Nathan Hara

(Université de Genève)

https://s3-seminar.github.io/seminars/nathan-hara/

Title — An optimal exoplanet detection criterion

Abstract — Over 4000 exoplanets have been detected so far. They have deeply transformed our understanding of planetary system formation, and expanded our possibilities to search for life outside of Earth. The smaller and the more distant to their host stars exoplanets are, the harder they are to detect. Earth « twins » orbiting solar-type stars are still out of reach because of very complex astrophysical and instrumental noises. To overcome this difficulty, we need new methods to analyse unevenly sampled, multi-variate time series: better models, computational methods and decision rules to claim detections. In this talk I will mostly focus on the latter aspect. Exoplanet detections are claimed based on the value of a statistical significance metric: if it is greater than a certain threshold, a detection is claimed. I will address the question of the optimal significance metric in the general setting of detection of parametric signals, and advocate for a Bayesian hypothesis testing framework where hypotheses are indexed by continuous variables.

Biography — Nathan Hara is a research fellow at the university of Geneva since 2017, which he joined after a PhD with Jacques Laskar and Gwenaël Boué at Paris Observatory. He works on statistical techniques to detect exoplanets, in particular Earth twins, which are prime candidates for the detection of life outside of Earth, and observational programs to unveil multi-planetary systems.

Transcript

1. An optimal detection criterion
   
   for parametric signals
   Nathan Hara
   
   Université de Genève
   GPR
   V
   
   Oxfor
   d
   
   29 March 2022
   With Thibault de Poyferré, Jean-Baptiste Delisle, Marc Ho
   ff
   mann
   Nicolas Unger, Rodrigo Díaz, Damien Ségransan
   

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2. Radial
   velocity
   Star
   Spectrograph
   Observer
   2
   

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3. Detecting exoplanets in RV data SCMA VII
   -3
   -2
   -1
   2
   0
   1
   ×10 -6
   z (AU)
   2
   3
   Motion of the star in the observational reference frame
   0 4
   ×10 -6
   y (AU)
   2
   ×10 -6
   x (AU)
   -2 0
   -2
   -4 -4
   Motion of the star
   To observer
   0 200 400 600 800 1000
   Time (days)
   -0.08
   -0.06
   -0.04
   -0.02
   0
   0.02
   0.04
   0.06
   0.08
   Velocity along z axis (m/s)
   Radial velocity as a function of time
   2
   10 1
   0
   5
   ×10 -5
   y (AU)
   -1
   -2
   x (AU)
   ×10 -6
   0
   Motion of the star in the observational reference frame
   -5
   1.5
   ×10 -5
   z (AU)
   1
   0.5
   0
   -0.5
   -1
   Motion of the star
   To observer
   0 200 400 600 800 1000
   Time (days)
   -0.5
   -0.4
   -0.3
   -0.2
   -0.1
   0
   0.1
   0.2
   0.3
   0.4
   0.5
   Velocity along z axis (m/s)
   Radial velocity as a function of time
   Motion of the star Radial velocity
   Decomposition of the signal
   in periodic components
   
   
   The amplitude of a periodic
   component is proportional
   to the planet projected mass
   Radial velocities
   3
   

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4. Detecting exoplanets in RV data SCMA VII
   Signal shape
   The signal shape depends on the orbital eccentricity
   
   
   Nearly circular
   
   
   Eccentric
   
   
   Very eccentric
   
   
   Orbit
   
   
   Signal
   
   
   Star
   
   
   Planet
   
   
   Credit: Perryman 2011
   Radial velocity
   4
   

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5. Detecting exoplanets in RV data SCMA VII
   Data model
   Signal
   
   
   Credit: Perryman 2011
   Radial velocity
   
   
   Time-series Sum of Keplerian components Other deterministic terms
   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
   y(t) =
   np
   X
   i=1
   Ki(cos(vi(t) + !i) + ei cos !i) + + noise
   Radial velocity
   5
   Stochastic term
   

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6. Detecting exoplanets in RV data SCMA VII
   Noise I
   6
   Photon noise Instrumental systematics
   Example: SOPHIE drift as a function of time
   Credit: François
   Bouchy
   Nominal error bars
   
   
   + jitter (instrumental and/or stellar)
   Error on measurement =
   AAACxXicjVHLSsNAFD2Nr1pfVZdugkVwVZIi6rLoQpdVbCvUIsl0WofmxWRSKEX8Abf6a+If6F94Z0xBLaITkpw5954zc+/1k0CkynFeC9bc/MLiUnG5tLK6tr5R3txqpXEmGW+yOIjlte+lPBARbyqhAn6dSO6FfsDb/vBUx9sjLlMRR1dqnPBu6A0i0RfMU0RditJtueJUHbPsWeDmoIJ8NeLyC27QQwyGDCE4IijCATyk9HTgwkFCXBcT4iQhYeIc9yiRNqMsThkesUP6DmjXydmI9tozNWpGpwT0SlLa2CNNTHmSsD7NNvHMOGv2N++J8dR3G9Pfz71CYhXuiP1LN838r07XotDHsalBUE2JYXR1LHfJTFf0ze0vVSlySIjTuEdxSZgZ5bTPttGkpnbdW8/E30ymZvWe5bkZ3vUtacDuz3HOglat6h5WaxcHlfpJPuoidrCLfZrnEeo4RwNN8u7jEU94ts6s0FLW6DPVKuSabXxb1sMHh6+PiQ==
   i 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
   q
   2
   i
   + 2
   J
   Nominal jitter
   Not on all spectrographs!
   

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7. Detecting exoplanets in RV data SCMA VII
   Credit: NASA/SDO
   Convection cells on the surface of
   the star (granulation)
   Creates noise at the time-scale of the stellar
   rotation period
   Creates correlated noise
   Noise II: stellar activity
   7
   Saar & Donahue 1997, Meunier et al. 2010, Boisse et al.
   2011, Dumusque et al. 2014
   See also Cegla 2019
   From Dumusque et al. 2011
   Stochastic apparition of spots and faculae on
   the surface
   
   
   + Inhibition of the convective blueshift
   Approaching
   limb
   Receding
   
   
   limb
   Super
   
   
   granulation
   Meso
   
   
   granulation
   granulation
   P-modes
   

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8. Detecting exoplanets in RV data SCMA VII
   Stellar activity effect
   0 200 400 600 800 1000 1200 1400 1600
   Time (days)
   -20
   -10
   0
   10
   20
   RV (m/s)
   Ideal and noisy RV signals
   Observations
   Ideal signal
   10 0 10 1 10 2 10 3 10 4
   Period (days)
   0
   0.2
   0.4
   0.6
   Normalized RSS
   Ideal and noisy RV signals, Generalized Lomb-Scargle periodogram
   Observations
   Ideal signal
   8
   Simulated observations:
   
   
   System 1, RV
   fi
   tting challenge
   
   
   Dumusque et al. 2017
   Periods of the injected planets
   Low frequency structures
   Stellar
   
   rotation
   

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9. Detecting exoplanets in RV data SCMA VII
   • Unevenly sampled
   
   Close to ~1 day sampling
   step with missing samples
   
   • ~ 40 - 1000 data points
   
   • Corrupted by uncorrelated
   and complex correlated
   noise
   
   -15
   -10
   -5
   0
   5
   10
   15
   3000 3200 3400 3600 3800 4000 4200 4400 4600 4800 5000
   ΔRV [m/s]
   Date (BJD - 2,450,000.0) [d]
   Radial velocity time-series: summary
   Main characteristics
   Objectives
   • How many planets?
   
   • With what orbital elements?
   9
   

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10. Detecting exoplanets in RV data SCMA VII
    Sun radial velocities
    
    
    observed by HARPS-N
    Expected signal due to
    the Earth
    The sun is observed as a planet hosting star
    Challenge II: dealing with the complex noises
    10
    Dumusque et al. 2021
    
    
    Collier-Cameron et al. 2019
    Credit: Annelies Mortier
    We need to deal with these
    complex noises to detect
    
    
    exo-Earths
    Credit: NASA/SDO
    + instrumental
    

    effects
    

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11. RV data analysis in a nutshell
    p(θ, η ∣ y) ≈ p(θ, η ∣ I, ̂
    RV ) =
    p(I, ̂
    RV ∣ θ, η)p(θ, η)
    p(I, ̂
    RV )
    ̂
    RV = RVcenter of mass
    + RVcontam
    + measurement error
    Planet parameters
    
    
    Other parameters
    
    
    Data:
    
    
    Shape variation indicators:
    θ
    η
    y
    I
    How to reduce the
    spectrum?
    What model do I use?
    How do I compute
    everything?
    Based on this, how
    do I take a decision
    on the number of
    planets?
    RV_center of mass is a pure Doppler shift
    
    
    Other effects also affect the spectral shape
    Upcoming review with Eric Ford
    

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12. RV data analysis in a nutshell
    reduce model compute decide
    (reduce model decide)
    compute
    

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13. Decision: how many planets?
    13
    Periodograms
    10 0 10 1 10 2 10 3 10 4
    Period (days)
    0
    0.1
    0.2
    0.3
    0.4
    0.5
    0.6
    Normalized RSS
    Generalized Lomb-Scargle periodogram 3 sines with SNR 10
    Periodogram
    True spectrum
    Tallest peak
    More precise but
    Fast, numerically stable but
    
    Looks for one planet at a time
    Much heavier computational
    workload, convergence not trivial to
    ensure, not giving information on the
    period
    
    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
    En planets =
    Z
    p(y|✓, n)p(✓|n)d✓
    Bayesian techniques
    Lomb 1976, Ferraz-Mello 1981, Scargle 1982, Baluev 2008,
    2009, 2013, 2015, Zechmeister & Küster 2009, Sulis 2016
    Gregory 2007, Gregory & Ford 2007,
    Tuomi et al. 2011, Diaz et al. 2016
    FAP < 0.1 %
    Evidence n + 1 planets
    Evidence n planets
    > 150
    

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14. Take 1: sparse recovery
    

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15. periodogram
    ℓ1
    Radial velocity data analysis with compressed
    sensing techniques Hara, Boué, Laskar Correia 2017
    10 0 10 1 10 2 10 3 10 4
    Period (days)
    0
    0.1
    0.2
    0.3
    0.4
    0.5
    0.6
    Normalized RSS
    Generalized Lomb-Scargle periodogram 3 sines with SNR 10
    Periodogram
    True spectrum
    Tallest peak
    10 0 10 1 10 2 10 3 10 4
    Period (days)
    0
    0.2
    0.4
    0.6
    0.8
    1
    RV (m/s)
    l1-periodogram 3 sines with SNR 10
    l1-periodogram
    True spectrum
    Interprétation
    Based on sparse recovery techniques
    
    (Chen & Donoho 1998)
    Nelson, Ford et al 2020
    
    6 systems with 200 points in 22s
    15
    Analytical estimate of false alarm
    probability
    E
    ff i
    cient modelling of correlated noise I,
    
    Delisle, Hara, Ségransan, 2020
    

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16. periodogram
    ℓ1
    16
    

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17. periodogram
    ℓ1
    17
    

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18. periodogram
    ℓ1
    18
    

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19. periodogram
    ℓ1
    19
    

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20. periodogram
    ℓ1
    20
    

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21. periodogram
    ℓ1
    21
    

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22. periodogram
    ℓ1
    22
    

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23. periodogram
    ℓ1
    23
    

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24. Application
    Interprétation
    The SOPHIE search for northern extrasolar planets. XVI. HD 15829:
    A compact planetary system in a near-3:2 mean motion resonance
    chain, Hara et al. 2020, A&A
    Six transiting planets and a chain of Laplace resonances in
    TOI-178, Leleu, Alibert, Hara et al. 2021, A&A
    HD 158259 l1 periodogram, noise model with best cross validation score
    HD 158259: 5 to 6 planets
    TOI 178: 6 planets
    24
    Période (jours)
    Périodogramme classique
    Do not transit
    Transits
    
    (TESS)
    Close to a 3:2 mean
    
    motion resonance
    SOPHIE radial velocities
    ESPRESSO (PI) + CHEOPS GTO
    Equilibrium temperature (K)
    Density (g/cm3)
    https://github.com/nathanchara/l1periodogram
    5 outer planets in a chain of Laplace resonances
    

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25. Works well, but what would work best?
    
    Take 2: optimal detection criterion
    

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26. Question
    • What do we mean by optimal detection criterion
    ?
    
    • What is the optimal solution
    ?
    
    • How does it perform?
    Hara, de Poyferré, Delisle, Hoffmann 2022 (submitted, arXiv:2203.04957
    )
    
    Hara, Unger, Delisle, Díaz, Ségransan 2021 (A&A, accepted)
    

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27. What do we mean by « optimal detection
    criterion »?
    

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28. Definition of a detection
    p(y ∣ (θj
    )j=1..n
    , η)
    data Vector of orbital elements
    
    of planet j
    n planets in the model
    Other parameters
    
    (O
    ff
    sets, trends,
    
    hyperparameters of
    
    a Gaussian process)
    General likelihood model
    We de
    fi
    ne a detection claim as
    
    
    « There are n planets, one planet with
    orbital elements , …, one planet
    with orbital elements »
    
    
    s are regions of the parameter spac
    e
    
    θ ∈ Θ1
    θ ∈ Θn
    Θi Parameter space
    Θ1
    Θ2
    

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29. General framework
    p(y ∣ (θj
    )j=1..n
    , η)
    data Vector of parameters
    
    of pattern j
    n patterns in the model
    Nuisance parameters
    General likelihood model
    We de
    fi
    ne a detection claim as
    
    
    « There are n patterns, one pattern with
    parameters , …, one pattern with
    parameters »
    
    
    s are regions of the parameter spac
    e
    
    θ ∈ Θ1
    θ ∈ Θn
    Θi
    Θ1
    Θ2
    Parameter space
    

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30. Definition of a detection: RV case
    Orbital
    
    frequency
    p(y ∣ (θj
    )j=1..n
    , η)
    Time-series of spectra
    
    Or RV
    Kj
    , ej
    , ϖj
    , M0j
    and ωj
    = 2π/Pj
    Example

    we claim the detection of two planets with a certain accuracy on their frequencies
    There is one planet with orbital frequency between
    
    and
    ω1
    −
    Δω
    2
    ω1
    +
    Δω
    2
    There is one planet with orbital frequency between
    
    and
    ω2
    −
    Δω
    2
    ω2
    +
    Δω
    2
    

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31. False and missed detections
    Correct detectio
    n
    
    I claimed that there is a planet with orbital frequency between and ,
    and there is one
    ω1
    −
    Δω
    2
    ω1
    +
    Δω
    2
    There are truly three planets at these frequencies
    Orbital
    
    frequency
    False detection
    I claimed that there is one planet with orbital frequency between and ,
    but there is none.
    ω1
    −
    Δω
    2
    ω1
    +
    Δω
    2
    Missed detections
    I claimed that there are two planets, but there are three (one missed detection)

    Alternately: I missed two planets truly present
    

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32. What do we mean by optimal detection criterion?
    A decision rule selecting the maximizing the expected value of the utilit
    y
    
    (Von Neumann and Morgenstern 1947) or equivalently minimizing
    Cost = Number of false detections + Number of missed detection
    s
    
    • As a function of
    Θi
    , i = 1..n
    γ ×
    γ
    « There are n planets, one planet with orbital elements , …, one planet
    with orbital elements »
    
    
    θ ∈ Θ1
    θ ∈ Θn
    Or
    
    
    Minimizing (Expected number of missed detections) with constraint Expected number of false
    detections <
    • As a function of
    x
    x
    Expectation is taken on the posterior probability p((θj
    )j=1..n
    , η ∣ y)
    

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33. Computing the cost function « There are n planets, one planet with
    orbital elements , …, one planet
    with orbital elements »
    
    
    θ ∈ Θ1
    θ ∈ Θn
    

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34. What is the optimal solution?
    

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35. Orbital
    
    frequency
    The complicated case: overlapping detections
    This case concentrates most of the theoretical complications
    Orbital
    
    frequency
    Everything becomes simple
    Δω
    Δω 2Δω
    If the prior forbids two signals to be too close
    

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36. Solution for exoplanets I
    Orbital
    
    frequency
    Δω
    Δω
    2Δω
    Minimize Cost function = Number of false detections + Number of missed detection
    s
    
    As a function of
    Minimizing (Number of missed detections)
    
    
    with constraint on the expected Number of false detections <
    As a function of
    γ ×
    γ
    x
    x
    Both problems have the same solution
    

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37. Solution for exoplanets II
    Simply compute the posterior probability to have a planet in a frequency interval
    Orbital
    
    frequency
    Δω
    TIP = p
    (
    planet with frequency in interval [ω −
    Δω
    2
    , ω +
    Δω
    2 ] y
    )
    FIP = 1 − TIP
    TIP = True inclusion
    probabilit
    y
    
    FIP = False inclusion
    probability
    Data from Lovis et al.
    2011
    

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38. Computing the TIP/FIP
    Orbital
    
    frequency
    Δω
    TIP = p
    (
    planet with frequency in interval [ω −
    Δω
    2
    , ω +
    Δω
    2 ] y
    )
    =
    nmax
    ∑
    k=1
    p
    (
    planet with frequency in interval [ω −
    Δω
    2
    , ω +
    Δω
    2 ] y, k planets
    )
    p(k planets|y)
    p(k planets|y) =
    p(y|k planets)p(k planets)
    ∑nmax
    j=1
    p(y| j planets)p(j planets)
    Simply compute the posterior probability to have a planet in a frequency interval
    By-products of Bayesia
    n
    
    evidence calculations
    We use Polychord (Handley et al. 2015a, b)
    Δω =
    2π
    Tobs
    

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39. Computational trick
    Gaussian mixture prior on x knowing θ
    RV1 planet
    = A cos ν + B sin ν
    y = M(θ)x
    Radial velocity
    P(y ∣ θ) =
    ∫
    P(y ∣ θ, x)p(x ∣ θ)dx
    has an analytical expression
    Need to explore 3 parameters per planet instead of 5
    Parameters on which the model depends
    non-linearly (eccentricity, period…)
    Gaussian mixture: possibility to have a multimodal prior on mass
    

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40. How do we decide on n_max?
    TIP =
    nmax
    ∑
    k=1
    p
    (
    planet with frequency in interval [ω −
    Δω
    2
    , ω +
    Δω
    2 ] y, k planets
    )
    p(k planets|y)
    For
    fi
    xed n_max:
    

    several runs
    Increment n_max:
    

    Does the
    
    
    FIP periodogram change?
    p(k planets|y) =
    p(y|k planets)p(k planets)
    ∑nmax
    j=1
    p(y| j planets)p( j planets)
    

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41. How does our new detection criterion
    perform?
    

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42. FIP: performances
    Simulation: 1000 systems with 0,1 ou 2 planets generated on 80 time-stamp
    s
    
    (Circular, Log-uniform period, Rayleigh prior on K, uniform on phase
    )
    
    Search for planets with different methods with correct priors
    Periodogram or periodogram
    
    + false alarm probability (FAP) or 

    Bayes factor
    
    FIP periodogram
    
    + FIP, FAP or Bayes factor
    ℓ1
    10 0 10 1 10 2 10 3 10 4
    Period (days)
    0
    0.1
    0.2
    0.3
    0.4
    0.5
    0.6
    Normalized RSS
    Generalized Lomb-Scargle periodogram 3 sines with SNR 10
    Periodogram
    True spectrum
    Tallest peak
    10 0 10 1 10 2 10 3 10 4
    Period (days)
    0
    0.2
    0.4
    0.6
    0.8
    1
    RV (m/s)
    l1-periodogram 3 sines with SNR 10
    l1-periodogram
    True spectrum
    There is one planet with
    
    orbital frequency between
    
    and
    ω1
    − Δω ω1
    + Δω
    False detection
    True detection
    Hara et al. 201
    7
    
    github.com/nathanchara/l1periodogram
    

    View Slide

43. FIP: performances Simulation: 1000 systems with 0,1 or 2 planets
    White noise simulation Red noise simulation (exponential kernel)
    

    View Slide

44. Interpretation
    Simulation: 1000 systems with 0,1 or 2
    planets
    On average, among N independent
    detections with TIP = p, pN detections
    are correct
    TIP: True inclusion probability
    

    View Slide

45. Robustness to a prior change We analyse the data
    with the wrong prior
    Dashed lines: FIP periodogram +
    Bayes factor
    Plain lines: FIP
    periodogram + FIP
    Data generated with
    
    
    • Periods log-uniform on 1-100 day
    s
    
    • Semi-amplitude Rayleigh prior with σ
    

    View Slide

46. HD 10180
    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
    Planets log(E) log(E)
    PNP Runtime
    0 -882.45 0.23 3.82e-108 17s
    1 -839.36 0.19 1.08e-93 1 min 56 s
    2 -789.03 0.24 3.72e-76 6 min 57 s
    3 -736.95 0.04 1.19e-57 17 min 39 s
    4 -677.56 0.15 7.27e-36 38 min 41 s
    5 -603.42 0.13 1.12e-07 1 h 33 min 31 s
    6 -590.14 0.30 6.60e-02 4 h 46 min 46 s
    7 -587.49 0.22 9.34e-01 14 h 59 min 57 s
    Favours 7 planets and the evidence keeps increasing
    

    View Slide

47. How do I compute all this?
    

    View Slide

48. Detecting exoplanets in RV data SCMA VII
    Computationally heavy
    Numerical aspects
    48
    

    View Slide

49. Detecting exoplanets in RV data SCMA VII
    Matrix inversions are typically in but faster for certain matrices
    For semi-separable matrices, the inversion is in
    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
    O(N
    3)
    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
    O(N)
    Numerical methods: S+LEAF matrices
    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
    k(ti, tj) = k( t) =
    X
    s(as cos(⌫s t) + bs sin(⌫s t)) e s t, (1)
    CELERITE kernels yield semi-separable covariance matrices (Foreman Mackey et al. 2017)
    Inversion still in 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
    O(N)
    Delisle, Hara, Ségransan 2020 b
    49
    S+LEAF matrices
    = semi-separable matrix + Leaf matrix
    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
    V
    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
    V = diag(A) + tril UST + triu SUT
    

    View Slide

50. Detecting exoplanets in RV data SCMA VII 50
    Complexity
    AAADH3icjVHNThsxGByW/tAF2gDHXqxGSEFI0QZV0AsSbS89gmADglDk3TiJhfdHXm8lFOVheBNuvVXcKl6gKqf2EfrZNagFVdSr3R3PNzP2ZyelkpWJoqupYPrR4ydPZ56Fs3Pzz180Fha7VVHrVMRpoQp9kPBKKJmL2EijxEGpBc8SJfaT0/e2vv9J6EoW+Z45K8Vxxoe5HMiUG6JOGoddtsl6GTcjnY37kg8nrbcrbPWWMlqqSU+JgWnFux/3eloOR+aOoPaCXRbfKsIwPGk0o3bkBrsPOh404cd20fiKHvookKJGBoEchrACR0XPETqIUBJ3jDFxmpB0dYEJQvLWpBKk4MSe0ndIsyPP5jS3mZVzp7SKoleTk2GZPAXpNGG7GnP12iVb9l/ZY5dp93ZG/8RnZcQajIh9yHej/F+f7cVggDeuB0k9lY6x3aU+pXanYnfO/ujKUEJJnMV9qmvCqXPenDNznsr1bs+Wu/p3p7SsnadeW+Pa7pIuuHP3Ou+D7lq7s95e23nd3Hrnr3oGL/EKLbrPDWzhA7YRU/YFvuEHfgbnwefgS3D5WxpMec8S/hrB1S+EFLIz
    V = diag(A) + tril UST + triu SUT
    AAAC+3icjVHLThsxFD0ZCoR3CsturEZILFA0QRWwRGXDMlUbEomgyjNxghXPo7anahTlT9h1V7HtD3Rb9hV/AH/Ra3eoClEFHs3Muefec+zrG+VKGhuGN5Vg7sX8wmJ1aXlldW19o/Zy89RkhY5FO85UprsRN0LJVLSttEp0cy14EinRiUbHLt/5LLSRWfrBjnNxnvBhKgcy5paoj7X996xnxRc7YTztsylrs7+xFhSnPSsTYZjeZbqnxCejeGoZCethI/SLzYJmCeooVyur/UIPfWSIUSCBQApLWIHD0HOGJkLkxJ1jQpwmJH1eYIpl0hZUJaiCEzui75Cis5JNKXaexqtj2kXRq0nJsE2ajOo0Ybcb8/nCOzv2f94T7+nONqZ/VHolxFpcEPuU7r7yuTrXi8UAh74HST3lnnHdxaVL4W/FnZz905Ulh5w4h/uU14Rjr7y/Z+Y1xvfu7pb7/K2vdKyL47K2wJ07JQ24+Xics+B0r9Hcb+y9e1M/eluOuopXeI0dmucBjnCCFtrkfYkf+InrYBp8Db4FV39Kg0qp2cKDFXz/DalCpD4=
    S and U are n ⇥ r, r 6 n
    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
    O(( ¯
    b2 + r¯
    b + r2)N)
    Semi-separable component
    Leaf component
    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
    bi non-zero extra diagonal coe
    ff
    i
    cients
    Inversion cost of a S+LEAF matrix
    Leaf matrices can model calibration noise
    

    View Slide

51. Detecting exoplanets in RV data SCMA VII
    Semi-separable + Leaf matrices
    Generated
    
    
    quasi-periodic signal
    Gaussian process prediction
    
    
    With quasi-periodic kernel
    
    
    Calibration noise ignored
    Simulated data
    
    
    With calibration noise
    
    
    Gaussian process prediction
    
    
    with quasi-periodic kernel
    
    
    and calibration component
    51
    Calibration noise
    Calibration
    
    
    Component
    
    
    Important for
    
    
    densely
    
    
    sampled data
    
    
    

    View Slide

52. Detecting exoplanets in RV data SCMA VII
    X is a Gaussian
    process
    Radial velocity
    Spectroscopic
    
    
    Indicators
    Gaussian processes
    52
    From Rajpaul et al 2015
    Wavelength lag
    Relative intensity
    Schematic CCFs
    Granular region
    Inter-granular region
    Wavelength lag
    Relative intensity
    Sum of CCF and bisector
    CCFs sum
    bisector
    FWHM
    BIS
    bottom
    BIS
    top
    AAADNnicjVHdahQxGP12qm1df7raS2+Ci7UiLLNFVJBCWb1o0Yta3e1CpyyZbLodmpmMmUyhDPtevknxxjvR3vUFCp7EWbQW0Qwzc3K+c07yJXGuksKG4edGMHft+vzC4o3mzVu37yy17t4bFLo0QvaFVtoMY15IlWSybxOr5DA3kqexkrvx0StX3z2Wpkh09sGe5HI/5ZMsOUgEt6BGrY8r0WupLGc7A7bOBiPBhqv2MXsCaFg01rYaTkG8ZCyKmiuR0hO2M6o230wfQf52Jve1lNtDk1a9rfdT1Hq/onqXokatdtgJ/WBXQbcGbarHtm6dUkRj0iSopJQkZWSBFXEq8OxRl0LKwe1TBc4AJb4uaUpNeEuoJBQc7BG+E8z2ajbD3GUW3i2wisJr4GT0EB4NnQF2qzFfL32yY/+WXflMt7cT/OM6KwVr6RDsv3wz5f/6XC+WDuiF7yFBT7lnXHeiTin9qbids9+6skjIwTk8Rt0AC++cnTPznsL37s6W+/p3r3Ssm4taW9KZ2yUuuPvndV4Fg7VO91ln7d3T9kavvupFuk8PaBX3+Zw2aJO2qY/sU7pozDcWgk/Bl+Br8O2nNGjUnmW6NILzH1CdszI=
    RV = V
    c
    X(t) + V
    r
    ˙
    X(t);
    log R0
    HK
    = L
    c
    X(t)
    BIS = B
    c
    X(t) + B
    r
    ˙
    X(t)
    Augmented data:
    
    
    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
    0
    @
    RV
    log R0
    HK
    BIS
    1
    A
    

    View Slide

53. Detecting exoplanets in RV data SCMA VII
    X is a Gaussian process
    
    
    Radial velocity
    Indicators
    Gaussian processes: a data driven approach
    53
    From Jones et al 2017
    Wavelength lag
    Relative intensity
    Schematic CCFs
    Granular region
    Inter-granular region
    Wavelength lag
    Relative intensity
    Sum of CCF and bisector
    CCFs sum
    bisector
    FWHM
    BIS
    bottom
    BIS
    top
    Let the data select which are
    
    
    Non zero with BIC, AIC, cross validation…
    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
    aij
    

    View Slide

54. Detecting exoplanets in RV data SCMA VII
    Multivariate Gaussian processes
    S+LEAF still in O(N) for multivariate timeseries
    
    
    See Delisle et al. 2022
    54
    https://gitlab.unige.ch/Jean-Baptiste.Delisle/spleaf
    

    View Slide

55. Back to the general case
    p(y ∣ (θj
    )j=1..n
    , η)
    data
    Vector of parameters of pattern j
    n patterns in the model
    Nuisance parameters
    General likelihood model
    We de
    fi
    ne a detection claim as
    
    
    « There are n patterns, one pattern with
    parameters , …, one pattern with
    parameters »
    
    
    s are regions of the parameter spac
    e
    
    θ ∈ Θ1
    θ ∈ Θn
    Θi
    Parameter space
    Θ1
    Θ2
    

    View Slide

56. Conclusion
    p(y ∣ (θj
    )j=1..n
    , η)
    Have n discrete hypotheses Hi
    , i = 1..n
    How many are true?
    s are indices
    θ θj
    = (i)i=1..m
    

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57. General context
    Works for y = Ax + ϵ
    p(y ∣ (θj
    )j=1..n
    , η) s are indices and amplitudes
    θ θj
    = (i, xi
    )i=1..m
    Generalises the Barbieri & Berger 2004 framework
    

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58. Conclusion
    Bayes factors are cool but why settle for second best?
    Just like Bayes factor, the result heavily depend on the mode
    l
    
    -> average over models, check residuals
    Probability(what I’m interested in | data)
    

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