similarity in cbr
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Similarity in CBR. Sources: Chapter 4 www.iiia.csic.es/People/enric/AICom.html www.ai-cbr.org. similarity. similarity. Computing Similarity. Similarity is a key (the key?) concept in CBR. We saw that a case consists of:. Problem Solution Adequacy. - PowerPoint PPT PresentationTRANSCRIPT
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 concentrating on one such aspect or aggregating several such aspects
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 “Lupita”• “Lupitas” 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 x is
similar 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:
1. Reflexive: S(x,x,u,v)
2. Symmetry: S(x,y,y,x)
3. Transitivity: S(x,y,u,v) & S(u,v,s,t) S(x,y,s,t)
4. Symmetry: 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:
sim(x,y) sim(u,v)
sim(x,y) sim(x,z)
• sim provides a quantitative value for similarity:
0 1y1 y2 y3 y4sim(x, yi)
Thus y4 is more similar to x
For all x, y, u, v: Ssim (x,y,u,v) holds if
For all x, y, z: y x z if
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
Other Similarity Metrics
• Suppose that we have cases represented as attribute-value pairs (e.g., the restaurant domain)
• Suppose initially that the values are binary
• We want to define similarity between two cases of the form:
X = (X1, …, Xn) where Xi = 0 or 1
Y = (Y1, …,Yn) where Yi = 0 or 1
PreliminariesLet:
A = (i=1,n)Xi•Yi
B = (i=1,n)Xi•(1-Yi)
C = (i=1,n)(1-Xi)•Yi
D = (i=1,n)(1-Xi) •(1-Yi)
Then, A + B + C + D =
(number of attributes for which Xi =1 and Yi = 1)
(number of attributes for which Xi =1 and Yi = 0)
(number of attributes for which Xi =0 and Yi = 1)
(number of attributes for which Xi =0 and Yi = 0)
n A+D =B+C=
“matching attributes”“mismatching attributes”
Hamming Distance
H(X,Y) = n – (i=1,n)Xi•Yi – (i=1,n)(1-Xi)•(1-Yi)
Properties:
Range of H:H counts the mismatch between the attribute valuesH is a distance metric:
H((1-X1, …, 1-Xn), (1-Y1, …,1-Yn)) =
[0,n]
• H(X,X) = 0• H(X,Y) = H(Y,X)
H((X1, …, Xn), (Y1, …,Yn))
Simple-Matching-Coefficient (SMC)
H(X,Y) = n – (A + D) = B + C
• 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 – (A+D))/n) = (A+D)/n = 1- ((B+C)/n)
Proportion of the difference
# of mismatches
Simple-Matching-Coefficient (SMC) (II)
• If we use on simH(X,Y) = (A+D)/n =1- ((B+C)/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)
Variations of the SMC• The hamming similarity assign equal value to matches (both 0 or
both 1)
• There are situations in which you want to count different when both match with 1 as when both match with 0
Thus, sim((1-X1, …, 1-Xn), (1-Y1, …,1-Yn)) = sim((X1, …, Xn), (Y1, …,Yn)) may not hold
Example: Two symptoms of patients are similar if they both have fever (Xi = 1 and Yi = 1) but not similar if neither have fever (Xi = 0 and Yi = 0)
Specific attributes may be more important than other attributesExample: manufacturing domain: some parts of the workpiece are more important than others
Variations of SMC (III)
• We introduce a weight, , with 0 < < 1:
• simH(X,Y) = (A+D)/n = (A+D)/(A+B+C+D)
sim(X,Y) = ((A+D))/ ((A+D) + (1 - )(B+C))
For which is sim(X,Y) = simH(X,Y)? = 0.5
sim(X,Y) preserves the monotonic and symmetric conditions
The similarity depends only from A, B, C and D (3)
• What is the role of ? What happens if > 0.5? If < 0.5?
sim(X,Y) = ((A+D))/ ((A+D) + (1 - )(B+C))
1
00 n
= 0.5 > 0.5
< 0.5
• If > 0.5 we give more weights to the matching attributes
• If < 0.5 we give more weights to the miss-matching attributes
Discarding 0-match
• Thus, sim((1-X1, …, 1-Xn), (1-Y1, …,1-Yn)) = sim((X1, …, Xn), (Y1, …,Yn)) may not hold
• Only when the attribute occurs (i.e., Xi = 1 and Yi = 1 ) will contribute to the similarity
Possible definition of the similarity:sim = A / (A+ B+C)
Specific Attributes may be More Important Than Other Attributes
• Significance of the attributes varies
• Weighted Hamming distance:
HW(X,Y) = 1 – (i=1,n) i • Xi•Yi – (i=1,n) i • (1-Xi)•(1-Yi)
There is a weight vector: (1, …, n) such that
(i=1,n) i = 1
• Example: “Process planning: some features are more important than others”
Non Monotonic Similarity
• The monotony condition in similarity, formally, says that:
sim(A,B) sim(A’,B) always holds if A counts the number of matches and A A’
• Informally the monotony condition can be expressed as: For any X, Y, X’ attribute-value vectors, If we obtain X’ by modifying X on the value of one attribute such that X’ and Y have the same value on that attribute then:sim(X,Y) sim(X’,Y)
Non Monotonic Similarity (2)
simH(X,Y) = (i=1,n)eq(Xi,Yi) / n
Is the hamming distance monotonic? Yes
Consider the XOR function:
(0,0) and (1,1) are on the same class (+)(0,1) and (1,0) are on the same class (-)Thus d((1,1),(1,0)) > d((1,1),(0,0))Is this monotonic? No
Non Monotonic Similarity (3)• You may think: “well that was mathematics, how about real
world?”
• Suppose that we have two interconnected batteries B and B’ and 3 lamps X, Y and Z that have the following properties:
If X is on, B and B’ work If Y is on, B or B’ work If Z is on, B works
1 0 1 1 Ok Fail2 0 1 0 Fail Ok3 0 0 0 Fail Fail
Situation X Y Z B B’ Thus:• sim(1,3) > sim(1,2)• Non monotonic!
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
A
S P
C B
Tversky Contrast Model (2)
• Tversky-distance:
• Where f: [0, )
• f, , , and are fixed and defined by the user
• Example: If f(A) = # elements in A = = = 1T 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)
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
sim(x,y) to increate monotonically with each simi(xi,yi).
• What requirements should we give to sim(x,y) in terms of the use of simi(xi,yi)?
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 : [0,1]n [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 sim:V 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))(X1,X2,…,Xn) = (X1+X2+…+Xn)/nExample:
Example• Cases may contain attributes of type:
– real number A: the voltage output of a device • define a local similarity metric, simvoltage()
– Integer B: revolutions per second • define a local similarity metric, simrps()
– A bunch of symbolic attributes m = (C1,..,Cm): front light blinking or none, year of manufacture, etc • define a Hamming similarity, simH(), combining all
these attributes• Define an aggregated similarity sim() metric:sim(C,C’) = 1 *simvoltage(A,A’) + 2 *simvoltage(A,A’) + 3 *simH(m, m’)
Homework (1 of 2)1. In Slide 12 we define the similarity relation Ssim(x,y,u,v).
Which of the 4 kinds of relations defined in Slide 9 are satisfied by Ssim(x,y,u,v)?
2. Let us define: SH(x,y,u,v) iff H(x,y) H(u,v)
where H is the Hamming distance (defined in Slide 20). Which of the 4 kinds of relations defined in Slide 9 are satisfied by SH(x,y,u,v)?
3. Let us define: ST(x,y,u,v) iff T(x,y) T(u,v)
where T is the Tversky Contrast Model (defined in Slide 31). Which of the 4 kinds of relations defined in Slide 9 are satisfied by ST(x,y,u,v)?
Homework (2 of 2)
4.
• X = (X1, …, Xn) where Xi Ti
• Y = (Y1, …,Yn) where Yi Ti
• Each Ti is finite
Define a formula for the Hamming distance when the attributes are symbolic but may take more than 2 values: