fda.usc (version 0.9.4)

metric.lp: Aproximates 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 p = 2 and weights w = 1.

Usage

metric.lp(fdata1,fdata2=NULL,p=2,w=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.

p

Lp norm, by default it uses p = 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

...

Further arguments passed to or from other methods.

Details

By default it uses the L2-norm with p = 2 and weights w with length=(m-1). $$Let \ \ f(x)= fdata1(x)-fdata2(x)\ \ and\ \ \Delta_i=t_{i+1}-t_i$$ {} $$\left\|f\right\|_p=\left ( \frac{1}{\int_{a}^{b}w(x)dx} \int_{a}^{b} \left|f(x)\right|^{p}w(x)dx \right)^{1/p}$$ $$\approx \Bigg( \sum_{i=1}^{m-1}\frac{\Delta_i({f(x_i)^p+f(x_{i+1})^p})}{2}{w(x_i)}\Bigg)^{1/p}$$ $$\approx \Bigg(\sum_{i=1}^{m-1} {\frac{f(x_i)^p+f(x_{i+1})^p}{2}{w(x_i)}}\Bigg)^{1/p}$$ The observed points on each curve are equally spaced (by default) or not.

See Also

See also semimetric.basis and semimetric.NPFDA

Examples

#	INFERENCE PHONDAT
data(phoneme)
mlearn<-phoneme$learn
mtest<-phoneme$test
glearn<-phoneme$classlearn
gtest<-phoneme$classtest
# Matrix of distances of curves of DATA1
mdist1<-metric.lp(mlearn)
# Matrix of distances beetween curves of DATA1 and curves of DATA2
mdist2<-metric.lp(mlearn,mtest,p=2)

# mdist with L1 norm and weigth=v
v=dnorm(seq(-3,3,length.out=dim(mlearn)[2]-1))
mdist3<-metric.lp(mlearn,mtest,p=1,w=v)
plot(1:250,mdist2[1,],type="l",ylim=c(1,max(mdist2[1,])+1))
lines(mdist3[1,],type="l",col="2")

# mdist with mlearn with diferents 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)

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