IDSS: Overview of Themes•AI
Introduction Overview
•IDTAttribute-Value Rep.Decision TreesInduction
•CBRIntroductionRepresentationSimilarityAdaptation
•Rule-based Inference & Expert Systems
•Computational Complexity•AI Method: Synthesis Tasks
AI Planning
•Uncertainty (MDP, Utility, Fuzzy logic)
•Applications to IDSS:Analysis Tasks
Help-desk systemsClassificationDiagnosisPredictionDesignTextual CBR
Synthesis TasksKBPPConfigurationSoftware Eng.
E-commerceKnowledge Management
Similarity in CBR
Sources:–Chapter 4
–www.iiia.csic.es/People/enric/AICom.html
–www.ai-cbr.org
Computing Similarity
•Similarity is a key (the key?) concept in CBR
We saw that a case consists of:
We saw that the CBR problem solving cycle consists of:
similarityProblemSolutionAdequacy
Retrieval ReuseReviseRetain
similarity
•We will distinguish between: Meaning of similarityFormal axioms capturing this meaning
Meaning of Similarity
Observation 1: Similarity always concentrates on one aspect or task:
There is no absolute similarityExample:
•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
Meaning of Similarity (2)
Observation 2: Similarity is not always transitive:
Example:
I define similar to mean “within walking distance”
•“Lehigh’s book store” is similar to “Café Havana”•“Café Havana” is similar to “Perkins”•“Perkins” is similar to “Monrovia book store”•…•But: “Lehigh’s book store” is not similar to “Best Buy” in Allentown !
The problem is that the property “small difference” cannot be propagated
Meaning of Similarity (3)
Observation 3: Similarity is not always symmetric:
Example:
The problem is that in general the distance from an element to a prototype of a category is larger than the other way around
• “Mike Tyson fights like a lion”
• But do we really want to say that “a lion fights like Mike Tyson”?
Similarity and Utility in CBR
•Utility: measure of the improvement in efficiency as a result of a body of knowledge (We’ll 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
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
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)
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)
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 >x z iff (y x z and ¬ z x y)
y ~x z iff (y x z and z x y)
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 holdsFor 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”
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)
•sim provides a quantitative value for similarity:
0 1y1 y2 y3 y4
sim(x, yi)
Thus y4 is more similar to x
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 holdsFor 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 = yFor all x, y, z in M: d(x,z) + d(z,y) d(x,y)
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
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)
d(x,y) d(u,v)
d(x,y) d(x,z)
Relation between Similarity and Distance Metric (2)
Property: Let f: [0,) (0,1]Be a bijective and order inverting (if u< v then f(v) < 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) < d(u,v) then sim(x,y) > sim(u,v)
f(d(x,y)) > f(d(u,v))
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/(x+1)) 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
Relation between Similarity and Distance Metric (3)
•The function f(x) = 1 – (x/(x+1)) is a bijective function from [0,) into (0,1]:
0
1
Homework (Oct 23)
• Find another order-inverting function and prove it
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