extracat (version 1.7-6)

approx.dcor: Distance Correlation Approximation

Description

Computes the distance correlation for two variables using an approximation based on binning and wdcor.table. The approximation underestimates the true value by a small error depending on the number of bins. (In simulations with the default of 50 bins the average error was about 0.001.)

Usage

approx.dcor(x, y, n = 50, ep = 1, bin = "eq")

Arguments

x

A numeric vector.

y

A numeric vector.

n

The number of bins per variable.

ep

The euclidean distances are taken to the power of ep.

bin

Either "eq" or "q" for equidistant breakpoints or quantile breakpoints.

Value

The correlation value which is between 0 and 1.

References

Szekely, G. J. Rizzo, M. L. and Bakirov, N. K. (2007). "Measuring and testing independence by correlation of distances", Annals of Statistics, 35/6, 2769-2794

Examples

# NOT RUN {
# The straightforward way of approximating the distance correlation fails:
# for instance the computation of dcor for a random sample with 4000 observations
# takes pretty long but drawing samples of 500, 1000 or even 2000 observations
# leads to relatively big errors.
# The approximation via approx.dcor is very fast and for 
# n = 50 or n=100 the results are very close to the truth

require(energy)
x<- rnorm(4000,mean=10,sd=3)
y <- rnorm(1,sd=0.01)*(x-10)^3 + rnorm(1,sd=0.1)*(x-10)^2
 + rnorm(1)*(x-10)+rnorm(4000,sd=4)

system.time(dd <- dcor(x,y))
system.time(dd0 <- wdcor(x,y))[[3]]
dd - dd0


system.time(da100 <- approx.dcor(x,y,100))[[3]]
da100-dd0
	
# For a smaller sample size we can try out how good the approximation really is:
test<-replicate(100,{
N <- 1000
x<- rnorm(N,mean=10,sd=3)
y <- rnorm(1,sd=0.01)*(x-10)^3 + rnorm(1,sd=0.1)*(x-10)^2 
y <- y + rnorm(1)*(x-10)+rnorm(N,sd=4)

dd <- wdcor(x,y)
dd25 <- approx.dcor(x,y,25)
dd50 <- approx.dcor(x,y,50)
dd100 <- approx.dcor(x,y,100)
dd75 <- approx.dcor(x,y,75)

dd25c <- approx.dcor(x,y,25,correct = TRUE)
dd50c <- approx.dcor(x,y,50,correct = TRUE)
dd100c <- approx.dcor(x,y,100,correct = TRUE)
dd75c <- approx.dcor(x,y,75,correct = TRUE)
c(2*dd, dd25, dd50, dd75, dd100, dd25c, dd50c, dd75c, dd100c)-dd
})

rm<-apply(test,1,mean)

plot( seq(25,100,25), rm[2:5],type="l",
 ylim= c(min(rm),abs(min(rm))), xlab = "No. of bins per axis",ylab = "error")
points( seq(25,100,25), rm[2:5],pch=19)
points( seq(25,100,25), rm[6:9],type="l", col=2)
points( seq(25,100,25), rm[6:9],pch=19,col=2)	
abline(h=0,lwd=3)
legend( 25,abs(min(rm)),legend=c("raw value after binning","corrected value"),
col=1:2,lwd=3)
# }