A sample of *n* data points *y*(*t*_{i})
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

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

- 1.
- Divide the data into
*k*separate*parts*with a length of . Compute the LSS spectrum within each part using a frequency step , i.e. the number of tested frequencies is . Use the average of all spectra for obtaining the crude periodicity estimates. - 2.
- Compute the fine grid search spectra forApply a frequency step of . A suitable width for the tested frequency interval is about , where is the mean frequency step that was used during the 1st stage. With this particular width, the number of tested frequencies is .
*all data*within narrow frequency intervals centered at these crude estimates. - 3.
- 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

The value of gives the maximum resolution for the spectrum, because 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

Although 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.

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