# astrobase.periodbase.zgls module¶

Contains the Zechmeister & Kurster (2002) Generalized Lomb-Scargle period-search algorithm implementation for periodbase.

`astrobase.periodbase.zgls.``generalized_lsp_value`(times, mags, errs, omega)[source]

Generalized LSP value for a single omega.

The relations used are:

```P(w) = (1/YY) * (YC*YC/CC + YS*YS/SS)

where: YC, YS, CC, and SS are all calculated at T

and where: tan 2omegaT = 2*CS/(CC - SS)

and where:

Y = sum( w_i*y_i )
C = sum( w_i*cos(wT_i) )
S = sum( w_i*sin(wT_i) )

YY = sum( w_i*y_i*y_i ) - Y*Y
YC = sum( w_i*y_i*cos(wT_i) ) - Y*C
YS = sum( w_i*y_i*sin(wT_i) ) - Y*S

CpC = sum( w_i*cos(w_T_i)*cos(w_T_i) )
CC = CpC - C*C
SS = (1 - CpC) - S*S
CS = sum( w_i*cos(w_T_i)*sin(w_T_i) ) - C*S
```
Parameters: times,mags,errs (np.array) – The time-series to calculate the periodogram value for. omega (float) – The frequency to calculate the periodogram value at. periodogramvalue – The normalized periodogram at the specified test frequency omega. float
`astrobase.periodbase.zgls.``generalized_lsp_value_withtau`(times, mags, errs, omega)[source]

Generalized LSP value for a single omega.

This uses tau to provide an arbitrary time-reference point.

The relations used are:

```P(w) = (1/YY) * (YC*YC/CC + YS*YS/SS)

where: YC, YS, CC, and SS are all calculated at T

and where: tan 2omegaT = 2*CS/(CC - SS)

and where:

Y = sum( w_i*y_i )
C = sum( w_i*cos(wT_i) )
S = sum( w_i*sin(wT_i) )

YY = sum( w_i*y_i*y_i ) - Y*Y
YC = sum( w_i*y_i*cos(wT_i) ) - Y*C
YS = sum( w_i*y_i*sin(wT_i) ) - Y*S

CpC = sum( w_i*cos(w_T_i)*cos(w_T_i) )
CC = CpC - C*C
SS = (1 - CpC) - S*S
CS = sum( w_i*cos(w_T_i)*sin(w_T_i) ) - C*S
```
Parameters: times,mags,errs (np.array) – The time-series to calculate the periodogram value for. omega (float) – The frequency to calculate the periodogram value at. periodogramvalue – The normalized periodogram at the specified test frequency omega. float
`astrobase.periodbase.zgls.``generalized_lsp_value_notau`(times, mags, errs, omega)[source]

This is the simplified version not using tau.

The relations used are:

```W = sum (1.0/(errs*errs) )
w_i = (1/W)*(1/(errs*errs))

Y = sum( w_i*y_i )
C = sum( w_i*cos(wt_i) )
S = sum( w_i*sin(wt_i) )

YY = sum( w_i*y_i*y_i ) - Y*Y
YC = sum( w_i*y_i*cos(wt_i) ) - Y*C
YS = sum( w_i*y_i*sin(wt_i) ) - Y*S

CpC = sum( w_i*cos(w_t_i)*cos(w_t_i) )
CC = CpC - C*C
SS = (1 - CpC) - S*S
CS = sum( w_i*cos(w_t_i)*sin(w_t_i) ) - C*S

D(omega) = CC*SS - CS*CS
P(omega) = (SS*YC*YC + CC*YS*YS - 2.0*CS*YC*YS)/(YY*D)
```
Parameters: times,mags,errs (np.array) – The time-series to calculate the periodogram value for. omega (float) – The frequency to calculate the periodogram value at. periodogramvalue – The normalized periodogram at the specified test frequency omega. float
`astrobase.periodbase.zgls.``specwindow_lsp_value`(times, mags, errs, omega)[source]

This calculates the peak associated with the spectral window function for times and at the specified omega.

NOTE: this is classical Lomb-Scargle, not the Generalized Lomb-Scargle. mags and errs are silently ignored since we’re calculating the periodogram of the observing window function. These are kept to present a consistent external API so the pgen_lsp function below can call this transparently.

Parameters: times,mags,errs (np.array) – The time-series to calculate the periodogram value for. omega (float) – The frequency to calculate the periodogram value at. periodogramvalue – The normalized periodogram at the specified test frequency omega. float
`astrobase.periodbase.zgls.``pgen_lsp`(times, mags, errs, magsarefluxes=False, startp=None, endp=None, stepsize=0.0001, autofreq=True, nbestpeaks=5, periodepsilon=0.1, sigclip=10.0, nworkers=None, workchunksize=None, glspfunc=<function _glsp_worker_withtau>, verbose=True)[source]

This calculates the generalized Lomb-Scargle periodogram.

Uses the algorithm from Zechmeister and Kurster (2009).

Parameters: times,mags,errs (np.array) – The mag/flux time-series with associated measurement errors to run the period-finding on. magsarefluxes (bool) – If the input measurement values in mags and errs are in fluxes, set this to True. startp,endp (float or None) – The minimum and maximum periods to consider for the transit search. stepsize (float) – The step-size in frequency to use when constructing a frequency grid for the period search. autofreq (bool) – If this is True, the value of stepsize will be ignored and the `astrobase.periodbase.get_frequency_grid()` function will be used to generate a frequency grid based on startp, and endp. If these are None as well, startp will be set to 0.1 and endp will be set to times.max() - times.min(). nbestpeaks (int) – The number of ‘best’ peaks to return from the periodogram results, starting from the global maximum of the periodogram peak values. periodepsilon (float) – The fractional difference between successive values of ‘best’ periods when sorting by periodogram power to consider them as separate periods (as opposed to part of the same periodogram peak). This is used to avoid broad peaks in the periodogram and make sure the ‘best’ periods returned are all actually independent. sigclip (float or int or sequence of two floats/ints or None) – If a single float or int, a symmetric sigma-clip will be performed using the number provided as the sigma-multiplier to cut out from the input time-series. If a list of two ints/floats is provided, the function will perform an ‘asymmetric’ sigma-clip. The first element in this list is the sigma value to use for fainter flux/mag values; the second element in this list is the sigma value to use for brighter flux/mag values. For example, sigclip=[10., 3.], will sigclip out greater than 10-sigma dimmings and greater than 3-sigma brightenings. Here the meaning of “dimming” and “brightening” is set by physics (not the magnitude system), which is why the magsarefluxes kwarg must be correctly set. If sigclip is None, no sigma-clipping will be performed, and the time-series (with non-finite elems removed) will be passed through to the output. nworkers (int) – The number of parallel workers to use when calculating the periodogram. workchunksize (None or int) – If this is an int, will use chunks of the given size to break up the work for the parallel workers. If None, the chunk size is set to 1. glspfunc (Python function) – The worker function to use to calculate the periodogram. This can be used to make this function calculate the time-series sampling window function instead of the time-series measurements’ GLS periodogram by passing in _glsp_worker_specwindow instead of the default _glsp_worker_withtau function. verbose (bool) – If this is True, will indicate progress and details about the frequency grid used for the period search. This function returns a dict, referred to as an lspinfo dict in other astrobase functions that operate on periodogram results. This is a standardized format across all astrobase period-finders, and is of the form below: ```{'bestperiod': the best period value in the periodogram, 'bestlspval': the periodogram peak associated with the best period, 'nbestpeaks': the input value of nbestpeaks, 'nbestlspvals': nbestpeaks-size list of best period peak values, 'nbestperiods': nbestpeaks-size list of best periods, 'lspvals': the full array of periodogram powers, 'periods': the full array of periods considered, 'method':'gls' -> the name of the period-finder method, 'kwargs':{ dict of all of the input kwargs for record-keeping}} ``` dict
`astrobase.periodbase.zgls.``specwindow_lsp`(times, mags, errs, magsarefluxes=False, startp=None, endp=None, stepsize=0.0001, autofreq=True, nbestpeaks=5, periodepsilon=0.1, sigclip=10.0, nworkers=None, glspfunc=<function _glsp_worker_specwindow>, verbose=True)[source]

This calculates the spectral window function.

Wraps the pgen_lsp function above to use the specific worker for calculating the window-function.

Parameters: times,mags,errs (np.array) – The mag/flux time-series with associated measurement errors to run the period-finding on. magsarefluxes (bool) – If the input measurement values in mags and errs are in fluxes, set this to True. startp,endp (float or None) – The minimum and maximum periods to consider for the transit search. stepsize (float) – The step-size in frequency to use when constructing a frequency grid for the period search. autofreq (bool) – If this is True, the value of stepsize will be ignored and the `astrobase.periodbase.get_frequency_grid()` function will be used to generate a frequency grid based on startp, and endp. If these are None as well, startp will be set to 0.1 and endp will be set to times.max() - times.min(). nbestpeaks (int) – The number of ‘best’ peaks to return from the periodogram results, starting from the global maximum of the periodogram peak values. periodepsilon (float) – The fractional difference between successive values of ‘best’ periods when sorting by periodogram power to consider them as separate periods (as opposed to part of the same periodogram peak). This is used to avoid broad peaks in the periodogram and make sure the ‘best’ periods returned are all actually independent. sigclip (float or int or sequence of two floats/ints or None) – If a single float or int, a symmetric sigma-clip will be performed using the number provided as the sigma-multiplier to cut out from the input time-series. If a list of two ints/floats is provided, the function will perform an ‘asymmetric’ sigma-clip. The first element in this list is the sigma value to use for fainter flux/mag values; the second element in this list is the sigma value to use for brighter flux/mag values. For example, sigclip=[10., 3.], will sigclip out greater than 10-sigma dimmings and greater than 3-sigma brightenings. Here the meaning of “dimming” and “brightening” is set by physics (not the magnitude system), which is why the magsarefluxes kwarg must be correctly set. If sigclip is None, no sigma-clipping will be performed, and the time-series (with non-finite elems removed) will be passed through to the output. nworkers (int) – The number of parallel workers to use when calculating the periodogram. glspfunc (Python function) – The worker function to use to calculate the periodogram. This is used to used to make the pgen_lsp function calculate the time-series sampling window function instead of the time-series measurements’ GLS periodogram by passing in _glsp_worker_specwindow instead of the default _glsp_worker function. verbose (bool) – If this is True, will indicate progress and details about the frequency grid used for the period search. This function returns a dict, referred to as an lspinfo dict in other astrobase functions that operate on periodogram results. This is a standardized format across all astrobase period-finders, and is of the form below: ```{'bestperiod': the best period value in the periodogram, 'bestlspval': the periodogram peak associated with the best period, 'nbestpeaks': the input value of nbestpeaks, 'nbestlspvals': nbestpeaks-size list of best period peak values, 'nbestperiods': nbestpeaks-size list of best periods, 'lspvals': the full array of periodogram powers, 'periods': the full array of periods considered, 'method':'win' -> the name of the period-finder method, 'kwargs':{ dict of all of the input kwargs for record-keeping}} ``` dict
`astrobase.periodbase.zgls.``probability_peak_exceeds_value`(times, peakval)[source]

This calculates the probability that periodogram values exceed the given peak value.

This is from page 3 of Zechmeister and Kurster (2009):

```Prob(p > p_best) = (1 − p_best)**((N−3)/2)
```

where:

```p_best is the peak value in consideration
N is the number of times
```

Note that this is for the default normalization of the periodogram, e.g. P_normalized = P(omega), such that P represents the sample variance (see Table 1).

Parameters: lspvals (np.array) – The periodogram power value array. peakval (float) – A single peak value to calculate the probability for. prob – The probability value. float
`astrobase.periodbase.zgls.``analytic_false_alarm_probability`(lspinfo, times, conservative_nfreq_eff=True, peakvals=None, inplace=True)[source]

This returns the analytic false alarm probabilities for periodogram peak values.

The calculation follows that on page 3 of Zechmeister & Kurster (2009):

```FAP = 1 − [1 − Prob(z > z0)]**M
```

where:

```M is the number of independent frequencies
Prob(z > z0) is the probability of peak with value > z0
z0 is the peak value we're evaluating
```
Parameters: lspinfo (dict) – The dict returned by the `pgen_lsp()` function. times (np.array) – The times for which the periodogram result in `lspinfo` was calculated. conservative_nfreq_eff (bool) – If True, will follow the prescription given in Schwarzenberg-Czerny (2003): http://adsabs.harvard.edu/abs/2003ASPC..292..383S and estimate the effective number of independent frequences M_eff as: ```min(N_obs, N_freq, DELTA_f/delta_f) ``` peakvals (sequence or None) – The peak values for which to evaluate the false-alarm probability. If None, will calculate this for each of the peak values in the `nbestpeaks` key of the `lspinfo` dict. inplace (bool) – If True, puts the results of the FAP calculation into the `lspinfo` dict as a list available as `lspinfo['falsealarmprob']`. The calculated false alarm probabilities for each of the peak values in `peakvals`. list