R: fcenLRu transformation (functional)

fcenLRu {robCompositions} | R Documentation

## fcenLRu transformation (functional)

### Description

fcenLR[u] transformation: mapping from B2(P) into unweigted L2(lambda)

### Usage

    fcenLRu(z, z_step, density, p)

### Arguments

`z` | grid of points defining the abscissa
`z_step` | step of the grid of the abscissa
`density` | grid evaluation of the P-density
`p` | density of the reference measure P

### Value

``out`` | grid evaluation of the P-density in unweighted L2(lambda)

### Author(s)

R. Talskatalskarenata@seznam.cz, A. Menafoglio, K. Hronkarel.hron@upol.cz, J. J. Egozcue, J. Palarea-Albaladejo

### References

Talska, R., Menafoglio, A., Hron, K., Egozcue, J. J., Palarea-Albaladejo, J. (2020). Weighting the domain of probability densities in functional data analysis.Stat(2020). https://doi.org/10.1002/sta4.283

### Examples

    # Common example for all transformations - fcenLR, fcenLRp, fcenLRu
    # Example (log normal distribution under the reference P)
    t = seq(1,10, length = 1000)
    t_step = diff(t[1:2])

    # Log normal density w.r.t. Lebesgue reference measure in B2(lambda)
    f = dlnorm(t, meanlog = 1.5, sdlog = 0.5)

    # Log normal density w.r.t. Lebesgue reference measure in L2(lambda)
    f.fcenLR = fcenLR(t,t_step,f)

    # New reference given by exponential density
    p = dexp(t,0.25)/trapzc(t_step,dexp(t,0.25))

    # Plot of log normal density w.r.t. Lebesgue reference measure
    # in B2(lambda) together with the new reference density p
    matplot(t,f,type="l",las=1, ylab="density",cex.lab=1.2,cex.axis=1.2,
      col="black",lwd=2,ylim=c(0,0.3),xlab="t")
    matlines(t,p,col="blue")
    text(2,0.25,"p",col="blue")
    text(4,0.22,"f",col="black")

    # Log-normal density w.r.t. exponential distribution in B2(P)
    # (unit-integral representation)
    fp = (f/p)/trapzc(t_step,f/p)

    # Log-normal density w.r.t. exponential distribution in L2(P)
    fp.fcenLRp = fcenLRp(t,t_step,fp,p)

    # Log-normal density w.r.t. exponential distribution in L2(lambda)
    fp.fcenLRu = fcenLRu(t,t_step,fp,p)

    # Log-normal density w.r.t. exponential distribution in B2(lambda)
    fp.u = fcenLRinv(t,t_step,fp.fcenLRu)

    # Plot
    layout(rbind(c(1,2,3,4),c(7,8,5,6)))
    par(cex=1.1)

    plot(t, f.fcenLR, type='l', ylab=expression(fcenLR[lambda](f)),
      xlab='t',las=1,ylim=c(-3,3),
      main=expression(bold(atop(paste('(a) Representation of f in ', L^2, (lambda)),'[not weighted]'))))
    abline(h=0,col="red")

    plot(t, f, type='l', ylab=expression(f[lambda]),
      xlab='t',las=1,ylim=c(0,0.4),
      main=expression(bold(atop(paste('(b) Density f in ', B^2, (lambda)),'[not weighted]'))))

    plot(t, fp, type='l', ylab=expression(f[P]), xlab='t',
      las=1,ylim=c(0,0.4),
      main=expression(bold(atop(paste('(c) Density f in ', B^2, (P)),'[weighted with P]'))))

    plot(t, fp.fcenLRp, type='l', ylab=expression(fcenLR[P](f[P])),
      xlab='t',las=1,ylim=c(-3,3),
      main=expression(bold(atop(paste('(d) Representation of f in ', L^2, (P)),'[weighted with P]'))))
    abline(h=0,col="red")

    plot(t, fp.u, type='l', ylab=expression(paste(omega^(-1),(f[P]))),
      xlab='t',las=1,ylim=c(0,0.4),
      main=expression(bold(atop(paste('(e) Representation of f in ', B^2, (lambda)),'[unweighted]'))))

    plot(t, fp.fcenLRu, type='l', ylab=expression(paste(fcenLR[u](f[P]))),
      xlab='t',las=1,ylim=c(-3,3),
      main=expression(bold(atop(paste('(f) Representation of f in ', L^2, (lambda)),'[unweighted]'))))
    abline(h=0,col="red")

[Package robCompositions version 2.3.1 Index]