# LombScarglePeriodogram¶

class lightkurve.periodogram.LombScarglePeriodogram(*args, **kwargs)

Subclass of Periodogram representing a power spectrum generated using the Lomb Scargle method.

Attributes Summary

 frequency_at_max_power Returns the frequency corresponding to the highest peak in the periodogram. max_power Returns the power of the highest peak in the periodogram. period Returns the array of periods, i.e. period_at_max_power Returns the period corresponding to the highest peak in the periodogram.

Methods Summary

 bin([binsize, method]) Bins the power spectrum. copy() Returns a copy of the Periodogram object. flatten([method, filter_width, return_trend]) Estimates the Signal-To-Noise (SNR) spectrum by dividing out an estimate of the noise background. from_lightcurve(lc[, minimum_frequency, …]) Creates a Periodogram from a LightCurve using the Lomb-Scargle method. plot([scale, ax, xlabel, ylabel, title, …]) Plots the Periodogram. show_properties() Prints a summary of the non-callable attributes of the Periodogram object. smooth([method, filter_width]) Smooths the power spectrum using the ‘boxkernel’ or ‘logmedian’ method. to_table() Exports the Periodogram as an Astropy Table.

Attributes Documentation

frequency_at_max_power

Returns the frequency corresponding to the highest peak in the periodogram.

max_power

Returns the power of the highest peak in the periodogram.

period

Returns the array of periods, i.e. 1/frequency.

period_at_max_power

Returns the period corresponding to the highest peak in the periodogram.

Methods Documentation

bin(binsize=10, method='mean')

Bins the power spectrum.

Parameters: binsize : int The factor by which to bin the power spectrum, in the sense that the power spectrum will be smoothed by taking the mean in bins of size N / binsize, where N is the length of the original frequency array. Defaults to 10. method : str, one of ‘mean’ or ‘median’ Method to use for binning. Default is ‘mean’. binned_periodogram : a Periodogram object Returns a new Periodogram object which has been binned.
copy()

Returns a copy of the Periodogram object.

This method uses the copy.deepcopy function to ensure that all objects stored within the Periodogram are copied.

Returns: pg_copy : Periodogram A new Periodogram object which is a copy of the original.
flatten(method='logmedian', filter_width=0.01, return_trend=False)

Estimates the Signal-To-Noise (SNR) spectrum by dividing out an estimate of the noise background.

This method divides the power spectrum by a background estimated using a moving filter in log10 space by default. For details on the method and filter_width parameters, see Periodogram.smooth()

Dividing the power through by the noise background produces a spectrum with no units of power. Since the signal is divided through by a measure of the noise, we refer to this as a Signal-To-Noise spectrum.

Parameters: method : str, one of ‘boxkernel’ or ‘logmedian’ Background estimation method passed on to Periodogram.smooth(). Defaults to ‘logmedian’. filter_width : float If method = ‘boxkernel’, this is the width of the smoothing filter in units of frequency. If method = logmedian, this is the width of the smoothing filter in log10(frequency) space. return_trend : bool If True, then the background estimate, alongside the SNR spectrum, will be returned. snr_spectrum : Periodogram object Returns a periodogram object where the power is an estimate of the signal-to-noise of the spectrum, creating by dividing the powers with a simple estimate of the noise background using a smoothing filter. bkg : Periodogram object The estimated power spectrum of the background noise. This is only returned if return_trend = True.
static from_lightcurve(lc, minimum_frequency=None, maximum_frequency=None, minimum_period=None, maximum_period=None, frequency=None, period=None, nterms=1, nyquist_factor=1, oversample_factor=None, freq_unit=None, normalization='amplitude', **kwargs)

Creates a Periodogram from a LightCurve using the Lomb-Scargle method.

By default, the periodogram will be created for a regular grid of frequencies from one frequency separation to the Nyquist frequency, where the frequency separation is determined as 1 / the time baseline.

The min frequency and/or max frequency (or max period and/or min period) can be passed to set custom limits for the frequency grid. Alternatively, the user can provide a custom regular grid using the frequency parameter or a custom regular grid of periods using the period parameter.

The sampling of the spectrum can be changed using the oversample_factor parameter. An oversampled spectrum (oversample_factor > 1) is useful for displaying the full details of the spectrum, allowing the frequencies and amplitudes to be measured directly from the plot itself, with no fitting required. This is recommended for most applications, with a value of 5 or 10. On the other hand, an oversample_factor of 1 means the spectrum is critically sampled, where every point in the spectrum is independent of the others. This may be used when Lorentzians are to be fitted to modes in the power spectrum, in cases where the mode lifetimes are shorter than the time-base of the data (which is sometimes the case for solar-like oscillations). An oversample_factor of 1 is suitable for these stars because the modes are usually fully resolved. That is, the power from each mode is spread over a range of frequencies due to damping. Hence, any small error from measuring mode frequencies by taking the maximum of the peak is negligible compared with the intrinsic linewidth of the modes.

The normalization parameter will normalize the spectrum to either power spectral density (“psd”) or amplitude (“amplitude”). Users doing asteroseismology on classical pulsators (e.g. delta Scutis) typically prefer normalization="amplitude" because “amplitude” has higher dynamic range (high and low peaks visible simultaneously), and we often want to read off amplitudes from the plot. If normalization="amplitude", the default value for oversample_factor is set to 5 and freq_unit is 1/day. Alternatively, users doing asteroseismology on solar-like oscillators tend to prefer normalization="psd" because power density has a scaled axis that depends on the length of the observing time, and is used when we are interested in noise levels (e.g. granulation) and are looking at damped oscillations. If normalization="psd", the default value for oversample_factor is set to 1 and freq_unit is set to microHz. Default values of freq_unit and oversample_factor can be overridden. See Appendix A of Kjeldsen & Bedding, 1995 for a full discussion of normalization and measurement of oscillation amplitudes (http://adsabs.harvard.edu/abs/1995A%26A…293…87K).

The parameter nterms controls how many Fourier terms are used in the model. Setting the Nyquist_factor to be greater than 1 will sample the space beyond the Nyquist frequency, which may introduce aliasing.

The freq_unit parameter allows a request for alternative units in frequency space. By default frequency is in (1/day) and power in (amplitude (ppm)). Asteroseismologists for example may want frequency in (microHz) in which case they would pass freq_unit=u.microhertz.

By default this method uses the LombScargle ‘fast’ method, which assumes a regular grid. If a regular grid of periods (i.e. an irregular grid of frequencies) it will use the ‘slow’ method. If nterms > 1 is passed, it will use the ‘fastchi2’ method for regular grids, and ‘chi2’ for irregular grids.

Caution: this method assumes that the LightCurve’s time (lc.time) is given in units of days.

Parameters: lc : LightCurve object The LightCurve from which to compute the Periodogram. minimum_frequency : float If specified, use this minimum frequency rather than one over the time baseline. maximum_frequency : float If specified, use this maximum frequency rather than nyquist_factor times the nyquist frequency. minimum_period : float If specified, use 1./minium_period as the maximum frequency rather than nyquist_factor times the nyquist frequency. maximum_period : float If specified, use 1./maximum_period as the minimum frequency rather than one over the time baseline. frequency : array-like The regular grid of frequencies to use. If given a unit, it is converted to units of freq_unit. If not, it is assumed to be in units of freq_unit. This over rides any set frequency limits. period : array-like The regular grid of periods to use (as 1/period). If given a unit, it is converted to units of freq_unit. If not, it is assumed to be in units of 1/freq_unit. This overrides any set period limits. nterms : int Default 1. Number of terms to use in the Fourier fit. nyquist_factor : int Default 1. The multiple of the average Nyquist frequency. Is overriden by maximum_frequency (or minimum period). oversample_factor : int Default: None. The frequency spacing, determined by the time baseline of the lightcurve, is divided by this factor, oversampling the frequency space. This parameter is identical to the samples_per_peak parameter in astropy.LombScargle(). If normalization=’amplitude’, oversample_factor will be set to 5. If normalization=’psd’, it will be 1. These defaults can be overridden. freq_unit : astropy.units.core.CompositeUnit Default: None. The desired frequency units for the Lomb Scargle periodogram. This implies that 1/freq_unit is the units for period. With default normalization (‘amplitude’), the freq_unit is set to 1/day, which can be overridden. ‘psd’ normalization will set freq_unit to microhertz. normalization : ‘psd’ or ‘amplitude’ Default: 'amplitude'. The desired normalization of the spectrum. Can be either power spectral density ('psd') or amplitude ('amplitude'). kwargs : dict Keyword arguments passed to astropy.stats.LombScargle() Periodogram : Periodogram object Returns a Periodogram object extracted from the lightcurve.
plot(scale='linear', ax=None, xlabel=None, ylabel=None, title='', style='lightkurve', view=None, unit=None, **kwargs)

Plots the Periodogram.

Parameters: scale: str Set x,y axis to be “linear” or “log”. Default is linear. ax : matplotlib.axes._subplots.AxesSubplot A matplotlib axes object to plot into. If no axes is provided, a new one will be generated. xlabel : str Plot x axis label ylabel : str Plot y axis label title : str Plot set_title style : str Path or URL to a matplotlib style file, or name of one of matplotlib’s built-in stylesheets (e.g. ‘ggplot’). Lightkurve’s custom stylesheet is used by default. view : str {‘frequency’, ‘period’}. Default ‘frequency’. If ‘frequency’, x-axis units will be frequency. If ‘period’, the x-axis units will be period and ‘log’ scale. kwargs : dict Dictionary of arguments to be passed to matplotlib.pyplot.plot. ax : matplotlib.axes._subplots.AxesSubplot The matplotlib axes object.
show_properties()

Prints a summary of the non-callable attributes of the Periodogram object.

Prints in order of type (ints, strings, lists, arrays and others). Prints in alphabetical order.

smooth(method='boxkernel', filter_width=0.1)

Smooths the power spectrum using the ‘boxkernel’ or ‘logmedian’ method.

If method is set to ‘boxkernel’, this method will smooth the power spectrum by convolving with a numpy Box1DKernel with a width of filter_width, where filter width is in units of frequency. This is best for filtering out noise while maintaining seismic mode peaks. This method requires the Periodogram to have an evenly spaced grid of frequencies. A ValueError exception will be raised if this is not the case.

If method is set to ‘logmedian’, it smooths the power spectrum using a moving median which moves across the power spectrum in a steps of

log10(x0) + 0.5 * filter_width

where filter width is in log10(frequency) space. This is best for estimating the noise background, as it filters over the seismic peaks.

Periodograms that are unsmoothed have multiplicative noise that is distributed as chi squared 2 degrees of freedom. This noise distribution has a well defined mean and median but the two are not equivalent. The mean of a chi squared 2 dof distribution is 2, but the median is 2(8/9)**3. (see https://en.wikipedia.org/wiki/Chi-squared_distribution) In order to maintain consistency between ‘boxkernel’ and ‘logmedian’ a correction factor of (8/9)**3 is applied to (i.e., the median is divided by the factor) to the median values.

In addition to consistency with the ‘boxkernel’ method, the correction of the median values is useful when applying the periodogram flatten method. The flatten method divides the periodgram by the smoothed periodogram using the ‘logmedian’ method. By appyling the correction factor we follow asteroseismic convention that the signal-to-noise power has a mean value of unity. (note the signal-to-noise power is really the signal plus noise divided by the noise and hence should be unity in the absence of any signal)

Parameters: method : str, one of ‘boxkernel’ or ‘logmedian’ The smoothing method to use. Defaults to ‘boxkernel’. filter_width : float If method = ‘boxkernel’, this is the width of the smoothing filter in units of frequency. If method = logmedian, this is the width of the smoothing filter in log10(frequency) space. smoothed_pg : Periodogram object Returns a new Periodogram object in which the power spectrum has been smoothed.
to_table()

Exports the Periodogram as an Astropy Table.

Returns: table : An AstroPy Table with columns ‘frequency’, ‘period’, and ‘power’.