fda.usc (version 2.1.0)

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

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. https://www.jstatsoft.org/v51/i04/

See Also

See also semimetric.basis and semimetric.NPFDA

Examples

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)
}