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