Chapter 6 Similarity Stand 20.12.00. - 2 - (c) 2000 Dr. Ralph Bergmann and Prof. Dr. Michael M....

Chapter 6

Similarity

Stand 20.12.00

- 2 - (c) 2000 Dr. Ralph Bergmann and Prof. Dr. Michael M. Richter, Universität Kaiserslautern

Recommended References

• Stefan Wess (1995). Fallbasiertes Problemlösen in wissensbasierten Systemen zur Entscheidungsunterstützung und Dianostik. DISKI 126, infix-Verlag. Kapitel 6.2 (S. 43-68 und S. 124-132).

• Bergmann, R. & Stahl, S. (1998). Similarity Measures for Object-Oriented Case Representations. Proceedings of the European Workshop on Case-

Based Reasoning, EWCBR'98. • Althoff, K.-D, Richter, M. M. (1999).: Similarity and Utility in Non-Numerical

Domains, Mathematische Methoden der Wirtschaftswissenschaften, Physika-Verlag, pp. 403 – 413.http://www.cbr-web.org/documents/RichterSimilarity99.pdf

• Richter, M. M. (1997).: A Note on Fuzzy Set Theory and Case Based Reasoning, Technical Report, International Computer Science Institute, University of California


The General Idea of Similarity (1)• Similarity can be defined on any set U and it relates any

two objects of U.• The idea is when one is interested in some x U and this

x is not available (for whatever reason) then some y U which is similar to x might do as well. That this is really the case needs of course a suitable definition of the specific notion of similarity used.

• This means: One has always to define the semantics of the similarity notion in use.

• This semantics has is always related to what the similarity is used for.

• Hence similarity is related to utility.


The General Idea of Similarity(2)

• Although similarity was first used in CBR to make use of previous experiences it applies to many more situations.

• The uses of applications of similarity often have little or nothing to do with the intuitive meaning of similarity !!

• Similarity is closely connected to search and retrieval: One starts a search for x and may end up with some “similar” y.

• This search is knowledge based and the knowledge is contained in the definition of the specific similarity notion.


Similarity in E-C

• In E-C similarity enters the scenario always when inexact

matching is involved. Therefore similarity will be defined between– products and products

– functionalities and products

– demands and products

– demands and demands

– functionalities and functionalities

– customers and customers

• The purpose is always to replace the exact solution of some

problem by some sufficiently good approximation.


The Intention of: Demand and Product are Similar (1)

• The basic intention: The product realizes the demand.

• The relation „Product P satisfies demand F“ is however

not a yes-or-no relation but a more-or-less property.

• The notion of similarity is intended to represent the

degree of satisfaction.

• The relation „Product P satisfies demand F“ requires

additional knowledge which has to be represented in the

similarity notion.

• The difficulty involved is there is a gap between demand

and product in the descriptions (see chapter 9).


The Intention of: Demand and Product are Similar (2)

• The demands are driven by the utility of the customer. Hence customer prefers a product which is more useful.

• The similarity notion has at run time no access anymore to the users utility function and is therefore an a-priori approximation of the utility (see chapter 9).

• A consequence of similarity based offers: The

customer obtains always an answer! (Because there

is always a nearest neighbor...)

• Restriction: One can use the method of rough sets in order to describe acceptable and non-acceptable offers.


Other Applications of Similarity in E-C

• The statement that two objects are similar is always followed by some action, e.g.:– some product may be replaced by a similar product– some product may realize a “similar” functionality– what was good for some demand may also be good to a

similar demand– if a product realizes some functionality it may also realize a

similar functionality– what was appropriate to some customer may also be

appropriate to a similar customer

• In each situation the notion of being similar has to carry some knowledge about the intended actions.


Similarity in CBR

• Purpose of similarity in CBR is analogue to the one in E-C: – Select cases that can be adapted easily to the current problem– Select cases that have (nearly) the same solution than the current

problem

• Basic goal: similar problems have similar solutions• Degree of similarity = utility / reusability of solution• Similarity is an a-priori approximation of utility / reusability• Goal of similarity modeling: provide a good approximation

– close to real reusability– easy to compute


Properties of Similarity (I)

• Observation 1:

Similarity is not necessarily transitive.• Example 1:

+ Kaiserslautern and Zürich are similar because both cities have (good) soccer teams.

+ Zürich and Basel are similar because both cities are located in Switzerland.

- But: Kaiserslautern and Basel are not similar!

• Problem:

Similarity between different aspects.

~


Properties of Similarity (II)

• Observation 2:

Similarity is not necessarily transitive.

• Example 2:

+ $ 10.- is similar to $ 12.-

+ $ 12.- is similar to $ 14.-

......

+ $ 1000.- is similar to $ 1002.-

- But: $ 10.- is not similar to $ 1002.-!

• Problem:

The property “low difference” is not a transitive notion.

…

~


Properties of Similarity (III)• Observation 3:

Similarity does not have to be symmetrical.• Example:

– “Peter is fighting like a lion.” With respect to certain properties (e.g., courage), Peter (as an element) is similar to a lion (as a prototype).

– “A lion is fighting like Peter.” This is an untypical statement. Peter is not a prototype for courage.

• Problem: Similarity of one element to a prototype of a category is greater than vice versa.

~


Possible Ways to Define Similarity

• Three ways to define similarity:

i) binary similarity predicate:“x and y are similar”

ii) binary dissimilarity predicate:

“x and y are dissimilar”

iii) similarity as an order relation:

“x is at least as similar to y as to z” For simplicity we assume that all elements are taken from the same

underlying set. Remarks:– i) and ii) are equivalent– iii) is more powerful in expressiveness:

different degrees of similarity can be distinguished


Similarity Relations (1)

• Aim: Definition of a relation R:R(x,y,z) “x is at least as similar to y as x to z”.

• First step: Consider the relation S:S(x,y,u,v) “x to y is at least as similar as u to v”.

• Definition of R: R(x,y,z) : S(x,y,x,z)• Possible requirements to the relation S:

for all x,y,z,u,v,s,t holds:– S(x,x,u,v) every object is at least as similar to itself as two

other arbitrary objects could be to each other (reflexivity)

– S(x,y,u,v) S(u,v,s,t) S(x,y,s,t) transitivity

– S(x,y,u,v) S(y,x,u,v) S(x,y,v,u) symmetry



• If one focuses on a fixed object x, (the current problem, the current demand):

• Look for:“y, which is most similar to x” for all z holds: R(x,y,z)

• For a fixed x, we regard the relations: x, >x, ~x

– y x z R(x,y,z)

“y is at least as similar to x as to z”

– y >x z (y x z) zx y)

“y is more similar to x than to z” (strict part of x )

– y ~x z (y x z) zx y)

“y and z are indistinguishable”



• For fixed x some y which satisfies

y is most similar to x, i.e: for all z holds: R(x,y,z)

is called a nearest neighbor of x.

Notation : NN(x,y)

• A nearest neighbor is not necessarily uniquely defined, there may be several (indistinguishable) such elements.

• The nearest neighbor is the element which has to be approximated.


Similarity Measures (1)• In its most abstract form a measure is defined for two

arguments taken from two arbitrary sets P (called queries) and S (called answers):

• A similarity measure is a mapping sim: P x S [0,1] (real interval)

Major specializations are:(1) S is a subset of P:

E.g. P = Description of all possible products, S = Description of all available products.

(2) If there is a case base CB of cases (p,s) and P´ is the set of all p s.t. there is some (p,s) CB then

sim: P x P’ [0,1]This is the traditional setting in CBR (here P´ is the set of problems with solutions in CB).


Similarity Measures (2)

• Idea: Numerical modeling of similarity (more or less similar)

• Along with ordinal information there is also a quantitative statement about the degree of similarity:

• Definition: A similarity measure on a set M is a real function sim: M2 [0,1]. The following properties can hold:– x M: sim(x,x) = 1 (Reflexivity)

A similarity measure is symmetrical iff:– x,yM: sim(x,y) = sim(y,x)(Symmetry)

• Induced similarity relation:

– sim(x,y) sim(x,z) “x is more similar to y than to z”

0 1sim(x,yi)

y1y2y3y4


Distance Functions / Measures• Instead of using similarity functions one often takes distance functions,

the dual notion.• Definition: A distance measure on a set M is a real valued function

d: M2 IR+.

• The following properties can hold:

– xM d(x,x) = 0 (Reflexivity)

A distance measure is symmetrical iff:

– x, yM d(x,y) = d(y,x) (Symmetry)

A distance measure is a metric iff:

– x, yM d(x,y) = 0 x = y

– x, y, zM d(x,y) + d(y,z) d(x,z) (Triangle Inequality)

• Induced similarity relation:

– d(x,y) d(x,z) “x is more similar (less distant) to y than to z”


Relation between Similarity and Distance Measures

• Definition: A similarity measure sim and a distance

measure d are called compatible if and only if

– x, y, u, vM Sd(x,y,u,v) Ssim(x,y,u,v)

(the same similarity relation is induced)

• Transformation of measures: If a bijective, order

inverting mapping f: [0,1] [0,1] exists with – f(0) = 1

– f(d(x,y)) = sim(x,y)

then sim and d are compatible.


Language Aspects

• If for the argument sets of sim we have P = S or S P, then we can define the similarity function by referring to the same representation language for both arguments.

• If objects are defined in terms of attributes then one can first define measures on the domains of the attributes and then use some constructors in order to define the total similarity on the whole object.

• The measures on the domains of attributes are called local measures and measure for the total similarity is called the global measure.

• If the languages for P and S are very different then additional actions have to be taken, see chapter 9


Some Simple Similarity Measures

• Representation language: Objects (Examples)

are described by attribute-value pairs

(feature vectors).

• At first: restriction to binary features.

• Regarding similarity between two

objects x and y:

x = (x1,...,xn) xi {0,1}

y = (y1,...,yn) yi {0,1}


Hamming Distance

• Properties: – H(x,y) [0,n] n is the maximum distance

– H(x,y) is the number of distinguishing attribute values

– H is a distance measure:

• H(x,x) = 0

• H(x,y) = H(y,x)

– H( (x1,...,xn), (y1,....,yn) ) = H( (1-x1,...,1-xn), (1-y1,....,1-yn) )

H x y n x y x yi ii

n

i ii

n( , ) ( ) ( )

1 11 1


Real-Valued Attributes

• Real-valued attributes xi , yi IR for all i

• Generalization of the Hamming Distance to the

City-Block-Metric:

• Alternative metrics

Euclidean Distance:

Weighted Euclidean Distance:

d x y x yi i

i

n

( , ) | |

1

d x y x yi i

i

n

( , ) ( ) 2

1

d x y x yi i i

i

n

( , ) ( ) 2

1


Arbitrary Attribute Types

• Description by n attributes A1,...,An, each attribute Ai has type Ti.

• Local similarity:

– one similarity function for each attribute: simAi(xi,yi): Ti x Ti [0..1]

– local measures depend on the type of the attribute

• Global similarity:

– sim(x,y)= sim( (x1, ..., xn), (y1, ..., yn) )

= F( simA1(x1,y1),..., simAn(xn,yn) )

– F: [0..1]n [0..1] is an amalgamation (or constructor) function

– For F, usually we claim that

• F is monotonous in each argument

• F(0,...0) = 0

• F(1,...,1) = 1

• This will later be extended to a general local-global principle.


Examples for F

• Weighted average:

• Generalization:

• Maximum:

• Minimum:

• k-Maximum:

• k-Minimum:

1 with ),...,(11

1

n

ii

n

iiin wswssF

1 and with )(),...,(11

1

n

ii

n

iiin wswssF IR

F s s s xn n( ,..., ) max{ ,..., }1 1

F s s s xn n( ,..., ) min{ ,..., }1 1

niiikin ssssssF ... with ),...,( 211

niiikin ssssssF ... with ),...,( 211


The General Local - Global Principle (1)

• Each object is described – from a global point of view – in a detailed view– Global measures are defined on the whole object– Local measures are defined on details, e.g. the

domains of some attribute

• General idea: – The global view reflects the task and has a

pragmatic character– The local details are of technical and domain

character and task independent


The Local - Global Principle (2)• The local measures should represent domain properties. • Example: Water where temperature is an attribute with the

domain e.g. T(water) = [-100, +200]• In numerical domains as here: Landmarks are numbers

which separate the real axis into intervals (temperature of water: 0 and 100 degrees). These landmarks are facts from physics and not personal relevances.

• Each interval is a qualitative region (where the material behaves in some sense very similar)

• The measure is not uniform: numbers inside a region have a high degree of similarity but numbers in different regions have low degree of similarity.

• The relevance of temperature for some problem is reflected by a weight.


The Local - Global Principle (3)

• The weights of the global measure (or more general: The amagalmation function) reflect – Importance– relevance– utility

• They depend on the pragmatics and are ultimately determined by the user.

• The local-global principle is a good guideline for defining measures for some application.


The Formal Local - Global Principle• Each object A is constructed from so-called “parts” Ai by some

construction process

• C(Ai |i I) = A;

• where we C is the construction operator. Often the Ai are also called the components. In an attribute-value representation the attributes are the components.

• The local-global principle for similarities now is stated in the following way:

• There are measures simi on the possible values for the Ai and some amalgamation function f such that

sim(A, B) = f(simi(Ai , Bi) |i I )

• The simi are local measures and sim is the global one.


The Monotonicity Axiom

• With the local-global principle a property of measures is connected called the monotonicity axiom:

• If sim(A, B) > sim(A, C) then there is at least one i I such that simi(Ai , Bi) > simi(Ai , Ci).

• This axiom can be violated if the measure depends on certain relations between local attribute values.

• In such situations there may even be no measure with this properties. Hence the axiom is also a demand on the representation language.

• The purpose of introducing virtual attributes is usually to establish that this axiom holds.


Similarities and Functional Dependencies (1)

• Attribute A is called functionally dependent from the attributes A1,..., An if the values of a can be computed by a function f from the values of the A1,..., An. The functional dependency may be conditional, i.e., it may only hold if certain other attributes have specific values.

• Functionally dependent attributes require special attention:– if two objects agree on the some attributes, they will also

agree on any functional dependent attribute.– However, if they agree on a functionally dependent attribute,

they do not necessarily agree on the base attributes.


Similarities and Functional Dependencies (2)

• It is therefore an important task of knowledge acquisition to detect functional dependencies for the similarity computation.

• In addition, there maybe unknown virtual attributes on which the measure is easier to define, e.g., as a linear weighted sum.

• Example: Similarity depends on the difference of a1 and a2 rather than on a1 and a2 themselves, e.g., bank application: a1=income; a2=spending

sim((a1,...,an),(b1,...,bn))

= sim’((a1-a2, a3,...,an),(b1-b2,b3,...,bn))


Similarities and Unknown Values

• If some value is unknown in two objects C and D this should not contribute anything to the measure.

• If C is in the case base and its value is known, Q the query and the value is unknown then one of the choices could be taken:– default value– a value which contributes to the similarity

• a maximal– or

• a minimal amount


Similarities and Redundant Values

• Redundant values are those which are known in the query and unknown in the answer from the case base. They can be handled in two ways:– Ignore them if it can be ensured that the case from the

base have all values necessary for the solution.– For CBR: Omit the case C from the base and replace it

by the query case D if C and D have different solutions. Even if C had provided a correct solution in the past it was only a good guess and the attribute value in the past was different from the value in D.


Further Measures: Local Similarity Measures for Unordered Symbols

• Similarity table: simA(x,y) = s[x,y]

• Symbol type TA={v1,...,vk}

• For reflexive similarity measures: diagonal entries = 1• For symmetric similarity measures:

upper triangle matrix = lower triangle matrix

s[x,y] v1 v2 ... vk

v1 s[1,1] s[1,2] s[1,k]

v2 s[2,1] s[2,2] s[2,k]

...

vk s[k,1] s[k,2] s[k,k]


Local Similarity Measures for Integer/Real

• Similarity often based on difference:– linearly scaled value ranges: simA(x,y) = f(x-y)

– exponentially scaled value ranges: simA(x,y) = f(log(x)-log(y))

• Generally, for f we claim:– f:IR [0..1] or Z [0..1]– f(0) = 1 (Reflexivity)– f(x): monotonously decreasing for x>0 and

monotonously increasing for x<0– Examples:

1

f

symmetric

1f

asymmetricx: query; y: case

query is minimum demand

asymmetricx: query; y: case

query is maximum demand

1

f


Similarity Characterization

• Goal: Find general properties of the similarity measures for every simple attribute of the domain model.

local similarity measures

• Basic Properties:– Symmetry: symmetric vs. asymmetric– Exact match preference– Continuous vs. sudden decrease– Kind of decrease


Property: Symmetry• Symmetric:

– if it is not important for the similarity computation which attribute value belongs to the query and which one belongs to the case

– Example: for numeric values (INTEGER, REAL) use the absolute distance d = |query value - case value|

sim = f(d) with f:[IR+] [0;1]

• Asymmetric:– if it is important which value belongs to the query and

which one belongs to the case– similarity can be computed with two different similarity

functions: d = (query value - case value)

]1;0[][2

]1;0[][1 with

0 )(2

0 1

0 )(1

IRf

IRf

dif df

dif

difdf

sim


Property: Exact Match Preference

• Often it can be useful to weight the exact match of the query and the case attribute value much higher than an inexact match

define a maximal possible similarity simmax < 1

• Example for a numeric attribute

sim

d

1

maxsim

a linear similarity function

d = |query value - case value|


Property:Continuous or Sudden Decrease

• Depending on the distance between two values, distinguish between decrease

– at a point ( Example a))– in an interval ( Example b))

• Example:

a) decrease of the whole similarity at the distance X

b) decrease in the interval [xmin;xmax]

sim

dX xmaxmin

x

a)

b)


Property: Kind of Decrease

• Different forms:– linear: There is a linear relationship between the

distance and the similarity of two values. – downward convexity: A small difference between two

values causes a big decrease of similarity. – upward convexity: A small difference between two

values causes only a small decrease of similarity. – combined

sim

d

linear

upward convexity

downward convexity

sim

d

upwardconvexity

downwardconvexity


Symmetric Similarity Measures for Numeric Attributes (I)

• Step-Function:

If there is only the possibility that an attribute-value of a case c is useful or not useful with respect to the query-attribute-value q:

q-cd(q,c)else

Scqdifcqsim

with 0

),( 1),(

d(q,c)

sim(q,c)

1

S


Symmetric Similarity Measures for Numeric Attributes (II)

• Linear Function:

The idea is that the similarity increases linearly in an interval with the decrease of distance between the two values.

– The begin of the interval is a user-defined distance min 0.– The end of the interval can be the greatest possible value

for the actual attribute (which we call Max) or it can be a user-defined distance max < Max.

maxmin0

;with

max),(

max),(min

min),(

0minmax

),(max1

),(Max

q-cd(q,c)

cqdif

cqdif

cqdifcqdcqsim

d(q,c)

sim(q,c)

1

min max


Symmetric Similarity Measures for Numeric Attributes (III)

• Asymptotic Function:Small differences between the two values cause a big decrease of similarity.

IRaacqdsim

with ),(exp

sim

d(q,c)

1

a3

2a

1a 1

a2

a a3

< <

The parameter a influences the gradient of the function


Symmetric Similarity Measures for Numeric Attributes (IV)

• S-Function:

If we tolerate differences between the two values even to a defined distance x1 we can use the following similarity function:

IRaaxsim

with 3exp

sim

d(q,c)

10,9

0,1

x1 x2


Local Similarity Measures for Ordered Symbols

• For ordered symbols, an order is given in addition to the value range.

Example: qualitative values: {small, medium, large} small < medium < large

• Local similarity measures like for numeric types.

• Therefore: Define an Integer value for each symbol so that the order is kept.

Example:– small 1– medium 2– large 3


Local Similarities for Taxonomies (I)

• Assumption: considered objects can be ordered in a Taxonomy or set

of notions (tree or graph structure)

• Example:

The ELSA 2000 is more similar to the Stealth 3D than to one of the S3 Trio adapters or MGA adapters.

Graphics Card

S3 Graphics Card MGA Graphics Card

ELSA 2000 Stealth 3D200 Miro Video

Matrox Mill. 220 Matrox Mystique220

VGA V64

S3 Virge Card S3 Trio Card


Local Similarities for Taxonomies (II)

• Definition of similarity for leaf nodes:– assignment of a similarity value to each inner node– similarity values for successor nodes become larger– similarity between two leaf nodes is computed by the

similarity value at the deepest common predecessor

• Example:

sim (ELSA 2000, Stealth 3D) = 0,7

sim (ELSA 2000, VGA V64) = 0,5

sim (ELSA 2000, Matrox Mill.) = 0,2

Graphics Card




VGA V64

S3 Virge Card S3 Trio Card0.7 0.9

0.5 0.8

0.2


Similarities for Taxonomies (III) Examples

• Example 1: Product Search

A customer wants a computer system with a S3 graphics adapter.

Which one (model) does not matter to the customer.

current problem: graphics adapter: S3_Graphics_Card

• Example 2: After Sales Support - Diagnosis system for PCs.

An error case holds for all S3 graphics adapters.

(generalised) case: graphics adapter: S3_Graphics_Card

• Example 3: After Sales Support - Diagnosis system for PCs.

The user has a S3 graphics adapter, but which one is not known.

current problem: graphics adapter: S3_Graphics_Card


Local Similarities for Taxonomies (IV) Similarity of Inner Nodes

• Semantics of inner nodes:

– Inner node stands for the set of the underlying leaf nodes

Example: S3 Virge = {ELSA 2000, Stealth 3D}

– Semantics of inner nodes (sets) in a case and in the current

problem can be:

• unknown: an element of the set exists, but it is not known.

• arbitrary: it does not matter, which element of the set is

considered.

– Semantics of inner nodes determines the way to calculate

similarity


Local Similarities for Taxonomies (V) Semantics of Inner Nodes: “Arbitrary”(1)

• Inner node occurs in case:

• Inner node occurs in problem (query):

otherwise

if 1}|),(max{),(

,CqC S

CqLccqsimCqSIM

SIM Q c sim q c q Lc Q

SQQ c

( , ) max{ ( , )| },

1 if

otherwise


Local Similarities for Taxonomies (V) Semantics of Inner Nodes: “Arbitrary” (2)

• Inner node occurs in case and problem:

SIM Q C sim q c q L c LC Q Q C

SQ CQ C

( , ) max{ ( , )| , },

1 if or

otherwise

Graphics Card




VGA V64


0.5 0.8

0.2 Notation:LQ, LC: Leaf node after inner node Q, C

C < Q: C lies under Q in taxonomy (C is successor of Q)

S<Q,C>: Similarity value of the common predecessor node of Q and C


Local Similarities for Taxonomies (VI) Semantics of Inner Nodes: “Unknown”

• Uncertainty can be interpreted in different ways:– optimistic strategy: presume the highest similarity, which is

still possible.

– pessimistic strategy: presume the lowest similarity, which is

still possible.

– strategy of the expected value: presume the expected value

for the similarity based on known information.


• Pessimistic Strategy:– inner node (uncertainty) occurs in case:

– inner node (uncertainty) occurs in problem (query):

– inner node (uncertainty) occurs in case and in problem:

Local Similarities for Taxonomies (VII)Semantics of Inner Nodes: “Unknown”

}|),(min{),( ,CqC SLccqsimCqSIM

SIM Q c sim q c q L SQ Q c( , ) min{ ( , )| } ,

SIM Q C sim q c c L q L Sc Q Q C( , ) min{ ( , )| , } , Graphics Card




VGA V64


0.5 0.8

0.2


Object-Oriented Representation

• Extension of the similarity calculation for attribute-value pairs

• Regard objects as attribute-value pairs• Similarity becomes a property of the object class

– slot level (comparison of single attribute values):local similarity measures simi(vi,wi)

– class level (comparison of objects):determination of object similarity by amalgamation (e.g., weighted sum) of the

• local similarities of typed slots• object similarity of sub-objects in relational slots


Basic Approach to Similarity Computation(cf. Wess, 1995)

• Bottom-up computation of similarity• Local similarity measures for typed attributes• Object similarity measures by aggregation (e.g.,

average)– results from local similarity measures– results from object similarity measures from related objects

Co

mp

utatio

n


Example

Object instance: Lanzarote

(Object class: Journey)

Location: Lanzarote

Duration: 14

Accommodation: Hotel_Playa

Sports facilities: Tennis

Object instance: Hotel_Playa(Object class: Hotel)

Category: 3-StarsDistance_to_Beach: medium

Object instance: Tennis(Object class: Sports)

Equipment: goodPrice_per_hour: 15,50

Case0.7

Object instance: Dream_Journey

(Object class: Journey)

Location: Tenerife

Duration: 10

Accommodation: Dream_Hotel

Sports facilities: Dream_Sports

Object instance: Dream_Hotel(Object class: Hotel)

Category: 2-StarsDistance_to_Beach: short

Object instance: Tennis(Object class: Sports)

Equipment: very goodPrice_per_hour: 20,00

Query (Current Problem)

0.50.20.20.1

0.50.5

0.80.2

Wei

ghts

0.8

0.7

0.75

0.81

10.5

0.96

0.75

0.96

0.786


Problems of Similarity for OO Cases

• OO cases to be compared may have different

structure

• Instances from different classes must be compared

– e.g., a CD-Writer and a Hard Disk

• Instances of classes from different abstraction levels

of the class hierarchy have to be compared

– e.g., a CD-Writer and a Magnetic Storage Device


Example 1a: Sales Support for PCs• Cases: Available PCs from stock• Query: Customer’s requirements

C1 C2

Q

C2 is more similar to Q than C1 because C2 and Q are both optical storage devices (CD-Media)

Storage Devicemanufacturer: Symbolcapacity: Realaccess time: Real

Optical Storage Dev.read-speed: Integertype-of-laser: Symbol

Magnetic Storage Devicebuild-in-size: {3.5 ; 5.25}type-of-magnetic-surface: Symbol

Streamer Floppy Disk Hard Disk

CD-ROM

CD-Writer CD-RW

Writeable O. S. D.write-speed: Integer

optional storage:


Example 1b: Sales Support for PCs

C2 is perfectly suited:

– C2 is more similar to Q than C1

– sim(C2,Q)=1 sim(C1,Q)<1

• Different Query: Customer wants a PC with an Optical Storage Device He does not care which one.





CD-ROM

CD-Writer CD-RW


optional storage:

C1 C2

Q


Example 2a: PC Trouble-Shooting



– sim (C1,Q)=1 sim(C2,Q)<1

• Case: Previous diagnostic situations (fault descriptions)

• Query: Observed symptoms for current problem





CD-ROM

CD-Writer CD-RW


optional storage:

C1

C2

Q

Fault occurs with every Optical Storage Device


Example 2b: PC Trouble-Shooting



– sim(C1,Q)= 1 BUT sim(C2,Q)< 1





CD-ROM

CD-Writer CD-RW


optional storage:

C1

C2

Q

User only knows that he has a writeable OSD

but doesn’t know what kind


Framework for Object Similarities• Intra-class similarity

– compute the similarity of two objects based on the shared attributes of the most specific parent class

– apply standard approach: aggregation of local similarity measures

• Inter-class similarity– compute the similarity of the object classes (taxonomy approach!)

– represents the highest possible similarity that can be achieved

• Combined to object similarity:

)).,.(),...,.,.((),(intra nAAAAA AcqsimcqsimcqSIMnn111

Aggregation function Local Similarity

),(inter CQSIM

Class of Query Object Class of Case Object

),())(),((),( intrainter cqSIMcclassqclassSIMcqsim


Semantics of Objects

• A class at a leaf node of the hierarchy represents a set of concrete

objects, described in a particular way:

– e.g., class Hard Disk represents concrete objects, namely particular hard

disks

• A class at an inner node of the hierarchy represents a set of abstract

objects, described in a particular way:

– e.g., instances of class Optical Storage Device are not concrete objects

• Abstract objects are sets of concrete objects:

– e.g., Optical Storage Device is an abbreviation for the set of all instances

from the subclasses CD-ROM, CD-Writer, CD-RW.


Semantics of Abstract Objects• What is the semantics of

optional storage: <instance of Optical Storage Device> ?

• Different semantics in the three examples 1b, 2a, 2b:

– any value in the query: Looking for a case that has an arbitrary object

from the set in the respective attribute

• Example 1b: User is looking for a PC with an optical storage device

– any value in the case: The case is valid for all objects from the set

• Example 2a: Fault occurs with every optical storage device

– uncertainty: The case (or query) is valid for one object from the set,

but we don’t know for which one

• Example 2b: User just knows that he has a writeable optical storage

device but doesn’t know which one


Example

Object class ApartmentSuperclass Accommodation

Rooms: IntegerKitchen: {yes, no}

Object class HotelSuperclass Accommodation

Food: SymbolBalcony: Boolean

Similarity Value: 0.6Semantic: “arbitrary”

sim*(o1,o2) sim<class(o1),class(o2)>

Hotel Apartment 0.6 simAccommodation

Hotel Hotel 1 simHotel

Hotel Accommodation 1 simAccommodation

Apartment Accommodation 1 simAccommodation

Accommodation Hotel 1 simAccommodation

Product Query

Situation of Comparison Used Similarity Measure

Object class AccommodationOrt: Taxonomy_of_RegionsPrice: RealClose_to_Sea: Symbol


• Suppose (P1, S1) and (P2, S2) are two pairs (problem, solution).

• Guiding principle:If sim(P, P1) sim(P, P2) then one would not prefer S2 over S1 as a solution for P.

• In particular: Nothing is preferred over a nearest neighbor.• Ordinal aspect: Preference order (partial)• Cardinal aspect: Meaning of numbers

Semantics of Similarity: Similarity Measures and Preferences


Preference Relations and Utility Functions

• Both concepts come from economics (v. Neuman-Morgenstern Theory)

• A preference relation over a universe U is just a binary relation where:– a b means that a preferred over b in a choice situation

• A utility function is a mapping u: U IR:– u realizes the preference relation if

u(a) u(b) iff a is preferred over b

• The utility function adds a cardinality aspect to the relation aspect of the preference, i.e., one can say how useful the choice is.


Semantics of Similarity (III): Utility

The similarity measure should reflect the utility function !

ProblemP

ProblemP1

SolutionS

SolutionS1

Utilityfor P: good

Utilityfor P1 ?

similar Copy,adapt


• Humans are better in estimating - Relationsrather then- numerical values

• Consequence: Weakening ofutility = similaritytoThey have the same monotonicity behavior

Comparing Similarity and Utility


• Given: An actual problem Q.

• Define for cases (P,S) CB: fsim,Q(S) := sim(P,Q)

• Now can be compared with a utility function because both are real valued functions defined on solutions.

• If the similarity function compares directly “demand and product” or “functionality and product” then similarity and utility can be directly compared.

Formal Comparison


• Ideal situation: The similarity measure coincides exactly with the intended utility function.

• A weaker demand: The utility function u and the function fsim,Q are similarly ordered.

• Definition:Two functions f,g : X -> R are similarly ordered ifD(f, g, x, y) : = (f(x) - f(y))*(g(x) - g(y)) 0 for any x,y X

Semantic Comparison


Approach: Use the correspondence

Similarities --- Utilities

for the building process over various stages

local --- global

This can be used top-down as well as bottom-up

.

Similarities and Utilities (1)


Basic ideas:

1) Use a language which describes similarities as well as

utilities

2) Do it in such a way that • local similarities correspond to preferences • global similarities correspond to utility functions

3) Take advantage of the well established methods for

building similarity measures

4) In particular, use the local - global view by applying the

same constructors for obtaining the global objects from the

local ones in both situations.

Similarities and Utilities (2)


Properties of Similarity Revisited

• The utility interpretation of similarity throws a new light on the properties of similarity. We consider two cases (P,S) and (P’,S’).

• Symmetry of sim: sim(P,P’) = sim(P’,P) would mean S’ is as useful for P as S for P’. This is obviously not always the case.

• Reflexivity sim(P,P) = 1 would mean: The solution S is at least as good for P than any other solution. This is a demand on the case base not satisfied if non-optimal solutions are stored.


Sensitivity Analysis

• Utility (and quality) functions u depend in general on several parameters in different ways.

• Sensitivity analysis asks for the dependency of u from the arguments:– When do small changes in an argument result in a big change of u?

• Main problem: The arguments are not independent in the sense that certain combinations are responsible for the changes of the values of u.

• For this purpose one needs to – decompose the preference relations (top-down)– define certain new (virtual) attributes (bottom-up).


Informal Utilities

• A problem with utilities and preferences is that they are often informally stated.

• First problem: When is a formal utility function equivalent to the informal one?

This cannot be proven but established by a Turing test.• Second problem: How to obtain a practicable formal

analogue of the informal utility?

This will be done by decomposition techniques.


Informal - Formal : The Ordering Principle A Turing test

How to compare a formal and an informal concept ?Use orderings !Suppose there is an ordering < with the concept C associated:

Informal humanversion of C

Formalversion

of C

< two versions: formal and informalThe goal is that whenvariations of the arguments of < are presented:The human says „up“if and only if the formalsystem says „up“

goal


preferences,quality aspects

Decomposition of user preferences until measurable quantitiesare obtained.

Decomposition : Preferences


Complex Preference Relation

Decomposepreference relation

meet

··

· ·

·

· · ·

Virtual attributeUse SensitivityAnalysis

Primary attributes

Bridge the Gap

Goal: Similarly Ordered

· · ·


Example (1): Decomposition

Quality

scope of applicability

resistanceagainst heat

nice appearance

Can bemeasured

• Product built from material m1, ...,m8


Example (2) Virtual Attributes

a=% of material2 b=% of material4 c =% of material6

material code:a (b + c)

virtualattribute



Example (3): Bridge the Gap

heat resistance

material code

heat_resistance (product) and material_code(product) are similarly ordered



Similarity and Fuzzy-Sets

• Fuzzy subsets of A of U are denoted by A f U

• µA is the membership function of A.

• A fuzzy partition of U into n fuzzy subsets is given by

membership functions µ1(x),...,µn(x) such that

(µi | 1in) = 1

• A fuzzy classifier for such a partition is mapping

cf: U [0, 1]n

such that for

cf(x) = (µ1(x),..., µn(x)) we have (µi(x) | 1in) = 1.


From Similarities to Fuzzy Sets

•A similarity measure sim on U defines a fuzzy subset SIM f V: = U x U

by µSIM (x1, x2): = sim (x1, x2)

•We associate to each x U a fuzzy subset Fx f U by µx (y) = sim(x,y)

•µ x(y) = sim (x,y) and symmetry of sim imply µx (y) = µy (x)

•So we obtain from sim for each x some fuzzy subset which can beregarded as how U is structured by sim from the viewpoint of x.

•The nearest neighbor relation now reads asNN (x, y) µx (y) µx (z), z CB,

or equivalently

NN (x, y) µSIM (x, y) µSIM (x, z), z CB


From Fuzzy Sets to Similarities (1)

• Suppose we have a fuzzy subsetS f V = U x U with µs (x, x) = 1

then S defines obviously a similarity measure.

• If we have simply a fuzzy set K f U

then we would need in addition a reference object (a prototype) in order to define a measure.

• Such a reference object has to satisfyµK (x) = 1.

In this case we can define a similarity measure bysim(x, y) = µK (y)


From Fuzzy Sets to Similarities (2)

• Suppose there is a subset CB U such that for each

x CB we have some fuzzy subset Kx f U with

membership functions µx (y), for which µx (x) = 1 holds.

• Then we can again define a measure on U x CB by

sim(x, y) = µx (y) for y U, x CB.

• This is sufficient to define the nearest neighbor notion

NN(y, x), y U, x CB.


Summary

• The similarity concept is the most important concept in inexact matching and reasoning.

• Similarity is closely related to utility.• Similarity can be defined by relations or measures.• There are different ways to define measures and they

can have several properties. These properties are dependent by the intended use.

• A basic building principle is the local-global principle.• Similarity measures can be regarded as fuzzy

relations.

Chapter 6 Similarity Stand 20.12.00. - 2 - (c) 2000 Dr. Ralph Bergmann and Prof. Dr. Michael M....

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