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SOSTAT 2021: Time Series Analysis

SOSTAT 2021: Time Series Analysis

Introduction lecture to time series analysis in astronomy for the 2nd Severo Ochoa School on Statistics, Data Mining, and Machine Learning in Granada, Spain: https://www.granadacongresos.com/sostat2021

Tutorial in Jupyter notebooks: https://github.com/abigailStev/timeseries-tutorial

More on Dr. Abbie Stevens: https://abigailstevens.com/

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Dr. Abbie Stevens

December 01, 2021
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  1. Time series analysis Dr. Abbie Stevens Michigan State University &

    University of Michigan alstev@msu.edu @abigailStev github.com/abigailstev
  2. Outline •Intro to the Fourier domain •Evenly-spaced and irregular time

    series •Power spectra/periodograms •Coffee break and tutorial time •Wavelets, Hilbert-Huang transform •Spectrograms/dynamical power spectra •Examples of astronomical signals •More coffee and more tutorial time IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 2
  3. The Fourier series Any function can be represented as a

    sum of sines and cosines (with some coefficients, which may also be functions) IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 3 f(x) = ∑ An cos(n 𝜋 x) + ∑ Bn sin(n 𝜋 x) n=0 n=1 ∞ ∞ Slide adapted from J. McIver
  4. The Fourier series IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU &

    UMich 4 Image credit: J. Belk via Wikimedia Example: first four Fourier approximation terms for a square wave The more terms you add, the closer the approximation gets
  5. The Fourier transform Take a periodic or well-bounded function (of

    time or space) by projecting f(t) onto an orthogonal basis of sines and cosines IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 5 F(𝜈) = ∫ f(t) e-i2𝜋𝜈t dt f(t) ⇾ F(𝜈) [or f(𝜈)] ^ f(t) = ∫ F(𝜈) ei2𝜋𝜈t d𝜈 Fourier transform Inverse Fourier transform Slide adapted from J. McIver
  6. The Fourier transform IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU &

    UMich GIF: L. Vieira via Wikimedia 6 Slide adapted from J. McIver f(t) ⇾ F(𝜈) [or f(𝜈)] ^ Think of it like decomposing the time series function into its component frequencies
  7. The Fourier transform IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU &

    UMich Problem solution solve (hard) Transformed problem Transformed solution solve (easy) 8 Fourier transform inverse Fourier transform The Fourier transform and inverse Fourier transform make “Fourier pairs”
  8. Sampling effect: Nyquist frequency IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU

    & UMich 9 The more terms you add, the closer the approximation gets
  9. Sampling effect: Nyquist frequency •In practice, cannot add terms forever!

    •Highest frequency you can sample: 𝜈Nyquist = 1/(2*dt) = d𝜈/2 IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 10 The more terms you add, the closer the approximation gets
  10. Positive and negative Fourier frequencies • Mirrored about 0 Hz

    • In an array, often arranged: (0, pos., Nyquist, neg.) IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 11 Image: J. VanderPlas 2018
  11. Evenly-sampled time series • Signal period << observation length •

    In X-rays and gamma-rays, we count photons. It’s very possible to have zero counts -- “sparse” light curves are common • Instead of saving light curves with lots of zeroes, we use event lists – can select a dt multiple of the detector’s dt • In optical, bright-enough sources mean you detect flux above background in every time bin (e.g., every 30 seconds) • Kepler, K2, TESS can give evenly sampled time series IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 1700 1702 1704 1706 1708 1710 2000 4000 6000 8000 10 4 1.2×10 4 Count/sec Time (s) Start Time 12339 7:28:14:566 Stop Time 12339 7:29:32:683 Bin time: 0.7812Eï02 s 12
  12. Applying Fourier transforms to data IAA-SOSTAT 2021 ☆ Abbie Stevens,

    MSU & UMich 1016 1018 1020 1022 1024 5000 104 1.5×104 Count/sec Time (s) Start Time 10168 18:16:52:570 Stop Time 10168 18:17:08:180 Bin time: 0.1562Eï01 s Time domain Light curve Frequency/Fourier domain Power density spectrum FOURIER TRANSFORM2 Light curve broken into equal-length chunks or segments, take power spectrum of each chunk, average those together x(t)→X(ν) P(ν)=X(ν)X*(ν) =|X(ν)|2 13
  13. Irregularly sampled time series • Many (most?) astronomical time series

    will be irregularly sampled Ø Signal period > observation length Ø Observing cadence Ø Weather Ø Visibility IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 14 Reference: J. VanderPlas 2018, “Understanding the Lomb-Scargle Periodogram”
  14. Lomb-Scargle periodogram saves the day! • Multiplies your Fourier signal

    X(ν) with the Fourier transform of the sampling window ⇾ “convolution” IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 15 Reference: J. VanderPlas 2018, “Understanding the Lomb-Scargle Periodogram” When interpreting: beware many noisy peaks and harmonics due to the sampling window convolved with noise in the data
  15. X-ray variability: Hard to see by eye 1014 1016 1018

    1020 1022 5000 104 1.5×104 Count/sec Time (s) CYGNUS_Xï1 Start Time 10168 18:16:52:578 Stop Time 10168 18:17:02:547 Bin time: 0.3125Eï01 s IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 1696 1698 1700 1702 1704 4000 5000 6000 7000 Count/sec Time (s) GRS1915+105 Start Time 12339 7:28:14:582 Stop Time 12339 7:28:24:542 Bin time: 0.4000Eï01 s Light curves Power density spectra Noise: Cygnus X-1 Signal: GRS 1915+105 16
  16. QPOs → Damped harmonic oscillators IAA-SOSTAT 2021 ☆ Abbie Stevens,

    MSU & UMich y = cos(⍵t) 17 I will discuss a little QPO physics in the next lecture later today
  17. QPOs → Damped harmonic oscillators IAA-SOSTAT 2021 ☆ Abbie Stevens,

    MSU & UMich y = cos(⍵t) x e-bt b=0 b=0.08 18
  18. IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich y =

    cos(⍵t) x e-bt b=0 b=0.08 b=0.22 19 QPOs → Damped harmonic oscillators
  19. QPOs → Damped harmonic oscillators IAA-SOSTAT 2021 ☆ Abbie Stevens,

    MSU & UMich y = cos(⍵t) x e-bt b=0 b=0.08 b=0.22 b=0.5 20
  20. QPOs → Damped harmonic oscillators IAA-SOSTAT 2021 ☆ Abbie Stevens,

    MSU & UMich y = cos(⍵t) x e-bt b=0 b=0.08 b=0.22 b=0.5 b=1.0 The stronger the damping, the wider the peak 21
  21. Poisson noise (“white noise”) Poisson noise from counting photons; power-law

    slope=0 IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 22
  22. Poisson noise + Lorentzian QPO IAA-SOSTAT 2021 ☆ Abbie Stevens,

    MSU & UMich 23
  23. Sampling effect: Aliasing IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU &

    UMich 24 Resonance between signal freq. and sample freq. gives false feature Video credit: Honda Windowing: similar false feature due to segment (“window”) length
  24. Tutorial time •In the SOSTAT2021 Jupyter hub: tutorials/abigail_timeseries/timeseries_workbook.ipynb •On my

    GitHub: https://github.com/abigailStev/timeseries-tutorial IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 25
  25. Wavelets • Fourier products don’t have an intrinsic way to

    tell time resolution (i.e., when in the light curve the signal is present) • Wavelets easily represent a signal in the time domain and in the frequency domain Resource: “A really friendly guide to wavelets”, C. Valens, 1999 IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 26
  26. Wavelets aren’t great for everything •Averaged power spectra (~50+ segments)

    follow a chi-squared distribution with 2 degrees of freedom, about the underlying true power spectrum • Errors are statistically well-defined and well-understood (and easy to compute!) •Wavelets do not follow such a well-defined and well- known distribution •No clear, easy way to assess statistical significance of a signal (which is one of the things we often want to do) IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 27
  27. Wavelets are fantastic for gravitational waves! IAA-SOSTAT 2021 ☆ Abbie

    Stevens, MSU & UMich 28 See also: “Q-transform”
  28. Time (s) 40 41 42 43 44 45 46 47

    48 49 50 Frequency (Hz) 1 2 3 4 5 6 7 8 9 10 Gaussian smoothing amplitude 0 5 10 15 20 25 30 Hilbert-Huang transform • Frequency-domain product designed for data that are non- stationary and non-linear • Like an instantaneous Fourier transform → gives (some) time localization! • Error from standard deviation of (1000+) simulations IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich See Su+15 & refs therein for application to black hole data 0.01 0.1 1 10 100 10 100 Frequency(Hz) Power (Leahy) Hilbert spectrum from Su+15 of a 4Hz QPO 29
  29. Elapsed time (in 64 s segments) Power (rms2/Hz) Elapsed time

    (in 64 s segments; not continuous) Spectrogram (dynamical power spectrum) •Evolution of a power spectrum in time •Instead of averaging together, plot in colormap IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 30
  30. Elapsed time (in 64 s segments) Power (rms2/Hz) Elapsed time

    (in 64 s segments; not continuous) Good Time Intervals (GTIs) • Jumps in the spectrogram below are cut out • Looks like tophat windows in big light curve, but the segments we use for the periodogram and averaging are much smaller than the window length IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 31
  31. Things to sometimes worry about • Deadtime occurs with X-ray

    detectors if your (bright) source is emitting photons faster than your detector can handle. Once some chip of the detector has detected a photon, it cannot detect another photon until it reads out its existing photon detection through the electronics. ØMeasurable as deviation from expected Poisson noise power- law at high frequencies in the power spectrum ØAccumulates over an observation; for a few ks observation, could have several seconds of deadtime to adjust the exposure time by • Pile-up is a sibling of deadtime that affects spectra IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 32
  32. Things to sometimes worry about IAA-SOSTAT 2021 ☆ Abbie Stevens,

    MSU & UMich 33 Slide adapted from D. Huppenkothen; credit: NuSTAR Observatory Guide ‘Deadtime’ ⇾ the detector is effectively dead for the brief readout period. NuSTAR
  33. Things to sometimes worry about IAA-SOSTAT 2021 ☆ Abbie Stevens,

    MSU & UMich 34 Slide adapted from D. Huppenkothen; credit: Huppenkothen & Bachetti 2021 (in press) Huppenkothen & Bachetti (under review)
  34. Poisson noise (a form of “white noise”) • In power

    spectra: P~ν𝛼, 𝛼=0 • Not correlated on any timescale IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 35
  35. Flicker noise • In power spectra: P~ν𝛼, 𝛼=-1 • Correlated

    on medium-short timescales (short “memory”) IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 36
  36. Red noise (random walk) • In power spectra: P~ν𝛼, 𝛼=-2

    • Correlated on long timescales (long “memory”) • Ornstein-Uhlenbeck: ~red noise + friction: tends towards a mean value over long time IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 37
  37. Beware of red noise! • Cannot apply standard peak-finding algorithms,

    since those assume white noise (see Vaughan & Uttley ‘06) • Bigger issue for SMBHs than stellar BHs due to timescales White/Poisson noise Red noise Smith+18b Fourier frequency IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 38
  38. Red noise vs signals −2 0 2 4 6 8

    10 12 time (yr) V mag • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 15.2 15.0 14.8 (a) −2 0 2 4 6 8 10 12 time (yr) V mag • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 15.2 15.0 14.8 (b) • • • • • • • • • • • • • • • • • • • • • (c) PG 1302-102, CRTS data Vaughan+16 (figure); Liu+18 Data looks periodic! IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 39
  39. Red noise vs signals −2 0 2 4 6 8

    10 12 time (yr) V mag • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 15.2 15.0 14.8 (a) −2 0 2 4 6 8 10 12 time (yr) V mag • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 15.2 15.0 14.8 (b) • • • • • • • • • • • • • • • • • • • • • (c) PG 1302-102, CRTS data Vaughan+16 (figure); Liu+18 Uneven sampling, gappy data, only ~1.5 cycles Data looks periodic! IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 40
  40. Red noise vs signals IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU

    & UMich −2 0 2 4 6 8 10 12 time (yr) V mag • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 15.2 15.0 14.8 (a) −2 0 2 4 6 8 10 12 time (yr) V mag • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 15.2 15.0 14.8 (b) • • • • • • • • • • • • • • • • • • • • • (c) Vaughan+16 (figure); Liu+18 PG 1302-102, CRTS data Also including LINEAR data (but it isn’t) Sampling a random red noise process in same way can look like a “periodic” signal Uneven sampling, gappy data, only ~1.5 cycles Data looks periodic! 41
  41. Red noise vs signals IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU

    & UMich −2 0 2 4 6 8 10 12 time (yr) V mag • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 15.2 15.0 14.8 (a) −2 0 2 4 6 8 10 12 time (yr) V mag • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 15.2 15.0 14.8 (b) • • • • • • • • • • • • • • • • • • • • • (c) Vaughan+16 (figure); Liu+18 Data looks periodic! Uneven sampling, gappy data, only ~1.5 cycles PG 1302-102, CRTS data Sampling a random red noise process in same way can look like a “periodic” signal Also including LINEAR data When in doubt, simulate! Also: claimed periodicity in J0045+41 disproven by Barth & Stern ‘18 42
  42. Cross spectra IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich

    x(t)→X(ν) for a narrow energy band y(t)→Y(ν) for a broad-energy reference band As you average segments together: signal adds, noise cancels CXY (ν)=X(ν)Y*(ν) real imaginary 43 real imaginary
  43. Cross spectra IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich

    x(t)→X(ν) for a narrow energy band y(t)→Y(ν) for a broad-energy reference band Cospectrum: real part of the cross spectrum (see Bachetti+15, Bachetti+Huppenkothen 18 and Huppenkothen+Bachetti 18 for statistical details) Note: for X(ν)=Y(ν), cospectrum = cross amplitude = power spectrum 44 Also used: amplitude of the cross spectrum
  44. What other science cases might use these timing analysis tools?

    IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich
  45. Strohmayer 2001 0.1 1 10 100 2 4 6 8

    Type-A 0.1 1 10 100 101 Leahy Power Type-B 0.1 1 10 100 Frequency [Hz] 101 Type-C Stevens Mo1a+17a QPOs in black holes and neutron stars • High-frequency: 100’s Hz • Hot Keplerian blobs in inner disk? • Low-frequency: ~0.01-10’s Hz • Precession of corona/hot flow? Magnetic warps in disk? IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 46
  46. • Short-timescale variability changes on long timescales (spectral state- dependent)

    • Short-timescale variability is energy- dependent IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich Corona-dominated state Mostly-corona-dominated state Disk-dominated state Few months for full spectral state transition Mostly-disk-dominated state If you want to read more, see the power colours paper by Heil+15a 47 QPOs in black holes and neutron stars
  47. • Short-timescale variability changes on long timescales (spectral state- dependent)

    • Short-timescale variability is energy- dependent IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich A B C D 2-60 keV 6.5-13.1 keV 13.1-60 keV 2-6.5 keV Homan+01 48 QPOs in black holes and neutron stars
  48. Pulsations in neutron stars • Spin-down: decreasing spin frequency (e.g.,

    losing rotational energy to the environment) • Spin-up: increasing spin frequency (e.g., accreting material and thus increasing angular momentum) • Glitch: sudden change in spin frequency (due to superfluid NS core?) • Seen in residuals of frequency or pulse timing IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich -400 -300 -200 -100 0 Timing residuals (ms) (a) -100 -50 0 50 100 Days from MJD = 53067.1 -3.750 -3.745 -3.740 -3.735 ν (10-10 Hz s-1) • (d) -100 -50 0 50 100 0 1 2 3 4 5 6 Δν (μHz) (c) -100 -50 0 50 100 -200 -100 0 100 200 Timing residuals (ms) (b) Espinoza+11 49
  49. Stellar pulsations • Cepheid variables, RR Lyrae stars, Delta Scuti

    variables, Blahzko effect (long-period modulation of the periodicity) • Period-luminosity relation makes them standard candles used as “cosmic distance ladder” • Slow enough (periods > hours) that time-domain photometry is often used IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich Image via APOD, credit: R. Vanderbei, ESA/Gaia/DPAC 50
  50. Asteroseismology (“starquakes”) • Understanding the internal structure of stars using

    their brightness oscillations • Convective zone excites oscillations • Fourier analysis of light curves: often see many harmonics IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich Info thanks to online slides by T. Bedding and refs therein Aerts+19 51
  51. QPOs in active galactic nuclei (AGN) • ~1 hr “periodicity”,

    91ks observation • RE J1034+396 is a narrow-line Seyfert 1 AGN • Saw 16 ‘cycles’ (periods) in one uninterrupted observation! • Evenly-sampled time bins • Signal attributed to high-freq. QPO • If at innermost stable circular orbit, MBH ~7x106-1x107 M☉ IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich Gierlinski+08 Alston+14 52
  52. Smith+18b • 44 day low-freq. QPO in KIC 9650712 •

    NLS1 in original Kepler field • 30-minute cadence over 3.5 years: ~30 cycles • Tested periodicity via simulations (Uttley+02) and Lomb- Scargle periodogram IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 53 QPOs in active galactic nuclei (AGN)
  53. • Open-source timing and spectral-timing software (Astropy affiliated package) Øhttps://docs.stingray.science

    • Stingray: Python library of analysis tools • HENDRICS: shell scripting interface • DAVE: graphical user interface • Tutorials in Jupyter notebooks • Huppenkothen, Bachetti, Stevens et al. 2019, ApJ & JOSS • Google Summer of Code students in 2016-2021 Stingray Please remember to cite software in your papers! IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 54
  54. Time series resources Evenly-sampled time series: • Uttley et al.

    2014, “X-ray Reverberation Around Accreting Black Holes” Irregularly-sampled time series: • VanderPlas 2018, “Understanding the Lomb-Scargle Periodogram” Wavelets: • C. Valens 1999, “A really friendly guide to wavelets” Software tools: • Stingray, Lightkurve, GWpy, astropy.timeseries IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 55
  55. IAA-SOSTAT 2021 ☆ Abbie Stevens, MSU & UMich 56