Information Technology Reference
In-Depth Information
The loci of the two-dimensional point
y
satisfying (2.28) for different values of
κ
define
a set of ellipses. In the case of
κ
=
2 we have the sample concentration ellipse and when
κ
=
1wehavethe
indicator ellipse
(Le Roux and Rouanet, 2004). Equation (2.28) then
implies the following if the configuration of two-dimensional points can be considered
to be a random sample from a bivariate normal distribution:
•
approximately 39.35% of the points in the configuration lie within the boundary
of the sample indicator ellipse;
2
2
1
/
2
,where
2
2
•
choosing
κ
=
(χ
−
α
)
χ
denotes the (1 -
α)
100th percentage point
;
1
;
1
−
α
2
of the
χ
2
distribution, results in an ellipse covering approximately 100
α
%ofthe
configuration of two-dimensional points - that is, choosing
κ
=
2
.
4477 results in
an ellipse covering approximately 95% of the points in the configuration; choosing
κ
=
3
.
0349 results in an ellipse covering approximately 99% of the points in the
configuration.
2
2
1
/
2
κ
κ
=
(χ
−
α
)
We will use the term
-
ellipse
to denote the choice of
resulting in
;
1
the ellipse covering approximately 100
% of the configuration of two-dimensional
points. In Figure 2.33 we show the results of a call to our function
ConCentrEllipse
with arguments
kappa = 2
,
sqrt(qchisq(0.95,2))
and
sqrt(qchisq(0.99,2)
respectively.
α
Concentration ellipse
0.05 kappa ellipse
0.01 kappa ellipse
45
50
55
60
65
70
75
LGAN
Figure 2.33
Concentration ellipse together with 0.05 and 0.01 kappa ellipses enclos-
ing respectively approximately 86.5%, 95% and 99% of the configuration of two head
dimensions variables.
