This section contains the detailed formulation
of a weighted three stage period analysis
for the simple model of Sect. 2.1.
The input data
i.e. the weights are ,
free parameters (
for our K:th order model
The order K (Eq. 1) determines the PSch "model''
(Eq. (2) below).
The PSch searches for the best period "candidates'' over a long
Typical correlation lengths in time
and phase ()
The PSch uses four sets of
The first three,
are independent of f,
The PSch periodogram determined at all integer multiples of
This algorithm divides the data into J bins with respect to the time differences ti,j between Dmin and Dmax. The original ti,j, yi,j and wi,j data are replaced by the J averages t'q, y'q and w'q within these bins. The algorithm is efficient, because the Z function in Eq. (6) has to applied only to J () values at each tested f.
The yi,j differences closer than in are smaller for a good f candidate, i.e. the data form a continuous curve. Such f minimize the PSch periodogram (Eqs. 3 or 6), the case being opposite for poor f candidates. Because W(ti,j) (Eq. 4) excludes yi,j too far in ti,j, phase shifts during time intervals longer than Dmax do not influence the periodogram. The combination of and would include all data (Eq. 4), but Dmin is applied, because yi,j do not contain significant information when ti,j goes below Pmin. Adjusting Dmin and Dmax determines the number of yi,j selected with W(ti,j). The function selects only closer than in (Eq. 5). For example, determines a sinusoidal model (Eqs. 1 and 2). The yi,j on a sinusoid correlate within , or those on a double wave () within . Reduction of enables detection of more complex variation, but reduces the number of yi,j selected with , i.e. requires more data.
|Figure 1: The second order TSPA for SET=42 (): a) (Eq. 6) is the PSch periodogram between and with correlation lengths , and . The number of independent frequencies is (Eq. 9). The diamonds on mark the five best periods P1, ..., P5. b-f) The with these P1, ..., P5. g-k) The diamonds on (Eq. 7: ) indicate the more accurate P1, ..., P5 obtained with the GSch. l-p) RSch determines the final P1, ..., P5, and their (Eq. 10). The continuous lines connect each to the closest point of the model . q-u) The versus (see end of Sect. 5) of each P1, ..., P5 for given by (Eq. 15). The critical levels for the linear correlations between and are|
|Figure 2: The RSch bootstrap of SET=42: a) The model () with already shown in Fig. 1l. b) The FS(u) and F(u) (Eqs. 12 and 13) for the residuals confirm a Gaussian distribution (i.e. HG is not rejected with in Eq. 14). A dark rectangle indicates the location and height of the Kolmogorov-Smirnov test statistic ( ). c and d) The bootstrap M estimates (open squares) also follow a Gaussian distribution. e-j) The same as in c and d) for the P, A and tmin,1 estimates|
|Figure 3: Same as in Fig. 1 for SET=114|
|Figure 4: Same as in Fig. 2 for SET=114. Note: e and f) is rejected for P (Eq. 14). k) Only estimates are obtained for tmin,2. The bootstrap samples with no tmin,2 are marked with vertical lines|
The PSch over a long f interval
usually detects numerous frequency "candidates'' f'.
More accurate values for
at least five best f' are determined with the GSch.
The TSPA applications for real data may sometimes require
testing of a much larger number of f' detected with the PSch.
Here the limit of testing only the five best f' was chosen,
because it is convenient for the graphical representation of the results,
like in Figs. 1, 3, 5 and 7.
The PSch "model''
can not fully constrain
the modelling order K (Eqs. 1 and 2),
because yi,j closer than
correlate in several different models.
For example, the PSch could
detect a box function or a sinusoid with
The model is fixed in the GSch, e.g. if
GSch proceeds with a
model (Eq. 1).
If PSch detects f', then
GSch tests all integer multiples of
is called the "overfilling factor''
The discrete tested f set within this narrow interval is denser than in the
Standard linear least squares fits to
with for every tested f in Eq. (1) (i.e. f is not a free parameter)
determine the GSch periodogram
|Figure 5: The fifth order TSPA () for the V magnitudes of the cepheid variable BL Her, otherwise as in Fig. 1|
|Figure 6: The fifth order RSch bootstrap with for the V magnitudes of BL Her, otherwise as in Fig. 2. Note: k) Only estimates are obtained for tmin,2|
|Figure 7: The first order TSPA () between and for the V magnitudes of SAO50205 during subsets SET=111, 112, 113 and 114, otherwise as in Fig. 1|
|Figure 8: The first order RSch bootstrap with for the V magnitudes of SAO50205, otherwise as in Fig. 2|
The model (Eq. 1) is nonlinear when f is a free parameter.
The RSch performs a standard Marquardt iteration
(e.g. Press et al. 1988)
But the result of this iterative refinement
depends on the trial solution (
Large discrete f sets are tested with PSch and GSch only to
provide a reliable
Combining the f' detected in the GSch to
(Eq. 7) gives
While the PSch and GSch test discrete f sets,
the RSch is
continuous in f, because it utilizes the analytical properties of
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