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Why Not Store Everything in Main Memory? Why use disks?

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FAUST Using std or rankK instead of just gap size to determine the best gap and/or using multiple attribute cutting can improve accuracy. We need a pTree ALGEBRA ... – PowerPoint PPT presentation

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Title: Why Not Store Everything in Main Memory? Why use disks?


1
FAUST Using std or rankK instead of just gap size
to determine the best gap and/or using multiple
attribute cutting can improve accuracy. We
need a pTree ALGEBRA (already well started with
the pTree Algebra paper - involves the pTree
operators, AND, OR, COMP, XOR, ... and their
algebraic properties (communtativity,
associativity, distributivity, ...) We need a
pTree CALCULUS (functions that produce the pTree
mask for just about any pTree-defining
predicate). Note that in FAUSTdiv,gap, we cut
perpendicular on the single attributes line,
which contains the maximum consecutive-class-mean
gap and use the midpoint of that gap (or the
confluence point of maximal stds) as cut_point to
separate the entire remaining space of pixels
into 2 big boxes, one containing one partition of
the remaining classes and the other, the balance
of classes. We never cut oblique to attribute
lines! Then we do it again on one of those
sub-partitions until we reach a single class. Can
that be improved?  With respect to speed,
probably not. Can accuracy be improved without
sacrificing speed (too much)?  Here is a way at
about the same speed.  As motivation, think
about a blue-red cars class. (Define, e.g., as 2
parts red, 1 part blue). We want to do a cut (at
the midpoint of the maximal gap) maximizing over
all oblique directions, not just along dimensions
(since the dimensions form a measure zero set of
all possible directions).  E.g., a blue-red cut
would define a line at a 30 degree angle from the
red axis toward the blue points in the blue-red
direction. If D is any unit vector, X dot D
?i1..nXiDi. X dot D gt cut_point defines an
oblique big box.  We ought to consider all
D-lines (noting that dimension lines ARE
D-lines).  For this we will need
an Multi-attribute "EIN-Oblique" mask pTree
formula P(X dot D)gta where X is any
vector and D is an oblique vector (NOTE if
Dei(0,...,1,...0) then this is just the
existing EIN formula for the ith dimension, PXi
gt a ). The pTree formula for the dot product
is in the pTree book, pages 133-134 (and Mohammad
is developing a better one?). We would like a
recursive, exhaustive search for the vector D
that gives us the maximal gap among the
consecutive training class means for the classes
that remain (not just over all attribute
directions, but all combination directions).
How can we find it? 1st examples
2
Using a quadratic hyper-surface? (instead of a
hyper-plane)
Suppose there are just 2 attributes (red and
blue) and we (r,b)-scatter plot the 10
reddish-blue class training points and the 10
bluish-red class training points                 
                                          
                                                 
                                                 
                                                  
     b       b                                    
                                                  
   b      b                                     
                                                  
           b     b      b                         
                                                  
     r               b                            
                                     
                      r         
b                                                 
                                  r       r
b                                                
                             r        
r                                                 
                             r      r
                                                  
                                  r     r
  -----------------------------------------------
--------------gt
D-line mean for the b class
D-line mean for the r class
Take the r and the b points that project closest
to the D-line as the "best" support
pair. similarly for the "next best" or "second
best" support pair similarly for the "third best"
pair. Form the quadratic support curve from the
three r-support points for class-r Form the
quadratic support curve from the three b-support
points for class-b (or move each point in each
pair 1/3 of the way toward the other and then do
the above) or ????.
3
Suppose there are just 2 attributes (red and
blue) and we (r,b)-scatter plot the 10
reddish-blue class training points and the 10
bluish-red class training points                 
                                          
                                                 
                                                 
                                                  
     b       b                                    
                                                  
   b      b                                     
                                                  
           b     b      b                         
                                                  
     r               b                            
                                     
                      r         
b                                                 
                                  r       r
b                                                
                             r         r
                                        
                                     r      r
               
                                                 
                  r     r   --------------------
-----------------------------------------gt
Fitting a parabolic hyper-surface
4
APPENDIX on FAUST is a Near Neighbor Classifier.
It is not a Voting NNC like pCkNN (where for
each unclassified sample pCkNN builds around that
sample, a neighborhood of TrainingSet voters, who
then classify sample through majority, plurality
or weighted (in PINE) vote. pCkNN classifies one
unclassified sample at a time. FAUST is meant
for speed and therefore FAUST attempts to
classify all unclassified samples at one time.
FAUST builds a Big Box Neighborhood (BBN) for
each class and then classifies all unclassified
samples in the BBN into that class (constructing
said class-BBNs with one EIN pTree calculation
per class).
The BBNs can overlap, so the classification needs
to be done one class at a time sequentially, in
maximum gap, maximum number of std's in gap, or
minimum rankK in gap order.) The whole process
can be iterated as in k-means classification
using the predicted classes or subsets of as
the new training set. This can be continued
until convergence.
A BBN can be a coordinate box for coord R,
cb(R,class,aR,bR) is all x such that
aRltxRltbR Either or both of the lt can be ? or ?.
aR and bR are what were called cut_points of the
class.
Or BBNs can be multi-coordinate boxes, which are
INTERSECTIONs of the best k (k?n-1,
assuming n classes) cb's for a given class
("best" can be wrt any of the above
maximizations). And instead of using a fixed
number of coordinates, k, we could use only those
in which the "quality" of its cb is higher than a
threshold, where "quality" might be measured
involving the dimensions of the gaps (or other
ways?).
FAUST could be combined with pCkNN (probably in
many ways) as follows FAUST multi-coordinate
BBN could be used first to classify the "easy
points" (that fall in an intersection of high
quality BBNs and are therefore fairly certain to
be correctly classified). Then for the remaining
"difficult points" could be classified using the
original training set (or the union of each
original TrainingSet class with the new "easy
points" of that same class) and using L? or Lp ,
p 1 or 2.
5
A Multi-attribute EIN Oblique (EINO) based
heuristic Instead of finding the best D, take
the vector connecting a class mean to another
class means as D To separate r from v
D(mv?mr) and amvvr/2
To separate r from b D(mb?mr) and ambvr/2
Question What's the best as cutpt? mean,
vector_of_medians, outermost, outermost_non-outlie
r?
P(mb?mr)oXgtmrmb/2
By "outermost, I mean the "furthest points away
from the means in each class (in terms of their
projections of the D-line) By "outermost
non-outlie" I mean the furthest non-outlier
points Other possibilities the best rankK
points, the best std points, etc. Comments on
where to go from here (assuming we can do the
above) I think the "medoid-to-mediod" method on
this page is close to optimal provided the
classes are convex. If they are not convex, then
some sort of Support Vector Machines, SVMs, would
be the next step. In SVMs the space is translated
to higher dimensions in such a way that the
classes ARE convex. The inner product in that
space is equivalent to a kernel function in the
original space so that one need not even do the
translation to get inner product based results
(the genius of the method). Final note I should
say "linearly separable instead of convex
(slightly weaker condition).
6
A Multi-attribute EIN Oblique (EINO) based
heuristic Instead of finding the best D, take
the vector connecting a class mean to another
class mean as D (d D/D)
Where a can be calculated either as (let d
D/D 1. a ( domr domv )/2 2. Letting
armaxdor avmindov (if domr lt domv, else
reverse max and min). For r take a av 3. Using
variance fits.(or rankK fits). (Notes Apply to
all other classes or only those for which there
is a positive gap.)
FAUST For isolating classi 1. Create table,
TBLclassi, meanvectori( classj, meanvectorj
) 2. Apply the pTree mask formula at left. Note
If we take the fastest route and just pick the
one class which when paired with r, gives the max
gap, then we can use max gap or maximum-std-point
is used instead of midpoint of max gap, then we
need stdj (or variancej) in TBL.
P(mr?mv)/mr?mvoXlta
    r   r r v v        r
 mr   r      v v v       r    r  
    v mv v      r   
v v     r           
v                    
D mr?mv
For classes r and b
7
Suppose there are just 2 attributes (red and
blue) and we (r,b)-scatter plot the 10
reddish-blue class training points and the 10
bluish-red class training points                 
                                          
blue                                              
                                                
                                                 
          rb      rb                              
                               
                           rb     rb    
                                                 
                                           rb 
  rb     rb                                       
                                        br       
       rb                                         
                                              br
        rb                                       
                                           br
     br rb                                  
                                           br
        br                                   
                                          br
    br                                            
                                        br    br
  ------------------------------------
-------------------------------------------------
------gtred
D-line mean for the rb class
D-line mean for the br class
etc.
Consecutive class mean mid-point Cut_Point
Cut-HyperPlane, CHP (what we are after)
Clearly we would want to find a 45 degree unit
vector, D, then calculate the means of the
projections of the two training sets onto the
D-line then use the midpoint of the gap between
those two means as the cut_point (erecting a
perpendicular bisector "hyperplane" to D there -
which separates the space into the two class big
boxes on each side of the hyperplane. Can it
an be masked using one EIN formula??)
                                                  
           blue                                 
                                                 
          rb      rb                              
                               
                           rb     rb    
                                                 
                                           rb 
  rb     rb                                       
                                        br       
       rb                                         
                                              br
        rb                                       
                                           br
     br rb                                  
                                           br
        br                                   
                                          br
    br                                            
                                        br    br
  ------------------------------------
--------------------------------gtred
The above "diagonal" cutting produces a perfect
classification (of the training points). If we
had considered only cut_points along coordinate
axes, it would have been very imperfect!
8
How do we search through all possible angles for
the D that will maximize that gap? We would have
to develop the formula (pTree only formula) for
the class means for any D and then maximize the
gap (distance between consecutive D-projected
means). Take a look at the formulas in the book,
think about it, take a look at Mohammads
formulas, see if you can come up with the mega
formula above. Let D (D1, , Dn) be a unit
vector (our cut_line direction vector) D dot X
D1X1 DnXn is the length of the perpendicular
projection of X on D (length of the high noon
shadow that X makes on the D line, as if D were
the earth). So, we project every training point,
Xc,i (classc, i1..10), onto D (i.e., D dot
Xc,i). Calculate D-line class means, (1/n)?(D
dot Xc,i), select the max consecutive mean gap
along D, (call it best_gap(D)bg(D). Maximize
bg(D) over all possible D. Harder? Calculate it
for a polar grid of Ds! Maximize over that
grid. Then use continuity and hill climbing to
improve it.
etc.
Cut_point
More likely the situation would be rb's are more
blue than red and br's are more red than blue.
Suppose there are just 2 attributes (red and
blue) and we (r,b)-scatter plot the 10
reddish-blue class training points and the 10
bluish-red class training points                 
                                          
                                                 
                                                 
                                                  
    rb      rb                                    
                                                  
  rb     rb                                     
                                                  
          rb    rb     rb                         
                                                  
    br              rb                            
                                     
                     br         rb               
                                                  
                 br      br
rb                                                
                             br        
br                                                
                             br     br
                                                  
                                 br    br
  -----------------------------------------------
--------------gt red
blue
D-line mean for the rb class
D-line mean for the br class
What if the training points are shifted away from
the origin? This should convince you that it
still works.
9
In higher dimensions, nothing changes (If there
are "convex" clustered classes,
FAUSTdiv,oblique_gap can find them (consider
greenish-redish-blue and bluish-greenish-red)
r
Before considering the pTree formulas for the
above, we note again that any pair of classes
(multi-classes, as in divisive) that are convex,
can be separated by this method. What if they
are not convex? A 2-D example
A couple of comments. FAUST resembles the SVD
(Support Vector Machine) method in that it
constructs a separating hyperplane in the
"margin" between classes. The beauty of SVD
(over FAUST and all other methods) is that it is
provable that there is a transformation to a
higher dimensions that renders two non-hyperplane
seperable classes to being hyperplane seperable
(and you don't actually have to do the
transformation - just determine the kernel that
produces it.). The problem with SVD is that that
it is computationally intensive. I think we want
to keep FAUST simple (and fast!). If we can do
this generalization, I think it will be a real
winner!
How do we search over all possible Oblique
vectors, D, for the one that is "best"? Of if we
are to use multi-box neighborhoods, how do we do
that? A heuristic method follows
10
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1
1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0
0 0 0 1 0
1 1 1 1 0 1 1 1 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 1 0
0 1 1 0 0
0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 0 1 1 0 0 1 0 1 1
0 0 0 0 1
0 0 0 0 0 1 0 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 1
1 0 0 1 0
1 1 1 1 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 0 0 0 0 0 0
1 1 0 1 0
0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 1 0 1 1 1 1 0 0 0
1 1 0 1 1
0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0
0 1 1 1 1
0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0
1 1 1 1 1
0 1 0 1 1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 1 1
1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 0 0
1 0 0 0 0
0 1 1 1 1 1 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 1 1 0 1
0 0 0 1 0
0 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0
0 0 1 0 0
1 1 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 0
1 1 1 0 1
1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 1 0 0 0 1 1 1 0
1 1 1 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 0 0
1 0 1 0 0 0 0 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1
1 1 1 0 0
1 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1
1 1 1 0 0
1 0 1 1 1 0 0 1 1 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 1
0 1 0 1 0
1 0 1 0 1 1 1 0 1 0 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1
0 0 0 0 0
0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 1 0 1 1 0 1 0 1 0 0
1 1 1 0 0
0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1
0 1 1 1 1
1 0 0 1 1 0 1 0 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 0 0
1 0 0 0 0
0 1 1 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 1
0 1 0 0 0
1 0 1 1 0 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 1
0 1 1 0 1
FAUST_pdq_std (using std's) 1.1 Create
attribute tables with clclass, mn, std,
nmax__stds_in_gap, cpcut_point (value in the
gap which allows the max of stds, n, to fit
forward from mean (using its std) and backward
from next mean (using its std)). n satisfies
meannstdmeanG-nstdG so n(mnG-mn)/(stdstdG
)
TpLN cl mn std n cp se 15 1.0 4.5 19 01
0011 ve
TA rec with max n
0 0 1 0 0 1 1 19
Note, since there is also a case with n4.1 which
results in the same partition (into se and
ve,vi) we might use both for improved accuracy
- certainly we can do this with sequential!
1 1 1 1 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 0 0 0 0 0 0
1 1 0 1 0
1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 1 0 0 0 1 1 1 0
1 1 1 1 0
1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0 1 1 1 0
1 1 1 1 0
0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0
0 1 1 1 1
0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0
1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1
TsLN cl mn std n cp se 49 3.5 0.9 53 ve 59
6.9 0.5 62 vi 66 7.6
TsWD cl mn std n cp ve 28 3.9 0.3 29 vi 29
3.1 1.3 33 se 33 3.1
TpLN cl mn std n cp se 15 1.0 4.5 19 ve 43
5.1 1.3 49 vi 57 6.0
TpWD cl n std n cp se 2 0.7 4.1 5 ve 13
2.0 1.5 16 vi 20 2.3
se_means 49.3 33.3 14.6 2.2 se_std 3.5 3.1
1.0 0.7 se_ve_n 0.9 -0.8 4.5 4.1
se_vi_n 1.5 -0.6 6.0 5.8 se_ve_cp 52.6 30.7
19.2 5.3 se_vi_cp 54.5 31.3 20.8 6.5
ve_means 59.0 27.5 42.5 13.4 ve_std 6.9
3.9 5.1 2.0 ve_vi_n 0.5 0.3 1.3 1.5
ve_se_n -0.9 0.8 -4.5 -4.1 ve_vi_cp 62.3 28.5
49.1 16.4 ve_se_cp 52.6 30.7 19.2 5.3
vi_means 65.9 29.3 56.8 19.9 vi_std 7.6
3.1 6.0 2.3 vi_se_n -1.5 0.6 -6.0 -5.8
vi_ve_n -0.5 -0.3 -1.3 -1.5 vi_se_cp 54.5 31.3
20.8 6.5 vi_ve_cp 62.3 28.5 49.1 16.4
Remove se from RC (ve, vi now) and TA's
11
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1
1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0
0 0 0 1 0
1 1 1 1 0 1 1 1 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 1 0
0 1 1 0 0
0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 0 1 1 0 0 1 0 1 1
0 0 0 0 1
0 0 0 0 0 1 0 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 1
1 0 0 1 0
1 1 1 1 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 0 0 0 0 0 0
1 1 0 1 0
0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 1 0 1 1 1 1 0 0 0
1 1 0 1 1
0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0
0 1 1 1 1
0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0
1 1 1 1 1
0 1 0 1 1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 1 1
1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 0 0
1 0 0 0 0
0 1 1 1 1 1 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 1 1 0 1
0 0 0 1 0
0 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0
0 0 1 0 0
1 1 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 0
1 1 1 0 1
1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 1 0 0 0 1 1 1 0
1 1 1 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 0 0
1 0 1 0 0 0 0 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1
1 1 1 0 0
1 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1
1 1 1 0 0
1 0 1 1 1 0 0 1 1 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 1
0 1 0 1 0
1 0 1 0 1 1 1 0 1 0 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1
0 0 0 0 0
0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 1 0 1 1 0 1 0 1 0 0
1 1 1 0 0
0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1
0 1 1 1 1
1 0 0 1 1 0 1 0 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 0 0
1 0 0 0 0
0 1 1 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 1
0 1 0 0 0
1 0 1 1 0 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 1
0 1 1 0 1
FAUST_pdq using std's 1.2 Use the 4
Attribute tables with rvmean, stds and
max__stds_in_gapn, cut value, cp (cpvalue in
gap which allows max of stds, n, to fit forward
from that mean (using its std) and backward from
next mean, meanG, (using stdG). n satisfies
mean nstd meanG - nstdG
so n(meanG-mean)/(stdstdG).
TpWD cl mn std n cp ve 13 2.0 1.5 16 vi
TA rec with max n
16 1 0 0 0 0
Pvi PpWDgt16
1 1 1 1 0 1 1 1 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 1 0
0 1 1 0 0
0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 0 1 1 0 0 1 0 1 1
0 0 0 0 1
0 0 0 0 0 1 0 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 1
1 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1
1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1
1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0
0 0 0 1 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1
1 1 1 1 1
Note that we get perfect accuracy with one epoch
using stds this way!!!
TsLN cl mn std n cp se 49 3.5 0.9 53 ve 59
6.9 0.5 62 vi 66 7.6
TsWD cl mn std n cp ve 28 3.9 0.3 29 vi 29 3.1
1.3 33 se 33 3.1
TpLN cl mn std n cp se 15 1.0 4. 19 ve 43 5.1
1.3 49 vi 57 6.0
TpWD cl mn std n cp se 2 0.7 4. 5 ve 13
2.0 1.5 16 vi 20 2.3
12
FAUST_pdq
SUMMARY We conclude that FAUST_pdq will be fast
(no loops, one pTree mask per step, may converge
with 1 or just a few epochs?? and is fairly
accurate (completely accurate in this example
using the std method!). FAUST_pdq is improved
(accuracy-wise) by using standard_deviation-based
gap measurements and choosing the maximum number
of stds as the attribute relevancy choice.
There may be many other such improvements one
can think of, e.g., using an outlier
identification method (see Dr. Dongmei Ren's
thesis) to determine the set of non-outliers in
each attribute and class. Within each attribute,
order by means and define gaps to be between the
maximum non-outlier value in one class and the
minimum non-outlier value in the next (allowing
these gap measurements to be negative if the max
of one exceeds the minimum of the next). Also
there are many ways of defining representative
values (means, medians, rank-points, ...) In
Conclusion, FAUST_pdq is intended to be very fast
(if raw speed is the need - as it might be for
initial processing of the massive and numerous
image datasets that the DoD has to categorize and
store). It may be fairly accurate as well,
depending upon the dataset, but since it uses
only one attribute or feature for each division,
it is not likely to be of maximal accuracy
compared to other methods (such as the FAUST_pms
coming up). Next look at FAUST_pms (pTree-based,
m-attribute cut_points, sequential (1 class
divided off at a time) so we can explore the
various choices for m (from 1 to the table width)
and alternate distance measures.
13
For i4..0 crc(PcPatt,i) if
(c?ps) rankK 2i Pc(PcPatt,i)
rank(n-K1)2i else psps-c
PcPcP'att,i
1 1 1 1
1 1 1 1
K10
4
0 6 0 0
25
pWD_vi_LO16 ? pWD_se_HI0, pWD_ve_HI0. So the
highest pWD_se_HI and pWD_ve_HI can get is 15 and
lowest pWD_vi_LO will ever be is 16. So cutting
?16 will separate all vi from se,ve. This is,
of course, with reference to the training set
only and it may not carry over to the test set
(much bigger set?) especially since the gap may
be small (1). Here we will use pWDcutpt?16 to
peal off vi! We need a theorem proof here!!!
1 1 1 1
4 2 0 0
25
26
1 1 1 1
7 2 1 10
25
26
24
26
sLN1 sWD2 pLN3 pWD4
10 10 10 10
0 6 0 0
10 10 10 10
4 2 0 0
10 10 10 10
7 2 1 10
24
sLN1 sWD2 pLN3 pWD4
14
For i4..0 crc(PcPatt,i) if
(c?ps) rankK 2i Pc(PcPatt,i)
rank(n-K1)2i else psps-c
PcPcP'att,i
1 1 1 1
1 1 1 1
3
10 6 0 0
0 6 0 0
25
24
25
1 1 1 1
pLN_ve_LO32 ? pLN_se_HI0. So the highest
pLN_se_HI can get is 31 and lowest pLN_ve_LO will
ever be is 32. So cutting ?32 will separate all
ve from se! Greater accuracy can be gained by
continuing the process for all i and for all K
then looking for the best gaps! (all gaps?) (all
gaps weighted?)
4 2 0 0
0 2 10 9
24
25
25
23
26
10 10 10 10
4
10 0 0 0
0 6 0 0
25
10 10 10 10
6 8
6 0 10 9
4 2 0 0
25
25
15
APPENDIX
FAUSTpdq,mrk (FAUSTpdq w max rank_k)
rank_k(S) is smallest kth largest value in S.
FAUSTpdq,gap divisive, quiet (no noise) with
gaps
? attr, A TA(class, md, k, cp) its attribute
table ordered on md asc, where
0. ?attr, A TA(class, rv, gap) ord on rv asc
(rvcls rep, gapdis to next rv.
k s.t. it's max k value s.t. set_rank_k of class
and set_rank_(1-k)' of the next class. (note
the rank_k for k1/2 is median, k1 is maximum
and k0 is the min. Same alg can clearly be used
as pms FAUSTpms,mrk
1. Find the TA record with maximum gap
WHILE RC not empty, DO
2. PAgtc (crvgap/2) to div RC at c into LT, GT
(pTrees, PLT and PGT).
3. If LT or GT singleton remove class)
END_DO
FAUSTpdq,std (FAUSTpdq using of gap
standard devs)
0. For each attribute, A TA(class, mn, std, n,
cp) is its attribute table ordered on n asc, where
cpval in gap allowing max of stds, n. n
satisfies meannstdmeanG-nstdG so
n(mnG-mn)/(stdstdG)
1. Find the TA record with maximum n
WHILE RC not empty, DO
2. Use PAgtcp to divide RC at cpcutpoint into LT
and GT (pTree masks, PLT and PGT).
3. If LT or GT singleton remove that class from
RC and from all TA's
END_DO
FAUSTpms,gap (FAUSTp m attr cut_pts, seq
class separation (1 class at time, m1
0. For each A, TA(class, rv, gap, avgap), where
avgap is avg of gap and previous_gap (if 1st
avgap gap). If x classes.
1. Find the TA record with maximum avgap
DO x-1 times
2. cLrv-prev_gap/2. cGrvgap/2, masks
PclassPAgtcLPA?cGPRC PRCP'classPRC (If 1st
in TA (no prev_gap), PclassPA?cGPRC. Last,
PclassPAgtcLPRC.
END_DO
3. Remove that class from RC and from all TA's
FAUSTpms,std (FAUSTpms using gap std
0. ?attr, A TA(class, mn, std, n, avgn, cp)
ordered avgn asc
cpcut_point (value in gap which allows max of
stds, n, (n satisfies mnnstdmnnext-nstdnext
so n(mnnext-mn)/(stdstdt)
1. Find the TA record with maximum avgn
DO x-1 times
2. cLrv-prev_gap/2. cGrvgap/2 and pTree masks
PclassPAgtcL PA?cGPRC PRC P'classPRC
(If class 1st in TA (has no prev_gap), then
Pclass PA?cGPRC. If last, Pclass PAgtcLPRC.)
3. Remove that class from RC and from all TA's
END_DO
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