Basic Astrometry Fit
Here is a worked example of a one-planet model fit to relative astrometry (positions measured between the planet and the host star).
Start by loading the Octofitter and Distributions packages:
using Octofitter, DistributionsSpecifying the data
We will create a likelihood object to contain our relative astrometry data. We can specify this data in several formats. It can be listed in the code or loaded from a file (eg. a CSV file, FITS table, or SQL database).
astrom_like = PlanetRelAstromLikelihood(
(epoch = 50000, ra = -505.7637580573554, dec = -66.92982418533026, σ_ra = 10, σ_dec = 10, cor=0),
(epoch = 50120, ra = -502.570356287689, dec = -37.47217527025044, σ_ra = 10, σ_dec = 10, cor=0),
(epoch = 50240, ra = -498.2089148883798, dec = -7.927548139010479, σ_ra = 10, σ_dec = 10, cor=0),
(epoch = 50360, ra = -492.67768482682357, dec = 21.63557115669823, σ_ra = 10, σ_dec = 10, cor=0),
(epoch = 50480, ra = -485.9770335870402, dec = 51.147204404903704, σ_ra = 10, σ_dec = 10, cor=0),
(epoch = 50600, ra = -478.1095526888573, dec = 80.53589069730698, σ_ra = 10, σ_dec = 10, cor=0),
(epoch = 50720, ra = -469.0801731788123, dec = 109.72870493064629, σ_ra = 10, σ_dec = 10, cor=0),
(epoch = 50840, ra = -458.89628893460525, dec = 138.65128697876773, σ_ra = 10, σ_dec = 10, cor=0),
)PlanetRelAstromLikelihood Table with 6 columns and 8 rows:
epoch ra dec σ_ra σ_dec cor
┌────────────────────────────────────────────
1 │ 50000 -505.764 -66.9298 10 10 0
2 │ 50120 -502.57 -37.4722 10 10 0
3 │ 50240 -498.209 -7.92755 10 10 0
4 │ 50360 -492.678 21.6356 10 10 0
5 │ 50480 -485.977 51.1472 10 10 0
6 │ 50600 -478.11 80.5359 10 10 0
7 │ 50720 -469.08 109.729 10 10 0
8 │ 50840 -458.896 138.651 10 10 0In Octofitter, epoch is always the modified Julian date (measured in days). If you're not sure what this is, you can get started by just putting in arbitrary time offsets measured in days.
In this case, we specified ra and dec offsets in milliarcseconds. We could instead specify sep (projected separation) in milliarcseconds and pa in radians. You cannot mix the two formats in a single PlanetRelAstromLikelihood but you can create two different likelihood objects, one for each format.
You can also specify it in separation (mas) and positon angle (rad):
astrom_like_2 = PlanetRelAstromLikelihood(
(epoch = 50000, sep = 505.7637580573554, pa = deg2rad(24.1), σ_sep = 10, σ_pa =deg2rad(1.2), cor=0),
# ...etc.
)Another way we could specify the data is by column:
astrom_like = PlanetRelAstromLikelihood(Table(;
epoch= [50000, 50120, 50240, 50360,50480, 50600, 50720, 50840,],
ra = [-505.764, -502.57, -498.209, -492.678,-485.977, -478.11, -469.08, -458.896,],
dec = [-66.9298, -37.4722, -7.92755, 21.6356, 51.1472, 80.5359, 109.729, 138.651, ],
σ_ra = fill(10.0, 8),
σ_dec = fill(10.0, 8),
cor = fill(0.0, 8)
))PlanetRelAstromLikelihood Table with 6 columns and 8 rows:
epoch ra dec σ_ra σ_dec cor
┌────────────────────────────────────────────
1 │ 50000 -505.764 -66.9298 10.0 10.0 0.0
2 │ 50120 -502.57 -37.4722 10.0 10.0 0.0
3 │ 50240 -498.209 -7.92755 10.0 10.0 0.0
4 │ 50360 -492.678 21.6356 10.0 10.0 0.0
5 │ 50480 -485.977 51.1472 10.0 10.0 0.0
6 │ 50600 -478.11 80.5359 10.0 10.0 0.0
7 │ 50720 -469.08 109.729 10.0 10.0 0.0
8 │ 50840 -458.896 138.651 10.0 10.0 0.0Finally we could also load the data from a file somewhere. Here is an example of loading a CSV:
using CSV # must install CSV.jl first
astrom_like = CSV.read("mydata.csv", PlanetRelAstromLikelihood)You can also pass a DataFrame or any other table format.
Creating a planet
We now create our first planet model. Let's name it planet b. The name of the planet will be used in the output results.
In Octofitter, we specify planet and system models using a "probabilistic programming language". Quantities with a ~ are random variables. The distributions on the right hand sides are priors. You must specify a proper prior for any quantity which is allowed to vary.
We now create our planet b model using the @planet macro.
@planet b Visual{KepOrbit} begin
a ~ truncated(Normal(10, 4), lower=0.1, upper=100)
e ~ Uniform(0.0, 0.5)
i ~ Sine()
ω ~ UniformCircular()
Ω ~ UniformCircular()
θ ~ UniformCircular()
tp = θ_at_epoch_to_tperi(system,b,50420)
end astrom_likeIn the model definition, b is the name of the planet (it can be anything), Visual{KepOrbit} is the type of orbit parameterization (see the PlanetOrbits.jl documentation page).
After the begin comes our variable specification. Quantities with a ~ are random variables aka. our priors. You must specify a proper prior for any quantity which is allowed to vary. "Uninformative" priors like 1/x must be given bounds, and can be specified with LogUniform(lower, upper).
Make sure that variables like mass and eccentricity can't be negative. You can pass a distribution to truncated to prevent this, e.g. M ~ truncated(Normal(1, 0.1),lower=0).
Priors can be any univariate distribution from the Distributions.jl package.
For a KepOrbit you must specify the following parameters:
a: Semi-major axis, astronomical units (AU)i: Inclination, radianse: Eccentricity in the range [0, 1)ω: Argument of periastron, radiusΩ: Longitude of the ascending node, radians.tp: Epoch of periastron passage
Many different distributions are supported as priors, including Uniform, LogNormal, LogUniform, Sine, and Beta. See the section on Priors for more information. The parameters can be specified in any order.
You can also hardcode a particular value for any parameter if you don't want it to vary. Simply replace eg. e ~ Uniform(0, 0.999) with e = 0.1. This = syntax works for arbitrary mathematical expressions and even functions. We use it here to reparameterize tp.
tp is a date which sets the location of the planet around its orbit. It repeats every orbital period and the orbital period depends on the scale of the orbit. This makes it quite hard to sample from. We therefore reparameterize using θ parameter, representing the position angle of the planet at a given reference epoch. This parameterization speeds up sampling quite a bit!
After the variables block are zero or more Likelihood objects. These are observations specific to a given planet that you would like to include in the model. If you would like to sample from the priors only, don't pass in any observations.
For this example, we specify PlanetRelAstromLikelihood block. This is where you can list the position of a planet relative to the star at different epochs.
When you have created your planet, you should see the following output. If you don't, you can run display(b) to force the text to be output:
Planet b
Derived:
ω, Ω, θ, tp,
Priors:
a Truncated(Distributions.Normal{Float64}(μ=10.0, σ=4.0); lower=0.1, upper=100.0)
e Distributions.Uniform{Float64}(a=0.0, b=0.5)
i Sine()
ωy Distributions.Normal{Float64}(μ=0.0, σ=1.0)
ωx Distributions.Normal{Float64}(μ=0.0, σ=1.0)
Ωy Distributions.Normal{Float64}(μ=0.0, σ=1.0)
Ωx Distributions.Normal{Float64}(μ=0.0, σ=1.0)
θy Distributions.Normal{Float64}(μ=0.0, σ=1.0)
θx Distributions.Normal{Float64}(μ=0.0, σ=1.0)
Octofitter.UnitLengthPrior{:ωy, :ωx}: √(ωy^2+ωx^2) ~ LogNormal(log(1), 0.02)
Octofitter.UnitLengthPrior{:Ωy, :Ωx}: √(Ωy^2+Ωx^2) ~ LogNormal(log(1), 0.02)
Octofitter.UnitLengthPrior{:θy, :θx}: √(θy^2+θx^2) ~ LogNormal(log(1), 0.02)
PlanetRelAstromLikelihood Table with 6 columns and 8 rows:
epoch ra dec σ_ra σ_dec cor
┌────────────────────────────────────────────
1 │ 50000 -505.764 -66.9298 10.0 10.0 0.0
2 │ 50120 -502.57 -37.4722 10.0 10.0 0.0
3 │ 50240 -498.209 -7.92755 10.0 10.0 0.0
4 │ 50360 -492.678 21.6356 10.0 10.0 0.0
5 │ 50480 -485.977 51.1472 10.0 10.0 0.0
6 │ 50600 -478.11 80.5359 10.0 10.0 0.0
7 │ 50720 -469.08 109.729 10.0 10.0 0.0
8 │ 50840 -458.896 138.651 10.0 10.0 0.0
Creating a system
A system represents a host star with one or more planets. Properties of the whole system are specified here, like parallax distance and mass of the star. This is also where you will supply data like images, astrometric acceleration, or stellar radial velocity since they don't belong to any planet in particular.
@system Tutoria begin
M ~ truncated(Normal(1.2, 0.1), lower=0.1)
plx ~ truncated(Normal(50.0, 0.02), lower=0.1)
end bTutoria is the name we have given to the system. It could be eg PDS70, or anything that will help you keep track of the results.
The variables block works just like it does for planets. Here, the two parameters you must provide are:
M: Gravitational parameter of the central body, expressed in units of Solar mass.plx: Distance to the system expressed in milliarcseconds of parallax.
M is always required for all choices of parameterizations supported by PlanetOrbits.jl. plx is needed due to our choice to use Visual{...} orbit and relative astrometry. The prior on plx can be looked up from GAIA for many targets by using the function gaia_plx:
plx ~ gaia_plx(;gaia_id=12345678910)After that, just list any planets that you want orbiting the star. Here, we pass planet b.
This is also where we could pass likelihood objects for system-wide data like stellar radial velocity.
You can display your system object by running display(Tutoria) (or whatever you chose to name your system).
Prepare model
We now convert our declarative model into efficient, compiled code:
model = Octofitter.LogDensityModel(Tutoria)LogDensityModel for System Tutoria of dimension 11 and 11 epochs with fields .ℓπcallback and .∇ℓπcallback
This type implements the julia LogDensityProblems.jl interface and can be passed to a wide variety of samplers.
Sampling
Now we are ready to draw samples from the posterior:
# Provide a seeded random number generator for reproducibility of this example.
# This is not necessary in general: you may simply omit the RNG parameter if you prefer.
using Random
rng = Random.Xoshiro(1234)
chain = octofit(rng, model)Chains MCMC chain (1000×28×1 Array{Float64, 3}):
Iterations = 1:1:1000
Number of chains = 1
Samples per chain = 1000
Wall duration = 3.24 seconds
Compute duration = 3.24 seconds
parameters = M, plx, b_a, b_e, b_i, b_ωy, b_ωx, b_Ωy, b_Ωx, b_θy, b_θx, b_ω, b_Ω, b_θ, b_tp
internals = n_steps, is_accept, acceptance_rate, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size, is_adapt, loglike, logpost, tree_depth, numerical_error
Summary Statistics
parameters mean std mcse ess_bulk ess_tail r ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Floa ⋯
M 1.1865 0.1002 0.0040 625.4657 541.8007 1.0 ⋯
plx 49.9997 0.0205 0.0006 1016.0046 625.3696 0.9 ⋯
b_a 11.7251 2.2734 0.1658 176.8311 291.6126 1.0 ⋯
b_e 0.1475 0.1067 0.0059 259.7011 148.5169 1.0 ⋯
b_i 0.6356 0.1523 0.0099 237.2781 432.7115 1.0 ⋯
b_ωy -0.0024 0.7180 0.0582 178.0170 557.4011 1.0 ⋯
b_ωx 0.1062 0.6982 0.0681 154.7313 566.9249 1.0 ⋯
b_Ωy 0.0717 0.7679 0.1557 30.4876 305.2511 1.0 ⋯
b_Ωx 0.0657 0.6492 0.0870 64.2592 217.3366 1.0 ⋯
b_θy 0.0743 0.0102 0.0003 995.9775 730.2656 1.0 ⋯
b_θx -0.9980 0.0966 0.0032 904.5252 674.5299 1.0 ⋯
b_ω -0.0191 1.8258 0.1308 297.4007 385.9136 1.0 ⋯
b_Ω -0.3881 1.7332 0.2166 87.1673 263.8360 1.0 ⋯
b_θ -1.4965 0.0073 0.0002 1033.5310 639.8899 1.0 ⋯
b_tp 42881.3135 4939.7223 335.9965 227.3572 458.9549 1.0 ⋯
2 columns omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
M 0.9907 1.1194 1.1873 1.2535 1.3793
plx 49.9609 49.9855 50.0000 50.0137 50.0396
b_a 8.3615 10.1076 11.2992 12.9234 17.2657
b_e 0.0054 0.0593 0.1254 0.2189 0.3772
b_i 0.3039 0.5390 0.6484 0.7441 0.9054
b_ωy -1.0523 -0.7306 -0.0025 0.7166 1.0227
b_ωx -1.0371 -0.5412 0.2036 0.7706 1.0688
b_Ωy -1.0775 -0.7510 0.1885 0.8405 1.0949
b_Ωx -1.0102 -0.4822 0.1132 0.6516 1.0481
b_θy 0.0553 0.0672 0.0739 0.0812 0.0951
b_θx -1.2023 -1.0630 -0.9929 -0.9287 -0.8276
b_ω -3.0152 -1.8424 0.3700 1.3193 2.9863
b_Ω -3.0111 -2.2226 0.1652 0.8861 2.8397
b_θ -1.5123 -1.5011 -1.4963 -1.4917 -1.4828
b_tp 30241.7350 40193.0650 43761.9364 46333.1373 49927.6700
You will get an output that looks something like this with a progress bar that updates every second or so. You can reduce or completely silence the output by reducing the verbosity value down to 0.
Once complete, the chain object will hold the sampler results. Displaying it prints out a summary table like the one shown above. For a basic model like this, sampling should take less than a minute on a typical laptop.
Diagnostics
The first thing you should do with your results is check a few diagnostics to make sure the sampler converged as intended.
A few things to watch out for: check that you aren't getting many (any, really) numerical errors (ratio_divergent_transitions). This likely indicates a problem with your model: either invalid values of one or more parameters are encountered (e.g. the prior on semi-major axis includes negative values) or that there is a region of very high curvature that is failing to sample properly. This latter issue can lead to a bias in your results.
One common mistake is to use a distribution like Normal(10,3) for semi-major axis. This left tail of this distribution includes negative values, and our orbit model is not defined for negative semi-major axes. A better choice is a truncated(Normal(10,3), lower=0.1) distribution (not including zero, since a=0 is not defined).
You may see some warnings during initial step-size adaptation. These are probably nothing to worry about if sampling proceeds normally afterwards.
You should also check the acceptance rate (mean_accept) is reasonably high and the mean tree depth (mean_tree_depth) is reasonable (~4-8). Lower than this and the sampler is taking steps that are too large and encountering a U-turn very quicky. Much larger than this and it might be being too conservative.
Next, you can make a trace plot of different variabes to visually inspect the chain:
using CairoMakie
lines(
chain["b_a"][:],
axis=(;
xlabel="iteration",
ylabel="semi-major axis (AU)"
)
)
And an auto-correlation plot:
using StatsBase
using CairoMakie
lines(
autocor(chain["b_e"][:], 1:500),
axis=(;
xlabel="lag",
ylabel="autocorrelation",
)
)
This plot shows that these samples are not correlated after only about 5 iterations. No thinning is necessary.
To confirm convergence, you may also examine the rhat column from chains. This diagnostic approaches 1 as the chains converge and should at the very least equal 1.0 to one significant digit (3 recommended).
Finaly, you might consider running multiple chains. Simply run octofit multiple times, and store the result in different variables. Ten you can combined the chains using chainscat and run additional inter-chain convergence diagnostics:
using MCMCChains
chain1 = octofit(model)
chain2 = octofit(model)
chain3 = octofit(model)
merged_chains = chainscat(chain1, chain2, chain3)
gelmandiag(merged_chains)Gelman, Rubin, and Brooks diagnostic
parameters psrf psrfci
Symbol Float64 Float64
M 1.0009 1.0018
plx 1.0017 1.0061
b_a 1.0036 1.0132
b_e 1.0007 1.0026
b_i 1.0008 1.0036
b_ωy 1.0001 1.0015
b_ωx 1.0011 1.0051
b_Ωy 1.0277 1.0923
b_Ωx 1.0233 1.0816
b_θy 1.0045 1.0147
b_θx 1.0021 1.0038
b_ω 1.0010 1.0049
b_Ω 1.0272 1.0863
b_θ 1.0033 1.0133
b_tp 1.0014 1.0049
As an additional convergence test.
Analysis
As a first pass, let's plot a sample of orbits drawn from the posterior. The function octoplot is a conveninient way to generate a 9-panel plot of velocities and position:
using CairoMakie
octoplot(model,chain)
This function draws orbits from the posterior and displays them in a plot. Any astrometry points are overplotted.
Pair Plot
A very useful visualization of our results is a pair-plot, or corner plot. We can use the octocorner function and our PairPlots.jl package for this purpose:
using CairoMakie
using PairPlots
octocorner(model, chain, small=true)
Remove small=true to display all variables.
In this case, the sampler was able to resolve the complicated degeneracies between eccentricity, the longitude of the ascending node, and argument of periapsis.
Saving your chain
Variables can be retrieved from the chains using the following sytnax: sma_planet_b = chain["b_a",:,:]. The first index is a string or symbol giving the name of the variable in the model. Planet variables are prepended by the name of the planet and an underscore.
You can save your chain in FITS table format by running:
Octofitter.savechain("mychain.fits", chain)You can load it back via:
chain = Octofitter.loadchain("mychain.fits")