


Discovering Multi-Level Association Rules using Fuzzy Hierarchies

Usha Rani1, R. Vijaya Prakash2, A. Govardhan3
  1. Research Scholar, JNTU, Hyderabad, India
  2. SR Engineering College, Warangal, India
  3. Dept. of Computer Science and Engineering, JNTU, Hyderabad, India
Related article at Pubmed, Scholar Google



In this paper, Fuzzy concept hierarchies are used for multi-level association rule mining from large datasets via Attribute-Oriented Induction approach [1]. In this the process of fuzzy hierarchical induction approach is used and extends it with two new characteristics which improve applicability of the original approach in data mining. The proposed drilling-down approach of fuzzy induction model allows user to retrieve estimated explanations of the generated abstract concept. An application to discovery of multi-level association rules from environmental data stored in a Toxic Release Inventory is presented.


Fuzzy Hierarchies, Propogation, Association Rules


The need for generalization mechanisms has been proven to be a critical factor for many data mining tasks. To efficiently analyze voluminous data sets, which are currently stored in databases of almost all companies, it is often necessary to start from pruning and reducing of the size of these repositories.
Decision makers are usually not interested in time consuming extraction of technical details (i.e. serial numbers, time of transactions with precision in seconds, detailed GPS locations, etc.) stored originally in large databases. Instead, they want to obtain knowledge at a certain level of abstraction, as dictated by their professional goals or by the character of the analysed dataset. The data conceptually represent information at multiple levels . For instance, an item, represented by a bar code: 040101000 is a Milky Way Bar, which is a Chocolate Bar, pertaining to a Snack, or even a Food group, etc.
Attribute-Oriented Induction (AOI) [1] allows compression of the original set of data into a generalized relation to provide data analysts with concise and summarize information about the original, massive set of task-relevant data. The AOI process employs background knowledge represented in the form of concept hierarchies, separately declared for each of the attributes in the analyzed database table.
Induction of the original relation is performed one step at a time on an attribute-by-attribute basis. The method has rather straightforward character [1]:
(1) Gather task-relevant data into the initial relation
(2) Generalize the data by removal or generalization of derived attributes. An attribute can be removed if it has a large set of distinct values and no generalization hierarchy for the attribute is available or if the abstract concepts are reflected by other attributes in the initial relation. Attribute values can be generalized if a concept hierarchy for them has been defined.
(3) Aggregate the data by merging identical tuples and accumulating their respective counts. New attribute COUNT has to be added to the relation, to keep track of original records, which are gradually merged during the transformation of attribute values from one level of abstraction to another. Value stored in the COUNT column reflects the number of original records merged together into a generalized tuple.
(4) Present generated (generalized) table to the user. Additional data mining algorithms can be further applied on this relation.
The hierarchical character of AOI provides analysts with ability to view the original information at multiple levels of abstraction, allowing them to progressively discover interesting data concentration points. In contrast to the flat summarization a gradual process of attribute-oriented induction through concept hierarchies allows detailed tracking of all records, and can lead to the discovery of interesting patterns among data at the lowest abstraction level of their occurrence. In the AOI approach the tuples, which initially (at a low level of abstraction) do not match the requirement of minimum support assigned to eliminate seldom occurring regularities, rather than being dropped, are gradually further aggregated and so given a chance to reach a meaningful count at a higher abstraction level. Such an approach improves accuracy of generalization, since now only those records that are truly infrequent at each of the abstraction levels are eliminated from further analysis.


The idea of applying concept hierarchies to generalize database records was proposed by Han [1-4] and developed further by other researchers [5-6]. Recently, Hsu [7] extended the basic AOI algorithm for generalization of numeric values. The majority of the mentioned work focuses on attribute oriented induction with utilization of crisp concept hierarchies where each attribute value can have only one direct abstract to which it fully belongs.
However, regular concept hierarchy are not the best representation for usually complex dependencies occurring in the attribute domains due to their crisp character. Recently several independent groups of researchers have investigated applications of a fuzzy concept hierarchy to AOI. Lee and Kim [8] used fuzzy ISA hierarchies, from the area of data modelling, to generalize database records to more abstract concepts. Lee [9] applied fuzzy generalization hierarchies to mine generalized fuzzy quantitative association rules. Cubero, Medina, Pons, and Vila [10] presented fuzzy gradual rules for data summarization. Raschia, and Mouaddib [11] implemented the SaintEtiq system for data summarization through extended concept hierarchies.
Fuzzy concept hierarchies (Figure 1) seem to better capture human approaches to induction. Trivi al examples supporting this conclusion can be found everywhere, e.g. how to generalize three colors, such as black, gray and white to two abstract terms: dark, light, or to which continent should we assign the Russian Federation, taking under consideration a fact that almost one fourth of this country lies west of the Urals. In fuzzy concept hierarchies, lower - level concepts can belong to more than one descriptor placed on the directly higher level of abstraction. Unfortunately, none of the above-mentioned fuzzy induction models focuses on guaranteeing the preservation of exact dependencies among low-level (original) tuples after transforming them to more general descriptors.


The main difference between a crisp concept hierarchy and a fuzzy one is the type of generalization relation between the concept and its direct abstract these structures are capable of modeling. In crisp hierarchies, each lower-level concept is fully assigned to only one direct abstract, and a single abstract can have many direct specializes. In fuzzy concept hierarchies a generalization relation is allowed to have a many-tomany character; a single lower-level concept can have multiple abstracts at the next abstraction level, related to each of these concepts to a different extent.
Our fuzzy concept hierarchy is an augmented rooted tree structure. Each link l in the fuzzy concept hierarchy is a bottom -up directed arc (edge) between two nodes (i.e. concepts) with a certain weight assigned to it. Such a structure reflects a fuzzy induction triple.
image, where image are endpoints of image represents the strength of conviction that the lower-level concept image should be qualified as concept image when the data generalization process moves to the next abstraction level. During AOFI the value of image dictates what fraction of a vote, representing original attribute values generalized already to the concept image ck, is propagated to its higher-level representation image
A. Consistency of Vote Propagation And Completeness Of The Model
Other approaches to the attribute-oriented fuzzy generalization do not concentrate on precise and consistent preservation of all original dependencies among the data at each level of induction. It limits their usage in many data mining projects, which perform exhaustive analysis of scientific data. To guarantee exactness of the data summarization via AOFI we need to preserve the number of original tuples and the relation between them in the identical proportions at each level of abstraction. In other words, to provide a consistent AOFI we need to assure that each record from the original database relation will be counted exactly once at each of the levels of the AOI hierarchy. We call this an exact vote propagation dilemma, which follows the vote propagation for crisp AOI.
This problem can be resolved by preservation of completeness in an AOFI model. The fuzzy generalization hierarchy is complete and consistent when for all adjacent hierarchy levels, image and image (where image is a direct abstraction level of image), the following relationship is satisfied:
In other words, the sum of weights assigned to the links leaving any single node in a fuzzy concept hierarchy needs to be 1.0 for completeness of the AOFI model.
Preservation of the above property prevents any attribute value (or its abstract) from being counted more or less than once at each step of the attribute-oriented fuzzy induction (i.e. consistency). It also guarantees that a set of abstracts concepts at each level of hierarchy will cover all of the attribute values that occurred in the original dataset (i.e. completeness of the model). So we are guaranteed to not lose correct count of the number of tuples when generalizing their attribute values.
Such normalized representation of uncertainty, which is a common element of many generalization processes, should not be confused with representation of probabilistic dependencies. First of all, fuzzy concept hierarchies do not reflect probabilities with which lower-level concepts may be assigned to their direct abstracts – the hierarchies reflect actual degrees with which the lower-level concepts belong to the generalized terms (allowing for more adequate reflection of natural human reasoning). Secondly, the only purpose of hierarchy normalization in our approach is to assure accurate representation of original tuples during the AOI process. Formally, each concept hierarchy (also the un -normalized one) can be utilized to perform consistent and completeAOI. Normalization of hierarchy can be performed via simple normalization of membership values in all outgoing links for every concept placed in the hierarchy.
B. Drilling Down Fuzzy Concept hierarchies To Provide Explanations Of Abstract Concepts
Since generalized tuples, which reached their count value above the minimal support threshold, may appear at multiple levels of abstraction, a data analyst may want to drill down the definition of each of the reported generalized descriptors. This can be achieved by recursive extraction of lower -level components of the abstract concept.
To obtain an explanation of what is behind a particular abstract concept, which characterizes a significant part of original data set generalized to the related abstraction level, we backtrack the components (lower-level concepts) of the concept. We are able to do so through the analysis of the induction paths, which were activated during AOFI and led to the particular abstract. In general, such an explanation can be either (1) Simplified and based only on the background knowledge (only knowledge about activated generalization paths from concept hierarchy is utilized) or have (2) a detailed, knowledge-driven, but data distribution based character (when we preserve the count of votes grouped at each node during the AOI process). In both of the cases we utilize a concept of ―descending memberships‖, which are derived from the original (i.e. ascending) membership values included in the concept hierarchies and used during the AOFI.
The second approach has a more comprehensive character but requires more time for computations and considerably larger memory for preservation of all intermediate data tables, generated during the AOI process. These characteristics make it inapplicable for mining very large databases.
Approximate explanations are based only on the concept hierarchies and the generalized output table, requiring much less memory than the original, massive dataset.
Using a concept hierarchy, we can explain each abstract concept by tracking down its members to the 0-abstraction level (original attribute values). In Figure 1 the descriptor Any contains two concepts Light and Dark. Since both of these descriptors have identical ascending membership degrees, presented in the links of the hierarchy, we would intuitively conclude that they both characterize their direct generalizer equally, therefore
Derivation of the descending memberships from fuzzy concept hierarchies has a traightforward character. Each concept cki in the fuzzy hierarchy can be explained by a set of its direct specializers (i.e. a subset of image The membership values are then normalized to allow consistent representation for the fuzzy abstract class cki . So now we can provide the explanation for each abstract class as follows:
image (2)
and h ≥ K > 0 , where h is the height of the fuzzy concept hierarchy.
In other words, we explain each abstract concept by providing the set of its direct specializes, and the degrees of their ―descending memberships‖, reflecting the knowledge -based participation (contribution) of these terms in the final abstract’s construction.
Drilling down further in Figure 1, based on the distribution of bottom-up membership values in the hierarchy links, we can explain Light as {Whitish(2/3), Grayish(1/3)} and consequently Dark as {Grayish(1/3), Blackish(2/3)}. We can then provide the user with a more detailed definition of the abstract concept any, which is
To make To make this explanation clearer we can merge overlapping components. To preserve completeness of the derived definition we applied the sum operator over the algebraic product when merging all overlapping concepts
So we can see that the approximated explanation of abstracts, via drilling down fuzzy concept hierarchies, has a transitive character. If we explain, concept image with the subset of image , and then each of the elements from this set with the concepts from the set CK-2 , then image can be also fully explained by the merger of definitions based on the set CK-2.
This approach to explanation of abstract concepts has a rather trivial character when none of the abstract’s components (direct specializes) managed to reach a count of the votes greater than the given generalization threshold (denoted further as T) and so was reported as a separate entity in the generalized relation. However, in the other case, we have to make certain that the concepts that reached a significant count at the lower level of abstraction (and were already placed in the final generalized relation) do not occur in descriptions of higher-level concepts. Otherwise the explanations would have a confusing character, since the client could be under impression that the same tuples are reported (and counted) multiple times at different levels of abstraction. We have to remember that final users of data mining applications are decisions makers who, in contrasts to experts and data analysts, usually do not have time to gather detailed knowledge about techniques which were applied to generalize the data. Therefore the last equation should include the restriction:
where T is the threshold value representing the minimal value of COUNT which is recognized by the client as significant aggregation of original data (e.g. one may wish all generalized tuples which describe more than T = 5% of the original dataset to be reported separately in the output relation); and image c represents the total number of original data records generalized to the abstract concept ckj
This property assures that each lower-level concept, employed in derivation of a hierarchical explanation, was not separately reported in the final generalized relation


A data-mining task, for which we have chosen to empirically test the AOFI method, was to provide a concise and exact summary of toluene emission in the air of Louisiana in 2001 (the most current dataset provided by U.S. Environmental Protection Agency). We have chosen this particular toxin since it is a chemical most commonly reported to the Toxics Release Inventory [13] by the local industrial facilities.
Toxics Release Inventory (TRI) is a major EPA database, which has been created under the Emergency Planning and Community Right-t o-Know Act (EPCRA) in 1986. The 2001 TRI dataset contains information on releases of approximately 650 toxic chemicals. The data was gathered from over 21,000 manufacturing facilities located within the US.
When performing our method over the dataset we concentrate our analysis on three main aspects: (1) Localization of the toluene emitters (i.e. discovery of regions in Louisiana which have high air emission of the chemical), (2) Amount of the toluene air emission in the state (i.e. estimation of the toxicity level in different regions of Louisiana), (3) Industrial sectors which are responsible for the local toxification (i.e. types of businesses which are emitting toluene in the air of Louisiana), represented by the federal SIC (i.e. Standard Industry Code) classification.
The number of abstract descriptors at each level of concept hierarchies for the analyzed TRI dataset is presented in Table 1. The second row of this table reflects Pre-Generalization [1], where attribute values are transformed to the concepts placed at the leaves of the concept hierarchies. Since this phase is performed without an increase of abstraction level (denoted as 0- abstraction level in Table 1), the descriptors placed in the leaves of the concept hierarchies are actually the original attribute values.
As with regular AOI, we are able to extract separate generalization paths for each attribute value from the concept hierarchies. However the fuzzy generalization paths can not be implemented in the form of lists but need to be represented in the form of trees.
As in the crisp approach, we are allowed to order steps of generalization according to our preferences. Each stage of AOI can be characterized by a vector representing the current abstraction level for each of the generalized attributes, where the order of its elements reflects the order of attributes in the generalized relation. In our example, since we generalize three columns of the table including task relevant data, we can describe Pre-Generalization as (0, 0, 0), which reflects the case where all three attributes are at the 0-abstraction level. So the final possible stage (all attribute values in each column are characterized by a single, most general concept) is a vector (4, 4, 3), which indicates that attributes FACILITY_LOCALIZATION and SIC_CODE are generalized to the 4th level of abstraction, and the last attribute, i.e. TOTAL_AIR_EMISSION, is abstracted to the 3rd level (all these levels ar e final, since they reflect roots of the utilized concept hierarchies).
The final, generalized relation, which provides the user with information about only significant clusters of data at the abstract level, is presented in the Table 2. Each of the generated tuples can be interpreted as a conjunctive rule, characterizing release of toluene in Louisiana (i.e. the tuples stored in the initial relation). Obviously, the order of AOFI steps could reflect different data mining goals. Since we were particularly interested in analyzing what types of facilities release toluene into the air in Louisiana we decided to generalize the SIC codes as late as possible.
When mapping generated abstract tuples into characteristic rules at a common level of abstraction with quantitative information about support of these characteristics, we must ensure preservation of the distribution of COUNT according to the background knowledge as reflected in the fuzzy concept hierarchies.
For instance, if we are particularly interested in the business sector classified by government as 51—(i.e. Wholesale Trade of the Non-durable Goods), we can merge the 1st, 2nd and 11th record of the generalized relation to build the following characteristic rule:
(South ∨ Cajun Country∨ Crossroads) ∧ (51-- ∨ 5169 ∨ 5171) ∧ (Low ∨ Below and about 603).
This rule can be further simplified to the form: (South ∨ Crossroads) ∧ 51-- ∧ Low and fit to the abstract description of 18.99% (i.e.4.52%+4.45%+10.02%) of the originally stored records. However if we want to transform this characterization to a more general form, such as: South ∧ 51-- ∧ Low , we have to remember that according to the fuzzy concept hierarchy presented in the Figure 2 only 25% of the Crossroads country lies in the southern Louisiana, therefore in this generalization a summarization of the COUNT values needs to be appropriately modified:
10.02% + 4.52% + 0.25*4.45% = 15.65%.
Now we can conclude that almost 19% of facilities, which officially reported a release of toluene into the air of Louisiana, are the wholesale traders of nondurable goods and that over 82% of them (i.e. 15.65 / 18.99 = 0.8241) are located in the Southern part of the state. All these ―51-type‖ facilities released only low amounts of toluene.


In this paper we first introduced a consistent model of fuzzy induction and then applied it to mine generalized association rules from environmental data. In addition, we presented how the generated results can be explained to users via extraction of their approximate definitions.
There are many aspects by which we may judge the technique presented here. The ordinary (crisp) AOI will usually perform with higher computational efficiency then AOFI. However utilization of fuzzy concept hierarchies provides more flexibility in reflecting expert knowledge and so allows better modeling of real-life dependencies among attribute values, which will lead to more satisfactory overall results for the induction process. The drawback of the computational cost may additionally decline when we notice that, in contrast to many other data mining algorithms, hierarchical induction algorithms need to run only once through the original (i.e. massive) dataset. We are continuing an investigation of computational costs of our approach for large datasets.

Tables at a glance

Table icon Table icon
Table 1 Table 2

Figures at a glance

Figure 1
Figure 1