Dear all,
given two numeric vectors x and y, the ACF(x) at lag k is
cor(x(t),x(t+k)) while the CCF(x,y) at lag k is cor(x(t),y(t-k)). See
below for a simple example.
> set.seed(1)
> x <- rnorm(10)
> y <- rnorm(10)
> x
[1] -0.6264538 0.1836433 -0.8356286 1.5952808 0.3295078 -0.8204684
0.4874291 0.7383247 0.5757814 -0.3053884
> y
[1] 1.51178117 0.38984324 -0.62124058 -2.21469989 1.12493092
-0.04493361 -0.01619026 0.94383621 0.82122120 0.59390132
> x <- x - mean(x);
> y <- y -mean(y);
## ACF OF x(t)
> print(acf(x,lag=2,plot=F),digits=5))
Autocorrelations of series 'x', by lag
0 1 2
1.00000 -0.26486 -0.25352
## For Instance ACF(x) at lag 1 is Corr(x(t),x(t+1)):
> sum(x[1:9]*x[2:10])/sum(x^2)
[1] -0.2648633
## whereas CCF(x,y) at lag 1 is Corr(x(t),y(t-1))
> ccf(x,y,lag=2,plot=F)
Autocorrelations of series 'X', by lag
-2 -1 0 1 2
-0.139 0.593 -0.377 -0.382 0.116
> sum(x[1:9]*y[2:10])/sqrt(sum(x^2)*sum(y^2))
[1] 0.5930216
> sum(x[2:10]*y[1:9])/sqrt(sum(x^2)*sum(y^2))
[1] -0.3822885
from a quick survey on textbooks (Brockwell and Davis, Chatfield,
Fuller) it looks like different authors use different conventions so
that I think that it would be nice to clarify this in the
documentation.
Ciao
Simone
> R.version
_
platform i386-pc-mingw32
arch i386
os mingw32
system i386, mingw32
status beta
major 2
minor 4.0
year 2006
month 09
day 20
svn rev 39421
language R
version.string R version 2.4.0 beta (2006-09-20 r39421)
> Sys.getlocale()
[1] "LC_COLLATE=English_United States.1252;LC_CTYPE=English_United
States.1252;LC_MONETARY=English_United
States.1252;LC_NUMERIC=C;LC_TIME=English_United States.1252"
______________________________________________________
Simone Giannerini
Dipartimento di Scienze Statistiche "Paolo Fortunati"
Universita' di Bologna
Via delle belle arti 41 - 40126 Bologna, ITALY
Tel: +39 051 2098262 Fax: +39 051 232153
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