Modeling Kepler systematics using Gaussian processes

Transcript

1. Dan Foreman-Mackey     CCPP@NYU dan.iel.fm dfm exoplaneteer
2. I'm an engineer.
3. I used to study galaxies.
4. Now I study exoplanets?
5. I do data analysis.
6. it's hammer time! emceethe MCMC Hammer introducing arxiv.org/abs/1202.3665 dan.iel.fm/emcee
7. Credit: NASA
8. Credit: Carter Roberts
9. Credit: NASA
10. Reference: kepler.nasa.gov 2 1study the population of Earth-sized planets in the "habitable zone" learn a whole lot about stars
11. Reference: kepler.nasa.gov 2 1study the population of Earth-sized planets in the "habitable zone" learn a whole lot about stars
12. Reference: kepler.nasa.gov 2 1study the population of Earth-sized planets in the "habitable zone" learn a whole lot about stars Earth-like
13. Reference: kepler.nasa.gov 2 1study the population of Earth-sized planets in the "habitable zone" learn a whole lot about stars Earth-like of Sun-like stars
14. Kepler data are awesome for finding planets!
15. Kepler data are awesome for finding planets!and planet candidates!
16. Earth-sized? Sure. Earth-like? Not yet!
17. Data from: NASA Exoplanet Archive
18. Kepler's photometry is really impressive!
19. relative aperture photometry [ppm] Object: Kepler-37
20. "corrected" aperture photometry [ppm] Object: Kepler-37
21. "corrected" aperture photometry [ppm] Object: KIC 3641858
22. "corrected" aperture photometry [ppm] Object: KIC 8415109
23. You can do a lot with photometry.
24. you can measure find planets confirm planets orbital parameters eccentricities masses and so much more…
25. What does all this have in common?
26. model parameters expected data observed (noisy) data physical model gravity, limb darkening, etc. noise model likelihood function
27. relative aperture photometry [ppm] Object: Kepler-62 how do you get from here...
28. Credit: NASA Kepler-62f …to here?
29. Credit: NASA Kepler-37b
30. Credit: NASA Kepler-37b
31. Systematics. Noise, de-trending, and the rest.
32. 2 astrophysical: variable stars, binary stars, star spots, etc. 1 instrumental: pointing drift, temperature variations, electronic noise, etc.
33. Credit: NASA
34. Object: Kepler-37 linear counts
35. Object: Kepler-37 log counts
36. Object: Kepler-37 pixel time series time flux
37. Object: Kepler-37 pixel time series time flux
38. Object: Kepler-32 linear counts
39. Object: Kepler-32 log counts
40. Object: Kepler-32 time flux
41. Object: Kepler-32 time flux
42. Pre-search Data Conditioning (PDC) (yeah it's a strange name) attempts to remove instrumental effects without affecting astrophysical signals (nearby targets are correlated) (nearby targets are uncorrelated)
43. Pre-search Data Conditioning (PDC) (yeah it's a strange name) run PCA on nearby light curves fit the target with the top few components
44. "corrected" aperture photometry [ppm] Object: Kepler-37 Cool story. I can haz planetz? still have astrophysical "noise" have introduced correlated systematic noise
45. What next? 2 preprocessing hacks to "remove" systematics 1 "sophisticated" model of stellar activity + transit
46. What next? 2 preprocessing hacks to "remove" systematics 1 "sophisticated" model of stellar activity + transit EXPENSIVE
47. What next? 2 preprocessing hacks to "remove" systematics 1 "sophisticated" model of stellar activity + transit EXPENSIVE IT'S A HACK
48. model parameters expected data observed (noisy) data physical model gravity, limb darkening, etc. noise model likelihood function
49. model parameters expected data observed (noisy) data physical model gravity, limb darkening, etc. noise model likelihood function + infinite time…
50. model parameters expected data observed (noisy) data physical model gravity, limb darkening, etc. noise model likelihood function + infinite time… in theory…
51. Which question are you asking?
52. Search versus Characterization (parameter estimation)
53. Search versus Characterization (parameter estimation) in practice…
54. Search 2 box least-squares: an efficient hypothesis test 1 running windowed median filter
55. Data from: NASA Exoplanet Archive
56. Characterization 2 independent noise or Carter + Winn (2009) 1 wide median filter + piecewise polynomial
57. Object: KIC 2301306 + injected super-Earth
58. Object: KIC 2301306 + injected super-Earth
59. Gaussian Processes. A non-parametric, probabilistic noise model.
60. 2 = N X n=1 [ yn ˆ y ( xn)]2 2 n
61. 2 = N X n=1 [ yn ˆ y ( xn)]2 2 n p(Y | ✓) = N Y n=1 N(yn; ˆ yn, 2 n )
62. 1 p 2 ⇡ 2 n exp ✓ 1 2 [ yn ˆ yn] 2 2 n ◆ 2 = N X n=1 [ yn ˆ y ( xn)]2 2 n p(Y | ✓) = N Y n=1 N(yn; ˆ yn, 2 n )
63. ln p(Y | ✓) = 1 2 [y ˆ y]TV 1[y ˆ y] 1 2 ln |V| + C ln p(Y | ✓) = 1 2 N X n=1  [yn ˆ yn]2 2 n + ln 2 n + C 1 2 n yn ˆ yn yn ˆ yn
64. ln p(Y | ✓) = 1 2 [y ˆ y]TV 1[y ˆ y] 1 2 ln |V| + C ln p(Y | ✓) = 1 2 N X n=1  [yn ˆ yn]2 2 n + ln 2 n + C 1 2 n yn ˆ yn yn ˆ yn A Gaussian Process
65. All the data are drawn from one huge Gaussian! (N-dimensional) but it's trivial.
66. correlated noise periodic noise Generalizations Vn,m = k ( xn, xm)
67. correlated noise k ( xn, xm ) = ↵ 2 e ( xn xm)2 / 2 ` 2
68. correlated noise k ( xn, xm ) = ↵ 2 e ( xn xm)2 / 2 ` 2
69. periodic noise k(xn, xm) = ↵ 2 e x 2 / 2 ` 2 + 2 e x 2 / 2 ` 2 2 cos 2⇡ | x | T
70. periodic noise k(xn, xm) = ↵ 2 e x 2 / 2 ` 2 + 2 e x 2 / 2 ` 2 2 cos 2⇡ | x | T
71. Measurement uncertainties?
72. Measurement uncertainties? k ( xn, xm) = 2 n nm (the sum of 2 valid kernels is a kernel)
73. physics transit LC observed (noisy) data GP
74. physics transit LC observed (noisy) data GP = posterior probability of "physics" optimize or marginalize
75. Isn't this too expensive? ln p(Y | ✓) = 1 2 [y ˆ y]TV 1[y ˆ y] 1 2 ln |V| + C
76. A Gaussian process framework for modelling instrumental systematics: application to transmission spectroscopy Gibson et al. (2012)
77. Credit: NASA/ESA & Evans et al. (2013) HD 189733b Evans et al. (2013): an "Azure" planet
78. Credit: NASA/ESA & Evans et al. (2013) HD 189733b Evans et al. (2013): an "Azure" planet – 6 – −1500 0 +1500 0 +300 λ = 290–450 nm λ = 450–570 n Relative Flux (ppm) flux time
79. relative aperture photometry [ppm] Object: Kepler-37 What about Kepler? Too much data?
80. I never throw away any data! David W. Hogg
81. I never throw away any data! David W. Hogg Throw Away All The Data™
82. relative aperture photometry [ppm] Object: Kepler-37
83. relative aperture photometry [ppm] Object: Kepler-37
86. Does all this work?
87. Object: KIC 2301306 + injected super-Earth
88. Object: KIC 2301306 + injected super-Earth
89. figure generated using: github.com/dfm/triangle.py
90. figure generated using: github.com/dfm/triangle.py This only took 1.5 minutes to sample!
91. Time for an unfair comparison.
92. Object: KIC 2301306 + injected super-Earth
93. Object: KIC 2301306 + injected super-Earth
94. Object: KIC 2301306 + injected super-Earth
95. figure generated using: github.com/dfm/triangle.py
96. figure generated using: github.com/dfm/triangle.py
97. figure generated using: github.com/dfm/triangle.py
98. figure generated using: github.com/dfm/triangle.py !!
99. No one would do that !!! Right?
100. What about search?
101. Object: KIC 3641858 ln " ptransit(data | t(k) 0 , P(k) . . .) pnull(data | t(k) 0 , P(k) . . .) #
102. import kplr client = kplr.API() # Find a KOI. koi = client.koi(952.01) print(koi.koi_period) # This KOI has an associated star. star = koi.star print(star.kic_teff) # Download the lightcurves for this KOI. lightcurves = koi.get_light_curves() for lc in lightcurves: hdus = lc.open() a Python interface to Kepler data kplr dan.iel.fm/
103. p e B A R T bart github.com/dfm/ Python/C framework for modeling exoplanet observations WORK IN PROGRESS flexible + fast limb darkening (C) Gaussian process likelihood (C++) expressive model building syntax (Python) simple interface to fitting software (Python)
104. import numpy as np import emcee def lnprob(x, ivar): return -0.5 * np.sum(ivar * x ** 2) ndim, nwalkers = 10, 100 ivar = 1. / np.random.rand(ndim) p0 = [np.random.rand(ndim) for i in range(nwalkers)] sampler = emcee.EnsembleSampler(nwalkers, ndim, lnprob, args=[ivar]) sampler.run_mcmc(p0, 1000) The MCMC Hammer: user-friendly MCMC in Python emcee dan.iel.fm/ it's hammer time!
105. THANKS! any questions?