Nonparametric Prediction and the Exoplanet Mass-Radius Relationship - - PowerPoint PPT Presentation

Nonparametric Prediction and the Exoplanet Mass-Radius Relationship

Bo Ning1

with Angie Wolfgang2,3 and Sujit Ghosh1,4

1North Carolina State University 2Pennsylvania State University 3NSF Astronomy & Astrophysics Postdoctoral Fellow 4Statistical and Applied Mathematical Sciences Institute

May 8, 2017

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Collaborators

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 2

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Outline

1

Background

2

Model

3

Estimating Mass-Radius Relations

4

Future work

5

Concluding remarks

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 3

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Section 1

1

Background

2

Model

3

Estimating Mass-Radius Relations

4

Future work

5

Concluding remarks

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 4

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Introduction

Since 1992, astronomers have discovered thousands of exoplanets Mass or Radius can be measured for those discovered exoplanets Measuring Mass and Radius is important because they tell us about the planets’ compositions. Measuring mass: radial velocity Measuring radius: transits

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 5

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Introduction

Since 1992, astronomers have discovered thousands of exoplanets Mass or Radius can be measured for those discovered exoplanets Measuring Mass and Radius is important because they tell us about the planets’ compositions. Measuring mass: radial velocity Measuring radius: transits

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 5

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Introduction

Since 1992, astronomers have discovered thousands of exoplanets Mass or Radius can be measured for those discovered exoplanets Measuring Mass and Radius is important because they tell us about the planets’ compositions. Measuring mass: radial velocity Measuring radius: transits

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 5

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Introduction

Since 1992, astronomers have discovered thousands of exoplanets Mass or Radius can be measured for those discovered exoplanets Measuring Mass and Radius is important because they tell us about the planets’ compositions. Measuring mass: radial velocity Measuring radius: transits

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 5

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Introduction

Since 1992, astronomers have discovered thousands of exoplanets Mass or Radius can be measured for those discovered exoplanets Measuring Mass and Radius is important because they tell us about the planets’ compositions. Measuring mass: radial velocity Measuring radius: transits

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 5

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Measuring mass and radius

Photo credit: Leslie Rogers

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 6

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Kepler planet candidates

Photo credit: Angie Wolfgang

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 7

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Goal

Most planets only have either a mass or a radius measurement, very few have both Measurements for planets radius are more precise Goal: To create a statistical tool for predicting planets mass given their radii.

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 8

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Section 2

1

Background

2

Model

3

Estimating Mass-Radius Relations

4

Future work

5

Concluding remarks

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 9

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

A hierarchical Bayesian power-law model

HBM [Wolfgang, Rogers and Ford, 2016]: Mobs

i

∼ N(Mi, σobs

M,i ),

Robs

i

∼ N(Ri, σobs

R,i ),

Mi|Ri,C, γ, σM ∼ N(CRγ

i , σM)

Mi is the planet mass divided by the Earth’s mass, Ri is the planet radius divided by the Earth’s radius.

RV only < 4R⊕: Mi|Ri ∼ N(2.7R1.3

i

, 1.9) RV only < 8R⊕: Mi|Ri ∼ N(1.6R1.8

i

, 2.9) (Data: NASA Exoplanet Archive) Why is intrinsic scatter normally distributed? Why is intrinsic scatter a constant, not σM(Ri)? σ =

• σ2

M + β(Ri − 1) [WRF16]

Why use a power law?

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 10

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

A hierarchical Bayesian power-law model

HBM [Wolfgang, Rogers and Ford, 2016]: Mobs

i

∼ N(Mi, σobs

M,i ),

Robs

i

∼ N(Ri, σobs

R,i ),

Mi|Ri,C, γ, σM ∼ N(CRγ

i , σM)

Mi is the planet mass divided by the Earth’s mass, Ri is the planet radius divided by the Earth’s radius.

RV only < 4R⊕: Mi|Ri ∼ N(2.7R1.3

i

, 1.9) RV only < 8R⊕: Mi|Ri ∼ N(1.6R1.8

i

, 2.9) (Data: NASA Exoplanet Archive) Why is intrinsic scatter normally distributed? Why is intrinsic scatter a constant, not σM(Ri)? σ =

• σ2

M + β(Ri − 1) [WRF16]

Why use a power law?

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 10

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

A hierarchical Bayesian power-law model

HBM [Wolfgang, Rogers and Ford, 2016]: Mobs

i

∼ N(Mi, σobs

M,i ),

Robs

i

∼ N(Ri, σobs

R,i ),

Mi|Ri,C, γ, σM ∼ N(CRγ

i , σM)

Mi is the planet mass divided by the Earth’s mass, Ri is the planet radius divided by the Earth’s radius.

RV only < 4R⊕: Mi|Ri ∼ N(2.7R1.3

i

, 1.9) RV only < 8R⊕: Mi|Ri ∼ N(1.6R1.8

i

, 2.9) (Data: NASA Exoplanet Archive) Why is intrinsic scatter normally distributed? Why is intrinsic scatter a constant, not σM(Ri)? σ =

• σ2

M + β(Ri − 1) [WRF16]

Why use a power law?

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 10

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

A hierarchical Bayesian power-law model

HBM [Wolfgang, Rogers and Ford, 2016]: Mobs

i

∼ N(Mi, σobs

M,i ),

Robs

i

∼ N(Ri, σobs

R,i ),

Mi|Ri,C, γ, σM ∼ N(CRγ

i , σM)

Mi is the planet mass divided by the Earth’s mass, Ri is the planet radius divided by the Earth’s radius.

RV only < 4R⊕: Mi|Ri ∼ N(2.7R1.3

i

, 1.9) RV only < 8R⊕: Mi|Ri ∼ N(1.6R1.8

i

, 2.9) (Data: NASA Exoplanet Archive) Why is intrinsic scatter normally distributed? Why is intrinsic scatter a constant, not σM(Ri)? σ =

• σ2

M + β(Ri − 1) [WRF16]

Why use a power law?

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 10

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

A hierarchical Bayesian power-law model

HBM [Wolfgang, Rogers and Ford, 2016]: Mobs

i

∼ N(Mi, σobs

M,i ),

Robs

i

∼ N(Ri, σobs

R,i ),

Mi|Ri,C, γ, σM ∼ N(CRγ

i , σM)

Mi is the planet mass divided by the Earth’s mass, Ri is the planet radius divided by the Earth’s radius.

RV only < 4R⊕: Mi|Ri ∼ N(2.7R1.3

i

, 1.9) RV only < 8R⊕: Mi|Ri ∼ N(1.6R1.8

i

, 2.9) (Data: NASA Exoplanet Archive) Why is intrinsic scatter normally distributed? Why is intrinsic scatter a constant, not σM(Ri)? σ =

• σ2

M + β(Ri − 1) [WRF16]

Why use a power law?

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 10

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

A hierarchical Bayesian power-law model

HBM [Wolfgang, Rogers and Ford, 2016]: Mobs

i

∼ N(Mi, σobs

M,i ),

Robs

i

∼ N(Ri, σobs

R,i ),

Mi|Ri,C, γ, σM ∼ N(CRγ

i , σM)

Mi is the planet mass divided by the Earth’s mass, Ri is the planet radius divided by the Earth’s radius.

RV only < 4R⊕: Mi|Ri ∼ N(2.7R1.3

i

, 1.9) RV only < 8R⊕: Mi|Ri ∼ N(1.6R1.8

i

, 2.9) (Data: NASA Exoplanet Archive) Why is intrinsic scatter normally distributed? Why is intrinsic scatter a constant, not σM(Ri)? σ =

• σ2

M + β(Ri − 1) [WRF16]

Why use a power law?

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 10

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Moving to nonparametric approach

Bernstein polynomials Bernstein polynomials are a special case of B-spline function [Turnbull and Ghosh, 2014] For any function f with support on [0, 1], it can be approximated using Bernstein Polynomials: Bd(x, f) =

d

• k=1

f k − 1 d − 1

• xk−1(1 − x)d−k,

To estimate a density function, it can be expressed as fN(x|w) =

d

• k=1

wkβk(x; k, d − k + 1), with d

k=1 wk = 1 and wk ≥ 0.

[Ghoshal, 2001] If underlying distribution itself is a Bernstein density, the convergence rate is n−1/2(log n)1/2, if the true density is not a Bernstein density, the rate is n−1/3(log n)1/3

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 11

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Moving to nonparametric approach

Bernstein polynomials Bernstein polynomials are a special case of B-spline function [Turnbull and Ghosh, 2014] For any function f with support on [0, 1], it can be approximated using Bernstein Polynomials: Bd(x, f) =

d

• k=1

f k − 1 d − 1

• xk−1(1 − x)d−k,

To estimate a density function, it can be expressed as fN(x|w) =

d

• k=1

wkβk(x; k, d − k + 1), with d

k=1 wk = 1 and wk ≥ 0.

[Ghoshal, 2001] If underlying distribution itself is a Bernstein density, the convergence rate is n−1/2(log n)1/2, if the true density is not a Bernstein density, the rate is n−1/3(log n)1/3

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 11

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Moving to nonparametric approach

Bernstein polynomials Bernstein polynomials are a special case of B-spline function [Turnbull and Ghosh, 2014] For any function f with support on [0, 1], it can be approximated using Bernstein Polynomials: Bd(x, f) =

d

• k=1

f k − 1 d − 1

• xk−1(1 − x)d−k,

To estimate a density function, it can be expressed as fN(x|w) =

d

• k=1

wkβk(x; k, d − k + 1), with d

k=1 wk = 1 and wk ≥ 0.

[Ghoshal, 2001] If underlying distribution itself is a Bernstein density, the convergence rate is n−1/2(log n)1/2, if the true density is not a Bernstein density, the rate is n−1/3(log n)1/3

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 11

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Moving to nonparametric approach

Bernstein polynomials Bernstein polynomials are a special case of B-spline function [Turnbull and Ghosh, 2014] For any function f with support on [0, 1], it can be approximated using Bernstein Polynomials: Bd(x, f) =

d

• k=1

f k − 1 d − 1

• xk−1(1 − x)d−k,

To estimate a density function, it can be expressed as fN(x|w) =

d

• k=1

wkβk(x; k, d − k + 1), with d

k=1 wk = 1 and wk ≥ 0.

[Ghoshal, 2001] If underlying distribution itself is a Bernstein density, the convergence rate is n−1/2(log n)1/2, if the true density is not a Bernstein density, the rate is n−1/3(log n)1/3

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 11

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Moving to nonparametric approach

Bernstein polynomials Bernstein polynomials are a special case of B-spline function [Turnbull and Ghosh, 2014] For any function f with support on [0, 1], it can be approximated using Bernstein Polynomials: Bd(x, f) =

d

• k=1

f k − 1 d − 1

• xk−1(1 − x)d−k,

To estimate a density function, it can be expressed as fN(x|w) =

d

• k=1

wkβk(x; k, d − k + 1), with d

k=1 wk = 1 and wk ≥ 0.

[Ghoshal, 2001] If underlying distribution itself is a Bernstein density, the convergence rate is n−1/2(log n)1/2, if the true density is not a Bernstein density, the rate is n−1/3(log n)1/3

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 11

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Bernstein polynomial

fN(x|w) =

d

• k=1

wkβk(x; k, d − k + 1),

• k

wk = 1, wk ≥ 0

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 12

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Coarsened Bernstein polynomial

The Bernstein polynomials may need to use a lot of “beta distributions” (polynomials) to estimate the true function f The number of weights w becomes large Weights are sparse To reduce the number of parameters, one can block the polynomials and assign the same weight within each block To be more explicit, take d2 as the total degrees, we treat 1, . . . , d-th polynomials as block 1, the d + 1, . . . , 2d-th polynomials as block 2, and so on. Theorem has shown that the coarsened Bernstein polynomials has a similar convergence rate as the original one. [Kruijer and van der Vaart, 2008] (But what about in practice?)

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 13

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Coarsened Bernstein polynomial

The Bernstein polynomials may need to use a lot of “beta distributions” (polynomials) to estimate the true function f The number of weights w becomes large Weights are sparse To reduce the number of parameters, one can block the polynomials and assign the same weight within each block To be more explicit, take d2 as the total degrees, we treat 1, . . . , d-th polynomials as block 1, the d + 1, . . . , 2d-th polynomials as block 2, and so on. Theorem has shown that the coarsened Bernstein polynomials has a similar convergence rate as the original one. [Kruijer and van der Vaart, 2008] (But what about in practice?)

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 13

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Coarsened Bernstein polynomial

The Bernstein polynomials may need to use a lot of “beta distributions” (polynomials) to estimate the true function f The number of weights w becomes large Weights are sparse To reduce the number of parameters, one can block the polynomials and assign the same weight within each block To be more explicit, take d2 as the total degrees, we treat 1, . . . , d-th polynomials as block 1, the d + 1, . . . , 2d-th polynomials as block 2, and so on. Theorem has shown that the coarsened Bernstein polynomials has a similar convergence rate as the original one. [Kruijer and van der Vaart, 2008] (But what about in practice?)

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 13

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Coarsened Bernstein polynomial

The Bernstein polynomials may need to use a lot of “beta distributions” (polynomials) to estimate the true function f The number of weights w becomes large Weights are sparse To reduce the number of parameters, one can block the polynomials and assign the same weight within each block To be more explicit, take d2 as the total degrees, we treat 1, . . . , d-th polynomials as block 1, the d + 1, . . . , 2d-th polynomials as block 2, and so on. Theorem has shown that the coarsened Bernstein polynomials has a similar convergence rate as the original one. [Kruijer and van der Vaart, 2008] (But what about in practice?)

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 13

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Coarsened Bernstein polynomial

The Bernstein polynomials may need to use a lot of “beta distributions” (polynomials) to estimate the true function f The number of weights w becomes large Weights are sparse To reduce the number of parameters, one can block the polynomials and assign the same weight within each block To be more explicit, take d2 as the total degrees, we treat 1, . . . , d-th polynomials as block 1, the d + 1, . . . , 2d-th polynomials as block 2, and so on. Theorem has shown that the coarsened Bernstein polynomials has a similar convergence rate as the original one. [Kruijer and van der Vaart, 2008] (But what about in practice?)

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 13

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Coarsened Bernstein polynomial

The Bernstein polynomials may need to use a lot of “beta distributions” (polynomials) to estimate the true function f The number of weights w becomes large Weights are sparse To reduce the number of parameters, one can block the polynomials and assign the same weight within each block To be more explicit, take d2 as the total degrees, we treat 1, . . . , d-th polynomials as block 1, the d + 1, . . . , 2d-th polynomials as block 2, and so on. Theorem has shown that the coarsened Bernstein polynomials has a similar convergence rate as the original one. [Kruijer and van der Vaart, 2008] (But what about in practice?)

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 13

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

The model

The model: Mobs

i ind

∼ N(Mi, σobs

Mi ),

Robs

i ind

∼ N(Ri, σobs

Ri ),

(Mi, Ri) ∼ f(m, r|w), f(m, r|w) =

d

• k=1

d

• l=1

wkl 1 d2 ˜ βk( m−M

M−M )

M − M ˜ βl( r−R

R−R )

R − R where ˜ βj(a) = jd

j′=(j−1)d+1 βj′(a), w = (w11, . . . , wdd), d k=1

d

l=1 wkl = 1,

wkl > 0, Frequentist: maximizing the likelihood Bayesian: putting a prior on w, i.e. Dirichlet distribution Choice of values M, M, R, R matters.

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 14

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Section 3

1

Background

2

Model

3

Estimating Mass-Radius Relations

4

Future work

5

Concluding remarks

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 15

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

M-R relations: benchmarking to WRF16

< 8REarth Nonparam < 8REarth WRF16 < 4REarth WRF16

20 40 60 80 2 4 6 8

Radius (REarth) Mass (MEarth) Mass−Radius Relations

WRF16: RV only < 4R⊕: Mi|Ri ∼ N(2.7R1.3

i

, 1.92) RV only < 8R⊕: Mi|Ri ∼ N(1.6R1.8

i

, 2.92)

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 16

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Prediction intervals using coarsened Bernstein polynomial

16% and 84% prediction intervals of f(M|R) (25 degree for mass and radius) Upper bound: log(Mi) > log(MpureFe

i

) [Fortney et al., 2007]

20 40 60 80 2 4 6 8

Radius (REarth) Mass (MEarth) Mass−Radius prediction intervals Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 17

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Prediction intervals using Bernstein polynomial

Degree: 25 for Mass and 25 for Radius

20 40 60 80 2 4 6 8

Radius (REarth) Mass (MEarth) Mass−Radius prediction intervals

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 18

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Moving to using the full Kepler dataset

Plotting Kepler dataset in original scale (left) and log scale (right)

• ●
• ●
• ●
• ●
• ●
• ●●●● ●
• ● ●
• 2000

4000 6000 5 10 15 20

Radius (REarth) Mass (MEarth)

• ●
• ●
• −1

1 2 3 4 0.0 0.5 1.0

log(Radius (REarth)) log(Mass (MEarth))

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 19

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Modeling the Kepler data in log scale

Benefit of modeling with log scale for the Kepler dataset: To stabilize the variances making it not increase with mean Helps to show the power law more succinctly (as the relation would appear more linear) Log transformed variables are computationally stable We can always apply the Jacobian method to obtain the joint distribution in original scale The model becomes: Mobs

i ind

∼ N(Mi, σobs

Mi ), Robs i ind

∼ N(Ri, σobs

Ri ),

(log(Mi), log(Ri)) ∼ f(log(m), log(r)|w), f(log(m), log(r)|w) =

d

• k=1

d

• l=1

wkl 1 d2 ˜ βk( log(m)−M

M−M

) M − M ˜ βl( log(r)−R

R−R )

R − R

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 20

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

M-R relations in log scale

Plotting M-R relations: f(M|R) with its 16% and 84% quantiles Degree: 49 for Mass and 49 for Radius

10−1 1 10 102 103 104 0.5 1 2 3 4 5 10 20 30

Radius (REarth) Mass (MEarth) Kepler data: Mass−Radius Relations

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 21

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

M-R relations in log scale

Do uncorsened Bernstein polynomials produce conditions are less flat? (Curties and Ghosh, 2011)

10−1 1 10 102 103 104 0.5 1 2 3 4 5 10 20 30

Radius (REarth) Mass (MEarth) Kepler data: Mass−Radius Relations

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 22

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Conditional pdf for f(R|M) given different values of Mass

Radius = 1 Radius = 3 Radius = 5 Radius = 10 Radius = 15

0.0 0.5 1.0 1.5 10−1 1 10 102 103 104

Mass (MEarth) Conditional pdf for mass given radius

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 23

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Radius given Mass: the R-M relation (in log scale)

Conditional pdf of Radius given Mass: f(R|M) = f(M,R)

f(M)

Coarsened Bernstein polynomial model (left) and original Bernstein polynomial model (right)

0.5 1 2 3 4 5 10 20 30 10−1 1 10 102 103 104

Mass (MEarth) Radius (REarth) Kepler data: f(Radius|Mass)

0.5 1 2 3 4 5 10 20 30 10−1 1 10 102 103 104

Mass (MEarth) Radius (REarth) Kepler data: f(Radius|Mass)

The slopes of f(M|R) and f(R|M) are not inverses of each other: f(M|R) = f(MR)

f(R)

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 24

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Radius given Mass: the R-M relation (in log scale)

Conditional pdf of Radius given Mass: f(R|M) = f(M,R)

f(M)

Coarsened Bernstein polynomial model (left) and original Bernstein polynomial model (right)

0.5 1 2 3 4 5 10 20 30 10−1 1 10 102 103 104

Mass (MEarth) Radius (REarth) Kepler data: f(Radius|Mass)

0.5 1 2 3 4 5 10 20 30 10−1 1 10 102 103 104

Mass (MEarth) Radius (REarth) Kepler data: f(Radius|Mass)

The slopes of f(M|R) and f(R|M) are not inverses of each other: f(M|R) = f(MR)

f(R)

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 24

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Section 4

1

Background

2

Model

3

Estimating Mass-Radius Relations

4

Future work

5

Concluding remarks

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 25

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Future work

Simulation studies to compare Bernstein polynomials and the coarsened Bernstein polynomials Computationally, two R programming codes will be provided:

A pre-calculated M-R function: input radius and obtain predicted mass using the full Kepler dataset A programming which allows to input your own dataset and calculates the M-R relations

How many power-laws in this exoplanet population? Adding planet composition information?

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 26

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Future work

Simulation studies to compare Bernstein polynomials and the coarsened Bernstein polynomials Computationally, two R programming codes will be provided:

A pre-calculated M-R function: input radius and obtain predicted mass using the full Kepler dataset A programming which allows to input your own dataset and calculates the M-R relations

How many power-laws in this exoplanet population? Adding planet composition information?

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 26

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Future work

Simulation studies to compare Bernstein polynomials and the coarsened Bernstein polynomials Computationally, two R programming codes will be provided:

A pre-calculated M-R function: input radius and obtain predicted mass using the full Kepler dataset A programming which allows to input your own dataset and calculates the M-R relations

How many power-laws in this exoplanet population? Adding planet composition information?

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 26

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Future work

Simulation studies to compare Bernstein polynomials and the coarsened Bernstein polynomials Computationally, two R programming codes will be provided:

A pre-calculated M-R function: input radius and obtain predicted mass using the full Kepler dataset A programming which allows to input your own dataset and calculates the M-R relations

How many power-laws in this exoplanet population? Adding planet composition information?

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 26

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Section 5

1

Background

2

Model

3

Estimating Mass-Radius Relations

4

Future work

5

Concluding remarks

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 27

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Conclusion

We proposed a nonparametric method to estimate M-R relations for Kepler datasets The method is used to predict Mass given Radius or Radius given Mass Compared with WRF16, the power-law assumption is valid However, there are at least two power-law relations: planets radii < 4R⊕, and 4R⊕ < R < 8R⊕ We modeled the Kepler data in log scale, the dataset has planets radius from 0R⊕ to 22R⊕ We plotted the conditional densities f(M|R) and f(R|M), their slopes are not inverse of each other

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 28

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Thank you! and Questions?

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 29

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Standard deviation

Plotting the s.d.(M|R) for the power-law and the nonparametric model (Right)

< 8REarth Nonparam < 8REarth WRF16 < 4REarth WRF16

20 40 60 80 2 4 6 8

Radius (REarth) Mass (MEarth) Mass−Radius Relations

0.0 2.5 5.0 7.5 2 4 6 8

Radius (REarth) sd of f(Mass| Radius) Standard deviation of f(M|R) Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 30

Background Model Estimating Mass-Radius Relations Future work Concluding remarks

Conditional density for f(R|M)

Mass = 1 Mass = 10 Mass = 50 Mass = 100 Mass = 500 1 2 3 4 0.5 1 2 3 4 5 10 20

Radius (REarth) Conditional pdf for radius given mass

Bo Ning — Nonparametric Prediction and the Exoplanet Mass-Radius Relationship 31