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## Similarity in CBR Contd

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### Another distance-similarity compatible function is ... Define a formula for the Hamming distance in this context. Tversky Contrast Model ... – PowerPoint PPT presentation

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Title: Similarity in CBR Contd

1
Similarity in CBR (Contd)
• Sources
• Chapter 4
• www.iiia.csic.es/People/enric/AICom.html
• www.ai-cbr.org

2
Simple-Matching-Coefficient (SMC)
n (A D) B C
• H(X,Y)
• Another distance-similarity compatible function
is
• f(x) 1 x/max (where max is the maximum
value for x)
• We can define the SMC similarity, simH

((BC)/n)
Solution (I) Show that f(x) is order inverting
if x lt y then f(x) gt f(y)
3
Simple-Matching-Coefficient (SMC) (II)
• If we use on simH(X,Y) 1- ((BC)/n) factor(A,
B, C, D)
• Monotonic
• If A ? A then
• If B ? B then
• If C ? C then
• If D ? D then

factor(A,B,C,D) ? factor(A,B,C,D)
factor(A,B,C,D) ? factor(A,B,C,D)
factor(A,B,C,D) ? factor(A,B,C,D)
factor(A,B,C,D) ? factor(A,B,C,D)
• Symmetric
• simH (X,Y) simH(Y,X)

Solution(II) Show that simH(X,Y) is monotonic
4
Variations of SMC (III)
• We introduce a weight, ?, with 0 lt ? lt 1

• For which ? is sim?(X,Y) simH(X,Y)?

? 0.5
• sim?(X,Y) preserves the monotonic and symmetric
conditions

Solution(III) Show that sim?(X,Y) is monotonic
5
Homework (Part IV) Attributes May Have multiple
Values
• X (X1, , Xn) where Xi ? Ti
• Y (Y1, ,Yn) where Yi ? Ti
• Each Ti is finite
• Define a formula for the Hamming distance in this
context

6
Tversky Contrast Model
• Defines a non monotonic distance
• Comparison of a situation S with a prototype P
(i.e, a case)
• S and P are sets of features
• The following sets
• A S ? P
• B P S
• C S P

7
Tversky Contrast Model (2)
• Tversky-distance
• Where f Sets ? 0, ?), ?, ?, and ? are
constants
• f, ?, ?, and ? are fixed and defined by the
user
• Example
• If f(A) elements in A
• ? ? ? 1
• T counts the number of elements in common minus
the differences
• The Tversky-distance is not symmetric

T(P,S) ?f(A) - ?f(B) - ?f(C)
8
Local versus Global Similarity Metrics
• In many situations we have similarity metrics
between attributes of the same type (called local
similarity metrics). Example

For a complex engine, we may have a similarity
for the temperature of the engine
• In such situations a reasonable approach to
define a global similarity sim?(x,y) is to
aggregate the local similarity metrics
simi(xi,yi). A widely used practice
• What requirements should we give to sim?(x,y) in
terms of the use of simi(xi,yi)?

sim?(x,y) to increate monotonically with each
simi(xi,yi).
9
Local versus Global Similarity Metrics (Formal
Definitions)
• A local similarity metric on an attribute Ti is a
similarity metric simi Ti ? Ti ? 0,1
• A function ? 0,1n ? 0,1 is an aggregation
function if
• ?(0,0,,0) 0
• ? is monotonic non-decreasing on every argument
• Given a collection of n similarity metrics sim1,
, simn, for attributes taken values from Ti, a
global similarity metric, is a similarity metric
simV ? V ? 0,1, V in T1 ? ? Tn, such that
there is an aggregation ? function with
• sim(X,Y) sim?(X,Y) ?(sim1(X1,Y1),
,simn(Xn,Yn))

Homework provide an example of an aggregation
function and a non-aggregation function and prove
it. Show a global sim. metric
10
Solution
• Suppose that cases use an object oriented
representation
• Suppose that cases use a taxonomical
representation, describe how you would measure
similarity and give a concrete example
illustrating the process you described to measure
similarity
• Suppose that cases use a compositional
representation, describe how you would measure
similarity and give a concrete example
illustrating the process you described to measure
similarity
• Suggestion look at the book!

11
Frontiers of Knowledge
• Dealing with numerical and non numerical values
• Aggregation of local similarity metrics into a
global similarity metric helps
• but sometimes we dont have local similarity
metrics

12
Homework (II)
• From Chapter 5, what is the difference between
completion and adaptation functions? What si
their role on adaptation? Provide an example
• Show that Graph coloring is NP-complete
• Assume that Constraint-SAT is NP complete
• Definition. A constraint is a formula of the
form
• (x y)
• (x ? y)
• Where x and y are variables that can take values
from a set (e.g., yellow, white, black, red, )
• Definition. Constraint-SAT given a conjunction
of constraints, is there an instantiation of the
variables that makes the conjunction true?