2. What is Fuzzy?
Vague, Blurred, Indistinct
• Fuzzy Logic may be viewed as an
attempt to deal with imprecision, just
as probability theory deals with
uncertainty.
• Covers the wide gulf between
Newtonian Mechanics and Statistical
Mechanics.
3. Fuzzy set theory:
• Unlike classical set theory, the boundaries are
not sharp.
• The membership function of a fuzzy set A has
the form
• So, For each x ϵ X, µA(x) express the degree of
membership of element x of X in fuzzy set A.
4. • An Example of Fuzzy set in classification of
sedimentary particles :
• The trapezoids here represent membership
functions.
5. Fuzzy Relations
• Fuzzy Relations are Cartesian products of
two or more sets, defined on the Universal
set. So a fuzzy relation R is thus defined by a
membership function of the general form
R(x): X1 x X2 x……..x Xn → [0,1]
• Individual sets in the Cartesian product are
called dimensions of the relation. Here it is
n-dimensional.
6. To implement Fuzzy Logic….
• Establishment of connection between degrees
of membership in fuzzy sets and degrees of
truth of fuzzy propositions.
• When X denote the Universal set of the
common objects, then a proposition takes the
form
where χ is a variable whose range is X and A is
a fuzzy set.
7. • Let pA(x) denote the degree of truth of the
respective proposition and is called as a Truth
Assignment function.
• So for a fuzzy set A its membership degree
A(x) for each x ϵ X, is the degree of truth of
the proposition
pA : χ is a member of A
for the same x ϵ X.
i.e. now pA (x) = A(x)
• Degree of membership and Degree of truth for
the same objects are numerically equal, when
they are connected.
8. • As a consequence logic operations can be
defined:
NEGATIONComplementation { Not, Neither
nor, Not both}
CONJUNCTIONIntersection { And, And then,
However}
DISJUNCTIONUnion {Or, Either or}
• Fuzzy System is a system whose variables (or
at least some of them) range over states that
are fuzzy sets.
9. Applications of Fuzzy Systems
Sendai (JAPAN) subway system
Other applications include Washing machines, Cooker, Air
Conditioner, Camera etc.
10. Applications to Geological Systems
• Stratigraphic modeling
• Hydro-climatic modelling.
• Earthquake research
Some instances: Bivalent logic is necessary &
sufficient.
Fuzzy logic is generalization of multi-valued
logic.
“Crisp sets are special fuzzy sets”
11. Formal Concept Analysis(FCA)
• Analysis of OBJECT-ATTRIBUTE data.
• Set X = Objects
Set Y= Attributes
Relation I between X & Y
• I(x,y) is Truth degree ; x ϵ X and y ϵ Y
• The triple <X,Y,I> is the Input data that can be
represented by a table.
12. Implementations of FCA
1. Discovers natural concepts hidden in the
data.
2. Determines hierarchy of discovered concepts.
3. Attribute dependencies.
4. Fuzziness & Similarity issues.
FCA generates only basic implications.
13. Concept lattice
• Previously it was seen that <X,Y,I> is the Input data for
FCA.
So ß(X,Y,I) is the concept lattice.
[ Lattice is a partially ordered set in which every two
elements have a unique Least upper bound (join) and a
unique Greatest lower bound(meet).]
Say, for a lattice L={a1,a2,….,an}
˅L= a1˅……..˅an and
˄L= a1˄……..˄an
are the join and meet of all elements respectively.
• So (L,˅, ˄) is a lattice in algebraic sense.
• A Bounded Lattice has a greatest element 1 and least
element 0.
14. Morphism of lattices
• Given 2 lattices (L,˅L,˄L) & (M,˅M,˄M) ; a
function f from L to M is a function
f : L→M such that
for all a,b ϵ L
f(a ˅L b )= f(a) ˅M f(b) & f(a ˄L b)= f(a) ˄M f(b)
Thus f is homomorphism of the lattices L & M.
• If L and M are bounded lattice then
f(0L)= 0M and f(1L)= 1M
should also be satisfied for homomorphism.
15. A case study
Taken 9 fossils belonging to some general category say C. Suppose we have no previous
knowledge about their natural categories(subcategories of C) exist.
16. 9 fossils
i.e. Objects= 9; assigned numbers 1 to 9
Size of Spine(X)
ss (short spine)
sb (big spine)
Shape of body(Y)
cs (circle shaped)
os (oval shaped)
Attributes :
Now an appropriate set of truth values should be taken and equip it with an
appropriate structure defined on L.
22. Belemnite Evolution
• Belemnite taxonomy is done based on
- Structure of the alveolar end.
- Shape & size of the rostrum.
- Internal structures at alveolar end.
- External characteristics of the rostrum.
-Structure of the apex.
23. a,b,…, α,β,…λ are attributes. And 1,2….,26 are different Belemnite species
Formal context
24. Conclusions from analyzed data:
• Belemnites are polyphyletic. (contradiction)
• Three parallel evolutions are inferred.
(contradiction)
• Genus origin : Goniocamax. (confirmed later
by phylogenetic analysis)
25. Reference:
• Robert V. Demicco and George J. Klir ; Fuzzy
Logic in Geology 2004
• L.A. Zadeh (1965); Fuzzy sets
• R. Belohlavek, M Kostak, P. Osicka; FCA with
background knowledge : a case study in
paleobiological taxonomy of belemnites
• W. Weaver (1948); Science and complexity