My mistake was a missing factor in the integral (I used Chatfield "The
Analysis of Time Series", 1980 and not V&R...) and the omission to turn off
tapering. (Also detrending should be turned off which did not make much of
a difference in my case though)
Don Percival also supplied a method for correcting the spectrum when
tapering is used.
Further reading:
"Spectral Analysis for Physical Applications," Percival &
Walden, Cambridge University Press, 1993
WRONG:
> x <- arima.sim(n=1000, model=list(ar=0.7)) # simulate AR process
> spec <- spectrum(x, span=20) # calculate smoothed spectrum
> f <- 10^(spec$spec/10) # convert dB into real numbers
> n <- length(f)
> w <- 2*pi*spec$freq # convert frequency into angular frequency
> sum(f[-n]*diff(w)) # simple integration
[1] 5.442544
> var(x)
[1] 2.021829
CORRECT:
x <- arima.sim(n=1000, model=list(ar=0.7)) # simulate AR process
spec <- spectrum(x, span=20, detrend=T, taper=0.0) # calculate smoothed
spectrum
f <- 10^(spec$spec/10) # convert dB into real numbers
2*(sum(f)-f[1])*diff(spec$freq[1:2]) # simple integration
[1] 2.153871
var(x)
[1] 2.167155
-----------------------------------------------------------------
Jan Schelling
Institute of Process Engineering
ETH Zurich, 8092 Switzerland
phone: ++41-1-633 62 63
fax: ++41-1-633 11 19
email: schelling@ivuk.mavt.ethz.ch
internet: http://www.ivuk.ethz.ch/staff/schelling/
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