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
=
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
Search WWH ::




Custom Search