fda.usc (version 2.1.0)

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.

Author

Manuel Febrero-Bande, Manuel Oviedo de la Fuente

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 \(\left(f_i(t)=0\ \& \ g_j(t)=0\right) \Longrightarrow metric.kl(f(t),g(t))=0\).

If \(\left|f_i(t)-g_i(t) \right|\leq \epsilon \Longrightarrow f_i(t)=f_i(t)+\epsilon\), 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

if (FALSE) {   
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)  
}