This section contains the detailed formulation
of a weighted three stage period analysis
for the simple model of Sect. 2.1.
The input data
at
are
(time),
(observation)
and
(error),
i.e. the weights are ,
where
.
The
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
interval between
P_{min} (
f_{max}^{-1})
and
P_{max} (
f_{min}^{-1}).
Typical correlation lengths in time
(
D_{min},
D_{max})
and phase ()
are
,
,
and
The PSch uses four sets of
pairs:
t_{i,j},
y_{i,j},
w_{i,j} and
(
and
).
The first three,
,
and
,
are independent of f,
but the
are not.
The PSch periodogram determined at all integer multiples of
between
f_{min} and
f_{max} is
(5) |
This algorithm divides the data into J bins with respect to the time differences t_{i,j} between D_{min} and D_{max}. The original t_{i,j}, y_{i,j} and w_{i,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 y_{i,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(t_{i,j}) (Eq. 4) excludes y_{i,j} too far in t_{i,j}, phase shifts during time intervals longer than D_{max} do not influence the periodogram. The combination of and would include all data (Eq. 4), but D_{min} is applied, because y_{i,j} do not contain significant information when t_{i,j} goes below P_{min}. Adjusting D_{min} and D_{max} determines the number of y_{i,j} selected with W(t_{i,j}). The function selects only closer than in (Eq. 5). For example, determines a sinusoidal model (Eqs. 1 and 2). The y_{i,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 y_{i,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 P_{1}, ..., P_{5}. b-f) The with these P_{1}, ..., P_{5}. g-k) The diamonds on (Eq. 7: ) indicate the more accurate P_{1}, ..., P_{5} obtained with the GSch. l-p) RSch determines the final P_{1}, ..., P_{5}, 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 P_{1}, ..., P_{5} 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 F_{S}(u) and F(u) (Eqs. 12 and 13) for the residuals confirm a Gaussian distribution (i.e. H_{G} 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 t_{min,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 t_{min,2}. The bootstrap samples with no t_{min},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 y_{i,j} closer than
in
may
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
between
and
,
where
is called the "overfilling factor''
and
.
The discrete tested f set within this narrow interval is denser than in the
PSch (i.e.
).
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 t_{min,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)
to compute
.
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
for RSch.
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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