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to confidence intervals of the form [canonical mean
−
z
α/
2
, canonical mean
+
z
α/
2
]or
[canonical mean -
z
α/
2
/
√
n
i
, canonical mean
+
z
α/
2
/
√
n
i
], where
z
α/
2
is defined by
P
(
Z
≤
z
α/
2
)
=
1
−
α/
2 with
Z
denoting the standard normal distribution. Equation (4.1)
defines
W
as an SSP matrix, whereas the assumption of normality requires variances.
Therefore, care has to be taken to scale down the radii by a factor
√
n
−
K
,where
W
has
n
-
K
degrees of freedom.
4.3 Geometric interpretation of the transformation
to the canonical space
Let us consider three classes with ellipsoidal shaped distributions as shown in Figure 4.5.
Each ellipsoid represents a set of sample points (not shown individually) and the whole
is assumed, without loss of generality, to be mean-centred around the origin.
In step I we obtain the nonsingular matrix
L
such that the Mahalanobis distances
between the class means in Figure 4.5 are Pythagorean distances in the canonical space.
Let us consider geometrically how this is accomplished. Any matrix can be written in
terms of its SVD,
L
=
PDQ
. If we perform the transformation piecewise with three
consecutive matrix multiplications, we have the transformations
X
→
XP
→
XPD
→
XPDQ
=
XL
. We notice that since both
P
and
Q
are orthogonal matrices, these transfor-
mations are rotations while
D
is a diagonal matrix resulting in a stretching or contracting
of each dimension. This is displayed in Figure 4.6.
In the top left panel of Figure 4.6 we have the original data similar to Figure 4.5. The
transformation implies
XW
−
1
X
=
XLL
X
=
XPDQ
QDP
X
(
P
X
=
(
XP
)
D
2
)
,which
Figure 4.5
Three-class data set for illustrating the two-step transformation to the canon-
ical space.
