1. Contours: Y R f R* f(x) Y R f S
 f:R(A1..An)  Y
A1 A An Af
x1 x xn f(x1..xn) :
. . . R*
Equivalently,  derived attribute, Af, with domain=Y
(equivalence is x.Af = f(x) xR)
A1 A An
x1 x xn :
. . .
f(x) f R Y x
and  SY, the f-contour(S) = f-1(S) Equiv., Af-contour(S) = Select x1..xn From R* Where x.Af=f(x1..xn)
A1 A An : : S f R Y
graph(f) = { ( x, f(x) ) | xR }
f-contour(S)
If S={a}, we use f-Isobar(a) equiv. Af-Isobar(a)
If f is a local density and {Sk} is a partition of Y, {f-1(Sk)} partitions R. (eg, In OPTICS, f=reachability distance, {Sk} is the partition produced by intersections of graph-f wrt to a walk of R and a horizontal line.
A Weather map use equiwidth interval partition of S=Reals (barometric pressure or temperature contours).
A grid is the intersection partition with respect to the dimension projection functions (next slide).
A Class is a contour under f:RC, the class map.
An L -disk about a is the intersection of the -dimension_projection contours containing a.

2. f:R(A1..An)Y SY The (uncompressed) Predicate-tree 0Pf, S is : 0Pf,S(x)=1(true) iff f(x)S
The Compressed P-tree, sPf,S is the compression of 0Pf,S with equi-width leaf size, s, as follows
1. Choose a walk of R (converts 0Pf,S from bit map to bit vector)
2. Equi-width partition 0Pf,S with segment size, s (s=leafsize, the last segment can be short)
3. Eliminate and mask to 0, all pure-zero segments (call mask, NotPure0 Mask or EM)
4. Eliminate and mask to 1, all pure-one segments (call mask, Pure1 Mask or UM)
(EM=existential aggregation UM=universal aggregation)
Compressing each leaf of sPf,S with leafsize=s2 gives: s1,s2Pf,S Recursivly, s1, s2, s3Pf,S s1, s2, s3, s4Pf,S
(builds an EM and a UM tree)
BASIC P-trees
If Ai Real or Binary and fi,j(x)  jth bit of xi ; {(*)Pfi,j ,{1} (*)Pi,j}j=b are basic (*)P-trees of Ai, *= s1..sk
If Ai Categorical and fi,a(x)=1 if xi=a, else 0; {(*)Pfi,a,{1} (*)Pi,a}aR[Ai] are basic (*)P-trees of Ai
Notes:
The UM masks (e.g., of 2k,...,20Pi,j, with k=roof(log2|R| ), form a (binary) tree.
Whenever the EM bit is 1, that entire subtree can be eliminated (since it represents a pure0 segment), then a 0-node at level-k (lowest level = level-0) with no sub-tree indicates a 2k-run of zeros. In this construction, the UM tree is redundant.
We call these EM trees the basic binary P-trees.

3. = xRd=1..n(xd2 - 2adxd + ad2) i,j,k bit slices indexes
A useful functional: TV(a) =xR(x-a)o(x-a) If we use d for an index variable over the dimensions,
= xRd=1..n(xd adxd ad2) i,j,k bit slices indexes
= xRd=1..n(k2kxdk)2 -
2xRd=1..nad(k2kxdk) +
|R||a|2
= xd(i2ixdi)(j2jxdj) -
2xRd=1..nad(k2kxdk) +
|R||a|2
= xdi,j 2i+jxdixdj
2 x,d,k2k ad xdk
|R||a|2
= x,d,i,j 2i+j xdixdj
|R||a|2
2 dad x,k2kxdk
= x,d,i,j 2i+j xdixdj
|R|dadad
2|R| dadd +
= x,d,i,j 2i+j xdixdj
dadad )
|R|( -2dadd +
TV(a) = i,j,d 2i+j |Pdi^dj| -
|R||a|2
k2k+1 dad |Pdk| +
Note that the first term does not depend upon a. Thus, the derived attribute, TV-TV() (eliminate 1st term) is much simpler to compute and has identical contours (just lowers the graph by TV() ).
We also find it useful to post-compose a log to reduce the number of bit slices.
The resulting functional is called the High-Dimension-ready Total Variation or HDTV(a).

4. TV(a) = x,d,i,j 2i+j xdixdj + |R| ( -2dadd + dadad )
From equation 7,
f(a)=TV(a)-TV()
TV(a) = x,d,i,j 2i+j xdixdj
+ |R| ( -2dadd +
dadad )
= |R| ( -2d(add-dd) +
d(adad- dd) )
+ dd2 )
= |R|( dad2
- 2ddad
= |R| |a-| so f()=0
g(a) HDTV(a) = ln( f(a) )=
ln|R| + ln|a-|2
The length of g(a) depends only on the length of a-, so isobars are hyper-circles centered at 
The graph of g is a log-shaped hyper-funnel:
For an -contour ring (radius  about a)
go inward and outward along a- by  to the points;
inner point, b=+(1-/|a-|)(a-) and
outer point, c=-(1+/|a-|)(a-).  x1 x2
g(a)=HDTV(x)
g(b)
g(c)
Then take g(b) and g(c) as lower and upper endpoints of a vertical interval.
Then we use EIN formulas on that interval to get a mask P-tree for the -contour
(which is a well-pruned superset of the -neighborhood of a)
-contour
(radius  about a) a b c

5. contour of dimension projection f(a)=a1
If the HDTV circumscribing contour of a is still too populous,
use circumscribing Ad-contour (Note: Ad is not a derived attribute at all, but just Ad, so we already have its basic P-trees).
As pre-processing, calculate basic P-trees for the HDTV derived attribute
(or another hypercircular contour derived attribute).
To classify a
1. Calculate b and c (Depend on a, )
2. Form mask P-tree for training pts with HDTV-values[HDTV(b),HDTV(c)]
3. User that P-tree to prune out the candidate NNS.
If the count of candidates is small, proceed to scan and assign class votes using Gaussian vote function, else prune further using a dimension projections).
(Use voting function, G(x) = Gauss(|x-a|)-Gauss(), where Gauss(r) is (1/(std*2)e-(r-mean)2/2var
(std, mean, var are wrt set distances from a of voters i.e., {r=|x-a|: x a voter} )
HDTV(x) 
-contour
(radius  about a) a
HDTV(c)
We can also note that HDTV can be further simplified (retaining same contours) using h(a)=|a-|. Since we create the derived attribute by scanning the training set, why not just use this very simple function?
Others leap to mind, e.g., hb(a)=|a-b|
HDTV(b) x1
contour of dimension projection f(a)=a1 b c x2

6. Graphs of functionals with hyper-circular contours
TV()=TV(x33)
TV(x15) 1 2 3 4 5 X Y TV 
HDTV
Graphs of functionals with hyper-circular contours 
h(a)=|a-| 1 2 3
TV(x15)-TV() 4 5 X Y
TV-TV() 
hb(a)=|a-b| b

7. = (1/|a|)d(xxdad) factor out ad
Angular Variation functionals: e.g., AV(a)  ( 1/|a| ) xR xoa d is an index over the dimensions, 
COS(a)
= (1/|a|)xRd=1..nxdad
= (1/|a|)d(xxdad) factor out ad
= (1/|a|)d=1..n(xxd) ad
= |R|/|a|d=1..n((xxd)/|R|) ad
= |R|/|a|d=1..n d ad
COS(a)  a
= ( |R|/|a| ) 
o a
COS(a)  AV(a)/(|||R|) = oa/(|||a|) = cos(a)
COS (and AV) has hyper-conic isobars center on 
COSb(a)?  a b
COS and AV have -contour(a) = the space between two hyper-cones center on  which just circumscribes the Euclidean -hyperdisk at a.
Intersection (in pink) with HDTV -contour.
Graphs of functionals with hyper-conic contours:
E.g., COSb(a) for any vector, b

8. f(a)x = (x-a)o(x-a) d = index over dims,
= d=1..n(xd adxd ad2) i,j,k bit slices indexes
= d=1..n(k2kxdk)
2d=1..nad(k2kxdk)
|a|2
= d(i2ixdi)(j2jxdj) -
2d=1..nad(2kxdk)
|a|2
=di,j 2i+jxdixdj
2 d,k2k ad xdk
|a|2
f(a)x = i,j,d 2i+j (Pdi^dj)x
|a|2
k2k+1 dad (Pdk)x +
β exp( -f(a)x ) = βexp(-i,j,d 2i+j (Pdi^dj)x)
*exp( -|a|2 )
* exp( k2k+1 dad (Pdk)x )
β exp( -f(a)x ) = β (exp(-i,j,d 2i+j (Pdi^dj)x)
exp( -|a|2 )
* exp( k2k+1 dad (Pdk)x ) )
Adding up the Gaussian weighted votes for class c:
xcβ exp( -f(a)x ) = β xc (exp(-i,j,d 2i+j (Pdi^dj)x)
exp( -|a|2 )
* exp( k2k+1 dad (Pdk)x ) )
xc exp((-i,j,d 2i+j (Pdi^dj)x)
k,d2k+1 ad (Pdk)x )
Collecting diagonal terms inside exp
xc exp( ij,d -2i+j (Pdi^dj)x
+ i=j,d(ad2i+1-22i ) (Pdi)x )
i,j,d inside exp we have coefs which do not involve x multiplied by a 1-bit or a 0-bit, depending on x
thus for fixed i,j,d we either have the x-indep coef (if 1-bit) or we don't (if 0-bit)
xc ( ij,d exp(-2i+j (Pdi^dj)x)
* i=j,d exp((ad2i+1-22i)(Pdi)x) )
 ( ij,d:Pdijx=1 exp(-2i+j )
* i=j,d:Pdijx=1 exp((ad2i+1-22i)) ) (eq1)