Discussion of Results relating to Orthogonal Subspace Projection:
(11/17/99)



1.  Connect with past via PC1, for full data set, length adjusted,
raw rescaling.  Earlier display (showing poor separation) is in
column 1 of:  ccf21d3sp1p2.ps

Show AxisAnim_PC1AllRaw.nb

Connects with expected?




2.  Show that Fisher Linear discriminant doesn't work here.  Again
for full data set, length adjusted, note from  ccf21d3sp1p340.ps
that get excellent separation (using 40 dim eigen subspace).
But, look at direction vector:

AxisAnim_FLDallNvsS40.nb

shows that direction is driven by boundary irregularities,
not "consistent shape".




3.  More extreme, but related case is, for females only,  Again
length adjusted, note from  ccf21d3sp3p340.ps
that get PERFECT separation (using 40 dim eigen subspace).
But, again a look at direction vector:

AxisAnim_FLDfemNvsS40.nb

shows that direction is driven by boundary irregularities,
not "consistent shape".




4.  Now try subspace projection method.  Basic idea is 
illustrated in:   
EgSubProj1Raw.ps  (shows raw data)
EgSubProj1.ps     (shows how projection works)

For full population, length adjusted,  note from   ccf25d3sp1p1.ps
that mad resacling appears to find directions with significant
structure.  Views of the two directions found are:


a.  1st row of the second column, in ccf25d3sp1p1.ps,
PC1 for Schizo's, orthogonal to subspace generated by Normals:

AxisAnim_ProjSSubN.nb

Looks like some serious change in "shape".  Interpretable????


b.  2nd row of the second column, in ccf25d3sp1p1.ps,
PC1 for Normals, orthogonal to subspace generated by Schizos:

AxisAnim_ProjNSubS.nb

Looks like some serious change in "shape".  Interpretable????




5.  Try the subspace projection method in another case.  For
Schizos only, Male vs. Female, seemed to find important structure
in the Spherical direction, as shown in ccf25d3sp5p1.ps.

Views of the two directions found are:


a.  1st row of the second column, in ccf25d3sp5p1.ps,
PC1 for Females, orthogonal to subspace generated by Males:

AxisAnim_SProjFSubM.nb

Here seems to be again just boundary artifacts, and not real changes
in "shape".


b.  2nd row of the second column, in ccf25d3sp5p1.ps,
PC1 for Males, orthogonal to subspace generated by Females:

AxisAnim_SProjMSubF.nb

Again seems to be again just boundary artifacts, and not real changes
in "shape".  Or is there a change in shape?




6.  Note:  Length rescaling is very important.  Get different answer
to (4) if do area rescaling:  ccf25d4sp1p1.ps




7.  Possible Future Directions

a.  Look at Bootstrap stability of PC1 vectors:
  i.  In generated subspaces
  ii.  In orthogonal subspaces

b.  Restrict first subspace, to lower dim eigenspace (eliminating 
"spurious noise" directions).

c.  Measure "stability" in terms of "dimensionality" or
"epsilon dimensionality".

d.  Do "leave one out" analysis of discrimination performance.
(leave out one of each, and project both onto ortho subspace,
and look at resulting spreads)

e.  Get at "area as a driving feature", by working with first 20 
Fourier coefficients, and then throwing in all cross terms (gives
20+400 dim feature space).

f.  Work on dimensions away, in particular studying effect of 
normal distribution, supported on lower dim'al subspace. (e.g. 
2-dim).  Can this be used to develop notion of "effective 
dimension of support"?

???