Projecting into spline spaces

The projection of splines or functional data into the linear spline space spanned over a given set of knots.

Usage

project(
  fdsp,
  knots = NULL,
  smorder = 3,
  periodic = FALSE,
  basis = NULL,
  type = "spnt",
  graph = FALSE
)

Arguments

fdsp

Splinets-object or a n x (N+1) matrix, a representation of N functions to be projected to the space spanned by a Splinets-basis over a specific set of knots; If the parameter is a Splinets-object containing N splines, then it is orthogonally projected or represented in the basis that is specified by other parameters. If the paramater is a matrix, then it is treated as N piecewise constant functions with the arguments in the first column and the corresponding values of the functions in the remaining N columns.

knots

vector, the knots of the projection space, together with smorder fully characterizes the projection space; This parameter is overridden by the SLOT basis@knots of the basis input if this one is not NULL.

smorder

integer, the order of smoothness of the projection space; This parameter is overridden by the SLOT basis@smorder of the basis input if this one is not NULL.

periodic

logical, a flag to indicate if B-splines will be of periodic type or not; In the case of periodic splines, the arguments of the input and the knots need to be within [0,1] or, otherwise, an error occurs and a message advising the recentering and rescaling data is shown.

basis

Splinets-object, the basis used for the representation of the projection of the input fdsp;

type

string, the choice of the basis in the projection space used only if the basis-parameter is not given; The following choices are available

'bs' for the unorthogonalized B-splines,
'spnt' for the orthogonal splinet (the default),
'gsob' for the Gramm-Schmidt (one-sided) OB-splines,
'twob' for the two-sided OB-splines.

The default is 'spnt'.

graph

logical, indicator if the illustrative plots are to be produced:

the splinet used in the projection(s) on the dyadic grid,
the coefficients of the projection(s) on the dyadic grid,
the input function(s),
the projection(s).

;

Value

The value of the function is a list made of four elements

project$input – fdsp, when the input is a Splinets-object or a matrix with the first column in an increasing order, otherwise it is the input numeric matrix after ordering according to the first column,
project$coeff – N x (n-k+1) matrix of the coefficients of representation of the projection of the input in the splinet basis,
project$basis – the spline basis,
project$sp – the Splinets-object containing the projected splines.

Details

The obtained coefficients $\mathbf A = (a_{ji})$ with respect to the basis allow to evaluate the splines $S_j$ in the projection according to $$ S_j=\sum_{i=1}^{n-k-1} a_{ji} OB_{i}, \,\, j=1,\dots, N, $$ where $n$ is the number of the knots (including the endpoints), $k$ is the spline smoothness order, $N$ is the number of the projected functions and $OB_i$'s consitute the considered basis. The coefficient for the splinet basis are always evaluated and thus, for example, PFD=project(FD,knots); ProjDataSplines=lincomb(PFD$coeff,PFD$basis) creates a Splinets-object made of the projections of the input functional data in FD. If the input parameter basis is given, then the function utilizes this basis and does not need to build it. However, if basis is the B-spline basis, then the B-spline orthogonalization is performed anyway, thus the computational gain is smaller than in the case when basis is an orthogonal basis.

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>

Examples

#-------------------------------------------------#
#----Representing splines in the spline bases-----#
#-------------------------------------------------#
k=3 # order
n = 10 # 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) 
#> 
#> Using  method RRM to correct the derivative matrix entries.
#> 
#> 
#> DIAGNOSTIC CHECK of a SPLINETS object
#> 
#> THE KNOTS:  
#> 
#> 
#> THE SUPPORT SETS:  
#> 
#> The support sets for the splines are equal to the entire range of knots.
#> 
#> 
#> THE DERIVATIVES AT THE KNOTS:  
#> 
#> The boundary zero conditions are not satisfied for spline 1 in the input 'Splinets' object.
#> Correction of the first and last rows of the derivative matrices are made in the output 'Splinets' object.
#> 
#> Spline 1 's highest derivative is not symmetrically defined at the center (the values at the two central knots should be equal).
#> Spline 1 's highest derivative values at the two central knots have been made equal by averaging the two central values in SLOT 'der'.
#> 
#> The matrix of derivatives at the knots for spline 1 does not satisfy the conditions 
#> 
#>           required for a spline (up to the accuracy SLOT 'epsilon').
#> 
#>           One of the reasons can be that SLOT 'taylor' is not correctly given.
#> The computed standard error per matrix entry is 1.383861 .
#> 
#> 
#> Correction of the LHS part of the matrix
#> Correction of the RHS part of the matrix
#> Correction of the LHS part of the matrix
#> Correction of the RHS part of the matrix
#> The output object Spline 1  has the derivative matrix corrected by the RRM method
#>  given that SLOT 'taylor' is properly given.
#> The matrix derivative is now corrected by method RRM .

plot(spl) # plotting a spline

spls=rspline(spl,5) # a random sample of splines

Repr=project(spl) #decomposition of splines into the splinet coefficients

Repr=project(spl, graph = TRUE) #decomposition of splines with the following graphs
                             #that illustrate the decomposition: 
                             # 1) The orthogonal spine basis on the dyadic grid; 
                             # 2) The coefficients of the projections on the dyadic grid;
                             # 3) The input splines;
                             # 4) The projections of the input.
                
Repr$coeff       #the coefficients of the decomposition
#>            [,1]        [,2]       [,3]     [,4]        [,5]       [,6]
#> [1,] -0.1006816 -0.02484691 -0.1980802 -0.17089 -0.08691961 -0.2024501
#>            [,7]      [,8]
#> [1,] -0.3437466 -0.124082
plot(Repr$sp) #plot of the reconstruction of the spline

plot(spls)
Reprs=project(spls,basis = Repr$basis) #decomposing splines using the available basis
plot(Reprs$sp) 

Reprs2=project(spls,type = 'gsob') #using the Gram-Schmidt basis


#The case of the regular non-normalized B-splines:
Reprs3=project(spls,type = 'bs') 
plot(Reprs3$basis)
gramian(Reprs3$basis,norm_only = TRUE) #the B-splines follow the classical definition and 
#> [1] 0.04357864 0.04357864 0.04357864 0.04357864 0.04357864 0.04357864 0.04357864
#> [8] 0.04357864
                                       #thus are not normalized
plot(spls)

plot(Reprs3$basis) #Bsplines
plot(Reprs3$sp) #reconstruction using the B-splines and the decomposition

#a non-equidistant example
n=10; k=3
set.seed(5)
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) 
#> 
#> Using  method RRM to correct the derivative matrix entries.
#> 
#> 
#> DIAGNOSTIC CHECK of a SPLINETS object
#> 
#> THE KNOTS:  
#> 
#> 
#> THE SUPPORT SETS:  
#> 
#> The support sets for the splines are equal to the entire range of knots.
#> 
#> 
#> THE DERIVATIVES AT THE KNOTS:  
#> 
#> The boundary zero conditions are not satisfied for spline 1 in the input 'Splinets' object.
#> Correction of the first and last rows of the derivative matrices are made in the output 'Splinets' object.
#> 
#> Spline 1 's highest derivative is not symmetrically defined at the center (the values at the two central knots should be equal).
#> The spline 1 'ths highest derivative at the two central knots has been made equal by averaging SLOT 'der'.
#> 
#> The derivative matrix for spline 1 does not satisfy the smoothness conditions (up to the accuracy SLOT 'epsilon').
#> The standard error per matrix entry is 1.177526 .
#> 
#> 
#> Correction of the LHS part of the matrix
#> There are less than 5 knots, the first 2 entries of the 5 nd row counting from the end in the input will be changed in the output.
#> 
#> 
#> Correction of the RHS part of the matrix
#> There are less than 5 knots, the first 2 entries of the 5 nd row counting from the end in the input will be changed in the output.
#> 
#> 
#> Correction of the LHS part of the matrix
#> Correction of the RHS part of the matrix
#> The output object has the derivative matrix corrected by the RRM method.
#> 
#> The matrix derivative is now corrected by method RRM .
plot(spl)
spls=rspline(spl,5) # a random sample of splines
plot(spls)
Reprs=project(spls,type = 'twob') #decomposing using the two-sided orthogonalization
plot(Reprs$basis)
plot(Reprs$sp) 

#The case of the regular non-normalized B-splines:
Reprs2=project(spls,basis=Reprs$basis) 
plot(Reprs2$sp) #reconstruction using the B-splines and the decomposition

#-------------------------------------------------#
#-----Projecting splines into a spline space------#
#-------------------------------------------------#
k=3 # order
n = 10 # 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) 
#> 
#> Using  method RRM to correct the derivative matrix entries.
#> 
#> 
#> DIAGNOSTIC CHECK of a SPLINETS object
#> 
#> THE KNOTS:  
#> 
#> 
#> THE SUPPORT SETS:  
#> 
#> The support sets for the splines are equal to the entire range of knots.
#> 
#> 
#> THE DERIVATIVES AT THE KNOTS:  
#> 
#> The boundary zero conditions are not satisfied for spline 1 in the input 'Splinets' object.
#> Correction of the first and last rows of the derivative matrices are made in the output 'Splinets' object.
#> 
#> Spline 1 's highest derivative is not symmetrically defined at the center (the values at the two central knots should be equal).
#> Spline 1 's highest derivative values at the two central knots have been made equal by averaging the two central values in SLOT 'der'.
#> 
#> The matrix of derivatives at the knots for spline 1 does not satisfy the conditions 
#> 
#>           required for a spline (up to the accuracy SLOT 'epsilon').
#> 
#>           One of the reasons can be that SLOT 'taylor' is not correctly given.
#> The computed standard error per matrix entry is 1.383861 .
#> 
#> 
#> Correction of the LHS part of the matrix
#> Correction of the RHS part of the matrix
#> Correction of the LHS part of the matrix
#> Correction of the RHS part of the matrix
#> The output object Spline 1  has the derivative matrix corrected by the RRM method
#>  given that SLOT 'taylor' is properly given.
#> The matrix derivative is now corrected by method RRM .

plot(spl) #the spline

knots=runif(8)
Prspl=project(spl,knots)
#> The knots were not given in the strictly increasing order, which is required. The ordered knots with removed ties are replacing the input knot values.

plot(Prspl$sp) #the projection spline
Rspl=refine(spl,newknots = knots) #embedding the spline to the common space
plot(Rspl)
RPspl=refine(Prspl$sp,newknots = xi)  #embedding the projection spline to the common space
plot(RPspl)
All=gather(RPspl,Rspl) #creating the Splinets-object with the spline and its projection
Rbasis=refine(Prspl$basis,newknots = xi) #embedding the basis to the common space
plot(Rbasis)

Res=lincomb(All,matrix(c(1,-1),ncol=2))
plot(Res)
gramian(Res,Sp2 = Rbasis) #the zero valued innerproducts -- the orthogonality of the residual spline
#>               [,1]          [,2]          [,3]         [,4]
#> [1,] -1.183883e-16 -4.617433e-16 -8.097876e-16 4.333287e-16

spls=rspline(spl,5) # a random sample of splines
Prspls=project(spls,knots,type='bs') #projection in the B-spline representation
#> The knots were not given in the strictly increasing order, which is required. The ordered knots with removed ties are replacing the input knot values.
plot(spls)
lines(Prspls$sp) #presenting projections on the common plot with the original splines

Prspls$sp@knots
#> [1] 0.2344320 0.4557002 0.4724108 0.5744543 0.6254065 0.7207331 0.9202377
#> [8] 0.9283003
Prspls$sp@supp
#> list()

plot(Prspls$basis)   #Bspline basis


#An example with partial support

Bases=splinet(xi,k)

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

knots=runif(13)
Prspl=project(spl,knots)
#> The knots were not given in the strictly increasing order, which is required. The ordered knots with removed ties are replacing the input knot values.
plot(Prspl$sp)
Prspl$sp@knots
#>  [1] 0.2304824 0.2505103 0.5214470 0.5321001 0.5742837 0.5957515 0.6999059
#>  [8] 0.7039371 0.7107075 0.7107501 0.8330875 0.8375177 0.9653198
Prspl$sp@supp
#> list()

#Using explicit bases 

k=3 # order
n = 10 # 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) 
#> 
#> Using  method RRM to correct the derivative matrix entries.
#> 
#> 
#> DIAGNOSTIC CHECK of a SPLINETS object
#> 
#> THE KNOTS:  
#> 
#> 
#> THE SUPPORT SETS:  
#> 
#> The support sets for the splines are equal to the entire range of knots.
#> 
#> 
#> THE DERIVATIVES AT THE KNOTS:  
#> 
#> The boundary zero conditions are not satisfied for spline 1 in the input 'Splinets' object.
#> Correction of the first and last rows of the derivative matrices are made in the output 'Splinets' object.
#> 
#> Spline 1 's highest derivative is not symmetrically defined at the center (the values at the two central knots should be equal).
#> Spline 1 's highest derivative values at the two central knots have been made equal by averaging the two central values in SLOT 'der'.
#> 
#> The matrix of derivatives at the knots for spline 1 does not satisfy the conditions 
#> 
#>           required for a spline (up to the accuracy SLOT 'epsilon').
#> 
#>           One of the reasons can be that SLOT 'taylor' is not correctly given.
#> The computed standard error per matrix entry is 1.383861 .
#> 
#> 
#> Correction of the LHS part of the matrix
#> Correction of the RHS part of the matrix
#> Correction of the LHS part of the matrix
#> Correction of the RHS part of the matrix
#> The output object Spline 1  has the derivative matrix corrected by the RRM method
#>  given that SLOT 'taylor' is properly given.
#> The matrix derivative is now corrected by method RRM .
spls=rspline(spl,5) # a random sample of splines

plot(spls)

knots=runif(20)
base=splinet(knots,smorder=k)
#> Knots were not given in the strictly increasing order, which is required.
#> 
#>           Ordered  knots with removed ties are replacing the input values.
plot(base$os)

Prsps=project(spls,basis=base$os)
plot(Prsps$sp) #projection splines vs. the original splines
lines(spls)

#------------------------------------------------------#
#---Projecting discretized data into a spline space----#
#------------------------------------------------------#
k=3; n = 10; 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); spls=rspline(spl,10) # a random sample of splines
#> 
#> Using  method RRM to correct the derivative matrix entries.
#> 
#> 
#> DIAGNOSTIC CHECK of a SPLINETS object
#> 
#> THE KNOTS:  
#> 
#> 
#> THE SUPPORT SETS:  
#> 
#> The support sets for the splines are equal to the entire range of knots.
#> 
#> 
#> THE DERIVATIVES AT THE KNOTS:  
#> 
#> The boundary zero conditions are not satisfied for spline 1 in the input 'Splinets' object.
#> Correction of the first and last rows of the derivative matrices are made in the output 'Splinets' object.
#> 
#> Spline 1 's highest derivative is not symmetrically defined at the center (the values at the two central knots should be equal).
#> Spline 1 's highest derivative values at the two central knots have been made equal by averaging the two central values in SLOT 'der'.
#> 
#> The matrix of derivatives at the knots for spline 1 does not satisfy the conditions 
#> 
#>           required for a spline (up to the accuracy SLOT 'epsilon').
#> 
#>           One of the reasons can be that SLOT 'taylor' is not correctly given.
#> The computed standard error per matrix entry is 1.383861 .
#> 
#> 
#> Correction of the LHS part of the matrix
#> Correction of the RHS part of the matrix
#> Correction of the LHS part of the matrix
#> Correction of the RHS part of the matrix
#> The output object Spline 1  has the derivative matrix corrected by the RRM method
#>  given that SLOT 'taylor' is properly given.
#> The matrix derivative is now corrected by method RRM .

x=runif(50)
FData=evspline(spls,x=x) #discrete functional data

matplot(FData[,1],FData[,-1],pch='.',cex=3)
                   #adding small noise to the data
noise=matrix(rnorm(length(x)*10,0,sqrt(var(FData[,2]/10))),ncol=10)

FData[,-1]=FData[,-1]+noise

matplot(FData[,1],FData[,-1],pch='.',cex=3)

knots=runif(12) 

DatProj=project(FData,knots)
#> The knots were not given in the strictly increasing order, which is required. The ordered knots with removed ties are replacing the input knot values.
#> The input numerical data were not given in the increasing order.
#>  The data are ordered and returned in the ordered form as the first value in the output list.
#> The range of the input data is larger than the range of knots in the projection space.
#> 
#> The  7  values at the arguments outside the projection range will not affect the projection.

lines(DatProj$sp) #the projections at the top of the original noised data

plot(DatProj$basis) #the splinet in the problem

#Adding knots to the projection space so that all data points are included
#in the range of the knots for the splinet basis

knots=c(-0.1,0.0,0.1,0.85, 0.9, 1.1,knots)

bases=splinet(knots)
#> Knots were not given in the strictly increasing order, which is required.
#> 
#>           Ordered  knots with removed ties are replacing the input values.

DatProj1=project(FData,basis = bases$os)
#> The input numerical data were not given in the increasing order.
#>  The data are ordered and returned in the ordered form as the first value in the output list.


matplot(FData[,1],FData[,-1],pch='.',cex=3)
lines(DatProj1$sp) 

#Using the B-spline basis

knots=xi

bases=splinet(knots,type='bs')

DatProj3=project(FData,basis = bases$bs)
#> The input numerical data were not given in the increasing order.
#>  The data are ordered and returned in the ordered form as the first value in the output list.

matplot(FData[,1],FData[,-1],pch='.',cex=3)
lines(DatProj3$sp) 

DatProj4=project(FData,knots,k,type='bs') #this includes building the base of order 4
#> The input numerical data were not given in the increasing order.
#>  The data are ordered and returned in the ordered form as the first value in the output list.

matplot(FData[,1],FData[,-1],pch='.',cex=3)
lines(DatProj4$sp) 
lines(spls) #overlying the functions that the original data were built from

#Using two-sided orthonormal basis

DatProj5=project(FData,knots,type='twob')
#> The input numerical data were not given in the increasing order.
#>  The data are ordered and returned in the ordered form as the first value in the output list.
matplot(FData[,1],FData[,-1],pch='.',cex=3)
lines(DatProj5$sp)
lines(spls)

#--------------------------------------------------#
#-----Projecting into a periodic spline space------#
#--------------------------------------------------#
#generating periodic splines
n=1# number of samples
k=3
N=3

n_knots=2^N*k-1 #the number of internal knots for the dyadic case
xi = seq(0, 1, length.out = n_knots+2)

so = splinet(xi,smorder = k, periodic = TRUE) #The splinet basis
stwo = splinet(xi,smorder = k,type='twob', periodic = TRUE) #The two-sided orthogonal basis

plot(so$bs,type='dyadic',main='B-Splines on dyadic structure') #B-splines on the dyadic graph 

plot(stwo$os,main='Symmetric OB-Splines') #The two-sided orthogonal basis 

plot(stwo$os,type='dyadic',main='Symmetric OB-Splines on dyadic structure')

# generating a periodic spline as a linear combination of the periodic splines 
A1= as.matrix(c(1,0,0,0.7,0,0,0,0.8,0,0,0,0.4,0,0,0, 1, 0,0,0,0,0,1,0, .5),nrow= 1)
circular_spline=lincomb(so$os,t(A1))

plot(circular_spline)

#Graphical visualizations of the projections 

Pro_spline=project(circular_spline,basis = so$os,graph = TRUE)
plot(Pro_spline$sp)

#---------------------------------------------------------------#
#---Projecting discretized data into a periodic spline space----#
#---------------------------------------------------------------#
nx=100 # number of discritization 
n=1# number of samples
k=3
N=3

n_knots=2^N*k-1 #the number of internal knots for the dyadic case
xi = seq(0, 1, length.out = n_knots+2)

so = splinet(xi,smorder = k, periodic = TRUE)

hf=1/nx 
grid=seq (hf , 1, by=hf) #grid 
l=length(grid)
BB = evspline(so$os, x =grid)
fbases=matrix(c(BB[,2],BB[,5],BB[,9],BB[,13],BB[,17], BB[,23], BB[,25]), nrow = nx)

#constructing periodic data 
f_circular=matrix(0,ncol=n+1,nrow=nx)
lambda=c(1,0.7,0.8,0.4, 1,1,.5)
f_circular[,1]= BB[,1]
f_circular[,2]= fbases%*%lambda

plot(f_circular[,1], f_circular[,2], type='l')

Pro=project(f_circular,basis = so$os) 
plot(Pro$sp)