Graphics Reference
In-Depth Information
. Determine the shortest path between frames using singular value decomposi-
tion.
A
′
′
a
A
z
=
V
a
Λ
V
z
,Λ
=
diag
(
λ
ċċċ
λ
d
)
, and the principal directions in
each plane are
B
a
A
z
V
z
,a within-plane rotation of the descriptive
bases
A
a
,
A
z
,respectively. heprincipal directions are the framesdescribing the
starting and target planes that have the shortest distance between them. he ro-
tation is defined with respect to these principal directions. he singular values,
λ
i
, i
=
A
a
V
a
,
B
z
=
,...,d, define the smallest angles between the principal directions.
. Orthonormalize
B
z
on
B
a
,giving
B
, to create a rotation framework.
. Calculate the principal angles, τ
i
=
cos
−
λ
i
, i
,...,d.
. Rotate theframesbydividing theanglesintoincrements, τ
i
=
=
,and
create the ith column of the new frame,
b
i
,fromtheith columns of
B
a
and
B
,
by
b
i
(
t
)
,fort
(
,
]
(
t
)=
cos
(
τ
i
(
t
))
b
ai
+
sin
(
τ
i
(
t
))
b
i
.Whent
=
, the frame will be
B
z
.
′
. Project the data into
A
(
t
)=
B
(
t
)
V
a
.
. Continue the rotation until t
=
.Setthecurrentprojection tobe
A
a
and go back
to step
.
Choosing the Target Plane
2.2.3
Grand Tour
he grand tour method for choosing the target plane is to use random selection.
A frame is randomly selected from the space of all possible projections.
A target frame ischosen randomly bystandardizing arandom vector froma stan-
dard multivariate normal distribution: sample p values from a standard univariate
normaldistribution,resultinginasamplefromastandardmultivariatenormal.Stan-
dardizing this vector tohavelength equal toone gives arandomvalue froma
-
dimensional sphere, that is, a randomly generated projection vector. Do this twice to
get a
-D projection, where the second vector is orthonormalized on the first.
Figure
.
illustrates the tour path, using GGobi to look at itself. Using GGobi, we
recorded the sequence of
D projections displayed of
-D data. his tour path
is a set of
points on a
-D sphere,where each point corresponds to a projection.
We use a tour to view the path (top let plot). he starting projection is
A
(
p
−
)
′
=(
)
,
indicated by a large point (
•
) or solid circle in the display. It is at the center right in
the plot, a projection in the first two variables. he corresponding data projection is
shown at top right. he grand tour path zigzags around the
-D sphere. he grand
tourcanbeconsideredasaninterpolatedrandomwalkoverthespaceofallplanes.
With enough time it will entirely cover the surface of the sphere. he bottom row
of plots shows two views of a grand tour path of
-D projections of
-dimen-
sional data.
Projection Pursuit Guided Tour
In a guided tour (Cook et al.,
) the next target basis is selected by optimizing
a PP index function. he index function numerically describes what is interesting