Chapter 06 - Changing Education Paradigm

22 CHAPTER 6. MINING ASSOCIATION RULES IN LARGE DATABASES
The Manhattan (city block) distance between t1 and t2 is
M anhattan d(t1;t2)=
mX
i=1
jx1i , x2ij: (6.24)
The diameter metric assesses the closeness of tuples. The smaller the diameter of S[X] is, the \closer" its tuples
are when projected on X. Hence, the diameter metric assesses the density of a cluster. A cluster CX is a set of tuples
de ned on an attribute set X, where the tuples satisfy a density threshold, dX 0 , and a frequency threshold, s0,
such that:
d(CX) d X 0 (6.25)
jCXj s0: (6.26)
Clusters can be combined to form distance-based association rules. Consider a simple distance-based association
rule of the form CX ) CY . Suppose that X is the attribute set fageg and Y is the attribute set fincomeg. Wewant
to ensure that the implication between the cluster CX for age and CY for income is strong. This means that when
the age-clustered tuples CX are projected onto the attribute income, their corresponding income values lie within
the income-cluster CY , or close to it. A cluster CX projected onto the attribute set Y is denoted CX[Y ]. Therefore,
the distance between CX[Y ] and CY [Y ]must be small. This distance measures the degree of association between CX
and CY . The smaller the distance between CX[Y ] and CY [Y ] is, the stronger the degree of association between CX
and CY is. The degree of association measure can be de ned using standard statistical measures, such as the average
inter-cluster distance, or the centroid Manhattan distance, where the centroid of a cluster represents the \average"
tuple of the cluster.
Finding clusters and distance-based rules
An adaptive two-phase algorithm can be used to nd distance-based association rules, where clusters are identi ed
in the rst phase, and combined in the second phase to form the rules.
A modi ed version of the BIRCH 5 clustering algorithm is used in the rst phase, which requires just one pass
through the data. To compute the distance between clusters, the algorithm maintains a data structure called an
association clustering feature for each cluster which maintains information about the cluster and its projection onto
other attribute sets. The clustering algorithm adapts to the amount ofavailable memory.
In the second phase, clusters are combined to nd distance-based association rules of the form
CX1CX2 ::CXx ) CY1CY2 ::CYy ,
where Xi and Yj are pairwise disjoint sets of attributes, D is measure of the degree of association between clusters
as described above, and the following conditions are met:
1. The clusters in the rule antecedent each are strongly associated with each cluster in the consequent. That is,
D(CYj [Yj];CXi [Yj]) D0, 1 i x; 1 j y, where D0 is the degree of association threshold.
2. The clusters in the antecedent collectively occur together. That is, D(CXi [Xi];CXj [Xi]) d Xi
0
3. The clusters in the consequent collectively occur together. That is, D(CYi [Yi];CYj [Yi]) d Yi
0
is the density threshold on attribute set Yi.
8i 6= j.
8i 6= j, where dYi
0
The degree of association replaces the con dence framework in non-distance-based association rules, while the
density threshold replaces the notion of support.
Rules are found with the help of a clustering graph, where each node in the graph represents a cluster. An edge
is drawn from one cluster node, nCX , to another, nCY ,ifD(CX[X];CY[X]) dX 0 and D(CX [Y ];CY[Y]) dY 0 . A
clique in such a graph is a subset of nodes, each pair of which is connected by an edge. The algorithm searches for
all maximal cliques. These correspond to frequent itemsets from which the distance-based association rules can be
generated.
5 The BIRCH clustering algorithm is described in detail in Chapter 8 on clustering.