Gwendolyn Eadie: A New Method for Time Series Analysis in Astronomy with an Application to Kepler Data*
- 1. Gwendolyn Eadie eScience & DIRAC Postdoctoral Fellow A New Method for Time Series Analysis in Astronomy … with an Application to Kepler Data
- 2. But first, something with globular clusters
- 3. Hierarchical Bayesian Model for estimating the mass of the Milky Way
- 4. Hierarchical Bayesian Model Likelihood ... Each GC has Individual parameters: ● True Position ● True Velocity
- 5. Hierarchical Bayesian Model Likelihood Prior ... Each GC has Individual parameters: ● True Position ● True Velocity Shared population parameters for galaxy: ● Spatial density of GCs ● Gravitational potential ● Velocity anisotropy
- 6. Hierarchical Bayesian Model Likelihood Prior Hyperprior ... Each GC has Individual parameters: ● True Position ● True Velocity Shared population parameters for galaxy: ● Spatial density of GCs ● Gravitational potential ● Velocity anisotropy Hyperparameters: ● bounds for model parameters ● mean and variance for parameters
- 7. 2018 APRIL 25 → Gaia DR2 Gaia Collaboration + HSTPROMO Catalogue → Helmi et al 2018, Sohn et al 2018 https://www.astro.rug.nl/~ahelmi/research/dr2-dggc/ Vasiliev catalogue → Vasiliev 2019 10.1093/mnras/stz171
- 8. Posterior Distribution for M200 0.71 (0.52, 1.07) x 1012 M๏ Eadie & Jurić (2019) ApJ , 875 159
- 9. Eadie & Jurić (2019) ApJ , 875 159
- 10. Published on the first anniversary of the DR2 release!
- 11. An improved method for time-domain astronomy A New Method for Time Series Analysis in Astronomy … with an Application to Kepler Data
- 12. An improved method for time-domain astronomy (time series analysis)
- 13. An improved method for time-domain astronomy (time series analysis) (spectral analysis)
- 14. Credit: NASA Our Sun: a medium/low mass star American Museum of Natural History
- 15. American Museum of Natural History chromosphere photosphere Convection Zone Radiative Zone CoreHydrostatic equilibrium → radiative and thermal pressure outward, and gravitational force inward
- 16. American Museum of Natural History chromosphere photosphere Convection Zone Radiative Zone CoreHydrostatic equilibrium → radiative and thermal pressure outward, and gravitational force inward Region of convection + different density zones → boundary conditions → standing waves
- 17. Illustration: European Space Agency (ESA) Helioseismology
- 18. Illustration: European Space Agency (ESA) Helioseismology Pressure Mode (p-mode) Oscillations Gravity Mode (g-mode) Oscillations
- 19. Pressure-mode (p-mode) oscillations: standing acoustic waves
- 20. Pressure-mode (p-mode) oscillations: standing acoustic waves
- 21. Pressure-mode (p-mode) oscillations: standing acoustic waves Thus, radial modes are when ℓ = 0 n = # of radial nodes ℓ = # of nodes around azimuth m = # nodes around equator
- 22. (Highly Exaggerated) Non-radial Stellar Vibration Modes By Grotsch - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=25075592 ℓ = 1 m = 0 ℓ = 2 m = 0 ℓ = 2 m = 1 ℓ =4 m = 2
- 23. Observe the surface of our Sun over time → time series data Credit: NASA
- 24. Observe the surface of our Sun over time → time series data Credit: NASA Look for signals in the frequency domain → Power Spectral Density → (Power Spectrum)
- 25. Power Spectral Density (PSD, or Power Spectrum) For a continuous time series, the PSD describes how the power is distributed across different frequencies.
- 26. Power Spectrum Estimate for the Sun Figure 1 in Bedding & Kjeuson (2003), PASA, 20, 203 Full-disk solar oscillations measured in intensity by the VIRGO instrument on the SOHO spacecraft
- 27. Power Spectrum for the Sun Full-disk solar oscillations measured in intensity by the VIRGO instrument on the SOHO spacecraft Figure 2 in Bedding & Kjeuson (2003), PASA, 20, 203
- 28. What can we measure? Spacing of adjacent modes: Δν ∝ ρ1/2 Peak mode frequency: Mean density
- 29. What can we measure? Spacing of adjacent modes: Δν ∝ ρ1/2 Peak mode frequency: νmax ∝ M R-2 Teff 1/2 Mean density Mass Radius Effective Temperature
- 30. What can we measure? Spacing of adjacent modes: Δν ∝ ρ1/2 Peak mode frequency: νmax ∝ M R-2 Teff 1/2 Mean density Mass Radius Effective Temperature With these two equations, and Teff , we can solve for the mass and radius of the star!
- 31. Asteroseismology The study of pressure and gravity waves inside other stars ceps.spacescience.org → measure stellar properties → test theories about stellar evolution
- 32. Asteroseismology The study of pressure and gravity waves inside other stars → measure stellar properties → test theories about stellar evolution He H-shell fusion Red Giant Star Observe star’s brightness over time → time series data These data allow us to estimate the power spectrum → periodogram ceps.spacescience.org
- 33. Power Spectrum estimate of KIC 5786154 (a red giant star) Gaulme et al (2016), The Astrophysical Journal, 832:121
- 35. Challenges in Asteroseismology 1. Cannot resolve stars Credit: NASA/Hubble
- 36. Challenges in Asteroseismology 1. Cannot resolve stars 2. Classical periodogram is a poor estimator of the power spectrum
- 37. 1. Cannot resolve stars 2. Classical periodogram is a poor estimator of the power spectrum 3. Unevenly sampled time series data Challenges in Asteroseismology
- 38. Challenges in Asteroseismology 1. Cannot resolve stars 2. Classical periodogram is a poor estimator of the power spectrum 3. Unevenly sampled time series data
- 39. Example time series Time Signal Autoregressive process
- 40. True Power Spectrum of AR(4) Time Signal
- 41. Periodogram: estimator of the power spectrum → Fourier transform of the time series’ autocorrelation function
- 42. Two statistical issues with the Periodogram 1. Biased estimator for small n n = 1000
- 43. Two statistical issues with the Periodogram 1. Biased estimator for small n n = 10000 n = 1000
- 44. Two statistical issues with the Periodogram 1. Biased estimator for small n 2. Inconsistent (variance does not go to 0 as n→ ∞) n = 10000 n = 1000
- 45. Two statistical issues with the Periodogram 1. Biased estimator for small n 2. Inconsistent (variance does not go to 0 as n→ ∞) n = 10000 n = 1000n = 50000
- 46. Spectral Leakage Frequency domainTime domain
- 47. Spectral Leakage Frequency domainTime domain
- 48. Spectral Leakage Frequency domainTime domain
- 49. Spectral Leakage Frequency domainTime domain
- 50. Spectral Leakage Frequency domainTime domain
- 51. Figure 6 in VanderPlas (2018), ApJS, 236:16. Spectral Leakage Frequency domainTime domain
- 52. Spectral Leakage Figure 6 in VanderPlas (2018), ApJS, 236:16. Common approach to mitigate spectral leakage: “Taper” the data Frequency domainTime domain
- 53. Thomson Multitaper Estimator Thomson, D. J. (1982) "Spectrum estimation and harmonic analysis." Proceedings of the IEEE, 70, 1055–1096
- 54. http://thomas-cokelaer.info/software/spectrum/html/user/ref_mtm.html Discrete Prolate Spheroidal SequencesThomson Multitaper Estimator Thomson, D. J. (1982) "Spectrum estimation and harmonic analysis." Proceedings of the IEEE, 70, 1055–1096
- 55. http://thomas-cokelaer.info/software/spectrum/html/user/ref_mtm.html Discrete Prolate Spheroidal SequencesThomson Multitaper Estimator Thomson, D. J. (1982) "Spectrum estimation and harmonic analysis." Proceedings of the IEEE, 70, 1055–1096 ● reduces spectral leakage between adjacent frequencies ● reduces bias and variance
- 56. → Multitaper reduces both bias and variance Back to our AR(4) example:
- 57. Back to our AR(4) example → Multitaper reduces both bias and variance This example originally appears in Percival & Walden (1993), Cambridge University Press.
- 58. Back to our AR(4) example → Multitaper reduces both bias and variance These estimators assume regularly- sampled time series data This example originally appears in Percival & Walden (1993), Cambridge University Press.
- 59. Challenges in Asteroseismology 1. Cannot resolve stars 2. Classical periodogram is a poor estimator of the power spectrum 3. Unevenly sampled time series data
- 60. Time Series of Red Giant Star Kepler-91
- 61. Lomb-Scargle Spectral Estimator (Lomb 1976, Scargle 1982)
- 62. Lomb-Scargle Spectral Estimator (Lomb 1976, Scargle 1982) Periodogram for unevenly-sampled time series
- 63. Lomb-Scargle Spectral Estimator (Lomb 1976, Scargle 1982) Periodogram for unevenly-sampled time series Not so good properties: 1. Spectral leakage even worse than the classical periodogram 2. Identification of anything other than the dominant frequency is very challenging
- 64. Multitaper for unevenly sampled time series?
- 65. Multitaper for unevenly sampled time series?
- 66. Multitaper for unevenly sampled time series? Bronez (1988) ● Generalized solution to the multitaper approach for unevenly-sampled times ● difficult and expensive to compute.
- 67. Multitaper for unevenly sampled time series? Springford (2017) ● Approx. slepian weights with splines ● Combine with Lomb-Scargle Bronez (1988) ● Generalized solution to the multitaper approach for unevenly-sampled times ● difficult and expensive to compute.
- 69. The result is not optimal for arbitrary sampling times, but it reduces the leakage and variance of the LS estimator. It is identical to the MT estimator for regular sampling. It is very fast to compute. Multitaper + Lomb-Scargle (MTLS)
- 70. The result is not optimal for arbitrary sampling times, but it reduces the leakage and variance of the LS estimator. It is identical to the MT estimator for regular sampling. It is very fast to compute. Multitaper + Lomb-Scargle (MTLS)
- 71. The result is not optimal for arbitrary sampling times, but it reduces the leakage and variance of the LS estimator. It is identical to the MT estimator for regular sampling. It is very fast to compute. Multitaper + Lomb-Scargle (MTLS)
- 72. Time Series (aka lightcurve) of Red Giant Star Kepler-91
- 73. Springford (2017), Queen’s University. Springford, Eadie, & Thomson (2019, in prep.) Data pre-processed by Tim Bedding (Sydney Institute for Astronomy) Time Series (aka lightcurve) of Red Giant Star Kepler-91
- 76. Lomb-Scargle (LS) vs. Multitaper LS Multitaper-LS Solar-like oscillations Springford (2017), Doctoral Thesis, Queen’s University. Springford, Eadie, & Thomson (2019, in prep.)
- 77. Lomb-Scargle (LS) vs. Multitaper LS Multitaper-LS Springford (2017), Doctoral Thesis, Queen’s University. Springford, Eadie, & Thomson (2019, in prep.) Δν ∝ ρ1/2 νmax ∝ M R-2 Teff 1/2
- 78. Challenges in Asteroseismology 1. Cannot resolve stars 2. Periodogram is a poor estimator of the power spectrum 3. Unevenly sampled time series data These are challenges for time-domain astronomy in general!
- 79. Periodic phenomena in astronomy ● Asteroseismology ● Stellar rotation ● Binary systems of stars (eclipsing systems) ● Variable stars (e.g. cepheids, RR Lyrae) ● Black hole accretion ● Stellar spots
- 80. Archival and New Time Series Data Kepler Spacecraft NuSTAR TESS
- 81. Actual Data from TESS (via “tess.casino”) https://github.com/benmontet/tess.casino#tesscasino
- 82. Actual Data from TESS (via “tess.casino”) https://github.com/benmontet/tess.casino#tesscasino
- 83. Time-domain astronomy is growing! Large Synoptic Survey Telescope 2022 – 2032 Zwicky Transient Facility 2018 – 2021
- 85. Thank you! References and Links Helmi et al (2018), A&A, Volume 16 Paper: https://doi.org/10.1051/0004-6361/201832698 Data: https://www.astro.rug.nl/~ahelmi/research/dr2-dggc/ Sohn et al (2018), ApJ 862 52 10.3847/1538-4357/aacd0b/meta Vasiliev (2019)): https://arxiv.org/abs/1807.09775 Work on the Milky Way’s mass with hierarchical Bayes: Eadie, Harris, & Widrow (2015), ApJ, 806 54 doi: 10.1088/0004-637X/806/1/54 Eadie & Harris (2016), ApJ, 829 108 doi: 10.3847/0004-637X/829/2/108 Eadie, Springford, & Harris (2017), ApJ, 835 167 10.3847/1538-4357/835/2/167 Eadie, Springford, & Harris (2017), ApJ 838 76 10.3847/1538-4357/aa64db Eadie, Keller, & Harris (2018), ApJ, 865 72, doi: 10.3847/1538-4357/aadb95 Eadie & Jurić (2019), ApJ, 875 159 , doi: 10.3847/1538-4357/ab0f97 eadieg@uw.edu @gweneadie