Mathematics of Incidence (part 4): Lattice dependencies
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Transcript of Mathematics of Incidence (part 4): Lattice dependencies
Mathematics of Incidencepart 4: Lattice dependencies
Benjamin J. Kellerbjkeller.github.iolinkedin.com/in/bjkeller
v.1.1, 7 November 2014
Creative Commons Attribution-ShareAlike 4.0 International License
ma1 a2 a3
g1
g2 g3 g4
g5 g6 g7
g8 g9 g10
a u1 u2 uk
s1s2sr(a) tsl
un(a)
�
⟘
⟙
�
� �
Recall: CF and dependency
• Recommend t to a by composing formal concepts for selection of users {ui } and items { si }
• Correspond to filter and ideal that capture dependencies
• Intuition: good recommendation determined by “strong” dependency
• [note: ideal and filter are dual, so good enough to define for one b/c other will be same]
Independence in concept lattice
apples doughnuts
Abby XDavid X
doughnutsapples
DavidAbby
no shared objects/attributes
Definition: anti-chain
doughnutsapples
DavidAbby
An anti-chain in lattice is a subset of elementsL S ⇢ L
p, q 2 S p 6 q q 6 pwhere for all both and
Dependenciesapples
Abby XBrian X
apples
Abby, Brian
apples cherries
Abby XBrian X X
apples
Abby
cherries
Brian
apples cherries grapes
Abby X XBrian X X
apples
grapes
Abby
cherries
Brian
Definition: chain
A chain in lattice is a subset of elementsL S ⇢ L
p, q 2 Swhere for all either orp q q p
apples
Abby
cherries
Brian
apples
grapes
Abby
cherries
Brian
bananas
doughnutsapples
eggscherries
Charles
DavidBrian
Abby
Definition: join and meet
For lattice elements p, q 2 L
the join of and is the least elementp q p _ q 2 L
p p _ q q p _ qsatisfying both and
the meet of p ^ q 2 Land is the greatest elementp q
p ^ q p p ^ q qsatisfying both and
(In)dependence
apples
grapes
Abby
cherries
Brian
�(Abby) _ �(Brian)
µ(grapes) ^ µ(cherries)
(dependent)
(independent)
Is there a shared object/attribute?
Strong Dependency
• Not clear what "strong" means
• Intuitively, Abby and Brian would have a "stronger" dependency if they shared "enough" common likes
• What does lattice look like if we add more common likes?
apples
grapes
Abby
cherries
Brian
Strong Dependency
• Adding common likes just increases number of items labeling top of lattice
• Lattice structure doesn't change
• These attributes are redundant from the perspective of constructing the lattice
applesbananas
doughnutseggs
flautas
grapes
Abby
cherriesBrian
Repeating is redundant
apples
Abby
apples
Abby, Brian
apples
Abby, Brian,Charles
apples
Abby XBrian X
apples
Abby XBrian XCharles X
apples
Abby X
apples cherries
Abby XBrian X X
apples bananas cherries
Abby X XBrian X X X
apples,bananas
Abby
cherries
Brian
apples
Abby
cherries
Brian
cherries
Abby
Brian X
Abby
cherries
Brian
Strong Dependency
• FCA is the wrong math to define strength of dependency
• Concept lattice only tells us independent or not
• Can construct a minimal (reduced) context that maintains lattice structure without redundancies
• "Strength" basically means mutually informative
• Information constrained by lattice structure, but determined by counts
• [Topic for a different set of slides…]
Attribute implications
A ! B A,B ✓ MDefine implication forµA � µBwhenever
Rules defined this way capture same relationships as lattice – some trivial, some more interesting
Can be read as “every object with attributes in A, also has attributes in B”
apples bananas cherries
Abby XBrian X XCharles X X X
apples
Abby
bananas
Charles
cherries
Brian
cherries ! cherries
cherries ! bananas
cherries ! apples
bananas ! bananas
bananas ! apples
apples ! apples
cherries, bananas ! apples
m a1 a2 a3
g1
g2 g3 g4 g5
m a1 a2 a3
g1 X X X Xg2 X X Xg3 X X Xg4 X X Xg5 X X X
A = {a1, a2, a3}
B ! ai
B [m ! m
Let
then have implications
for
B ✓ Afor
ai 2 B ✓ A
m a1 a2 a3
g1 X X X Xg2 X X Xg3 X X Xg4 X X X
ai, aj 2 A, i 6= j
aj ! aj
ai, aj ! aj
m ! m
aj ,m ! m
ai, aj ,m ! m
a1, a2, a3 ! m
A = {a1, a2, a3}Let
then have implications
but also have
m
g1
g2 g3 g4
m a1 a2 a3
g1 X X X Xg2 X X Xg3 X X Xg4 X X Xg5 Xg6 Xg7 Xg8 X Xg9 X Xg10 X X
m a1 a2 a3
g1
g2 g3 g4
g5 g6g7
g8 g9 g10
Yields the same implications including
a1, a2, a3 ! m
Flavors of implicationsImplication is
bananas , cherries ! bananas
cherries ! bananas
A ! B
B ✓ A
B 6✓ A
A,B ✓ Mfor where either
read as people who like bananas and something else, like bananas
read as people who like cherries like bananas
First is boring, second corresponds to association rules
Pointer: Association Rules
A,B ✓ MA ) BAn association rule is a pair of setsB 6= ; A \B = ;such that and
| A [ B|| B|
Evaluated in terms of confidence:
FCA used for frequent item set mining to find association rules
Defintion: Interval
Let be a poset, then the interval foris
(P,) p, q 2 P
[p, q] = {r 2 P | p r q}
[ponder: how would you write a principal filter or ideal as an interval?]
Intervals and implicationsa1, a2, a3 ! mThe implication corresponds to
[µ{a1, a2, a3}, µm] whereµ{a1, a2, a3} = µ{a1, a2, a3,m}interval
m a1 a2 a3
g1
g2 g3 g4
Abby and those pesky doughnuts
bananas
doughnutsapples
eggscherries
Charles
DavidBrian
Abby
cherries ! apples
apples ! bananas
doughnuts ! bananas
cherries ! doughnuts
eggs ! doughnuts
A minimal set of implications
Shorter chains seem better, but are these rules enough to say what is a good recommendation?
Questions to ponder
• What information does a formal concept carry? Can we use a measure of it to define strong dependency?
• How does strength of dependency relate to implications?
• We can recommend doughnuts and cherries to Abby, but which first?
Corrections
11.7.14: had drawn lattice on slide 16 incorrectly
About me and these slides
I am Ben(jamin) Keller. I learn and, sometimes, create through explaining. I had been involved in a big (US) federally funded project that had the goal of helping biomedical scientists tell stories about their experimental observations. The project is long gone, but I’m still trying to grok how such a thing would work. Much of biological data comes in the form of observations that are distilled to something that looks like an incidence relation, which brings us to this series of presentations. My goal for the slides is to deal with the mathematics and analysis of incidence in an approachable way, but the intuitive beginnings will eventually allow us to embrace the more complex later.
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0
International License.