Mining Confident Minimal Rules with Fixed

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Mining Confident Minimal Rules with Fixed-Consequents
Imad Rahal, Dongmei Ren, Weihua Wu, and William Perrizo
Department of Computer Science and Operations Research
North Dakota State University
[email protected]
Abstract
Association rule mining (ARM) finds all the
association rules in data that match some measures of
interest such as support and confidence In certain
situations where high support is not necessarily of
interest, fixed-consequent association-rule mining for
confident rules might be favored over traditional ARM.
The need for fixed consequent ARM is becoming more
evident in a number of applications such as market
basket research (MBR) or precision agriculture.
Highly confident rules are desired in all situations;
however, support thresholds fluctuate with the
applications and the data sets under study as we shall
show later. In this paper1, we propose an approach for
mining minimal confident rules in the context of fixedconsequent ARM that relieves the user from the burden
of specifying a minimum support threshold. We show
that the framework suggested herein is efficient and
can be easily expanded by adding new pruning
conditions pertaining to specific situations.
1. Introduction
Association rule mining [1] is considered one of the
most important applications, by far, to ever emanate
from the data mining community. It was initially
proposed by Agrawal et al in 1993 [1] for market
basket data in what is known as Market Basket
Research (MBR). MBR data is usually in the form of
transactions each containing a list of items bought by a
customer during a single visit to a store. An association
rule of the form XY relates two sets of items, X and
Y, where X is called the antecedent and Y the
consequent of the rule. The underpinning relation
follows the direction of the arrow (from X to Y) and is
then described as “customers who buy X are very likely
1
This work was partially supported by the GSA grant ACT#:
K96130308
to buy Y”. The degree of likeliness of an association
rule is determined by external measures associated
with it. Two ubiquitously used measures are support
and confidence [1] [2]. The former measures the
number of transactions containing both the antecedent
and consequent sets of items (i.e. statistical
significance of the rule); meanwhile, the latter is used
to measure the proportion of the transactions
containing the antecedent that also contain the
consequent (i.e. strength of the rule). The originally
proposed goal in [1] for association rule mining is to
find all the rules with support and confidence
surpassing some minimum user-specified thresholds.
A number of approaches such as [6] and [14] use
extra parameters to code user’s rule preferences, thus,
reducing the total number of produced rules. Some of
the suggested parameters in the literature include lift
[6], conviction [6] [19] and reliability [3]. It is worth
noting that the naming used for the parameters is not
standard. A point made against those measures is that
while they attempt to capture user preferences more
fully, they obviously overburden the user with more
parameter tuning. Other algorithms take a different
approach for coding user preferences by requiring the
rules to have special formats through the integration of
item constraints [5]. One popular type of item
constraints is the consequent constraint where all the
rules are required to have the consequent specified by
the user. Numerous applications exist for consequentconstraint-based ARM (aka fixed-consequent ARM)
such as in MBR where one is interested in finding
which items are sold with a specific set of items of
interest for the purpose of running promotions on it.
Though this information can also be obtained through
the use of the general ARM scheme proposed by
Agrawal et al and then the application of a “postARM” processing step to retain only those rules that
have the specified item set as consequents, we believe
that since users are not interested in the rest of the rules
it would not be optimal to burden them with more
processing time for generating additional rules that will
eventually be removed through the “post-ARM”
processing step.
In this paper, we motivate the need for fixedconsequent ARM and present an algorithm based on
set enumeration trees [17] and P-trees [10] [16] for this
purpose bearing in mind the importance of alleviating
the burden of user parameter tuning. Our algorithm
produces confident minimal rules in an efficient
manner as we shall experimentally demonstrate.
This paper is organized as follows: In section 2 we
explain the notion of fixed-consequent, confidencebased ARM and motivate its need. Section 3 discusses
the set enumeration tree structure which is the
underpinning framework for our algorithm. In section
4 we present our proposed algorithm. Section 5
discusses details about the implementation devised for
this work in addition to some performance analysis.
We finally end this paper with some concluding
remarks in section 6.
2. Towards mining confident, minimal,
fixed-consequent rules
As aforementioned, the algorithm proposed herein
is based on the fixed-consequent constraint. Users
specify the set of items they wish to have as a
consequent for all the produced rules and then the
algorithm returns the rules that match. Some
applications that motivate this type of environment
include MBR and agricultural data analysis.
2.1. Fixed-consequent ARM
As mentioned in our introduction, using MBR,
some stores may want to find all the rules that have a
certain item or set of items as a consequent for the
purpose of running some promotions on those specific
items. In this context, a rule of the form XY, where
X and Y are item sets and Y is user specified, can
inform the analyst that since people that buy X are
inclined to buy Y then to promote the sales of Y we
can run some promotion on Y that also includes X.
Perhaps the use of fixed-consequent ARM is more
evidently needed in the context of precision
agriculture. Agricultural data is usually described by
visible reflectance bands (Blue, Green and Red),
infrared reflectance bands (e.g., NIR, MIR1, MIR2 and
TIR) and possibly some other bands of data gathered
from ground sensors such as yield quantity, yield
quality, and soil attributes like moisture and nitrate
levels. For simplicity and without loss of generality,
we will consider the following relation, R, as a
prototypic relation for this type of data: R (Red, Green,
Blue, NIR, …, Yield quality). In this context, analysts
are usually interested in finding high yield quality
given other properties like Red, Green, Blue, etc…; as
a result, using high yield quality as our fixed
consequent, we get rules describing the best reflectance
bands combinations that produce high yield [11]. Some
applications in medical research investigate patient
records to analyze the characteristics that lead to the
death of patients (or the existence of chronic diseases);
such a task can be accomplished in a similar manner.
2.2. Confidence-based ARM
We take confidence-based ARM to mean that users
only specify the confidence threshold of interest. In
general, users are always interested in high-confidence
rules. The higher the confidence of a rule, the lower the
error rate of generalizing it over the data set. A rule
with a confidence value of 95% tells us that we are
tolerating an error rate of 5% when we assume the
validity of the corresponding rule. We believe it should
be totally up to the user to specify the error rate that
can be tolerated.
In spite of the fact that we are always interested in
rules with high confidence values, we argue that
support fluctuates depending on the data set we operate
on. In some data sets, high support values are always
desired. This is the case with MBR data because store
managers always like to see high support values for
their rules. However, this is not the case with other
data sets. Considering our medical research example
again, let us suppose that we have a large database of
patient records. We are interested in finding what
combinations of attributes stored in those records that
imply the death of the patient. Ideally, we would
require our ARM application to detect such rules early
on so that we could attempt to diagnose the problem –
if any – before the support of those rules increases in
the database. Similar considerations apply to fraud and
copy detection.
In [9], Cohen et al motivates the need for lowsupport, high-confidence rules by observing that
usually high support rules are obvious and well-known
unlike low-support ones which provide “interesting”
insights into the data. Example applications for lowsupport high-confidence ARM mentioned in [9]
include copy detection [18] (for identifying similar
documents) and collaborative filtering (for making
recommendations to users based on behavior similarity
to others) [20].
We have devised our algorithm to generate the
highest support rules that match the user’s specified
minimum threshold without having the user to specify
any support threshold. By doing so, we would reduce
the number of parameters supplied on behalf of the
user relieving her from the burden of determining an
optimal value for the support threshold in order to
produce meaningful and interesting rules.
2.3. Rule minimality property
We will assume that the minimum confidence
threshold c (usually high) provided by the user
represents the threshold of interest; as a result, any two
rules having confidence values exceeding c, where the
antecedent of one of them is a superset of the
antecedent of the other, are to a great extent equally
interesting to the user. Support comes into the picture
to tip the scales in the direction of the rule with higher
support. For example, suppose we generate two rules,
R1 and R2, with confidence values greater than the
confidence threshold where R1 is “formula milk” 
“diapers” and R2 is “formula milk”, “baby shampoo”
 “diapers”. R1 and R2 are considered very equally
interesting to the user, and thus we select only the one
with greater support value and prune the other. Since
R2’s antecedent is a superset of R1’s, then R1’s support
is surely greater than or equal to R2’s. A more
contextual justification for our choice would be by
arguing that since the user knows that “formula milk”
 “diapers” from R1 no new knowledge is given by
R2:“formula milk”, “baby shampoo”  “diapers”.
In this context, we refer to R1 as a minimal rule and
R2 as non-minimal. Formally speaking, a rule, R, is
said to be non-minimal, if there exists at least one other
rule, S, such that both rules have confidence values
greater than the minimum confidence threshold, have
the same consequent, and the antecedent of R is a
superset of that of S. Producing minimal rules can be
viewed as an extension over previous approaches in the
literature such as [9] and [12] which restrict their work
to rules of item pairs only (i.e. one item in the
antecedent and one item is the consequent).
The work in [4] [7] [15] support our claim of the
importance of minimal rules. [4] uses the notion of
non-redundant or minimal rules to refer to rules with
the smallest possible antecedents and largest possible
consequents. This matches our rule minimality
definition to a large extent since we also require the
antecedent to be as small as possible; however, we
relax the requirement on the consequent because we
are dealing with fixed consequents. [7] emphasizes the
importance of “small” rules in the context of genome
analysis while [15] extends it to medical data.
3. Set enumeration trees
The algorithm presented herein is modeled after a
well-known set enumeration structure that supports
complete and non-redundant search of an item space,
the Set Enumeration (SE) tree [17]. The SE tree
framework provides us with a scheme for enumerating
all possible subsets of an item set without redundancy
by imposing an ordering on the set of items, and for
presenting them graphically in a tree structure thus
shifting our problem from a random subset search
problem to an organized tree search problem. The
reader is referred to [17] for more details on SE trees.
In the general context of ARM, a number of
publications in the literature have suggested using SE
trees for mining association rules. [8] proposes two
new tree data structures based on SE trees along with
algorithms that demonstrate significant speed and
storage savings. [5] presents an interesting approach
for mining association rules with fixed consequents at
very low support thresholds. As we shall demonstrate
later in this paper, the structure of the SE tree can be
utilized for our problem too as it provides a framework
which facilitates easy augmentation of pruning
conditions along its branches thus reducing the space
of considered item subsets and focusing only on what
adheres to our notion of “interestingness”.
4. The proposed approach
We say that transaction, t, supports an item set, I, if
every item in I exists in t. As in [1] and [2], the support
of an item set is defined by the proportion of
transactions supporting that item set. The support of a
rule is the support of the union of all item sets existing
on both sides of the rule. The confidence of a rule is
defined as the proportion of transactions supporting the
antecedent of the rule that also supports the consequent
of the rule and is calculated as the ratio between the
support of the rule and the support of the antecedent of
the rule.
4.1 Pruning
After the user specifies a minimum confidence
threshold, minconf, and a consequent item set, C, our
algorithm proceeds by the creating an SE-tree
structure. Every node in the tree represents the
antecedent item set of the rule IC. Whenever we
create a new node, n, in the tree and label it with an
item set, I, we apply the pruning conditions listed next
to check whether we can terminate the processing at n
pre-maturely thus reducing the space. By terminating a
node, we stop further processing at that node and no
longer list any item sets under it. We exploit the
following three pruning conditions:
 Zero-confidence pruning: If the confidence
(IC) = 0, then terminate n.

One-support pruning: If the support (IC) = 1,
then terminate n.
 Minimality pruning: If the confidence (IC) >=
minconf, then terminate n
Zero-confidence pruning is a simple and straight
forward pruning condition based on the observation
that if the support of a certain item set, I, is zero then
every superset of I has a zero support; as a result, any
rule formed by replacing the antecedent of a rule
having a zero confidence by a superset of that
antecedent will also have a zero confidence. To justify
that, suppose a rule IC has a support of zero (i.e.
support(I U C) = 0); as a result, the confidence of IC
is also zero – recall that confidence(IC) =
support(IC) / support(I). Suppose that I’ is a superset
of I; we know that support of (I’ U C) is zero since (I’
U C) is a superset of (I U C). As a result, support of
(I’C) is zero and thus confidence of I’C is also
zero. In short, if we know that a rule I C has a zero
confidence, then there is no need to process the
supersets, I’, of I and calculate the confidence values of
the rules I’C.
Rules with support equal to 1 are based on a single
sample from the database and thus are not rules per se.
Note that if the support of any rule IC is 1, then any
non-minimal rule formed from this rule can not have a
support greater than 1. In practice we can easily
combine the previous two pruning conditions by
pruning a node if its support is less than or equal to 1;
this is because the confidence of a rule is equal to zero
if and only if its support is equal to zero. Hereinafter,
we shall refer to those two pruning conditions
combined as the support-less-than-two pruning.
The minimality pruning condition is based on our
previous argument presented in section 2.3 for the need
to remove non-minimal rules. Once we generate a rule
R: IC with confidence value exceeding minconf, we
terminate the node, n, that generated R because all
successor nodes of n may only generate non-minimal
rules of R according to our definition of minimality.
As noted earlier, other pruning conditions that
pertain to special situations can be easily added to our
set of conditions and tested in a similar manner. Each
pruning condition when satisfied results in node
termination.
Our algorithm employs the two pruning conditions,
support-less-than-two pruning and minimality pruning,
to prune the SE tree. Every time we generate a new
item set, I, at some node, n, three scenarios may
transpire:
 Support of (IC) is less than 2; we terminate n
 Confidence of (IC) is in the range (0, minconf)
exclusively; we continue processing without
terminating n

Confidence of (IC) is greater or equal to
minconf; we terminate n and output IC as a rule
The formal algorithm is depicted in Figure 2. Note
that our algorithm is complete in terms of producing all
confident minimal rules. We omit the proof due to
space limitations.
Input: an ordered list of items, minconf, and the
consequent item set
Output: set of minimal rules satisfying minconf
Algorithm:

Create the root node (the empty set)

For every item, I, in our ordered list of items, do
the following:
o Generate a new node, n
o Insert n under the root from right to left
o Label n with I
o Depending on the resulting confidence and
support of (IC), proceed with one of the
following:

If support of (IC) is less than 2 then
terminate n

If confidence of (IC) is greater or equal
to minconf then terminate n and output
IC as a rule

If confidence of (IC) is in the range (0,
minconf) exclusively then for every unterminated node, Unter-n, to the right n
(working from highest level in the tree
down, and at each level processing nodes
right to left)
 Generate a new node, nn
 Insert nn under n from right to left,
 Label nn with the label of Unter-n.
(Note that the item set represented by nn
is the union of the items in n and Untern)
 Recursively repeat the confidence
testing process for nn
Figure 2. The formal algorithm.
4.2 Ensuring rule minimality
Suppose we have a set of items {1, 2, 3, …, 11} with
11 being the fixed consequent. Suppose further that we
are currently working with node 1 and that to the right
of 1 we have two un-terminated nodes, 9 and 8,9. Both
nodes will be considered with 1 because node 8,9 falls
under node 8 in the tree and is thus independent of the
result of the join of 1 and 9. A possible scenario would
be for both rules, 1,911 and 1,8,911, to succeed
thus we end up generating a non-minimal rule,
1,8,911 in this case.
Another scenario we need to consider is when rule
1,911 has support 0 or 1, then there is no need to try
rule 1,8,911 because its support is also going to be 0
or 1. In this scenario, our algorithm will not produce
undesired rules but will test for rules whose outcome is
known beforehand which undesirable because this
involves the time-consuming process of scanning a
large database to compute the support and confidence
of the rule – in cases where a non-vertical database
layout is adopted.
To rectify this problem, we associate a temporary
taboo list (TL) with every item, I, which will save all
the nodes that are pruned under I. For example, if
1,911 is not a confident rule or its support is less
than 2, then we append 9 to 1’s taboo list (TL1 for
short). In later steps, before we test for a new potential
rule under node 1, 1,X11, we check if any subset of
X is in TL1. If so, we directly terminate X under 1 and
add X to TL1. This may seem unfeasible at first;
however, an efficient implementation has been devised
as we shall discuss in more details in section 5.2.
parent node. In practice, we do not directly store the
number of 1s, but a 1 if the node contains only 1s and a
0 otherwise. Nodes containing only 1s or only 0s are
considered pure (otherwise they are mixed) and are not
partitioned further – called pure-1 and pure-0,
respectively. This aspect is referred to as P-tree
compression [10] [16] and is one of the most important
characteristics of the P-tree technology. Note that we
can easily differentiate between pure-0 nodes and
mixed nodes by the fact that pure-0 nodes have no
children (i.e. they are leaf nodes). Figure 4 shows the
P1-trees corresponding to the P-trees in Figure 3.
P1-trees are manipulated using operations such as
AND, OR, NOT and ROOTCOUNT (the count of the
number of “1”s in the bit group represented by the tree)
in order to query the underlying data. The reader is
referred to [10] [16] for more details on P-trees and
their logical operators.
Column1 Column2
0
1
0
1
0
1
0
1
1
0
0
0
1
1
1
1
5. Implementation & performance analysis
5.1. The P-tree technology2
Our implementation for this work is based on an
efficient data structure, the P-tree (Predicate or Peano
tree). P-trees are tree-like data structures that store
relational data in a loss-less compressed column-wise
format by splitting each attribute into bits, grouping
bits at each bit position, and representing each bit
group by a P-tree. To create P-trees from transactional
binary data, we store all bit values in each binary
attribute for all the transactions separately. In other
words, we group all bits, for all transactions t in the
table, in each binary attribute, separately. Each such
group of bits is called a bit group. Figure 3 shows the
process of creating P-trees from a binary relational
table. Part b) of the figure shows the creation of two bit
groups, one for each attribute in a). Parts c) and d)
show the resulting P-trees, P1 and P2, one for each of
the bit groups in b). P1 and P2 are constructed by
recursively partitioning the bit groups into halves.
Each P-tree will record the total number of 1s in the
corresponding bit group on the root level. The second
level in the tree gives the number of 1s in each of the
halves of the bit group. The first node from the left on
the second level will give the number of 1s in the first
half of the bit group; similarly, the second node will
give the number of 1s in the second half of the bit
group. This logic is continued throughout the tree with
each node recording the number of 1s in either the first
or the second half (depending on whether it is the left
or right node) of the bit group represented by the
2
Patents are pending for the P-tree technology.
a) A 2-column table
0
0
0
0
1
0
1
1
1
1
1
1
0
0
1
1
b) The resulting two
bit groups
6
3
0
3
1
4
2
2
0
2
1 0
d) P2
c) P1
Figure 3. Construction of basic P-trees.
0
0
0
0
0
1
1
1
0
0
1
1 0
a) P1
b) P2
Figure 4. Pure-1 Trees.
Note that by using P-trees, we can compute the
confidence of rules in a quick and efficient manner
(without any database scans). Each item i will be
represented by a P-tree, Pi; to get the support of an
item, we issue a ROOTCOUNT operation on the P-tree
of that item. To get the support of an item set of size
more than one, we AND the P-trees of the items in the
item set and then issue a ROOTCOUNT operation on
the result.
5.2. Taboo lists
For every new item I, we create a new node under
the root and test for potential rules with all possible unterminated nodes lying to I’s right. The actual node
creation order of the SE tree is the lexicographic subset
ordering of the items in the given item list. For
example, the node creation order for the set {1,2,3}
would be: {1}, {2}, {1,2} ,{3}, {1,3}, {2,3}, {1,2,3}.
This ensures that every item set has all its subsets
before it.
For every item, I, we maintain a TLI which saves
the item sets whose outcomes, when joined with I, are
known beforehand and thus need not be tested. In our
implementation, each TLI is a P-tree having a size
equal to the number of un-terminated nodes to the right
of I (i.e. all nodes that I will be tested with). A value 1
is used for nodes which, when joined with I, result in
the firing of some pruning condition; thus, none of
their supersets need to be tested with I. The remaining
TLI entries will be 0s. For example, for the set of items
{1,2,3,4}, suppose that the nodes created so far are:
{1}, {2}, {1,2} ,{3}, {1,3}, {2,3}, {1,2,3}. For node 4,
we initialize a TL4 having 7 entries all containing 0s
initially. If the joining of item 4 with node {2} results
in firing at least one pruning condition, then the second
entry pointing to item set {2} is flagged with 1, and so
are all entries containing 2 (i.e. entries pointing to
{1,2}, {2,3} and {1,2,3}). But how can we efficiently
tell which other entries contain item 2? We maintain
for each item an index list as a P-tree (called an index
P-tree) that has a 1 value for every position that this
item exists in. For example, item 1 will have the
following index list (it will be stored as a P-tree but we
are just listing the entries in a list for convenience):
1,0,1,0,1,0,1. In other words, viewing the nodes of the
SE tree in node creation order, item 1 occurs in node
positions 1, 3, 5 and 7. Every new node added to the
SE tree results in the expansion of all index P-trees by
either a 1, if the corresponding item is in the new node
added, or a 0, otherwise.
Going back to the previous scenario where the
joining of item 4 with node {2} results in firing at least
one pruning condition and thus node {2} need to be
added to TL4, we simply OR the index P-tree of item 2
with TL4. In general, if we want to add node {x,
y,…,z} to some taboo list, TLI, we AND the index Ptrees for all items in the node (i.e. AND index P-tree of
x with that of y … with that of z). The result will give
us where item set {x, y,…,z} occurs. We then OR the
resulting P-tree index with TLI which results in
appending node {x, y,…,z} to TLI.
We maintain the taboo lists and index lists as Ptrees as this will give us faster logical operations and
compression. In addition, in the case of taboo lists, it
could speed the node traversal especially in cases
where there many consecutive 1’s. For example,
suppose the entries in a taboo list are: 1111 0011.
Figure 5 below shows the corresponding P-tree of the
given taboo list.
0
1
0
0
1
Figure 5. The resulting taboo list in P-tree
format.
In this example, instead of going through the first
four nodes sequentially and then skipping them
because they are flagged, using a P-tree to represent
the taboo list, we can directly skip the first 4 entries
because they form a pure-1 node on 2nd level of the Ptree.
It is worth mentioning that our traversal through the
itemset space using taboo lists is very similar to a
popular approach used in AI literature and known as
Tabu search [13]. The idea in Tabu search is to traverse
the space in a more effective manner by avoiding
moves that result in revisiting points in the space
previously visited whose outcome is known not to be
acceptable (hence the name "tabu”). The fact that the
union of I and X produces an infrequent itemset implies
that future joins of I with any superset of X will
produce an infrequent itemset; a scenario similar in
essence to revisiting a point in the search space whose
outcome is known to be unsatisfactory and which
could be circumvented by putting the point on a Tabu
list. Because of the difference in context and problem
definition, we refer to our lists as taboo lists instead of
Tabu lists.
5.3. Comparison analysis
To the best of our knowledge, no previous work has
attempted to mine the type of rules we are considering.
We find [5] to be particularly interesting as it proposes
an algorithm called Dense-Miner which is capable of
mining association rules with fixed consequents at very
low support thresholds. For the lack of a better
benchmark, we will compare our approach with DenseMiner; however, we have to emphasize that a number
briefly describes the two datasets by listing the number
of transactions, items, and items per transaction for
each data set.
Connect-4 Dataset
250
Number of rules
200
150
100
50
99
99
.5
80
85
70
75
90
95
91
.5
90
91
74
80
85
Table 3.
49046
7117
70
75
PUMSB
60
65
43
50
55
129
40
45
67557
30
35
Connect-4
160000
140000
120000
100000
80000
60000
40000
20000
0
20
25
Items per
trans.
PUMSB-4 Dataset
Number of rules
Items
60
65
Confidence Threshold (%)
Table 2. Data sets description
Trans.
50
55
40
45
30
35
0
20
25
of fundamental differences exists between the two
approaches which we briefly outline next:
 Dense-Miner mines all association rules while we
only mine minimal, confident rules
 Dense-Miner uses support as a pruning mechanism
while this is not the case in our work (In reality we
also use support pruning, but in our case, the
support threshold is always set to 2 – as an
absolute threshold)
In terms of rule overlap between the two
approaches, all rules produced by our approach that
have a support value greater than the minimum support
threshold used for Dense-Miner will be produced by
Dense-Miner also.
Confidence Threshold (%)
P-tree based
Connect-4 Dataset
Figure 7. Number of rules produced.
Dense Miner
3500
3000
Time (s)
2500
2000
1500
1000
500
0
20
30
40
50
60
70
80
90 100
Confidence Threshold (% )
P-tree based
PUMSB Dataset
Dense Miner
2500
Time (s)
2000
1500
1000
500
0
40
50
60
70
80
90
100
Confidence Threshold (% )
Figure 6. Speed comparison.
All experiments were conducted on a P-II 400 with
128 SDRAM running Red hat Linux 9.1. C++ was
used for coding. We experimented on two real-life
dense data sets, Connect-4 and PUMSB, which are
available at the UCI data repository. Table 2 below
Figure 6 shows the time in seconds needed to mines
rules at different confidence thresholds by our
approach (P-tree based) and Dense-Miner. As
aforementioned, Dense-Miner mines all rules using
support pruning. It uses a variant of support referred to
as coverage and defines the minimum coverage
threshold as the minimum support divided by support
of the fixed consequent. Results for Dense-Miner are
observed with minimum coverage threshold fixed at
1% and 5%, respectively.
Our approach, on the other hand, mines only
minimal rules without using any support pruning. It is
very clear from the figure that users interested in
minimal rules without support would prefer our
approach as the time needed is many orders of
magnitude less than that of Dense-Miner.
The user might notice from both parts of Figure 6
how the two approaches differ in the way they produce
the rules. Dense-Miner takes more time at lower
confidence while our approach takes more time at
higher confidence thresholds. This is mainly because
our approach mines minimal rules using only
confidence pruning and the higher the confidence
threshold, the more difficult it would be to get
confident rules high in the SE tree; as a result, the SE
tree grows deeper and thus requires more time to
traverse. Dense-Miner, on the other hand, mines all
rules and it is very logical to have more rules satisfying
lower confidence thresholds (and vice versa) which
obviously requires more time to mine.
The number of rules produced by Dense-Miner
ranges from around 500,000 rules to less than 10 rules
over both data sets. Figure 7 shows the number of rules
produced by our approach over the two data sets at the
different confidence thresholds. The same discussion
presented in the previous paragraph applies here
regarding the larger (smaller) number of rules
produced at higher (lower) confidence thresholds.
6. Conclusion
In this paper we proposed a framework based on
SE-trees and the P-tree technology for extracting
minimal, confident rules using fixed-consequent ARM.
Our methodology relieves the user from the burden of
specifying a minimum support threshold by extracting
the highest support rules that satisfy user confidence
threshold. Albeit, to the best of our knowledge, no
previous work has attempted to mine minimal,
confident rules with fixed consequents, we provide a
comparison analysis study showing how well we
compare to other close approaches in the literature.
In terms of limitations, we acknowledge that our
approach suffers in situations where the desired rules
lie deep in the tree because a large number of nodes
and levels need to be traversed then. A future direction
in this area targets finding measures for estimating the
probability of rule availability along certain branches
and quitting early in cases where such probability is
low.
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