fda.usc (version 1.2.3)

metric.kl: 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 $$ metric.kl(f(t),g(t))= \int_{a}^{b} {f(t) log\left(\frac{f(t)}{g(t)}\right)dt} $$ 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, $$metric.kl(f(t),g(t))\neq metric.kl(g(t),f(t))$$ A symmetry version of K--L distance (by default) can be obtained by $$0.5\left(metric.kl(f(t),g(t))+metric.kl(g(t),f(t))\right)$$

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

If $abs(f(t)-g(t))<\epsilon, then="" f(t)="f(t)+\epsilon$," where="" $\epsilon$="" 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.

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:    
# int.simpson2<-fda.usc:::int.simpson2
# 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)