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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

simH(X,Y) 1 ((n (AD))/n) (AD)/n 1-
((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)
  • simH(X,Y) (AD)/n (AD)/(ABCD)
  • We introduce a weight, ?, with 0 lt ? lt 1

sim?(X,Y) (?(AD))/ (?(AD) (1 - ?)(BC))
  • 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?
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