Two Dimensional Gaussian Contour Levels

Calculates the contours of a two dimensional normal distribution of a desired amount of probability. The idea is to transform the ellipse to a circle and calculate the integral of the PDF, set it equal to a quantile and tranform it back. The main steps are as follows:

$$(\vec{x}-\vec{\mu})^\intercal \Sigma^{-1} (\vec{x}-\vec{\mu}) = d^2 \\$$ $$\Sigma^{-1} = U \Lambda U^{\intercal}$$ $$(\vec{x}-\vec{\mu})^\intercal U \Lambda U^{\intercal} (\vec{x}-\vec{\mu}) = d^2$$ $$a = d \sqrt{\lambda_1} \text{ and } b = d \sqrt{\lambda_2}$$ $$q = \int_A P(x,y) d\vec{x}$$ $$q = \int_0^{R} r\,e^{-r^2/2} dr = 1-e^{-R^2/2}$$ $$R = \sqrt{-2\ln{(1-q)}}$$ $$a = R \sqrt{\lambda_1} = \sqrt{-2 \lambda_1\ln{(1-q)}} \text{ and } b = R \sqrt{\lambda_2} = \sqrt{-2 \lambda_2\ln{(1-q)}}$$

Get Started

Get a dictionary containing semi-major axis, semi-minor axis and the angle of the ellipse w.r.t to the major axis from example_ellipse_prop.py by the arguments of confidence as list and covariance matrix of your probability distribution function.

from numpy.random import randint
from numpy import dot, array

#  A random covariance matrix
off_diag = randint(10)
a, b = randint(1, 10), randint(1, 10)
A = array([
    [a, off_diag],
    [off_diag, b]
])
cov = dot(A, A.T)

cov can now be used in make_ellipse_parameter_dict:

from contourlevels import make_ellipse_parameter_dict

confidence_regions = [.5,.9,.99]  # Your desired confidence regions
ell_props = make_ellipse_parameter_dict(cov, confidence_regions)

The variable cov can be manipulated to any desired covariance matrix as long the mathmatical necessities are full filled.

For a single ellipse use get_ellipse_props(cov, confidence).

cov = #  Your covariance matrix
confidence = 0.5

major_axis, minor_axis, angle = get_ellipse_props(cov, confidence)

Visualization

For visualization use example_ellipse_plot.