, CR-FC, to adjust random error matrix so it corresponds to a valid spline;

Value

A Splinets-object that contains N generated splines constituting an iid sample of splines;

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

is.splinets for diagnostics of the Splinets-objects; construct for constructing a Splinets-object; gather for combining Splinets-objects into a bigger object; subsample for subsampling Splinets-objects; plot,Splinets-method for plotting Splinets-objects;

Examples

#-----------------------------------------------------#
#-------Simulation of a standard random splinet-------#
#-----------------------------------------------------#
n=17; k=4; xi=sort(runif(n+2)); xi[1]=0; xi[n+2]=1 
S=matrix(rnorm((n+2)*(k+1)),ncol=(k+1))
spl=construct(xi,k,S) #Construction of the mean spline

RS=rspline(spl)
graphsp=evspline(RS) #Evaluating the random spline
meansp=evspline(spl)
RS=rspline(spl,5) #Five more samples
graphsp5=evspline(RS)

m=min(graphsp[,2],meansp[,2],graphsp5[,2:6])
M=max(graphsp[,2],meansp[,2],graphsp5[,2:6])

plot(graphsp,type='l',ylim=c(m,M))
lines(meansp,col='red',lwd=3,lty=2) #the mean spline
for(i in 1:5){lines(graphsp5[,1],graphsp5[,i+1],col=i)}

#-----------------------------------------------------#
#------------Different construction method------------#
#-----------------------------------------------------#
RS=rspline(spl,8,mthd='CRLC'); graphsp8=evspline(RS)

m=min(graphsp[,2],meansp[,2],graphsp8[,2:6])
M=max(graphsp[,2],meansp[,2],graphsp8[,2:6])

plot(meansp,col='red',type='l',lwd=3,lty=2,ylim=c(m,M)) #the mean spline
for(i in 1:8){lines(graphsp8[,1],graphsp8[,i+1],col=i)}

#-----------------------------------------------------#
#-------Simulation of with different variances--------#
#-----------------------------------------------------#
Sigma=seq(0.1,1,n+2);Theta=seq(0.1,1,k+1)
RS=rspline(spl,N=10,Sigma=Sigma) #Ten samples
RS2=rspline(spl,N=10,Sigma=Sigma,Theta=Theta) #Ten samples
graphsp10=evspline(RS); graphsp102=evspline(RS2)

m=min(graphsp[,2],meansp[,2],graphsp10[,2:10])
M=max(graphsp[,2],meansp[,2],graphsp10[,2:10])

plot(meansp,type='l',ylim=c(m,M),col='red',lwd=3,lty=2) 
for(i in 1:10){lines(graphsp10[,1],graphsp10[,i+1],col=i)}

m=min(graphsp[,2],meansp[,2],graphsp102[,2:10])
M=max(graphsp[,2],meansp[,2],graphsp102[,2:10])

plot(meansp,type='l',ylim=c(m,M),col='red',lwd=3,lty=2) 
for(i in 1:10){lines(graphsp102[,1],graphsp102[,i+1],col=i)}

#-----------------------------------------------------#
#-------Simulation for the mean spline to be----------#
#------=----defined on incomplete supports------------#
#-----------------------------------------------------#
n=43; xi=seq(0,1,by=1/(n+1)); k=3; xi=sort(runif(n+2)); xi[1]=0; xi[n+2]=1;
support=list(matrix(c(2,14,25,43),ncol=2,byrow = TRUE))
ssp=new("Splinets",knots=xi,supp=support) #with partial support
nssp=is.splinets(ssp)$robject
nssp@smorder=3 #changing the order of the 'Splinets' object
m=sum(nssp@supp[[1]][,2]-nssp@supp[[1]][,1]+1) #the number of knots in the support
nssp@der=list(matrix(rnorm(m*(k+1)),ncol=(k+1)))  #the derivative matrix at random
spl=is.splinets(nssp)$robject
RS=rspline(spl,Sigma=0.05,Theta=c(1,0.5,0.3,0.05))
graphsp=evspline(RS);
meansp=evspline(spl)

m=min(graphsp[,2],meansp[,2],graphsp5[,2:6])
M=max(graphsp[,2],meansp[,2],graphsp5[,2:6])

plot(graphsp,type='l',ylim=c(m,M))
lines(meansp,col='red',lwd=3,lty=2) #the mean spline
[Package Splinets version 1.5.0 Index]