Detecting Periodicities in RR Lyrae Stars

Detecting Periodicities inRR Lyrae Stars

R. F. StellingwerfStellingwerf ConsultingJanuary, 24, 2011

RR Lyrae Stars, Metal-Poor Stars and the GalaxyPasadena, CA

Stellingwerf 1/25/2011 2

RR Lyrae Special Attributes

• Highly non-sinusoidal light variations in many cases• Changes in the period and amplitude due to evolution• Changes in amplitude due to Blazhko effect• Possible multiple modes with nonlinear modal interaction• Other variations caused by companions, eclipses, etc.• Gaps in data sets (yearly, nightly, historic)• Multiple overlapping data sets (different colors, observers)


Review of Methods

1. Fourier (discrete, FFT, etc)

2. Periodogram (Vanicek/Lomb/Scargle)

3. PDM / PDM2

Spectral:

Period Search:

Stellingwerf 1/25/2011

Discrete Fourier Method

4

• Complex amplitude of transform produces power spectrum• Mathematically well understood• FFT available• Problems with unequally spaced data and data gaps• Limitations in sampling and frequency coverage

• Xk are coefficients of a Fourier decomposition of {xn}


Periodogram / Least Squares Fit to Sin()

• Find optimal fit to a sin() function at each frequency• Must adjust sin() amplitude, phase,

average value and frequency


Least Squares Periodogram Formula

2))cos((),( ii tAxS

• Sum of squares of deviations:

• Pick A to minimize sum of squares… • Differentiate wrt A, set to 0, solve for A, get…

)(cos/)cos(),( 2max iii ttxA

• Note, min S usually implies max A. So, one approach is to vary (and possibly looking for large values of(or A2)

•Alternate view: minimize 2)(/),( ixS


Periodogram Variations

• Classic Form (Fourier form)

• Lomb/Scargle Form (least squares form)

• Possible questions about normalization, smoothing, and bandwidth effects


Phase Dispersion Minimization

• Minimize => least squares fit to mean curve through bin means• Very simple and efficient to compute• Additivity of the variances allows generalizations and variations

…where s is sj summed over 10 bins

Mean curve is determined by the data.

Compute scatter in each bin.

sj


PDM Calculation

phase = frac( ti* fj )

bin = int( 10 * phase )

…where s is sj summed over 10 bins, all segments, all data sets

U

B

Time

1 2 3

4 5 6

• Use separate phases for time segments to get a quick trial estimate, use single phase across all times to refine the period


PDM Window Function• Standard test case: 101 points cover 10 cycles of a pure sine wave, time 0->10, amplitude 1, f = 0->2, 200 frequency points

Top: wide (5/2) bins

Bottom: narrow (10/1) bins

Frequency scans from 0 ->F will detect signals at allhigher frequencies. (multi-

periodic cases, or“subharmonics”).


PDM Treatment of Sigmas (error estimate)

• Deviation from mean is taken from edge of error bar• If sigma > distance from mean, point does not contribute to

variance

Bin points shown, radii are estimated sigmas.

Deviations used in variance shown.

More accurate points contribute more to result.

Tests show this gives cleaner results than 1/sigma2 weighting.


PDM2 Options

• Classic PDM – step function fit 10 bins, double wide option for small data sets.

• Linear mean curve fit. Only used for significant frequencies.

• Spline mean curve fit.

• Subharmonic Averaging (uses PDM window transform)

• Beta Distribution and Non-Parametric significance tests.


Sin() Wave Test Case

• 10 cycles of sin() variation plus Gaussian noise

• Lomb shows clear peak at f = 1.0

• PDM shows mins at f = 1.0, 1/2, 1/3, etc.


Sawtooth Test Case

• 10 cycles of saw() variation - No noise

- Gap inserted in data

• Lomb shows peak at f = 1.0, side lobes

• PDM shows similar patterns at f = 1, ½, 1/3

• Main peak stronger and narrower


Sawtooth – Big Gap

• 10 cycles of saw() variation - No noise

- Larger Gap inserted in data

• Lomb shows peak at f = 1.0, many side lobes

• PDM shows similar patterns at f = 1, ½, 1/3

Subharmonic averaging helps here.

• This pattern is typical of a long time-scale data set


Pulse Test Case

• 3 Pulses at t = 3, 6, 9

• Lomb shows peak at f = 2/3, but not a clear winner.

• PDM shows the f = 2/3 result more clearly.

Note narrowness of main result.


Double Peak Test Case

• Alternating peaks, period = 2, f = 0.5

• Lomb shows major peak at f = 1, wrong frequency.

• PDM shows f = 0.5 correctly, as well as a possible hit at

f = 1.


Significance Testing

• PDM2 now includes the incomplete beta distribution (Schwarzenberg-Czerny, 1997, 1999) with multiple trial correction. This is the correct distribution for PDM in noise dominated cases.

• The Lomb analysis follows an exponential distribution.

• Note, however, that the “noise” present in many variable star signals is due to other modes, Blazhko variability, etc., and is not Gaussian, as assumed in any parametric statistical criteria.

• Furthermore, the “bandwidth correction” depends on the data distribution in time, and may not be accurate. Used for Lomb also.

• A more accurate estimate is now included using Fisher Randomization / Monte Carlo re: Nemec & Nemec (1985). This uses multiple permutations to derive a specific “noise” distribution unique to each data set and analysis parameters.


Significance Testing – Compare with LombTest Case 1 : 51 points of Gaussian noise, random phases

Lomb

PDM


Significance Testing – Compare with LombTest Case 2 : 511 points of Gaussian noise, random phases

Lomb

PDM

Result better for PDM Much worse for Lomb

Conf Lev = 0.79Monte Carlo Conf Lev = 0.94

Conf Lev = 0.18Monte Carlo Conf Lev = 0.19


Significance Testing – PDMTest Case 1: 51 points of Gaussian noise, random phases, wide bins

Right curve = Beta distribution.

Left curve = Beta with bandwidth correction.

Arrow = data range.

Noise distribution extends to Theta = 0.65 at the 0.01 significance level. Any values lower than this are not likely due to noise.

Right curve = Randomized Monte-Carlo derived distribution for Theta.

Left curve = Distribution of Theta_min over 500 separate analyses. Results agree for both curves.

5/2 binning, 220 frequency points

Correction is large

Extreme value dist


Significance Testing – PDM - 2Test Case 1: 51 points of Gaussian noise, random phases, narrow bins

Right curve = Beta distribution.

Left curve = Beta with bandwidth correction.

Arrow = data range.

Noise distribution extends to Theta = 0.62 at the 0.01 significance level.

Right curve = Randomized Monte-Carlo derived distribution for Theta.

Left curve = Distribution of Theta_min over 500 separate analyses. Results no longer agree. The narrower bins have caused a much larger noise level, extending to Theta = 0.50 at the 0.01 level.

10/1 binning, 220 frequency points

Beta changes slightly

Actual


Significance Testing - LombTest Case 1: 51 points of Gaussian noise, random phases, Lomb

Left curve = Exp confidence distribution.

Right curve = Exp with bandwidth correction.

Noise distribution extends to Power = 7 at the 0.01 significance level.

Arrow = data range.

Left curve = Randomized Monte-Carlo derived distribution for Power.

Right curve = Distribution of Power_max over 500 separate analyses.

Good agreement, Above noise at Pwr > 7

Correction is very large

Power Max Dist


IC4499 - v04

• V Data, colored on sigma

• PDM - First do a “rough cut” with nine data segments, wide frequency range. Best

value is at f = 1.603.

• Zero in on the main candidate with full data set.

Get f = 1.60355.(published value = 1.60354)

Data courtesy of G. Bono/ A. Kunder

Stellingwerf 1/25/2011

Comparison of Beta Dist and Monte Carlo for IC4499 v04 (V data - 340 pts)

25

ParametricBeta distribution andCorrected values

Monte Carlo distrib and Extreme value distrib250 or 500 iterations

0.91

0.85


IC4499 - v04

Best period, data plotted versus phase. Colored on point number

(note red points offset to right on descending branch)


IC4499 - v04 PDM2 Variable Period Option

• Redo the analysis with a range of beta, whereP = P0 + t

• Agrees with O-C result (0.18 +/- 0.05, shown)• Get best alignment at about + 0.15 d/My


IC4499 - v04

Phase plot with period increasing at 0.20 d/My.Note much cleaner descending branch.


IC4499 – v48 (Blazhko)


• First do a “rough cut” with six data segments, wide

frequency range. Best value is at f = 1.924.


Get f = 1.92385.(O-C frequency = 1.92385)




Best period, data plotted versus phase Colored on point number .Note red offset to left on rising branch.


PDM2 Variable Period Option

• This gives a period change of -0.13 +/- 0.05• Based on full light curve, so applicable to Blazhko variables.



Phase plot with period decreasing at -0.13 d/My.Note much cleaner rising branch.

Also note offsets of the max light phases.





• First do a “rough cut” with six data segments, wide

frequency range. Best value is at f = 1.5936.


Get f = 1.593572.(O-C value = 1.593570)



IC4499 – v83 (Blazhko - ?)

Best period, data plotted versus phase Colored on point number .Doesn’t look like a Blazhko…



PDM2 Variable Period Option

• This gives a period change of -0.20 +/- 0.10• Based on full light curve, so applicable to Blazhko variables.


Phase plot with period decreasing at -0.20 d/My.Cleaner rising branch.

But Blazhko variations are still missing.



• Part of S_tran.exe application on www.stellingwerf.com• Typical PDM2 script shown here

#---read the data---read_lab data.csv // specify csv data filefor_i = 1 100000 read_dat Xdat[i] Ydat[i] //[Sdat[i]] if eof breakend_iN = i-1

#---PDM ANALYSIS---pdm_lpoints 3 // frequency finenesspdm_f_range 0 4 // frequency range

pdm2 // do a PDM analysis

lomb // do a Lomb analysis

• Source C code for PDM2 routine also available for download

PDM2 / Lomb Application Available

Data.csv

Documents

Detecting Periodicities in RR Lyrae Stars