semimetric.projec {fsemipar}

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)

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.

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.

Value

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

Author(s)

German Aneiros Perez [e-mail]

Silvia Novo Diaz [e-mail]

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, doi: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))