Raphael Roth - 2 months ago 20
Python Question

# multidimensional confidence intervals

I have numerous tuples (par1,par2), i.e. points in a 2 dimensional parameter space obtained from repeating an experiment multiple times.

I'm looking for a possibility to calculate and visualize confidence ellipses (not sure if thats the correct term for this). Here an example plot that I found in the web to show what I mean:

source: blogspot.ch/2011/07/classification-and-discrimination-with.html

So in principle one has to fit a multivariate normal distribution to a 2D histogram of data points I guess. Can somebody help me with this?

It sounds like you just want the 2-sigma ellipse of the scatter of points?

If so, consider something like this (From some code for a paper here: https://github.com/joferkington/oost_paper_code/blob/master/error_ellipse.py):

``````import numpy as np

import matplotlib.pyplot as plt
from matplotlib.patches import Ellipse

def plot_point_cov(points, nstd=2, ax=None, **kwargs):
"""
Plots an `nstd` sigma ellipse based on the mean and covariance of a point
"cloud" (points, an Nx2 array).

Parameters
----------
points : An Nx2 array of the data points.
nstd : The radius of the ellipse in numbers of standard deviations.
Defaults to 2 standard deviations.
ax : The axis that the ellipse will be plotted on. Defaults to the
current axis.
Additional keyword arguments are pass on to the ellipse patch.

Returns
-------
A matplotlib ellipse artist
"""
pos = points.mean(axis=0)
cov = np.cov(points, rowvar=False)
return plot_cov_ellipse(cov, pos, nstd, ax, **kwargs)

def plot_cov_ellipse(cov, pos, nstd=2, ax=None, **kwargs):
"""
Plots an `nstd` sigma error ellipse based on the specified covariance
matrix (`cov`). Additional keyword arguments are passed on to the
ellipse patch artist.

Parameters
----------
cov : The 2x2 covariance matrix to base the ellipse on
pos : The location of the center of the ellipse. Expects a 2-element
sequence of [x0, y0].
nstd : The radius of the ellipse in numbers of standard deviations.
Defaults to 2 standard deviations.
ax : The axis that the ellipse will be plotted on. Defaults to the
current axis.
Additional keyword arguments are pass on to the ellipse patch.

Returns
-------
A matplotlib ellipse artist
"""
def eigsorted(cov):
vals, vecs = np.linalg.eigh(cov)
order = vals.argsort()[::-1]
return vals[order], vecs[:,order]

if ax is None:
ax = plt.gca()

vals, vecs = eigsorted(cov)
theta = np.degrees(np.arctan2(*vecs[:,0][::-1]))

# Width and height are "full" widths, not radius
width, height = 2 * nstd * np.sqrt(vals)
ellip = Ellipse(xy=pos, width=width, height=height, angle=theta, **kwargs)

return ellip

if __name__ == '__main__':
#-- Example usage -----------------------
# Generate some random, correlated data
points = np.random.multivariate_normal(
mean=(1,1), cov=[[0.4, 9],[9, 10]], size=1000
)
# Plot the raw points...
x, y = points.T
plt.plot(x, y, 'ro')

# Plot a transparent 3 standard deviation covariance ellipse
plot_point_cov(points, nstd=3, alpha=0.5, color='green')

plt.show()
``````

Source (Stackoverflow)