time series in R

Martin Maechler Martin Maechler <maechler@stat.math.ethz.ch>
Tue, 20 Jul 1999 11:32:37 +0200

    BDR> On Tue, 20 Jul 1999, Ross Ihaka wrote:
    >> On Mon, 19 Jul 1999, Prof Brian D Ripley wrote:
    >> > > 3. On the definition question: The existing FFT implementation >
    >> > uses a particular definition for the discrete transform which > >
    >> is pretty standard.  (Edwards "Fourier Series", Brillinger > > "Time
    >> Series" etc.) Using another definition may complicate > >
    >> documentation.
    >> > 
    >> > That's the simple part.  But what is the divisor in the
    >> periodogram? Which > way do lags go in acfs of bivariate series, and
    >> which sign is the phase for > bivariate spectra?
    >> Once you settle the forward discrete transform much of this is
    >> settled.

    BDR> Isn't that the tail wagging the dog?

I think Ross meant to say that  fft()'s definition can't (shouldn't) be changed
anymore, and I agree on that [and that *is* S compatible anyway].

    >> No constant in the dft implies Periodogram = |dft|^2/(2*pi*T)

    BDR> Why is the 2*pi there? It is in Bloomfield, but not Brockwell &
    BDR> Davis, for example. And S-PLUS divides by the fequency to make the
    BDR> periodogram estimate the spectral density.

We have to decide if we go for S-plus compatibility here, see below.

    >> Phases in spectra also fall out if you take a +ve exponent in the
    >> dft.
    BDR> But you have to decide which of series i and j gets the complex
    BDR> conjugate.  (Same issue with acfs: second relative to first or
    BDR> first relative to second?  It's like defining `lags'.)

    >> I don't much mind about these choices, but its probably a good idea
    >> to be consistent.

    BDR> Consistent with whom?

Mostly "with your own notation and definitions",
maybe even ``consistent with one book's author''.

I'd prefer Brockwell & Davis,
however I agree with Brian that S compatibility -- to some extent -- is a
goal here; on the other hand, S-plus uses definitions that are not widely
used in standard texts (my limited experience).

This really must must be settled now.
If nobody can show why S-plus definitions are `wrong by design' here,
I'd vote for adopting them, inspite of all...
[if only to keep V&R's chapter on time-series coherent :-)]

Something which should be discussed however is spectrum(0);
Several of us think that S-plus does the wrong thing, at least in some
cases. If  demean=T (mean removed),  should have periodogram(0) = 0,
and maybe even spectrum(0) = 0 [and hence dB-spec. = -Inf ..]
Another possibility would be to leave it NA
and maybe provide methods for estimating it specifically, if desired.

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