FrequencyFrequency AnalysisAnalysisMethods and Programs

LiteratureLiterature

Methods : www.helas-eu.org�Education ���� Lecture Notes ���� Asteroseismology (Conny Aerts)

(cross-references are given there)

PERIOD04: www.univie.ac.at/tops/period04 ����Downloads

AstronomicalAstronomical time time seriesseries

• Variables: brightness, colours, radial velocity, spectralline profiles

• Classical theory of TSA assumes uninterruptedmeasurements of evenly spaced data

• Astronomical time series contain large gaps and unevenly spaced data

• The data may occur in quasi-evenly spaced groupsdivided by days, months, or years

P = 1.12345 dt = 0,1,2,…

P = 1.12 d P = 1.123 d

NonNon --parametricparametricfrequencyfrequency analysisanalysis toolstools

• Non-parametric means without a priori assump-tions about some model function like sinusoidsetc. as in case of harmonic analysis or Fouriertransform based methods

• Methods are useful in case of strongly non-sinusoidal variations but suffer from the large number of alias and subharmonic peaks

String String lengthlength methodsmethods

- Based on least squares fit

- Fold the measured values with some period

- Sum up the distances between neighbouring datapoints in the phase diagram

- Compare this with the sum of the distances fromthe mean value

(Lafler & Kinman 1965, Clarke 2002)

f-4 f-3 f-2 f-1 f f+1 f+2

1d aliases + subharmoics

Lecture Notes by Conny Aerts

The origin of alias frequencies

Check the pase diagrams!

Lecture Notes by Conny Aerts

Phase Dispersion Phase Dispersion MinimizationMinimization (PDM)(PDM)

• Work in the phase diagram folded with some period

• Compute an average curve from the mean values computedwithin phase bins

• Minimize the spread of data around the average curve

• Vary the period

f-4 f-3 f-2 f-1 f f+1 f+2 f+3

f-4 f-3 f-2 f-1 f f+1 f+2

String length

PDM � improved S/N

Lecture Notes by Conny Aerts

HarmonicHarmonic analysisanalysis byby least least squaressquares

Determine a sum of harmonic functionsthat best describe the data in the least-squares sense

PeriodogramsPeriodograms

We are searching for

- a minimum in the residuals or- a maximum in the variance reduction

Multiple Multiple frequenciesfrequencies and and successivesuccessive prewhiteningprewhitening

• Least squares fitting is linear in the coefficients a,b,c but not in the frequency f

• After finding one frequency fk we subtract the found contribution from the dataand repeat the frequency search in the prewhitened data

• For a given set of (fixed) frequencies we can determin al l the coefficients ak,bk,ckfor all frequencies fk

• This is equivalent to an optimization of amplitudes and phases

FourierFourier techniquestechniques

• Continous FT of infinite TS• Continous FT of finite TS• Discrete FT• The classical periodogram• The Lomb-Scargle periodogram• Significance criteria• Handling of alias problems

EffectsEffects of finite, of finite, discretediscrete time time seriesseries

Noiseless data

1000,000 data pointsin 1000 days

10,000 data pointsin 10 days

4472 data pointsin 10 days + gaps

Resolution criterion: |f1-f2| > 1/T

No interference for: |f1-f2| > 2/T

Lecture Notes by Conny Aerts

Spectral window

Time sampling

Amplitude spectrum of anoise-free sinusoid withthe time sampling asabove (f = 5.1234567/d)

Nyquist frequency

fNy = 1/(2dt)

for evenly spaced data

Lecture Notes by Conny Aerts

fNy

periodiccontinuation

Lecture Notes by Conny Aerts

SignificanceSignificance criteriacriteria

Hypotheses

A) The observed peak in the periodogramdoes not origin from noise

B) The data contain a specific periodicity

A) Monte Carlo A) Monte Carlo simulationssimulations

• Use the time sampling of observed data• Assume a certain noise level and distribution• Compute thousands of randomly distributed data sets conta ining only noise• Compute the periodograms and determine the maximum peak he ights• Derive the empirical probability distribution

Lehmann & Mkrtichian (2004)

• Use the amplitude periodogram of the residuals after subtr acting all periodic contributions

• Determine the average level Pmean around the frequency of interest

• Compare the peak Pmax of the test frequency with Pmean:

S/N = Pmax /Pmean

Breger et al. (1993): S/N = 4.0 ���� 99.9% confidence level

S/N = 3.6 ���� 95.0%S/N = 3.0 ���� 80.0%

B) B) EmpiricalEmpirical S/N S/N levellevel criteriacriteria

Error Error estimationestimation

If• the times of measurements are error-free,• we have only white, uncorrelated noise in the data and• there is no interference between true frequencies and noi se peaks

then

Two-frequency model :

P1 = 21.7 d, f1 = 0.04608/d, A1 = 1.0P2 = 10.3 d, f2 = 0.09709/d, A2 = 0.5

Time sampling :

data set 1: N = 512 equidistant valuesT = 512 d, d t = 1 dfNy = 0.5/d, d f = 0.0020/d

data set 2: data set 1 + noise

data set 3: N = 70, 7 nights a 10 spectraa 1 h starting at the same timeT = 154 hr, d t = 1 hrfNy = 0.5/hr, d f = 0.0065/hr

(PERIOD04 by Patrick Lenz, Uni Wien)

PERIOD04 PERIOD04 –– an an exampleexample

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