Classical Lomb-Scargle and FFT

Using the classes presented below, several periodograms can be calculated:

• Fourier transform

• Lomb Scargle

All the presented classes inherit from the PeriodBase class.

The Fourier Transform

class PyAstronomy.pyTiming.pyPeriod.Fourier(lc, norm='Leahy')

This class computes the Fast Fourier Transform (FFT) of the input data.

This is actually a wrapper around numpy’s FFT routines. The constructor takes the light curve, lc (TimeSeries instance), as input. The optional argument specifies the normalization of the Fourier Power. Currently, only the normalization according to [Leahy83] is supported, which in the case of purely Poissonian noise results in a mean power of 2.

Parameters

lcTimeSeries instance

The light curve to be analyzed.

normoptional, string

Normalization method; currently, only default (“Leahy”) is supported.

Methods

FAP(Pn)	Obtain the false-alarm probability (FAP).
Prob(Pn)	Returns the probability to obtain a power larger than the threshold, Pn.
plot(*args, **kwargs)	Creates a matplotlib figure and axes class instance to visualize the result.
powerLevel(FAPlevel)	Power threshold for FAP level.
stats(Pn)	Obtain basic statistics for power threshold.

prob
probInv

FAP(Pn)

Obtain the false-alarm probability (FAP).

The FAP denotes the probability that at least one out of M independent power values in a prescribed search band of a power spectrum computed from a white-noise time series is as large as or larger than the threshold, Pn. It is assessed through

\[FAP(Pn) = 1 - (1-Prob(P>Pn))^M \; ,\]

where “Prob(P>Pn)” depends on the type of periodogram and normalization and is calculated by using the prob method; M is the number of independent power values and is computed internally.

Parameters

Pnfloat

Power threshold.

Returns

FAPfloat

False alarm probability.

Prob(Pn)

Returns the probability to obtain a power larger than the threshold, Pn.

In the calculations, the noise is assumed to be Poisson-distributed.

Note

According to [vdK] the probability to obtain a power larger than a given threshold from the noise is given by

\[Prob(p>Pn) = Q(M \times W \times Pn, 2 \times M \times W)\]

where \(Q(\chi^2, \nu)\) is the cumulative \(\chi^2\) distribution with \(\nu\) d.o.f.

Parameters

Pnfloat

Power threshold.

Returns

FAPfloat

The probability to obtain a power larger than the specified threshold from noise.

plot(*args, **kwargs)

Creates a matplotlib figure and axes class instance to visualize the result.

Parameters:

• FAPlevels - optional, List of false-alarm probability (FAP) levels

• *args - optional, Arguments passed to plot method of axes class.

• **kwargs - optional, Keyword arguments passed plot method to axes class.

This method provides a quick and simple way to visualize the results of the a periodogram calculation.

Returns:

The created Figure and Axes class instances.

powerLevel(FAPlevel)

Power threshold for FAP level.

Parameters

FAPlevelfloat or array

“False Alarm Probability” threshold

Returns

Thresholdfloat or array

The power threshold pertaining to a specified false-alarm probability (FAP). Powers exceeding this threshold have FAPs smaller than FAPlevel.

stats(Pn)

Obtain basic statistics for power threshold.

Parameters

Pnfloat

Power threshold.

Returns

Statisticsdictionary

A dictionary containing {‘Pn’: Pn, ‘FAP’: FAP(Pn) , ‘Prob’: Prob(Pn)} for the specified power threshold, Pn.

The Lomb-Scargle-Periodogram (fast)

class PyAstronomy.pyTiming.pyPeriod.LombScargle(lc, ofac, hifac)

Calculate the Lomb-Scargle periodogram.

The constructor of LombScargle takes a TimeSeries instance, i.e., a light curve object, as first argument. It then computes the usual Lomb-Scargle periodogram using a fast algorithm. The frequency array is constructed on the fly based on the oversampling keywords, which are mandatory in this case. The power is normalized according to the prescription of [HB86].

The result, i.e., the power, is stored in the class attribute power.

Note

Adapted from routine of the same routine in [NR] , based on period.pro by Han Wen, August 1996.

Parameters

lcTimesSeries instance

The light curve to be analyzed.

ofacint

Oversampling factor.

hifacfloat

Maximum frequency freq = hifac * (average Nyquist frequency).

Methods

FAP(Pn)	Obtain the false-alarm probability (FAP).
Prob(Pn)	Outdated -- use "prob" instaed.
ProbInv(Prob)	Outdated -- use "probInv" instaed
plot(*args, **kwargs)	Creates a matplotlib figure and axes class instance to visualize the result.
powerLevel(FAPlevel)	Power threshold for FAP level.
prob(Pn)	Returns the probability to obtain a power larger than the threshold.
probInv(Prob)	Returns the minimum power for a given probability level Prob.
stats(Pn)	Obtain basic statistics for power threshold.

FAP(Pn)

Obtain the false-alarm probability (FAP).

The FAP denotes the probability that at least one out of M independent power values in a prescribed search band of a power spectrum computed from a white-noise time series is as large as or larger than the threshold, Pn. It is assessed through

\[FAP(Pn) = 1 - (1-Prob(P>Pn))^M \; ,\]

where “Prob(P>Pn)” depends on the type of periodogram and normalization and is calculated by using the prob method; M is the number of independent power values and is computed internally.

Parameters

Pnfloat

Power threshold.

Returns

FAPfloat

False alarm probability.

Prob(Pn)

Outdated – use “prob” instaed.

ProbInv(Prob)

Outdated – use “probInv” instaed

plot(*args, **kwargs)

Creates a matplotlib figure and axes class instance to visualize the result.

Parameters:

• FAPlevels - optional, List of false-alarm probability (FAP) levels

• *args - optional, Arguments passed to plot method of axes class.

• **kwargs - optional, Keyword arguments passed plot method to axes class.

This method provides a quick and simple way to visualize the results of the a periodogram calculation.

Returns:

The created Figure and Axes class instances.

powerLevel(FAPlevel)

Power threshold for FAP level.

Parameters

FAPlevelfloat or array

“False Alarm Probability” threshold

Returns

Thresholdfloat or array

The power threshold pertaining to a specified false-alarm probability (FAP). Powers exceeding this threshold have FAPs smaller than FAPlevel.

prob(Pn)

Returns the probability to obtain a power larger than the threshold.

Compute the probability of obtaining a power value as large or larger the the threshold based on the noise, which is assumes to be Gaussian.

Note

LombScargle calculates the quantity (N-1)/2.*p=p’ (in the formalism of [ZK09]), which is de facto the normalization prescription of [HB86]. In this scheme the probability P(p’>Pn) is given by the following statement:

\[P(p'>Pn) = \left(1 - 2 \frac{Pn}{N-1} \right)^{(N-3)/2}\]

If properly normalized to the population variance of the time series, which must be known a priori (usually not the case), the power \(p/p_n=p"\) is a direct measure of the SNR as proposed by [Scargle82]:

\[P(p">Pn) = exp(-Pn) \; .\]

This formula is often used erroneously in this context.

Parameters

Pnfloat

Power threshold

Returns

FAPfloat

False alarm probability.

probInv(Prob)

Returns the minimum power for a given probability level Prob.

This is the inverse of prob(Pn).

Parameters

Probfloat

Probability

Returns

MPLfloat

Minimum power level required.

stats(Pn)

Obtain basic statistics for power threshold.

Parameters

Pnfloat

Power threshold.

Returns

Statisticsdictionary

A dictionary containing {‘Pn’: Pn, ‘FAP’: FAP(Pn) , ‘Prob’: Prob(Pn)} for the specified power threshold, Pn.