IDSS: Overview of Themes

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IDSS Overview of Themes

• AI
• Introduction
• Overview
• IDT
• Attribute-Value Rep.
• Decision Trees
• Induction
• CBR
• Introduction
• Representation
• Similarity
• Adaptation
• Rule-based Inference Expert Systems
• Computational Complexity
• AI Method Synthesis Tasks
• AI Planning

• Uncertainty (MDP, Utility,
• Fuzzy logic)

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• Applications to IDSS
• Analysis Tasks
• Help-desk systems
• Classification
• Diagnosis
• Prediction
• Design
• Textual CBR
• Synthesis Tasks
• KBPP
• Configuration
• Software Eng.
• E-commerce
• Knowledge Management

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Similarity in CBR

• Sources
• Chapter 4
• www.iiia.csic.es/People/enric/AICom.html
• www.ai-cbr.org

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Computing Similarity

• Similarity is a key (the key?) concept in CBR

• We saw that a case consists of

• Problem
• Solution
• Adequacy

• We saw that the CBR problem solving cycle
  consists of

• Retrieval
• Reuse
• Revise
• Retain

• We will distinguish between
• Meaning of similarity
• Formal axioms capturing this meaning

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Meaning of Similarity

• Observation 1 Similarity always concentrates on
  one aspect or task
• There is no absolute similarity
• Example

• Two cars are similar if they have similar
  capacity (two compact cars may be similar to each
  other but not to a full-size car)
• Two cars are similar if they have similar price
  (a new compact car may be similar to an old
  full-size car but not to an old compact car)

• When computing similarity we are doing some sort
  of abstraction of the cases

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Meaning of Similarity (2)

• Observation 2 Similarity is not always
  transitive
• Example

• I define similar to mean within walking
  distance
• Lehighs book store is similar to Café Havana
• Café Havana is similar to Perkins
• Perkins is similar to Monrovia book store
• But Lehighs book store is not similar to
  Best Buy in Allentown !

• The problem is that the property small
  difference cannot be propagated

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Meaning of Similarity (3)

• Observation 3 Similarity is not always
  symmetric
• Example

• Mike Tyson fights like a lion

• But do we really want to say that a lion fights
  like Mike Tyson?

• The problem is that in general the distance from
  an element to a prototype of a category is larger
  than the other way around

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Similarity and Utility in CBR

• Utility measure of the improvement in efficiency
  as a result of a body of knowledge (Well come
  back to this point)

• The goal of the similarity is to select cases
  that can be easily adapted to solve a new problem

Similarity Prediction of the utility of the case

• However
• The similarity is an a priori criterion
• The utility is an a posteriori criterion

• Ideal Similarity makes a good prediction of the
  utility

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Axioms for Similarity

• There are 3 types of axioms
• Binary similarity predicate x and y are similar
• Binary dissimilarity predicate x and y are
  dissimilar
• Similarity as order relation x is at least as
  similar to y as it is to z

• Observation
• The first and the second are equivalent
• The third provides more information grade of
  similarity

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Similarity Relations

• We want to define a relation
• R(x,y,z) iff x is at least
  as similar to y as it is to z
• First lets consider the following relation
• S(x,y,u,v) iff x is at least as
  similar to y as u is similar to v

• Definition of R in terms of S

R(x,y,z) iff S(x,y,x,z)
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Similarity Relations (2)

• Possible requirements on the relation S
• S(x,x,u,v)
• S(x,y,y,x)
• S(x,y,u,v) S(u,v,s,t) ? S(x,y,s,t)
• S(x,y,u,v) iff S(y,x,u,v) iff S (x,y,v,u)

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Similarity Relations (3)

• In CBR we have an object x fixed when computing
  similarity. Which x?

The new problem

• We are looking for a y such that y is the most
  similar to x. In terms of R this be seen as

? z R(x,y,z)

• Given a problem x we can define an ordering
  relation ?x as follows
• y ?x z iff R(x,y,z)
• y gtx z iff (y ?x z and z ?x y)
• y x z iff (y ?x z and z ?x y)

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Similarity Metric

• We want to assign a number to indicate the
  similarity between a case and a problem

• Definition A similarity metric over a set M is a
  function
• sim M ? M ? 0,1
• Such that
• For all x in M sim(x,x) 1 holds
• For all x, y in M sim(x,y) sim(y,x)

the closer the value of sim(x,y) to 1, the more
similar is x to y
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Similarity Metric (2)

• Given a similarity metric sim M ? M ? 0,1, it
  induces a similarity relation Ssim (x,y,u,v) and
  ?x as follows

Ssim(x,y,u,v) iff sim(x,y) ? sim(u,v) y ?x z
iff sim(x,y) ? sim(x,z)
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Distance Metric

• Definition A distance function over a set M is a
  function
• d M ? M ? 0,?)
• Such that
• For all x in M d(x,x) 0 holds
• For all x, y in M d(x,y) d(y,x)

• Definition A distance function over a set M is a
  metric if
• For all x, y in M d(x,y) 0 holds then x y
• For all x, y, z in M d(x,z) d(z,y) ? d(x,y)

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Relation between Similarity and Distance Metric

• Given a distance metric, d, it induces a
  similarity relation Sd(x,y,u,v), ?x as follows

• For all x, y, u, v S(x,y,u,v) holds if
• For all x, y, z y ?x z if

d(x,y) ? d(u,v)
d(x,y) ? d(x,z)
Definition A similarity metric sim and a
distance metric d are compatible iff for
all x,y, u, v Sd(x,y,u,v) iff Ssim(x,y,u,v)
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Relation between Similarity and Distance Metric
(2)

• Property Let
• f 0,?) ? (0,1
• Be a bijective and order inverting (if ult v then
  f(v) lt f(u)) function such that
• f(0) 1
• f(d(x,y)) sim(x,y)
• then d and sim are compatible

If d(x,y) lt d(u,v) then sim(x,y) gt sim(u,v)
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Relation between Similarity and Distance Metric
(3)

• F(x) can be used to construct sim giving d.
  Example of such a function is
• if you have the Euclidean distance
• d((x,y),(u,v))
  sqr((x-u)2 (y-v)2)
• Since f(x) 1 (x/(x1)) meets the property
  before
• Then
• sim((x,y),(u,v))) f(d((x,y),(u,v)))
• 1
  (d((x,y),(u,v)) /(d((x,y),(u,v)) 1))
• is a similarity metric

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Relation between Similarity and Distance Metric
(3)

• The function f(x) 1 (x/(x1)) is a bijective
  function from 0,?) into (0,1

1
0
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Homework (Oct 23)

• Find another order-inverting function and prove
  it