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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.
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