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Chapter: Artificial Intelligence

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

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, dp 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


= XC-1Y


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 ux and uy

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 Ci , P(Ci| X), the conditional probability of Ci 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 ={o1, o2, . . . } be the universe of objects of interest


Let Ai be the set of attributes used to represent oi


A similarity measure s which is a function of three disjoint sets of attributes common to any two objects Ai and Aj is given as


s(Ai, Aj) = F(Ai & Aj, Ai - Aj, Aj - Ai)


Where Ai & Aj is the set of features common to both oi and oj


Where Ai - Aj is the set of features belonging to oi and not oj


Where Aj - Ai is the set of features belonging to oj and not oi


The function F is a real valued nonnegative function


s(Ai, Aj) = af(Ai & Aj) – bf(Ai - Aj) – cf(Aj - Ai) 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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