[vsnet-chat 260] Re: period determination errors

[Fraser Farrell <fraserf@dove.net.au>]

G'day all,

I said:
   >> Fernie's method does a least-squares fit of a parabola to the
   >>"valley"  (PDM plot) or "peak" (Fourier Transform plot).  For a
   >>parabola of the  form:  P = a*f*f + b*f + c   (where P="power",
   >>f=frequency), the  relevant PASCAL equations are:
   >> true_frequency := -b/(2*a);    {the min or max of the parabola}
   >> sigma := sqrt( abs( (4*a*c-b*b)/(4*a*a*(N-1)) ) );

Taichi replied:
   >Although I have not yet read that paper, the equation seems to be
   >understood when we try to solve P(f) = 0.  The denoted "sigma" is
   >likely to be a measure of the half width of the fitted parabola,
   >extrapolated to the zero signal level (P=0), then normalized by the
   >degree of freedom (probably assuming the deviation from the
   >periodic signal is random and uncorrelated).

If I have understood Fernie's mathematics, "sigma" represents its usual
meaning of "one standard deviation".  Apparently he assumes the errors
are random & uncorrelated; which would be true for a dataset with many
observers.

   >yet ...). The case of PDM seems to be harder to understand.  Since
   >in PDM plot (theta plot) the "zero" signal level corresponds to
   >P(f)=1, it should at least be solved P(f)=1, yielding
   >sigma: = sqrt( abs( (4*a*(c-1)-b*b)/(4*a*a*(N-1)) ) );

No; the item of interest is the x-value (frequency) of the peak or
valley, calculated from -b/(2*a) which is just the standard equation to
find the x-value of where the parabola turns around.  The y-value
(power) of the peak/valley is not really significant in this method.

Fernie's method can also be applied to some lightcurves to determine
times of maximum/minimum.

   >The second point I can not understand is what determines the width
   >of the valley of PDM plot.  Is ther the same rule as in Fourier
   >transform?

I'm not aware of any rule.  From my playing around with test data,
however, the peak/valley widths in both methods:

(1) decreases for longer time spans of data - doubling the time span
seems to halve the width.
(2) decreases for less noisy data, but even a pure sinusoidal signal will
produce peaks (rather than a spike) if it is sampled discontinuously.
(3) increases if more than one frequency is present.
(4) increases if the sampling rate is infrequent.
(5) increases if the main frequency is not constant.

There must be good mathematical reasons for these relationships, but my
math knowledge is best described as a few islands of understanding in an
ocean of ignorance!

I tested a 1200 day span of R Car (M  3.9-10.5v 309d) visual
observations (from our local database) with both PDM and Discrete
Fourier.  It's a well-observed star and the lightcurve suggests
near-constant period over the time span.

PDM spectra showed a deep valley centered on ~310 days with a half-width
of about 30 days, a slight dip at 150-160 days, and not much else.
Fourier showed a big spike at 310 days with a half-width of about 25
days, and smaller spikes at 155 and 29.5 days.  The 155 day period may
be genuine - there are some intriguing bumps on the lightcurve - but the
29.5 day is of course the Full Moons.

Neither method picked up any significant yearly signal because R Car is
circumpolar from here.  Adjustment for systematic errors between
observers noticeably improved the PDM spectrum but had lesser effect on
the Fourier.  Using 10-day mean magnitudes actually worsened both
spectra by "reducing" the data points.  Removing observers from the
dataset - or arbitrarily cutting out 100 day segments - also worsened
both plots.

In summary: many observations for many cycles is the best way to sharpen
up any type of periodogram...
              


cheers,
Fraser Farrell

http://vsnet.dove.net.au/~fraserf/   email: fraserf@dove.net.au
traditional: PO Box 332, Christies Beach, SA  5165, Australia

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