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?