Up: Three stage period analysis
A sample of n data points y(ti)
is fitted to a model
in our period search.
The Q free parameters are
The most probable periodicity is searched for by determining
the absolute minimum for the sum of the squared residuals
Let us assume that the model can be expressed as a linear sum
of separate model components
which depend nonlinearly on the periods (frequencies) or other
parameters (e.g. decay rates, frequency shifts).
In this case the
RSS can be evaluated as a function
of the nonlinear parameters only,
because the linear parameters
be estimated by ordinary regression methods for any fixed
set of nonlinear parameters
This gives the Least Squares Spectrum
where the linear parameter estimates
are also functions of the nonlinear parameters.
LSS minima reveal the most probable
nonlinear parameter combinations.
If the model is correctly specified,
LSS minimum should give just the set of parameter
estimates searched for.
In this way the problem of model parameter estimation
can be converted to the problem of solving the global minimum or minima
This approach has been applied to astronomical data already
by Vanicek (1969, 1971)
and Taylor & Hamilton (1972),
or more recently by
Martinez & Koen (1994),
Wilcox & Wilcox (1995) and
Bossi & La Franceschina (1995).
LSS analysis is not trivial for semirandomly
like the astronomical data that often contain periodic gaps.
We formulate a flexible and computationally efficient approach to
this problem by dividing the analysis into several stages,
where the resolution of the spectra increases at each subsequent stage.
2.1 Multistage search
The first stage of the analysis, the Pilot Search,
provides crude estimates for the nonlinear parameters.
A standard Grid Search on a fine mesh centered at
these crude estimates is performed during the second stage.
Finally, traditional parameter refinement techniques are applied to
obtain the highest possible precision (Refined Search).
While the grid search and the refinement of the model parameters
represent commonly used astronomical data processing techniques,
the pilot search method is not so well known.
As a particular case of the general problem,
we consider the
LSS of the simple model
where the frequency f is the only nonlinear free parameter.
The linear free parameters for the mean ()
and the amplitudes
are functions of f.
The grid search over the full nonlinear parameter space
tests a discrete frequency set
fmax limits may represent
the physically possible frequency range or some other interval of interest.
The fixed frequency step
must be chosen carefully.
If the data cover an interval of
it is reasonable constrain the maximum phase shift
can occur within
when any tested fl
changes by .
Sufficient values are
like the overfilling factor of
in Sect. 3.2.
Hence the corresponding number of tested frequencies
If the tested periods are much shorter than ,
then L becomes large.
one can reduce L by shortening
The following general scheme of a simple multistage
on the first,
on the second stage,
Divide the data into k
with a length
LSS spectrum within each part using a frequency
i.e. the number of tested frequencies is
Use the average of all spectra for obtaining the crude periodicity
Compute the fine grid search spectra for all data
within narrow frequency intervals centered at these crude estimates.
Apply a frequency step of
A suitable width for the tested frequency interval
the mean frequency step that was used during the 1st stage.
With this particular width, the number of tested frequencies is
Refine the precision of the frequency estimates
with the standard Marquardt iteration
(see e.g. Press et al. 1988
The simplest multistage search techniques were
first described by Evans (1961).
Since then the computation of long spectra with powerful
computers has become more popular than the
application of complex multistage algorithms.
2.2 Nonparametric dispersion estimation
The average of the local
LSS spectra gave the crude
frequency estimates during the first stage of the simple
multistage search described above.
Pelt (1983, 1997)
has introduced an alternative to compute these estimates.
The idea is that if the data form a continuous periodic curve,
the dispersion around this curve can be estimated without solving
the curve itself.
This dispersion is measured with the nonparametric test statistic
deviates from zero only if
test statistic is a good approximation of
A spectral window can be introduced like in
the standard Fourier estimation.
This is achieved with the more general formulation
that allows additional smoothing with the data window
The value of
gives the maximum resolution for the
for all pairs of observations.
Adjusting this algorithm parameter to
enables a faster computation of a smoother spectrum.
This reduces the number of tested frequencies during the
first stage of the analysis,
because a smaller
Dmax allows a longer step
in the tested frequencies.
involves double summation for every tested frequency,
this apparently complex test statistic can be computed effectively
by rearranging these sums and using trigonometric approximations.
Such computational techniques for different realizations of the
algorithm can be found in (Pelt 1980, 1992).
The application of this method speeds up the complete multistage
analysis, because the
spectrum for a full grid of
tested frequencies can usually be computed as efficiently as
the fast Fourier transform.
One particular version of a multistage analysis that can
also utilize the additional information of the measurement errors
is presented below.
Up: Three stage period analysis
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