Visualizing Association Rules in Hierarchical Groups
Michael Hahsler and Sudheer Chelluboina
May 19, 2011
Abstract
Association rule mining is one of the most popular data mining methods. However, mining
association rules often results in a very large number of found rules, leaving the analyst with the
task to go through all the rules and discover interesting ones. Sifting manually through large sets
of rules is time consuming and strenuous. Visualization has a long history of making large amounts
of data better accessible using techniques like selecting and zooming. However, most association
rule visualization techniques are still falling short when it comes to a large number of rules. In this
paper we present a new interactive visualization technique which lets the user navigate through a
hierarchy of groups of association rules. We demonstrate how this new visualization techniques can
be used to analyze a large sets of association rules with examples from our implementation in the
R-package arulesViz.
1 Introduction
Many organizations are generating a large amount of transaction data on a daily basis. For example,
a department store like \Macy's" stores customer shopping information at a large scale using check-out
data. Association rule mining is one of the major techniques to detect and extract useful information
from large scale transaction data. Mining association rules was st introduced by Agrawal et al. (1993)
and can formally be dened as:
Let I = fi1; i2; : : : ; ing be a set of n binary attributes called items. Let D = ft1; t2; : : : ; tmg be a set
of transactions called the database. Each transaction in D has a unique transaction ID and contains a
subset of the items in I. A rule is dened as an implication of the form X ) Y where X;Y  I and
X \Y = ;. The sets of items (for short itemsets) X and Y are called antecedent (left-hand-side or LHS)
and consequent (right-hand-side or RHS) of the rule. Often rules are restricted to only a single item in
the consequent.
Association rules are rules which surpass a user-specied minimum support and minimum condence
threshold. The support supp(X) of an itemset X is dened as the proportion of transactions in the
data set which contain the itemset and the condence of a rule is dened conf(X ) Y ) = supp(X [
Y )=supp(X). Therefore, an association rule X ) Y will satisfy:
supp(X [ Y )  
and
conf(X ) Y )  
where  and  are the minimum support and minimum condence, respectively. Note that both minimum
support and minimum condence are related to statistical concepts. Finding itemsets which surpass
a minimum support threshold can be seen as a simplication of the unsupervised statistical learning
problem called \mode nding" or \bump hunting" (Hastie et al., 2001) where each item is seen as a
variable and the goal is to nd prototype values so that the probability density evaluated at these values
is suciently large. Minimum condence can be interpreted as a threshold on the estimated conditional
42nd Symposium on the Interface: Statistical, Machine Learning, and Visualization Algorithms (Interface 2011)
1

probability P (Y jX), the probability of nding the RHS of the rule in transactions under the condition
that these transactions also contain the LHS (see e.g., Hipp et al., 2000).
Another popular measure for association rules used throughout this paper is lift (Brin et al., 1997).
The lift of a rule is dened as
lift(X ) Y ) =
supp(X [ Y )
supp(X)supp(Y )
and can be interpreted as the deviation of the support of the whole rule from the support expected under
independence given the supports of both sides of the rule. Greater lift values ( 1) indicate stronger
associations. Measures like support, condence and lift are generally called interest measures because
they help with focusing on potentially more interesting rules.
For example, let us assume that we nd the rule fmilk, breadg ) fbutterg with support of 0.2,
condence of 0.9 and lift of 2. Now we know that 20% of all transactions contain all three items together,
the estimated conditional probability of seeing butter in a transaction under the condition that the
transaction also contains milk and bread is 0.9, and we saw the items together in transactions at double
the rate we would expect under intependence between the itemsets fmilk, breadg and fbutterg. For
a more detailed treatment of association rules we refer the reader to the introductory paper for the
R-package arules (Hahsler et al., 2005) and the literature referred to there.
Association rules are typically generated in a two-step process. First, minimum support is used to
generate the set of all frequent itemsets for the data set. Frequent itemsets are itemsets which satisfy
the minimum support constraint. Then, in a second step, each frequent itemsets is used to generate all
possible rules from it and all rules which do not satisfy the minimum condence constraint are removed.
Analyzing this process, it is easy to see that in the worst case we will generate 2n n 1 frequent
itemsets with more than two items from a database with n distinct items. Since each frequent itemset
will in the worst case generate at least two rules, we will end up with a set of rules in the order of O(2n).
Typically, users increase the minimum support threshold  to keep the number of association rules found
at a manageable size. However, this has the disadvantage that it removes potentially interesting rules
with lower support. Therefore, the need to deal with large sets of association rules is unavoidable when
applying association rule mining in a real setting.
Many researchers introduced visualization techniques like scatter plots, matrix visualizations, graphs,
mosaic plots and parallel coordinates plots to analyze association rules. However, most existing visual-
ization techniques are not suitable for displaying large sets of rules. This paper introduces a completely
new method called \grouped matrix-based visualization" which is based on a novel way of creating nested
groups of rules (more specically antecedent itemsets) via clustering. The nested groups form a hierarchy
which can be interactively explored down to the individual rule.
The rest of the paper is organized as follows. In Section 2 we discuss existing visualization techniques
for association rules. In Section 3 we introduce matrix-based visualization and illustrate with an example
its shortcomings. The new grouped matrix-based visualization is introduced and illustrated in Section 4.
Section 5 concludes the paper.
2 Related Work
In the last decade several visualization techniques have been developed. In the following we review the
most important types. A more thorough overview is provided by Bruzzese and Davino (2008).
A straight-forward visualization of association rules is to use a scatter plot with two interest measures
(typically support and condence) on the axes. Such a presentation can be found already in an early
paper by Bayardo, Jr. and Agrawal (1999) when they discuss sc-optimal rules. Unwin et al. (2001)
introduced a special version of a scatter plot called Two-key plot. Here support and condence are used
for the x and y-axes and the color of the points is used to indicate \order," i.e., the number of items
contained in the rule. Scatter plots work well for very large sets of association rules and zooming into
the plot can be easily implemented. The main drawback is there is not enough space in the plot the
display the labels of the items in the rules. In an interactive version of this plot this information can be
obtained by selecting a single or a small set of rules. However, this still makes exploring a large set of
rules time-consuming.
2

Graph-based techniques (Klemettinen et al., 1994; Rainsford and Roddick, 2000; Buono and Costabile,
2005; Ertek and Demiriz, 2006) visualize association rules using vertices and directed edges where vertices
typically represent items or itemsets and edges connect antecedents and consequents in rules. Interest
measures are typically added to the plot as labels on the edges or by color or width of the arrows
displaying the edges. Graph-based visualization oers a very clear representation of rules but they tend
to easily become cluttered and thus are only viable for very small sets of rules. To explore large sets of
rules with graphs, dierent layout mechanisms and advanced interactive features like zooming, ltering,
grouping and coloring nodes are vertices. Such features are available in interactive visualization and
exploration platforms for networks and graphs like Gephi (Bastian et al., 2009).
Parallel coordinates plots are designed to visualize multidimensional data where each dimension is
displayed separately on the x-axis and the y-axis is shared. Each data point is represented by a line
connecting the values for each dimension. Parallel coordinates plots were used previously to visualize
discovered classication rules (Han et al., 2000) and association rules (Yang, 2003). Yang (2003) displays
the items on the y-axis as nominal values and the x-axis represents the positions in a rule, i.e., rst item,
second item, etc. Instead of a simple line an arrow is used where the head points to the consequent item.
Arrows only span enough positions on the x-axis to represent all the items in the rule, i.e., rules with
less items are shorter arrows. Parallel coordinates plots are prone to clutter caused by crossing lines.
Reordering the items on the x and y-axes can alleviate some of the problems but it still makes large sets
of rules hard to analyze.
A double decker plot is a variant of a mosaic plot which displays a contingency table using tiles
created by recursively vertical and horizontal splitting a rectangle. The size of each tile is proportional
to the value in the contingency table. Double decker plots use only a single horizontal split. Hofmann
et al. (2000) introduced double decker plots to visualize a single association rule. Here the displayed
contingency table is computed for a rule by counting the occurrence frequency for each subset of items
in the antecedent and consequent in the original data set. The items in the antecedent are used for the
vertical splits and the consequent item is used for horizontal highlighting. Although this visualization is
powerful for analyzing a single rule, it cannot be used for large sets of rules.
Another popular method is matrix-based visualization. We introduce it in more detail in the following
section since it is related to the new technique presented in this paper.
3 Matrix-based Visualization
Matrix-based visualization techniques organize the antecedent and consequent itemsets on the x and
y-axes, respectively. A selected interest measure is displayed at the intersection of the antecedent and
consequent of a given rule. If no rule is available for an antecedent/consequent combination the inter-
section area is left blank.
Formally, the visualized matrix is constructed as follows. We start with the set of association rules
R = fhX1; Y1; 1i; : : : ; hXi; Yi; ii; : : : ; hXn; Yn; nig
where Xi is the antecedent, Yi is the consequent and i is the selected interest measure for the i-th rule,
i = 1; : : : ; n. In R we identify the set of A unique antecedents and C unique consequent. We create a
A  C matrix M = (mac), a = 1; : : : ; A and c = 1; : : : ; C, with one column for each unique antecedent
and one row for each unique consequent. We populate the matrix by setting mac = i where i = 1; : : : ; n
is the rule index, and a and c correspond to the position of Xi and Yi in the matrix. Note that M will
contain many empty cells since many potential association rules will not meet the required minimum
thresholds on support and condence.
Ong et al. (2002) presented a version of the matrix-based visualization technique where a 2-
dimensional matrix is used and the interest measure is represented by color shading of squares at the
intersection. An alternative visualization option is to use 3D bars at the intersection (Wong et al., 1999;
Ong et al., 2002).
For this type of visualization the number of rows/columns depends on the number of unique itemsets
in the consequent/antecedent in the set of rules. Since large sets of rules typically have a large number
3

of dierent itemsets as antecedents (often not much smaller than the number of rules themselves), the
size of the colored squares or the 3D bars gets very small and hard to see.
We illustrate matrix-based visualization using the R-package arulesViz (Hahsler and Chelluboina,
2011) an extension for package arules (Hahsler et al., 2010). For the examples in this paper we load the
\Groceries" data set which is included in arules.
> library("arulesViz")
> data("Groceries")
> Groceries
transactions in sparse format with
9835 transactions (rows) and
169 items (columns)
Groceries contains sales data from a local grocery store with 9835 transactions and 169 items (product
groups). The data sets most popular item is \whole milk" and the average transaction contains less than
5 items. Next we mine association rules using the Apriori algorithm implemented in arules. We use
 = 0:001 and  = 0:5.
> rules <- apriori(Groceries, parameter = list(support = 0.001,
+ confidence = 0.5), control = list(verbose = FALSE))
> rules
set of 5668 rules
The result is a set of 5668 association rules. The top three rules with respect to the lift measure are:
> inspect(head(sort(rules, by = "lift"), 3))
lhs rhs support confidence lift
1 {Instant food products,
soda} => {hamburger meat} 0.001220132 0.6315789 18.99565
2 {soda,
popcorn} => {salty snack} 0.001220132 0.6315789 16.69779
3 {flour,
baking powder} => {sugar} 0.001016777 0.5555556 16.40807
These rules represent easy to explain purchasing patterns. However, it is clear that going through all
the 5668 rules manually is not a viable option. Therefore, we create a matrix-based visualization using
shaded squares and 3D bars.
> plot(rules, method = "matrix", measure = "lift")
> plot(rules, method = "matrix3D", measure = "lift")
The resulting plots are shown in Figures 1 and 2. The rules contain 4097 unique antecedent and
25 unique consequent itemsets. Since there is not much space for long labels in the plot, we only show
numbers as labels for rows and columns (x and y-axis) and the complete itemsets are printed to the
terminal for look-up. We omit the complete output here, since this plot print several thousand labels in
the following form to the screen.
Itemsets in Antecedent (lhs)
[1] "{liquor,red/blush wine}"
[2] "{curd,cereals}"
[3] "{yogurt,cereals}"
[4] "{butter,jam}"
[5] "{soups,bottled beer}"
4

Matrix with 5668 rules
1000 2000 3000 4000
5
10
15
20
25
Antecedent (LHS)
Co
nse
que
nt (R
HS)
5
10
15
lift
Figure 1: Matrix-based visualization with colored squares.
(lines omitted)
[343] "{tropical fruit,root vegetables,rolls/buns,bottled water}"
[344] "{tropical fruit,root vegetables,yogurt,rolls/buns}"
Itemsets in Consequent (rhs)
[1] "{bottled beer}" "{whole milk}" "{other vegetables}"
[4] "{tropical fruit}" "{yogurt}" "{root vegetables}"
(lines omitted)
Obviously matching the labels to the entries on the x and y-axis is cumbersome. In order to be
able to print the complete labels on the axes we would have to reduce the number of rules signicantly
to typically less than 100 rules. Alternatively, rules in the plot can be interactively selected to reveal
the rule's antecedent and consequent itemsets, but the plot is so crowded, that it is almost impossible
to select a specic rule. In Hahsler and Chelluboina (2011) we experimented with several reordering
strategies to improve the plots usefulness for large number of rules, but only with very limited success.
This illustration clearly shows that the usefulness of simple matrix-based visualization is very limited
when facing large rule sets.
4 Grouped matrix-based visualization
Matrix-based visualization is limited in the number of rules it can visualize eectively since large sets
of rules typically also have large sets of unique antecedents/consequents. Here we introduce a new
visualization techniques that enhances matrix-based visualization by grouping rules via clustering to
handle large sets of rules. Groups of rules are presented by aggregating rows/columns of the matrix.
The groups are nested and organized hierarchically allowing the user to explore them interactively by
zooming into groups.
A direct approach to clustering itemsets (and rules) is to dene a distance metric between two itemsets
Xi and Xj . Distance between two sets can be measured for example by the Jaccard distance dened as
dJaccard(Xi; Xj) = 1
jXi \Xj j
jXi [Xj j
:
This distance is based on the number of items that Xi and Xj have in common divided by the
number of unique items in both sets and was called for clustering association rules conditional market-
basket probability by Gupta et al. (1999). For a set of m rules we can calculate the m(m 1)=2 distances
between the sets of all items in each rule and use them as the input for clustering. However, using
clustering on the itemsets creates several problems. First of all, data sets typically mined for association
rules are high-dimensional, i.e., contain many dierent items. This high dimensionality carries over to
the mined rules and leads to a situation referred is as the \course of dimensionality" where, due to the
exponentially increasing volume, distance functions lose their usefulness. The situation is getting worse
5

Matrix with 5668 rules
0 5 10 15 20 25
0

5
10
15
20
0
1000
2000
3000
4000
5000
Consequent (RHS)
An
tec
ede
nt (L
HS)
lift
Figure 2: Matrix-based visualization with 3D bars.
since minimum support used in association rule mining leads in addition to relatively short rules resulting
in extremely sparse data.
Several approaches for clustering association rules and itemsets to address the dimensionality and
sparseness problem were proposed in the literature. Toivonen et al. (1995) and Berrado and Runger
(2007) propose clustering association rules by looking at the number of transactions which are covered
by the rules. A transaction is covered by a rule if it contains all the items in the rule's antecedent. Using
common covered transactions avoids the problems of clustering sparse, high-dimensional binary vectors.
However, it introduces a strong bias towards clustering rules which are generated from the same frequent
itemset. By denition two subsets of a frequent itemset will cover many common transactions. This bias
will lead to just rediscovering the already known frequent itemset structure from the set of association
rules.
Here we pursue a completely dierent approach. We start with the matrix M dened in Section 3
which contains the values of a selected interest measure of the rules in set R. The columns/rows are the
unique antecedents/consequents in R, respectively. Now grouping rules becomes the problem of grouping
columns or rows in M.
Since for most applications the consequents in mined rules are restricted to a single item there is no
problem with combinatorial explosion and we can restrict our treatment to only grouping antecedents
(i.e., columns in M). However, note that the same grouping method can be used also for consequents.
We use the interest measure lift here, but other interest measures can be used as well. The idea for lift
is that antecedents that are statistically dependent on the same consequents (i.e., have a high lift value)
are similar and thus should be grouped together. Compared to other clustering approaches for itemsets,
this method enables us to even group antecedents containing substitutes (e.g., butter and margarine)
which are rarely purchased together since they will have a similar dependence relationship with the
same consequents (e.g., bread). Clustering based on shared items or common covered transaction cannot
uncover this type of relationship.
For grouping we propose to split the set of antecedents into a set of k groups S = fS1; S2; : : : ; Skg
6

while minimizing the within-cluster sum of squares
argminS
kX
i=1
X
mj2Si
jjmj ijj
2;
where mj , j = 1; : : : ; A, is a column vector representing all rules with the same antecedent and i is
the center (mean) of the vectors in Si. Minimizing the stated loss function is known as the k-means
problem which is NP-hard (Aloise et al., 2009). However, several good and fast heuristics exist which
do not require a precomputed distance matrix. We use the k-means algorithm by Hartigan and Wong
(1979) and restart it 10 times with random initialized centers to nd a suitable solution.
To visualize the grouped matrix we use a balloon plot with antecedent groups as columns and con-
sequents as rows (see Figure 3). The color of each balloon represents the aggregated interest measure
in the group and the size of the balloon shows the aggregated support. Aggregation in groups can be
achieved by several aggregation functions (e.g., maximum, minimum, average, median). In the examples
in this paper we use the median to represent the group since it is robust against outliers. The number of
rules and the most important (frequent) items in the group are displayed as the labels for the columns
followed by the number of other items in that antecedent group. Furthermore, the columns and rows in
the plot are reordered such that the aggregated interest measure is decreasing from top down and from
left to right, directing the user to the most interesting group in the top left corner.
To allow the user to explore the whole set of rules we can create a hierarchical structure of subgroups.
This is simply achieved by creating for each group Si; i = 1; : : : ; k, a submatrix Mi which only contains
the columns corresponding to the elements in Si. Now we can use the use same grouping process again
on a submatrix selected by the user. This allows the user to recursively \drill down" into the rule set.
An advantage of this process is that we only need to run the k-means algorithm on demand when the
user wants to explore a group further.
A chellange with using the k-means algorithm is that M contains many missing values for rules which
are not included in R since they do not pass the minimum support or minimum condence threshold.
Since most values will be missing, marginalization (i.e., remove antecedents/consequents with missing
values) is not an option and we use imputation. Imputation strategies typically assume that the values
are missing randomly which is not the case here. Values miss in our case systematically when rules do
not meet the support and condence thresholds and thus are deemed not interesting. This means that
we would like to group antecedents when they have many missing values with the same set of consequents
in common. To achieve this we replace all missing lift values with 1, a value indicating that antecedent
and consequent of the rule are statistically independent. This ensures that matching missing values will
contribute positively for grouping while it will help to separate them from existing rules with most likely
larger lift values.
The matrix visualization with grouped antecedents for the set of 5668 rules mined earlier can be easily
created with arulesViz by
> plot(rules, method = "grouped", control = list(k = 20))
The resulting visualization uses k = 20 groups and is shown in Figure 3. The group with the most
interesting rules according to lift (the default measure) are shown in the top-left corner of the plot.
There are 3 rules which contain \Instant food products" and up to 2 other items in the antecedent and
the consequent is \hamburger meat."
In the interactive version an antecedent group can be inspected by selecting a column. The result for
the left-most group is:
Selected rules:
lhs rhs support confidence lift
1 {Instant food products,
soda} => {hamburger meat} 0.001220132 0.631579 18.995654
2 {whole milk,
Instant food products} => {hamburger meat} 0.001525165 0.500000 15.038226
7

Grouped matrix for 5668 rules
size: support
color: lift
3 (I
nsta
nt fo
od
pr
od
uct
s +
2)
32
8 (w
hole
mil
k +7
3)
4 (p
roc
ess
ed
che
ese
+2
)
59
4 (o
the
r ve
ge
tab
les
+8
5)
74
2 (o
the
r ve
ge
tab
les
+9
3)
77
(bu
tte
r +
24
)
47
(ot
her
veg
eta
ble
s +
20
)
27
2 (o
the
r ve
ge
tab
les
+5
0)
30
(tro
pica
l fru
it +
12
)
31
0 (y
og
urt
+5
0)
23
7 (w
hole
mil
k +3
8)
35
3 (r
oot
veg
eta
ble
s +
51
)
46
7 (t
rop
ical
fru
it +
51
)
36
7 (o
the
r ve
ge
tab
les
+5
6)
29
2 (o
the
r ve
ge
tab
les
+6
5)
36
(tro
pica
l fru
it +
21
)
46
(so
da
+34
)
91
4 (r
oot
veg
eta
ble
s +
69
)
49
5 (w
hole
mil
k +8
4)
54
(bo
ttled
wa
ter
+3
3)
{whole milk}{rolls/buns}
{soda}{other vegetables}
{yogurt}{bottled water}
{root vegetables}{tropical fruit}
{shopping bags}{pastry}
{sausage}{citrus fruit}
{whipped/sour cream}{pip fruit}
{fruit/vegetable juice}{domestic eggs}
{bottled beer}{butter}
{curd}{beef}
{white bread}{cream cheese }
{sugar}{salty snack}
{hamburger meat}
LH
S
RHS
Figure 3: Grouped matrix-based visualization.
3 {whole milk,
Instant food products} => {other vegetables} 0.001525165 0.500000 2.584078
The rst two rules with rather large lift values are represented in Figure 3 by the upper balloon.
While the third weaker rules is the second balloon in the gure.
The grouped matrix visualization can be used interactively to zoom into groups and inspect rules.
Figure 4 shows the interactive version zoomed into the 4th group in Figure 3. This group contains 594
rules and the most common item in the antecedents is other vegetables. In Figure 4 we see that all
antecedents have whole milk as a possible consequent. This can be easily explained by the fact that milk
is the most frequent item in the whole data set. Most groups are very similar and they are only split into
several groups because we require the algorithm to split the antecedents into k = 20 groups. However,
there are a few dierent antecedent groups which have curd, citrus fruit and sausage also as consequents.
These groups contain more interesting rules since they have a higher lift (darker color of the balloon)
and they are displayed therefore closer to the top-left corner. Using the inspect button we look at the 2
rules in the the rst antecedent group.
Selected rules:
lhs rhs support confidence lift
8

Figure 4: Interactive grouped matrix-based visualization (zoomed into the 4th group in Figure 3).
1 {other vegetables,
yogurt,
whipped/sour cream,
cream cheese } => {curd} 0.001016777 0.5882353 11.040638
2 {other vegetables,
yogurt,
whipped/sour cream,
cream cheese } => {whole milk} 0.001220132 0.7058824 2.762576
Although the rst rule has a lower support and condence, it is the more important rules with an
extremely high lift. The whole rule set can be explored by zooming in and out of groups and by inspecting
dierent subsets of rules.
9

5 Conclusion
In this paper we introduced a completely new visualization method for association rules. The method
addresses the problem that sets of mined association rules are typically very large by grouping antecedents
and allowing the user to interactively explore a hierarchy of nested groups. Coloring and the position
of elements in the plot guide the user automatically to the most interesting groups/rules. Finally, the
ability to interpret the grouped matrix-based visualization can be easily acquired since it is based on the
easy to understand concepts of matrix-based visualization of association rules and grouping.
Grouped matrix-based visualization is unique in the way that most other visualization methods (see
Bruzzese and Davino, 2008) are not able to eciently deal with very large sets of association rules and
that no existing method can handle complementary items.
Interesting areas for future research are to explore dierent other ways to group antecedents and to
look at grouping antecedents and consequents simultaneously (i.e., by co-clustering/two-mode clustering).
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