fsemipar (version 1.1.1)

semimetric.projec: Projection semi-metric computation

Description

Computes the projection semi-metric between each curve in data1 and each curve in data2, given a functional index \(\theta\).

Usage

semimetric.projec(data1, data2, theta, order.Bspline = 3, nknot.theta = 3,
  range.grid = NULL, nknot = NULL)

Value

A matrix containing the projection semi-metrics for each pair of curves.

Arguments

data1

Matrix containing functional data collected by row.

data2

Matrix containing functional data collected by row.

theta

Vector containing the coefficients of \(\theta\) in a B-spline basis, so that length(theta)=order.Bspline+nknot.theta.

order.Bspline

Order of the B-spline basis functions for the B-spline representation of \(\theta\). This is the number of coefficients in each piecewise polynomial segment. The default is 3.

nknot.theta

Number of regularly spaced interior knots of the B-spline basis. The default is 3.

range.grid

Vector of length 2 containing the range of the discretisation of the functional data. If range.grid=NULL, then range.grid=c(1,p) is considered, where p is the discretization size of data (i.e. ncol(data)).

nknot

Number of regularly spaced interior knots for the B-spline representation of the functional data. The default value is (p - order.Bspline - 1)%/%2.

Author

German Aneiros Perez

Silvia Novo Diaz

Details

For \(x_1,x_2 \in \mathcal{H}, \), where \(\mathcal{H}\) is a separable Hilbert space, the projection semi-metric in the direction \(\theta\in \mathcal{H}\) is defined as $$d_{\theta}(x_1,x_2)=|\langle\theta,x_1-x_2\rangle|.$$

The function semimetric.projec computes this projection semi-metric using the B-spline representation of the curves and \(\theta\). The dimension of the B-spline basis for \(\theta\) is determined by order.Bspline+nknot.theta.

References

Novo S., Aneiros, G., and Vieu, P., (2019) Automatic and location-adaptive estimation in functional single--index regression. Journal of Nonparametric Statistics, 31(2), 364--392, tools:::Rd_expr_doi("https://doi.org/10.1080/10485252.2019.1567726").

See Also

See also projec.

Examples

data("Tecator")
names(Tecator)
y<-Tecator$fat
X<-Tecator$absor.spectra

#length(theta)=6=order.Bspline+nknot.theta 
semimetric.projec(data1=X[1:5,], data2=X[5:10,],theta=c(1,0,0,1,1,-1),
  nknot.theta=3,nknot=20,range.grid=c(850,1050))