Graphics Reference
In-Depth Information
Mathematically, a projection of data is computed by multiplying an n
p data ma-
trix,
X
,havingn sample points in p dimensions, by an orthonormal p
d projection
matrix,
A
, yielding a d-dimensional projection. For example, to project a
-D object
(
columns, or variables, of data) onto a
-D plane (the shadow of the object), we
would use an orthonormal
matrix.
Here is a concrete example. Suppose our data matrix and projection were these:
X
=
and
A
=
then
XA
=
is the first two columns of the data matrix. If instead
.
.
.
.
.
.
.
.
.
.
.
−
A
=
then
XA
=
.
.
.
.
.
.
−
is a combination of all three variables.
heseprojectionsareillustratedinFig.
.
.hetoprowshowsthedataprojections,
XA
and
XA
, respectively. he bottom row displays the projection coe
cients,
A
and
A
.Arowin
A
can also be interpreted as the projection of the coordinate axis
(p-dimensional to d-dimensional) for each variable, and it is represented by a line in
this display. he length and direction of the line displays the contribution each vari-
able makes to the projected data view. In
A
the data projection isconstructed purely
from variable
in the horizontal direction and variable
in the vertical direction. In
A
variables
and
share the horizontal direction, and variable
makes no contri-
bution horizontally. Vertically all three variables make a contribution, but variable
has twice the contribution of the other two variables. his type of axis display is used
to match structure in a data projection with the variable dimensions of the data and,
hence, enable to the analyst to interpret the data.
We also commonly use
-D projections in data analysis. With a
-D projection we
typically use a histogram or density plot to display the data. Consider the
-D data
in Fig.
.
(letplot) and two
-Dprojections (middle,right). heprojection matrices
are:
−
A
and
A
=
=
−
respectively.
