Visualization of Weighted Lattices for
Data Analysis
Tim Hannan, Lance Miller, Alex Pogel
Physical Science Laboratory
New Mexico State University
apogel@psl.nmsu.edu
Weighted lattices
• A weighted lattice is a pair (L, w:L[0,1]), where
L is a (join semi-) lattice and
w is an order-preserving map
(level of generality to be explained)
• (for us) L is always finite
• This talk: provide motivation (from applications)
for defining criteria by which to judge methods of
drawing weighted lattices
Our Motivation
General task: provide analysis of time-series data
Need: a visualization of time-series data, e.g. as an
interval is moved across a timeline
t0
tf
t0
tf
goal: near real-time analysis tool
Formal Concept Analysis (FCA)
FCA background: input binary relation
I  GM
Galois correspondence P(G) P(M) : closure system
Result: the concept lattice (aka Galois lattice, etc), is
labeled and used for analysis
The concept lattice is an example of a weighted lattice,
with relative cardinality of domain sets used to
define weight (let w(C) be relative size of extent)
Application Problems
1. the lattice representation of data (binary relation)
can be highly sensitive (in terms of cardinality) to
minor variation in the data
2. order-theoretic presentations of lattices are often
ineffective for the task of finding weak implications
(naturally occurring or introduced by noise)
Problem 1: highly sensitive
5
5
a1
o1
o2
o3
o4
o5
a2
1
1
0
0
1
5
a4
0
0
1
0
1
a5
0
0
1
1
1
1
1
1
1
1
5
a1
o1
o2
o3
o4
o5
a3
1
0
0
0
1
a2
1
1
0
0
1
a3
1
0
0
0
1
a4
0
0
1
0
1
a5
0
0
1
1
1
1
1
1
1
0
Proposed Solution
1. Extend the usual order-based drawings of lattices to
include two factors: order and weight values
2. Compare various weight functions, some involving
order only and some with support, use to evaluate
3. Introduce various criteria for judging a weight
function with respect to the variation it introduces in
the face of minor variation in data
OUTLINE
• basic constructions of Formal Concept Analysis,
Plus attribute logic and association rules
• The problems and a new drawing tool
• theoretical examples and existing data sets
• criteria
Basic construction
Input binary relation:
Maps
I  GM
(-)’:P(G)P(M)
H’ = {m in M : for all h in H, h I m },
(-)’:P(M)P(G)
dually
yield a Galois correspondence between P(G) and P(M)
Which induces a closure system, a complete lattice
More directly: the complete meet subsemilattice of P(G) that is generated by {{m}’: m in M}
(aka smallest topped intersection structure in P(G) generated by {{m}’: m in M} )
Basic construction
Domain and codomain closure systems (on G and M)
are dually isomorphic,
so we consider one lattice, represented by closed set
pairs (H,N)
[H’=N, N’=H]
where H is in P(G), N is in P(M)
and ordered to reflect one of the closure systems:
inclusion in 1st coordinate
Important labeling
The lattice is labeled:
({g}’’,{g}’) is labeled g
({m}’,{m}’’) is labeled m
I is preserved in the lattice:
g I m iff ({g}’’,{g}’) < ({m}’,{m}’’)
Animal Context
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
es
ha
s_
fu
r
ha
s_
sc
al
d
4le
gg
e
-b
lo
od
ed
w
ar
m
ai
r_
br
ea
th
er
es
_i
n_
w
at
er
liv
liv
es
to
ck
m
on
_p
et
co
m
an
-e
at
er
m
no
la
ys
_e
gg
animal1
animal2
animal3
animal4
animal5
animal6
animal7
animal8
animal9
animal10
animal11
animal12
animal13
animal14
animal15
animal16
animal17
animal18
animal19
animal20
animal21
animal22
animal23
animal24
animal25
animal26
animal27
animal28
animal29
animal30
animal31
animal32
animal33
animal34
animal35
ct
ur
na
l
11
s
35
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Binary relation is preserved
Weight function from “extent”
The concept lattice is an example of a weighted
lattice (L, w:L[0,1]),
where we use the relative cardinality of domain sets
to define w for C=(H,N):
|  1 (C ) | | extent(C ) | | H |
w(C ) 


|G|
|G|
|G|
View weight (via color) for global view
100
88.5
75
66
50
45
33
25
12.5
10-chain, uniform weights
10-chain, mostly low weights
10-chain, mostly low weights
100
88.5
75
66
50
45
33
25
12.5
10-chain, mostly low weights
100
88.5
75
66
50
45
33
25
12.5
Implications in the labeled lattice
Interpreting: view G as a set of objects, M as a set of
attributes, and I as the satisfaction relation; then
For n, m in M,
nm
iff
({n}’,{n}’’) < ({m}’,{m}’’)
{n}'  {m}': “Every object that satisfies n also satisfies m”
And for A,B subsets of M,
AB
iff
(A’,A’’) < (B’,B’’)
Example: Animals
A Subinterval of the lattice
fourlegged implies airbreather
pet implies warm-blooded
(iguana?)
and
pet and nocturnal implies fur
Association rules = Weak Implications
Oft-seen topic in data mining is the mining of association rules
from a data set (binary relation I)
An association rule is a pair (A,B), with A,B subsets of M,
interpreted to say
“in cases where A holds, B also holds” (weakened implication)
“in the event of A, event B also occurs” (conditional event)
important additional information is needed to evaluate a pair:
Confidence (A,B)
and
Support(A,B)
Support
the function supp:P(M)xP(M)[0,1] outputs
| A'B' |
Supp(( A, B)) 
|G|
[the percent of overall evidence for which the rule is positively
witnessed]
Grocery Example: with G being shoppers and M being
items they may have bought,
supp(beer  pretzels) = 0.22
means that of all grocery shoppers, 22% bought both
beer and pretzels
Confidence
the function conf:P(M)xP(M)[0,1] outputs
| A' B' |
conf (( A, B)) 
| A' |
[the percent of those instances where the hypothesis holds for
which the conclusion also holds]
Grocery Example: conf(beer  pretzels) = 0.84
means that of those shoppers who bought beer,
84% of them also bought pretzels
Use of support & confidence
• user indicates support and confidence thresholds, to
filter the massive output (22|M| rules)
• In practice, setting supp and conf may require trial
and error to find values that give presentable info
• like FCA, this is an exploratory data analysis tool
• e.g., a Boston University CS (Gnu P.L.) tool:
ARMiner (short for Association Rule Miner)
livestockfur, 80% confidence
Identified because of the
similarity in color
between “livestock” and
the concept node below it
Support = 11%
Support of a concept
• Define the support of a concept C=(H,N) (as for w) by
| extent(C ) | |  1 (C ) |
s(C ) 

|G|
|G|
• So that the support of an implication (viewed as an
association rule with 100% confidence) is the support
of its premise
• And the support of an association rule is the support
of the concept through which it is expressed
| A' B' |
Supp(( A, B)) 
 s(C )
|G|
A
B
C
SARS data: 43 x13… 105 concepts
NOTE: 2^13 = 8192
Violent Deaths (MA,2000) data
5% cutoff threshold,
to battle screen
bottleneck
Violent Deaths (MA, 2000) data:
towards OR
Utility as part of the KDD Process
• Needs attention given to data preparation (work)
• Need more attention to training/testing, for built-in
verification of discovered rules
• No domain-specific constructions (advantage ?)
• Does not scale without clustering (universal ?)
Potential Over-sensitivity
Problem: the cardinality of the concept lattice arising
from a given data set can vary drastically with only
minor changes in the data set
This is a problem for applications because
1. noise is often present in data (for many reasons)
2. Changes in data are introduced for analysis
Focus: on lattice diagrams that do not vary much, then
definitions of what that means, when minor
changes occur in the data set
Desire: an idea of the degree to which this is possible
3. Worst case analysis: exponential
a1 a2 a3
g1 1 1 1
g2 1 1 1
g3 1 1 1
g4 1 1 1
g5 1 1 1
g6 1 1 1
..
.
. . . an
..
1
1
1
1
1
1
a1 a2 a3
g1 0 1 1
g2 1 1 1
g3 1 1 1
g4 1 1 1
g5 1 1 1
g6 1 1 1
yields 1-elt lattice
..
.
.
gn 1 1 1
1
gn 1 1 1
. . . an
..
1
1
1
1
1
1
yields 2-chain
.
1
here each change of 1/n2 induces a doubling of lattice cardinality
a1 a2 a3
g1 0 1 1
g2 1 0 1
g3 1 1 1
g4 1 1 1
g5 1 1 1
g6 1 1 1
..
.
gn 1 1 1
. . . an
..
1
1
1
1
1
1
.
1
yields 22
a1 a2 a3
g1 0 1 1
g2 1 0 1
g3 1 1 0
g4 1 1 1
g5 1 1 1
g6 1 1 1
..
.
gn 1 1 1
. . . an
..
1
1
1
1
1
1
.
0
 2n = P({1,2,…,n})
Countries: orgs & weapons
243
13
N
seekN
Afghanistan
0
Albania
0
Algeria
0
Angola
0
Argentina
0
Austria
0
Azerbaijan
0
Bahrain
0
Bangladesh
0
Belarus
0
Belgium
0
Benin
0
Brazil
0
Brunei
0
Bulgaria
0
BurkinaFaso
0
Cameroon
0
Canada
0
Chad
0
Chile
0
China
1
Comoros
0
Cuba
0
CzechRepublic
0
Denmark
0
Djibouti
0
Egypt
0
ElSalvador
0
Ethiopia
0
Finland
0
France
1
Gabon
0
Gambia
0
Germany
0
Greece
0
Guinea
0
Guinea-Bissau
0
Guyana
0
Hungary
0
Iceland
0
India
1
Indonesia
0
Iran
0
Iraq
0
Ireland
0
Israel
1
Italy
0
IvoryCoast-Coted'Ivoire
0
Japan
0
Jordan
0
Kazakhstan
0
Korea(North)
1
Korea(South)
0
Kuwait
0
Kyrgyzstan
0
Laos
0
Lebanon
0
Libya
0
Luxembourg
0
Malaysia
0
Maldives
0
Mali
0
Mauritania
0
Mexico
0
Morocco
0
Mozambique
0
Myanmar
0
Netherlands
0
Nicaragua
0
Niger
0
C
0
0
1
0
0
0
0
0
0
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
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
B
0
0
0
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
1
1
1
0
1
0
0
0
1
1
1
0
1
0
0
0
0
0
0
0
0
0
1
1
1
1
0
1
0
0
0
0
0
1
1
0
0
1
0
1
0
0
0
0
0
0
0
1
1
0
1
0
M
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
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
0
1
0
0
0
0
0
1
1
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
NATO
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
0
0
1
0
0
0
0
0
0
0
0
0
1
0
1
1
0
1
0
0
0
0
0
1
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
OPEC
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
1
1
0
0
0
0
0
1
0
0
1
1
0
0
0
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
EU
0
0
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
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
UNSC
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
1
0
0
1
1
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
OIC
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
1
1
0
0
0
0
0
0
0
0
0
1
0
0
1
0
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
0
0
1
0
0
0
0
0
0
G8
1
1
1
0
0
0
1
1
1
0
0
1
0
1
0
1
1
0
1
0
0
1
0
0
0
1
1
0
0
0
0
1
1
0
0
1
1
1
0
0
0
1
1
1
0
0
0
1
0
1
1
0
0
1
1
0
1
1
0
1
1
1
1
0
1
1
0
0
0
1
SST
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
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
AL
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
0
0
0
0
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
1
0
0
1
1
0
0
0
0
1
0
1
0
0
0
0
0
3159 = 243 x 13 entries,
comparing original with
1% noise, 2% noise
data
# of concepts
Original
65
1% noise
100
= 32 changes
2% noise
140
= 64 changes
animal1
animal2
animal3
animal4
animal5
animal6
animal7
animal8
animal9
animal10
animal11
animal12
animal13
animal14
animal15
animal16
animal17
animal18
animal19
animal20
animal21
animal22
animal23
animal24
animal25
animal26
animal27
animal28
animal29
animal30
animal31
animal32
animal33
animal34
animal35
1
1
1
1
1
data
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
es
ha
s_
fu
r
ha
s_
sc
al
d
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
35
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
17%
44%
1
1
1
1
1
1
1
1
1
1
59
1
1
1
1
1
1
1
1
1
1
1
2% noise
1
1
1
1
# of concepts
41
1
1
1
1% noise
1
1
1
1
1
1
4le
gg
e
-b
lo
od
ed
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Original
1
1
1
1
w
ar
m
ai
r_
br
ea
th
er
liv
liv
es
to
ck
m
on
_p
et
co
m
m
an
-e
at
er
l
ct
ur
na
no
es
_i
n_
w
at
er
Animal context
11
la
ys
_e
gg
s
35
3% noise
1
1
1
1
1
1
62
1
1
1
1
1
1
1
1
1
1
1
5%
77%
Problem 2.
usual presentations of lattices can
obscure weak implications, as node
placement uses covering relations
Recall:
conf(livestockfur)=0.8
Same rule, new diagram
Same rule,
new diagram
Zoom in, two methods
80% of livestock have fur:
confidence(livestockfur) = 0.8
Connected spaces
• Counterexamples in Topology (28 spaces, 10 props.)
• Table of connected spaces modified at 1 posn.
GraphWin: Experimental tool
• Details on this drawing program (largely unfinished)
• constructed to quickly generate multiple alternatives
• to test the range of possibilities and be able to
generate and evaluate criteria for the problem
• Two other excellent drawing programs (also java):
– ConExp (Sergey Yevtushenko): nice interface, useful
controls, available at SourceForge
– LatDrawWin (Ralph Freese): control over attraction and
repulsion, avail. at www.math.hawaii.edu/~ralph/LatDraw
Choices in GraphWin
Vector sums chosen vs. level-wise criteria optimization
Choose representation method, to govern vector sums
• Additive line diagram (ALD) – use positions in
intent(C)
• Use vectors of upper covers, and symmetry (or some
other rule) in meet irreducible cases
Algorithm then uses the covering relation to march down
through the lattice, treating new elements only after
their upper covers are placed
Additive Line diagram for poset P
• Introduce a set representation: order-embedding
(or dual order embedding)
rep : P  P( X )
Can set X=G (X=M), or use irreducible versions
• And a placement of the representation set
vec : X  R 3
• And then position each vector based on that
information:

 
pos( p)    vec( x)  n
 xrep( p )

Upper covers method for P
Place maximal elements on a circle in z = 0,
centered at (0,0,0)
For further lower covers,
1. if element has more than one upper cover, use
vector sum of their positions
2. If element has only one upper cover x, find all
such lower covers of x, and distribute all
uniformly about a circle d units below
Choices in GraphWin
Choose a height dampening method, to provide a zvalue at each vector placement
To each element a we assign
• No dampening
height s to be the length of the
• Longest path layer
longest path to the top (negate)
• Balanced height (Freese)
To a we assign height r-s+k,
• Support
where r is s(0) in [0,a] and k is
s(0) in L
• Log(support+1)
• Weight value (from (L, w:L[0,1])), which includes
f(support) for any order preserving f:[0,1][0,1]
Notice that each choice can be converted into some o.p. w:L[0,1]
Apply weight values
Once all vector positions are determined
w( p)
 (xp , yp , z p )
replace coordinates (xp,yp,zp) by
| zp |
In particular, this forces the new z-value to be w(p)
Choices in GraphWin
Choose level improvements:
Once a level has all its positions given,
• Shift level so that its center of mass matches the center of
mass of all the previous levels’ points, projected into one z=c
plane – this amounts to the xy-position of 1L, (0,0)
•
Dilate (expand/contract) the elements about the center of
mass (e.g. to preserve density)
•
Other local optimizations are available
4. Theoretical examples
To see the difference between
order-theoretic drawing and order+support drawing
• 25 and 24
• A near- Boolean algebra
ConExp on 2^5
LatDrawWin on 2^5
GraphWin on 2^5
distension
ALD vs Upper Covers, no dampening
GraphWin on 2^5
ALD vs Upper Covers, LPL
ALD vs Upper Covers, Freese
GraphWin on 2^5:
ALD vs Upper Covers, support
2^3 x 2^2
ALD vs Upper Covers, LOG(support)
GraphWin on 2^5:
2^3 x 2^2
ALD vs Upper Covers, support
2^4: LatDrawWin
2^4: GraphWin
2^4: GraphWin
2^4: GraphWin
Various forms of BA3
ALD-support
UC-support
Various forms of BA3
ALD-support
UC-support
Various forms of BA3
ALD-support
UC-support
4. Isolated Cluster
As input
No dampening
LatDrawWin
4. Isolated Cluster: ALD, Freese & LPL
4. Isolated Cluster: ALD & UppCov,
Freese
4. Isolated Cluster: ALD, Support &
Log(Support)
4. Isolated Cluster: ALD & UppCov, no
height control
4. Isolated Cluster: UppCov, Support &
Log(Support)
Slight node movement
5. Using the tool on real data sets
Now we consider real data sets
original vs. noisy versions
• Animal context
• Countries: organizations and weapons
some time shifting
• Stock exchanges, 1986
To see how the different lattice drawing methods fare
with the sensitivity problem
Animal context
data
# of concepts
Original
35 x 11 = 385
1% noise
= 4 changes
2% noise
= 8 changes
3% noise
= 12 changes
35
17
41
44
77%
59
5
62
Animal Comparisons UC-BH, orig.
Animal Comparisons UC-BH, 1% noise
Animal Comparisons UC-BH, 2% noise
Animal Comparisons UC-BH, 3% noise
Animal Comparisons
UC-BH: orig, vs 1%, vs 2%
Animal Comparisons
UC-BH: orig, vs 1%, vs 2%
Animal Comparisons
ALD-LPL: orig, vs 1%, vs 2%
Animal Comparisons
ALD-LPL: orig, vs 1%, vs 2%
Animal Comparisons
LatDrawWin: orig, vs 1%, vs 2%
Animal Comparisons
LatDrawWin: orig, vs 1%, vs 2%
Animal Comparisons
ALD-Support: orig, vs 1%, vs 2%
Animal Comparisons
ALD-Support: orig, vs 1%, vs 2%
Animal Comparisons: ALD-Support, orig
Animal Comparisons: ALD-Support, 1%
Animal Comparisons: ALD-Support, 2%
Animal Comparisons: ALD-Support, 3%
Countries: orgs & weapons
243
13
N
seekN
Afghanistan
0
Albania
0
Algeria
0
Angola
0
Argentina
0
Austria
0
Azerbaijan
0
Bahrain
0
Bangladesh
0
Belarus
0
Belgium
0
Benin
0
Brazil
0
Brunei
0
Bulgaria
0
BurkinaFaso
0
Cameroon
0
Canada
0
Chad
0
Chile
0
China
1
Comoros
0
Cuba
0
CzechRepublic
0
Denmark
0
Djibouti
0
Egypt
0
ElSalvador
0
Ethiopia
0
Finland
0
France
1
Gabon
0
Gambia
0
Germany
0
Greece
0
Guinea
0
Guinea-Bissau
0
Guyana
0
Hungary
0
Iceland
0
India
1
Indonesia
0
Iran
0
Iraq
0
Ireland
0
Israel
1
Italy
0
IvoryCoast-Coted'Ivoire
0
Japan
0
Jordan
0
Kazakhstan
0
Korea(North)
1
Korea(South)
0
Kuwait
0
Kyrgyzstan
0
Laos
0
Lebanon
0
Libya
0
Luxembourg
0
Malaysia
0
Maldives
0
Mali
0
Mauritania
0
Mexico
0
Morocco
0
Mozambique
0
Myanmar
0
Netherlands
0
Nicaragua
0
Niger
0
C
0
0
1
0
0
0
0
0
0
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
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
B
0
0
0
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
1
1
1
0
1
0
0
0
1
1
1
0
1
0
0
0
0
0
0
0
0
0
1
1
1
1
0
1
0
0
0
0
0
1
1
0
0
1
0
1
0
0
0
0
0
0
0
1
1
0
1
0
M
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
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
0
1
0
0
0
0
0
1
1
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
NATO
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
0
0
1
0
0
0
0
0
0
0
0
0
1
0
1
1
0
1
0
0
0
0
0
1
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
OPEC
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
1
1
0
0
0
0
0
1
0
0
1
1
0
0
0
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
EU
0
0
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
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
UNSC
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
1
0
0
1
1
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
OIC
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
1
1
0
0
0
0
0
0
0
0
0
1
0
0
1
0
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
0
0
1
0
0
0
0
0
0
G8
1
1
1
0
0
0
1
1
1
0
0
1
0
1
0
1
1
0
1
0
0
1
0
0
0
1
1
0
0
0
0
1
1
0
0
1
1
1
0
0
0
1
1
1
0
0
0
1
0
1
1
0
0
1
1
0
1
1
0
1
1
1
1
0
1
1
0
0
0
1
SST
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
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
AL
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
0
0
0
0
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
1
0
0
1
1
0
0
0
0
1
0
1
0
0
0
0
0
3159=243x13 entries,
comparing original with
1% and 2% noise versions
table
# of concepts
original
65
1% noise
100
2% noise
140
ALD-no dampening
ALD-no dampening, 1% change
ALD-no dampening, 2% change
ALD-no dampening
Orig
65
1% change
100
2% change
140
ALD-support
ALD-support, 1% change
ALD-support, 2% change
ALD-support
Orig
65
1% change
100
2% change
140
ALD-LPL vs BH
Lung Cancer, Bird Keeping
orig
10% noise
Stock Exchanges UP & DOWN ’86-’87
• Now we view six month periods, with only one
week shifts, yielding max. 4% change (5/125)
• ALD-Support (resp ALD-LOG(Support)) does
well in preserving structure
Six month intervals, 1 week shifts
Weeks shifted
0
# concepts
122
1
2
3
122
120
120
4
5
6
118
121
120
7
8
9
10
120
123
128
128
11-14
122
1/1986-6/1986, 0 shifts
100
88.5
75
66
50
45
25
12.5
1/1986-6/1986, 1 shifts
100
88.5
75
66
50
45
25
12.5
1/1986-6/1986, 2 shifts
100
88.5
75
66
50
45
25
12.5
1/1986-6/1986, 3 shifts
100
88.5
75
66
50
45
25
12.5
1/1986-6/1986, 4 shifts
100
88.5
75
66
50
45
25
12.5
1/1986-6/1986, 5 shifts
100
88.5
75
66
50
45
25
12.5
1/1986-6/1986, 6 shifts
100
88.5
75
66
50
45
25
12.5
1/1986-6/1986, 7 shifts
100
88.5
75
66
50
45
25
12.5
1/1986-6/1986, 8 shifts
100
88.5
75
66
50
45
25
12.5
1/1986-6/1986, 9 shifts
100
88.5
75
66
50
45
25
12.5
1/1986-6/1986, 10 shifts
100
88.5
75
66
50
45
25
12.5
1/1986-6/1986, 11 shifts
100
88.5
75
66
50
45
25
12.5
1/1986-6/1986, 12 shifts
100
88.5
75
66
50
45
25
12.5
Compact spaces, noised up
# of concepts
350
300
250
200
# of concepts
150
100
50
0
0
2
4
6
8
10
12
# of errors # of concepts
0
101
1
107
2
114
3
127
4
135
5
140
6
151
7
177
8
225
9
230
10
237
11
311
Compact – ALD-support, 0 errors
Compact – ALD-support, 1 errors
Compact – ALD-support, 2 errors
Compacct – ALD-support, 3 errors
Compact – ALD-support, 4 errors
Compact – ALD-support, 5 errors
Compact – ALD-support, 6 errors
Compact – ALD-support, 7 errors
Compact – ALD-support, 8 errors
Compact – ALD-support, 9 errors
Compact – ALD-support, 10 errors
Compact – ALD-support, 11 errors
Compact spaces UC-support 0 errors
Compact spaces UC-support 1 errors
Compact spaces UC-support 2 errors
Compact spaces UC-support 3 errors
Compact spaces UC-support 4 errors
Compact spaces UC-support 5 errors
Compact spaces UC-support 6 errors
Compact spaces UC-support 7 errors
Compact spaces UC-support 8 errors
Compact spaces UC-support 9 errors
Compact spaces UC-support 10 errors
Compact spaces UC-support 11 errors
Compact spaces, noised up
ALD 0,1,2,3
Compact spaces, noised up
ALD 4,5,6,7
Compact spaces, noised up
ALD 8,9,10,11
Compact spaces, noised up
ALD 12,13,14,15
Compact spaces
UC-support 0,1,2,3
Compact spaces
UC-support 4,5,6,7
Compact spaces
UC-support 8,9,10,11
Compact spaces
UC-support 12,13,14,15
6. Need Criteria
• Need a distance function to measure the overall change in the
lattices when the dataset is changed, drawn using weight
functions
• Idea: take max of radii that can be placed around all nodes of
one lattice so that all nodes of the other are captured within
(compute in both directions)
• Count the balls only according to values of sim:L1xL2[0,1]
• OR: Use these balls to discuss edges; every edge in one lattice
must be between balls with edges in the other
• A function of the radius: the percent of larger lattice covered
by balls of the smaller lattice, in combination with the radii
Shift right lattice onto left lattice
References
R. Agrawal, T. Imielinski, and A. Swami. Mining association rules between sets of
items in large databases. In ACM SIGMOD Intl. Conf. Management of Data,
May 1993.
R. Freese, LatDrawWin.java, at http://www.math.hawaii.edu/~ralph/LatDraw/
B. Ganter and R. Wille, Formal Concept Analysis: Mathematical Foundations,
Springer 1999.
G. Stumme, R. Taouil, Y. Bastide, N. Pasquier, and L. Lakhal, Computing iceberg
concept lattices with Titanic, In Data & Knowledge Engineering, 42 (2002), pp.
189--222.
S. Yevtushenko, http://sourceforge.net/projects/conexp, Release 1.0 (2002); now
Release 1.1 is available (May 2003).
M. Zaki and M. Ogihara, Theoretical Foundations of Associations Rules, In
Proceedings of 3rd SIGMOD'98 Workshop on Research Issues in Data Mining
and Knowledge Discovery (DMKD'98), Seattle, Washington, USA, June 1998.