[Rd] CCF and ACF

[Gabor Grothendieck]

Using "ts" objects these can be written in the following consistent
manner:

> sum(ts(x) * lag(ts(x)) / sum(ts(x)^2))
[1] -0.2648633

> sum(ts(x) * lag(ts(y)) / sqrt(sum(ts(y)^2) * sum(ts(x)^2)))
[1] 0.5930216

On 10/2/06, Simone Giannerini <sgiannerini at gmail.com> wrote:
> 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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