[R] Spectral Coherence

[Rolf Turner]

On 12/07/11 09:04, Joseph Park wrote:
>     Greetings,
>     I would like to estimate a spectral coherence between
>     two timeseries. The stats : spectrum() returns a coh matrix
>     which estimates coherence (squared).
>     A basic test which from which i expect near-zero coherence:
>     x = rnorm(500)
>     y = rnorm(500)
>     xts = ts(x, frequency = 10)
>     yts = ts(y, frequency = 10)
>     gxy = spectrum( cbind( xts, yts ) )
>     plot( gxy $ freq, gxy $ coh )
>     yields a white spectrum of 1. Clearly i'm not using
>     this correctly... or i mis-interpret the coh as a cross-spectral
>     density estimate of coherence |Gxy|^2/(Gxx Gyy)
>     Thanks in advance!
By default spectrum() calls spec.pgram() with spans=NULL.  The result is 
that
it calculates the coherence between x and y as

         |I_{xy}(omega)|^2 / (I_{xx}(omega) * I_{yy}(omega))

where I_{xy}() is the cross periodogram and I_{xx}() and I_{yy}() are 
the respective periodograms.

This quantity will indeed be identically 1 --- see equation 10.1 in 
Bloomfield, second ed.,
page 203.

It would be nice if the help on spectrum mentioned this.

According to Bloomfield what is needed is not the periodograms but 
rather estimated spectra,
smoothed  versions of the periodograms, s_{xy}() etc.  Equation 10.4 in 
Bloomfield, second
ed., page 206 indicates that the spectral estimates satisfy an *inequality*

     |s_{xy}(omega)|^2 <= s_{xx}(omega) * s_{yy}(omega)

whence the coherence is always between 0 and 1.

To resolve the problem you need to specify the spans argument in the 
call to spectrum,
e.g.

     gxy <- spectrum(cbind(xts,yts),spans=c(5,7))

HTH

     cheers,

         Rolf Turner