metric.kl: Kullback-Leibler distance

In fda.usc: Functional Data Analysis and Utilities for Statistical Computing

Kullback–Leibler distance

Description

Measures the proximity between two groups of densities (of class fdata) by computing the Kullback–Leibler distance.

Usage

metric.kl(fdata1, fdata2 = NULL, symm = TRUE, base = exp(1), eps = 1e-10, ...)

Arguments

fdata1	Functional data 1 (fdata class) with the densities. The dimension of fdata1 object is (n1 x m), where n1 is the number of densities and m is the number of coordinates of the points where the density is observed.
fdata2	Functional data 2 (fdata class) with the densities. The dimension of fdata2 object is (n2 x m).
symm	If TRUE the symmetric K–L distance is computed, see details section.
base	The logarithm base used to compute the distance.
eps	Tolerance value.
...	Further arguments passed to or from other methods.

Details

Kullback–Leibler distance between f(t) and g(t) is

dist(f(t),g(t))= \int_{a}^{b} f(t)* log(f(t)/g(t)) dt

where t are the m coordinates of the points where the density is observed (the argvals of the fdata object).

The Kullback–Leibler distance is asymmetric,

dist(f(t),g(t))!=dist(g(t),f(t))

A symmetry version of K–L distance (by default) can be obtained by

0.5*(dist(f(t),g(t))+dist(g(t),f(t)))

If (f(t)=0 and g(t)=0), then metric.kl(f(t),g(t))=0.

If abs(f(t)-g(t))<ε, then f(t)=f(t)+ε, where ε is the tolerance value (by default eps=1e-10).

The coordinates of the points where the density is observed (discretization points t) can be equally spaced (by default) or not.

Author(s)

Manuel Febrero-Bande, Manuel Oviedo de la Fuente [e-mail]

References

Kullback, S., Leibler, R.A. (1951). On information and sufficiency. Annals of Mathematical Statistics, 22: 79-86

See Also

See also metric.lp and fdata

Examples

## Not run:    
n<-201                                                                                       
tt01<-seq(0,1,len=n)                                                                         
rtt01<-c(0,1)  
x1<-dbeta(tt01,20,5)                                                                           
x2<-dbeta(tt01,21,5)                                                                           
y1<-dbeta(tt01,5,20)                                                                           
y2<-dbeta(tt01,5,21)                                                                           
xy<-fdata(rbind(x1,x2,y1,y2),tt01,rtt01)
plot(xy)
round(metric.kl(xy,xy,eps=1e-5),6)  
round(metric.kl(xy,eps=1e-5),6)
round(metric.kl(xy,eps=1e-6),6)
round(metric.kl(xy,xy,symm=FALSE,eps=1e-5),6)  
round(metric.kl(xy,symm=FALSE,eps=1e-5),6)

plot(c(fdata(y1[1:101]),fdata(y2[1:101])))                       
metric.kl(fdata(x1))  
metric.kl(fdata(x1),fdata(x2),eps=1e-5,symm=F)       
metric.kl(fdata(x1),fdata(x2),eps=1e-6,symm=F)       
metric.kl(fdata(y1[1:101]),fdata(y2[1:101]),eps=1e-13,symm=F)  
metric.kl(fdata(y1[1:101]),fdata(y2[1:101]),eps=1e-14,symm=F)  

## End(Not run)   

fda.usc documentation built on Oct. 17, 2022, 9:06 a.m.