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(
# MJD mas mas mas mas
(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),
instrument_name = "GPI" # optional -- name for this group of data
)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 0You 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.
instrument_name = "GPI" # optional -- name for this group of data
)Another way we could specify the data is by column:
astrom_like = PlanetRelAstromLikelihood(
Table(
epoch= [50000, 50120, 50240, 50360,50480, 50600, 50720, 50840,], # MJD
ra = [-505.764, -502.57, -498.209, -492.678,-485.977, -478.11, -469.08, -458.896,], # mas
dec = [-66.9298, -37.4722, -7.92755, 21.6356, 51.1472, 80.5359, 109.729, 138.651, ], # mas
σ_ra = fill(10.0, 8),
σ_dec = fill(10.0, 8),
cor = fill(0.0, 8),
),
instrument_name = "GPI" # optional -- name for this group of data
)Finally 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_data = CSV.read("mydata.csv", Table)
astrom_like = PlanetRelAstromLikelihood(
astrom_data,
instrument_name="GPI"
)You can also pass a DataFrame or any other table format.
In 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.
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.
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 for the total mass of the system, expressed in units of Solar mass.plx: Distance to the system expressed in milliarcseconds of parallax.
Make sure to truncate the priors to prevent unphysical negative masses or parallaxes.
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.
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.
Initialize starting points for chains
Run the initialize! function to find good starting points for the chain. You can provide guesses for parameters if you want to.
init_chain = initialize!(model) # No guesses provided, slower global optimization will be usedinit_chain = initialize!(model, (;
plx = 50,
M = 1.21,
planets = (;
b=(;
a = 10.0,
e = 0.01,
# note! Never initialize a value on the bound, exactly 0 eccentricity is disallowed by the `Uniform(0,1)` prior
)
)
))Chains MCMC chain (1000×16×1 Array{Float64, 3}):
Iterations = 1:1:1000
Number of chains = 1
Samples per chain = 1000
Wall duration = 1.18 seconds
Compute duration = 1.18 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 = logpost
Summary Statistics
parameters mean std mcse ess_bulk ess_tail r ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Floa ⋯
M 1.2021 0.0771 0.0025 945.7619 904.5700 0.9 ⋯
plx 50.0001 0.0227 0.0007 947.5717 992.3671 0.9 ⋯
b_a 11.0798 1.0734 0.0308 1193.7284 998.9087 0.9 ⋯
b_e 0.1938 0.0805 0.0026 939.5029 876.3151 0.9 ⋯
b_i 0.6335 0.1209 0.0037 1043.4855 1004.6833 0.9 ⋯
b_ωy 0.5533 0.6286 0.0191 1112.6303 868.6575 0.9 ⋯
b_ωx 0.3457 0.4084 0.0124 1060.9354 927.7721 0.9 ⋯
b_Ωy 0.7399 0.3578 0.0114 967.9296 893.1140 0.9 ⋯
b_Ωx 0.3874 0.4217 0.0140 914.2489 985.6804 0.9 ⋯
b_θy 0.0741 0.0089 0.0003 922.0427 835.5932 1.0 ⋯
b_θx -0.9773 0.0729 0.0024 893.1805 706.4381 0.9 ⋯
b_ω 0.6543 1.0287 0.0335 1107.7191 957.7191 1.0 ⋯
b_Ω 0.5075 0.5860 0.0191 927.0425 988.0099 1.0 ⋯
b_θ -1.4951 0.0064 0.0002 1042.6948 867.5731 0.9 ⋯
b_tp 43102.2933 2620.5422 85.8354 894.2458 895.0928 0.9 ⋯
2 columns omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
M 1.0435 1.1688 1.2125 1.2395 1.3380
plx 49.9593 49.9878 49.9968 50.0166 50.0400
b_a 8.6875 10.9533 11.3261 11.7166 12.6266
b_e 0.0184 0.1476 0.1894 0.2333 0.3255
b_i 0.4170 0.5101 0.6658 0.7285 0.8288
b_ωy -1.0241 0.6685 0.7894 0.9176 1.0410
b_ωx -0.6057 0.2492 0.4384 0.6374 0.8444
b_Ωy -0.0144 0.5071 0.9332 0.9785 1.1420
b_Ωx -0.1255 0.0529 0.3258 0.8132 1.0610
b_θy 0.0596 0.0689 0.0742 0.0819 0.0883
b_θx -1.1091 -1.0255 -0.9773 -0.9479 -0.8006
b_ω -0.8341 0.2851 0.6069 0.7948 2.8935
b_Ω -0.1247 0.0576 0.3249 1.0220 1.5855
b_θ -1.5087 -1.4994 -1.4965 -1.4900 -1.4844
b_tp 40667.6779 41054.1065 42090.6675 44753.6968 49035.4725
Visualize the starting points
Plot the inital values to make sure that they are reasonable, and match your data. This is a great time to confirm that your data were entered in correctly.
using CairoMakie
octoplot(model, init_chain)
The starting points for sampling look reasonable!
The return value from initialize! is a "variational approximation". You can pass that chain to any function expecting a chain argument, like Octofitter.savechain or octocorner. It gives a rough approximation of the posterior we expect. The central values are probably close, but the uncertainties are unreliable.
Sampling
Now we are ready to draw samples from the posterior:
chain = octofit(model)Chains MCMC chain (1000×28×1 Array{Float64, 3}):
Iterations = 1:1:1000
Number of chains = 1
Samples per chain = 1000
Wall duration = 3.08 seconds
Compute duration = 3.08 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 rh ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float ⋯
M 1.1811 0.0959 0.0036 704.2649 648.3496 1.00 ⋯
plx 49.9993 0.0187 0.0006 907.4223 593.8112 1.00 ⋯
b_a 11.7780 2.2752 0.1402 245.7569 313.0496 1.00 ⋯
b_e 0.1576 0.1138 0.0065 307.8376 419.0989 1.00 ⋯
b_i 0.6327 0.1540 0.0095 283.6325 261.1875 1.00 ⋯
b_ωy -0.0431 0.7495 0.0579 200.0304 781.2544 1.00 ⋯
b_ωx -0.0540 0.6751 0.0551 166.6325 757.9030 1.00 ⋯
b_Ωy 0.0608 0.7448 0.1102 55.1621 385.3888 1.00 ⋯
b_Ωx 0.0833 0.6773 0.0984 57.5994 241.0397 1.00 ⋯
b_θy 0.0749 0.0109 0.0004 759.6095 306.3232 0.99 ⋯
b_θx -1.0052 0.0987 0.0035 836.8297 560.7032 0.99 ⋯
b_ω -0.2901 1.8579 0.1386 231.8360 650.7426 1.00 ⋯
b_Ω -0.2650 1.7502 0.2173 96.8815 317.6240 1.00 ⋯
b_θ -1.4965 0.0074 0.0003 813.6123 641.3347 1.00 ⋯
b_tp 42629.5788 5214.0891 329.0673 246.9617 642.8424 1.00 ⋯
2 columns omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
M 1.0131 1.1128 1.1804 1.2451 1.3823
plx 49.9606 49.9869 49.9996 50.0108 50.0356
b_a 8.1027 10.0905 11.3544 13.2475 16.8267
b_e 0.0067 0.0627 0.1326 0.2429 0.4096
b_i 0.2797 0.5401 0.6487 0.7436 0.9009
b_ωy -1.0628 -0.7656 -0.1987 0.7528 1.0796
b_ωx -1.0546 -0.6855 -0.1096 0.5695 1.0296
b_Ωy -1.0772 -0.7069 0.1283 0.8148 1.0964
b_Ωx -1.0340 -0.5315 0.2036 0.6881 1.0733
b_θy 0.0560 0.0671 0.0744 0.0819 0.0980
b_θx -1.2069 -1.0713 -0.9999 -0.9338 -0.8258
b_ω -2.9793 -2.1048 -0.1922 1.0944 3.0030
b_Ω -2.9719 -2.0871 0.2641 1.0562 2.9208
b_θ -1.5111 -1.5015 -1.4963 -1.4914 -1.4825
b_tp 30235.5670 39389.3753 43629.4224 46367.4052 49877.5208
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.
Sampling can take much longer when you have measurements with very small uncertainties.
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 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. Then you can combine 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 1.0005 1.0028
plx 1.0001 1.0005
b_a 1.0015 1.0061
b_e 1.0011 1.0046
b_i 1.0085 1.0137
b_ωy 1.0007 1.0032
b_ωx 1.0009 1.0031
b_Ωy 1.0002 1.0020
b_Ωx 1.0019 1.0064
b_θy 0.9998 1.0003
b_θx 1.0010 1.0042
b_ω 1.0016 1.0064
b_Ω 1.0010 1.0050
b_θ 1.0002 1.0018
b_tp 1.0007 1.0038
This will check that the means and variances are similar between chains that were initialized at different starting points.
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.
You can control what panels are displayed, the time range, colourscheme, etc. See the documentation on octoplot for more details.
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")Saving your model
You may choose to save your model so that you can reload it later to make plots, etc:
using Serialization
serialize("model1.jls", model)Which can then be loaded at a later time using:
using Serialization
using Octofitter # must include all the same imports as your original script
model = deserialize("model1.jls")Serialized models are only loadable/restorable on the same computer, version of Octofitter, and version of Julia. They are not intended for long-term archiving. For reproducibility, make sure to keep your original model definition script.
Comparing chains
We can compare two different chains by passing them both to octocorner. Let's compare the init_chain with the full results from octofit:
octocorner(model, chain, init_chain, small=true)