evspline {Splinets}

Evaluating splines at given arguments.

Description

For a Splinets-object S and a vector of arguments t, the function returns the matrix of values for the splines in S. The evaluations are done through the Taylor expansions, so on the ith interval for t\in [\xi_i,\xi_{i+1}]:

S(t)=\sum_{j=0}^{k} s_{i j} \frac{(t-\xi_{i})^j}{j!}.

For the zero order splines which are discontinuous at the knots, the following convention is taken. At the LHS knots the value is taken as the RHS-limit, at the RHS knots as the LHS-limit. The value at the central knot for the zero order and an odd number of knots case is assumed to be zero.

Usage

evspline(object, sID = NULL, x = NULL, N = 250)

Arguments

object

Splinets object;

sID

vector of integers, the indicies specifying splines in the Splinets list to be evaluated; If sID=NULL, then all splines in the Splinet-object are evaluated. The default value is NULL.

x

vector, the arguments at which the splines are evaluated; If x is NULL, then the splines are evaluated over regular grids per each interval of the support. The default value is x=NULL.

N

integer, the number of points per an interval between two consequitive knots at which the splines are evaluated. The default value is N = 250;

Value

The length(x) x length(sID+1) matrix containing the argument values, in the first column, then, columnwise, values of the subsequent 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 diagnostic of Splinets-objects; plot,Splinets-method for plotting Splinets-objects;

Examples

#---------------------------------------------#
#-- Example piecewise polynomial vs. spline --#
#---------------------------------------------#
n=20; k=3; xi=sort(runif(n+2))
sp=new("Splinets",knots=xi) 

#Randomly assigning the derivatives -- a very 'wild' function.
S=matrix(rnorm((n+2)*(k+1)),ncol=(k+1))
sp@supp=list(t(c(1,n+2))); sp@smorder=k; sp@der[[1]]=S

y = evspline(sp)
plot(y,type = 'l',col='red')

#A correct spline object
nsp=is.splinets(sp)
sp2=nsp$robject 
y = evspline(sp2)
lines(y,type='l')

#---------------------------------------------#
#-- Example piecewise polynomial vs. spline --#
#---------------------------------------------#
#Gathering three 'Splinets' objects using three different
#method to correct the derivative matrix
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)) # generate a random matrix S

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
y = evspline(spl, sID= 1)
plot(y,type = 'l',col='red')

y = evspline(spl, sID = c(1,3))
plot(y[,1:2],type = 'l',col='red')
points(y[,c(1,3)],type = 'l',col='blue')

#sID = NULL
y = evspline(spl)
plot(y[,1:2],type = 'l',col='red',ylim=range(y[,2:4]))
points(y[,c(1,3)],type = 'l',col='blue')
points(y[,c(1,4)],type = 'l',col='green')

#---------------------------------------------#
#--- Example with different support ranges ---#
#---------------------------------------------#
n=25; k=3; 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,],'CRFC')
spl=gather(spl,nspl) #the second and the first ones
nspl=construct(xi,k,SS3,supp[3,],'CRLC')
spl=gather(spl,nspl) #the third is added

y = evspline(spl, sID= 1)
plot(y,type = 'l',col='red')

y = evspline(spl, sID = c(1,3))
plot(y[,1:2],type = 'l',col='red')
points(y[,c(1,3)],type = 'l',col='blue')

#sID = NULL -- all splines evaluated
y = evspline(spl)
plot(y[,c(1,3)],type = 'l',col='red',ylim=c(-1,1))
points(y[,1:2],type = 'l',col='blue')
points(y[,c(1,4)],type = 'l',col='green')