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The problem of comparing structures without the use of pattern matching variables. This requires consideration of measures used to determine the likeness or similarity between two or more structures

**Measure for Matching**

The problem of comparing structures without the use
of pattern matching variables. This requires consideration of measures used to
determine the likeness or similarity between two or more structures

The similarity between two structures is a measure
of the degree of association or likeness between the object’s attributes and
other characteristics parts.

If the describing variables are quantitative, a
distance metric is used to measure the proximity

Distance Metrics

Ø For all
elements x, y, z of the set E, the function d is metric if and only if d(x, x) = 0

d(x,y) ≥ 0

d(x,y) = d(y,x)

d(x,y) ≤ d(x,z) + d(z,y)

The Minkowski metric is a general distance measure
satisfying the above assumptions

It is given by

For the case p = 2, this metric is the familiar
Euclidian distance. Where p = 1, d_{p} is the so-called absolute or
city block distance

Probabilistic measures

The representation variables should be treated as
random variables

Then one requires a measure of the distance between
the variates, their distributions, or between a variable and distribution

One such measure is the Mahalanobis distance which
gives a measure of the separation between two distributions

Given the random vectors X and Y let C be their
covariance matrix

Then the Mahalanobis distance is given by

= X^{’}C^{-1}Y

Where the prime (‘) denotes transpose (row vector)
and C^{-1} is the inverse of C

The X and Y vectors may be adjusted for zero means
bt first substracting the vector means u_{x} and u_{y}

Another popular probability measure is the product
moment correlation r, given by

Where Cov and Var denote covariance and variance
respectively

The correlation r, which ranges between -1 and +1,
is a measure of similarity frequently used in vision applications

Other probabilistic measures used in AI
applications are based on the scatter of attribute values

These measures are related to the degree of
clustering among the objects

Conditional probabilities are sometimes used

For example, they may be used to measure the
likelihood that a given X is a member of class C_{i} , P(C_{i}|
X), the conditional probability of C_{i} given an observed X

These measures can establish the proximity of two
or more objects Qualitative measures

Measures between binary variables are best described using contingency tables in the below Table

The table entries there give the number of objects
having attribute X or Y with corresponding value of 1 or 0

For example, if the objects are animals might be
horned and Y might be long tailed. In this case, the entry a is the number of
animals having both horns and long tails

Note that n = a + b + c + d, the total number of
objects

Various measures of association for such binary
variables have been defined

For example

Contingency tables are useful for describing other
qualitative variables, both ordinal and nominal. Since the methods are similar
to those for binary variables

Whatever the variable types used in a measure, they
should all be properly scaled to prevent variables having large values from
negating the effects of smaller valued variables

This could happen when one variable is scaled in
millimeters and another variable in meters

Similarity measures

Measures of dissimilarity like distance, should
decrease as objects become more alike

The similarities are not in general symmetric

Any similarity measure between a subject
description A and its referent B, denoted by s(A,B), is not necessarily equal

In general, s(A,B) ≠ s(B,A) or “A is like B” may
not be the same as “B is like A”

Tests on subjects have shown that in similarity
comparisons, the focus of attention is on the subject and, therefore, subject
features are given higher weights than the referent

For example, in tests comparing countries,
statements like “North Korea is similar to Red China” and “Red China is similar
to North Korea” were not rated as symmetrical or equal

Similarities may depend strongly on the context in
which the comparisons are made

An interesting family of similarity measures which
takes into account such factors as asymmetry and has some intuitive appeal has
recently been proposed

Let O ={o_{1}, o_{2}, . . . } be
the universe of objects of interest

Let A_{i} be the set of attributes used to
represent o_{i}

A similarity measure s which is a function of three
disjoint sets of attributes common to any two objects A_{i} and A_{j}
is given as

s(A_{i,} A_{j}) = F(A_{i}
& A_{j}, A_{i} - A_{j}, A_{j} - A_{i})

Where A_{i} & A_{j} is the set
of features common to both o_{i} and o_{j}

Where A_{i} - A_{j} is the set of
features belonging to o_{i} and not o_{j}

Where A_{j} - A_{i} is the set of
features belonging to o_{j} and not o_{i}

The function F is a real valued nonnegative
function

s(A_{i,} A_{j}) = af(A_{i}
& A_{j}) – bf(A_{i} - A_{j}) – cf(A_{j} - A_{i})
for some a,b,c ≥ 0

Where f is an additive interval metric function

The function f(A) may be chosen as any nonnegative
function of the set A, like the number of attributes in A or the average
distance between points in A

When the representations are graph structures, a similarity
measure based on the cost of transforming one graph into the other may be used

For example, a procedure to find a measure of
similarity between two labeled graphs decomposes the graphs into basic
subgraphs and computes the minimum

cost to transform either graph into the other one,
subpart-by-subpart

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