refine: Refining splines through adding knots

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

Refining splines through adding knots

Description

Any spline of a given order remains a spline of the same order if one considers it on a bigger set of knots than the original one. However, this embedding changes the Splinets representation of the so-refined spline. The function evaluates the corresponding Splinets-object.

Usage

refine(object, mult = 2, newknots = NULL)

Arguments

object	Splinets-object, the object to be represented as a Splinets-object over a refined set of knots;
mult	positive integer, refining rate; The number of the knots to be put equally spaced between the existing knots.
newknots	m vector, new knots; The knots do not need to be ordered and knots from the input Splinets-object knots are allowed since any ties are resolved.

Details

The function merges new knots with the ones from the input object. It utilizes deriva()-function to evaluate the derivative at the refined knots. It removes duplications of the refined knots, and account also for the not-fully supported case. In the case when the range of the additional knots extends beyond the knots of the input Splinets-object, the support sets of the output Splinets-object account for the smaller than the full support.

Value

A Splinet object with the new refined knots and the new matrix of derivatives is evaluated at the new knots combined with the original ones.

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

deriva for computing derivatives at selected points; project for an orthogonal projection into a space of splines;

Examples

#-------------------------------------------------#
#----Refining splines - the full support case-----#
#-------------------------------------------------#
k=3 # order
n = 16 # number of the internal knots (excluding the endpoints)
xi = seq(0, 1, length.out = n+2)
set.seed(5)
S=matrix(rnorm((n+2)*(k+1)),ncol=(k+1))

spl=construct(xi,k,S) 

plot(spl) # plotting a spline


rspl=refine(spl) # refining the equidistant by doubling its knots
plot(rspl)
rspl@equid  # the outcome is equidistant

#a non-equidistant case
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) 
plot(spl)
mult=3 #adding two knots between each subsequent pair of the original knots
rspl=refine(spl,mult) 
is.splinets(rspl)
plot(rspl)

#adding specific knots
rspl=refine(spl,newknots=c(0.5,0.75))
rspl@knots
is.splinets(rspl)
plot(rspl)

#----------------------------------------------------#
#----Refining splines - the partial support case-----#
#----------------------------------------------------#

Bases=splinet(xi,k)
plot(Bases$bs)
Base=Bases$bs

BS_Two=subsample(Bases$bs,c(1,length(Base@der)))
plot(BS_Two)
A=matrix(c(1,-1),ncol=2)
spl=lincomb(BS_Two,A)

rspl=refine(spl) #doubling the number of knots
plot(rspl)
is.splinets(rspl)
rspl@supp #the support is evaluated 
spl@supp

#The case of adding knots explicitely
BS_Middle=subsample(Bases$bs,c(floor(length(Base@der)/2)))
spls=gather(spl,BS_Middle)
plot(spls)

rspls=refine(spls, newknots=c(0.2,0.5,0.85)) #two splines with partial support sets 
                                             #by adding three knots to B-splines
plot(rspls)

#----------------------------------------------------#
#------Refining splines over the larger range--------#
#----------------------------------------------------#

k=4 # order
n = 25 # number of the internal knots (excluding the endpoints)
xi = seq(0, 1, length.out = n+2)
S=matrix(rnorm((n+2)*(k+1)),ncol=(k+1))

spl=construct(xi,k,S) 

plot(spl) # plotting a spline
newknots=c(-0.1,0.4,0.6,1.2)  #the added knots create larger range 
rspl=refine(spl,newknots=newknots)
spl@supp #the original spline has the full support
rspl@supp #the embedded spline has partial support
spl@equid
rspl@equid
plot(rspl)

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