Using Skew Normal Distributions for Asymmetric Uncertainties¶
Many exoplanet parameters often have asymmetric errorbars, expressed as something like $$\mu^{+\sigma_{upper}}_{-\sigma_{lower}}$$ where $\mu$ is some estimate of central tendency, and the $\sigma$s are the uncertainties. In exoatlas, we follow Pineda et al. (2021) to use skew normal distributions to approximate the probability distributions of table quantities with unequal upper and lower uncertainties.
Creating a Samples with a Skew Normal¶
For some quantity $\mu^{+\sigma_{upper}}_{-\sigma_{lower}}$, we can make samples from a skew normal that treats $\mu$ as the mode of the distribution and $\mu - \sigma_{lower}$ to $\mu + \sigma_{upper}$ as the central $68\%$ confidence interval. Behind the scenes, this interpolates from a table of skew normal coefficients depending on the ratio $\sigma_{upper}/\sigma_{lower}$, generates some samples, and renormalizes them to the requested $\mu$, $\sigma_{upper}$, and $\sigma_{lower}$.
from exoatlas.populations.pineda_skew import *
To see how this looks for different levels of asymmetry, you can run this test function, which shows the generated samples on a violin plot.
from exoatlas.tests import test_skew
test_skew()
Understanding the Skew Normal¶
The skew normal depends on a location $\mu$, a scale $\sigma$, and an asymmetry parameter $\alpha$, which can be positive or negative.
plot_skewnormal(mu=0, sigma=1, alpha=5)
Setting Up Interpolation Tables¶
Pineda et al. provide code to numerically solve for the parameters of a skew normal distribution, given a set of asymmetric error bars. Since this takes a little time and can occasionally be a little unstable, for exoatlas we derive a table of coefficients and interpolate from it based on how asymmetric the errors are using $\sigma_{upper}/\sigma_{lower}$.
Here we derive those tables, assuming iterating a few times, using a smoothed version of the previous results as guesses to the subsequent iterations. We derive tables assuming $\mu$ represents either the mode or the median of the distribution, and that the range of $\mu - \sigma_{lower}$ to $\mu + \sigma_{upper}$ represents the central $68\%$ confidence interval. We save the resulting table into the code, so most folks will never actually need to run this code directly.
tables = make_skewnormal_parameters_to_interpolate()
saved parameter interpolating table to skewnormal_parameters_for_interpolating_mode.ecsv
saved parameter interpolating table to skewnormal_parameters_for_interpolating_median.ecsv
Practically, we use the mode (= "Peak") because it's impossible to achieve errorbars more than about 50% asymmetric if we treat $\mu$ as representing the median.