metric.lp: Approximates Lp-metric distances for functional data.

Description

Measures the proximity between the functional data and curves approximating Lp-metric. If w = 1 approximates the Lp-metric by Simpson's rule. By default it uses lp = 2 and weights w = 1.

Usage

metric.lp(fdata1, fdata2 = NULL, lp = 2, w = 1, dscale = 1, ...)

Arguments

fdata1: Functional data 1 or curve 1. If fdata class, the dimension of fdata1$data object is (n1 x m), where n1 is the number of curves and m are the points observed in each curve.
fdata2: Functional data 2 or curve 2. If fdata class, the dimension of fdata2$data object is (n2 x m), where n2 is the number of curves and m are the points observed in each curve.
lp: Lp norm, by default it uses lp = 2
w: Vector of weights with length m, If w = 1 approximates the metric Lp by Simpson's rule. By default it uses w = 1
dscale: If scale is a numeric, the distance matrix is divided by the scale value. If scale is a function (as the mean for example) the distance matrix is divided by the corresponding value from the output of the function.
...: Further arguments passed to or from other methods.

Author

Manuel Febrero-Bande, Manuel Oviedo de la Fuente manuel.oviedo@usc.es

Details

By default it uses the L2-norm with lp = 2. $$Let \ \ f(x)= fdata1(x)-fdata2(x)$$ $$\left\|f\right\|_p=\left ( \frac{1}{\int_{a}^{b}w(x)dx} \int_{a}^{b} \left|f(x)\right|^{p}w(x)dx \right)^{1/p}$$
The observed points on each curve are equally spaced (by default) or not.

The L$\infty$-norm is computed with lp = 0. $$d(fdata1(x),fdata2(x))_{\infty}=sup \left|fdata1(x)-fdata2(x)\right|$$

References

Febrero-Bande, M., Oviedo de la Fuente, M. (2012). Statistical Computing in Functional Data Analysis: The R Package fda.usc. Journal of Statistical Software, 51(4), 1-28. http://www.jstatsoft.org/v51/i04/

Examples

Run this code

if (FALSE) {
#	INFERENCE PHONDAT
data(phoneme)
mlearn<-phoneme$learn[1:100]
mtest<-phoneme$test[1:100]
glearn<-phoneme$classlearn[1:100]
gtest<-phoneme$classtest[1:100]
# Matrix of distances of curves of DATA1
mdist1<-metric.lp(mlearn)

# Matrix of distances between curves of DATA1 and curves of DATA2
mdist2<-metric.lp(mlearn,mtest,lp=2)
# mdist with L1 norm and weigth=v
v=dnorm(seq(-3,3,len=dim(mlearn)[2]))
mdist3<-metric.lp(mlearn,mtest,lp=1,w=v)
plot(1:100,mdist2[1,],type="l",ylim=c(1,max(mdist3[1,])))
lines(mdist3[1,],type="l",col="2")

# mdist with mlearn with different discretization points.
# mlearn2=mlearn
# mlearn2[["argvals"]]=seq(0,1,len=150)
# mdist5<-metric.lp(mlearn,mlearn2)
# mdist6<-metric.lp(mlearn2,mlearn) 
# sum(mdist5-mdist6)
# sum(mdist1-mdist6)

x<-seq(0,2*pi,length=1001)
fx<-fdata(sin(x)/sqrt(pi),x)
fx0<-fdata(rep(0,length(x)),x)
metric.lp(fx,fx0)
# The same
integrate(function(x){(abs(sin(x)/sqrt(pi))^2)},0,2*pi)
}

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