Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach Bo Ning 1 with Angie Wolfgang 2 , 3 and Sujit Ghosh 1 1 North Carolina State University 2 Pennsylvania State University 3 NSF Astronomy & Astrophysics Postdoctoral Fellow September 24, 2017 Project is supported by SAMSI and NSF.
Collaborators Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 2/24
Outline Background 1 Bernstein polynomials 2 3 Building a model for estimating M-R relation Results 4 Conclusion 5 Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 3/24
Section 1 Background 1 Bernstein polynomials 2 Building a model for estimating M-R relation 3 4 Results 5 Conclusion Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 4/24
Background Astronomers have discovered thousands of exoplanets with either Mass or radius measurements Knowing a planet’s mass and radius is important for understanding its compositions Only small portion planets have both mass and radius measurements Mass: radial velocity; Radius: transits To estimate the mass-radius relation (M-R relation) and use it to predict other planets’ mass or radius Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 5/24
Background Astronomers have discovered thousands of exoplanets with either Mass or radius measurements Knowing a planet’s mass and radius is important for understanding its compositions Only small portion planets have both mass and radius measurements Mass: radial velocity; Radius: transits To estimate the mass-radius relation (M-R relation) and use it to predict other planets’ mass or radius Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 5/24
Background Astronomers have discovered thousands of exoplanets with either Mass or radius measurements Knowing a planet’s mass and radius is important for understanding its compositions Only small portion planets have both mass and radius measurements Mass: radial velocity; Radius: transits To estimate the mass-radius relation (M-R relation) and use it to predict other planets’ mass or radius Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 5/24
Background Astronomers have discovered thousands of exoplanets with either Mass or radius measurements Knowing a planet’s mass and radius is important for understanding its compositions Only small portion planets have both mass and radius measurements Mass: radial velocity; Radius: transits To estimate the mass-radius relation (M-R relation) and use it to predict other planets’ mass or radius Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 5/24
Background Astronomers have discovered thousands of exoplanets with either Mass or radius measurements Knowing a planet’s mass and radius is important for understanding its compositions Only small portion planets have both mass and radius measurements Mass: radial velocity; Radius: transits To estimate the mass-radius relation (M-R relation) and use it to predict other planets’ mass or radius Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 5/24
A hierarchical Bayesian power-law model Model (HBM, WRF16) ind M obs ∼ N ( M i , σ obs M , i ) , i R obs ∼ N ( R i , σ obs ind R , i ) , i M i | R i , C , γ, σ M ∼ N ( CR γ i , σ M ) M i is the planet mass divided by the Earth’s mass, R i is the planet radius divided by the Earth’s radius. Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 6/24
A hierarchical Bayesian power-law model Figure: M-R relation using power-law model. (Copy from WRF16) Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 7/24
A hierarchical Bayesian power-law model Model (HBM, WRF16) ind M obs ∼ N ( M i , σ obs M , i ) , i ind R obs ∼ N ( R i , σ obs R , i ) , i M i | R i , C , γ, σ M ∼ N ( CR γ i , σ M ) M i is the planet mass divided by the Earth’s mass, R i is the planet radius divided by the Earth’s radius. Normal distributed? Constant intrinsic scatter? Only one power-law? Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 8/24
Section 2 Background 1 Bernstein polynomials 2 Building a model for estimating M-R relation 3 4 Results 5 Conclusion Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 9/24
Nonparametric approaches Basis expansion, Gaussian process, Kernel methods, Dirichlet process, P´ olya tree .... Basis expansion: spline functions, Bernstein polynomials, wavelets, trigonometric polynomials .... Definition (Bernstein polynomial) For a continuous function F : [ 0 , 1 ] → R , the associated Bernstein polynomial is defined as � k d �� � d � x k ( 1 − x ) d − k . B ( x ; k , F ) = F d k k = 0 As d → ∞ , B ( x ; d , F ) converge to F (uniformly) by Weierstrass approximation theorem Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 10/24
Nonparametric approaches Basis expansion, Gaussian process, Kernel methods, Dirichlet process, P´ olya tree .... Basis expansion: spline functions, Bernstein polynomials, wavelets, trigonometric polynomials .... Definition (Bernstein polynomial) For a continuous function F : [ 0 , 1 ] → R , the associated Bernstein polynomial is defined as � k d �� � d � x k ( 1 − x ) d − k . B ( x ; k , F ) = F d k k = 0 As d → ∞ , B ( x ; d , F ) converge to F (uniformly) by Weierstrass approximation theorem Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 10/24
Nonparametric approaches Basis expansion, Gaussian process, Kernel methods, Dirichlet process, P´ olya tree .... Basis expansion: spline functions, Bernstein polynomials, wavelets, trigonometric polynomials .... Definition (Bernstein polynomial) For a continuous function F : [ 0 , 1 ] → R , the associated Bernstein polynomial is defined as � k d �� � d � x k ( 1 − x ) d − k . B ( x ; k , F ) = F d k k = 0 As d → ∞ , B ( x ; d , F ) converge to F (uniformly) by Weierstrass approximation theorem Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 10/24
Bernstein polynomial A Bernstein polynomial density can be obtained by taking derivative on B ( x ; k , F ) , such that � k d � k − 1 � � �� � − F β k ( x ; k , d − k + 1 ) , b ( x ; k , f ) = F d d k = 1 where β k ( x ; k , d − k + 1 ) is a beta density. One often estimates the density by rewriting it corresponding to a weight sequence w = ( w 1 , . . . , w d ) , such that d � � f N ( x | w ) ≡ b ( x ; k , f ) = w k β k ( x ; k , d − k + 1 ) , w k = 1 , w k ≥ 0 . k = 1 k Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 11/24
Bernstein polynomial A Bernstein polynomial density can be obtained by taking derivative on B ( x ; k , F ) , such that � k d � k − 1 � � �� � − F β k ( x ; k , d − k + 1 ) , b ( x ; k , f ) = F d d k = 1 where β k ( x ; k , d − k + 1 ) is a beta density. One often estimates the density by rewriting it corresponding to a weight sequence w = ( w 1 , . . . , w d ) , such that d � � f N ( x | w ) ≡ b ( x ; k , f ) = w k β k ( x ; k , d − k + 1 ) , w k = 1 , w k ≥ 0 . k = 1 k Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 11/24
Bernstein polynomial (cont’d) d � � f N ( x | w ) = w k β k ( x ; k , d − k + 1 ) , w k = 1 , w k ≥ 0 . k = 1 k Degree d = 1 Degree d = 2 Degree d = 3 1.0 1.0 1.0 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Degree d = 5 Degree d = 10 Degree d = 20 1.0 1.0 1.0 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 12/24
Connections Connection to mixture models: f N ( x | w ) = � d k = 1 w k β k ( x ; k , d − k + 1 ) Clustering: number of power-laws d is not the number of clusters Gaussian mixture models: requires to estimate parameters in each Gaussian component Connection to a multivariate density estimation. For x , y ∈ [ 0 , 1 ] , a bivariate Bernstien polynomial density is, d 1 d 2 � � w kl β k ( x ; k , d 1 − k + 1 ) β l ( y ; l , d 2 − l + 1 ) , f ( x , y ; k , F ) = k = 1 l = 1 d 1 d 2 � � w kl = 1 , w kl ≥ 0 k = 1 l = 1 Modeling the joint density: when both masses and radii have measurement errors. The conditional and marginal distributions are mixture of beta distributions Ning, Wolfgang & Ghosh — Predicting Exoplanet Mass-Radius Relationship: a Nonparametric Approach 13/24
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