Time-resolved Analysis

Table 1: Weight functions for time-resolved SIGSPEC analysis. The beginning of the time interval associated with the referring subset is denoted $t_B$, whereas $t_C$ symbolises the centre of the time interval.

keyword	arguments	weight function
timeres:w:none		$1$
timeres:w:ipow	$\xi$	$0$ if $t=t_C$, $\left\vert\, t-t_C\right\vert ^{-\xi}$ else
timeres:w:gauss	$\sigma$	$\mathrm{e}^{-\left(\frac{ t-t_C}{\sigma}\right) ^2}$
timeres:w:exp	$\zeta$	$\mathrm{e}^{-\frac{\left\vert t-t_C\right\vert}{\zeta}}$
timeres:w:damp	$\zeta$	$\mathrm{e}^{-\frac{t-t_B}{\zeta}}$
timeres:w:cos	$\nu ,\Phi$	$\cos\left( 2\pi\nu\left\vert\, t-t_C\right\vert -\Phi\right)$
timeres:w:cosp	$\nu ,\Phi ,\xi$	$\cos\, ^{\xi}\left( 2\pi\nu\left\vert\, t-t_C\right\vert -\Phi\right)$

In time-resolved mode, SIGSPEC performs an analysis for a set of time intervals rather than for the entire time series. An interval of width given by the keyword timeres:range is moved in steps the width of which is given by the keyword timeres:step from the beginning of the time series to the end. Consecutive time intervals are free to overlap. Time series data within such an interval are used to form a subset for which the analysis is performed. In addition, statistical weights may be applied to the subset data, all with respect to the centre of the interval, which shall be denoted $t_C$.

The only exception is the keyword timeres:w:damp. In this case, the analysis is optimised for signal excited at the beginning of the time interval corresponding to the subset under consideration, $t_B$ and exponentially damped towards the end of the subset.

The weight functions of time are given in Table1. The normalisation of weights is automatically performed by SIGSPEC. Also the combination of a weight function for time-resolved mode with weights columns (keyword col:weights) is supported.

In time-resolved mode, the set of output files as given in ``Default Output'', p., is generated for each subset of the time series. This requires the introduction of an additional six-digit index, #interval#, in addition to #iteration#, and the annotation for the output files is

1. wts.#interval#.dat for the weight function vs. time in each subset,
2. s#iteration#.#interval#.dat for the spectra,
3. t#iteration#.#interval#.dat for the residuals after each step of prewhitening,
4. r#iteration#.#interval#.dat for the results after each step of prewhitening,
5. m#index#.#interval#.dat for the results after each step of prewhitening,
6. result.#interval#.dat for the result files, each with a list of significant signal components,
7. residuals.#interval#.dat for the final residuals after the prewhitening of all significant signal components,
8. resspec.#interval#.dat for the residual spectrum after the prewhitening of all significant signal components,

The column syntax is strictly consistent with the time-unresolved versions (see ``Default Output'', p.). The additional files, wts.#interval#.dat, are in two-column format. The first column represents the time values in the corresponding subset, the second column contains the weight function without normalisation.

Furthermore, SIGSPEC generates a file t000000.#interval#.dat, which contains the part of the original time series which is actually used as input.

Special functions - as introduced in ``Analysis of the Time-domain Sampling'' (p.), ``Preview'' (p.), and ``Correlograms'' (p.) - are also supplied with the #interval# index, i.e.

1. win.#interval#.dat for the amplitude windows,
2. profile.#interval#.dat for the sampling profiles,
3. sock.#interval#.dat for the Sock Diagrams,
4. phdist.#interval#.dat for the phase distribution diagrams,
5. preview.#interval#.dat for the previews,
6. c#iteration#.#interval#.dat for the correlograms after each step of prewhitening,
7. rescorr.#interval#.dat for the final correlograms after the prewhitening of all significant signal components.

Figure 19: Time series used for the sample project timeres, representing 14 nights of Strømgren $y$ photometry of the Delta Scuti star 4CVn, acquired in February and March, 2007.

Example. The sample project timeres illustrates the time-resolved analysis using Strømgren y photometry of the Delta Scuti star 4CVn acquired with the Vienna University Automatic Photoelectric Telescope (Strassmeier et al.1997). The data represent 16 nights from February 21 to March 16, 2007, and are displayed in Fig.19.

Figure 20: Time-resolved significance spectra for 14 subsets (from top to bottom) automatically generated in the sample project timeres. in each panel, the significance spectrum of the full dataset is displayed in grey colour for comparison.

The file timeres.ini contains the specifications

timeres:range 10
timeres:step 1

which provide a 10-day interval moving over the time base of 24 days, with one-day steps. The resulting 14 subsets are represented by the files timeres/t000000.000000.dat to timeres/t000000.000013.dat. Gaussian weight functions with a standard deviation of 5 days are applied:

timeres:w:gauss 5

The files timeres/wts.000000.dat to timeres/wts.000013.dat contain the weights applied to each datapoint within each subset. Further output files are

• timeres/s000000.######.dat for the significance spectra of the original time series without prewhitening (Fig.20,
• timeres/result.######.dat for the lists of significant signal components,
• timeres/residuals.######.dat for the residual time series after all prewhitening steps (divided into subsets according to the time intervals), and
• timeres/resspec.######.dat for the significance spectra of residuals.

Here ###### denotes six-digit numbers ranging from 000000 to 000013.