Ellipse Representing the Confidence Region of a Covariance Matrix

r

0

σ

1

1

σ

2

1

ρ

0.5

2

Δχ

1

best-fit vector r 0 = {1,1}
horizontal error σ 1 Δχ =	1
vertical error σ 2 Δχ =	1
slope of correlation axis ϕ =	45.

inverse covariance matrix

-1

V

=

1.33	-0.67
-0.67	1.33

covariance matrix V =

2 σ 1	ρ σ 1 σ 2
ρ σ 1 σ 2	2 σ 2

=

1.	0.5
0.5	1.

This Demonstration shows the confidence region defined by a covariance matrix. The equation of the contour of the region is given by

(r-

r

0

)·

-1

V

·(r-

r

0

)=

2

Δχ

, where

r

0

is the best-fit vector and

V

is the covariance matrix. The parameter

2

Δχ

is the large data sample limit corresponding to a coverage probability and characterizes the confidence level (e.g.

2

Δχ

=2.3

in the 2D case and 68.3% CL). The dashed blue line represents the direction of the parameters correlation.